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Vol. 73, No. 2

M a rch/ A pril 1 9 9 1

3 Should G o v ern m en t Spending on
C apital G oods Be Raised?
16 T ra d e Im b alan ces and E co n o m ic
T h e o ry : T h e C ase fo r a U .S .-Jap an
T ra d e D eficit
32 A n In tro d u ctio n to C om plete
M ark ets
5 8 A P rim e r on C o in teg ratio n w ith an
A pp lication to M oney and In co m e

THE
FEDERAL
RESERVE
RANK of
ST.IXMIIS

F e d e ra l R e s e rv e B an k of St. L o u is
R eview

March/April 1991

In This Issue . . .

A growing number of public policymakers and analysts argue that a
decline in government capital formation has reduced private sector out
put, productivity, profitability and international competitiveness. In their
view, a rise in public capital formation is essential to reverse these ef
fects. John A. Tatom explains this argument and addresses several
issues surrounding it in the first article in this Review, "Should Govern
ment Spending on Capital Goods Be Raised?”
Tatom notes that the size of the public capital stock is important for
broader reasons than its effect on private sector output. Thus, he says,
the case for raising or lowering public capital spending does not rest on
ly upon its effects on private sector production.
The author explains that, while there has been a slowing in the growth
of the public capital stock, this slowdown was consistent with changing
demographic and relative price trends that influence the public’s desired
stock of government-held assets. Tatom also explains that the slowdown
in public capital formation has taken place primarily at the state and
local government level. According to Tatom, the case for a sharp rise in
public capital spending is not supported by past trends.
* * *
The size of the U.S. trade deficit with Japan has given rise to con
siderable concern and discussion. In the second article in this issue, "A
Case for a Bilateral Trade Deficit,” Alison Butler examines the possible
causes of this bilateral trade deficit. She analyzes whether this deficit is
a natural implication of Japanese and U.S. macro and microeconomic
policies or, in fact, results from unfair Japanese trade practices. She
finds that much of the trade imbalance is due to the composition of
trade in the two countries as well as their savings and investment
behavior.
* * *
In this Review's third article, “An Introduction to Complete Markets,”
Mark D. Flood brings an important component of economic theory
down to earth. The theory of complete markets is the cornerstone of
modern mathematical economics. Its most fruitful applications have
been to models of economic equilibrium, but its range is more general.
One area in which the theory has important practical applications is in
the design and analysis of marketable securities: the theory makes sense
of complex "financial innovations,” such as options on a stock index
portfolio.
Flood begins by developing the context of complete markets in a sim
ple gambling example, demonstrating how the theory incorporates
uncertainty in a systematic fashion. The upshot is that a complete
system of markets is superior to one in which the opportunities for ex
change are restricted. Next, the context of dollar payoffs is generalized

MARCH/APRIL 1991

to uncertain payoffs in multiple commodities over many periods, the
starting point for the Arrow-Debreu model of general equilibrium. Final
ly, some financial market applications are considered: the markets for
cotton futures and stock index options.
Accompanying the article for classroom purposes is a set of six review
questions and answers.
* * *
In the fourth article in this Review, “A Primer on Cointegration with
an Application to Money and Income,” David A. Dickey, Dennis W.
Jansen and Daniel L. Thornton discuss the statistical or econometric
concept of “cointegration.” The authors outline the relationship between
cointegration and tests for cointegration and the concept of a unit root
and tests for a unit root, widely used in univariate analyses, and provide
a framework for relating the statistic concept to economic analysis.
Finally, the authors detail three commonly used tests for cointegration
and apply these tests to U.S. data on output, interest rates and various
monetary aggregates. Their empirical analysis suggests that there is a
stable long-run relationship between output, interest rates and several
monetary aggregates, including M l and the adjusted monetary base.

FEDERAL RESERVE BANK OF ST. LOUIS

* * *

John A. Tatom
John A. Tatom is an assistant vice president at the Federal
Reserve Bank of St. Louis. Kevin L. Kiiesen provided research
assistance.

Should G overnm ent Spending
on Capital Goods Be Raised?

GROWING BODY of public opinion and
analysis argues that government spending for
capital formation is deficient and should be in
creased. Such spending is largely for goods called
infrastructure. Adam Smith (1937) referred to
spending on infrastructure as the third rationale
for the state (behind the provision of defense
and justice).1 Today, these capital goods include
highways, mass transit systems, airports, electric
and gas facilities, wastewater treatment
facilities, water supply and distribution, in addi
tion to the facilities and equipment used in
governmental and judicial administration, police
and fire protection and health and educational
institutions.
In 1988 the National Council on Public Works
Improvement concluded in its final report to
Congress and the President that “the quality of
America's infrastructure is barely adequate to
fulfill current requirements, and insufficient to
meet the demands of future economic growth
and development” (1988, p. 1). It recommends a
’ See Smith (1937), Book V, part 3, especially p. 682.
Krueger (1990) attributes the principal comparative advan
tage of government to its provision of infrastructure.
Krueger’s focus is on global trends in government activity
not just trends in the United States. She argues that the
failure of government to promote economic development
has resulted from the diversion of specialized organiza
tional and administrative resources that provide infrastruc
ture to activities in which government does not have a
comparative advantage (e.g., manufacturing, regulating
markets or licensing).

national commitment to improve infrastructure
that could double spending on public works by
the year 2000.2
The dearth of infrastructure spending has also
been noted by others. For example, Benjamin
Friedman argues that the relatively large federal
budget deficits of the 1980s created pressure to
reduce infrastructure spending. He stresses the
ways in which infrastructure investment affects
the business sector:
“Government investment in roads, bridges, air
ports, port facilities, and other kinds of infrastruc
ture also has a direct bearing on how easy or dif
ficult it is, and also how cheap or costly, for many
com panies to do business."

In Friedman's view, the economic policies of the
1980s created a situation in which “we have
been cheating our future in all these respects—
not just in business capital formation but in
government infrastructure and education too.”3
2The National Council was established by the U.S. Con
gress in 1984 to assess the state of the nation’s
infrastructure.
3See Friedman (1988, p. 202 and p. 204, respectively).
Friedman includes educational spending in infrastructure
and argues that reduced educational spending lowers the
quality of labor and, hence, income and productivity. The
stock of human capital is not included in the analysis of
public capital accumulation conducted here. Public educa
tional facilities are included in the physical public capital
measures analyzed below, however.

MARCH/APRIL 1991

Table 1
The Composition of the Public Capital Stock: 1989 (millions of dollars,
1982 prices)
Federal
Non-military structures
Highways and streets
Educational buildings
Other buildings
Hospital buildings
Water supply facilities
Sewer systems structures
Conservation & development
Industrial buildings
Other structures
Non-military equipment

Total

$190,617

$1,321,477

$1,512,094

(90.9)

13,926
831
22,921
7,933

544,458
242,997
134,080
36,376
78,259
152,753
26,073

558,384
243,828
157,001
44,309
78,259
152,753
143,048
21,148
113,364
151,357

(33.6)
(14.7)
(9.4)
(2.7)
(4.7)
(9.2)
(8.6)
(1.3)
(6.8)
(9.1)

1,663,451

(100.0)

—
—

116,975
21,148
6,883
49,991

Non-military structures and equipment

240,608

Military equipment
Military structures

382,548
82,813

Total

State and Local

$705,970

—

106,481
101,366
(14.5)

1,422,843

(85.5)

382,548
82,813

___

—
(42.4)

$1,422,843

(85.5)

$2,128,813

(128.0)

NOTE: Shares of the non-military total are given in parentheses. Components may not add to total due to rounding.

Ratner (1983), and later, Aschauer (1989a, b)
and Munnell (1990) provide evidence that the
services of the public capital stock affect private
sector output. Aschauer and Munnell argue that
the slowdown in U.S. public capital formation
accounts for the slowing of U.S. productivity
and general economic growth since the early
1970s. Their arguments and evidence have fos
tered the view that there is a "third” deficit
(over and above the trade and government bud
get deficits) that threatens this nation’s future
standard of living.4 As Malabre (1990) notes,
Aschauer attributes up to 60 percent of the
"productivity slump” to the “neglect of our core
infrastructure.”
This is the first of two articles focusing on the
issue of whether a public capital deficiency ex
4This characterization is described in Malabre (1990) and
contained in Aschauer (1990). Reich (1991) expresses a
similar view of the critical role of public capital formation in
both accounting for past shortcomings in U.S. economic
performance and in facilitating future economic growth.

FEDERAL RESERVE BANK OF ST. LOUIS

ists. This article examines recent trends in public
capital formation and some of the factors that
might be responsible for these trends.5 This ar
ticle concludes that the decline in public capital
formation can be explained, in part, by funda
mental economic influences. Thus, while there
may be convincing reasons to boost public capital
formation, they are not found among those ex
amined here.

HOW MUCH CAPITAL DOES THE
PU BLIC WANT?
Table 1 provides a breakdown of the types of
public capital held by federal, state and local
governments in 1989. The data are the constant
dollar (1982 prices) net stocks of capital of vari
ous types estimated by the Bureau of Economic
sThe second article will focus on the empirical evidence
that links public capital formation with private sector
performance.

Analysis, U.S. Department of Commerce.6 Since
the public capital formation debate focuses on
non-military capital, military capital goods are
reported separately. These goods make up more
than 60 percent of the total federal capital stock
and more than 20 percent of the total govern
ment capital stock. Public capital includes
highways and streets, educational buildings,
hospitals, water supply and sewer facilities,
other structures (electric and gas production
and distribution facilities, transit systems, air
fields, etc.), conservation and development
structures (e.g., locks, dams) and non-military
equipment. Most non-military public capital is
owned by state and local governments; the
largest categories are highways and streets and
educational structures.
Public capital goods produce "public" goods or
services that are typically provided on a large
scale to many consumers. The use of these goods
or services is generally described as “non-rivalrous” or "non-exclusive,” indicating that use by
one consumer does not impinge on the ability of
others to use the same product.7 In addition, the
goods or services yielded by such capital goods
are sometimes considered insufficiently profitable
to be provided by the private sector. Therefore,
in the absence of government provision, these
goods or services would be produced in relative
ly small quantities or, perhaps, not at all.8

The Optimal Quantity o f Public
Capital
How much capital is it optimal for govern
ment to accumulate? The demand for public
capital, like that for private capital goods, is
6The net capital stock data used in this article were pro
vided by John Musgrave, U.S. Department of Commerce;
these data are described in U.S. Department of Commerce
(1987) and Musgrave (1988). The net stock series de
duct depreciation from gross stock measures that cumu
late gross investment less discards or assets that are
scrapped. The depreciation estimates use straight-line
depreciation for service lives equal to 85 percent of those
given in the U.S. Treasury Department’s Bulletin F. The
use of constant prices, instead of historical or current cost
valuation, results in a measure of the quantity of capital.
The focus on net capital stock measures follows the
literature on the public capital hypothesis. Tatom (1989)
discusses the tendency of these data to understate the
rise in the quantity of private capital, especially in the first
half of the 1980s.
H'hese descriptions are based on Krueger (1990). Similar
discussions of public goods and public choice models are
found in Musgrave and Musgrave (1984) and other public
finance texts.

derived from the value placed on the assets’ ser
vices by its users. The present value of an asset’s
current and future services is the maximum
that government would pay for the capital if it
seeks to promote an efficient allocation of the
nation’s resources. The principle of diminishing
returns indicates that, as the quantity of an
asset and its services rises, the value of a unit
of its services declines, so that the price of these
services and the maximum price of the asset
declines. The optimal quantity of capital goods
occurs when the price of public capital assets
equals this present value of current and future
goods and services.9
When "too little” public capital exists, the
value of the goods and services provided by the
asset is worth more than its price. Conversely,
when there is "too much” public capital, the
supply price exceeds the present value of the
asset. More importantly, the optimal quantity of
public capital will decline if the supply price of
public capital assets rises or if the demand for
these assets declines. Proponents of the public
capital hypothesis suggest, however, that the
quantity of public capital has fallen simply due
to neglect or budget pressures.

The Sources o f Dem and F o r Public
Capital
There are two types of goods or services pro
duced by public capital goods: final goods or
services which yield valuable benefits directly to
consumers or final users, and intermediate goods
or services which assist in producing other final
goods or services. In either case, the value of
Musgrave and Musgrave (1984) and other public finance
texts for a discussion of this issue and these models.
Moreover, many public capital assets are quite similar to
private capital goods and are used to produce goods and
services similar to private sector output. These similarities
are ignored below, however; all public capital goods are
assumed to produce public goods and services.
9According to public choice theorists, the use of an
economic efficiency criterion to determine an optimal stock
of public capital may be a poor guide to actual public
policy decisions. The quantity of public capital goods may
not be determined on purely economic grounds. Never
theless, the prices of resources used in the public sector
and the benefits of the goods and services produced with
these resources to consumers and firms are important,
even in public choice models. Thus, efficiency-based con
siderations of an optimal public capital stock offer some in
sight into the causes and implications of recent trends in
public capital formation.

8There is a vast and growing “ public choice” literature that
questions the existence of these public goods. See

MARCH/APRIL 1991

the capital good is derived from the value of the
goods and services produced directly or indirect
ly. Some examples include the public capital
that produces publicly provided gas, electric,
water or sewer services used in the home or in
the production of other goods and services.
Highways and roads are used to facilitate the
acquisition of raw materials, labor and other
capital services for private sector production
and distribution, as well as for recreational and
other consumer purposes. Public capital pro
vides educational services with both immediate
and future consumer benefits and affects the
productivity of labor services. Lock and dam
systems provide conservation and recreational
benefits as well as intermediate services to
business by lowering the cost of transporting
goods and services.

determined prior to drawing any conclusions
about whether to raise public capital spending.

The Quantity o f Public Capital
Could A ffect Private Sector
Perform an ce
Proponents of raising public capital spending
argue that public capital increases private sector
output both directly and indirectly. The direct
effect of public capital on private output depends
on whether public capital provides important in
termediate services to private sector firms. If so,
an increase in public capital would raise private
output just as an increase in the use of private
capital (or labor) would raise output.
The indirect effect arises, according to pro
ponents of this argument, because the use of a
larger quantity of public capital raises the rate
of return on private sector capital, providing an
incentive for firms to increase private capital
formation. Therefore, private sector output and
productivity, measured as output per worker or
per hour, rise further. In this case, public and
private capital are said to be "complements.”
Thus, according to the public capital hypothesis,
a decline in public capital reduces private sector
output (direct effect), productivity and the rate
of return to capital in the short run. Given time
to adjust, firms would reduce the amount of
private sector capital per worker and this would
further reduce private sector output (indirect
effect) and productivity.11

The demand for public capital goods, their
present value, is the sum of the values placed
on the services of the asset by both business
and consumers.10 Suppose that, initially, the pre
sent value of the services of a unit of public
capital equals its price. Given the current stock
of public capital, a decline in the value placed
on public capital goods by consumers will
reduce the total demand for public capital so
that the value of a unit of public capital is
lower than its price. If government decision
makers are concerned with promoting an effi
cient allocation of the nation's resources, they
will reduce the quantity of public capital accor
dingly. This decline would lower the quantity of
public capital available for use by firms even
though their demand for public capital has not
changed.

TRENDS IN TH E GROWTH OF
PU BLIC CAPITAL SPENDING

In addition, the optimal quantity of public
capital would decline if business sector demand
for public capital falls. Therefore, if the public
capital stock has fallen (either absolutely or
relative to some measure of private firms’ de
mand), the reason for the decline must be

Figure 1 shows the real (constant cost) net
stock of public capital (1982 prices) at the end
of each year from 1947 to 1989. The capital
stock displays a strong positive trend until 1970,
when, as the critics of public sector performance
point out, the growth of public capital slows.

10See Musgrave and Musgrave (1984), pp. 47-69, or other
public finance texts for a more extensive discussion of the
differences between public, or social, goods and private
goods. The extreme assumption of non-exclusivity for
public goods can be relaxed without affecting the analysis
here. In particular, there are “ mixed goods” which are
private but yield external benefits or costs, and public
goods whose use can entail congestion or other costs for
others. In either case, the analysis here applies so long as
the services of the public capital goods are not exclusive.
’ ’ While public capital may provide valuable services to
private firms, close substitutes for these same services

FEDERAL RESERVE BANK OF ST. LOUIS

could also be available from private producers. If this is
the case, a rise in the quantity of such public capital
would lower the value, or rate of return, of acquiring new
private capital goods. As a result, less private capital in
vestment would take place. In this event, public and
private capital are “ substitutes.” Public capital formation
would “ crowd out” private capital formation by lowering its
rate of return, instead of “ crowding in” private capital for
mation by raising its rate of return. Aschauer (1985), for
example, argues that public and private sector spending
are generally substitutes.

Figure 1
Net Stock of Nonmilitary Public Capital
Ratio Scale
Billions of Dollars (1982 Prices)
1750

Ratio Scale
Billions of Dollars (1982 Prices)
1750

1500

1250

1000

500
1947
53
NOTE: End-of-year data.

The quantity of public capital per member of
the civilian labor force, referred to as “per
worker” below, is useful for assessing the trend
in public capital formation.12 Growth of the
public capital stock per worker is plotted in
figure 2; the amount of public capital available
each year is measured by the amount in ex
istence at the end of the previous year. Federal,
state and local components of public capital per
worker are also shown.

Public Capital Stock P e r W orker
Figure 2 shows that the trend of public capital
per worker was sharply different after 1971
than it had been previously. From 1948 to 1971,
total public capital per worker rose at a 2.6 per
cent annual rate. Since 1971, the amount of
capital per worker has declined. Figure 2 sug
gests that public capital formation’s contribution
12The capital stock is measured relative to the civilian labor
force instead of employment to remove transitory varia
tions in capital per worker associated with the business cy-

500
1989

(if any) to the growth of the U.S. standard of
living disappeared after 1971; its contribution (if
any) since then has been negative.
The figure also shows that this trend shift
was concentrated in state and local government
capital spending. The trend in the stock of net
federal capital per worker has been nearly flat
or slightly declining throughout the whole period.
In particular, it declined slightly from 1967 to
1989 reaching about the same level as in the
early 1950s. Moreover, since federal capital
stock is a small fraction of the total stock of
public capital, relatively large changes in the
pace of federal capital formation would have lit
tle effect on the trend in the total amount of
capital per worker. Therefore, if the decline in
public capital per worker indicates that there is
a problem with public capital formation, its
source is not at the federal level, but rather at
the state and local level.
cle. Removing cyclical variations smooths out the resulting
series for capital per worker.

MARCH/APRIL 1991

Figure 2
Real Nonmilitary Government Capital Stock per Person
in the Civilian Labor Force
Thousands of Dollars per person
(1982 Prices)

Thousands of Dollars per person
(1982 Prices)

It could be argued that a decline in federal
financing might account for a decline in state
and local capital per worker. But this ignores
the evidence of substitutability between federal
and state and local financing of state and local
government spending.13 Moreover, despite federal
funding, the decision to invest in federally
assisted spending (e.g., highways) depends on
the willingness of state and/or local govern
ments to meet federal matching requirements.
Also, the decline in federal grants to state and
local governments relative to overall state and

local spending did not begin until 1979, well
past the time when the growth of capital per
worker began to decline.14

13See CBO (1988) or Eberts (1990) for a discussion of these
issues. The CBO (1986), p. 81-83, reviews 11 studies that
show a large degree of substitutability between federal in
vestment grants and state and/or local spending. It is
shown below that much of the decline in capital growth is
concentrated in highways. Even in this case, CBO (1986)
shows evidence of substitution between federal and state
spending.

from 50 percent to 70 percent in 1970 and to 75 percent
in 1978. He also argues that, despite relatively high mat
ching rates (the interstate highway matching rate is 90
percent), “ at the margin, states and localities are paying
the full costs of investment” because federal capital pro
grams are “ closed-end” matching grants where sharing
occurs up to a maximum dollar amount which is generally
less than almost all state and local governments spend.
Also see Gramlich (1990).

14Peterson (1991) notes that the matching rate on what are
referred to as non-interstate federal aid highways rose

FEDERAL RESERVE BANK OF ST. LOUIS

Public Capital Stock P e r Person
The trend in public capital relative to the
labor force measures the relative availability of
capital for both private production and final
goods and services only if the population and
labor force grew at about the same pace. How
ever, the baby boom created a disparity between
the growth of the labor force and the popula-

Figure 3
Real Nonmilitary Government Capital Stock per Person
Thousands of Dollars per Person
(1982 Prices)

1950
55
NOTE: End-of-year data.

Thousands of Dollars per Person
(1982 Prices)

tion. In the 15 years following World War II,
U.S. population growth was quite rapid relative
to the growth of the labor force. As the babyboom generation matured and entered the labor
force, the growth rate of the latter accelerated
sharply, while population growth slowed. Final
ly, after the baby-boomers had fully entered the
labor-force, growth of the labor force slowed
sharply. These movements in the size of the
labor force relative to the general population
imply that the public capital stock per person
will show a different pattern than public capital
measured relative to actual or potential
employment.
Figure 3 shows the nonmilitary real net stock
of capital and its components when measured
15End-of-year population and capital stock data are used for
each year. Thus, the data shown are comparable to the
data plotted one year later in figure 2, because beginningof-the-year capital stocks are measured relative to the an
nual average level of the civilian labor force in figure 2.

1989

relative to population.15 While the trend growth
for both the state and local stock and the total
stock per capita slowed after 1970 (similar to
the measures in figure 2), the stock available
per person at each level of government did not
decline. Instead, the per capita stock rose until
about 1975 and then leveled off. The public
capital stock per person has remained nearly
unchanged for about 15 years.16 The patterns in
figures 2 and 3 differ because of differences in
the growth of the labor force relative to that of
the population. Since the early 1970s, these dif
ferences reflect the faster growth in the labor
force relative to the population due, in part, to
the aging of baby boomers. The slowdown in
population growth associated with the end of
would be inversely related to the number of persons with
access to these public facilities. In this case, the capital
stock per person should fall over time as the population
grows.

16lf government capital or infrastructure were considered
“ social overhead capital,” so that a fixed quantity is re
quired for virtually any size population, then the quantity of
such capital per person, like other overhead measures,

MARCH/APRIL 1991

Figure 4
State & Local Net Capital Stock per Person
Thousands of Dollars per Person
(1982 Prices)
6

Thousands of Dollars per Person
(1982 Prices)

■■■■
■

" 'W

1 feL

1950
55
NOTE: End-of-year data.

the baby boom is also one reason for the slower
pace of capital formation shown in figure 1.

Trends in Com ponents o f State
and Local Capital Stochs
Further insight into the changing trends of
state and local government capital can be ob
tained by a closer inspection of the stocks’ com
ponents. Table 1 indicates that nearly half of
this capital consists of highways, streets and
educational buildings. These two types of capital
goods have been influenced by demographic
factors and trends in transportation.
Figure 4 shows the state and local govern
ment capital stock per person, its two major
components, (highways and streets, and educa
tional buildings) and the total excluding these
two components (labeled "other”). The break in
17Rubin (1990) has also noted the relationship between the
decline in school-age population and the slowdown in the
growth of public capital. She uses movements in the
population aged 5 to 15 to measure the demographic
variable and indicates that it fits movements in productivity

FEDERAL RESERVE BANK OF ST. LOUIS

1989

the trend shown in figures 2 and 3 is heavily
concentrated in these two sectors. As shown in
figure 4, the growth in the rest of the capital
stock per capita (labeled "other”) slowed after
1970, but remained positive. From 1970 to 1989
this category grew at a 1.7 percent rate. While
this growth was down from its 3 percent rate
from 1950 to 1970, it nearly matched the 1.8
percent growth rate of real GNP per capita over
the period. Thus, the slowdown in growth for
real non-military public capital stock is largely
concentrated in the quantities of highways,
streets and educational facilities per person.
Demographics have played a role in the
decline in the stock of per capita state and local
educational facilities. Figure 5 shows the stock
of these facilities per person and the share of
the school-age population (aged 5 to 24) in the
total population. 17 Due to the baby boom, the
growth about as well as those in the public capital stock.
She argues that this use of the demographic variable has
no conceptual basis, however, and suggests that either
relation is spurious.

Figure 5
Stock of Educational Facilities per Person and the
Share of Young People in the Total Population
Thousands of Dollars per Person
(1982 Prices)

Percent
38

1.2

Educational
Facilities

0.4
1950
’ Young people are those ages 5-24.
NOTE: Capital stock is end-of-year data. Population is mid-year data.

proportion of the population in that group rose
rapidly, as did the stock of educational facilities
per person. The share of youth in the popula
tion peaked in 1971 and has declined sharply
since then. The occurrence of this peak matches
that of total government real nonmilitary capital
stock per person in the labor force (figure 2)
and leads by five years the outright decline in
the stock of educational facilities per person.18

The baby boom and its associated population
trends influenced the demand for other capital
goods, including highways and streets. The

18Private sector educational facilities per person have also
declined. Such facilities peaked in 1972 at $217 per per
son after growing at a 3.7 percent rate from 1950 to 1972.
From 1972 to its lowest subsequent level, the per-capita
stock of private educational facilities declined at a 1.6 per
cent rate. From 1986 to 1989, this measure rose about 1
percent.
The stock of public educational facilities per school-aged
person (5-24) rose at a 3.5 percent rate from 1947 to 1969,
a 2.6 percent rate from 1969 to 1976 and a 0.4 percent
rate from 1976 to 1989. Thus, the decline in the level of

28
1989

growth rate of the population was relatively
rapid in the 1950s and began to slow sharply in
the 1960s. From 1955 to 1960, the population
grew at a 1.7 percent rate, and then slowed
steadily to a 1.1 percent rate from 1965 to 1970.
The population growth rate subsequently fluc
tuated between about 1.0 and 1.1 percent for
five-year intervals.
Other factors also slowed the growth of
highway and street capital per person.19 The in
terstate highway system program, funded begin
ning in 1956 and which was largely built be
tween 1963 and 1975, led to a sharp rise in

educational facilities per person in figure 5 did not arise
from a decline in this specific public capital stock per
young person. The latter did not decline, only its growth
rate fell.
19A CBO study of the nation’s public works (1988) indicates
that “ the capacity of the existing major network is broadly
sufficient for its traffic” (p. 5). The study does indicate that
some regions have relatively high urban traffic congestion,
but suggests that this is matched by a relatively large ex
cess capacity elsewhere.

MARCH/APRIL 1991

Figure 6
State and Local Stock of Highways and Streets per
Person and the Growth of Miles per Person
Thousands of Dollars per Person
(1982 Prices)

Percent

'Compounded annual rate for the five-year period ending in each year.
NOTE: Capital stock is end-of-year data.

highways and roads per person. As the program
neared completion, highway spending slowed
sharply. Changes in the price of fuel also in
fluenced the demand for highways and streets
by reducing the growth in road usage. Sharp in
creases in fuel costs in 1974 and 1979-80 reduced
the total miles driven per person. The number
of miles driven per capita grew at a 4.1 percent
rate from 1968 to 1973 (figure 6).20 The 5 year
growth rate subsequently declined to a 0.3 per
cent rate from 1978 to 1983. When oil prices
began to decline in 1981, this trend was revers
ed and miles per person grew at a 3.1 percent
rate from 1983 to 1989. Highway and street
capital per capita bottomed out in 1986.
While demographic and driving patterns may
not fully explain the slower growth of govern
ment capital, they suggest that the decline in
the growth of public capital was not inexplicably
20The mileage data are prepared by the U.S. Federal
Highway Administration and published in their “ Highway
Statistics, Summary to 1985." These and subsequent un-

FEDERAL RESERVE BANK OF ST. LOUIS

capricious, unplanned, or completely induced
by the federal budget deficit. The decline was
at the state and local level, where government
is considered to be more responsive to voter
demands, and was largely due to reductions in
the stock of highways, streets and educational
facilities per person. The latter reductions were
consistent with changes in demographics and
driving patterns.

The Relative P rice o f Public
Capital Goods
The supply price of public capital goods also
is important when determining the optimal quan
tity of public goods. Given the demand for public
capital, a rise in the price of public capital
goods and their services will reduce the optimal
quantity of such goods and services.
published data were obtained from the U.S. Federal
Highway Administration.

Figure 7
Price and Quantity of Public Capital Relative to
Private Capital
Ratio

Ratio

'R a tio o f the con stant d o lla r net stocks o f pu blic n o n m ilita ry capital and private nonresidential fixed
capital stocks.
2Ratio of the implicit price deflators for public nonmilitary investment and for private nonresidential fixed
investment. (1982=1.0)

Figure 7 shows the ratio of the implicit price
deflator for gross investment in non-military
public capital to the implicit price deflator for
private non-residential fixed investment and the
ratio of the non-military public capital stock to
the private non-residential capital stock from
1948 to 1989.21 The relative quantity of non
military public capital provides another in
dicator of the declining trend in non-military
public capital. When assessed relative to private
capital, the decline began in 1965 and accelerated
after 1972. This decline followed a general rise

in the relative price of public capital goods that
began in 1961 and became especially sharp
after 1968. There were declines in this price in
1975-77 and in 1981-82, however. A sharp
decline in the relative price of public capital
from 1952 to 1960 also was associated with a
rise in the relative quantity of the public capital
stock. While this rise began in 1952, it was
largest from 1957 to 1964.

21The price ratio (1982 = 1.00) measures the price of fixed
capital goods purchased by governments relative to the
fixed capital goods purchased by the private sector. The
deflator for government capital is constructed as the ratio
of current dollar non-military public gross fixed investment
to its constant dollar (1982 prices) counterpart.

streets and educational buildings. The correlation between
the relative quantity of non-military public capital and the
ratio of state and local highways, streets and educational
buildings to private capital is 0.92, while the correlation for
the relative price of all public capital and that of only the
highways, streets and educational buildings is 0.88. Both
correlation coefficients are statistically significantly dif
ferent from zero at a 95 percent confidence level.

22The patterns shown in figure 7 are nearly identical for the
relative quantity and price of state and local highways,

The relative stock of public capital is strongly
and inversely related to the relative price of
public capital.22 The correlation coefficient for

MARCH/APRIL 1991

this relative stock and relative price is - 0.88
(statistically significantly different from zero at
the 95 percent confidence level). The year-toyear changes in the relative price and the rela
tive quantities are also statistically significantly
and negatively correlated; their correlation coef
ficient is - 0 .3 7 (which also is significant at a 95
percent confidence level).23 Thus, the slowing
growth of the public capital stock has also been
associated with a rise in the price of such goods
relative to the price of private capital goods.

CONCLUSION
An increasing number of people advocate ad
ditional government capital spending as a means
of returning private sector output, productivity
and capital formation to form er trend levels.24
Even if public capital would have such effects,
the decline in the growth of public capital would
not provide a rationale for increasing the pace
of government capital formation. Most com
ponents of public capital yield direct benefits to
consumers as final users. Slower growth of such
capital can be justified in a variety of circum
stances, including the reduced growth of con
sumer demand for such capital or higher prices
for such capital goods.
Upon closer inspection, it is difficult to argue
that the decline in public capital growth per
worker has been unwarranted and adverse, or
that it should be reversed, simply because of its
occurrence or because it may have imposed
some costs on other sectors of the economy.
The decline in the quantity of public capital per
worker has been associated with a rise in the
labor force relative to the population and is
largely due to a decline, on a per capita basis,
in the two largest components of public capital:
highways and streets and educational facilities.
The reductions in these two components have
occurred for demographic reasons. The decline
in public capital growth has also been associated
with a rise in the prices of public capital goods
relative to private capital goods.
The view that federal policy should redress
the purported adverse effects of the decline in
the public capital stock per worker is misleading,
at best. The decline in the public capital stock
23For just the highways, streets and educational buildings
components, the correlation coefficient for the level of
relative prices and relative quantities is - 0 .7 4 and that for
their first-differences is -0 .3 0 ; again, both of these are
statistically significant at a 95 percent confidence level.

FEDERAL RESERVE BANK OF ST. LOUIS

per person has not been the result of changing
trends in federal spending, but rather reflects
decisions by a multitude of state and local
governments, principally in response to increased
relative prices and to reduced demand for high
ways, streets and public educational facilities.

REFEREN CES
Aschauer, David Alan. “ Infrastructure: Spending Trends and
Economic Consequences,” paper presented at the Western
Economic Association meetings, San Diego, CA, June 30,
1990.
________ “ Does Public Capital Crowd Out Private Capital?
Journal of Monetary Economics, (September 1989b), pp.
171-88.
________“ Is Public Expenditure Productive?” Journal of
Monetary Economics (March 1989a), pp. 177-200.
________“ Fiscal Policy and Aggregate Demand,” American
Economic Review (March 1985), pp. 117-27.
Congressional Budget Office. New Directions for the Nation’s
Public Works, September 1988.
________Federal Policies For Infrastructure Management,
June 1986.
Eberts, Randall W. “ Public Infrastructure and Regional
Economic Development,” Federal Reserve Bank of
Cleveland Economic Review (Quarter 1, 1990), pp. 15-27.
Friedman, Benjamin. Day of Reckoning (Random
House, 1988).
Gramlich, Edward M. “ Financing Infrastructure Investment:
Should Money Be Thrown at the Third Deficit?” paper
presented at the Federal Reserve Bank of Boston con
ference on “ The Third Deficit: The Shortfall in Public In
vestment,” Howard’s Port, MA, June 27-29, 1990.
Krueger, Anne O. “ Government Failures in Development,”
Journal of Economic Perspectives (Summer 1990),
pp. 9-23.
Malabre, Alfred L. “ Economic Roadblock: Infrastructure
Neglect,” Wall Street Journal, July 30, 1990.
Munnell, Alicia H. “ Why Has Productivity Growth Declined?
Productivity and Public Investment,” New England
Economic Review (January/February 1990), pp. 3-22.
Musgrave, John C. “ Fixed Reproducible Tangible Wealth
in the United States, 1984-87,” Survey of Current Business
(August 1988), pp. 84-87.
Musgrave, Richard A., and Peggy B. Musgrave. Public
Finance in Theory and Practice, 4th ed. (McGraw-Hill,
1984).
National Council on Public Works Improvement. Fragile Foun
dations: A Report on America's Public Works, final report of
the President and the Congress, February 1988.
Peterson, George E. “ Historical Perspectives on Infrastruc
ture Investment: How Did We Get Where We Are?” paper
presented at the American Enterprise Institute Conference,
“ Infrastructure Needs and Policy Options for the 1990s,”
Washington, DC., February 4, 1991.
2<1Tatom (1991) examines the shortcomings of existing pro
duction function estimates that find a statistically signifi
cant relationship between public capital and private output
and shows that the significant effect vanishes when the
relationship is appropriately estimated.

Ratner, Jonathan B. “ Government Capital and The Produc
tion Function for U.S. Private Output,” Economics Letters
(1983), pp. 213-17.
Reich, Robert B. “ The REAL Economy,” The Atlantic Monthly
(February 1991), pp. 35-52.
Rubin, Laura. “ Productivity and the Public Capital Stock:
Another Look,” paper presented at the Regional Analysis
Committee meeting of the Federal Reserve System, Oc
tober 4, 1990.
Smith, Adam. An Inquiry Into The Nature and Causes of The
Wealth of Nations, Edwin Canaan ed. (Modern Library,
1937).

Tatom, John A. “ Public Capital and Private Sector Perform
ance,” this Review, forthcoming.
________“ U.S. Investment in the 1980s: The Real Story,”
this Review (March/April 1989), pp. 3-15.
________“ Two Views of The Effects of Government Budget
Deficits in the 1980s,” this Review (October 1985),
pp. 5-16.
U.S. Department of Commerce, Bureau of Economic
Analysis. Fixed Reproducible Tangible Wealth in the United
States, 1925-85 (GPO, June 1987).

MARCH/APRIL 1991

Alison Butler
Alison Butler, currently a visiting scholar at the Swiss National
Bank, is an economist at the Federal Reserve Bank o f St.
Louis. Lora Holman provided research assistance.

Trade Im balances and
Econom ic Theory: The Case
fo r a U.S.-Japan Trade Deficit

A h e U.S. GOVERNMENT and members of the
media have exchanged heated rhetoric with
Japan regarding the existence and size of the
trade deficit between the two countries which,
according to the U.S. Department of Commerce,
stood at $42 billion in 1990.1 The rhetoric on
both sides is exemplified by books such as Trad
ing Places: How We Allow Japan to Take the
L ead in the United States by former U.S. trade
negotiator Clyde Prestowitz, Jr. and Japan is Not
to Blam e: It's America's Fault by Osamu
Shinomura, a government economist in Japan.
Each of these books blames the other country
for the large bilateral trade imbalance between
the two. This type of rhetoric assumes that the
existence of a bilateral trade deficit is prim a
fa c ie evidence that at least one country is an
unfair trader.
In addition, these types of books implicitly
endorse what is called the “mercantilist” view
of trade—that the bigger the trade surplus a
country runs (at the expense of trade deficits
for other countries), the better off that country

'This is seen by the increased call from many in Congress,
such as Sen. Richard Gephardt of Missouri and Rep. Helen
Bentley of Baltimore, for increased protectionist policies.
Similarly, Congress recently passed the Omnibus Trade
and Competitiveness Act of 1988, which greatly increased

FEDERAL RESERVE BANK OF ST. LOUIS

is. This view, however, was discredited long ago
because it denies that trade can be mutually
beneficial to both countries. Of course, if trade
were not mutually beneficial, it would not
occur.
Misperceptions about the nature of the trad
ing relationship and the cause of the trade
imbalance between the United States and Japan
exist on both sides of the Pacific. This article
looks at some of the underlying causes of bilat
eral trade imbalances. The paper first examines
this issue theoretically, then focuses specifically
on trade between the United States and Japan.
The purpose of this article is to determine
whether the U.S. trade deficit with Japan is a
natural consequence of the composition of trade
between the two countries, the result of "un
fair” trading practices, or some combination of
the two. The paper concludes with a discussion
of recent trade talks between the United States
and Japan, and some words of caution about in
terpreting the meaning of the bilateral trade
deficit with Japan.

the power of the U.S. trade representative and Congress
to bring charges of unfair trade practices against foreign
countries.

TRIANGULAR TRADE
In a world in which trade is conducted across
many countries, it is extremely unlikely that trade
would be balanced between all pairs of coun
tries, especially if there are significant differ
ences in the composition of their imports and
exports. As demonstrated below, it is far more
likely that a country will import goods from one
country and export goods to another. This pat
tern is called "triangular” trade.

The Composition o f U.S.-Japan
Trade

A Simple Example
Suppose there are three islands, A, B and C,
each of which produces one product. Island
A produces fuel, which is used to keep the peo
ple of Island A warm, and as an input in boat
production on Island B. Island B produces boats,
but needs fuel from Island A to do so. These
boats are used for fishing (the residents of
Island B eat only fish) or shipping food. Island
C produces fruits and vegetables for domestic
consumption, and can also sell them to Island A
(whose residents are all vegetarians) if it has
boats. The residents of Island C also desire boats
for recreational purposes.
If, for simplicity, the value of goods exported
is the same for each island and equals
$100,then trade can be described by the follow
ing table:
Exports (+)
Island

A B C

—

100 0
—

100 0

100
—

Imports (- )

Balance

A B C
—

100

0 100
0

—

0 100

(100(

100 ) = 0

1 0 0 - 100 ) = 0

(100 - 100 ) = 0

Although no island in this example runs an
overall trade deficit, each island runs bilateral
trade imbalances with the other two. As this
highly-simplified example suggests, only under
improbable circumstances will trade balance
between any pair of countries.
One implication of this example is that policies
which hinder trade in an attempt to reduce bi
lateral trade imbalances will generally make
both countries worse off. For example, a coun
try with few energy resources will import
significant amounts of oil from oil-exporting
countries. To run balanced trade with the oil-

exporting countries, the oil-importing country
would have to export an equivalent amount of
products (in terms of value) to the oil-exporting
countries. If, however, the oil-exporting coun
tries prefer products produced by other coun
tries, the oil-importing country can eliminate
its bilateral trade deficits only by importing less
oil, which could have negative repercussions on
both economies.

Japan has few natural resources, and conse
quently, as table 1 shows, relies heavily on im
ports of oil and raw materials. To pay for these
imports, Japan primarily exports finished manu
factured products to industrialized countries,
with the United States receiving the largest pro
portion of these exports.
The composition of U.S. trade differs consid
erably from that of Japan because the United
States primarily imports and exports finished
products. In fact, in 1988 (the last year for which
complete data is available), approximately 80
percent of U.S. trade (both imports and exports)
was in manufactured goods. This difference in
the composition of trade (seen in figure 1) for
the two countries is reflected in the bilateral
trade balances between the countries.
Table 2 shows bilateral merchandise trade
balances for the United States and Japan vis-avis each other and other groups of countries
for selected years between 1965-1989. While
there has been a substantial increase in the over
all U.S. trade deficit since 1975 (with some im
provement in recent years), the United States
generally runs a surplus (or smaller deficit) with
the western European countries (who import a
significant amount of manufactured products)
and a deficit against Japan and the developing
countries. This pattern existed even before 1976,
when the US. merchandise trade balance turned
negative. For example, although the United
States had a $2.2 billion trade surplus overall in
1975, its bilateral trade balance with Japan was
a $2.8 billion deficit.
Japan, on the other hand, generally runs a
trade deficit against the oil-exporting countries
and Canada (which exports raw materials and
food to Japan) and a surplus against the United
States; this surplus occurred even in 1980, when

MARCH/APRIL 1991

FEDERAL

Table 1

RESERVE

Trade by Commodity (as percent of total)

BANK OF ST. LOUIS

Japan
Crude materials
excluding
petroleum

Food,
beverages,
tobacco
Date

Im ports

1964
1970
1975
1980
1985
1986
1987
1988

17.5%
13.6
15.2
10.5
12.1
16.0
15.2
15.5

M achinery and
transport
equipm ent

Mineral
fuels

O ther
m anufactured
goods

Exports

Im ports

Exports

Im ports

Exports

Im ports

Exports

Im ports

Exports

4.8%
3.4
1.4
1.2
0.7
0.7
0.7
0.6

39.0%
35.4
20.1
17.0
14.2
14.8
15.2
15.0

3.2%
1.8
1.6
1.2
0.8
0.7
0.7
0.7

17.7%
20.7
44.3
50.0
43.8
30.9
26.8
20.6

0.4%
0.2
0.4
0.4
0.3
0.3
0.3
0.2

10.4%
11.3
6.6
6.3
8.7
11.1
11.8
13.2

29.3%
40.5
49.2
54.9
61.7
63.8
65.3
69.4

15.4%
19.0
13.7
16.2
21.2
27.2
31.0
35.8

62.3%
54.0
47.5
42.3
36.4
34.5
33.0
29.0
00

United States
Food,
beverages,
tobacco
Date

Im ports

1964
1970
1975
1980
1985
1986
1987
1988

21.4%
15.6
10.2
8.0
6.8
7.0
6.3
5.5

Crude m aterials
excluding
petroleum

O ther
m anufactured
goods

Exports

Im ports

Exports

Im ports

Exports

Im ports

Exports

Im ports

Exports

17.4%
11.9
15.8
14.2
10.7
9.8
9.3
10.1

15.9%
8.7
6.2
4.4
3.1
3.0
3.0
3.1

13.0%
12.0
10.1
11.8
8.8
8.8
8.6
8.7

10.7%
7.7
27.2
32.8
15.5
10.4
11.1
9.3

3.5%
3.7
4.2
3.7
4.9
4.0
3.2
2.7

11.8%
28.0
25.0
25.0
37.8
41.8
42.1
43.7

35.8%
42.0
43.1
40.3
47.6
48.3
46.8
46.1

40.1%
40.1
31.3
29.8
36.7
37.9
37.6
38.4

30.3%
30.4
26.8
30.0
28.1
29.1
32.1
32.4

SOURCE : OECD Economic Outlook: Historical Statistics. OECD Economic Surveys: Japan (1990)

M achinery and
transport
equipm ent

Mineral
fuels

Figure 1
United States and Japan’s Trade by Com m odity
for 1988
Percent of Total Trade

Percent of Total Trade

100

Exports

Imports

Exports

Imports

SOURCE: OECD Economic Outlook: Historical Statistics
1This includes food, beverages, tobacco, crude materials and mineral fuels.
2This includes machinery, transport equipment and other manufactured goods.

Japan’s overall trade balance was in deficit.2
Thus, Japan must run a surplus against other
countries to pay for its trade deficit with (pri
marily) oil-producing countries even i f it w ere to
reduce its trade surplus to zero. As a result, it
appears unlikely that U.S.-Japan trade will ever
balance. According to one study, triangular

trade patterns alone predetermine a U.S. bilat
eral deficit with Japan of approximately $11
billion annually.3
Recent U.S. bilateral trade deficits with Japan,
however, have substantially exceeded $11 bil
lion. The U.S. deficit with Japan reached a high

2The Japanese trade deficit during those years was primar
ily a result of the oil shock of the 1970s, hence the sub
stantial increase in its trade deficit with oil-producing
countries.
3Bergsten and Cline (1985, 1987). When exchange rate
effects are included, this estimate becomes larger.

MARCH/APRIL 1991

Table 2
Merchandise Trade Balances: Japan and the United States (in billions of dollars)
Japan’s Trade Balance With

Date

United S tates1

O il-exporting
developing
countries

1965
1970
1975
1980
1985
1986
1987
1988
1989

$ 0.16
0.46
-0 .3 7
7.34
40.58
52.52
53.06
47.98
45.70

$ -0 .6 7
-1 .8 6
-1 1 .6 0
-3 9 .6 4
-2 5 .9 7
-1 3 .7 8
-1 6 .7 4
-1 6 .8 4
-2 2 .1 6

Non-oil-exporting
developing
countries

Canada

W estern2
Europe

$ 0.52
1.45
5.65
13.77
19.11
24.76
24.09
26.64
26.18

$ -0 .1 4
-0 .3 7
-1 .3 5
-2 .3 0
-0 .2 4
0.63
-0 .4 5
-1 .8 7
-1 .8 3

$ 0.10
-0 .0 4
1.24
7.34
10.78
17.32
20.05
21.24
17.16

O verall trade3
balance
$

0.40
0.44
-2 .1 2
-1 0 .8 5
46.67
83.06
80.43
77.48
64.96

United States’ Trade Balance With

Date

Japan1

O il-exporting
developing
cou ntrie s

1965
1970
1975
1980
1985
1986
1987
1988
1989

$ -0 .5 3
-1 .6 0
-2 .7 8
-1 2 .1 8
-4 9 .7 5
-5 8 .5 8
- 59.83
-55.51
-5 2 .5 3

$ - 0 .3 9
0.11
-9 .5 0
-4 0 .1 9
-9 .6 6
-9 .3 6
-1 3 .6 4
-1 0 .2 9
-1 8 .2 8

Non-oil exporting
developing
cou ntrie s
$

1.89
2.26
6.03
1.00
-4 3 .2 7
-4 8 .7 2
- 58.03
-4 9 .8 0
-4 9 .2 7

Canada
$

0.40
-2 .7 0
-1 .0 1
-6 .6 0
-2 2 .1 8
- 23.33
-1 1 .7 0
-1 1 .7 5
-1 1 .2 8

W estern2
Europe
$

2.85
2.49
7.86
19.04
-2 5 .9 9
-3 0 .5 4
- 29.05
-1 5 .4 7
-2 .0 7

O verall trade3
balance
$

4.48
0.54
2.15
-3 6 .1 8
-1 4 8 .4 7
-1 6 9 .7 8
-1 7 1 .1 8
-1 4 0 .3 6
-1 2 9 .5 2

1The U.S.-Japan and Japan-U.S. trade balances do not sum to zero because import values include cost, insurance and freight
(cif), while export values only include free on board (fob) costs.
in c lu d e s Australia and New Zealand.
3The first five columns do not sum to the total trade balance because the Soviet Union and (former) eastern-bloc countries
are excluded due to the unreliability of the data.
SOURCE: International Monetary Fund, Direction of Trade.

FEDERAL RESERVE BANK OF ST. LOUIS

of $57 billion in 1987, and currently (in 1990)
stands at $42 billion.4 To explain deficits of this
magnitude, other factors besides triangular trade
patterns must be examined.

where C is private domestic consumption, I is
private domestic investment, G is government
spending, X is exports, and M is imports. This
can be rewritten as:7
(2) CA = (S - I) + (T - G),

O TH ER ECONOMIC FACTORS
THAT A FFEC T TH E TRADE
BALANCE
Several other factors play a role in deter
mining the magnitude of the trade deficit with
Japan. These factors are both macroeconomic
(such as the different savings and investment
rates) and microeconomic (such as industry
structure and barriers to trade).5

M acroeconom ic Factors
A nation’s savings and investment behavior
has a significant effect on its trade balance. The
balance on goods described in the previous sec
tion is called the merchandise trade balance; it
is the most commonly cited trade balance statis
tic. The current account, the most general mea
sure of a country’s trade balance, includes trade
in services and earnings on foreign investment
both in the United States and abroad (see shaded
insert on next page).6
The macroeconomic determinants of the cur
rent account can be generated from national in
come accounting identities. The gross national
product (GNP) of a country is defined as the
following:
(1) GNP = C + I + G + X - M ,

4These numbers are from the U.S. Department of Com
merce, which uses the same classification for both imports
and exports, and thus is a more consistent series. These
numbers are not used in the table, however, because a
similar measure of the trade balance for Japan is not
available.
5Another important macroeconomic factor is the behavior of
the dollar/yen exchange rate. However, the nature of the
relationship between exchange rates and the trade bal
ance is the source of a vast literature. The exact link be
tween these variables remains unresolved. Even studies
that argue that the exchange rate has been a substantial
factor in the trade imbalance between the United States
and Japan remain unable to completely explain the size of
the trade deficit. For simplicity, therefore, the relationship
between the dollar/yen exchange rate and the U.S.-Japan
trade balance is ignored here. For discussions of this rela
tionship, see, for example, Haynes, Hutchison and
Mikesell (1986), Bergsten and Cline (1985, 1987) and
Sakamoto (1988).

where CA is the balance on the current account,
T is tax revenue and S is private domestic sav
ings.8 Thus, any surplus (deficit) in the current
account must be due to an excess (shortfall) of
net domestic savings, either private, as shown
by (S—I), and/or public savings, as given by
(T-G ).
Table 3 presents the current account balance
for the United States and Japan for selected
years between 1965 and 1988, along with esti
mates of net domestic savings and investment
rates as a percentage of their respective gross
domestic products (GDP).9 Any shortfall in net
savings must be made up by importing foreign
savings; this relationship is measured in the cur
rent account. If a country has a negative cur
rent account balance, such as the United States,
it is a net debtor (that is, it owes more to the
rest of the world than it is owed).
As is seen in table 3, the savings/investment
differential is almost identical to the balance on
the current account.10 If the savings/investment
(or current account) balance is negative, as it
has been for the United States since 1982, then
that country is a net debtor, which simply means
that it spends more on government expen
ditures and private investment than it saves.
This indicates that U.S. citizens have chosen
higher levels of consumption now at the ex-

together over time. Although the current account is the
more general measure of trading activity, bilateral current
account figures are not available. For a more detailed
discussion of the balance of payments statistics, see U.S.
Department of Commerce (1990).
7For a derivation of Equation 2, see Chrystal and Wood
(1988).
8For details and a more complete discussion of national
income accounting in an open economy, see Dornbusch
(1980).
9The difference between GNP and GDP is that GDP ex
cludes net factor payments from abroad while GNP in
cludes them.
10ln fact, the only difference is due to measurement error,
because the current account is measured in terms of
goods, services and income flows rather than in terms of
savings and investment. The difference is analogous to the
statistical discrepancy between the expenditure and in
come approaches to calculating GNP.

6While the magnitude of the merchandise trade and current
accounts are not identical, they have moved fairly closely

MARCH/APRIL 1991

D ifferen t M easu res of th e T rad e B alan ce
The merchandise trade balance measures
the difference between the value of im
ported and exported goods. The merchan
dise trade figures do not account for other
forms of trade such as trade in services
and investment. These are included in the
current account, which is the more
general measure of the trade balance. In
1989, merchandise trade accounted for

73.4 percent ($360 billion) of total U.S. ex
ports ($491 billion), and 85.7 percent ($493
billion) of total U.S. imports ($575 billion). As
trade in services and foreign direct invest
ment increases internationally, the distinction
between the two measures of the trade
balance becomes more important. The dif
ference can be seen in the following
summaries:

C re d its to th e c u r r e n t a c c o u n t

D ebits to th e c u r r e n t a c c o u n t

Exports of goods such as food and
computers produced in the United States.

Imports of goods such as cars produced
in Japan.

Exports of services such as banking and
insurance sold to foreigners by U.S.
companies.

Imports of services such as a U.S. citizen
holding a Swiss bank account in
Switzerland.

Receipts of income on U.S. assets abroad
such as repatriated earnings of a Ford
assembly plant in Mexico.

Payments of income on foreign assets in the
United States such as profits earned
by a Japanese auto plant in Ohio.
Unilateral transfers such as U.S. government
transfer payments abroad.

B a la n ce o n M erch an d ise T ra d e = E x p o rts of G oods - Im p o rts of G oods.
B a la n ce o n th e C u rre n t A cco u n t = C red its - D ebits.

1 9 8 9 C u rre n t A cco u n t B a la n ce F o r T h e U nited S tates
(millions of dollars)
Exports of goods, services and income
Merchandise, excluding military
Services
Income receipts in investment

$603,169
360,465
115,169
127,536

Imports of goods, services and income
Merchandise, excluding military
Services
Income payments on investments

$698,483
475,329
94,706
128,448

Unilateral transfers

- $ 14,720

Balance on merchandise trade
Balance on current account

-$ 1 1 4 ,8 6 4
-$ 1 1 0 ,0 3 4

FEDERAL RESERVE BANK OF ST. LOUIS

Table 3
Savings, Investment and the Current Account (as percent of GDP)
Japan
Date

Investment

Savings

Savings-lnvestment

1965
1970
1975
1980
1985
1986
1987
1988

31.94%
39.02
32.77
32.24
28.52
28.12
29.15
30.96

33.34%
40.30
32.81
31.35
31.93
32.15
32.36
33.26

1.40%
1.28
0.04
-0 .9 0
3.41
4.03
3.21
2.31

Current Account
1.01%
0.98
-0 .1 4
-1 .0 1
3.71
4.38
3.67
2.80

United States
Date

Investment

Savings

1965
1970
1975
1980
1985
1986
1987
1988

20.16%
17.91
16.96
18.94
18.68
18.25
18.04
17.43

20.74%
18.04
17.81
18.36
15.69
14.94
14.65
15.22

Savings-lnvestment
0.57%
0.13
0.85
-0 .5 8
-2 .9 9
-3 .3 1
-3 .3 9
-2 .2 1

Current Account
0.77%
0.23
1.14
0.07
-3 .0 7
-3 .4 6
-3 .6 0
-2 .6 6

SOURCE: International Monetary Fund, International Financial Statistics.

pense of lower consumption later, instead of
saving more now and consuming more later.11
How does this affect U.S. trade with Japan?
Japan has a positive (and, until recently, an in
creasing) savings/investment balance.12 This
means its citizens have chosen to attain higher
levels of consumption in the future relative to
higher current consumption. As a result, Japan's
net savings are invested abroad, with much of
its savings flowing into the United States. In
deed, the United States has been a good place
for foreign citizens to invest their savings for
several reasons; until recently, the United States
"T h e re is much debate over whether the size of the U.S.
current account deficit is undesirable. For some arguments
as to why a current account deficit might have a positive
effect on an economy, see Chrystal and Wood (1988).
12The “ high” rate of savings in Japan is usually attributed
to many different factors, including the price of housing,
the demographics of the population and the tax system.

has had relatively higher interest rates than
many industrial countries, and it is a safe haven
for foreign investments since there is essentially
no possibility that the United States government
will default on its bonds.
As a result, the fundamental difference in
their net savings positions is a significant factor
in the size of the U.S.-Japan bilateral trade
deficit.

M icroecon om ic Factors
Microeconomic factors also affect the volume
of imports and exports and therefore the trade
For a discussion of the savings and investment rates and
their implications for Japan’s economy, see Bergsten and
Cline (1985, 1987) and Belassa and Noland (1988). For a
discussion of some problems in measuring the savings
rates in the two countries, see Christiano (1989) and
Hayashi (1989).

MARCH/APRIL 1991

balance of a country. It is often argued that
Japan has implicit and explicit trade barriers
and that its economy inherently has a more
protectionist structure than economies in other
industrialized countries.
There have been several attempts to measure
empirically how “open” the Japanese economy
is to imports. Such estimates vary widely due to
different assumptions and methodologies. In
general, however, these studies usually find lit
tle evidence that the composition of Japan’s ex
ports and imports deviates substantially from
what general economic trade theory would pre
dict, given Japan's comparative advantage and
location.13 Nevertheless, the perception that
Japan is more protectionist than other industri
alized countries still remains. As a result, there
have been several bilateral negotiations between
the United States and Japan such as the MarketOriented Selected Sector (MOSS) talks in the
mid-1980s directed at increasing trade in spe
cific sectors.14 There is some evidence that these
talks have had some success. In the past four
years, U.S. firms have increased their annual
sales in semiconductors by nearly $1 billion
(roughly a 4 percentage point increase).15 More
recently, such talks have expanded to include
more general policies that affect trade, as seen
in the recently-concluded Structural Impedi
ments Initiative (SII). These talks were an at
tempt to “identify and solve structural prob
lems in both countries that stand as impedi
ments to adjustments in trade and balance of
payments accounts, with the goal of contribu
ting to a reduction of trade imbalances.”16 The
SII is discussed in more detail below.
E x p licit T ra d e B a r r ie r s — In terms of its
explicit trade barriers, Japan's tariffs and subsi
dies are fairly low. Figure 2 shows the change
in tariffs before and after the Tokyo Round of
negotiations of the General Agreement of Trade
and Tariffs (GATT), the last completed round

13See, for example, Saxonhouse and Stern (1989), Bergsten
and Cline (1985,1987) and Belassa and Noland (1988). For
a critical discussion of these studies, see Cline (1990). For
a dissenting view, see Lawrence (1987).
14These talks focused on four areas: telecommunications,
electronics, forest products and medical equipment and
pharmaceuticals. These sectors (excluding forest products)
are imperfectly competitive, which creates the possibility
that profits are earned above economic costs (including
the cost of capital). Recent trade theory has suggested
that government can, through trade barriers, shift these
profits from foreign to domestic firms. This policy is called

FEDERAL RESERVE BANK OF ST. LOUIS

of multilateral trade negotiations, which demon
strates that Japan has reduced these barriers by
more than the European Community. In fact,
Japanese tariff rates on industrial goods are,
on average, lower than U.S. rates. Overall, most
studies agree that Japan’s explicit trade barriers
are not out of line with other industrialized
countries.
There are certain sectors, however, that re
main heavily protected in Japan. The most ex
treme example is rice, which cannot be import
ed by Japanese law. Japanese officials argue
that this ban is necessary for national secur
ity, because of Japan’s dependence on foreign
sources for much of its food stuffs. U.S. indus
try officials, on the other hand, argue that if
the ban were lifted, U.S. rice exports could rise
to as much as $656 million.17
Typically agriculture is heavily subsidized in
industrialized countries; the U.S. rice industry is
no exception. As of 1986, the U.S. rice industry
was the most heavily subsidized U.S. grain.18 As
a result, if current negotiations succeed in open
ing Japan’s rice market to foreigners, U.S. rice
subsidies will have a distorting effect on trade.
Studies that attempt to measure the effect of
eliminating this trade barrier with Japan must
take into account the price distortions resulting
from subsidies in other countries as well.
Im p licit T ra d e B a r r ie r s — A more con
tentious issue between these countries, how
ever, involves implicit trade barriers, which are
less clearly defined and therefore their effect is
more difficult to measure. Such barriers can
take various forms. Table 4 provides a descrip
tion of selected trade barriers, most of which
Japan has been accused of employing. The first
section lists explicit trade barriers, while the
lower half of the table lists implicit barriers. For
example, standards, testing and certification
procedures can be used as trade barriers if the
regulations discriminate against foreign firms.

“ strategic trade policy,” (although the term strategic refers
to the government actions, not national security).The prac
tical problems of trying to use this policy, however, appear
to override its theoretical justification. For further discus
sion, see Krugman (1987) and Coughlin and Wood (1989).
15Schlesinger (1990)
16Assistant trade secretary Charles Dallara in Rowen (1990).
17Office of the U.S. Trade Representative (1990).
18This is determined using producer subsidy equivalents.
For more detail, see Webb, Lopez and Penn (1990).

Figure 2
Rate of Average Tariffs
Before and After the Tokyo Round

Before

After

Before

After

SOURCE: OECD Economic Surveys: Japan

One example of this type of discriminatory prac
tice — taken from a recent book by form er U.S.
trade negotiator Clyde Prestowitz, Jr .— occurred
when U.S. firms attempted to sell baseball bats
in Japan.19 After a U.S. company finally received
approval to sell aluminum bats in Japan, new
standards for the required safety seal from the
government were introduced that necessitated
the use of a specific aluminum alloy as well as a
base plug not found in U.S.-produced bats.
After the U.S. filed a formal complaint through
procedures established by GATT, the standards
were revised to allow U.S. firms access to the
Japanese aluminum baseball bat market. New

restrictions, however, were then passed, requir
ing inspection of the factory and products to
take place in Japan. Because the bats were pro
duced in the United States, Japanese officials
individually inspected every lot of bats upon ar
rival in Japan. This slowed down imports and
increased the cost of importing bats, making
them less competitive. While this issue has since
been resolved and the restrictive requirements
have been eliminated, it provides a good exam
ple of how implicit trade barriers are used.20
Another example of implicit trade law restric
tions is seen in the Japanese Large Scale Retail
Store Law, which was recently modified as a re-

19See Prestowitz, Jr.(1988).
20For further discussion and analysis of implicit trade bar
riers in Japan, see Christelow (1985-86), Cline (1990) and
Belassa and Noland (1988).

MARCH/APRIL 1991

Table 4
Selected Non-Tariff Trade Barriers
Explicit Trade Barriers
1. Import quotas

Restrictions on quantity and/or value of imports of specific com
modities for a given time period; administered globally, selectively
or bilaterally.

2. Voluntary export
restraints

Restrictions imposed by importing country but administered by
exporting country; administered multilaterally and bilaterally; re
quires system of licensing; essentially similar to an orderly
marketing arrangement.

3. Domestic content and
mixing requirements

Requires that an industry use a certain proportion of domestically
produced components and/or materials in producing final products.

4. Antidumping duties

Imposition of a special import duty when the price of imports is
alleged to lie below some measure of foreign costs of production;
minimum prices may be established to “ trigger” antidumping in
vestigations and actions.

5. Countervailing duties

Imposition of a special import duty to counteract an alleged foreign
government subsidy to exports; normally required that domestic in
jury be shown.

Implicit Trade Barriers
1. Government
procurement policies

Preferences given to domestic over foreign firms in bidding on
public-procurement contracts, including informal procedures favour
ing procurement from domestic firms.

2. Macroeconomic
policies

Monetary/fiscal, balance-of-payments, and exchange-rate actions
which have an impact on national output, foreign trade and capital
movements.

3. Competition policies

Antitrust and related policies designed to foster or restrict competi
tion and which may have an impact on foreign trade and
investment.

4. Government industrial
policy and regional
development measure

Government actions designed to aid particular firms, industry
sectors, and regions to adjust to changes in market conditions.

5. Government financed
research and
development and
other technology
policies

Government actions designed to correct market distortions and aid
private firms; includes technological spillovers from government
programmes, such as defense and public health.

6. Health and sanitary
regulations and
quality standards

Actions designed for domestic objectives but which may
discriminate against imports.

7. Safety and industrial
standards and
regulations

Actions designed for domestic objectives but which may
discriminate against imports.

SOURCE: Deardorff and Stern, 1985.

FEDERAL RESERVE BANK OF ST. LOUIS

suit of the SII talks. This law required neighbor
hood approval in Japan for any store larger
than 5400 square feet. Neighborhood shop
owners were able to effectively block large
retailers from opening stores that could charge
lower prices because of their economies of scale
(for example, because of the volume of each
product bought by Toys "R” Us, it can charge
lower prices than a small neighborhood toy
store that purchases only a few of each
product).
This policy blocked foreign firms, such as Toys
"R” Us, from opening stores in Japan. Some
firms in the United States also argued that lim
iting the size of each shop reduced the likeli
hood of U.S. products being sold. Many Japan
ese, on the other hand, argued that these laws
serve to protect a way of life in Japan. Al
though changes in the retail store law that
weaken the power of neighborhood shop owners
may increase U.S. access to these markets, it
may also, in the eyes of some Japanese, have
adverse social consequences by altering the
structure of neighborhoods.21 This conflict be
tween domestic and trade policy goals underlies
much of the problems that arise in negotiating
trade disputes between Japan and the United
States.

IM PLICIT TRAD E BA R R IER S:
TRAD E PO LICY OR DOMESTIC
PO LICY?
One difficulty in determining the intent of
economic policies is the problem of separating
domestic and trade-related policies, such as in
the retail law described above. Another example
of this problem can be seen in the different
anti-trust legislation in the United States and
Japan. While it is illegal in the United States
for a company to require a distributor to sell
only the company’s line of products, this prac
tice is permitted under certain conditions in
Japan.22 As a result, many U.S. manufacturers
have had difficulty finding distributors for their

21See, for example, Sanger (1990).
22Under the Antimonopoly Act, restricting the business tran
sactions between the firm ’s trading partners and competi
tors is, in general, illegal if such conduct could result in
limiting the options of the firm ’s competitors. Whether the
business opportunities of the competitor have been re
duced is to be decided on a case-by-case basis, and there
are many exceptions to these regulations.

products because the existing distribution sys
tem is controlled by companies already in the
market. While U.S. policymakers have consid
ered this an unfair trade practice, new Japanese
firms trying to enter these same markets en
counter the same difficulty. Thus, referring to
these regulations as “unfair” trade practices may
be incorrect, because the policy treats both
foreign and domestic firms exactly the same. In
deed, at the heart of this problem is the debate
between the notion of reciprocity, the idea that
firms must be given the same opportunities in a
foreign country that foreign firms would have
in the domestic market, and the notion of na
tional treatment, which argues that foreign firms
should be treated the same as a nation’s
domestic firms.23
Another example of the problem of distin
guishing between domestic and trade policies is
the relationship between suppliers and pur
chasers. Companies in Japan typically have long
term implicit (and sometimes explicit) contracts
with their suppliers. While it has frequently
been alleged that these arrangements are in
tended to exclude foreign firms, they actually
serve several useful economic purposes unre
lated to foreign trade. For example, because
Japanese land prices are higher than those in
the United States, it is relatively more costly for
firms to hold inventories in Japan. As a result,
firms arrange for more frequent purchases of
inputs (referred to as “just in time” scheduling)
rather than maintaining sizable inventories of
raw materials. Long-term arrangements with
suppliers provide one way to economize on the
costs of frequent recontracting. Other Japanese
policies that are sources of trade disputes be
tween the United States and Japan include the
procedures for obtaining patents, governmentsupported research and development and public
expenditure on infrastructure (such as roads
and sewers).24
Criticism of domestic practices affecting the
flow of trade has also been leveled against the
United States. For example, the U.S. government

23This article assumes that national treatment is the appro
priate policy. For a recent discussion of this issue, see
Bhagwati and Irwin (1987).
24These policies are described in greater detail in Belassa
and Noland (1988).

MARCH/APRIL 1991

S tru c tu ra l Im p ed im en ts In itiativ e
The Structural Impediments Initiative (SII)
was an unusual trade agreement for several
reasons. In particular, the negotiations fo
cused primarily on domestic policies with im
plications for the trade deficit between the
United States and Japan. As a result,
negotiators were in the unusual position of
demanding changes that could benefit the
foreign country as well as the negotiators’
own country. For example, the U.S. wanted
the Japanese to “reform ” its distribution
system. In doing so, Japan might not only in
crease U.S. firms' access to these markets,
but also improve their own market
efficiency.1
These talks were also unusual in that they
addressed broader, more fundamental macroeconomic factors rather than focusing on
problems in specific sectors as in previous
talks. One problem with specific sector negot
iations is that agreements and their subse
quent enforcem ent do not necessarily imply a
reduction in the bilateral trade deficit. As
discussed by Frankel (1990), this "resultsoriented” approach has the unfortunate con
sequence that failures to reduce the trade
'This twist in trade negotiations is discussed in an aptly
titled paper, “ The Structural Impediments Initiative:
Japan Again Agrees to Become More Efficient,” by Jef
frey A. Frankel (1990).

deficit are seen as "proof” that the
agreements are not being honored.
One problem with SII is that much of the
"agreement” is not binding, particularly on
the United States. Rather, the language is
couched in terms of commitments to improv
ing these areas and not in terms of formal
obligations. Although many analysts believe
these talks will have little, if any, effect on
U.S.-Japan trade, there is some preliminary
evidence that both countries are at least
attempting to implement some of these
changes. The U.S. Congress has passed a new
budget-reduction plan; moreover, Japan's
budget calls for an increase in spending on
public infrastructure.2
S u m m a ry of th e SII a g re e m e n t
Among other things, Japan agreed to the
following:3
1. Expanding investment in social overhead
capital (e.g., water supply, sewers, hous
ing, parks), transportation infrastructure,
international ports and airports and cargo
and customs processing facilities.
3These summaries are taken from the Joint Report of
the U.S.-Japan Working Group on the Structural Im
pediments Initiative (1990).

2For a discussion of attitudes regarding the SII, see
U.S.International Trade Commission (1990).

budget deficit has been blamed for much of the
U.S. trade deficit, because it contributed to the
demand for foreign savings. As a result, the size
of the federal deficit has been an issue in re
cent negotiations to reduce the U.S. bilateral
trade deficit with Japan. Thus, issues that once
were considered purely domestic now begin to
enter trade negotiations. The most recent exam
ple of this phenomenon is the Structural Im
pediments Initiative (see shaded insert above).
25For a discussion on how to measure the effectiveness of
these types of policy declarations, see von Furstenberg
and Daniels (1990).

FEDERAL RESERVE BANK OF ST. LOUIS

W hether that initiative will be successful in
reducing the U.S.-Japan trade deficit is unclear.
Perhaps the biggest problem is that there are
few obligations for either country to implement
specific policies. Not surprisingly, as a result of
these talks there has been considerable debate
over the usefulness of this type of bilateral trade
negotiation.25 Recent evidence, particularly in
Japan, suggest that these talks have had some
effect. Aside from the changes to the Large-

2. Reviewing its land policies, including taxes,
use restrictions and zoning laws to more
fully utilize public lands.
3. Reviewing standards, testing and certifica
tion requirements, introducing greater
transparency in the issuance of official ad
ministrative guidance and in the opera
tions of industry advisory committees and
government study groups.
4. Improving import procedures and relaxing
laws and regulations that impede foreign
direct investment and restrict entry by
large retailers, liquor stores, truck
operators and pharmacies.
5. Examining and revising as necessary the
Japan Fair Trade Commission and other
government policies toward premium of
fers, advertisements and vertical business
practices affecting consumer goods (e.g.,
resale price maintenance, “suggested
prices,” exclusive dealerships or territories,
rebates and returns).
The United States also made a series of
concessions; however, these appear to be less

binding and generally emphasize “intention”
rather than "commitment.” Among other
things, the United States agreed to the
following:4
1. Reaffirming its goals to reduce the size
of the budget deficit.
2. Encouraging private savings and reducing
consumer debt by tightening access to
credit cards.
3. Reducing the cost of capital for corpora
tions through such mechanisms as a lower
capital gains tax.
4. Reducing U.S. export controls and liberal
izing import restrictions such as the volun
tary export restraint agreements on steel
and machine tools.
5. Increasing funds for research and devel
opment and spending on education (in par
ticular, for foreign language, mathematics
and science).
6. Maintaining non-discriminatory treatment
of Japanese investment in the United
States.

4These summaries are taken from the Joint Report of
the U.S.-Japan Working Group on the Structural Im
pediments lnitiative(1990).

Scale Retail Store Law already discussed, the
current Japanese budget calls for a 6.2 percent
increase in spending on public works (such as
roads and housing).26 In fact, some analysts have
suggested that these negotiations actually pro
vide a rationale for countries to implement un
popular domestic reform .27 For example, al
though regulations limiting the size of retail
stores are very popular, one effect of this law is
that consumers pay higher retail prices. For this
reason, there has been pressure (both inside
and outside Japan) to open these markets; the
SII included an agreement to relax these regula
tions. In the United States, the agreement to
maintain non-discriminatory treatment of
Japanese investment in the United States comes

26Thompson (1990). See also “ You W on’t Know it in 2000”
(1991).
27See, for example, Frankel (1990).

at a time when there is increasing concern
voiced in the media regarding recent Japanese
purchases in the United States of such high
profile items as Rockefeller Center in New York
City and Columbia Pictures in Hollywood.28 A
new round of talks on structural impediments
began January 17, 1991.
A more serious problem with bilateral trade
agreements is that the issues they attempt to
address are inherently multilateral rather than
bilateral. In general, a reduction in Japan’s trade
restrictions affects its trade with not only the
United States but virtually all other countries as
well. For this reason, there has historically been
more support, particularly in the United States,
for multilateral negotiations.

28See, for example, Smith (1990). For a discussion of
foreign investment in the United States and some of the
common misperceptions, see Tolchin and Tolchin (1988).

M AR CH/APRIL 1991

CONCLUSION
Many people assume that the existence of a
bilateral trade deficit is considered prim a fa c ie
evidence of unfair trade practices by the coun
try running the trade surplus. However, this ar
ticle has shown that many factors determine
the bilateral trade balance between two coun
tries. These factors include the composition of
trade, the net savings position of each country
and the types of anti-trust legislation and en
forcement policies. Much of these differences
are due to different economic characteristics, in
terms of natural resources and industrial struc
ture, as well as social and cultural differences.
None of these factors however, suggest that
trade is ever likely to balance bilaterally be
tween any pair of countries.
This conclusion certainly applies in the case
of U.S.-Japan trade. Much of the U.S.-Japanese
bilateral trade deficit can be attributed to dif
ferent savings/investment ratios and the dif
ferences in the composition of trade between
the two countries. Unless fundamental domestic
economic changes occur in both the United
States and Japan, it is unlikely that the trade
imbalance between the two countries will be
significantly reduced. As a result, U.S. policies
designed simply to reduce the trade imbalance
will likely be ineffective or even detrimental to
both countries’ economies.

Coughlin, Cletus C., and Geoffrey E. Wood. “An Introduction
to Non-Tariff Barriers to Trade,” this Review
(January/February 1989), pp. 32-46.
Deardorff, Alan V., and Robert M. Stern. “ Methods of
Measurement of Non-Tariff Barriers,” United Nations Con
ference on Trade and Development (Geneva: United Na
tions, 1985).
Dornbusch, Rudiger. Open Economy Macroeconomics (Basic
Books, Inc., 1980).
Frankel, Jeffrey A. “ The Structural Impediments Initiative:
Japan Again Agrees to Become More Efficient,” unpublish
ed paper, University of California, Berkeley (July 1990).
Hayashi, Fumio. “ Is Japan’s Savings Rate High?,” Federal
Reserve Bank of Minneapolis Quarterly Review (Spring
1989), pp. 3-9.
Haynes, Stephen E., Michael M. Hutchison and
Raymond F. Mikesell. “Japanese Financial Policies and the
U.S. Trade Deficit,” Essays in international Finance
(Princeton University, April 1986).
__________“ U.S.-Japanese Bilateral Trade and the YenDollar Exchange Rate: An Empirical Analysis,” Southern
Economic Journal (April 1986), pp. 923-32.
Joint Report of the U.S.-Japan Working Group on the
Structural Impediments Initiative (June 28, 1990).
Krugman, Paul R. “ Is Free Trade Passe?,” Economic
Perspectives (Fall 1987), pp. 131-44.
Lawrence, Robert Z. “ Imports in Japan: Closed
Markets or Minds?,” Brookings Papers on Economic Activity
(2:1987), pp. 517-54.
Office of the U.S. Trade Representative. 1990 National Trade
Estimate Report on Foreign Trade Barriers (GPO, 1990).
Organisation for Economic Co-Operation and Development.
OECD Economic Surveys: Japan (December 1990).
Prestowitz, Clyde V., Jr. Trading Places (Basic Books,
Inc., 1988).
Rowen, Hobart. “ Trade Talks Bought Only Time,"
Washington Post, April 15, 1990.

REFEREN CES

Sakamoto, Tomohiko. “ The Japan-U.S. Bilateral Trade,"
Federal Reserve Bank of San Francisco Economic Review
(Spring 1988), pp. 3-13.

Belassa, Bela, and Marcus Noland. Japan in the World
Economy (Institute for International Economics, 1988).

Sanger, David E. “Japanese Give In Grudgingly on a
New Way of Shopping,” New York Times, November 12,
1990.

Bergsten, C. Fred, and William R. Cline. The United StatesJapan Economic Problem, 13 (Institute for International
Economics, 1985, 1987).
Bhagwati, Jagdish N., and Douglas A. Irwin. “ The Return
of the Reciprocitarians— U.S. Trade Policy Today,” World
Economy (June 1987), pp. 109-30.

Saxonhouse, Gary R., and Robert M. Stern. “An Analytical
Survey of Formal and Informal Barriers to International
Trade and Investment in the United States, Canada, and
Japan,” in Robert M. Stern, ed., Trade and Investment
Relations Among the United States, Canada and Japan
(University of Chicago Press, 1989), pp. 293-353.

Christelow, Dorothy. “Japan’s Intangible Barriers to Trade
in Manufactures,” Federal Reserve Bank of New York
Quarterly Review (Winter 1985-86), pp. 11-18.

Schlesinger, Jacob M. “ U.S. Chip Makers Find ‘Quotas’
Help Them Crack Japan’s Market,” Wall Street Journal,
December 20, 1990.

Christiano, Lawrence J. “ Understanding Japan’s Savings
Rate: The Reconstruction Hypothesis,” Federal Reserve
Bank of Minneapolis Quarterly Review (Spring 1989),
pp. 10-25.
Chrystal, Alec, and Geoffrey E. Wood. “Are Trade Deficits a
Problem?,"this Review (January/February 1988), pp. 3-11.
Cline, William R. “Japan’s Trade Policies,” unpublished
paper, Institute for International Economics (May 1990).

FEDERAL RESERVE BANK OF ST. LOUIS

Smith, Lee. “ Fear and Loathing of Japan,” Fortune
(February 26, 1990).
Thompson, Robert. “Japanese budget plan provides for
6.2% increase in spending," Financial Times, December 27,
1990.
Tolchin, Martin, and Susan Tolchin. Buying Into America
(Time Books, 1988).
U.S. Department of Commerce, Bureau of Economic
Analysis. Survey of Current Business (June 1990).

U.S. International Trade Commission. “ Phase II: Japan’s
Distribution System and Options for Improving U.S. Ac
cess,” Report to the House Committee on Ways and
Means on Investigation No. 332-283 Under Section 332(g)
of the Tariff Act of 1930, Publication 2327 (October 1990).

Webb, Alan J., Michael Lopez, and Renata Penn, eds.
Estimates of Producer and Consumer Subsidy Equivalents:
Government Intervention in Agriculture, 1982-87, U.S. Depart
ment of Agriculture, Statistical Bulletin No. 803 (GPO,
1990).

von Furstenberg, George M., and Joseph P. Daniels.
“ Policy Undertakings by the Seven ’Summit' Countries:
Ascertaining the Degree of Compliance,” unpublished
paper, Indiana University (1990).

“ You won’t know it in 2000,” Economist (January 5, 1991),
p. 28.

MARCH/APRIL 1991

Mark D. Flood
Mark D. Flood is an economist at the Federal Reserve Bank of
St. Louis. David H. Kelly provided research assistance.

An Introd uction to Complete
M arkets

T
h e PAST t w o DECADES have seen a proliferation of new and often complex financial
securities and commodity contracts in the m ar
ketplace. Flipping through the financial pages of
the newspaper, one finds, for example, that an
investor can purchase the right to buy (at a fixed
price on a set future date) a futures contract on
10-year U.S. Treasury notes. Alternatively, one
reads of a con tract that pays off various dollar
amounts depending on the level of the markyen exchange rate at the end of October 1992
(see shaded insert at right). Are such arcana
practically useful? The theory of complete
m arkets—an important element of modern
theoretical economics—can provide some
insight.1
A complete system of markets is one in which
there is a m arket for every good. This simple
statement conceals the significance of the con
cept, however, by failing to specify what is
meant by a “good.” By carefully defining "good”
to include the date and environment in which a
commodity is consumed, economists are able to
consider consumption, production and invest
ment choices in a multiperiod, uncertain world.
Moreover, they can do so using largely the same
utility theory originally developed to analyze
timeless certainty. In particular, state-preference
theory, which was developed to analyze the
completeness of a system of markets, is a pow■'To be correct, one should refer to a “ complete system of
markets.” This phrase can be unwieldy at times, however,
and it is usually abbreviated to “ complete markets” or

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Goldman Sachs S tarts Selling New
W arran ts Betting Yen vs. Mark
B y a W A L L STREET JO U R N A L S t a f f R e p o r te r
NEW YORK—Goldman, Sachs &, Co. began selling five million
w arrants at $3.50 each that allow investors to bet that the
Japanese yen will strengthen against the German mark over the
next two years.
T he w arrants w ere designed by Goldman bu t they are b e
ing offered by A T& T Capital Corp., a finance arm of the com
munications company. Co-managers of the offering are
Oppenheimer & Co. and Dean W itter Reynolds Inc.
According to cu rrency option traders these are the first
cross-currency w arrants to be offered in the U.S., though such
warrants have been sold in Europe. The warrants started trading
yesterday on the New York Stock Exchange.
Most w arrants, like options, give holders the right but not
the obligation, to buy or sell securities, currencies and other
instrum ents at a specific price on or before a certain date. This
w arrant is more complicated because it allows holders to bet
on the relative strength of two foreign currencies with a payoff
in dollars.
The Wall Street Journal publishes a daily table of cross-rates
between various currencies. For example, the closing cross-rate
yesterday was 85.696 yen to buy one mark.
The warrants entitle an investor to a payment in U.S. dollars
if the num ber o f yen to purchase one m ark drops below 85.20.
The intrinsic value of the w arrant is calculated from a formula
specified in the w arrant’s prospectus. Each w arrant initially will
represent $50 of cu rrency and expires Oct. 30, 1992.
Holders of at least 1,000 w arrants will not have to wait un
til expiration to exercise them. "American style" w arrants can
be exercised on any business day. Holders will also be able to
sell them on the Big Board to other investors.

“ complete market.” These three terms are used inter
changeably here (see glossary),

A G lossary fo r C om p lete M ark ets
Arbitrage: A collection of payoff vectors with
zero net cost, whose net payoffs in all out
comes are non-negative and not all zero. In
other words, a set of transactions that re
sults in getting something for nothing.
Arrow-Debreu security: Same as pure
security.
Bet: A contract whose payoffs are a constant
amount in all outcomes that are elements
of a specified event, and zero in all other
outcomes. For example, a contract that
pays off $5 in the event "M wins” and zero
otherwise is a bet.
Call option: A contract giving the holder the
right to buy a particular good at a prespe
cified future date for a prespecified price
(see also put option).
Complete market: A system of markets in
which every agent is able to exchange
every good, either directly or indirectly,
with every other agent. Also called a com
plete system of markets.
Consistent odds: Betting odds on some col
lection of events that are inversely related
to a probability measure over those events,
according to the formula: Oc = (1/P(e)) - 1 ,
where Oe is the odds on the event e, and
P(e) is the probability of e occurring.
Efficient allocation: Same as Pareto-optimal
allocation.
Event: A collection of outcomes. For example,
“M wins” is an event that includes the
rankings M-T-C and M-C-T (see also
outcome).
Futures contract: A standardized contract to
purchase a commodity, which is to be de
livered at a specific date in the future.
General equilibrium: A feasible combination
of prices, consumption choices, and pro
duction choices such that there is no ex
cess supply or demand for any good.
Hedge: To arrange a combination of contracts
such that low payoffs for one contract in a
given state are offset by high payoffs for
another contract in the same state. The
result is a portfolio with reduced variabil

ity of payoffs across states (see also
speculate).
Incomplete market: A system of markets
which is not complete.
Linear combination: A vector produced by
adding (or subtracting), element by ele
ment, any scalar multiples of two or more
other vectors.
Linearly independent: A set of vectors is lin
early independent if no vector in the set
can be expressed as a linear combination
of the other vectors in the set.
Option: See call option and put option.
Outcome: A complete specification of the
realizations of all relevant variables over
the entire relevant time horizon (see also
event).
Pareto-optimal allocation: An allocation of
economic resources among individuals
such that no individual can be made better
off without making some other individual
worse off.
Payoff vector: A vector whose elements rep
resent the amount paid of each good in
each outcome under a particular contract.
Primitive security: Same as pure security.
Pure security: A contract that pays off one
unit (usually one dollar) in a particular out
come and nothing in all other outcomes.
Put option: A contract giving the holder the
right to sell a particular good at a prespeci
fied future date for a prespecified price
(see also call option).
Redundant bet: A bet whose payoff vector
can be constructed as a linear combination
of the payoff vectors of other available
bets.
Scalar multiple: A vector produced by scaling
another vector up or down, i.e., by
multiplying each element of the latter by
the same real number.

MARCH/APRIL 1991

Short sale: The sale of a borrowed good with
the intention of later repurchasing that
good at a profit.

gain accruing through use of the good (see
also hedge).
State: Same as outcom e.

Span a space: A collection of vectors is said
to span a space if the collection has the
property that any point in the space is
reached by some linear combination of
those vectors.

State-claim: A contract that pays off differing
amounts (or different goods) under differ
ent states of the world.

Speculate: To purchase (or sell) a good, in
tending to resell (repurchase) later, where
the motive is to profit from an antici
pated change in price rather than any

State of the world: Same as outcom e.

erful tool with which to study behavior under
uncertainty.
The purpose of this paper is to introduce the
non-specialist to the basic theory of complete
markets, providing the reader with an insight
into the nature of markets and recent financial
innovations in particular. The paper first intro
duces the major concepts of the state-preference approach to uncertainty, illustrating them
with a parimutuel gambling example. In this
framework, the notion of completeness arises
naturally as the extreme case in which bettors
have the greatest range of opportunities to bet
on the outcome of a race. The terminology is
then transferred to an economic context, where
again, complete markets provide consumers,
producers and investors the most flexibility in
allocating payoffs and planning for uncertain
contingencies. Particular attention is given to
the markets for futures and options.
Such securities are shown to improve the effi
ciency of the marketplace, a result with implica
tions for regulatory policy.
In the real world, systems of markets are not
complete, as we shall see.The notion of com
pleteness, however, is of interest for two rea
sons. First, it serves as a theoretical benchmark,
relative to which incompleteness can be assess
ed; such a comparison might, for example, sug
gest whether incompleteness implies inefficiency
2The theory can be traced to the work of Arrow (1964),
Debreu (1959), Arrow & Debreu (1954) and McKenzie
(1954) in the mid-1950s. Both Arrow and Debreu were
later awarded the Nobel Memorial Prize in Economics
(Arrow in 1972, Debreu in 1983), largely for their work in
developing the theory of complete markets and applying it
to the problem of general equilibrium. The theory is often
cited in the guise of its two most common avatars, the
Arrow-Debreu general equilibrium theory and statepreference theory; it also often appears as its implicit

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State-contingent-claim: Same as state-claim.
Vector: An ordered collection of numbers. A
vector can also be represented as a point
or an arrow in space.
in a particular model. Second, although the no
tion of market completeness appears most often
in theoretical discussions, the ideas involved can
also be applied to more realistic problems. For
example, in the state-preference context, mar
kets for so-called "derivative” securities — fu
tures and options — add value by providing
investors with flexibility in fashioning their
portfolios; thus, they make systems of markets
less incomplete. The popularity of such securi
ties can thus be explained from a theoretical
perspective that incorporates complete markets.
In some cases, the theory can even suggest
new markets that would alleviate existing
incompleteness.

THE TH EO RETICAL A PPA RA TU S
The tools and results of the theory of com
plete markets represent one the most significant
developments of theoretical economics in this
century.2 At the same time, the concepts embed
ded in the theory are very general and have
been used in many other economic contexts.
Thus, our first task is to explore the basic struc
ture of state-preference theory. We do this with
a simple gambling example, because it involves
a well-defined and relatively small collection
of outcomes and payoffs in an uncertain
environment.3
counterpart, the theory of incomplete markets. The litera
ture applying state-preference theory and the theory of
complete markets is lengthy; for a partial list, consult the
references in Radner (1982) and Debreu (1982).
3Previous articles have recognized the connection between
gambling and economic activity, especially financial
markets. See, for example, Gabriel & Marsden (1990) or
Asch, Malkiel & Quandt (1984), and the references therein.

State-Claims Defined
One dominant theme of state-preference
theory and complete markets is uncertainty.
State-preference theory incorporates uncertainty
by defining outcomes, or potential future states,
only one of which will ultimately be realized.
The theory has been fruitfully applied in many
areas of economics, especially in the study of
financial markets. For now, however, consider
an imaginary racetrack called Portfolio Downs.
We are bettors at Portfolio Downs, hoping to
win enough for an early retirement. To make
our life simpler, the track’s management has
done away with the tote board.4 Instead, all
bets are placed at fixed odds with the track’s of
ficial bookmaker. This allows us to confine our
uncertainty to the race itself, without worrying
that the posted odds might shift after we’ve
placed a bet. Furthermore, to keep the number
of contingencies to a minimum, only one race
will be run today.
Our first assignment at the track is to define
the set of relevant states and payoffs. A state
o f the world is defined as a complete specifica
tion of the values of all relevant variables over
the entire relevant time horizon. A state of the
world is also called an outcom e, or simply a
state. In an economic system, the relevant vari
ables might include the structure of the tax
code, global weather patterns, infant mortality
rates, scientific discoveries, etc. The relevant
time horizon might be stated in periods as long
as a decade or as short as a second; it might ex
tend into both the past and the future for a few
hours or many centuries.
At the racetrack, however, matters are much
simpler: a state of the world is a complete list
ing of the finishing position for every horse in
today’s race.5 For example, if there are only
three horses in the race:
1. Tricky Bond
2. Mastercharger
3. Charge Me Interest

(T)
(M)
(C)

T-M-C, T-C-M, M-T-C, M-C-T, C-T-M, C-M-T.6 Al
though we may have definite opinions at the
start of the day, we cannot know the state of
the world for sure until the race has been run.
An event is a collection of one or more states.
Thus, for example, "Mastercharger wins the
race” is an event. It includes two states, M-T-C
and M-C-T, both of which are consistent with
the stated criterion.
All bets are placed on events. If the state of
the world that ultimately occurs is an element
of the event, then the bet pays off at the fixed
odds; otherwise the bet pays nothing. Bets are a
special type of state-contingent claim (or simply
state claim). More generally, a state claim is a
contract that pays off differing amounts—per
haps even different goods—under different
states of the world.
A state claim can be represented as a p a y o ff
vector with one element for each state of the
world. The notion of a payoff vector, central to
the theory of complete markets, is stated in
terms of the mathematics of vectors, linear alge
bra.7 In our example, a $2 bet on Mastercharger
to win that pays off at 4-to-l odds can be repre
sented by the vector: (0,0,$10,$10,0,0), where
the positions in the vector are in the same
order as the states listed above. Alternatively, a
$2 trifecta bet on the ranking C-T-M that pays
off at 3-to-l odds has the payoff vector:
( 0 , 0 , 0 , 0 , $ 8 , 0 ).

In state-preference theory, a market is
equivalent to a payoff vector: a market repre
sents the ability to exchange goods or payoffs.
At the racetrack, we exchange pre-race dollars
for a state-contingent bundle of post-race dol
lars. Some exchanges are available directly: we
can exchange $2 pre-race for the post-race vec
tor (0,0,$10,$10,0,0). Other exchanges can be
constructed indirectly: we might exchange $4
pre-race for the post-race vector (0,0,$10,$10,
$4,$4) by placing two separate $2 bets.

then there are six possible states of the world,
which we can write in win-place-show order:

The key to the theory of complete markets is
to deal with such combinations in a systematic
fashion. A system of markets is complete when

4Some basic racing and betting terminology is defined in
the shaded insert on page 36.

6ln general, the number of permutations is given by the fac
torial function. E.g., 3! = 3-2«1 = 6.

5Note that if there were more than one race, a state of the
world would involve a complete listing of all horses in all
races. In other words, a single state of the world describ
ing all the day's races would be ultimately realized.

7See the shaded insert on pp. 37-38 for a quick introduc
tion to linear algebra.

MARCH/APRIL 1991

P a rim u tu e l B ettin g
The term parimutuel comes from the French
words pari meaning "stake” and mutuel mean
ing “mutual.” It describes the mechanics of
determining betting odds and payoffs in the
system commonly used at American race
tracks. Although Portfolio Downs uses a
bookmaker rather than a parimutuel system,
the terminology is the same with both
methods. In a parimutuel system, bets are
grouped together in pools corresponding to
the type of bet (e. g., win, place, show, quinella, etc.). Each pool is treated separately. The
track takes a cut off the top (the take), and
the remaining pool is pro-rated among the
winning bettors according to the amount bet.
Odds are stated as the ratio of profit to
wager; for example, a winning $2 bet at 3-to-l
odds pays back the $2 initial wager plus $3
for each dollar bet, making a total of $8. The
odds themselves are a function of the amount
bet on each event in the pool. In particular,
ignoring the take, the odds on a horse are
given by: Oh = (B/bh) - 1 , where Oh is the
odds on horse h in the pool, B is the total
size of the pool, and b h is the amount in
the pool that was bet on horse h. Odds are
posted on the tote board and fluctuate accor
ding to the relative volume of bets placed up
until the time of the race. Implicit in this ar
rangement is a sort of probability measure
representing the aggregate beliefs of the bet
tors. Note that Ph = 1/(1+ Oh) = b h/B. This
quantity can be treated as the implicit proba
bility that a bet on horse h will pay off. The
parimutuel system can have some curious
properties, however. For example, if many
people bet a speedy horse to win, ignoring
the place pool, the final odds on that horse to
place can exceed the odds on the same horse
to win, even though the place bet is safer
than the win bet. Although this clearly im
plies inconsistent odds, it can and occasion
ally does occur.
we can arrange a portfolio with any conceivable
payoff vector. We may not want certain payoff
vectors, and even among those we do find de
sirable, our decision on which portfolio
ultimately to arrange will depend both on their
prices and our resources. These issues of desir
ability and affordability are secondary for the
notion of completeness, however. The important

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Bettor’s Glossary
Daily Double: A bet that specifies the two
winning horses in two designated races;
i. e., both horses must win for the bet to
pay off.
Exacta: Same as a perfecta.
Odds: The ratio of net profits to amount
wagered for a winning ticket.
Parlay wager: A sequence of two or more
bets in which all contingencies are chosen
in advance, and in which the full proceeds
of earlier bets are wagered on the next bet
in the sequence. For example, a daily dou
ble bet is a parlay wager.
Perfecta: A bet that specifies the top two
horses, in order (see also quinella).
Place: To come in second in a race. Also, a
bet that a specified horse will at least place
(i. e„ win or place).
Quinella (or quiniela): A bet that specifies the
top two horses, but does not specify their
order (see also perfecta).
Show: To come in third in a race. Also, a bet
that a specified horse will at least show
(i. e., win, place or show).
Take: The cut from the betting pool that is
taken by the racetrack to cover taxes,
overhead, etc.
Totalizator or tote board: A system for repor
ting a horse’s current betting odds implicit
in the mutuel pool.
Trifecta: A bet that specifies the top three
finishers, in order.
Win: To come in first in a race. Also, a bet
that a specified horse will win.
characteristic of a complete system of markets
is that, without a wealth constraint, any con
ceivable payoff vector can be arranged. In
terms of linear algebra, a complete system of
markets is one in which the set of available bets
contains enough linearly independent payoff
vectors to span the space of all conceivable
payoff vectors.

T he A lgeb ra of P ay o ff V e cto rs
The mathematical concept of a spanned
vector space and the economic concept of a
complete system of markets are closely re
lated. Although the algebraic logic remains
the same for any number of states, we can
gain some intuition for the problem by con
sidering the special case in which there are
only two states of the world. For this case
we can graph the relationships involved. For
example, let the two states be boom and
bust, and consider two securities in which
we might invest, Abelard Abrasives and
Zwingli Swings. We don’t know yet whether
the economy will go boom or bust, but we
do know how the securities will perform in
each case. Security performance is represent
ed by the payoff vectors: PA = (7,1), and
Pz = (5,4). This situation is depicted in figure
1, where each state is assigned to an axis,
and the payoff vectors are represented by ar
rows extending from the origin.

We can add vectors by simply adding to
gether the corresponding components. Thus,
if we buy a portfolio of one share of each
security, our payoff vector will be PA+Z = PA
+ Pz = (7 + 5,1+ 4) = (12,5). Graphically, this
is achieved by connecting the tail of one vec
tor to the head of the other; the resulting
vector is a new arrow extending from the
origin to the location of the unconnected ar
rowhead. It does not matter which head is at
tached to which tail, because (7 + 5,1+4) =
(5+ 7,4+1). A little experimentation reveals
that buying portfolios of different numbers of
the two securities results in payoff vectors to
the purple points in the graph.
We can also calculate multiples of a vector
in an analogous fashion. For example, Pz + Pz
= 2PZ, which is obtained by multiplying each
element of the vector by two. More gen
erally, we can multiply a vector by any num
ber, simply by performing the multiplication

Security Payoff Vectors

Bust Payoffs

MARCH/APRIL 1991

on each of the elements.1 Graphically, this
results in the lengthening or shortening of
the original vector, without changing its
direction. In our securities example, if we are
allowed to purchase fractional amounts of
the securities for our portfolio, then we can
obtain a payoff vector that lies anywhere in
the purple-shaded wedge.
Scalar multiplication of a vector also works
with negative numbers. If we multiply a vec
tor by a negative number, the resulting arrow
in the graph lies along the same line as the
original, but points in the opposite direction.
For our securities example, this is the equiv
alent of a short sale, the sale of a borrowed
security. Short sales are generally made with
the intention of repurchasing the security
later at a (hopefully) lower price. If we sell a
security short, we are responsible for reim
bursing the lender the amount of the securi
ty’s payoff when the state of the world is
revealed — hence the negative payoffs. Mathe
matically, this is the same as adding a nega
tive security to our portfolio: Pz -P a = Pz +
( —PA). Graphically, we add the arrows in the
same way, making sure that we use the ( - PA)
vector in the addition. If we are allowed to
sell shares of Abelard and Zwingli short, we
can add the black points to our list of poten
tial portfolio payoffs.
Finally, if we are also allowed to sell frac
tional amounts short, we can add the entire
grey shaded region to our feasible set. We
can arrange any payoff combination in the
two-dimensional plane. In this case, when any
sort of addition and scalar multiplication is

allowed, we say we can use all linear co m
binations of the securities. The payoff vector
for a linear combination of the securities will
always take the form: Pp = x aPA + xzPz,
where xa and xz are real numbers. If there
are no restrictions on xa and xz, then all
linear combinations are allowed.
In our example, the payoff vectors are
chosen so they do not lie along the same line.
To describe this non-parallel quality, we say
the two vectors are linearly independent. If
they did lie along the same line, say instead
that Pz = 2Pa = (14,2), then the vectors would
be linearly dependent. In the case of linearly
dependent vectors, we cannot arrange port
folios with any payoff combination in the
plane; it's as if we only had a single security
to choose from. When the two vectors are
linearly independent, so that any payoff com
bination is accessible, the two vectors are
said to span the plane.
All of this generalizes directly to situations
with more than two states of the world. The
only difference is that it is more difficult to
graph three states of the world, and impossi
ble to graph four or more. Whatever the
number of dimensions, a collection of vectors
always spans something; the question of in
terest for us is what that something is. For
example, two vectors in a three-dimensional
space can span at worst a single point — the
origin (let Pz = PA = (0,0)). At most they span
a two-dimensional plane. In either case, there
remains an infinity of points in the threedimensional space that cannot be reached by
any combinations of the two vectors.

’ This is also called “ scalar multiplication,” to distinguish it
from “ vector multiplication,” in which two vectors are
multiplied.

An Incom plete Market
To illustrate why and how completeness might
be valuable to us, let's impose some more condi
tions on Portfolio Downs. This first example il
lustrates both an incomplete system of markets
and a redundant bet. A bet is redundant if its
payoffs in every outcome can be duplicated by

8Readers familiar with linear algebra should note that
redundancy is defined as the linear dependence of the set
of payoff vectors. Roughly speaking, if the market is

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buying or selling some combination of the
other available bets. The system of markets is
incom plete if the number of outcomes ex
ceeds the number of non-redundant bets.
Later on, we shall replace one of these
redundant bets with a non-redundant bet,
thus completing the market.8

incomplete, our payoff vectors are restricted to a
“ flatland” within the larger space of possible state claims,

Table 1
PAYOFFS BY OUTCOME
ODDS

PAYMENT

T-M-C

T-C-M

M-C-T

C-T-M

C-M-T

7-3

-$ 2

$6.67

T places

2-3

- $2

$3.33

M wins

4-1

- $2

M places

9-11

—$2

C wins

1-1

- $2

C places

3-17

- $2

BET
T wins

$3.33

$10.00

$3.64

$4.00

$2.35

0
$3.64

Suppose that the race is the three-horse affair
described earlier. To make things challenging,
further suppose that the resident bookmaker
will accept only win bets and place bets, refus
ing to take the seemingly more complicated
quinellas, trifectas, etc .9 Finally, to make com
pleteness an intriguing proposition, suppose
that we get a hot tip from our pal in the stables
that the race is fixed: the final outcome will be
M-T-C. Because we trust our pal, we now want
to bet only on this single state of the world.
The key question for market completeness is:
can we do it, using only win and place bets?
First, note that we aren’t satisfied with a bet
on Mastercharger to win: in this bet, we would
also be buying a payoff in state M-C-T, which
we’re convinced won’t occur. With consistent
odds, our payoff per dollar wagered can’t be as
high betting Mastercharger to win (i.e., betting
on the event, [M-T-C, M-C-T]), or betting Tricky
Bond to place (i.e., the event, [T-M-C, T-C-M,
M-T-C, C-T-M]), as it would be on the M-T-C trifecta bet .10 We want a portfolio of bets with a
payoff vector that looks like this: (0 ,0 ,x,0 ,0 ,0 ),
where x is some positive number, to make the
maximum profit on our bet. The number x
should equal exactly the payoff we would get
on the M-T-C trifecta bet, if only the bookmaker
would allow trifecta bets.
9Show bets are superfluous here, because in a three-horse
race, all show bets automatically pay off. Thus, they’re not
really wagers at all. Also note that, although a trifecta
might seem complex, it is in a sense the simplest bet
here, because it pays off in only one state. This would not
be true of trifecta bets if there were more than three
horses or more than one race.
10“ Consistent” simply means that the odds on any event
are inversely related to the probability, in the bookmaker’s
view, of it occurring — the more likely an event, the lower
its odds. Consistent odds are related to the probability of
an event, e, by the formula: Ot = (1/P (e))-1 , where Oc is
the odds ratio for the event e, and P(e) is the probability
that e will occur. With consistency, the probabilities of all
the individual outcomes add up to 100 percent.

M-T-C

It turns out we cannot achieve this payoff
vector from the available bets, an upshot of the
fact that this system of markets is not complete.
In practical terms, if the system of markets
were complete, then we could get higher odds
on a bet paying off in the event M-T-C than we
can get with the current incomplete set, as will
be demonstrated below.
To see more clearly what this means, let's ex
periment with the numbers. Suppose that the
bookmaker gives odds on the six allowable bets,
implying the six payoff vectors listed in table 1 .
These payoffs have been rounded off to the
nearest penny. They include the initial wager of
$ 2 plus any profit from the wager, which
depends on the odds.11
To say that one of the bets is redundant
means that its payoffs in the six outcomes can
be replicated by an appropriate portfolio of the
other bets .12 The number of outcomes (six) equals
the number of bets (six), but exceeds the
number of non-redundant bets (five); the system
of markets is thus incomplete. Any payoff vec
tor attainable by combinations of the six bets is
still attainable if we disallow one of the redun
dant bets and include combinations consisting
only of the other five.
11To convert between odds and payoffs, use the formula:
p = b(1 + Ot), where p is the payoff, b is the size of the
bet (e.g., $2), and O, is the odds ratio (e.g., 7-3 odds imp
ly Ot = 7/3 = 2.33).
12Redundancy requires that we be able to take bets as well
as place them. By taking bets, we can arrange for a nega
tive payoff in certain events. For convenience, we assume
that the bookmaker at Portfolio Downs meets this need by
placing bets at his posted odds. It is also convenient if we
assume that bets can be both taken and placed in any
fractional amount, allowing us to fine-tune our portfolio.

MARCH/APRIL 1991

Table 2
PAYOFFS BY OUTCOME
BET
T wins

ODDS

PAYMENT

7-3

-$ 2 4

$80.00

-$ 4 0 .0 0

T places

2-3

$24

M wins

4-1

-$ 1 6

M places

9-11

C wins

1-1

Total:

$22

T-M-C

T-C-M

0
-$ 4 0 .0 0

M-T-C
0
-$ 4 0 .0 0

M-C-T
0
0

$80.00

- $40.00

-$ 4 0 .0 0

C-T-M
0
-$ 4 0 .0 0

C-M-T
0
0

-$ 4 0 .0 0

-$ 4 0

$80.00

-$ 3 4

$40.00

For example, we can construct the sixth bet
as a portfolio of the other five. Consider first a
$34 bet on Charge Me Interest to place . 13 Ignor
ing the rounding error, 17($2.35) = $40.00, and
our payoffs would be (0,$40,0,$40,$40,$40) un
der the six possible outcomes. Now consider a
portfolio that consists of different amounts of
the other five bets, either taken (negative pay
offs) or placed (positive payoffs). In particular,
let the amounts be those listed in table 2. For
example, we take 12 $2 bets on Tricky Bond to
place, while placing 1 2 $ 2 bets on the same
horse to win. The payoffs to our portfolio are
given by totaling the six columns in the table.
The portfolio yields the same result as if we
had placed a $34 bet on Charge Me Interest to
place. In fact, since we placed $80 in bets while
taking $46 worth, our net investment in the
portfolio is $ ( - 8 0 + 46) = -$ 3 4 . The fact that
equivalent payoff vectors require the same in
vestment reveals that our bookmaker has set
the odds (i.e., payoffs) in a consistent way.
One way to look at the r e d u n d a n c y o f o n e o f
our bets is that we cannot synthesize the trifecta bet that we’re after. Even though we would
place such a bet at any consistent positive odds
(because we’re convinced we know the outcome)
that bet is neither offered directly nor can we
synthesize it from the others. In other words,
we can’t get there from here.

Completing The Market
Another way to look at this is that the book
maker can drop a redundant bet from the list
of allowable bets without affecting our oppor
13Using 17 two-dollar tickets ($34) simply keeps all the
following payoffs in terms of nice round numbers. We
could just as well scale all the amounts and payoffs down
by a factor of 17 to show the same result.
14This is not necessarily the case with redundant bets. Con
sider, for example, three redundant bets on two outcomes:
(0,1), (1,1) and (2,2). If (2,2) is dropped, it can be

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tunities. It turns out that, in our example, the
six bets are mutually redundant: any bet that is
omitted can be reconstituted from the remain
ing five. The same procedure that was just used
to reconstitute the sixth bet could be applied to
generate any of the individual bets from the
other five .14 Let’s drop a bet then and replace it
with a non-redundant bet.
For example, suppose the bookmaker does not
offer a bet on Mastercharger to place. Instead
of that bet, he gives 3-1 odds on a trifecta bet
on the ranking C-M-T. The odds and the implicit
payoffs now available to us are given in table 3.
The fourth row of the original payoff array has
been replaced by the payoffs to the new tri
fecta bet. The question is still whether we can
arrange a portfolio of bets that pays off only
when the ranking is M-T-C. That is, can we syn
thesize an M-T-C trifecta bet? More generally, is
the system of markets complete? The answer to
both questions is yes. The system of markets is
complete, which implies that we can achieve
any payoff vector, including the payoff vector
that corresponds to an M-T-C trifecta bet.
To synthesize that bet, combine the bets as
described in table 4. The result is a net twodollar investment that only pays off if the final
outcome of the race is M-T-C. The payoff in
that case is $40, implying 19-to-l odds. 15 By
creating a bet that pays off under such narrow
circumstances, we have maximized the return
on our $ 2 investment (assuming our pal in the
stables is trustworthy!). With a complete system
of markets, of course, we can also generate a
safer portfolio—that is, one that pays off in
reconstituted from (1,1). If (0,1) is dropped, however, it
cannot be synthesized from (1,1) and (2,2).
15Again, the scale of the payoffs is unimportant here. Forty
dollar amounts are used to keep the numbers consistent
with the previous example. The important result is that the
portfolio pays off only in one specific state of the world.

Table 3
PAYOFFS BY OUTCOME
ODDS

PAYMENT

T-M-C

T-C-M

M-T-C

M-C-T

C-T-M

C-M-T

T wins

7-3

-$ 2

$6.67

T places

2-3

- $2

$3.33

$10.00

BET

- $2

$10.00

M wins

4-1

C-M-T trifecta

3-1

-$ 2

C wins

1-1

- $2

$4.00

C places:

3-17

- $2

$2.35

ODDS

PAYMENT

T-M-C

T-C-M

M-T-C

M-C-T

C-T-M

C-M-T

$8.00

Table 4
PAYOFFS BY OUTCOME
BET

7-3

$12

-$ 4 0 .0 0

T places

2-3

-$ 2 4

$40.00

M wins

4-1

C-M-T trifecta

3-1

-$ 1 0

- $40.00

-$ 4 0 .0 0

T wins

$40.00
0

0
0
$40.00

C wins

1-1

$20

C places

3-17

- $2

$40.00

Total:

more states—but this would be more costly. For
example, to get the same payoff in one addi
tional outcome, M-C-T (i.e., to achieve the payoff
vector (0,0,$40,$40,0,0)), would require an $ 8
bet on Mastercharger to win, a four-fold in
crease from the investment required for the
trifecta bet.
It is no accident that the number of bets in
the portfolio equals the number of possible out
comes . 16 Every portfolio we construct is a sys
tem of six linear equations in some unknowns,
namely the amounts to put into each of the
allowable bets. In other words, we start with
six state-dependent payoffs (the desired payoff
vector), and we seek a combination of weights
(i.e., investment amounts) for the available bets
that yield those six payoffs. To ensure that such
a combination exists, we need at least six un
knowns (i.e., available bets) to work with. Fur
thermore, some collection of six of those avail
able bets must have payoff vectors that are lin
early independent.

What if there are more unknowns than equa
tions (i.e., more available bets than elements in
the desired payoff vector )?17 In that case, at
least some of the bets must be redundant. Thus,
in determining whether the system of markets
is complete, it is not safe simply to count equa
tions and unknowns; we must find the largest
collection of non-redundant bets. In our exam
ple, if we can find six non-redundant bets, then
every payoff vector is the result of a unique
combination of these six bets. Thus, in our ex
ample, the only way to achieve the payoff (0 , 0 ,
$40,0,0,0) is to combine the bets in the amounts
($ 1 2 ,-$ 2 4 ,0 ,-$ 1 0 ,$ 2 0 ,0 ).
Finally, there is one last bit of terminology
which appears frequently in state-preference
theory. In our example, a trifecta bet has a posi
tive payoff in one state only; in all other states,
its payoff is zero. Notice that all other bets con
sist of various collections of trifecta bets. Clear
ly then, a set of six different trifecta bets would
produce a complete system of markets. This

16An allowable bet is included in the count, even if the
amount wagered on that bet is zero. The decision to
wager nothing on a particular allowable bet is an implicit
portfolio decision.
17We shall see below that this situation is not usually a prac
tical consideration. The normal problem is that there are
not enough unknowns (available bets) rather than too
many.

M AR CH/APRIL 1991

leads to the notion of a reference bet or pure
security. A pure security is a normalized bet
that pays off in only one state. By normalized,
we mean that the payoff in the selected state is
one unit .18 To get a normalized payoff, we must
adjust the amount invested in that bet. The
wager amount that implies the normalized
payoff is called the price o f the pure security.
Any state-contingent claim can be regarded as a
collection of pure securities. A system of mar
kets is complete if and only if the number of
attainable pure securities (either directly or
through combination of other securities) equals
the number of outcomes.

oped a simple context in which to introduce the
terminology of complete markets. An analogy
can be made between Portfolio Downs and real
markets. Bookmakers manage their risk by lay
ing off bets, while investors manage their risk
by hedging their portfolios; bettors are unsure
whether Tricky Bond will win the race, while
investors are unsure if pet rocks will be
popular with consumers; etc .20 Our next task is
to flesh out this analogy to see what importance
state-preference theory can have for more gen
eral economic analysis. This is done with a
series of examples.

Multiple Periods, Multiple
Commodities

SOME ECONOMIC APPLICATIONS
Historically, the theoretical economics
literature has generally followed two distinct,
but entirely compatible lines in interpreting and
applying the theory of complete markets. First,
led by the initiators of the theory, there were
applications to the problem of general equilib
rium. Most work in this area now starts with
the notion that markets are not complete and
proceeds to analyze the nature of equilibrium
(or disequilibrium) with incomplete markets .19
The other line of research has focused on the
relative efficiency of financial markets at alloca
ting risk by providing greater investment flex
ibility to investors, in the same way that a com
plete system of markets makes bettors better
off at Portfolio Downs. In practice, the general
equilibrium applications tend to de-emphasize
uncertainty and concentrate on production, con
sumption and intertemporal optimization. Con
versely, the financial market applications tend
to ignore real resource constraints and temporal
factors, in order to concentrate on uncertainty.
The first section of this paper considered the
properties of betting odds at an imaginary race
track. Our goal in that example was to find
some conditions that would ensure complete in
vestment flexibility. In the process, we devel
18At Portfolio Downs everything is measured in dollars, so a
pure security would be a trifecta bet that paid $1 if the
outcome was the ranking specified in the bet. Pure secur
ities are also called Arrow-Debreu securities or primitive
securities.
19See Geanakoplos (1990) for some examples.
20The terms “ laying off bets,” “ covering a position,” and
“ hedging or reinsuring a portfolio” all refer to the same
basic process of reversing a transaction with one party
by making a countermanding transaction with a third party.

FEDERAL RESERVE BANK OF ST. LOUIS

One of the primary insights of state-preference
theory is that the traditional notion of what
constitutes an economic good can be readily ex
tended in a way that allows us to examine
economic behavior across time and under un
certainty. At the same time, this extension
forces us to think of "goods” in a very different
way. Our notion of a good is broadened in two
directions: time and state.
Examining the time dimension first, consider a
simple example involving a banana and an apple.
A textbook would tell us that each consumer
has a set of preferences such that she either
prefers one to the other or is indifferent be
tween them. In our new way of looking at
things, however, we must include a time dimen
sion in a description of the commodity. From
this new perspective, "apple” is not sufficient to
describe a good; one must specify either "apple
today” or “apple tomorrow.” Merely specifying
“apple” is analogous to stating “red” without
specifying “red tricycle” or “red Ferrari.”
To make the example more specific, suppose
that our consumer generally prefers apples at
time t, A„ to bananas, B„ but that variety is also
valuable to her. In this case, the following pre
ferences for consumption over two consecutive
days might hold:
For example, a bookmaker who takes a large bet on some
event from a bettor can lay it off by placing a bet on the
same event with another bookmaker.

Ranking Bundle
1
2
3
4

(1,0,0,1)
(0,1,1,0)
(1,1,0,0)
(0,0,1,1)

Sequence of fru its
apple today, banana tom orrow
banana today, apple tom orrow
apple today, apple tom orrow
banana today, banana tomorrow,

where the bundles (i.e., payoff vectors) are of
the form: (A0,A 1,B0,B1). With our new definition
of commodities in mind, the four sequences
ranked above can be treated under standard
utility theory exactly as four commodity bun
dles, each composed of a pair of the four dif
ferent commodities. One might consider, for
example, the indifference curves between ap
ples today and bananas today, or budgeting
between apples today and apples tomorrow.
An infinite number of other bundles could
also be described and ranked, of course. For
example, (15,0,3,7) is a possible bundle, one that
would be preferable to any of those listed
above. Restricting the time dimension to two
days and considering the resulting four goods,
we see that the system of markets defined by
the four bundles here is incomplete: we can get
bundle 1 by buying bundles 3 and 4 while sell
ing bundle 2. Thus, there are at most three
non-redundant markets for four goods.21

Uncertainty
The same approach is used to incorporate
uncertainty. To do so, the definition of a com
modity is expanded to include the state of the
world. Thus, for example, an umbrella in the
rain is now a fundamentally different commod
ity from an umbrella on a sunny day.22 Simi
larly, "bananas in peacetime” are different from
“bananas in wartime.” This new application,
however, fundamentally expands our notion of
a good. Because states of the world, by defini
tion, are mutually exclusive, we must separate
the notion of economic consumption from that
of physical consumption. For example, if we
purchase the bundle of goods consisting of “an
apple in peacetime tomorrow” and "a banana in
wartime tomorrow,” we shall not be eating both
21There is, of course, no reason to limit the time dimension
to two days. By extending the time dimension indefinitely,
we get an infinite number of goods, which would require
an infinite number of markets for a complete system.
22Street vendors in New York, for example, charge $3 for an
umbrella under clear skies, and $5 for an umbrella when
it’s raining.

apple and banana tomorrow; we shall eat only
one or the other.
Because the additional consideration of time
and uncertainty forces such a radical shift in
our conception of commodities, it is worthwhile
to consider it further. The distinctions between
apples and bananas, today and tomorrow, and
peace and war serve as a simple basis for an
example. Suppose that, today, the political situa
tion is peaceful, but tomorrow’s situation is un
certain. This implies two possible states of the
world: peace today, peace tomorrow, and peace
today, war tomorrow. We thus have six com
modities, abbreviated as follows :23
C o m m od ity
apple (peace) today
banana (peace) today
apple peace tomorrow
apple war tomorrow
banana peace tomorrow
banana war tomorrow

A b b rev iatio n
A - » -0
B -*-0
A -P -l
A -W -l
B -P -l
B -W -l

Suppose now that the local wholesaler, Whim
sical Fruits, sells the "fruit baskets” or state
claims described in table 5.
This admittedly contrived example (note, for
example, that basket No. 4 consists of 364 ap
ples today and 364 apples tomorrow, plus 364
“bonus” bananas to be delivered only in case of
war tomorrow) allows for ready interpretation,
because of its similarity to an earlier example.
The array of goods here is essentially identical
to the payoff array for Portfolio Downs; only
the labels and the scale have been changed. At
Portfolio Downs, the only distinction between
goods was the state of the world; the physical
commodity in all cases was cash, and time dif
ferences did not exist. At Whimsical Fruits, on
the other hand, the goods have different inter
pretations, and all the amounts are scaled up by
a factor of 1 0 0 ; otherwise, the payoff array is
identical to that in table 1. We can therefore
conclude immediately that the fruit market here
is incomplete. For example, a buyer wanting on
ly to buy apples to be delivered tomorrow in
commodities dated today, the true state of the world is
uncertain, although the current political situation is known
to be peace. In principle, therefore, we should distinguish
between apple-peace tomorrow-today and apple-war
tomorrow-today. As a practical matter, however, we cannot
observe this distinction until it is too late for it to affect our
behavior. For convenience, the distinction is not made in
the example.

23Note that there are only two states of the world, distin
guished by the political situation tomorrow. Thus, for the

MARCH/APRIL 1991

Table 5
NUMBER OF FRUIT IN EACH BASKET
PAYMENT

A -« -0

B -.-O

-$ 2 0 0

667

-$ 2 0 0

333

-$ 2 0 0

364

-$ 2 0 0

BASKET

case of peace cannot arrange such a transaction
as a mixture of the baskets offered at Whim
sical Fruits.

D o Complete Markets Really Exist?
We now have a sufficient context to ask
whether markets are complete in reality. The
answer from economists who have considered
this question is a resounding “No.”24 The prob
lem is the huge number of markets that would
be required for completeness to hold. Even with
the roughest distinctions (e.g., measuring time
in one-year intervals, considering all automo
biles in a given year as perfect substitutes, etc.),
an astonishing number of states of the world
must still be considered. For example, every
conceivable future invention must be included.
Furthermore, the timing of invention is signifi
cant; if a cure for cancer comes in 1997 instead
of 1998, that implies a different state of the
world. Some would have us distinguish goods
by geographical location and even by the iden
tity of the final consumer !25 We need to mul
tiply the number of states by the number of
periods and then by the number of physical
goods and services. Finally, we need a market
with which to exchange every one of these
many goods with every other one.
The absence of many markets from a real
economy may be explained by transaction costs.
For example, a contract for "an apple dated

24Radner (1982), p. 930, for example, states that "... it clear
ly requires that the economic agents possess capabilities
of imagination and calculation that exceed reality by many
orders of magnitude.” Geanakoplos (1990), p. 2, states
that, ‘‘There is little doubt that permitting the incomplete
ness of asset markets is a step in the direction of
realism.”
25See, for example, Geanakoplos (1987), p. 116. This last
distinction is made in analyzing public goods and
externalities.

FEDERAL RESERVE BANK OF ST. LOUIS

B-P-1

B-W-1

333

1000

364

400

235

A-P-1
0

A-W-1

2006 if a cure for cancer is discovered in 1997”
is too costly to arrange, relative to the marginal
benefit of such a transaction. Such a commodity
is too narrowly defined to be of interest.
Turning to a more plausible example, the
International Monetary Market (IMM) of the
Chicago Mercantile Exchange (CME) established
an organized marketplace for several commod
ity futures contracts in 1973. Among these were
contracts for (1) bagged silver Canadian coins
worth C$ 5,000 at face value, and (2) 12,500,000
Japanese yen .26 While futures trading in silver
coins was discontinued several months later
after trading volume dwindled, yen futures still
trade today with thousands of contracts chang
ing hands every business day.
In hindsight, it seems clear that the conve
nience provided by a market for bagged Cana
dian silver coins was outweighed by transaction
costs of some form .27 On the other hand, the
successful introduction of a market for yen fu
tures suggests that investors would have faced
significant portfolio constraints in their absence.
The contrast between these two raises an inter
esting question: are the incomplete systems of
markets observed in the real world efficient in
the sense that the missing markets are absent
because they are not operationally cost-effective
given current trading technology? Or, are they
incomplete simply because no one has thought

26See Chicago Mercantile Exchange (1974, 1989). All futures
contracts specify a future time and place for delivery, in
addition to other standardizations. The state-contingent
nature of markets for futures and options is considered in
greater detail in later examples.
270n e possible explanation is that the payoffs to coin futures
lie in the space spanned by linear combinations of Cana
dian dollar futures and silver futures, making the coin
futures redundant.

to provide the services required to match buy
ers and sellers?
A full answer, unfortunately, is beyond the
scope of this paper, although we return briefly
to the issue below when considering futures
markets. It is not a simple task to identify be
forehand which missing markets impose the
most significant constraints by their absence,
and thus would be most likely to succeed with
investors. The theory of complete markets does
tell us, however, that, contingent on the level of
transaction costs, identifying and providing such
markets can make everyone involved better off.
Not surprisingly then, much of the theoretical
work in this area investigates the properties of
economies where markets are incomplete. Some
of the issues involved are presented in the fol
lowing examples.

Complete Markets and Efficient
Allocations
We first examine one of the most basic pro
blems in economics: how to arrange for the
best allocation of available resources. Consider
the nursery rhyme of Jack Sprat, but suppose
that Jack and his wife, Bubbles, are now di
vorced and living separately .28 For simplicity,
let's ignore uncertainty and the time dimension.
The local market has a special on pork chops,
which consist of one fat portion and one lean
portion, for $P per pork chop .29 This can be
represented as a single payoff vector:
COMMODITY
PRICE

FAT

LEAN

Now consider the allocation of a fixed set of
pork chops, say three chops, between Jack and
Bubbles. We represent this situation by the
modified Edgeworth box in figure 1. Given the
special preferences of Jack and Bubbles, there is
a unique optimal allocation in this example, rep
resented by the point A* in the lower-right-hand
corner of the box. This is the point at which
Jack gets all three lean portions of pork,
28Recall that: “ Jack Sprat could eat no fat / His wife could
eat no lean / And so betwixt them both / They licked the
platter clean.”
29Note:“ To market,to market,to buy a fat pig,/Home again,
home again, jiggety-jig.”

Figure 1
T h e A llo c a tio n o f P o rk
FAT (JACK)

BUBBLES

JACK

FAT (BUBBLES)

and Bubbles gets all three fat portions .30 Despite
their differences, Jack and Bubbles recognize
the value of trade. For example, the vector V, in
the figure represents a sale of one pork chop
from Bubbles to Jack, where Jack owns one
pork chop before the sale and two pork chops
afterward.
Because there are two commodities but only
one transaction (i.e., one payoff vector) in this
example, the set of markets is incomplete. The
single available market spans the 45-degree line
in the payoff space of both consumers; each
consumer can achieve any payoff where the
number of fat and lean portions is given by
some multiple of (1,1). Unfortunately, the opti
mal point, A*, does not lie in the space spanned
by this available market. The upshot is that the
system of markets is incomplete, implying a suboptimal allocation of resources. In other words,
the allocation of the six (fat and lean) portions
here ideally would be represented by A*, the
unique Pareto-optimal allocation. The available
market does not allow them to trade to this
be made better off without harming someone else. In this
particular example, there is only one Pareto-efficient
allocation, because of the extreme nature of the prefer
ences. In a less extreme case, there would be many
Pareto-efficient allocations, depending on preferences, in
itial endowments and relative prices.

30The optimal allocation is called Pareto-efficient by
economists. With a Pareto-efficient allocation, no one can

MARCH/APRIL 1991

allocation, however, and Jack’s fat goes to
waste, as does Bubbles’ lean. Now, let's complete
the system of markets by adding a second al
lowable transaction: say that Jack and Bubbles
agree to trim the fat from the chops and sell it
separately to one another at a price of $P/2 per
portion of fat. Thus the set of payoff vectors
becomes:
COMMODITY
FAT

LEAN

The vector V2 in the figure represents the sale
of a single fat portion from Jack back to Bub
bles. By thus completing the market, we have
made the optimal allocation attainable .31
This same principle of allocational efficiency
can be seen at work in more general settings.
The major contribution of Arrow and Debreu
was to show that an efficient allocation of all
commodities is feasible for an economy with
complete markets, even with many periods,
many physical goods and services, and uncer
tainty about the future .32 We next illustrate the
importance of complete markets in this context,
without considering a full general equilibrium
model.

Futures Markets and Risk-Shifting
Uncertainty is a salient concern in many mar
kets. It is common for contracts to require
alternative payoffs in different states. Commod
ity futures contracts, for example, specify the
exact physical characteristics of the commodity,
the date of delivery and the location of delivery;
moreover, they typically provide that “If deliv
ery or acceptance or any precondition or re
quirement of either is prevented by a strike,
fire, accident, action of government or act of
God,” the directors of the exchange will decide
the duties of buyer and seller .33
From a practical perspective, however, the im
plicit state-dependent nature of such contracts
is much more important than such explicit stip

31Starting from the same initial allocation, where Jack has
one chop and Bubbles two, Jack buys both chops from
Bubbles and then sells back to her the three fat portions.
32This is their proof of the existence of an efficient general
equilibrium in an economy with complete markets. See
Geanakoplos (1987) for an overview.

FEDERAL RESERVE BANK OF ST. LOUIS

ulations. Commodity futures contracts enable
owners of a physical commodity to hedge the
value of their inventory exactly against uncer
tain future fluctuations in price. The statepreference approach can provide useful insights
into the nature of such markets.
Futures markets allow people to contract to
day for future delivery of a specific commodity
at a specific price. To see why such a contract
is valuable, consider a hypothetical cotton
market without a futures market. In particular,
suppose that the current price of cotton is 71
cents per pound ($35.50 for a 50-lb. bale), and
there are two states of the world. In one state,
the price of cotton will increase to 80 cents per
pound; in the other, it will fall to 70 cents per
pound. Finally, assume borrowing and lending
are possible at an interest rate of 5 percent
over the period. Using the tools of statepreference theory, we can set up the payoff ar
ray in table 6 . There will ultimately only be two
cash markets for cotton here. We can either
transact now in the spot market, or we can
transact later in whichever of the two subse
quent spot markets is available.

The H edger
Now consider the situation of a cotton farmer
who will harvest 25 tons of cotton in the com
ing period. To restrict the number of contingen
cies, we treat the size of the crop as certain. In
terms of state-claims, this is the endowment
(not a transaction) described in table 7. Finally,
assume the farm er is risk-averse and wants his
cash receipts to be the same, regardless of the
state of the world. In other words, he wants
a final consumption bundle, after harvesting
and selling the crop, as described in table 8 .
Note that we must restrict the payoff $X to be
strictly greater than $35,000 here. Otherwise,
the farm er could achieve a certain payoff of
$35,000 by giving away $5,000 in the high-price
state of the world. The potential for arbitrage
implies that he should be able to do better than
this. Can he achieve his desired payoff with the
current set of markets?

33See Chicago Mercantile Exchange (1983), p. 8.

Table 6
STATE CLAIM PAYOFFS
TRANSACTION

CURRENT
COST

Cotton
Now

Buy now

- $35,500

25 tons

Cash
Later
@ 70*

Cash
Later
@ 80*

-$3 5,00 0

25 tons

- $40,000

$100

Cotton
Now

Cotton
Later
@ 70*

Buy @ 80*
Borrow cash

Buy @ 70*

Cotton
Later
@ 80*

Cotton
Later
@ 70*

25 tons

-$ 1 0 5

Table 7
STATE CLAIM PAYOFFS
CURRENT
COST

Endowment

25 tons

Cotton
Later
@ 80*

Cash
Later
@ 70*

Cash
Later
@ 80*

25 tons

Table 8
STATE CLAIM PAYOFFS

Desired Endowment

CURRENT
COST

Cotton
Now

Cotton
Later
@ 70*

Cotton
Later
@ 80*

Cash
Later
@ 70*

Cash
Later
@ 80*

CURRENT
COST

Cotton
Now

Cotton
Later
@ 70*

Cotton
Later
@ 80*

Cash
Later
@ 70*

Cash
Later
@ 80*

- 25 tons

Table 9
STATE CLAIM PAYOFFS
TRANSACTION

Forward sale

The answer is no: to convert his certain crop
into a certain dollar payoff, he requires a for
ward sale contract of the form described in
table 9. Such a contract does not exist here, nor
can it be synthesized as a combination of avail
able contracts. In terms of our earlier discus
sion, the system of markets is incomplete. In
this case, the incompleteness prevents the
farm er from arranging his desired payoff vec
tor. Let’s consider two ways of alleviating this
constraint by completing the system of markets.
First, let’s add storage (and leasing) of the
physical commodity to the list of allowable tran

sactions. In this example, no price is charged
to store or lease cotton, although such a price
could readily be included. In our simplified twostate, two-period example, this is sufficient to
complete the system of markets. The payoff
array is now given in table 10. To arrange the
desired bundle, the farm er can now arrange the
transactions in table 1 1 , converting his endow
ment into a certain dollar amount next period.
In other words, the farm er leases 25 tons of
cotton, sells it for cash today and invests the
proceeds, repaying the borrowed cotton when
his own crop is harvested. Thus, it is possible,
even in the absence of a futures market, to hedge

M AR CH/APRIL 1991

Table 10
STATE CLAIM PAYOFFS
Cash
Later
@ 70*

Cash
Later
@ 80*

-$ 3 5 ,0 0 0

- $40,000

TRANSACTION

CURRENT
COST

Cotton
Now

Cotton
Later
@ 70*

Cotton
Later
@ 80*

Buy now

-$ 3 5,50 0

25 tons

Buy @ 70*

25 tons

Borrow cash

$100

Store cotton

Buy @ 80"

- 25 tons

-$ 1 0 5

25 tons

-$ 1 0 5

Cash
Later
@ 70*

Cash
Later
@ 80*

Table 11
STATE CLAIM PAYOFFS
TRANSACTION

Lease cotton
Sell now

CURRENT
COST
0
$35,500

Cotton
Now

Cotton
Later
@ 70*

Cotton
Later
@ 80*

25 tons

- 2 5 tons

- 25 tons

Lend

-$3 5,50 0

$37,275

Total:

- 2 5 tons

- 25 tons

$37,275

Table 12
STATE CLAIM PAYOFFS
TRANSACTION

Futures

CURRENT
COST

Cotton
Now

the crop. This is achieved, however, through a
circuitous and potentially costly chain of three
transactions. Why shouldn't the farm er arrange
the desired transaction directly as a single
contract?
This is precisely the role of a futures contract,
our second means of completing this system of
markets. A futures contract would have just
this payoff vector, perhaps adjusted by a scale
factor. A cotton future is a standardized for
ward sale contract; i.e., a contract to pay a spe
cified price (PF) for a standardized quantity (50
lbs. of cotton) at a specific time (next period), re
gardless of the state of the world, as described
in table 12. The futures contract makes the
marketplace more flexible; in our simple exam
ple, it completes the system of markets. It
allows the farm er to transfer directly the price
risk associated with commodity ownership with
out transferring ownership of the commodity
per se.

F E D E R A L R E S E R V E B A N K O F S T. LO U IS

Cotton
Later
@ 70*
- 5 0 lb.

Cotton
Later
@ 80*
- 5 0 lb.

Cash
Later
@ 70*
PF

Cash
Later
@ 80*
PF

There are three lessons here. First and fore
most, completing the system of markets makes
everyone better off (or at least not worse off)
by allowing risk to be transferred from the
farmer to a speculator. Second, there is more
than one way to complete an incomplete system
of markets. Third, some means of completing a
system of markets may be more cost-effective
than others. One might even plausibly conjec
ture that all missing markets result from trans
action costs that render them cost-ineffective.
Confirming or refuting such a conjecture, how
ever, is beyond the scope of this paper.

The Speculator
The same transaction can also be considered
from the perspective of the speculator who
buys a futures contract from the farmer. A
speculator is someone who wagers that she can

Table 13
STATE CLAIM PAYOFFS
TRANSACTION

Buy now
Buy @ P l
Buy @ P h
Store
Invest

CURRENT
COST

Cotton
Now

- P 0C

0
0
0

0
0
-C

-$ 1

CURRENT
COST

Cotton
Now

0
PoC
0
0

Cotton
Later

Cash
Later

@ Pl

@ Ph

0
0

0
c
0
c
0

0
0

- P LC

0
0

C
C

- P HC

0
1 +R

1+ R

Table 14
STATE CLAIM PAYOFFS
TRANSACTION

Cotton
Later
@ Pl

Lease

Lend

- P HC /(1+R )

-c
0
0
0

Net:

C[P q - P H/(1 + R )]

Sell now
Buy @ P l
Buy @ P(-|

accurately predict the state of the world .34 To
see how speculation occurs in the absence of
futures markets, we start with a market that
allows for storage (and leasing) of cotton. If we
omit cotton storage, as in table 6 , then the sys
tem of markets is incomplete; the speculator is
thus prevented from arranging her desired pay
off vector. Once again we complete the market
in two different ways: with commodity market
speculation and with futures market
speculation.
To make the problem more general, we now
represent contract quantities (measured in lbs.
of cotton) by C, and prices (measured as cents
per lb. of cotton) by P. There are three prices,
the current spot price P0, low price PL and a
future high price PH(where P„ > P L). There is a
single fixed interest rate R, at which the
speculator can borrow and lend money. This
payoff array is given in table 13. Let’s say the
speculator believes the low-price state will oc
cur. In particular, she wants to arrange a trans
action that pays only cash in the low price
state. She can accomplish this by arranging the
bundle of transactions described in table 14.
34Note that the speculator is not necessarily a consumer of
cotton. Indeed, a cotton consumer (e. g., a clothing manu
facturer) is likely to be as risk-averse as the farmer. In

-C

Cotton
Later
@ PH
-C

0
c
0
0
0

0
0
C

Cash
Later

@ Pl

@ Ph

0
0
- P LC

0
0
0

- P HC

PHC

C(P h - P

PHC
l

)

This concentrates the speculator’s return on the
single state-claim, cash later in state PL.
In other words, to speculate on the low-price
outcome in this world, the speculator must sell
the physical commodity short. This requires
leasing the commodity, selling it in the spot
market and repurchasing the cotton later when
the price has changed. These four transactions
imply a profit if the cotton price falls and a
loss if it rises. Finally, the speculator invests
an amount equal to the discounted value of the
cotton in the high price state. If this state oc
curs, then the cost of repurchasing the cotton
will be exactly covered by the investment; if the
low price state occurs, the investment more
than covers the repurchase of the cotton, and
the difference is a speculative profit. The cur
rent cost of this basket is the proceeds of the
spot sale, P 0C, less the amount that must be in
vested, PHC/(1 + R). This must be negative—i.e., a
positive cost to the speculator—if an arbitrage
opportunity is to be avoided.
One must question the practicality of such a
transaction, however. Once again, there is more
than one way to complete a system of markets.
effect, a speculator sells insurance (i. e., bears risk) for a
living.

MARCH/APRIL 1991

Table 15
STATE CLAIM PAYOFFS
Cotton
Later

Cash
Later

@ Pl

@ PH

@ Pl

- 5 0 lb.

CURRENT
COST

Cotton
Now

Cotton
Later

Sell Standard Futures

Buy @ P|_

50 lb.

Buy @ P h

Cash Settlement Futures

TRANSACTION

Although short sales are commonplace in the
stock market, for example, such transactions
can be considerably more difficult when dealing
with physical commodities. If a market for cot
ton futures is available, however, the speculator
can arrange her desired bundle without ever
having to store or lease a physical commodity.
Indeed, most futures contracts are retired by a
process called cash settlem ent, which obviates
any transfer of the physical commodity.

0
50 lb.
0

Cash
Later
@ Ph

PF
-

Pf
0

0
P f -P l

ph
p f -p h

futures markets. Arrow (1964) showed that the
ability to reallocate risk without otherwise con
straining economic activity is a general property
of securities markets. This principle can also be
seen at work in the options m arket . 37 In con
sidering an options example, we abstract from
the issues of timing and consumption and
concentrate on uncertainty, to streamline the
example.

Options and Investor Flexibility

Options markets are especially useful for shift
ing risks, because of a special characteristic of
an option contract. In particular, a call option
specifies a strike price, labeled K, at which the
holder of the option can purchase the underly
ing commodity .38 By simply changing the strike
price in an option contract, we can create a
fundamentally different financial security. Thus,
a "single” options market provides the opportu
nity to exchange a multitude of state claims. To
see how this works, consider the following
example.

The preceding example illustrated how the
introduction of a futures contract, a paper
transaction, could improve the allocation of re
sources in an economy . 36 Such applications of
state-preference theory are not limited to the

The Chicago Board of Trade Options Exchange
(CBOE) trades options on the Standard and
Poors’ 500 stock index (S&.P 500), among other
things .39 A call option on the S&P 500 with
striking price of 295, theoretically gives the op-

In effect, a cash settlement futures contract
(table 15) is a bundle of transactions sold as a
single unit, where PFis the futures price. Note
that, to eliminate arbitrage here, it must be the
case that PF- P L > 0 > PF- P H. 35 Such a standar
dized contract facilitates the transfer of risk
from hedgers to speculators. Moreover, with
cash settlement, the speculator never has to
handle the physical commodity sold in the fu
tures contract, thus reducing transaction costs.

35ln general, in order to eliminate arbitrage opportunities,
every payoff vector must involve a trade-off, in which some
element (including the current cost) of every vector is
negative and some element of every vector is positive. In
other words, every vector must represent an exchange of
some sort rather than a unilateral gift.
36ln one sense, we have reallocated risk rather than
resources. Recall, however, that we have redefined goods,
so that cotton in the high-price state is a different resource
from cotton in the low-price state.
37For more thorough analyses of options and complete mar
kets, see Ross (1976), and Arditti and John (1980).
38The “ commodity” in most options markets is really a
share of a corporation’s common stock. However, there
are also organized options exchanges for cattle, copper,
crude oil, Canadian dollars, etc. See the shaded insert on
pp. 51-52 for a basic description of option contracts.

FEDERAL RESERVE BANK OF ST. LOUIS

39The S&P 500 is a weighted average of 500 common
stocks. The amount of each stock included in the index is
pro-rated according to the value of that stock, where the
value is defined as the price per share times the number
of shares outstanding. The value of the index is then scal
ed down to make the base period index (1941-43) worth
10 units; if the S&P 500 today is worth 295, for example, it
is worth 29.5 times as much as it was in 1943. Thus, the
price of the S&P 500 is not, strictly speaking, a dollar
value for the index.

Call O ption B a sics
A call option is a legal contract that gives its
owner the right to buy a specified asset at a
fixed price on a specified date . 1 Similarly, a
put option gives its owner the right to sell a
specified asset at a fixed price on a specified
date. Option contracts are usually sold by one
party to another .2 The person who owns an
option contract is called the holder of the op
tion. The person who sells an option contract
—that is, the person who will be compelled
to perform if the option holder invokes her
right as specified in the contract—is called
the writer of the option. The act of invoking
the contract is called exercising the option.
The fixed price identified by the option con
tract is called the striking price. The date at
which the option can be exercised is called
the expiration date of the option.
These legal contracts are probably best
known by the stock options that are bought
and sold by brokers in the trading pits of
organized options exchanges in Chicago, New
York and elsewhere. In addition to options on
common stock, there are active markets for
options on foreign currencies, on stock index
portfolios, on government securities, and on
futures for agricultural commodities, to name
a few. The definition of an option, however,
does not limit the term to those contracts ac
tively traded on the floors of organized finan
cial exchanges. By definition, an option is any
appropriately constructed legal contract bet
ween the writer and the holder, regardless of
whether it is actively traded.
Consider now the value to the holder of an
expiring call option, as illustrated in figure 1 .
The value of the underlying asset specified
by the contract, Ax, is given on the horizontal
axis, while the value of the option itself, Vc,
is given on the vertical axis. The point K on
the horizontal axis is the specified striking
price for the asset. If the value of the
underlying asset is below the striking price
on the expiration date, then the call option
will not be exercised; anyone who truly
wanted to buy the asset would do so outright
at the going price, rather than using the op
tion and paying the striking price. In this
case, the option expires worthless, and the
option holder experiences no gain or loss on
the expiration date.

Figure 1

Value of Call Option to Holder
Expiration Value of Option: Vc

°-|

...■■

......... f -------------------K

Value of Underlying Asset at Expiration: A t

On the other hand, if the value of the asset
is above the striking price, then the holder
will exercise his option and pay the striking
price for the asset. In this case, his net gain
on the expiration date will be (AT-K ), the dif
ference between the current price and the
striking price, since he can turn around and
resell the asset immediately, if he wants to.
Thus, the expiration value of the option and
the decision about whether to exercise are
contingent upon the value of the underlying
asset at that time:
State

A ctio n

O ption v alu e

AT < K

No exercise

Vc = 0

AT > K

Exercise

Vc = AT- K

For this reason, options are also referred to
as "contingent claims" on the underlying
assets.
The corresponding net payoffs to the writer
of the call option are given in figure 2. Notice
that his payoffs are exactly the inverse of
those for the option holder. Also note that
the payoff at expiration to the writer of an
option is never positive; at best it is zero. It
is for this reason that options are sold to the
holder, rather than being given away free

MARCH/APRIL 1991

Figure 2

Value of Call Option to Writer

To simplify our example, we limit the number
of possible outcomes as before and ignore the
time dimension. As at Portfolio Downs, there
are only two relevant times: before the true
outcome is revealed and afterward. Each state
of the world corresponds to a different value
for the S&P 500:

Expiration Value of Option: Vc

Value of Underlying Asset at Expiration:

A t

of charge. The price initially paid for the
option—the option price or option premium—
could be incorporated into the figures by
simply shifting the holder’s payoffs down and
the w riter’s payoffs up by the appropriate
amount.

'This definition is a paraphrase of the definition given by
Cox & Rubinstein (1985), p. 1. It describes a “ European”
option, which is distinguished from an “ American” option.
An American option gives its owner the right to buy at any
time on or before the specified date.
2They are sold, because options have a non-negative value;
because they are a right to buy (or sell) the asset, they
do not compel the owner of the contract to do anything.
Although they are valuable, nothing in the definition of an
option requires that they be offered for sale; that is, their
value does not depend on how they were obtained.

tion holder the right to purchase the S&P 500
index at 295.40 Because the weighting scheme
involved in constructing the index involves 500
stocks, many held in fractional quantities, all of
which must then be scaled down to a relative
value, however, it is impossible to purchase the
exact S&P 500 index. Instead, cash settlement is
used: at maturity, the holder of a call option

40The units for the option contract must be rescaled to give
a dollar value. In particular, option contracts are for $500
times the level of the index. For example, to exercise a
call option at strike price 295 would cost $500 x 295 =
$147,500.

F E D E R A L R E S E R V E B A N K O F ST. LO U IS

receives in cash $500 times the difference be
tween the value of the index and the striking
price, if this difference is positive, and nothing
otherwise .41

S tate
A
B
C
D
E
F

V alue of S& P 5 0 0
286
294
299
301
306
314

Thus, our option to buy at 295 is a state claim.
In the event that the value of the S&P 500 itself
is less than 295, then the option is worthless,
because the (approximate) index can be pur
chased in the open market for less than the
strike price. If the index is worth more than
295, then the option is worth $500 times the
difference between the actual price and the
strike price. This is summarized in table 16.
It should be clear from our earlier examples
that this system of markets is not complete.
There is no way, for example, to arrange a
portfolio that pays off exactly $50,000 if the
S&P 500 is below 300 and zero otherwise; i.e.,
($50000,$50000,$50000,0,0,0). The special char
acteristic of options markets, however, is that
linearly independent payoff vectors can be
achieved by changing the striking price alone.
Because of this, numerous options on the same
security are actively traded. Many of these op
tions differ only in their striking prices. Adding
some of these options to our example, we get
the payoff array in table 17.
This system of markets is complete. To achieve
the payoff vector ($50000,$50000,$50000,0,0,0),
for example, we can transact the amounts of
the six listed call options described in table 18.

41For example, an investor exercising a call option at strike
price 295 when the index itself is at 290 would receive
$500 x (295-290) = $2500.

Table 16
PAYOFFS BY STATE
SECURITY

Call option at K = 295

$2000

$3000

$5500

$9500

Call at K @ 285

$500

$4500

$7000

$8000

$10,500

$14,500

Call at K @ 290

$2000

$4500

$5500

$8000

$12,000

Call at K @ 295

$2000

$3000

$5500

$9500

Call at K @ 300

$500

$3000

$7000

Call at K @ 305

$500

$4500

Call at K @ 310

$2000

Table 17
PAYOFFS BY STATE
SECURITY

Table 18
PAYOFFS BY STATE
SECURITY

QUANTITY

Buy calls @ 285

-1 0 0

A
$50,000

$450,000

$700,000

$ 800,000

$1,050,000

$1,450,000

-$900,000

-$1,100,000

-$1,600,000

-$2,400,000

$250,000

$ 375,000

$ 687,500

$1,187,500

450,000

-$1,050,000

$ 312,500

$2,812,500

Sell calls @ 290

200

- $400,000

Buy calls @ 295

-1 2 5

Sell calls @ 300

150

Buy calls @ 305

-6 2 5

Sell calls @ 310

1000

-$2,000,000

$ 50,000

Totals:

0
$50,000

0
$ 50,000

75,000

Note that the numbers of the various options
bought and sold here are not dollar amounts;
the total dollar cost is the price of each option
times the respective quantity. Option pricing is
beyond the scope of this paper. The no-arbi
trage condition implies, however, that the total
cost of this portfolio of options must be

areas of research in the economics and finance
literature for almost 40 years. Despite their
theoretical importance, these topics have re
ceived little exposure elsewhere, doubtless
because of the technical nature of the argu
ment. This paper conveys the basic concepts of
the theory for a non-specialized audience.

s o m e w h e r e b e tw e e n z e r o a n d $ 5 0 ,0 0 0 .

CONCLUSIONS

The theory of complete markets sheds light
on many economic issues. It starts by redefining
goods to include attributes not normally con
sidered inherent: the time and state at which
something is consumed. An immediate implica
tion of this new definition of a good, however,
is that the system of markets available in the
real world is far from complete: taking all the
possible combinations of attributes into account
leaves the number of goods far in excess of the
number of actual markets. By the same token,
this relative dearth may provide a ready explan
ation for the recent proliferation of unusual
financial innovations.

The theory of complete markets and the par
allel state-preference theory have been active

One of the overriding themes appearing in all
the examples here is the efficiency of the alloca

In practice, of course, there are many more
than six possible prices for the S&.P 500 index
and far more states of the world than there are
possible prices for the index portfolio. Nonethe
less, the linearly independent payoffs of these
six options necessarily span more of the payoff
space than does the index alone. The result is
greater flexibility for investors in fashioning
their portfolios in an uncertain world.

MARCH/APRIL 1991

tion of goods. This can be seen in the allocation
of risks in futures and options markets, the abil
ity to refine payoffs at the racetrack, or the dis
tribution of fat and lean between Jack Sprat
and his wife, Bubbles. The common implication
in each example is that additional markets can
improve the welfare of all concerned. That is,
given the ability to reallocate, individuals will do
so: they will exchange relatively less desirable
commodity bundles for those that are, for them,
relatively more desirable. A complete system of
markets provides this ability. Thus, an economy
with greater flexibility in production, consump
tion and investment is uniformly preferable to
one with less.
A second recurring theme is uncertainty.

R eview Questions
(1) Using the payoff array in table 3, construct
a book of bets that amounts to a pure
security for the outcome T-C-M; i. e., make a
book with the payoff vector (0 ,$ 1 , 0 ,0 ,0 ,0 ).
What is the net investment required for this
book of bets?
(2) Using the payoff array in table 3, what is
the probability, implicit in the bookmaker’s
odds, that the outcome of the race will be
T-C-M?
(3) Suppose the bookmaker offers a redundant
bet with inconsistent odds. In particular,
suppose the last bet in the payoff array in
table 1 is changed, yielding the new array in
table Q l. Construct an arbitrage portfolio (a
portfolio that shows a positive net profit in
all outcomes) from these bets.
(4) Ross shows that a necessary condition for
a collection of securities to span the state
space is that, "for any two states there must
be some asset whose payoffs distinguish bet
ween them .”1 Suppose we have instead a
market in which no asset can distinguish
between two states. In particular, consider
1See Ross (1976), p. 81.

FEDERAL RESERVE BANK OF ST. LOUIS

State-preference theory and the theory of com
plete markets are one way to incorporate un
certainty systematically in an economic model.
This is central to a theory that includes an un
certain state of the world as a fundamental at
tribute of a good. One result is a recognition
of the value of the ability to reallocate risk
through financial transactions. While specula
tion in financial markets may or may not be un
fettered gambling, the implicit transfer of risk
from hedgers to speculators still produces eco
nomic value. The theory of complete markets
thus provides a systematic explanation for the
popularity and value of many so-called deriva
tive securities, such as futures and options.

the payoff array in table Q2 , in which states
E and F are indistinguishable by the avail
able assets. Show that this system of mar
kets is incomplete.
(5) Consider a speculator in the cotton market
who believes that the price will rise. Assume
that storage (and leasing) of cotton is not
practical, but that a cash settlement futures
market exists. In other words, assume the
allowable transactions are those in table Q3.
What sort of payoff vector might this specu
lator want? How could she construct it from
the available transactions?
(6 ) Consider a banker facing the transactions
available in table Q4 over the next 30 days
in the international financial markets, where
S is the spot foreign exchange rate (¥/$),
and R, and Rs are the foreign and domestic
30-day interest rates. Is this system of
markets complete? Suppose the banker
wants to arrange a forward exchange con
tract of the form presented in table Q5. Ar
range such a payoff from the available
transactions. What does the forward rate F
equal? What is the intuition behind this
value for the forward rate?

Table Q1
PAYOFFS BY OUTCOME
ODDS

PAYMENT

T-M-C

T-C-M

M-T-C

M-C-T

C-T-M

C-M-T

T wins

7-3

- $2

$6.67

T places

2-3

-$ 2

$3.33

-$ 2

$10.00

$3.64

BET

M wins

4-1

M places

9-11

-$ 2

$3.64

C wins

1-1

-$ 2

$4.00

C places

1-4

-$ 2

$2.50

Table 0 2
PAYOFFS BY STATE
SECURITY
Buy calls @ 285

QUANTITY
-1 0 0

$50,000

$450,000

$700,000

$ 800,000

$1,450,000

-$900,000

-$1,100,000

-$2,400,000

$250,000

$ 375,000

$1,187,500

75,000

-$1,050,000

Sell calls @ 290

200

- $400,000

Buy calls @ 295

-1 2 5

Sell calls @ 300

150

Buy calls @ 305

-6 2 5

$1,250,000

Sell calls @ 310

1000

-$2,000,000

Table 03
STATE CLAIM PAYOFFS
TRANSACTION

Sell Futures
Sell now

CURRENT
COST

Cotton
Now

@ Pl

Cotton
Later
@ Ph

-c

PoC

Cotton
Later

Cash
Later
@ Pl
PFC - PLC

Cash
Later
@

pfc

phc

Buy @ P|_

- P LC

Buy @ Ph

- P HC

-$ 1

Lend

1 +R

Table 0 4
PAYOFFS TO CONTRACTS
CONTRACT

$ NOW

¥ now

¥ 30-day

Buy spot ¥

-$ 1

Convert & lend ¥

-$ 1

S ( 1 + R ¥)

Lend $

-$ 1

$ 30-day
0
0
1 +R $

Table 0 5
PAYOFFS TO CONTRACTS
CONTRACT
Buy forward ¥

$ NOW

¥ now

¥ 30-day

$ 30-day
-$ 1

M AR C H /A PR IL 1991

REFEREN CES
Arditti, Fred D., and John Kose. “ Spanning the State Space
with Options,” Journal of Financial and Quantitative Analysis
(March 1980), pp. 1-9.
Arrow, Kenneth J. "The Role of Securities in the Optimal
Allocation of Risk Bearing,” Review of Economic Studies
(April 1964), pp. 91-96.
. "Insurance, Risk and Resource Allocation,” in
Essays in the Theory of Risk-Bearing (Markham, 1971),
pp. 134-43.
Arrow, Kenneth J., and Gerard Debreu. “ Existence of an
Equilibrium for a Competitive Economy,” Econometrica
(1954), pp. 265-90.
Asch, Peter, Burton G. Malkiel, and Richard E. Quandt.
“ Market Efficiency in Racetrack Betting,” Journal of
Business (April 1984), pp. 165-75.
Chicago Mercantile Exchange. International Monetary
Market Yearbook (1973-74).
_______ . Index and Option Market Yearbook 1982 (CME
Statistical Dept., 1983).
________ 1989 Statistical Yearbook, Futures.
Copeland, Thomas E., and J. Fred Weston. Financial Theory
and Corporate Policy (Addison-Wesley, 1983).
Cox, John C., and Mark Rubinstein. Options Markets
(Prentice-Hall, 1985).
Daily Racing Form (Midwestern Edition) (News America
Publications, July 20, 1990).

Geanakoplos, John. “Arrow-Debreu Model of General Equil
ibrium,” in J. Eatwell, M. Milgate, and P. Newman, eds.,
The New Palgrave, A Dictionary of Economics, Volume I
(London: MacMillan Press, 1987), pp. 116-24.
________“An Introduction to General Equilibrium With In
complete Asset Markets,” Journal of Mathematical Econom
ics (1990), pp. 1-38.
Hirshleifer, Jack. Time, Uncertainty, and Information
(Oxford: Basil Blackwell, 1989).
Hirshleifer, Jack, and John G. Riley. “ The Analytics of
Uncertainty and Information — An Expository Survey,”
Journal of Economic Literature (December 1979),
pp. 1375-421.
McKenzie, Lionel. “ On Equilibrium in Graham’s Model of
World Trade and Other Competitive Systems,” Econom
etrica (1954), pp. 147-61.
Mossin, Jan. The Economic Efficiency of Financial Markets
(Lexington Books, 1977).
Myers, Stewart C. “A Time-State-Preference Model of
Security Valuation,” Journal of Financial and Quantitative
Analysis (March 1968), pp. 1-33.
Radner, Roy. “ Equilibrium under Uncertainty,” in Handbook
of Mathematical Economics, Volume II (Amsterdam: NorthHolland, 1982), pp. 923-1006.
Ross, Stephen A. “ Options and Efficiency,” Quarterly Journal
of Economics (February 1976), pp. 75-89.
Rubinstein, Mark. “ Securities Market Efficiency in an
Arrow-Debreu Economy,” American Economic Review
(December 1975), pp. 812-24.

Debreu, Gerard. Theory of Value: An Axiomatic Analysis of
Economic Equilibrium (Cowles Foundation for Research in
Economics at Yale University, 1959).

Stoll, Hans R., and Robert E. Whaley. “ The New Option
Markets,” in A.E. Peck, ed., Futures Markets: Their
Economic Role (American Enterprise Institute for Public
Policy Research, 1985).

________“ Existence of Competitive Equilibrium,” in Hand
book of Mathematical Economics, Volume II (Amsterdam:
North-Holland, 1982), pp. 697-743.

Townsend, Robert M. “ On the Optimality of Forward
Markets,” American Economic Review (March 1978),
pp. 54-66.

Friesen, Peter H. “ The Arrow-Debreu Model Extended to
Financial Markets,” Econometrica (May 1979),
pp. 689-707.

“ Goldman Sachs Starts Selling New Warrants Betting Yen
vs. Mark,” Wall Street Journal, November 1, 1990.

Gabriel, Paul E., and James R. Marsden. “An Examination
of Market Efficiency in British Racetrack Betting,” Journal
of Political Economy (August 1990), pp. 874-85.

FEDERAL RESERVE BANK OF ST. LOUIS

Wilson, Charles. “ Incomplete Markets,” in J. Eatwell, M.
Milgate, and P. Newman, eds., The New Palgrave, A Dic
tionary of Economics, Volume II (London: MacMillan Press,
1987), pp. 759-61.

A n sw e rs to th e R ev iew Q u estion s
(1) Try:(30C, - 60C,20C, - 25C,100C, - 85<t).
The net investment is 20C.

there are six states and five non-redundant
securities—the system of markets is
incomplete.

(2) Odds are given by:
O =

1 .0 0 - . 2 0

4 -to-l

(5) She would want to purchase a payoff
Z > 1 in the high-price state, as described
in table A l. This could be obtained by the
transactions listed in table A2.

odds.

.20

O. s __ L - 1 = 4
P(e)

P(e) = .20 = 20%.

(6 ) Yes, it is complete. Combine the trans
actions listed in table A3. Thus, F =
S(l + R ¥) /(1+R j ). This is simply a statement
of the covered interest parity condition,
F(l + R$) = S(l + R ¥), which prevents arbi
trage between money markets in different
countries.

(3)Try: ($240, - $240,$160, - $220,$400, - $320).
(4) Compare: (0,0,0,0,1,0) with: (0,0,0,0,0,1),
where the elements of the vectors repre
sent the numbers of each option to buy or
sell. The last two options are redundant:
Table A1

STATE CLAIM PAYOFFS
TRANSACTION
Desired
Payoffs

CURRENT
COST
-$ 1

Cotton
Now

Cotton
Later

Cash
Later

@ Pl

@ Ph

@ Pl

@ Ph

Table A2
STATE CLAIM PAYOFFS
TRANSACTION

CURRENT
COST

Cotton
Now

Cotton
Later
@ Pl

Lend:

-C (P f - P l)/(1+R)

Net:

- C(PF - PL)/(1 + R)

0
0
0

Scaling this down, it becomes:
Net:
-$ 1

Buy futures:

Cotton
Later
@ Ph

Cash
Later
@ pl

Cash
Later
@ Ph

0
0
0

PLC-P f C

PFC-P|

PHC-PFC

Pf C-P|

C(P h -P l )
(1 + R) I-------------L(Pf - P l )

Table A3
PAYOFFS TO CONTRACTS
CONTRACT

$ NOW

Convert & lend ¥

¥ now
0

1+R$
Borrow $

¥ 30-day

„

$ 30-day

1+R¥
1 +R$

-$ 1

1 + R$
Net:

MARCH/APRIL 1991

David A. Dickey, Dennis W. Jansen
and Daniel L. Thornton
David A. Dickey is a professor of statistics at North Carolina
State University, Dennis W. Jansen is an associate professor of
economics at Texas A&M University and Daniel T. Thornton is
an assistant vice president at the Federal Bank of St. Louis.
The authors would like to thank Mark Watson, Tom Fomby and
David Small for helpful comments on an earlier draft of this
article. Lora Holman and Kevin L. Kliesen provided research
assistance.

A P rim e r On Cointegration

w ith an Application to Money
and Incom e

J . OR SOME TIME NOW, macroeconomists
have been aware that many macroeconomic
time series are not stationary in their levels and
that many time series are most adequately
represented by first differences . 1 In the parlance
of time-series analysis, such v a r ia b le s a r e said to
be integrated of order one and are denoted 1(1 ).
The level of such variables can become arbitrari
ly large or small so there is no tendency for
them to revert to their mean level. Indeed,
neither the mean nor the variance is a mean
ingful concept for such variables.
Nonstationarity gives rise to several economet
ric problems .2 One of the most troublesome

1That is, formal statistical tests often cannot reject the null
hypothesis of a unit root. The results of these tests,
however, are sensitive to how the tests are performed—
that is, whether the tests assume a non-zero mean or a
time trend, whether an MA or AR data generating pro
cesses is assumed [Schwert (1987)] and whether the test
is performed using classical or Bayesian statistical in
ference [Sims (1988), and Sims and Uhlig (1988)]. These
sensitivities are partly due to the lack of power these tests
have against an alternative hypothesis of a stationary but
large root.

FEDERAL RESERVE BANK OF ST. LOUIS

stems from a common prediction of macroeco
nomic theory that there should be a stable longrun relationship among the levels of certain eco
nomic variables. That is, theory often suggests
that some set of variables cannot wander too
far away from each other. If individual time
series are integrated of order one, however,
they may be "cointegrated.” Cointegration means
that one or more linear combinations of these
variables is stationary even though individually
they are not. If these variables are cointegrated,
they cannot move "too far” away from each
other. In contrast, a lack of cointegration sug
gests that such variables have no long-run link;

2lt can give rise to the possibility of a spurious relationship
among the levels of the economic variables. Also, the
parameter estimates from a regression of one such
variable on others are inconsistent unless the variables are
cointegrated.

in principle, they can wander arbitrarily far
away from each other .3
This article illustrates the salient features of
cointegration and tests for cointegration. The
discussion, initially motivated by the simple ex
ample of Irving Fisher’s “equation of exchange,”
draws an analogy between cointegration and
unit roots on the one hand and tests for cointe
gration among multiple time series and the usual
tests for unit roots in univariate time-series
analysis on the other. The article then addresses
the broader question of the economic inter
pretation of cointegration by contrasting it with
the usual linear, dynamic, simultaneous equa
tion model which is frequently used in
macroeconomics.
The article goes on to compare three recently
proposed tests for cointegration and outlines the
procedures for applying these tests. An applica
tion of these tests to U.S. time-series data using
alternative monetary aggregates, income and in
terest rates suggests that there is a stable longrun relationship among real output, interest
rates and several monetary aggregates, including
the monetary base.

TESTING FO R COINTEGRATION:
A GENERAL FRAM EW ORK
Because of the close correspondence between
tests for cointegration and standard tests for
unit roots, it is useful to begin the discussion by
considering the univariate time-series model

(1) yt-M = e(y,-,-M> + e„
where y, denotes some univariate time series, \i
is the series’ mean and et is a random error
with an expected value of zero and a constant,
finite variance. The coefficient g measures the
degree of persistence of deviations of yt from ju.
3At the present time, tests for cointegration deal only with
looking for stable linear relationships among economic
variables. Consequently, a failure to find cointegration
does not necessarily mean that there is no stable, long-run
relationship among the variables, it only suggests that
there is no stable, long-run, linear relationship among
them.

When g = l, these deviations are permanent. In
this case, y, is said to follow a random walk—
it can wander arbitrarily far from any given
constant if enough time passes .4 In fact, when
6 = 1 the variance of y, approaches infinity as t
increases and the mean of y„ \jl, is not defined.
Alternatively, when \q \ < 1, the series is said to
be mean reverting and the variance of yt is
finite.
Although there is a similarity between tests
for cointegration and tests for unit roots, as we
shall see below, they are not identical. Tests for
unit roots are performed on univariate time
series. In contrast, cointegration deals with the
relationship among a group of variables, where
(unconditionally) each has a unit root.
To be specific, consider Irving Fisher’s impor
tant equation of exchange, MV = Pq, where M is
a measure of nominal money, V is the velocity
of money, P is the overall level of prices and q
is real output.5 This equation can be rewritten
in natural logarithms as:
(2) InM + InV — InP — lnq = O.
In this form, the equation of exchange is an
identity. The theory of the demand for money,
however, converts this identity into an equation
by making velocity a function of a number of
economic variables; both the form of the func
tion and its arguments change from one
theoretical specification to another. In the theory
of money demand, V is unobservable and in ap
plied work it is proxied with some function of
economic variables, V*, lnV* = lnV + E, where E
denotes a random error associated with the use
of the proxy for V .6 The proxy is a function of
one or more observed variables, other than in
come and prices, that are hypothesized to deter
mine the demand for money. Hence, equation 2
is replaced with
ple, money holders might have a money illusion or money
demand might not be homogenous of degree one in real
income.
6For the classic discussion of velocity and a long list of its
potential determinants, see Friedman (1956). Empirical
proxies for velocity often contain one or more of these
determinants.

4That is, for any numbers C > 0 and 0 < p < 1 and for
any starting value Yo there is a time, T, such that, for all t
> T, Pr(|Y,| > C) > p. When le l< 1 the process
generating Y, is stationary in that it does not wander too
far from its mean, i.e., for any given probability p we can
find a constant C > 0 such that Pr (|Yt - / i| C) < p.
5The cointegrating vector could be different from the
hypothesized one for other reasons as well. For exam

MARCH/APRIL 1991

(2') InM + InV* — InP — lnq = E.
If the proxy is good, the expected value of E
should be zero. Furthermore, E should be sta
tionary, so that, V* might deviate from its true
value in the short-run, but should converge to it
in the long-run. Failure to find a stationary rela
tionship among these variables—that is, to find
that they are not cointegrated—implies either
that V* is a poor proxy for V or that the longrun demand for money does not exist in any
meaningful sense.
In essence, the Fisher relationship embodies a
long-run relationship among money, prices, out
put and velocity. In particular, it hypothesizes
that the cointegrating vector (1 , 1 , - 1 , - 1 ) exists.
This vector combines the four series into a
univariate series, E. Given this assumption, a
test for cointegration can be performed by ap
plying any conventional unit root test to E.
Using conventional unit root tests to test for
cointegration [such as tests prepared by Dickey
and Fuller (1979) and Phillips (1987)], however,
requires prior knowledge of the cointegrating
vector. And most often, this vector is unknown.
Therefore, some linear combination of these
variables, for example,
(3) b,lnM + b 2lnV* + b 3lnP + b 4lnq,
is hypothesized to be stationary, where the
cointegrating vector (b,, b 2, b 3, b4) is unknown
and must be estimated.

Locating Stationary Linear C om
bination o f Variables
Engle and Granger (1987), Stock and Watson
(1988), and Johansen (1988) have suggested
alternative tests for cointegration and methods
for estimating the cointegrating vectors. While
differing in a number of respects, all of these
procedures involve locating the "most stationary”
linear combinations (among all of the possible
ones) of the vector time series in question. If
the linear combinations being compared are not
chosen a priori, but are determined by choosing
among all possible vectors, tests for cointegra
tion encounter the type of distributional pro
blems associated with order statistics and multi
ple comparisons. Hence, it is useful to discuss
some of these problems in more detail.
7See Steel and Torrie (1980), p. 588, tor these tables.
8See Stock and Watson (1988) and Johansen (1988) for fur
ther discussion of the relationship to order statistics.

FEDERAL RESERVE BANK OF ST. LOUIS

In multiple comparison tests, an experimenter
is usually concerned with, say, comparing the
highest and lowest sample means among several.
He wants to find the pair of sample means with
the largest difference to see whether the dif
ference is statistically significant. When the
means to be compared are chosen ahead of
time, tests for a significant difference between
the means can be done using the usual tstatistics. If, however, the means to be com
pared are chosen simply because they are the
largest and smallest from a sample of, say, five
means, the rejection rate under the null
hypothesis will be much higher than that im
plied by the percentile of the t-distribution. In
order to control for the experimentwise error
rate, as it is called, tables of distributions of
highest mean minus lowest mean (standardized)
have been computed for the case of no true dif
ferences in the population means. These "tables
of the studentized range” are then used to test
for significant differences between the highest
and lowest means .7
The price paid for controlling the experimentwise error rate is a loss in power. That is, the
difference between the means must be much
larger than in the case of the standard t-test
before it can be declared significant at some
predetermined significance level. Thus, the
power of the test to detect significant dif
ferences is reduced.
In an analogous way, it is difficult to reject
the null hypothesis that there are no stationary
linear combinations when the observed data are
used to estimate the most stationary-looking
linear combination before testing for cointegra
tion. This loss of power is evident in the test
tables given by Stock and Watson or Johansen,
where the percentiles are shifted far away from
those of the standard unit root distribution of
Dickey and Fuller (1979).8 Consequently, detec
ting cointegrating relationships among variables
is relatively hard. Some power can be gained,
however, if economic theory is used to assign
values to some coefficients, a priori. Indeed, if
theory fully specifies the cointegrating vector,
as in our example of the Fisher equation, using
conventional unit root tests to test for cointegra
tion would be appropriate.

Multiple Cointegrating Vectors
Until now, the possibility that only one linear
combination of variables is stationary has been
considered; however, this need not be the case.
In cases where more than two time series are
being considered, more than one stable linear
combination can exist. Also, until now, cointe
gration was discussed without any explicit refer
ence to a dynamic specification of the levels of
the economic variables. Nevertheless, the fact
that cointegration is related to their dynamic
specification was implicit in the fact that all of
the univariate series are 1(1). While a number of
alternative multivariate representations could be
used, it is convenient to use the following multi
variate AR(1) representation:
(4) Yt = AY,., + £„
where Y„ an n by 1 vector, is Z, minus \x,
where Z, is a vector of economic time series (in
our example, M, V*, P and q) and ju is the vec
tor of the means of Z .9 A is an n by n matrix
and £, is a vector of independent random distur
bances that are stationary around zero—that is,
E(e,) = 0 and E(£,£,') = Q for all t.
The possibility of k cointegrating vectors
means there exists a k by n matrix ft', of rank k,
such that P'Y, is stationary in the sense that it is
mean reverting. It is assumed that all of the
elements of Y, are integrated of the same order,
1(1 ); however, for the sake of illustration (here
and elsewhere) we first consider the possibility
that the elements of Y, are 1(0). In this case, the
long-run stationary solution to equation 4 is
Y, = ( I - A r 1£t.10 But there is no need to ask
whether these variables are cointegrated,
because clearly any linear combination of Y, is
mean reverting.
Now consider the case where each element of
Y, is 1(1). Assume further that the elements of Y,
are mutually independent so that A = I. In this
case, no long-run equilibrium exists because the
matrix (I-A ) is of rank zero. Since any linear
combination of these independent 1(1 ) series
must itself be 1(1 ), these variables are not
cointegrated. No stationary linear combination
of these variables exists!
9Note that the system given by Y could be thought of as a
multivariate p-"-order AR system that has been rewritten as
an AR(1) system.
10(l —A)-1 exists if all of the eigenvalues of A are less than
one in absolute value.

Finally, consider the case where all of the
univariate series are 1(1), but A is not an identi
ty matrix. In this case, not every linear com
bination of Y, is stationary because (I-A ) is not
of full rank, that is, (I-A ) -1 does not exist.
However, as we will show, some linear com
binations of Y, may be stationary. The number
of such cointegrating vectors is determined by
the rank of (I-A ). From a purely statistical
point of view, cointegration places some restric
tions on the matrix A. From an economic
perspective, economic theory, which determines
the matrix A, places some restrictions on the
long-run behavior of Yt.
From a somewhat broader perspective then,
the objective of cointegration analysis is to find
an n by n matrix B', of rank n, such that B'Y,
decomposes Y, into its stationary and nonstationary components. This is accomplished by
obtaining a k by n sub-matrix of B', /3', of rank
k such that the transformed series /J'Y, is sta
tionary. The k rows of B' associated with these
stationary series are called "cointegrating vec
tors.” The remaining n - k unit root combina
tions are termed “common trends .”11

Tests f o r Cointegration and Their
Relation to Unit R o o t Tests
In illustrating tests for cointegration, we draw
the analogy between tests for cointegration and
tests for unit roots. Any autoregressive time
series of order p can be written in terms of its
first difference, one lag level and p - 1 lag dif
ferences. Consider first the univariate case,
(5) y| = a,y*_, + a 2y ;_2 + . . . + a„y;.p +e„
where y* = y, - /i. The equation can be
reparameterized as
(6) Ay; = b ,A y ;., + b 2Ay ;_2 + . . . + b p.,A y ; . p+i

-cy.-p + e„
where b, = - 1 + a, + a 2 +. . .+ a, and
c = l - a , - a2
. . - a„.
plying the transformed vector by the inverse of B. In
general, this spreads the unit root “ trend” processes
through all the original series depending on where the
zeros occur in B _1. See Stock and Watson (1988).

"T h is terminology stems from the fact that the original
series are retrieved from the transformed ones by multi-

MARCH/APRIL 1991

Alternatively, equation 5 can be reparameterized
as

(7) Ay; = d.Ay*., + d2Ay*_2 +. . .+ dp_,AylP+1
-cy,_, + e„
where dp_, = - a p, dp_2 = - a p - a ,.,, . .
d,= - a p- a p_, - . . . - a 2.
The Dickey and Fuller test uses the t-statistic
from the ordinary least squares regression to
test the null hypothesis that the coefficient c in
equation 7 is equal to zero. The t-statistic is the
likelihood ratio test of the null hypothesis of a
unit root, where the likelihood is conditional
upon an initial value, y 0. 12
Consider now the multivariate analogue to
equation 5:
(8 ) Y, = A,Yt., + A2Y, . 2 + A3Y,_3 + . . . + ApY,.p
where A,, A2, . . . Ap are n by n matrices. Equa
tion 8 , too, can be reparameterized as either
(9) AY, = r.AY,., + r 2AYt_2 + . . . + r p„ AY, . P+1
- ^Y.-p + t x,
or
(10) A Y , = f l . A Y , . , + 02A Y , . 2 + . . . + e p_, A Y , . P+1

+ £„

If the matrix y> - ( I - A l - A 2~. . . - A p) is full
rank, then any linear combination of Y, will be
stationary. If yj is a matrix of zeros, then any
linear combination of Y, will be a unit root pro
cess and, hence, nonstationary .13
This leaves an intermediate case where ip is
not a matrix of zeros, but is less than full rank.

12An initialization assumption is necessary for unit root pro
cesses. The critical values of the test statistics change
depending on whether n is equal to zero or whether a time
trend is included. Dickey and Fuller (1979) present ver
sions of both tests where n is assumed to be 0 and where
n is replaced with a linear time trend.
13For example, let p= 2, if A 2 = 0 and A, = I, then
( I - A , - A : ) = 0. In this case, all the elements of X, are
unit root processes and ( I - A , - A , ) is rank zero. But, if
A2= 0 and A, is (e, e2 • • ■e.)l. where |e,| < 1 for all i, i =
1,2.......... n, then (i - A , - A 2) is full rank and all the
elements of Xt are stationary AR(1) processes.
14To our knowledge, Theil and Boot (1962) were the first to
prepare a test for the statistical significance of the eigen
values of
In so doing, they were the first to develop a
test for cointegration.
15There are three versions of the univariate test for unit
roots (zero mean, constant, non-zero mean and a constant
linear time trend). Johansen (1988) presents the multivari
ate analogue of the likelihood ratio test when ^ is zero.

FEDERAL RESERVE BANK OF ST. LOUIS

The rank of yj, k, is the number of linearly in
dependent and stationary linear combinations of
Y, that can be found. In other words, it is the
number of linearly independent cointegrating
relations among the variables in Yt. Of course,
the estimate of xp, yj, will almost always be of
full rank in a numerical sense. The objective of
tests for cointegration is to test for the rank of
yj by testing whether the eigenvalues of yj are
significantly different from zero .14 From this
perspective, the tests for cointegration purposed
by Stock and Watson (1988) and Johansen
(1988) are multivariate analogues of the DickeyFuller test .15

IS TH ER E AN ECONOMIC INTER
PRETA TIO N O F COINTEGRATION
VECTORS?
It should be clear now that, if several 1(1)
variables are cointegrated, then one or more
linear combinations of them will have a finite
variance. But the broader question of the
economic interpretation of such cointegrating
vectors remains. In general, cointegrating vec
tors are obtained from the reduced form of a
system where all of the variables are assumed
to be jointly endogenous. Consequently, they
cannot be interpreted as representing structural
equations because, in general, there is no way
to go from the reduced form back to the struc
tu re .16 Nevertheless, they might be thought of
as arising from a constraint that an economic
structure imposes on the long-run relationship
among the jointly endogenous variables. For ex
ample, economic theory suggests that arbitrage

Johansen and Juselius (1990) extend this to the case
where \i is non-zero. In practice, the difference between Y,
and Y ] , the first sample observation, should be used if the
/i= 0 assumption is maintained; hence, it seems reason
able to use the unconstrained \i versions of the tests [See
Dickey, Bell and Miller (1986) for a discussion of which
test to use]. Note that the formulas for computing the test
statistics are straightforward least-squares formulas. It is
the distribution of the test statistics that are nonstandard.
16Of course, because equation 17 is essentially a multivar
iate VAR, in principle, it should be possible to give these a
“ structural” interpretation by imposing identifying restric
tions on the reduced-form parameters, as has been done
recently for VAR models. For example, see Bernanke
(1986) and Blanchard and Quah (1989).

will keep nominal interest rates—especially
those on assets with the same or similar maturi
ty—from getting too far away from each other.
Thus, it is not surprising that such interest
rates are cointegrated .17

Cointegration with Exogenous
Variables
The importance of cointegration in economics
can be highlighted by noting the close cor
respondence between cointegration and the
typical linear, dynamic, simultaneous equation
model used in macroeconomics and
econometrics,
(11) AY, = BY,., + CX, + u,.
Y, is a n by 1 vector of endogenous variables, X,
is a g by 1 vector of exogenous variables and u,
is a n by 1 vector of random disturbances, and
A, B and C are matrices of unknown
parameters. It is assumed that A -1 exists, so
that the dynamic reduced form can be written
as
(12) Y, = nY,_, + rx , + £„
where n ( = A~'B) and I” ( = A~'C) are matrices of
unknown reduced-form parameters. Equation
1 2 contains only predetermined variables on the
right-hand side, so that the dynamic response of
Y, to X, can be studied by recursive substitution
using equation 12. Letting L denote the lag
operator (LY, = Y,_,), the result is written as
(13) Y, = (I + ttL +

t i 2L 2

+ . . .)I~X,

+ (I + TlL + II2L 2 + . . .)£,.

The infinite series (I + tiL + ti2L2. . .), evaluated
at L = l, converges to (I-tt)- 1 if all of the eigen
values of ti are less than 1 in absolute value.
The expected value of equation 13, conditional
on X„ is
(14) E(Y,) = (I + nL + n2L 2 . . JfX ,
= ( I - ttL)-TX ,.
Equation 14 is used to investigate the long-run
response of Y, to a change in one or more of
the elements of X,. Assuming that X, is a vector

of zeros for all t < 0 and X, = 6 for all t > 0,
and that the system converges, then, in the
limit
(15) E(Y) = (I - 7i)-Tc5.
Thus, (I - 7z)_ T c5 gives the long-run response of
Y, to a permanent change, 6, in X. Theil and
Boot (1962) have termed (I - n J 'T the finalform multipliers.
The cointegration techniques of Johansen and
Stock and Watson start with an equation similar
to equation 12. There are two key differences,
however. First, all of the variables are explicitly
endogenous. Second, because n has unit eigen
values corresponding to the common trends in
the system, (I - t i) is not invertible, and, there
fore, the final-form multipliers in 15 are
undefined . 18
A related technical point is that the initial con
ditions are not transient. To see this in equation
12, let X, = 0 for all t and substitute recursively
to obtain
(16) Y, = £, + 7i£,_, + 7t2£,_2 + . . . + n,_1£,
+ n'Yo.
Because the matrix n' does not converge to a
matrix of zeros as t approaches infinity, the in
itial condition is not transient and must be speci
fied. Often, the initial vector, Y0, is assumed to
be a vector of zeros.
Can cointegration be used to determine the
long-run response to changes in exogenous
variables? In general, the answer is yes; how
ever, how this is done, the interpretation of the
estimated cointegrating vectors and the method
of estimation, all depend on the assumptions
made about X,. Consider, for example, the case
where the elements of X, are non-stochastic and
fixed in repeated samples. Rewrite equation 12
as
(17) AY, = —(I —tt)Y,_, + TX, + £,.
If Y, is cointegrated, the n by n matrix (I —tt) is
rank k < n , and t i can be represented as aft’,
where a and fi are n by k matrices of rank k.
Then
(18) p % = (I-/3'a)/3'Y,_, + /JTX, + 0'e,

17Stock and Watson (1988) find that the nominal federal
funds, the three-month Treasury bill and one-year Treasury
bill rates are cointegrated.
18Finding unit eigenvalues in n is equivalent to finding zero
eigenvalues in ip. This is so because yj = i - n so that
|i+»—Al| = |A(I —tt) —11 = |n -(1 -A )l| = |n —dl| = 0, SO A
= 0 is equivalent to 6 = 1.

MARCH/APRIL 1991

is stationary . 19 Equation 18 can be expressed as
the k-dimensional system, with p'Yt = Y*„
(19) Y*t =

+ r*X, + £*,.

ti'Y*,,,

Because the k by k matrix I —tt* is full rank,
(I - t t *)-1 exists. Thus, this lower-dimensional
system has the steady-state representation,
(20) Y*. = (I —tt*)- 1 T'X, + (I-TT* ) " 1 £*,.
The expected value of 20 gives the final-form
multipliers for the cointegrated subsystem of Yt.
Consequently, there is a representation for the
long-run average response of /3'Yt to a change in
X.
It must be remembered, however, that because
the estimated cointegrating vector is conditional
on the information set in the model, X, needs to
be included along with lagged differences of the
endogenous variables when estimating the coin
tegrating vectors. Moreover, the distribution of
the test statistic will not be invariant to Xr Con
sequently, the Monte Carlo experiments used to
derive the distributions of the test statistics used
by, say, Johansen (1988), would have to be
redone including Xt in the model.
In the case where X, is stochastic, the method
used to obtain estimated cointegrating vectors
and their interpretation changes with the
assumed structure of the model. A general form
of such a model is

where it is assumed that

[

a2 1
0

' '

a\ I

for all t. Again, assume that A " 1 exists, so 21
can be rewritten as
M

[V ,] .

[ A - .,

-A "C D ]

+ p .]

or
19More technically, the eigenvalues of i-p'a are less than one
in absolute value.
20A form of this model has been proposed recently by Hoff
man and Rasche (1990). This is the Stock-Watson formula
tion, except that X, is a set of latent variables. Conditional
on the unobserved X,, the endogenous variables are
assumed to be stationary.
21This so because
7 i-/ll
0

D-Al

= | n - A l | I D - A l l = 0.

FEDERAL RESERVE BANK OF ST. LOUIS

Cointegrating vectors and their interpretation
depend on the assumptions made about tt and
D. For example, if all of the eigenvalues of tt are
less than one, so that conditional on Xt, Yt is sta
tionary, the common trends in the system are
the result of X, being non-stationary. The num
ber of common trends will be equal to the
number of unit roots in D. If D = I, for example,
then there will be g common trends and n coin
tegrating vectors. In this case, there is a steadystate representation for the endogenous variables
conditional on the exogenous variables .20 That
is, the final-form multipliers of Theil and Boot
are conditional on the exogenous variables.
They do not exist, however, because the
variance of the exogenous variables is unbound
ed and, hence, so is the unconditional variance
of the endogenous variables.
If tt also has some unit eigenvalues, however,
then part of the nonstationarity of the system is
due to instability in the dynamics of the endog
enous variables in the system .21 In this case, the
conditional final-form multipliers, analogous to
those obtained by Theil and Boot, exist along
the stationary directions given by the cointegrat
ing vectors as before. An important aspect of
this formulation is that, given the proposed
structure of the system, the researcher can
identify whether the common trends stem from
the structural dynamics or are simply a mani
festation of the stochastic properties of the ex
ogenous variables. Under the assumption on
(etr7t)', estimating cointegrating vectors for the
system given by the equations in 23 is the same
as estimating them for the system given by the
equations in 8 . 22 Hence, any of the multivariate
methods discussed in the next section can be
employed. The reader is cautioned, however,
that these procedures would have to be
modified to impose the upper triangular struc
ture of the system given by the equations in 23.23

22lf these error terms are not independently distributed ob
taining consistent estimates of the cointegrating vector will
be more difficult. For an example of this case, see Stock
and Watson (1989).
23lndeed, both Stock and Watson (1988) and Johansen
(1988) allow for the possibility of exogenous variables in
the sense of equation 22; however, they do not impose the
exogeneity restriction ex ante.

Should There B e M any o r Few
Cointegrating Vectors?
Since it is possible to have n-1 cointegrating
vectors in the system given by 17, the question
naturally arises, is it better to have many or
few cointegrating vectors? Providing a general
answer to this question is difficult. Cointegrating
vectors can be thought of as representing con
straints that an economic system imposes on the
movement of the variables in the system in the
long-run. Consequently, the more cointegrating
vectors there are, the "more stable” the system.
Other things the same, it is desirable for an
economic system to be stationary in as many
directions as possible.
To see why, consider a model with no com
mon trends, so the system is stationary: Yt
never wanders “too far” from its steady-state
equilibrium value (in the model discussed in the
text, the vector of means). If there is one com
mon trend and n - 1 cointegrating vectors,
however, n - 1 of the variables must be solved
for in terms of the nth, and the structure of
these variables follows a single common trend.
Hence, there are only n - 1 directions (as oppos
ed to n in the previous example) where the
variance is finite and one direction in which it
is infinite. On the other hand, if there is only
one cointegrating vector, the nth variable must
be solved for in terms of the other n - 1
variables. The system can wander off in n - 1
independent directions; it is stable in only one
direction .24
To see this point from a geometric perspec
tive, consider the case were there are three en
dogenous variables that span R3. If these vari
ables are stationary, the system converges to a
steady-state equilibrium, a point in R3, and varia
tion around that point is finite. If the variables
are non-stationary, and there is one common

24 In terms of the model given by the equations in 22, when
(I —tt) is full rank, the more cointegrating vectors in D, the
more directions in which the “ final-form” multipliers will
exist.
250 f course, the subsystem defined by equation 20 con
verges to a point in R2. This point is given by the intersec
tion of this plane and the line given by the intersection of
the two cointegrating vectors.

trend, however, the system converges to a longrun equilibrium represented by a line, deter
mined by the intersection of the planes defined
by the two cointegrating vectors, in R 3.25 This is
a stationary equilibrium in the sense that the
variance about this line is finite. If there are
two common trends and one cointegrating vec
tor, the long-run equilibrium is represented by
a plane defined by the single cointegrating vec
tor. The variables are unbounded in the plane,
but cannot move too far from it. That is to say,
that the variance in the plane is infinite, but the
variance about the plane is finite. If there are
no cointegrating vectors, the variables are free
to wander anywhere in R3—they are unbounded!
Consequently, when non-stationary variables
are cointegrated, there exists a direction where
a meaningful long-run relationship among them
exists. The fewer the number of cointegrating
vectors, the less constrained is the long-run
relationship. Hence, all other things the same, it
seems desirable to have many cointegrating vec
tors .26 Alternatively stated, we prefer economic
models that have unique steady-state equilibria.
Accordingly, researchers are interested not only
in testing to see whether variables are
cointegrated, but in obtaining precise estimates
of the cointegrating vectors.

ALTERNATIVE TESTS FO R
COINTEGRATION
With a number of tests for cointegration be
ing available, it is important to understand their
similarities and their differences. The purpose
of this section is to discuss the salient features
of alternative tests for cointegration. Step-bystep instructions on how to perform two of the
more difficult of these (the Stock-Watson and
Johansen tests) are presented in a shaded insert
on the following pages.

vectors. At a more practical level, it is well known that
macroeconomic time-series data are highly correlated so
that, typically, the generalized variance of the matrix of
such variables is concentrated on relatively few principal
components. See footnote 27.

26Having said that, we should hasten to add that it is very
unlikely that these tests will indicate that there are a large
number of cointegrating vectors. These tests lack the
power to reject the null hypothesis of zero cointegrating

MARCH/APRIL 1991

Step-By-Step A p p lication Of T h e Jo h a n s e n And
S tock -W atson A p p ro a ch e s To C o in teg ratio n
Because the procedures developed by
Johansen, Stock and Watson are more dif
ficult to employ, this insert provides stepby-step procedures for applying these ap
proaches. Both of these procedures can be
illustrated with the multivariate model
(equation 8 of the text)
Y, = A .Y,., + A2Yt_2 + . . . + ApY ,.p +

MAX EIGENVALUE TEST =
- N l n ( l - e 2+1).
Compare the test statistic to the ap
propriate table in Johansen and
Juselius (1990).2
Note: The squared canonical correla
tions are the solution to the determinantal equation
lG? s kk“ S k0 S07

Step-by-Step Application o f
Johansen’s Approach to
Integration:
1.

Pick an autoregressive order p for the
model.
2. Bun a regression of AYt on AY,.,, AY, . 2 ,
. . . , AY, . p+1 and output the residuals, D,.
For each t, D, has n elements.
3. Begress Y,.p on AY,., , AY, . 2 , . . . ,
AY,_p+i and output the residuals, L t. For
each t, L, has n elements.
4. Compute squares of the canonical cor
relations between D, and L t , calling
these q\ > qI > . . , > qI
5a. Letting N denote the number of time
periods available in the data, compute
the trace test as
TBACE TEST = - N .2M n(l -g f).1

The null hypothesis is “there are k or
less cointegrating vectors.”
5b. You may choose to use the maximal
eigenvalue test (which really uses the
k + r h largest squared canonical correla
tion or eigenvalue) as follows:

’ Theoretically, a cointegrating vector is associated with
ef.e*. e2
,- O' 0- The theoretical counterpart of the trace
test"for’hlo: k = 3 or less cointegrating vectors is
- N l ln(1 °-0 ) = 0 so the test statistic would be within
sampling error of 0. For Ho: k = 2 or less cointegrating
vectors, the theoretical value of the test is
- N X In ( l- e f) = - N ln(1 - q*) > 0 and as N gets
large’ this diverges to +
Note that e; and ef (which
are both zero) do not contribute to the test statistic and
this is the motivation for the so-called “ maximal eigen
value” test of Johansen.

F E D E R A L R E S E R V E B A N K O F ST. LO U IS

S 'J = 0 where

Skk = N~‘ t L,L,' S00= N~‘ ID .D ,' and
N

S k0 = N_1Z L,D'and D, and L, are colt= 1

umn vectors of residuals from steps 2
and 3. The maximum likelihood
estimates of the k cointegrating vectors
(k columns of ft, for which e 2 §kk p. =
S k0 S oo^ko Pi)-

Step-by-Step Application o f the
Stock-Watson Approach to
Cointegration:
In the Stock-Watson approach the null
hypothesis is that there are m common trends
( n - k = m) against the alternative that there are
less than m, say m —q, common trends. There
are six steps:
1.
Pick the autoregressive order p for the
model.
2.
Compute the eigenvectors of SY,Y',
that is, do a principal components
analysis of Y,.

2The appropriate table as well as the handling of the in
tercept term in estimation depends on the role of the
intercept in the model. For a discussion of this, refer to
Dickey and Rossana (1990).

Using the m principal components with
highest variance, that is, largest eigen
values, fit a vector autoregression to the
differences. If Pt is the vector of m prin
cipal components (selected as described
in the text) then the autoregressive model
is denoted AP, = A 1AP,_, + . . . +
Ap.jAP,.,., +£t , where, as before, p stands
for the number of lags in the “original”
autoregression. This provides a filter to
use in step 4.
Compute a filtered version, Ft , of P, by
F , = P ,-A , P,_, - . . . - ApP,_p. This

The relationship among tests for cointegration
developed by Johansen (1988), Engle and
Granger (1987), Stock and Watson (1988) and
Fountis and Dickey (1989) can be illustrated by
considering the multivariate model
(24) Y, = A.Y,., + A2Y,_2 + . . . + Ap Y,_p + e,.
Johansen reparameterizes equation 24 as
(25) AY, = r , AY,., + r 2 a y , . 2 + . . . + r p_, ay,_p+,
-

ip Y ,.p + E„

where , as before, ip = (I - Ai - A2 . . . - Ap).
He then makes use of the fact that any n by n
matrix, ip, of rank k < n can be written as the
product of two n by k matrices of rank k—that
is, if) = afi', where a and P are n by k matrices
of rank k. He maximizes the likelihood function
for Yt conditional on any given ft using standard
least squares formulas for regression of AY, on
AY,.,, AYj_2, . . ., AY, . p+1 and /J'Y,_P. The solution
to this maximization problem gives estimates of
T,, l~2, . . . , r p_, and a conditional on p. Once
this is done, p, or more specifically, the row
space of p, is estimated.
In cointegration, it is only possible to deter
mine the rank of a/3' . 27 Specifically, obtaining
unique elements of a and p without imposing
arbitrary constraints is impossible. The rank of
yj can be obtained by computing canonical cor
relations between AY, and Y,_p, adjusting for all
intervening lags. Johansen chooses to put the
lag level at the largest lag but, as has been
shown earlier, this is not crucial.

5.
6.

reduces the multi-lag model to a one
lag model.
Regress AFt on F,_, getting coefficient
matrix B.
Compute the eigenvalues of B, normal
ize, and compare to the distributional
tables of Stock and Watson (1988). Re
jecting the null hypothesis of m com
mon trends in favor of the alternative
of m - q common trends means a
reduction in the number of common
trends by q and thus an increase of q
in the number of cointegrating vectors

In the Johansen approach, - a p ' is the coeffi
cient matrix on the lagged level. Upon pre
multiplying equation 25 by p', the last term in
equation 25 is P'aP'Y,_p. Note that P'a has no
zero eigenvalues so that P'Y, is a stationary vec
tor time series of dimension k. Thus, rows of p'
are the cointegrating vectors.
Now consider adding to the k rows of P', n - k
more rows orthogonal to the columns of a. De
note the resulting matrix B'. Note that B'a is sim
ply n - k rows of zeros appended to the bottom
of P'a. In equation 25, after this transormation,
the last n - k rows involve only differences.
Hence, C, = B'Y, is a column vector of k stationary
processes followed by n - k unit root processes
(common trends). In Johansen’s approach, then,
the matrix B' has the k cointegrating vectors as
its first k rows and coefficients yielding the
n - k common trends as its last n - k rows.
By standard results in unit root estimation, the
vector C, = B'Y, is such that N~2 X C,C,'(where
N is the series' length) converges to an n by n
matrix with zeros everywhere except in the
lower right n - k by n - k submatrix. This result
underlies the other approaches to cointegration.
For example, Stock and Watson point out that
if W, = (B ')"‘C„ then the sum of squares and
cross products matrix N" 2ZW,W' converges to
the same limit as (B')_ 1N ' 2ZC,C,' B~‘. This limit
matrix has rank n - k and, thus, has n - k nonzero
eigenvalues. Because the generalized variance of

27The space spanned by the columns of the a and f)'
matrices is important because a/3' can be obtained by
many choices of matrices. To see this, note that for any
nonsingular k by k matrix, H, (o-H- ')(H/3') = ap'.

MARCH/APRIL 1991

a sum-of-squares and cross-products matrix is
equal to the sum of the eigenvalues of this
matrix, they suggest a test based on computing
the largest eigenvalues of the sum-of-squares
and cross-products matrix .28

level. In this case, however, the test of the null
hypothesis that q = 1 is complicated by the pre
sence of a nuisance parameter, A. Regardless of
q , as noted before, the model can be
reparameterized as

In multivariate statistics, a vector random
variable Y can be transformed into a canonical
vector random variable C = TY by choosing T (a
matrix of constants) so that the elements of C
are uncorrelated. The elements of C are refer
red to as “principal components” of Y. Stock
and Watson reason that if a vector process has
n - k common trends (that is, the vector Ct = B'Yt
has n - k unit root processes and k stationary
ones), then the n - k principal components with
largest variance should correspond to the unit
root processes or "common trends.” This reason
ing is based on the previously Stated notion that
the normalized sum-of-squares and crossproduct matrix N~2ZY,Y,' converges to a singular
limit so the variances of the n - k common
trends correspond to principal components giv
ing the nonzero eigenvalues of the limit matrix.
The k principal components with smaller
variances correspond to the zero eigenvalues of
the limit matrix and each of the k rows is a
cointegrating vector.

(27) Ay,= - ( g - l) ( A - l) ( y t_, -fi) + AgAyt_, + et.

A Note about Distributions
Having more than one lag in the model intro
duces matrices of nuisance parameters (that is,
parameters which must be estimated but are
not of primary interest) that affect the form of
the test statistics. How the existence of these
parameters complicates the test procedures is il
lustrated for the univariate case .29 Consider a
simple version of the univariate model given by
equation 5:

(26) (yt- m) = (e+A) (y,_,-fi) -

+ e,.

If g = l, M drops out. The series yt is a unit root
process with no tendency to move toward any

28For example, let X be an n by k matrix and let J = (X'X)
be a k by k matrix of rank k. Then there exists a k by k
matrix T such that T' T = A, where A is a diagonal matrix
with the eigenvalues of J on the diagonal. The columns of
T are the eigenvectors of J. These eigenvectors are called
“ principal components” because the generalized variance
of J, |J |, is equal to the trace of A. Because often many of
the eigenvalues of J are close to zero, its generalized vari
ance can be closely approximated by a relatively few
“ principal components.” A good discussion of this can be
found in Dhrymes (1970), pp. 53-59.
29These ideas can also be found in Fuller’s text (1976) or in
Dickey, Bell and Miller (1986).

FEDERAL RESERVE BANK OF ST. LOUIS

Notice that the coefficient on y,_,
is now a
multiple of g - 1. In ordinary regression, multi
plying a regressor by a constant changes the
distribution of the regression coefficient by a
multiplicative constant, but does not change the
distribution of its t-statistic. The same holds
true asymptotically in cointegration .30
If A is known, the estimated coefficient on
could be divided by (1 —A) and the limit
distribution n(gM- l ) listed in Fuller (1976, sec
tion 8.5) could be obtained. If A is unknown,
however, an approximation must be obtained by
dividing by (1 —A), where X is estimated by
regressing Ay, on Ay,.,, suggested by imposing
the null hypothesis g = l.
Alternatively, the univariate model could be
written as
(28) A(y,-Ay,.,) = (e -l)(y t_,-Ay,_2- /i(l-A )) + e,.
Since A is a consistent estimate of A, it can be
used to filter y,. That is, F ,= y ,-X y ,.,. AF, can
then be regressed on F,_, (with an intercept) or
on F ,_ ,-F , where F is the mean of the series, F„
to get a test statistic asymptotically equivalent to
n(eM- l ) .

Other Approaches to Cointegration
The approach of Fountis and Dickey (1989) is
similar to that of Stock and Watson but only
allows for the possibility that there is one unit
root. As such, it is much less general than
either of the above approaches. It does, how
ever, provide estimates of the cointegrating vec
tors and common trend, based on the coeffi-

30Johansen, using a multivariate analogue of the DickeyFuller t statistic, only needs to have a correctly specified
likelihood function. Stock and Watson, using a multivariate
analogue of the normalized coefficient tests n(g —1),
n(e„-1), or n(e - 1 ) of Dickey and Fuller need to adjust for
this fact multiplier (corresponding to 1 - A in our univariate
example).

cient matrix of the lagged levels. Although there
is an asymptotic link between variances and lag
coefficients as established above, the actual
definition of a unit root process is in terms of
lag relationships. All approaches to cointegration
use lags in testing. However, in transform ing Y„
Stock and Watson use only the variancecovariance matrix while Fountis and Dickey
(and Johansen) use the lag information.
Finally, we mention an approach by Engle and
Granger which is especially easy to use and to
explain in the bivariate case. Consider two
univariate series y, and zt. The first step is to
check that each is a unit root process. Next
regress yt on z, (or zt on yt) getting the linear
combination yt- b z t with smallest variance.
Notice that this does not use lag information to
obtain the transformation y ,- b z t. The next step
is to test if the series y ,-b z , is stationary (if so,
the vector (1 ,-b ) is a cointegrating vector). Had
the parameter b been pre-specified instead of
estimated, an ordinary Dickey-Fuller test would
be appropriate. Since the data are used to
estimate b, however, the previously mentioned
problems analogous to the “multiple com
parisons1' problem exist. Engle and Granger
have provided the appropriately adjusted
percentiles. Other than using these special
tables, this approach uses only the standard
univariate unit root testing strategy.
One problem with this approach is that it re
quires the researcher to choose one of the joint
ly endogenous variables to put on the left-hand
side. While the test is asymptotically invariant
to this so-called direction normalization rule, the
test results may be very sensitive to it in finite
samples. Indeed, practical experience indicates
that the result of the test depends qualitatively
on which variable is chosen to be on the lefthand side. The alternative multivariate ap
proaches have the advantage that all of the
variables are explicitly endogenous, so the
researcher does not have to make such ar
bitrary normalization choices.

31Strictly speaking, the nonstationarity indicates only that the
cointegrating vector for M1 and nominal GNP is not (1, -1).
For example, see Nelson and Plosser (1982). Also, see
Engle and Granger (1987) who test for cointegration bet
ween M1 and income.

AN APPLICATION OF
COINTEGRATION: TH E DEMAND
FO R MONEY
One important macroeconomic relationship
that has received considerable attention is the
link between money and income. This relation
ship, embedded in the demand for money, is
commonly represented by the income velocity
of money. Since the income velocity of M l has
drifted upward over time, it does not appear to
be stationary. Furthermore, formal tests indicate
that the income velocity of M l is not stationary,
indicating that M l and income, in the form of
income velocity, are not cointegrated .31 In con
trast, the income velocity of M2 appears to
move around an unchanged mean, and formal
tests suggest that M2 and income are
cointegrated .32 These results have been inter
preted by some as evidence against the ex
istence of a stable long-run demand for M l and
for the existence of a stable long-run demand
for M2 .33
But if M l is the relevant measure of "money,”
the Fisher equation suggests that there should
be at least one cointegrating relationship be
tween M l, its velocity, real income and the
price level. The specification of tests for
cointegration depend on the specification of the
demand for money. Consequently, it is impor
tant to review the theory of money demand
before performing tests for cointegration.
A general specification for the long-run de
mand for money is
(29) Md = f(P, Q, Z),
where M and Q denote the nominal money
stock and the nominal income level, respective
ly, P denotes the level of prices and Z denotes
all other variables that affect money demand—
for example, current and expected future real
interest rates, the expected rate of inflation, etc.
Assuming that economic agents do not suffer

33For example, Hallman, Porter and Small (1989) have
predicated their P-star model on such a long-run stable
demand for M2.

32For example, see Engle and Granger (1987). However, the
results for M2 appear to be sensitive to the sample period
and how the test is performed. See Hallman, Porter and
Small (1989, 1990) and Hafer and Jansen (1991).

MARCH/APRIL 1991

from a money illusion, equation 29 can be writ
ten as
(30) Md/P = ma = f(Q/P, Z) = f(q, Z).
That is, the demand for real money balances,
md, is a function of real income, q, and some
other variables. Furthermore, it is commonly
assumed that the demand for money is homo
genous of degree one in real income, so that
equation 30 can be written as
(31) md/q = h(Z),
where h(Z) is the famous k in the Cambridge
cash balance equation—that is, the reciprocal of
the income velocity of money. In equilibrium,
the demand for real money equals the supply of
real money, ms, so that h(Z) is observed simply
as the ratio of real money stock to real income.

The Velocity o f M l and M2
Consider now two alternative monetary ag
gregates, M l and M2. In this framework, the
reciprocals of their velocities can be written as
(32) ml/q = hl(Z), and
(33) m2/q = h2(W),
respectively. The specification for M2 allows for
the possibility that there are factors that affect
its demand that do not affect the demand for
M l—that is, Z is a subset of W. Since M2 is
simply M l plus some other financial assets,
equation 33 can be written as
(34) ml/q + nmlm2/q = hl(Z) + v(W),
where nm lm 2 denotes the real non-Ml com

34These relationships hold for seasonally adjusted M2 as
well, because M1 and the non-M1 components of M2 are
seasonally adjusted separately and added together.
35Dickey-Fuller tests cannot reject the null hypothesis of a
unit root over some sample periods. Figure 2 suggests
that these results are likely driven by the low power of the
test against the alternative of a large, but stationary, root.
Hence, one might wish to rely on his eyes rather than the
formal test results.
36At a more formal level, the proposition that the decline in
reciprocal of M1 velocity is just offset by the rise in the
reciprocal of the velocity of NM1M2 was tested by a sim
ple linear regression the reciprocal of NM1M2 on the re
ciprocal of M1 velocity and testing the null hypothesis that
the coefficient is equal to - 1 . The estimated slope coeffi
cient was -1 .0 5 5 with a t-statistic of -41.47. While the
estimated coefficient was very close to - 1 , the null
hypothesis was rejected at the 5 percent significance level.
The t-statistic was 2.17. Nevertheless, a formal test for
cointegration using an augmented Dickey-Fuller test on the
residuals from this equation suggests that these variables
are cointegrated. This proposition also was tested by

F E D E R A L R E S E R V E B A N K O F ST. LO U IS

ponents of M2 and v(W) = h2(W )-hl(Z),
hereafter, called the reciprocal of NM1M2
velocity .34
The above analysis has two important implica
tions. First, because velocity is not directly and
independently observable, its proxy must be
specified to perform tests for cointegration. Se
cond, if, in fact, M l and income are not cointe
grated but M2 and income are, a long-run,
stable inverse relationship must exist between
hl(Z) and v(W). On average, movements in hl(Z)
must be offset by movements in v(W), that is,
hl(Z) and v(W) must be cointegrated.
A simple analysis of M l, M2 and GNP data is
consistent with this conjecture. Figures 1 and 2
show the observed income velocities of M l and
M2—that is, (q/ml) and (q/m2)—respectively, for
the period from 1953.1 to 1988.4. M l velocity
trends upward through the early 1980s and
then declines. In contrast, M2 velocity appears
to cycle with no apparent trend .35 Also, the
sharp break in the pattern of M l velocity in the
early 1980s is not as apparent in M2 velocity.
Figure 3 shows the reciprocals of M l, M2 and
NM1M2 velocities. As expected, the downward
trend in the reciprocal of M l velocity through
1980 appears to be matched by an upward
trend in the reciprocal of the velocity of
NM1M2.36 Also, a comparison of the series in
figure 3 reveals that much of the variability in
the reciprocal of M2 velocity is associated with
variability in the reciprocal of NM1M2 velocity,
rather than variability of the reciprocal of M l
velocity .37

estimating a simple linear time trend for the reciprocals of
the velocities of each M1 and NM1M2 and testing the
hypothesis that trends are equal and opposite in sign. The
estimated trend coefficients for M1 and NM1M2 were
-.00143 and .00154, respectively, and both were statisti
cally significant at well below the 5 percent level. But,
again, despite the closeness of the estimates, the null
hypothesis was rejected at the 5 percent significance level.
The F-statistic was 8.67.
37This can be illustrated by a simple linear regression of the
reciprocal of M2 velocity on the reciprocals of the veloci
ties of each M1 and NM1M2. These regressions indicate
that the reciprocal of the velocity of NM1M2 explains 19
percent of the variation in the reciprocal of the velocity of
M2, while the reciprocal of M1 velocity alone explains only
.03 percent of the variation.

Figure 1

The Income Velocity of M1
1/1953 to IV/1988

Figure 2

The Income Velocity of M2
1/1953 to IV/1988
Ratio

Ratio

Figure 3
T h e R e c ip ro c a ls o f th e In c o m e V e lo c itie s o f M 1,
M 2 an d th e N o n -M 1 C o m p o n e n ts o f M2
1/1953 to IV/1988

MARCH/APRIL 1991

The Velocity o f the M onetary Base
The concept of income velocity can be extended
directly to a broader or narrower range of
monetary aggregates. One such aggregate, the
monetary base, is particularly important because
the monetary authority can control it fairly
well. While the observed monetary base velocity
is simply the ratio of income to the monetary
base, the measure is only meaningful in the
context of the demand for money .38 Monetary
base velocity can be incorporated into the
money demand framework by noting that the
nominal money supply, M‘, can be expressed as
(35) Ms = mm(H)MB,
where MB denotes the adjusted monetary base
and mm denotes the money multiplier which is
a function of a set of variables H, for example,
interest rates, portfolio preferences of the
public, etc. Dividing both sides of equation 35
by the price level and substituting the result for
md in equation 31, yields
(36) mm(H)mb = qh(Z),
where mb denotes the real monetary base.
Because the money multiplier is not observed
independently, tests of cointegration involving
the monetary base must include not only q and
Z, but H.39

Empirical Results
The empirical work is presented in two parts.
The first presents tests for cointegration using
methodologies suggested by Johansen, StockWatson and Engle-Granger. The results in this
part are presented only for M l. In the second
part, the analysis is extended to a broader set
of monetary aggregates using the Johansen ap
proach. The Johansen methodology was chosen
because it is based on the well-accepted likeli

38For example, it is not reasonable to obtain this result by
simply assuming that the monetary base is the appropriate
measure for money because it is composed of currency
and bank reserves.
39lt is not necessary that H include variables that are not in
cluded in Z. If it does not, however, there is an identifica
tion problem. That is, one cannot tell the difference bet
ween the above model and simply treating the monetary
base as the appropriate monetary aggregate. Since this
possibility is difficult to conceive of, it is useful if H in
cludes variables that are not in Z, as in our empirical
work which follows.
40A simple linear regression of the multiplier (or its growth
rate) on K (or its growth rate) using quarterly data, indi

FEDERAL RESERVE BANK OF ST. LOUIS

hood ratio principle. Moreover, recent Monte
Carlo evidence by Gonzalo (1989) suggests that
Johansen’s maximum likelihood technique for
estimating and testing cointegrating relation
ships performs better than both single equation
methods and alternative multivariate methods.
Nevertheless, because the other two approaches
are widely known and the Engle-Granger ap
proach is especially widely used in the empirical
literature to date, we report test results for M l
using all three techniques.
In the second part, the analysis is extended to
other monetary aggregates. These aggregates
are the adjusted monetary base (calculated by
the Federal Reserve Bank of St. Louis), M2 and
the non-Ml component of M2, denoted NM1M2.
When the monetary base is used, the ratio of
currency to total checkable deposits, denoted K,
is also included because it is the most important
determinant of the money multiplier.40 In both
parts, the income and price level measures are
real GNP, q, and the GNP deflator, P, respective
ly. Two measures of nominal interest rates, R,
are used: the three-month Treasury bill rate,
R3M, and the yield on 10-year government
securities, R10Y. The data consist of quarterly
observations from 1953.2 to 1988.4 and all data
are transformed to natural logarithms.
T e s ts f o r t h e O r d e r o f I n t e g r a t io n : Before
testing for cointegration, the order of integra
tion of the individual time series must be deter
mined. Tests for unit roots are performed on all
of the data using the augmented Dickey-Fuller
test with three lagged differences. The null
hypothesis is that the variable under investiga
tion has a unit root, against the alternative that
it does not. The substantially negative values of
the reported test statistic lead to rejection of the
null hypothesis.
The tests are performed sequentially. The first
column in the top half of table 1 reports tests

cates that K alone explains 95 percent of the variation in
the level of the multiplier. Moreover, the growth rate of K
alone explains 84 percent of the variation in the growth
rate of the multiplier.

ly consistent with the hypothesis that the in
dividual time-series are individually 1(1). Because
these data appear to be stationary in first dif
ferences, no further tests are performed.

Table 1
Augmented Dickey-Fuller Test For
A Unit Root
‘m
3 lags

*T
3 lags

R3M
R10Y
K
NM1M2/P

-0.80 1
-0 .8 1 7
0.308
-0 .7 2 3
-2 .6 8 9
-1 .8 7 3
-0.56 1
-2 .1 5 0

-2 .4 5 6
-1 .4 4 4
-2 .2 8 5
-2 .1 6 7
-3 .8 8 4 *
-2 .4 4 4
-2 .2 7 4
-1 .5 8 0

A(M2/P)
A(M1/P)
A(MB/P)
Aq
AR3M
AR10Y
AK
A(NM1M2/P)

-4 .0 0 6 *
-3 .6 1 8 *
-3 .0 6 7 *
-5 .9 2 3 *
-6 .6 3 4 *
-5 .5 8 6 *
-5 .1 2 7 *
-3 .8 5 5 *

Variable
M2/P
M1/P
MB/P

"Indicate statistical significance at the 5 percent level.
Critical values t^ (T = 100) = -2 .8 9 , tT (T = 100) = -3 .4 5 .

of stationarity of the levels of the time series
about a non-zero mean. The critical values of
the test statistic [tM
] are tabulated in Fuller (1976)
and discussed in Dickey and Fuller (1979). The
reported test statistics indicate that the null
hypothesis cannot be rejected for any variable.
We then test for stationarity about a deter
ministic time trend, using the Dickey-Fuller
statistic [tj. The results of this test are given in
the second column in the top half of table 1 .
Critical values for this test statistic are tabulated
in Fuller (1976). With the exception of R3M, the
null hypothesis that the time series has a unit
root cannot be rejected.
The bottom half of table 1 reports results for
the augmented Dickey-Fuller test on first dif
ferences of the variables. The null hypothesis of
a unit root is rejected for all of the time series
using differenced data. These results are broad

41Johansen and Juselius (1990) note, however, “ One would,
however, expect the power of this procedure [the trace
test] to be low, since it does not use the information that
the last three eigenvalues have been found not to differ
significantly from zero. Thus one would expect the max-

T e s ts f o r C o in te g r a tio n U sing T h r e e
M e t h o d o l o g i e s : Tests for cointegration for real
M l, real income and either R3M or R10Y using
methodologies proposed by Johansen, StockWatson and Engle-Granger are presented in
table 2. For the Johansen and Engle-Granger
tests, three lagged differences were used. Both
the test statistics and the estimated
cointegrating vector (setting the coefficient on
Ml/P equal to one) are reported. The estimated
cointegrating vector is reported even when the
test does not indicate cointegration.
For the Johansen method, there are two test
statistics for the number of cointegrating vec
tors: the trace and maximum eigenvalue
statistics. In the trace test, the null hypothesis is
that the number of cointegrating vectors is less
than or equal to k, where k is 0, 1 or 2. In each
case the null hypothesis is tested against the
general alternative. The maximum eigenvalue
test is similar, except that the alternative
hypothesis is explicit. The null hypothesis k = 0
is tested against the alternative that k = l, k = l
against the alternative k = 2, etc. The critical
values for these tests are tabulated by Johansen
and Juselius (1990). For the trace test, the
hypotheses k < 1 and k < 2 cannot be rejected
for either of the two interest rates, while the
hypothesis k = 0 can be rejected .41 Consequently,
we conclude that there is one cointegrating
vector.
Turning to the maximum eigenvalue test, the
hypothesis k = 0 is uniformly rejected in favor of
the alternative k = l . Consequently, this test in
dicates that real M l is cointegrated with real in
come and either of the two nominal interest
rates. Moreover, there appears to be a single
cointegrating vector. The maximum eigenvalue
test of k = 1 vs. k = 2 fails to reject the null
hypothesis of k = l. Thus, there are two com
mon trends and one cointegrating vector.
The Johansen test produced results that were
markedly different from those obtained using

imum eigenvalue test to produce more clear cut results,”
(p. 19).

M AR CH/APRIL 1991

Table 2
Tests for Cointegration for M1
Cointegrating Vector

Test Statistics

R3M

R10Y

Johansen test1
Max. Eigenvalue

Trace
k=0

k< 1

k<2

k=0

k= 1

32.3*
28.4

5.2
6.0

1.2
1.8

27.1*
22.4*

4.0
4.2

.680
.845

.369

.398
.635

-.1 5 2
—

.353
.558
.671
.794
.730
.826

-.1 3 1
—
-.2 7 1

.570

Stock-Watson test
q<3,2)
-24 .9 1
-15 .6 1

—
-.3 5 3

Engle-Granger test3
Dependent Variable
(M1/P)
q
R

-2 .2 6
-2 .0 8
-3 .9 2
-3 .1 8
-4 .8 6 *
-3 .3 6

'Indicates statistical significance at the 5 percent level.
’ Critical values:
Trace

—

-.3 5 9
—

—
- .3 0 3
—
-.4 4 9
—

-.4 9 6

Max. Eigenvalue

k=0

k<1

k<2

o
II
JSC

k= 1

31.3

17.8

8.1

21.3

14.6

2Critical value for q(3,2) is - 31.5.
3Critical values for ADF taken from Engle and Yoo (1987)

100 observations

200 observations

-3.93

-3 .7 8

either the Engle-Granger or Stock-Watson
methodologies. Because the results of the EngleGranger can change with the variable chosen as
the dependent variable, the test was performed
with each variable on the right-hand side. Not
surprisingly, the test was sensitive to this
choice. The test indicated cointegration only
when R3M was the dependent variable (although
the test indicated cointegration at slightly higher

significance level if real GNP is the dependent
variable and R3M is used). It is interesting to
note, however, that the estimated cointegrating
vectors obtained from the Johansen and EngleGranger approaches are nearly identical when
both indicate cointegration. This is not the case,
however, when the Engle-Granger or StockWatson tests do not indicate cointegration .42

42The estimated cointegrating vector from the Engie-Granger
approach when the equation is normalized on real output
is (M1/P) = -.2 7 1 R + .671Q. While the estimated coeffi-

cient on the three-month Treasury bill rate is markedly different from that when cointegration is indicated, the
estimated coefficient on output is nearly identical.

FEDERAL RESERVE BANK OF ST. LOUIS

Table 3
Tests for Cointegration for the Broader Monetary Aggregates
Max. Eigenvalue

Trace Statistic

k=0

k= 1

33.7*
38.7*

13.9
17.3

1.6
4.4

19.8
21.4*

12.3
12.9

NM1M2

53.4*
39.7*

20.6*
16.3

2.7
3.7

32.8*
23.4*

17.9*
12.6

k<2

k<1

Aggregate

k=0

k< 1

k<2

k=0

k=1

31.3

17.8

8.1

21.3

14.6

Although evidence concerning cointegration
between real M l, interest rates and output is
sensitive to the method used, the results using
Johansen methodology are similar to those of
Hoffman and Rasche (1989) using monthly data
and the same methodology. Moreover, for the
Johansen results, the hypothesis that the nor
malized coefficient on output is unity is not re
jected using either interest rate, while the
hypothesis of a zero coefficient for the interest
rate is rejected for both interest rates (see table
4).43 Thus, the data appear to support the no
tion that there is a stable long-run relationship
between real M l, real income and interest rates.
C o in te g r a tio n U sing A lte r n a tiv e M o n e ta r y
A g g r e g a t e s : Tests for cointegration using M2
and NM1M2 are presented in table 3. Both the
trace and maximum eigenvalue tests indicate
one cointegrating vector for M2, and both in
dicate cointegrating vectors for NM1M2.44 The
previous discussion of the relationship between
M l, M2 and NM1M2 suggests that certain longrun relationships should exist between the
estimated cointegrating vectors using the
various aggregates. In order to examine this
suggestion, the estimated cointegrating vectors
normalized on the real value of the respective
aggregate are presented in table 4, along with
«T he x 2(1) statistics for the test of the coefficient on output
are 2.23 and .61, respectively, for the three-month and
10-year rates, and the test statistics for the interest rates
are 21.37 and 14.66, for the two rates, respectively.

Max. Eigenvalue

Trace

Critical Values

tests of the hypotheses that the coefficient on
output is unity and the coefficient on the in
terest rate is zero. These test statistics are
asymptotically distributed X2(l).
The apparent nonstationarity of M l velocity
and the stationarity of M2 velocity implies that
M l and NM1M2 must have compensating
nonstationary behavior. This suggests that the
sum of the income elasticities or interest
elasticities for M l and NM1M2 should equal that
of M2. While there are no formal tests of these
cross-equation restrictions, the point estimates
in table 4 indicate that, with the exception of
the income elasticity when R3M is used, these
restrictions do not do too much violence to the
data. For the three-month rate, the sum of the
interest elasticities for M l and NM1M2 is -.1 5 ,
compared with the estimated elasticity for M2
of .02; the sum of the income elasticities is 1.72,
compared with an estimated income elasticity
for M2 of .97. For the 10-year rate, the sum of
the elasticities is -.0 8 , compared with the
estimated elasticity of -.0 3 ; the sum of the in
come elasticities is .92, compared with an
estimated income elasticity of 1.04. Nevertheless,
the fact that the hypothesis that the income
elasticity is unity cannot be rejected for either
M l or M2 is troubling.
44|f a 10 percent significance level is used, the test indi
cates there are two cointegrating vectors for M2 when the
10-year bond rate is used.

MARCH/APRIL 1991

Table 4
Normalized Cointegrating Vectors and Hypothesis Tests
C ointegrating V ector

Aggregate

NM1M2

H ypothesis Test

R3M

R10Y

R3M2

R10Y2

0.680
0.845

-0 .3 6 9
—

-0 .5 7 0

2.23
0.61

21.37*
—

14.66*

0.971
1.044

0.016
—

-0.03 1

0.20
0.21

0.16
—

0.24

1.043
0.078

0.217
—

0.646

0.09
2.72

8.58*
---

4.30*

_
_

'Indicates statistical significance at the 5 percent level.
1Null hypothesis is the income elasticity is one.
2Null hypothesis is the interest elasticity is zero.

Cointegration and the M onetary
Base
As was noted earlier, the monetary base must
be regarded as a supply-side variable, and
cointegration of the monetary base with income
and interest rates arises due to the relationship
between the monetary base and the relevant
money stock measure. Consequently, it is neces
sary to include a proxy for the money
multiplier in an investigation of cointegration
for the monetary base. Because the primary
determinant of the multiplier is the currencydeposit ratio, K, it is included along with the
real monetary base, real income and an interest
rate in tests for cointegration.
The results, presented in table 5, indicate that
there are two cointegrating vectors linking the
real monetary base, real income, the nominal in
terest rate and K when B3M is used, but only
one cointegrating vector when B10Y is used.
Because a cointegrating vector merely represents
45Taking the log of equation 36 and letting the money
multiplier be a function of both K and the interest rate and
h(Z) be solely a function of the interest rate, results in
In mb = In q + In h(R )- In mm (R,K), where h '< 0 and
3 mm/3 R > 0 and 3 mm/3 K < 0.
This equation implies that the long-run elasticity of In mb
with respect to In q is unity, the elasticity of In mb with
respect to R is negative, but, smaller than the estimate for
the long-run demand for money, and that the elasticity
with respect to K is negative.

FEDERAL RESERVE BANK OF ST. LOUIS

a long-run, stable relationship among jointly en
dogenous variables, in general, they cannot be
interpreted as structural equations. Consequent
ly, neither of the estimated cointegrating vec
tors necessarily represents either the long-run
demand for or long-run supply of money. All
that can be said is that there are two linear
combinations for which the variance is bounded.
Nevertheless, it is interesting to note that the se
cond reported cointegrating vector is broadly
consistent with equation 36.45
Nevertheless, because a stable long-run de
mand for money implies that there is a stable
long-run relationship between real money, real
income and either a short- or long-term interest
rate. Consequently, these results are consistent
with the proposition that the long-run demand
for money is stable, even though they may not
be estimates of the long-run money demand
function itself.46
They also suggest that the reason M2 velocity
is stable is because it includes transactions and
^These results are both quantitatively and qualitatively similar to those obtained by Hoffman and Rasche (1989).

Table 5
Tests for Cointegration Using the Monetary Base
Trace test

Max. Eigenvalue

Cointegrating vector

I
CM

T“

k<3

65.5*

30.6

7.4

2.3

34.9*

23.2*

5.1

52.6*

22.5

5.3

1.4

30.1*

17.2

3.9

k<2

k<1

k=0

R3M

R10Y

.465
.883
1.239

.333
- .278
- .540

- .271
- .204
—

—
—
- .269

'Indicates statistical significance at the 5 percent level.

non-transactions components that are close
substitutes for each other in the long run. In
particular, the upward drift in M l velocity ap
pears to be largely due to a relatively steady
shift from M l to the non-transactions deposits
in M2. The magnitude of the trend movements
in these variables is approximately equal so that
M2 velocity is essentially trendless over the
estimation period.47

SUMMARY AND CONCLUSIONS
This paper reviews the concept of cointegra
tion, notes the relationship between tests for it
and common tests for unit roots and considers
its implications for the relationship among real
money balances, real income and nominal in
terest rates. We argue that if M2 and nominal
income are cointegrated, while M l and nominal
income are not, there necessarily exists a sta
tionary long-run relationship between M l and
the non-Ml components of M2. We also argue
that, if M l, real income and the nominal in
terest rate are cointegrated, the same could be
true for real income, the nominal interest rate,
the monetary base and a proxy for the
monetary base/money multiplier.
Tests for cointegration among real M l, real
income and one of two interest rates using
three alternative procedures show that the

results are sensitive to the method used. Never
theless, the technique proposed by Johansen in
dicates that there is a single cointegrating rela
tionship among these variables. While the
cointegrating vector cannot be interpreted as
the long-run demand for money, the estimated
long-run income and interest elasticities are con
sistent with those often hypothesized and
estimated for the long-run demand for money.
We also show that the hypothesized long-run
relationship for the cointegrating vectors for
M l, M2 and the non-Ml components of M2,
namely that the sum of the income and interest
elasticities for Ml and the non-Ml components
of M2 equal the income and interest elasticities
of M2, is supported by the data. Finally, we
show that if the currency-deposit ratio is used
to proxy the monetary base multiplier, the real
monetary base, real income, the interest rate
and the currency-deposit ratio are cointegrated.
The last two results are consistent with the
notion of a stable long-run relationship between
monetary aggregates and prices when both real
income and nominal interest rates are taken in
to account. Moreover, that there appears to be
a stable long-run relationship between real
money, real income and nominal interest rates
establishes the potential for achieving price level
stability by controlling the growth rates of
either M l or the monetary base.

47The stable long-run relationship between real income, the
real monetary base, nominal interest rates and the
currency-deposit ratio, is also consistent with the idea
recently put forth by McCallum (1987) that nominal GNP
can be controlled in the long-run by monetary base
targeting.

MARCH/APRIL 1991

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FEDERAL RESERVE BANK OF ST. LOUIS

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