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Foreign Exchange Operations
and the Federal Reserve
J. Alfred Broaddus, Jr., and Marvin Goodfriend

T

he operations of U.S. government agencies in foreign exchange markets
are probably regarded as arcane by most Americans. These operations
are, however, an important element of U.S. international economic policy. And from time to time they are highly visible to the public: for example,
when the United States and other major industrial countries intervene jointly
in the markets to influence exchange rates, or when they provide assistance to
particular countries such as the substantial aid extended to Mexico in 1995.
The Gold Reserve Act of 1934 gives the Treasury primary responsibility for
United States foreign exchange operations through its Exchange Stabilization
Fund (ESF). Although the Federal Reserve (Fed) had been active in foreign
exchange markets in the 1920s and early 1930s, its involvement ceased after
1934.1 There was relatively little need for official U.S. foreign exchange
operations in the early post-World War II period. Under the Bretton Woods
arrangements of 1944, foreign governments assumed responsibility for fixing
the value of their currencies against the dollar. For its part, the United States
managed its monetary policy in accordance with the Gold Reserve Act so as
to maintain the dollar’s convertibility into gold at $35 an ounce.
U.S. authorities, however, were reluctant to pursue sufficiently tight monetary policy to protect the country’s gold reserves following the resumption of

The authors are respectively president, and senior vice president and director of research. This
article originally appeared in this Bank’s 1995 Annual Report. The article benefited greatly
from stimulating discussions with Robert Hetzel and from presentations at the Macro Lunch
Group of the Wharton School, University of Pennsylvania; the Masters of International Business Lecture Series, College of Business, University of South Carolina; and the Norwegian
School of Management. Comments from Michael Dotsey, Thomas Humphrey, Robert King,
Bennett McCallum, and Alan Stockman are greatly appreciated. The views expressed are the
authors’ and not necessarily those of the Federal Reserve System.
1

See Chandler (1958) and Clarke (1967).

Federal Reserve Bank of Richmond Economic Quarterly Volume 82/1 Winter 1996

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Federal Reserve Bank of Richmond Economic Quarterly

full convertibility among the major currencies in the late 1950s. And the Fed
resumed foreign exchange operations in 1962, after a nearly 30-year hiatus,
to supplement and substitute for monetary tightening in defense of the dollar.
Although the Fed has consistently held that it has independent authority to
undertake foreign exchange operations, in practice the Fed works closely with
the Treasury in conducting them. Indeed, the Federal Open Market Committee’s
(FOMC’s) foreign currency directive requires that these operations be conducted “in close and continuous consultation and cooperation with the United
States Treasury.”2 So it seems fair to say that the Fed recognizes the Treasury’s
preeminence in foreign exchange policy.
The Treasury welcomed the Fed’s renewed participation in large part because the Fed brought with it resources to supplement those of the ESF. In
1962 the Fed established reciprocal currency agreements—commonly called
“swaps”—with nine central banks and the Bank for International Settlements.
Further, in 1963, the Fed agreed to “warehouse” foreign currencies held by the
ESF. The primary objective of these initiatives was to provide U.S. authorities
with a supply of foreign currencies to buy back dollars in order to help protect
U.S. gold reserves.3
FOMC discussions at the time made it clear that some Fed officials recognized how following the Treasury’s lead in foreign exchange operations could
compromise the Fed’s independence in conducting monetary policy.4 This risk
did not present serious operational problems at the time, however, because the
United States was committed to the Bretton Woods arrangements and monetary
policy was committed to defending the dollar.5 Thus, the Fed and the Treasury
were working toward the same general objectives, and the Fed’s independence
was not a pressing issue in practice.
We argue below that subsequent developments have undermined the favorable conditions that enabled the Fed to participate in foreign exchange operations without compromising either its independence or its monetary policy
goals. We make our case by developing several preliminary points. In Section
1 we explain how theoretical advances and practical experience in recent years
teach that the Fed’s longer-term low-inflation objective must be credible if
the Fed is to pursue this objective efficiently via monetary policy. Moreover,
the Fed’s independence is the cornerstone of this credibility. In Section 2 we
explain why Fed credibility based on independence is inherently fragile, and we

2

See the discussion in Humpage (1994), pp. 3–4.
Pauls (1990) details the evolution of U.S. exchange rate policy in the post-World War II
period.
4 See Hetzel (1996).
5 That either a fixed exchange rate or a fixed gold price commitment requires monetary
policy to be dedicated to that objective is emphasized, for example, by McCallum (1996b),
Chapters 4 and 7.
3

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

3

emphasize the crucial importance of the Fed’s off-budget status in supporting
its independence.
We take up the role of the Fed in foreign exchange operations in Section 3,
where we distinguish two broad types of official foreign exchange transactions:
unsterilized and sterilized. As explained there, unsterilized transactions are essentially monetary policy actions and therefore are carried out independently
by the Federal Reserve. Since sterilized transactions are not monetary policy
actions, the Fed can acknowledge the Treasury’s leadership regarding them
without directly compromising its independence.
Evidence accumulated over the past two decades suggests, however, that
sterilized intervention in exchange markets has at best only temporary effects on
exchange rates and must be supported by monetary policy actions to have lasting effects. Consequently, the Fed’s participation with the Treasury in sterilized
operations creates confusion as to whether monetary policy is dedicated to the
support of exchange rate or domestic objectives. Such confusion weakens the
public’s perception of the Fed’s independence and undermines the credibility
of the Fed’s low-inflation goal.
In Section 4 we lay out in more detail the inherent contradictions for monetary policy that arise when the Fed follows the Treasury’s lead on exchange
rate policy. And we argue in Section 5 that the Fed’s financing of even sterilized
foreign exchange operations constitutes a misuse of the Fed’s off-budget
status that risks undermining the public’s acceptance of the independence of
the Fed. We believe that the best way to resolve the conflict between foreign
exchange operations and monetary policy is for the Fed to disengage from
foreign exchange operations completely. The concluding section summarizes
our argument.

1. CREDIBILITY AND THE EFFECTIVENESS
OF MONETARY POLICY
Numerous disinflations since the early 1980s have taught central bankers around
the world that credibility—having a reputation for pursuing price level stability consistently and persistently—is the key to an effective anti-inflationary
monetary policy.6 We would even go so far as to say that the primary policy
problem facing the Fed during this period has been the acquisition and maintenance of credibility for its commitment to low inflation—so much so that
credibility concerns remain a motivating or restraining influence on monetary
policy actions today, even though the Federal Reserve’s low-inflation objective
has nearly been achieved.
6

See the accounts in Leiderman and Svensson (1995).

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Federal Reserve Bank of Richmond Economic Quarterly

As it happens, the growing practical appreciation of the importance of
credibility is supported by an improved scientific understanding associated with
game theory and the rational expectations approach to monetary theory. In many
ways, theory simply articulates what central bankers have learned from practical experience. Briefly, the theory recognizes that monetary policy involves
continuous interaction between a central bank and the public that introduces
a link, in the public’s mind, between current policy and future policy actions.
In the absence of credibility, expansionary current monetary policy tends to
generate expectations of expansionary policy—and possibly excessively expansionary policy—in the future. Such expectations trigger aggressive wage and
price increases that, in turn, neutralize the beneficial effects of the expansionary
current policy. The result is higher inflation with little, if any, sustained increase
in employment and output.
Theory supports the idea that the potential for future inflation, which can be
thought of as a punishment imposed collectively by wage- and price-setters on a
central bank, can discipline a central bank. In a reputational equilibrium, wageand price-setters keep their part of an implicit bargain by not inflating as long
as the central bank demonstrates its commitment to low inflation by eschewing
excessively easy policy. A central bank may be said to have credibility when an
implicit mutual understanding between the public and the central bank sustains
a low-inflation equilibrium.7
The key point is that a low-inflation equilibrium sustained by central bank
credibility is fragile. In such an equilibrium the public is very sensitive to any
central bank departure from the behavior it has come to anticipate; this expected
continued behavior, indeed, is the essence of the central bank’s credibility. The
public is particularly nervous about such departures when the central bank
has acquired credibility only recently. But there is evidence that low-inflation
equilibria sustained by credibility continue to be fragile even when a central
bank’s actions have repeatedly demonstrated its commitment to low inflation
over a period of years.
The fragility of the Fed’s credibility is evident in the behavior of long-term
bond rates.8 The real yield on the 30-year U.S. government bond probably
moves within a range of 2 percentage points or so around 3 percent per year.9
The remainder of the nominal long-term yield reflects inflation expectations. In
the early 1960s, for example, when inflation averaged between 1 and 2 percent

7 The introductory chapter in Persson and Tabellini (1994) contains a good survey of
research on the role of credibility in monetary and fiscal policy. Barro and Gordon (1983),
Cukierman (1992), and Sargent (1986) contain seminal analyses of credibility.
8 Goodfriend (1993) and King (1995), for example, interpret movements in long-term bond
rates as indicators of credibility for low inflation.
9 Ireland’s (1996) study of the ten-year bond rate provides some support for this view.

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

5

per year, the 30-year bond yielded roughly 4 percent.10 In 1981, when the
public’s confidence in the Fed’s commitment to controlling inflation was at its
low point, the long-term bond yield reached nearly 15 percent. The rate stood
at around 6 percent in late 1995, which indicated that the public expected about
3 percent inflation on average over the long term.
Doubts about a central bank’s credibility often surface as “inflation scares”
in the long-term bond market. Following a period of rising inflation in the
late 1970s, for example, the 30-year rate jumped 2 percentage points in the
first quarter of 1980, which signaled the most serious and sudden collapse
of confidence in the Fed on record. The fragility of the Fed’s credibility was
apparent again in 1984 when the bond rate, after falling to about 10 percent in
late 1982, registered another inflation scare by rising to around 13.5 percent,
even though the Fed had by then brought actual inflation down from over 10
percent to around 4 percent.
The swings in the bond rate over the past two years have been less dramatic
than in the early 1980s, but nonetheless substantial. Rising from a low of about
5.8 percent in October 1993, the bond rate peaked at around 8.2 percent in
November 1994. We interpret that wide swing as evidence that the Fed’s antiinflationary credibility remains exceedingly brittle despite years of sustained
progress in bringing the actual inflation rate down.
The fragile nature of the Fed’s credibility imposes a number of costs on the
economy. First, there is the direct cost of higher long-term interest rates with
their negative effects on economic performance. Second, with inflation expectations higher than they should be, the Fed is left with the difficult choice of
either accommodating these expectations and accepting higher rates of inflation
or failing to accommodate them and risking negative short-term effects on real
economic activity. Moreover, even hesitating to react can be costly because, by
suggesting indifference, the Fed may encourage workers and firms to ask for
wage and price increases to protect themselves from higher expected costs.
Finally—a related point—weak credibility makes it difficult for the Fed to
respond when employment considerations call for an easing of policy, as they
did in the 1990–91 recession and again in mid-1995. In such circumstances,
the Fed must balance the desirable short-term effects of lower short-term rates
against the risk of higher long-term rates.

2. FEDERAL RESERVE INDEPENDENCE
A number of prominent institutional mechanisms have been used to assist central banks in maintaining credibility for low-inflation objectives. Historically,
a national commitment to a gold or silver standard—that is, a commitment to
maintain a fixed currency price of gold or silver—was the most important. A
10

See Salomon Brothers and Hutzler (1968).

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Federal Reserve Bank of Richmond Economic Quarterly

second mechanism, more prominent in recent years, is for a country to commit
to fix its exchange rate against the currency of a trading partner that credibly
maintains the purchasing power of its currency. An important motivation for
the establishment of the European Monetary System (EMS), for example, was
the desire of some countries to import credibility for low inflation by pegging
their currencies to the deutsche mark (D-mark). Difficulties with fixing exchange rates, including the near collapse of the EMS in the early 1990s, have
led some countries to experiment recently with a third commitment device:
inflation targets.11 Finally, countries have relied on central bank independence
to supplement one of the other mechanisms or to substitute for them.
Broadly speaking, central bank independence implies a separation of bank
decisions from the regular decisions of the political system.12 At a minimum,
it means that a central bank is free to conduct monetary policy without interference from the Treasury. The degree of actual operational freedom enjoyed
by an independent central bank, however, has varied widely depending on the
circumstances. For instance, in the nineteenth century, when wide support for
central bank independence first developed, independent central banks were narrowly constrained by national commitments to various commodity standards.
Similarly, the Federal Reserve was established in 1913 as an independent central
bank mandated by the Federal Reserve Act to stabilize financial markets while
keeping the United States on the gold standard.
A central bank may be said to lack “goal independence” when its objective
is given by legislative mandate; however, one can still speak of a central bank
as having “instrument independence”—the freedom to use a short-term interest
rate or other monetary policy instrument to achieve its mandated goals.13 The
Fed has had full instrument independence, except for the World War II years and
the period from the end of the war to the 1951 Fed-Treasury Accord. During
that time the Fed was obliged to maintain low interest rates on government
securities to facilitate the Treasury’s finances. The Accord reasserted the principle that monetary policy should be used for macroeconomic stabilization,
the fiscal concerns of the Treasury notwithstanding. In terms of the above
definitions, the Accord fully restored the Fed’s instrument independence.14
The Accord did not give the Fed goal independence because monetary
policy was still committed under the Bretton Woods arrangements to support
the fixed dollar price of gold. When the Bretton Woods System collapsed
11 Leiderman and Svensson (1995) and McCallum (1996a) contain accounts of the experience with inflation targets in a number of countries. For an empirical study of exchange rate
credibility in the EMS, see Rose and Svensson (1994).
12 This definition is from Hetzel (1990), p. 165.
13 Fischer (1994), p. 292, distinguishes between goal and instrument independence.
14 The Fed actually abandoned its short-term interest rate peg in 1947; it gave up its longterm rate peg in 1951. Stein (1969) contains a good discussion of developments leading up to the
1951 Fed-Treasury Accord.

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

7

in 1973, however, the national consensus on the proper goal for monetary
policy collapsed with it, and the Fed has been operating without an explicit
congressional mandate since then.15 Thus, during this period the Fed has had
goal independence by default, as it were, and this independence is now arguably
the sole institutional mechanism supporting low inflation in the United States.
Independence and Credibility
A goal-independent Fed unrestrained by a legislative mandate is a particularly
deficient mechanism for maintaining low inflation. The reason is that in this
situation a low-inflation equilibrium must be supported entirely by credibility
that the Fed creates for itself—credibility that is inherently fragile as discussed
above. The unbridled discretion conferred on the Fed in this case only makes
the acquisition and maintenance of credibility for low inflation more difficult.
The Fed’s goal independence gives other government entities strong incentives
to attempt to influence its policies via such channels as congressional oversight
hearings, appointments of Federal Reserve governors, proposed changes in the
Fed’s regulatory role, and so forth. Moreover, such attempts at influence can
be of a conflicting nature, adding to the confusion. Knowing this, the public is
rightly suspicious of any potential conflict between the Fed, the Treasury, and
Congress. In this environment, any contact that Fed officials have with the rest
of the government risks creating credibility problems for monetary policy.
At the same time—and paradoxically—central bank goal independence
actually creates incentives for Fed officials to interact with the rest of the
government.16 The lack of clarity in the Fed’s mandate necessitates deeper
involvement in the legislative process by Fed officials who must see to it that
proposed legislation does not compromise its monetary policy mission. Finally,
the Fed’s independence confers upon it a nonpartisan aura which leads others
in government to seek its advice, certification, or arbitration in controversial
policy disputes.
Financial Independence
In principle, a healthy democracy requires full public discussion of expenditures
of public monies. The congressional appropriations process enables Congress
to evaluate competing budgetary programs and to establish priorities for the
allocation of public resources.
Congress has long recognized, however, that the pressure of budgetary
politics could tempt future Congresses to press the Fed at least implicitly to
15 It is true that the 1978 Humphrey-Hawkins law mandates the Fed to set monetary
aggregate targets as guides to short-run policy. But the Humphrey-Hawkins law instructs the
Fed to take account of so many potentially conflicting macroeconomic concerns in setting the
targets that it has exercised little restraint on the Fed’s freedom of action.
16 See Bradsher (1995).

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Federal Reserve Bank of Richmond Economic Quarterly

help finance federal expenditures through inflationary monetary policy. Consequently, the Fed has been made financially independent—its operations are
funded from the interest payments on its portfolio of securities—and the Fed
has wide discretion over the assets it holds. In short, the Fed is exempt from
the congressional appropriations process in order to keep the political system
from exploiting inflationary money creation. It is critically important that the
Fed not misuse this exceptional “off-budget” status so as not to undermine
public understanding of and support for its financial independence. This, in
turn, requires the Fed to understand clearly what activities are and are not
essential to its central banking mission.

3. THE ROLE OF THE FED IN FOREIGN
EXCHANGE OPERATIONS
The points about credibility and independence developed above will serve as
the basis for our assessment of the Fed’s role in foreign exchange operations
in what follows. Here we review the basic mechanics of foreign exchange
operations. We begin by making the important distinction between unsterilized
and sterilized transactions. Then we briefly discuss the means by which the
Fed finances foreign exchange operations for its own account and warehouses
foreign exchange for the ESF.17 Our analysis identifies in a preliminary way
the fundamental sources of conflict for monetary policy arising from the Fed’s
participation in foreign exchange operations.
Unsterilized and Sterilized Operations
The distinction between unsterilized and sterilized operations is straightforward:
unsterilized transactions involve changes in the monetary base, and sterilized
transactions do not. For example, the Fed could acquire foreign exchange in an
unsterilized purchase using newly created base money: that is, bank reserves or
currency. Such a transaction would be an expansionary monetary policy action
because it would increase the monetary base.
A foreign exchange purchase would be sterilized, in contrast, if the Fed offset its effect on the base by selling an equivalent amount of dollar-denominated
securities. Because the Fed controls the monetary base, it is in a position to
determine whether a foreign exchange operation is sterilized or not. In practice,
the Fed routinely sterilizes foreign exchange operations that it undertakes for its
own account and for the ESF. In sterilized operations the current federal funds
rate target (the key policy instrument indicating the current stance of monetary
17 A detailed description of the mechanics of foreign exchange operations using T-accounts
is found in Humpage (1994).

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

9

policy) is maintained. This point is important because it implies that—at least
as a mechanical matter—the Fed can follow the Treasury’s lead in sterilized
foreign exchange operations without relinquishing control of monetary policy.
Nevertheless, sterilized foreign exchange operations, or “intervention,”
pose significant problems for the Fed. For the most part, economists agree
that sterilized intervention by central banks in foreign exchange markets has
no lasting effect on exchange rates.18 In the absence of supporting monetary
policy actions, sterilized interventions can influence exchange rates temporarily,
especially when the interventions are unexpected. But obviously the ability of
authorities to surprise markets is very limited. Sterilized intervention can be
most effective when it signals a government’s resolve to follow up with monetary or fiscal policy actions that will powerfully influence the exchange rate
in the future.19 Consequently, Fed participation in sterilized foreign exchange
operations under the Treasury’s leadership creates confusion as to whether
monetary policy will support short-term exchange rate objectives or longerterm anti-inflationary objectives. Only occasionally will the monetary policy
actions required to pursue these two objectives coincide.
This confusion is compounded by a lack of consistency in U.S. exchange
rate policy in the post-1973 floating exchange rate regime. Officially, the objective of foreign exchange operations is to counter “disorderly market conditions,”
but that phrase has never been defined operationally. It was interpreted most
narrowly in the first Reagan administration, when U.S. operations were minimal. It was interpreted broadly between 1977 and 1979 when the dollar was
viewed as unacceptably low and again in 1985 when the dollar was unacceptably high. Intervention was undertaken in these periods to help push the dollar
into an acceptable range. Extensive interventions were carried out in the years
following the Louvre Accord of 1987 to help stabilize the exchange rate.20
Moreover, much U.S. intervention in recent years has been coordinated with
foreign governments. The Group of Seven finance ministers and central bank
governors meet regularly to discuss exchange rate objectives. The enormous
publicity surrounding these discussions, designed to underscore international
harmony on exchange rate policy, heightens uncertainty regarding whether the
Fed will support sterilized operations with monetary policy actions. The widespread coverage of internationally “coordinated” foreign exchange operations
is almost certainly harmful to the public’s perception of the Fed’s independence
and thereby weakens the credibility of the Fed’s low-inflation strategy.

18 A representative survey of the academic literature on this point would include Bordo and
Schwartz (1991), Edison (1993), and Obstfeld (1990), and references contained therein.
19 See Mussa (1981).
20 See Destler and Henning (1989), Funabashi (1989), and Pauls (1990) for discussions of
U.S. exchange rate policy.

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Federal Reserve Bank of Richmond Economic Quarterly

Financing Mechanisms
Federal Reserve acquisitions of foreign exchange are generally financed in one
of three ways. If the FOMC approves, the Fed can acquire foreign exchange for
its own account by creating additional bank reserves or currency—that is, via an
unsterilized transaction. Sterilized acquisitions, on the other hand, are financed
by selling Treasury securities from the Fed’s portfolio. Finally, the Fed has
the option of borrowing currencies from foreign central banks using reciprocal
currency agreements—the so-called “swap” network. Swap facilities are, in
effect, short-term lines of credit giving central banks access to one another’s
currencies. The facilities provide for the swap (simultaneous spot purchase and
forward sale) of each other’s currency by the Fed and the foreign central bank.
Swaps typically are not accompanied by any change in monetary policy—in
other words they are sterilized transactions.21 The Fed holds foreign exchange
in the form of short-term securities or interest-bearing deposits at foreign central
banks, so that sterilized transactions amount to substituting foreign-currencydenominated interest-earning assets for dollar-denominated securities in the
Fed’s portfolio.
The Fed bears the exchange rate revaluation risk—as well as the credit
risk—for any foreign-currency-denominated assets it holds for its own account.
Since the Fed marks its foreign currency assets to market monthly, a depreciation of the foreign exchange value of the dollar, for instance, raises the
dollar value of the Fed’s foreign holdings. Any such gains or losses eventually
show up as larger or smaller Fed payments to the Treasury after expenses.22
Whenever the Fed disperses foreign exchange acquired through a swap, it
bears the exchange risk involved in covering its forward commitment to reverse
the swap.
The Exchange Stabilization Fund
As mentioned above, the Treasury conducts foreign exchange operations
through its Exchange Stabilization Fund. When it was established by the Gold
Reserve Act, the ESF was capitalized with $2 billion derived from the proceeds
of the 1934 revaluation of the U.S. gold stock from $20.67 to $35 per ounce.
Later, $1.8 billion was transferred from the ESF as partial payment on the
U.S. subscription to the International Monetary Fund (IMF), which left $200
million as the remaining capital of the ESF. ESF capital has grown since then
21 The Fed drew on its swap lines in the 1960s to protect the Treasury’s gold stock by using
the borrowed currencies to buy back dollar reserves from foreign central banks. These transactions
effectively allowed the United States to assume a portion of other countries’ devaluation risk.
More recently, the United States has had sufficient foreign currency reserves and has not drawn
on its swap lines.
22 See the discussions in Goodfriend (1994) and Humpage (1994).

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

11

as a result of retained interest earnings, revaluations of gold, and profits on
foreign exchange acquisitions. 23
Since use of its funds is not subject to the appropriations process, the
ESF provides the Treasury with a degree of flexibility and discretion in its
foreign exchange operations. The ESF serves two broad purposes. First, it is
used to intervene in foreign exchange markets to influence dollar exchange
rates with major currencies such as the D-mark and the Japanese Yen. Second,
the ESF makes loans to foreign governments—frequently to heavily indebted
governments and often in association with IMF or other official assistance
programs. Typically such loans are made to deal with a serious balance-ofpayments problem or to assist a country managing its external debt. Often the
currencies of recipient countries are not fully convertible or are of secondary
importance.24 The recent loans to Mexico are a prominent example of this type
of assistance.
The ESF’s capacity for purchasing foreign currencies is limited, however,
as it has not received an appropriation from Congress since 1934. Apart from
the retained earnings on its investments mentioned above, the ESF has been
able to augment the resources at its disposal in three significant ways. First,
Congress has authorized advancing to the ESF foreign currencies borrowed
from the IMF. Second, the ESF receives the Special Drawing Rights (SDRs)
allocated to the United States by the IMF.25 Third, the Fed has provided the
ESF with additional resources, either by helping to finance operations on its
own account or by warehousing foreign exchange for the ESF. It was because
the ESF’s resources were limited that the Treasury encouraged the Fed in the
early 1960s to participate for its own account in foreign currency operations
and to warehouse foreign currencies. In 1990, the dollar value of U.S. net
foreign currency balances (the sum of acquisitions on the Fed’s and the ESF’s
accounts) exceeded $40 billion.26 The FOMC authorized warehousing of ESF
foreign currencies up to a limit of $15 billion in 1990.
Warehousing allows the ESF to finance purchases of foreign exchange
in much the same way that securities dealers use repurchase agreements with
banks to finance their portfolios. That is, warehousing allows the ESF to enlarge
its portfolio of foreign-currency-denominated assets with funds borrowed from
the Fed. Suppose, for example, that the ESF wishes to sell dollars for foreign
exchange to depreciate the dollar but has inadequate resources to do so. The
Fed can execute the transaction—warehouse the foreign exchange—by selling
a Treasury security from its portfolio in the open market and using the proceeds
23

U.S. Congress (1976), pp. 3–5.
See U.S. Department of the Treasury (1991). U.S. Congress (1976) details ESF operations
from 1968 to 1975. Todd (1992) presents a history of the ESF.
25 SDRs are monetized by transferring them to the Fed.
26 See Pauls (1990), pp. 894 and 904, and U.S. Congress (1976), pp. 3–5.
24

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Federal Reserve Bank of Richmond Economic Quarterly

to acquire the foreign-currency-denominated securities on behalf of the ESF.
Because the Fed executes the purchase of foreign exchange on behalf of the
ESF, the latter remains exposed to the revaluation gains or losses on the foreign
exchange warehoused. Interest earnings on the foreign currencies warehoused
accrue to the Fed. Note that the warehousing operation amounts to a sterilized
acquisition of foreign exchange.
Whether or not the Fed finances sterilized foreign exchange purchases
for its own account, or warehouses foreign currencies for the ESF, a sale of
Treasury securities to the public is the ultimate source of the funds. True, the
securities involved are not newly issued; they are sold from the Fed’s portfolio. The results, however, are equivalent in many ways to those of a new issue
since the Fed simply returns to the Treasury all of the interest it receives on the
Treasury securities that it holds, minus a small fraction that covers the Fed’s
operating expenses. The main difference between Fed financing and financing
by the Treasury itself is that the former is arranged between Treasury and Fed
officials without an explicit appropriation from Congress. A second difference
is that Fed financing does not show up as a measured increase in the federal
deficit, since it does not involve newly issued debt.
Although the Fed is the junior partner with the Treasury on foreign exchange policy, it is certainly an equal partner in terms of the resources provided.
It is able to make these resources readily available without a congressional appropriation because its financial independence puts its open market operations
in Treasury securities off-budget. The exchange operations arranged by the
Treasury, however, not infrequently involve broader foreign relationships in
ways that may be politically charged. Hence, the Fed’s involvement, especially
because it is outside the formal budget process, puts public support for its
financial independence at risk, and with it, the credibility of its low-inflation
policy.

4. THE CONFLICT BETWEEN EXCHANGE RATE
POLICY AND MONETARY POLICY
The national commitment to the Bretton Woods arrangements minimized the
risk of policy conflict between the Fed and the Treasury when the Fed resumed
its participation in foreign exchange operations in the early 1960s. But the
nation’s unwillingness to support that commitment with sufficiently restrictive
monetary policy led to the collapse of the fixed exchange rate system in 1973.
Several years of sharply rising inflation followed. Despite this, Congress was
unable to reach consensus on a new monetary policy mandate. Consequently,
in 1979 the Fed asserted its own commitment to restore low inflation.
We believe that these developments have undermined the Fed’s ability to
participate in exchange rate policy without compromising its independence and

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

13

its monetary policy goals. In particular, with the potential for Fed-Treasury
policy conflicts now significantly enlarged, it is no longer possible for the Fed
simply to follow the Treasury’s lead on exchange rate policy without endangering its monetary policy credibility. This is true even in the case of sterilized
interventions. Under the current arrangement, the Fed participates in sterilized
operations without committing to support the operations with future monetary
policy actions. This maintains the Fed’s independence by keeping its options
open. But such discretion increases the likelihood that particular operations
may fail because the Fed is not willing to support them with monetary policy.
Failed foreign exchange operations are costly because they give the impression that the authorities are either unable or unwilling to achieve a prominent
objective that they appear to be pursuing. For example, the failure of the June
24, 1994, intervention was reported in a front-page New York Times story carrying the headline: “16 Central Banks are Thwarted in Huge Effort to Prop
Up Dollar.”27 Nor was attention to the event confined to major money centers.
On the following day the Richmond Times-Dispatch reported the story with the
front-page headline: “Effort to Bolster Dollar a Failure.” Widely publicized policy failures undermine Fed credibility and thereby jeopardize the effectiveness
of overall monetary policy.
We believe that, to best protect the credibility of its low-inflation goal
and the independence of monetary policy more generally, the Fed should be
separated completely from the Treasury’s foreign exchange operations. In principle, the Fed could disengage unilaterally; however, there would be two major
practical obstacles to such an action. The most serious obstacle is that the
appointment process would make it difficult for the Fed to bind itself not to
participate, since appointments to the Federal Reserve Board could be made on
condition of cooperation with the Treasury. Congress might be able to block
such conditions in the confirmation process in particular cases if it were so
disposed, but legislation probably would be required to remove the Fed from
exchange market intervention definitively.
The second main obstacle to unilateral disengagement is that it would
deny the Treasury the benefit of the Fed’s advice on foreign exchange intervention and the certification that goes with it. Here, though, the Fed cannot
be indifferent to the use of its name in headlines that either box it in or harm
its credibility. Moreover, the act of certification itself creates a perception of
partisanship that erodes the value of that certification, even as it undermines
the public’s perception of the Fed’s independence.
In these circumstances, it is natural to look for a middle-of-the-road solution to the problems presented by the Fed’s involvement in exchange market
operations. One might, for example, try to specify particular circumstances in

27

See Friedman (1994).

14

Federal Reserve Bank of Richmond Economic Quarterly

which the Fed could participate. For instance, if the Fed routinely announced an
inflation target, it could agree to help the Treasury intervene if the inflation rate
were within a specified range of the target. Defining such conditions clearly,
however, would be difficult, and this approach would leave the door open to
many of the same problems the Fed faces currently.

5. THE CONFLICT BETWEEN FOREIGN
EXCHANGE OPERATIONS AND
THE FED’S FINANCIAL INDEPENDENCE
From the start, a major reason for the resumption of Federal Reserve foreign exchange operations in the 1960s was to make Fed resources available
to the ESF. The Fed’s financial independence gave it the discretion to allocate
resources to foreign exchange operations without an explicit congressional appropriation. Apparently there was then little concern about misuse of the Fed’s
off-budget status because Fed financing of foreign exchange operations at the
time seemed conformable with the nation’s commitment to the Bretton Woods
system. Such financing has become more problematic with the breakdown of
the national consensus on monetary and exchange rate policy in the aftermath
of the collapse of Bretton Woods.
Economists understand more clearly today than they did in the 1960s the
distinction between Federal Reserve monetary policy and credit policy.28 As
pointed out in Section 3, sterilized foreign exchange operations are not monetary policy since they leave the monetary base and the federal funds interest
rate target unchanged. Such operations do, however, constitute credit policy
since they amount to a substitution of loans to foreign authorities for dollardenominated securities in the Fed’s portfolio. In effect, sterilized operations are
extensions of Fed credit financed by selling Treasury debt from the Fed’s portfolio. Such extensions of credit are clearly fiscal policy, not monetary policy.
The extension of credit by U.S. authorities involves both market and credit
risk. Although the default or credit risk of the securities in which major foreign currency balances are held is negligible, the revaluation or market risk
is considerable. Credit risk, however, can also be substantial when a loan is
made to assist, say, a country managing its external debt or one with a serious
balance-of-payments problem. Provisions can be made to take collateral if the
borrowing country proves unable to make scheduled payments. But such provisions are not always feasible or entirely effective. When a borrowing country’s
financial problems prove persistent, the ESF and the Fed can be “taken out” by
longer-term funding arranged through international organizations such as the
28

(1994).

This distinction is developed in Goodfriend and King (1990) and used in Goodfriend

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

15

IMF.29 But to the extent that collateralization is incomplete or “take outs” are
not arranged in advance or are uncertain, taxpayers are at risk. Thus, in their
foreign exchange operations the Fed and the ESF assume risk—both market
risk and credit risk—on behalf of the U.S. taxpayer.
The national decision to put funds at risk in foreign exchange operations
is clearly an important fiscal policy matter. The presumption is that—as with
any fiscal action—Congress should authorize the expenditure and explicitly
appropriate the funds. Fed financing of foreign exchange operations through
its own account and by warehousing funds for the ESF sidesteps congressional
authorization and obscures the funding.
The Fed’s financing of foreign exchange operations without explicit direction from Congress exposes it to potentially harsh criticism if an initiative
goes badly. Unfavorable outcomes would obviously undermine public support
for the Fed’s financial independence. But there is a more subtle risk, even
if foreign initiatives funded by the Fed go well. Some will ask whether, if
Fed financing of credit extensions to foreigners is beneficial, it might also be
desirable for the Fed to support worthy domestic objectives. Any attempt to
exploit the Fed’s financial independence in this manner would almost guarantee
that its independence would be withdrawn over time.
Fed off-budget funding attracted substantial attention in the Mexican case
in 1995, as indicated by a remarkable headline in The New York Times: “Clinton
Offers $20 Billion to Mexico for Peso Rescue; Action Sidesteps Congress.”30
Should the Fed take comfort from the relative absence to date of significant
negative repercussions from its involvement in this initiative? We think not.
The publicity for the Mexican rescue put the Fed’s off-budget funding powers
on the radar screen, along with the potential risks described above. The Fed
appeared to receive the implicit support of the congressional leadership in this
instance, but Congress itself probably would not have voted to authorize the
funds, and the public at large did not seem to favor such generous support for
Mexico. Indeed, many Americans, including some prominent ones, viewed the
transaction as a bailout of big investors. If, over time, developments in Mexico
turn unfavorable, the result could be an erosion of public and congressional
support for the Fed’s financial independence.

29 To the extent that the funds are provided by the United States in the first place, the
possibility of such takeouts amounts to only a partial reduction of U.S. taxpayer risk. On some
occasions when U.S. authorities have drawn and dispersed foreign currencies through the swap
network, the U.S. Treasury has repaid the swap loans with foreign exchange borrowed on a longterm basis using so-called “Roosa,” or “Carter,” bonds. Such actions, however, only shift the
market risk from short to long term. See U.S. Congress (1976), pp. 4, 5, and 40.
30 See Sanger (1995). Folkerts-Landau and Ito et al. (1995) contains a thorough account of
the Mexican peso crisis.

16

Federal Reserve Bank of Richmond Economic Quarterly

In brief, Congress deliberately placed the Fed outside the appropriations
process in order to safeguard its independence. The Fed should not misuse its
off-budget status to finance initiatives that are unrelated to monetary policy
because there is very little to be gained and much to lose.

6. CONCLUSION
We have assessed the consequences of the Fed’s participation in foreign exchange operations. Our analysis was based on the idea that central bank credibility for low inflation is the cornerstone of an effective monetary policy and
that public support for Fed independence is the foundation of that credibility.
Distinguishing between sterilized and unsterilized foreign exchange operations, we recognized that as a mechanical matter the Fed can follow the
Treasury’s lead on sterilized operations without compromising its independence
on monetary policy. There is little evidence, however, that sterilized intervention alone can have a sustained effect on the exchange rate. Thus, the Fed’s
participation in foreign exchange policy with the Treasury creates doubt about
whether monetary policy will support domestic or external objectives, and this
doubt undermines the credibility of the Fed’s longer-term objective of reducing
and ultimately eliminating inflation.
Although the Fed is the junior partner with the Treasury on foreign exchange operations, it has been an equal partner when it comes to providing the
resources. The Fed can make these resources available without a congressional
appropriation because its financial independence puts its open market operations off-budget. Foreign exchange operations initiated by the Treasury involve
foreign relationships in ways that can be politically charged, especially when
they involve direct loans to foreign governments. We think that Fed financing of
such operations risks undermining public respect for its financial independence
and with it the credibility of its longer-term price level stability objective.
We argued that central bank independence alone is an inherently fragile basis for the credibility of monetary policy. In view of that fragility, we
recommended that the Fed be separated completely from foreign exchange
operations. We did not argue that the nation should forsake official foreign exchange operations—only that the Fed, as an independent central bank, should
not participate. The Treasury would be free to carry out sterilized operations.
Having made this point, we acknowledged that it would be difficult for the
Fed to disengage from foreign exchange operations unilaterally. Consequently,
some sort of congressional legislation would probably be required to remove
the Fed from foreign exchange operations permanently.
In our view, the problems created by the Fed’s involvement in foreign
exchange operations underscore the need for Congress to provide the Fed with
a mandate for price level stability, recognizing a concern for the stabilization

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

17

of employment and output. Such a mandate would constitute a long overdue replacement for the commitments made at Bretton Woods.31 Moreover,
firm congressional support is needed to strengthen the credibility of the Fed’s
anti-inflation strategy. By providing an overarching national goal for monetary
policy once again, a price stability mandate would greatly reduce the risk of
conflicts and credibility problems when the Fed works closely with the Treasury
and other parts of the government.

REFERENCES
Barro, Robert, and David Gordon. “Rules, Discretion, and Reputation in a
Model of Monetary Policy,” Journal of Monetary Economics, vol. 12 (July
1983), pp. 101–22.
Blinder, Alan. “Central Banking in Theory and Practice,” The Marshall
Lecture, Cambridge University. Federal Reserve Board, processed May
1995.
Bordo, Michael D., and Anna J. Schwartz. “What Has Foreign Exchange
Market Intervention since the Plaza Agreement Accomplished?” Open
Economies Review, vol. 2 (No. 1, 1991), pp. 39–64.
Bradsher, Keith. “Lately, A Much More Visible Fed Chief,” The New York
Times, January 17, 1995.
Chandler, Lester V. Benjamin Strong, Central Banker. Washington, D.C.: The
Brookings Institution, 1958.
Clarke, Stephen V. O. Central Bank Cooperation: 1924–31. New York: Federal
Reserve Bank of New York, 1967.
Cukierman, Alex. Central Bank Strategy, Credibility, and Independence.
Cambridge, Mass.: MIT Press, 1992.
Destler, I. M., and C. Randall Henning. Dollar Politics: Exchange Rate Policymaking in the United States. Washington, D.C.: Institute for International
Economics, 1989.
Edison, Hali J. “The Effectiveness of Central-Bank Intervention: A Survey
of the Literature after 1982.” Special Papers in International Economics,
No. 18. Princeton: Princeton University, Department of Economics,
International Finance Section, July 1993.
31 Our conclusion that a central bank should have its goal legislatively mandated is also the
recommendation, for example, of Blinder (1995), Lecture II, p. 16, Friedman (1962), pp. 224–43,
and Fischer (1994), p. 316; although the suggested mandates differ from ours and from each
other’s in certain respects.

18

Federal Reserve Bank of Richmond Economic Quarterly

“Effort to Bolster Dollar a Failure,” Richmond-Times Dispatch, June 25,
1994.
Fischer, Stanley. “Modern Central Banking,” in Forrest Capie, Stanley
Fischer, Charles Goodhart, and Norbert Schnadt, eds., The Future of
Central Banking: The Tercentenary Symposium of the Bank of England.
Cambridge: Cambridge University Press, 1994.
Folkerts-Landau, David, and Takatoshi Ito, et al., “Evolution of the Mexican
Peso Crisis,” in International Capital Markets: Developments, Prospects,
and Policy Issues. Washington, D.C.: International Monetary Fund, 1995,
pp. 53–69.
Friedman, Milton. “Should There Be an Independent Monetary Authority?”
in In Search of a Monetary Constitution, Leland Yeager, ed., Cambridge,
Mass.: Harvard University Press, 1962.
Friedman, Thomas. “16 Central Banks Are Thwarted in Huge Effort to Prop
Up Dollar,” The New York Times, June 25, 1994.
Funabashi, Yoichi. Managing the Dollar: From the Plaza to the Louvre, 2d
ed. Institute for International Economics. Washington, D.C., 1989.
Goodfriend, Marvin. “Why We Need an ‘Accord’ for Federal Reserve Credit
Policy,” Journal of Money, Credit, and Banking, vol. 26 (August 1994),
pp. 572–80.
. “Interest Rate Policy and the Inflation Scare Problem,” Federal
Reserve Bank of Richmond Economic Quarterly, vol. 79 (Winter 1993),
pp. 1–24.
, and Robert King. “Financial Deregulation, Monetary Policy, and
Central Banking,” in W. Haraf and R. Kushmeider, eds., Restructuring
Banking and Financial Services in America. Boston: Kluwer, 1990. Also in
Federal Reserve Bank of Richmond Economic Review, vol. 74 (May/June
1988), pp. 3–22.
Hetzel, Robert. “Sterilized Foreign Exchange Intervention: The Fed Debate in
the 1960s.” Federal Reserve Bank of Richmond, 1996.
. “Central Banks’ Independence in Historical Perspective,” Journal
of Monetary Economics, vol. 25 (January 1990), pp. 165–76.
Humpage, Owen F. “Institutional Aspects of U.S. Intervention,” Federal
Reserve Bank of Cleveland Economic Review, vol. 30 (Quarter 1, 1994),
pp. 2–19.
Ireland, Peter N. Long-Term Interest Rates and Inflation: A Fisherian
Approach,” Federal Reserve Bank of Richmond Economic Quarterly,
vol. 82 (Winter 1996), pp. 21-35.
King, Mervyn. “Credibility and Monetary Policy: Theory and Evidence,” Bank
of England Quarterly Bulletin, vol. 35 (February 1995), pp. 84–91.

Alfred Broaddus, Marvin Goodfriend: Foreign Exchange Operations

19

Leiderman, Leonardo, and Lars Svensson, eds. Inflation Targets. London:
Center for Economic Policy Research, 1995.
McCallum, Bennett. “Inflation Targeting in Canada, New Zealand, Sweden,
the United Kingdom, and in General,” forthcoming in the Bank of Japan
Seventh International Conference, 1996a.
. International Monetary Economics. London: Oxford University
Press, 1996b.
Mussa, Michael. “The Role of Intervention,” Occasional Paper, No. 6. New
York: Group of Thirty, 1981.
Obstfeld, Maurice. “The Effectiveness of Foreign-Exchange Intervention:
Recent Experience, 1985–1988” in William H. Branson, Jacob A. Frenkel,
and Morris Goldstein, eds., International Policy Coordination and
Exchange Rate Fluctuations. National Bureau of Economic Research.
Chicago: University of Chicago Press, 1990.
Pauls, Dianne. “U.S. Exchange Rate Policy: Bretton Woods to Present,”
Federal Reserve Bulletin, vol. 76 (November 1990), pp. 891–908.
Persson, Torsten, and Guido Tabellini, eds. Monetary and Fiscal Policy,
Vol. 1: Credibility, and Vol. 2: Politics. Cambridge, Mass.: MIT Press,
1994.
Rose, Andrew, and Lars Svensson. “European Exchange Rate Credibility
before the Fall,” European Economic Review, vol. 38 (June 1994), pp.
1185–1216.
Salomon Brothers and Hutzler. An Analytical Record of Yields and Yield
Spreads: 1950–1968. New York: Salomon Brothers and Hutzler, 1968.
Sanger, David. “Clinton Offers $20 Billion to Mexico for Peso Rescue; Action
Sidesteps Congress,” The New York Times, February 1, 1995.
Sargent, Thomas J. Rational Expectations and Inflation. New York: Harper
and Row, 1986.
Stein, Herbert. “The Liberation of Monetary Policy,” in The Fiscal Revolution
in America. Chicago: University of Chicago Press, 1969.
Todd, Walker F. “Disorderly Markets: The Law, History, and Economics
of the Exchange Stabilization Fund and U.S. Foreign Exchange Market
Intervention,” in George G. Kaufman, ed., Research in Financial Services:
Private and Public Policy, Vol. 4. Greenwich, Conn.: JAI Press, 1992.
U.S. Congress, House of Representatives, Committee on the Budget. Exchange
Stabilization Fund. Hearings before the Task Force on Tax Expenditures
and Off-Budget Agencies of the Committee on the Budget, 94 Cong. 2
Sess. Washington: Government Printing Office, 1976.
U.S. Department of the Treasury. Exchange Stabilization Fund Annual Report.
Washington: U.S. Department of the Treasury, 1991.

Long-Term Interest
Rates and Inflation: A
Fisherian Approach
Peter N. Ireland

I

n recent years, Federal Reserve (Fed) policymakers have come to rely
on long-term bond yields to measure the public’s long-term inflationary
expectations. The long-term bond rate plays a central role in Goodfriend’s
(1993) narrative account of Fed behavior, 1979–1992, which links policyrelated movements in the federal funds rate to changes in the yield on long-term
U.S. Treasury bonds. According to Goodfriend, Fed officials interpreted rapid
increases in long-term bond rates as the product of rising inflationary expectations, reflecting a deterioration in the credibility of their fight against inflation.
To restore that credibility, they responded by tightening monetary policy, that
is, by raising the federal funds rate. Mehra (1995) presents statistical results
that support Goodfriend’s view. Using an econometric model, he demonstrates
that changes in long-term bond rates help explain movements in the federal
funds rate during the 1980s.
While these studies provide convincing evidence of a link between Fed
policy and long-term bond rates, both start with the untested hypothesis that
movements in such rates primarily reflect changes in long-term inflationary
expectations. And while economic theory does identify expected inflation as
one determinant of nominal bond yields, it suggests that there are other determinants as well. Using theory as a guide, this article seeks to measure the
contribution each determinant makes in accounting for movements in longterm bond yields. By doing so, it attempts to judge the extent to which Fed
policymakers are justified in using these bond yields as indicators of inflationary
expectations.

The author thanks Bob Hetzel, Tom Humphrey, Yash Mehra, and Stacey Schreft for helpful
comments and suggestions. The opinions expressed herein are the author’s and do not necessarily represent those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 82/1 Winter 1996

21

22

Federal Reserve Bank of Richmond Economic Quarterly

Irving Fisher (1907) presents what is perhaps the most famous theory of
nominal interest rate determination.1 According to Fisher’s theory of interest,
movements in nominal bond yields originate in two sources: changes in real
interest rates and changes in expected inflation. Thus, Fisher’s theory provides
a guide for investigating the extent to which long-term bond yields serve as
reliable indicators of long-term inflationary expectations. Specifically, it implies
that movements in long-term bond yields provide useful signals of changes in
inflationary expectations if and only if their other determinant, the long-term
real interest rate, is stable.
Although Fisher acknowledges the potential importance of risk in outlining
his theory, he stops short of explicitly considering the effects of uncertainty in
his graphical and mathematical treatment of interest rate determination. Recognizing that risk can play a key role in determining interest rates, and exploiting
advances in mathematical economics made since Fisher’s time, Lucas (1978)
develops a model that extends the relationships obtained by Fisher to a setting
where future economic magnitudes are uncertain.
In addition to real interest rates and expected inflation, Lucas’s model
identifies a third determinant of nominal bond yields: a risk premium that
compensates investors for holding dollar-denominated bonds in a world of uncertainty. Thus, Lucas’s model provides a more exhaustive set of conditions
under which movements in long-term bond rates provide useful signals of
changes in long-term inflationary expectations: it indicates that the long-term
real interest rate must be stable and that the risk premium must be small.
This article draws on Fisherian theory to assess the practical usefulness of
long-term bond yields as indicators of long-term inflationary expectations. It
begins, in Section 1, by outlining Fisher’s original theory of interest. It then
shows, in Section 2, how Lucas’s model generalizes the relationships derived
by Fisher to account for the effects of uncertainty. Section 3 uses Lucas’s
model to decompose the nominal bond yield into its three components: the
real interest rate, the risk premium, and the expected inflation rate. Section
4 applies this procedure to estimate the relative importance of the expected
inflation component in explaining movements in long-term U.S. Treasury bond
rates. Finally, Section 5 concludes the article.

1. FISHER’S THEORY OF INTEREST
To derive a relationship between the yield on a nominal bond and its determinants, Fisher (1907) considers the behavior of an investor in a simple model
economy. The economy has two periods, labelled t 0 and t 1, and a single
1 Although Fisher is usually identified as the inventor of this theory, Humphrey (1983) argues
that its origins extend back to the eighteenth century writings of William Douglass.

P. N. Ireland: Long-Term Interest Rates and Inflation

23

consumption good. The consumption good sells for P0 dollars in period t 0
and is expected to sell for Pe dollars in period t 1.
1
Fisher’s investor chooses between two types of assets. The first asset, a
nominal bond, costs the investor one dollar in period t
0 and pays him a
gross return of R dollars in period t
1. The yield R on this nominal bond
measures the economy’s nominal interest rate. The second asset, a real bond,
costs the investor one unit of the consumption good in period t 0 and returns
r units of the good in period t 1. The gross yield r on this bond represents
the economy’s real interest rate.
In order to purchase a nominal bond in period t 0, the investor must first
acquire one dollar; he can do so by selling 1/P0 units of the consumption good.
When it matures in period t
1, the nominal bond returns R dollars, which
the investor expects will buy R/Pe units of the good. Measured in terms of
1
goods, therefore, the expected return on the nominal bond equals the investor’s
receipts, R/Pe , divided by his costs 1/P0 . Letting e Pe /P0 denote the econ1
1
omy’s expected gross rate of inflation, one can write this goods-denominated
return as R/ e .
In equilibrium, the goods-denominated returns on nominal and real bonds
must be the same. For suppose the return R/ e on the nominal bond were
to exceed the return r on the real bond. Then every investor could profit by
selling the real bond and using the proceeds to purchase the nominal bond. The
resulting decrease in the demand for real bonds would raise the return r, while
the increase in the demand for nominal bonds would depress the return R/ e ,
until the two were brought back into equality. Similarly, any excess in the return
r over R/ e would be eliminated as investors attempted to sell nominal bonds
and purchase real bonds. Thus, Fisher concludes that R/ e r or, equivalently,

R

r e.

(1)

Fisher’s equation (1) expresses the nominal interest rate R as the product of
two terms: the real interest rate r and the expected inflation rate e . It therefore
describes the circumstances under which the nominal bond yield serves as a
reliable indicator of inflationary expectations. In particular, it implies that one
can be sure that a movement in the nominal interest rate reflects an underlying
change in inflationary expectations if and only if the real interest rate is stable.
The nominal bond in Fisher’s model resembles a U.S. Treasury bond since,
upon maturity, it returns a fixed number of dollars. Thus, the yield on Treasury
bonds measures the economy’s nominal interest rate R. Unfortunately, assets
resembling Fisher’s real bond do not currently trade in U.S. financial markets.
As a result, it is not possible to directly observe the real interest rate r and then
use equation (1) to determine the extent to which movements in Treasury bonds

24

Federal Reserve Bank of Richmond Economic Quarterly

reflect movements in real interest rates rather than inflationary expectations. 2
However, Fisher’s theory also links an economy’s real interest rate to its growth
rate of consumption. Hence, the theory suggests that the real interest rate may
be observed indirectly using data on aggregate consumption.
To derive a relationship between the real rate of interest and the growth
rate of consumption, Fisher returns to his model economy and uses a graph
like that shown in Figure 1. The graph’s horizontal axis measures consumption
in period t 0, and its vertical axis measures consumption in period t 1.
Fisher’s investor receives an income stream consisting of y0 units of the
consumption good in period t
0 and y1 units of the consumption good in
period t 1. He continues to trade in real bonds, which allow him to borrow
or lend goods at the real interest rate r. In particular, if y1 is large relative to
y0 , the investor borrows by selling a real bond; this transaction gives him one
more unit of the good in period t
0 but requires him to repay r units of
the good in period t 1. Conversely, if y1 is small relative to y0 , the investor
lends by purchasing a real bond; this gives him one less unit of the good in
period t 0 but pays him a return of r units of the good in period t 1. Thus,
the real interest rate r serves as an intertemporal price; it measures the rate at
which financial markets allow the investor to exchange goods in period t 1
for goods in period t
0. In Figure 1, the investor’s budget line A, which
passes through the income point (y0 , y1 ), has slope r.
Fisher’s investor has preferences over consumption in the two periods that
may be described by the utility function
U(c0 , c1 )

ln(c0 )

ln(c1 ),

(2)

where c0 denotes his consumption in period t 0, c1 denotes his consumption
in period t 1,ln is the natural logarithm, and the discount factor
1 implies
that the investor receives greater utility from a given amount of consumption
in period t 0 than from the same amount of consumption in period t 1.3 In
Figure 1, these preferences are represented by the indifference curve U, which
traces out the set of all pairs (c0 , c1) that yield the investor a constant level of
utility as measured by equation (2).
The slope of the investor’s indifference curve is determined by his marginal
rate of intertemporal substitution, the rate at which he is willing to substitute
consumption in period t
1 for consumption in period t
0, leaving his
utility unchanged. Mathematically, the investor’s marginal rate of intertemporal
2 For exactly this reason, Hetzel (1992) proposes that the U.S. Treasury issue bonds paying
a fixed return in terms of goods. Until Hetzel’s proposal is implemented, however, only indirect
measures of the real interest rate will exist.
3 Although Fisher does not use a specific utility function to describe his investor’s preferences, equation (2) helps to sharpen the implications of his theory by allowing the relationships
shown in Figure 1 to be summarized mathematically by equations (3) and (4) below.

P. N. Ireland: Long-Term Interest Rates and Inflation
Figure 1

25

Fisher’s Diagram

c *
1

y1
Slope = r

U

A
c *

+

y0

0

Period t = 0 Consumption

substitution equals the ratio of his marginal utility in period t
marginal utility in period t 1:
∂U(c0 , c1 )/∂c0
∂U(c0 , c1 )/∂c1

c1
.
c0

0 to his

(3)

To maximize his utility, the investor chooses the consumption pair (c0 , c1 ),
where the budget line A is tangent to the indifference curve U. At (c0 , c1 ), the
slope of the budget line equals the slope of the indifference curve. The former
is given by r; the latter is given by equation (3). Hence,
r

x/ ,

(4)

where x c1 /c0 denotes the optimal growth rate of consumption.
Equation (4) shows how Fisher’s theory implies that even when the real
interest rate r cannot be directly observed, it can still be estimated by computing
the growth rate x of aggregate consumption and dividing by the discount factor
. With this estimate in hand, one can use equation (1) to assess the usefulness
of Treasury bond yields as indicators of expected inflation. Specifically, if the

26

Federal Reserve Bank of Richmond Economic Quarterly

estimated real rate turns out to be fairly stable, then equation (1) implies that
movements in Treasury bond yields primarily reflect changes in inflationary
expectations.

2. LUCAS’S GENERALIZATION OF FISHERIAN THEORY
While Fisher recognized that the presence of risk may affect interest rates in
important ways, he lacked the tools to incorporate uncertainty formally into
his analysis and therefore assumed that his investor receives a perfectly known
income stream and faces perfectly known prices and interest rates. More than
seventy years later, advances in mathematical economics allowed Lucas (1978)
successfully to generalize Fisher’s theory to account for the effects of risk.
Lucas’s model features an infinite number of periods, labelled t
0,1,
2, . . . , and a single consumption good that sells for Pt dollars in period t.
Lucas’s investor receives an income stream consisting of yt units of the consumption good in each period t and consumes ct units of the good in each
period t.
Lucas’s investor, like Fisher’s, trades in two types of assets. A nominal
bond costs Lucas’s investor one dollar in period t and returns Rt dollars in
period t 1. Hence, Rt denotes the gross nominal interest rate between periods
t and t 1. A real bond costs him one unit of the consumption good in period
t and returns rt units of the good in period t 1. Hence, rt denotes the gross
real interest rate between periods t and t 1. During each period t, the investor
purchases Bt nominal bonds and bt real bonds.
Unlike Fisher’s investor, however, Lucas’s investor may be uncertain about
future prices, income, consumption, interest rates, and bond holdings. That is,
he may not learn the exact values of Pt , yt , ct , Rt , rt , Bt , and bt until the
beginning of period t; before then, he regards these variables as random.
As sources of funds during each period t, the investor has yt units of
the consumption good that he receives as income and rt 1 bt 1 units of the
consumption good that he receives as payoff from his maturing real bonds. He
also has Rt 1 Bt 1 dollars that he receives as payoff from his maturing nominal
bonds; he can exchange these dollars for Rt 1 Bt 1 /Pt units of the consumption
good. As uses of funds, the investor has his consumption purchases, equal to ct
units of the good, and his bond purchases. His real bond purchases cost bt units
of the good, while his nominal bond purchases cost Bt /Pt units of the good.
During period t, the investor’s sources of funds must be sufficient to cover his
uses of funds. Hence, he faces the budget constraint
yt

rt

1 bt 1

Rt

1 Bt 1 /Pt

ct

bt

Bt /Pt ,

which is Lucas’s analog to the budget line A in Fisher’s Figure 1.

(5)

P. N. Ireland: Long-Term Interest Rates and Inflation

27

Lucas’s investor chooses ct , Bt , and bt in each period t to maximize the
utility function
j

Et

ln(ct j ) ,

(6)

j 0

subject to the budget constraint (5), where Et denotes the investor’s expectation at the beginning of period t. Equation (6) simply generalizes Fisher’s
utility function (2) to Lucas’s setting with an infinite number of periods and
uncertainty. The solution to the investor’s problem dictates that
1/rt

Et 1/xt

(7)

1

and
1/Rt

Et (1/xt

1 )(1/ t 1 )

,

(8)

in each period t
0,1,2, . . . , where xt 1
ct 1 /ct denotes the gross rate
of consumption growth and t 1 Pt 1 /Pt denotes the gross rate of inflation
between periods t and t 1.
Lucas’s equation (7) generalizes Fisher’s equation (4); it is analogous to
the tangency between the investor’s budget line and indifference curve shown
in Figure 1. As in equation (3), the investor’s marginal rate of intertemporal
substitution is xt 1 / . Hence, equation (7) shows that, under uncertainty, the
investor chooses his consumption path so that the expected inverse of his marginal rate of intertemporal substitution equals the inverse of the real interest
rate. Also, like Fisher’s equation (4), Lucas’s equation (7) suggests that while
the real interest rate cannot be directly observed, it can still be estimated using
data on aggregate consumption.
For any two random variables a and b,
E ab

Cov a, b

Ea Eb,

(9)

where Cov[a, b] denotes the covariance between a and b. Using this fact, one
can rewrite equation (8) as
1/Rt

Covt (1/xt

1 ),(1/ t 1 )

Et 1/xt

1

Et 1/

t 1

,

(10)

where Covt denotes the covariance based on the investor’s period t information.
In light of equation (7), equation (10) simplifies to
1/Rt

Covt (1/xt

1 ),(1/ t 1 )

(1/rt )Et 1/

t 1

.

(11)

Lucas’s equation (11) generalizes Fisher’s equation (1); it shows how, under
uncertainty, the nominal interest rate Rt depends on the real interest rate rt and
the expected inflation term Et [1/ t 1 ].
The covariance term in equation (11) captures the effect of risk on the
nominal interest rate. It appears because random movements in inflation make
the goods-denominated return on a nominal bond uncertain. To see this, recall

28

Federal Reserve Bank of Richmond Economic Quarterly

from Section 1 that the return on the nominal bond, measured in terms of the
consumption good, equals Rt / t 1 . Since the inflation rate t 1 remains unknown until period t 1, so too does Rt / t 1 . Hence, random inflation makes
the nominal bond a risky asset.
Equation (11) shows that inflation uncertainty may either increase or decrease the nominal interest rate, depending on whether the covariance term is
negative or positive. In particular, inflation uncertainty increases the nominal
interest rate if the covariance between 1/xt 1 and 1/ t 1 is negative, that is,
if periods of low consumption growth coincide with periods of high inflation.
In this case, high inflation erodes the nominal bond’s return Rt / t 1 precisely
when the investor, suffering from low consumption growth, finds this loss most
burdensome. Hence, the higher nominal yield Rt compensates the investor for
this extra risk. Conversely, uncertainty decreases the nominal interest rate if the
covariance term in equation (11) is positive, so that periods of high consumption
growth coincide with periods of high inflation.
Thus, Lucas’s model, like Fisher’s, identifies real interest rates and expected
inflation as two main determinants of nominal bond yields. Lucas’s model goes
beyond Fisher’s, however, by identifying a third determinant: a risk premium,
represented by the covariance term in equation (11), that compensates investors
for holding dollar-denominated bonds in the presence of inflation uncertainty.
According to Lucas’s model, therefore, movements in long-term bond yields
accurately reflect changes in expected inflation if and only if the real interest
rate is stable and the risk premium is small.

3. DERIVING BOUNDS ON EXPECTED INFLATION
Lucas’s equation (7), like Fisher’s equation (4), suggests that the unobservable real interest rate can be estimated using data on aggregate consumption.
Like the real interest rate, however, the risk premium component of nominal
bond yields cannot be directly observed. Hence, without further manipulation,
Lucas’s equation (11) cannot be used to assess the extent to which movements
in nominal bond yields reflect changes in expected inflation rather than changes
in their other two components.
Fortunately, as shown by Smith (1993), Lucas’s model also places bounds
on the plausible size of the risk premium. These bounds, together with estimates
of the real interest rate constructed from the consumption data, can be used
to determine the extent to which movements in nominal bond yields reflect
changes in inflationary expectations.4

4 Smith (1993) takes the opposite approach: he uses the bounds on risk premia, along with
estimates of expected inflation, to characterize the behavior of real interest rates.

P. N. Ireland: Long-Term Interest Rates and Inflation

29

Recall that the effects of risk enter into Lucas’s equation (11) through the
covariance term. Smith rewrites this term as
Covt (1/xt

1 ),(1/ t 1 )

t Stdt

1/xt

Stdt 1/

1

t 1

,

(12)

where
Covt (1/xt

t

1 ),(1/ t 1 )

/ Stdt 1/xt

Stdt 1/

1

t 1

(13)

denotes the correlation between 1/xt 1 and 1/ t 1 based on the investor’s
period t information and Stdt denotes the standard deviation based on period t
information.
Equation (12) conveniently decomposes the covariance term into three
components. The first component, the correlation coefficient t , can be negative
or positive but must lie between 1 and 1. Hence, this component captures
the fact that the covariance term may be of either sign. The second and third
components, the standard deviations of 1/xt 1 and 1/ t 1 , must be positive.
Hence, these terms govern the absolute magnitude of the covariance term,
regardless of its sign. Thus, equation (12) places bounds on the size of the
covariance term:
Stdt 1/xt

Stdt 1/

1

Covt (1/xt

t 1

Stdt 1/xt

1

Stdt 1/

1 ),(1/ t 1 )

,

t 1

(14)

where the upper bound is attained in the extreme case where t 1, the lower
bound is attained at the opposite extreme where t
1, and the tightness of
the bounds depends on the size of the standard deviations.
Evidence presented in the appendix justifies the additional assumption that
inflation volatility in the United States is limited in the sense that the coefficient
of variation of 1/ t 1 conditional on period t information is less than one:
Stdt 1/

t 1

/Et 1/

1.

t 1

(15)

This assumption allows equation (14) to be rewritten
Stdt 1/xt

1

Et 1/

Covt (1/xt

t 1

Stdt 1/xt

1

Et 1/

t 1

1 ),(1/ t 1 )

,

(16)

which, along with equations (7) and (11), implies
Stdt 1/xt

1

Et 1/

1/Rt

t 1

Stdt 1/xt

1

Et 1/xt
Et 1/

t 1

Et 1/

1

,

t 1

(17)

or, equivalently,
Rt Et 1/xt

1

Stdt 1/xt

Rt Et 1/xt

1

1

1/Et 1/

Stdt 1/xt

1

.

t 1

(18)

30

Federal Reserve Bank of Richmond Economic Quarterly

Since
1/Et 1/

Et

t 1

t 1

,

(19)

equation (18) places bounds on the expected inflation component that is embedded in the nominal interest rate Rt . Again, these bounds arise because the
covariance term in Lucas’s equation (11) may be negative or positive and
because the absolute magnitude of the covariance term depends on the standard deviation of 1/xt 1 . In particular, the bounds will be tight if this standard
deviation—and hence the magnitude of the risk premium—is small. Equation
(18) also indicates that these bounds may be estimated using data on aggregate
consumption.
Together, therefore, equations (7) and (18) show how one may use data
on aggregate consumption to assess the usefulness of nominal bond yields as
indicators of inflationary expectations. If the estimates provided by equation (7)
show that the real interest rate is stable, and if the bounds provided by equation
(18) indicate that the risk premium is small, then Lucas’s model implies that
most of the variation in the nominal bond yield reflects underlying changes in
expected inflation.

4. ESTIMATING THE REAL INTEREST RATE AND
BOUNDS ON EXPECTED INFLATION
In order to estimate the real interest rate and the bounds on expected inflation
using equations (7) and (18), one must first obtain estimates of the quantities Et [1/xt 1 ] and Stdt [1/xt 1 ]. Suppose, in particular, that the evolution of
gt 1 1/xt 1 , the inverse growth rate of aggregate consumption, is described
by the linear time series model
gt

(L)gt

1

t 1,

L2

(20)
Lk

where is a constant, (L)
...
is a polynomial in
0
1L
2
k
the lag operator L, and t 1 is a random error that satisfies
E

t 1

0, Std

, Et

t 1

t 1 t j

0, E

t 1 gt j

0

(21)

for all t 0,1,2, . . . and j 0,1,2, . . . . One may then use estimates of ,
(L), and to compute Et [1/xt 1 ] Et [gt 1 ] and Stdt [1/xt 1 ] Stdt [gt 1 ] as
Et g t

(L)gt

1

(22)

and
Stdt gt

1

.

(23)

Here, as in Mehra (1995), the long-term nominal interest rate is measured
by the yield on the ten-year U.S. Treasury bond. This choice for Rt identifies
each period in Lucas’s model as lasting ten years. In this case, gt 1 1/xt 1

P. N. Ireland: Long-Term Interest Rates and Inflation

31

corresponds to the inverse ten-year growth rate of real aggregate consumption
of nondurables and services in the United States, converted to per-capita terms
by dividing by the size of the noninstitutional civilian population, ages 16 and
over. The data are quarterly and run from 1959:1 through 1994:4.
Hansen and Hodrick (1980) note that using quarterly observations of tenyear consumption growth to estimate equation (20) by ordinary least squares
yields consistent estimates of
and the coefficients of (L). But since the
sampling interval of one quarter is shorter than the model period of ten years,
the least squares estimate of is biased. Thus, the results reported below are
generated using the ordinary least squares estimates of and (L) and Hansen
and Hodrick’s consistent estimator of , modified as suggested by Newey and
West (1987). The limited sample size and the extended length of the model
period imply that only one lag of gt 1 can be included on the right-hand side
of equation (20).
Finally, equation (18) indicates that the discount factor determines the
location of the bounds on expected inflation. Thus, may be chosen so that
the midpoint between the lower and upper bounds, averaged over the sample
period, equals the actual inflation rate, averaged over the sample period. This
procedure yields the estimate
0.856, which corresponds to an annual
discount rate of about 1.5 percent.
Figure 2 illustrates the behavior of the ten-year real interest rate, estimated
using equations (7) and (22).5 The real interest rate climbs steadily from 1969
until 1983 before falling sharply between 1983 and 1985. But despite these
variations, the long-term real interest rate remains within a narrow, 75 basis
point range throughout the entire 26-year period for which estimates are available. The average absolute single-quarter movement in the real interest rate is
just two basis points; the largest absolute single-quarter move occurs in 1973:4,
when the real interest rate increased by only eight basis points. Thus, Figure
2 suggests that the long-term interest rate in the United States is remarkably
stable.
Figure 3 plots the bounds on ten-year expected inflation estimated using
equations (18), (22), and (23). The bounds are very tight. Even at their widest,
in 1981:4, they limit the expected rate of inflation to a 28 basis point band,
implying that changes in the risk premium cannot account for movements in
the ten-year bond rate larger than 28 basis points. Thus, Figure 3 suggests that
the risk premium in the ten-year Treasury bond is very small.
Some intuition for the results shown in Figures 2 and 3 follows from
equations (7), (18), (20), (22), and (23). Equation (22), along with equation
(7), links variability in the long-term real interest rate to variability in the
5 Although the sample used to estimate equation (20) extends back to 1959:1, the ten-year
model period and the presence of one lag of gt 1 on the right-hand side imply that estimates of
the real rate can only be constructed for the period beginning in 1969:1.

32

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Ten-Year Real Interest Rate

3.2

3.0
3

2.8

2.6

2.4
69:1

71:1 73:1

7 5 : 1 7 7 : 1 79:1 81:1

8 3 : 1 8 5 : 1 8 7 : 1 8 9 : 1 93:1 95:1
91:1

+
predictable part of consumption growth, measured by
(L)gt in equation
(20). Equation (23), along with equation (18), links the size of the risk premium
to variability in the unpredictable part of consumption growth, measured by t 1
in equation (20). In the U.S. data, aggregate consumption growth varies little
over ten-year horizons. And since total consumption growth is quite stable, both
of its components—the predictable and unpredictable parts—are quite stable as
well. Thus, given the stability in aggregate consumption growth, Lucas’s model
implies that the long-term real interest rate must be quite stable and that the
risk premium must be quite small.
According to Lucas’s model, the stability of the real interest rate and the
small size of the risk premium shown in Figures 2 and 3 imply that most of the
variation in the ten-year Treasury bond rate reflects underlying changes in the
third component, expected inflation. Indeed, as the largest quarterly real interest
rate movement shown in Figure 2 is eight basis points, and as the bounds in
Figure 3 are at most 28 basis points wide, the results suggest that any quarterly
change in the ten-year bond rate in excess of 36 basis points almost certainly
signals a change in inflationary expectations.

P. N. Ireland: Long-Term Interest Rates and Inflation

33

Figure 3 Bounds on Ten-Year Expected Inflation

14
12
10
8
6
4
2

+

0
69:1 7 1 : 1 7 3 : 1 7 5 : 1 7 7 : 1 7 9 : 1 8 83:1 85:1 87:1 89:1 91:1 93:1 95:1
1:1

+
5. CONCLUSION
Although Federal Reserve officials use the yield on long-term Treasury bonds to
gauge the public’s inflationary expectations, contemporary versions of Fisher’s
(1907) theory of interest suggest that variations in bond yields can originate
in other sources as well. In particular, Lucas’s (1978) model indicates that
movements on long-term bond yields will accurately signal changes in longterm inflationary expectations if and only if long-term real interest rates are
stable and risk premia are small.
Unfortunately, neither real interest rates nor risk premia can be directly
observed. However, Lucas’s model also shows how these unobservable components of nominal bond yields can be estimated using data on aggregate
consumption.
This article lets Lucas’s model guide an empirical investigation of the
determinants of the ten-year U.S. Treasury bond yield. The results indicate
that, indeed, the ten-year real interest rate is quite stable and the ten-year
risk premium is quite small. Hence, according to Lucas’s model, movements
in the long-term bond rate primarily reflect changes in long-term inflationary
expectations. Evidently, the Federal Reserve has strong justification for using
long-term bond yields as indicators of expected inflation.

34

Federal Reserve Bank of Richmond Economic Quarterly

APPENDIX
The bounds on expected inflation given by equation (18) were derived in Section 3 under the extra assumption that equation (15) holds. Thus, this appendix
provides some justification for (15).
Consider the following linear time series model for the inverse inflation
rate qt 1 1/ t 1 :
qt
where the random error
E

t 1

0, Std

1

t 1

satisfies

t 1

(L)qt

,E

t 1 t j

t 1,

0, E

(24)

t 1 qt j

0

(25)

for all t 0,1,2, . . . and j 0,1,2, . . . . One can use this model to estimate
Et [1/ t 1 ] Et [qt 1 ] and Stdt [1/ t 1 ] Stdt [qt 1 ], just as equation (20) was
used to estimate Et [1/xt 1 ]
Et [gt 1 ] and Stdt [1/xt 1 ]
Stdt [gt 1 ]. In the
U.S. data, qt 1 corresponds to the inverse ten-year growth rate of the price
deflator for the aggregate consumption of nondurables and services.
Estimates of (24) using quarterly data from 1959:1 through 1994:4 reveal
that Stdt [qt 1 ]
0.0363. The smallest estimate of Et [qt 1 ] is 0.441, for
1969:1. Thus, for the entire sample period, estimates of Stdt [1/ t 1 ]/Et [1/ t 1 ]
never exceed 0.0823, which suggests that the upper bound of unity imposed
by equation (15) is an extremely conservative one.

P. N. Ireland: Long-Term Interest Rates and Inflation

35

REFERENCES
Fisher, Irving. The Rate of Interest. New York: MacMillan Company, 1907.
Goodfriend, Marvin. “Interest Rate Policy and the Inflation Scare Problem:
1979–1992,” Federal Reserve Bank of Richmond Economic Quarterly,
vol. 79 (Winter 1993), pp. 1–24.
Hansen, Lars Peter, and Robert J. Hodrick. “Forward Exchange Rates as
Optimal Predictors of Future Spot Rates: An Econometric Analysis,”
Journal of Political Economy, vol. 88 (October 1988), pp. 829–53.
Hetzel, Robert L. “Indexed Bonds as an Aid to Monetary Policy,” Federal
Reserve Bank of Richmond Economic Review, vol. 78 (January/February
1992), pp. 13–23.
Humphrey, Thomas M. “The Early History of the Real/Nominal Interest Rate
Relationship,” Federal Reserve Bank of Richmond Economic Review, vol.
69 (May/June 1983), pp. 2–10.
Lucas, Robert E., Jr. “Asset Prices in an Exchange Economy,” Econometrica,
vol. 46 (November 1978), pp. 1429–45.
Mehra, Yash P. “A Federal Funds Rate Equation,” Working Paper 95-3.
Richmond: Federal Reserve Bank of Richmond, May 1995.
Newey, Whitney K., and Kenneth D. West. “A Simple, Positive Semi-Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,”
Econometrica, vol. 55 (May 1987), pp. 703–08.
Smith, Stephen D. “Easily Computable Bounds for Unobservable Real Rates.”
Manuscript. Federal Reserve Bank of Atlanta, August 1993.

The Early History
of the Box Diagram
Thomas M. Humphrey

E

conomists hail it as “a powerful tool,” “a work of genius,” and “one
of the most ingenious geometrical constructions ever devised in economics.” It graces the pages of countless textbooks on price theory,
welfare economics, and international trade. It is associated with some of the
greatest advances ever made in economic theory. It elegantly depicts the two
fundamental welfare theorems that are absolutely central to modern economics.
In short, it ranks with the preeminent schematic devices of economics since it
illuminates the most important ideas economists have to offer. It is none other
than the celebrated box diagram used to illustrate efficiency in exchange and
resource allocation in hypothetical two-agent, two-good, two-factor models of
general economic equilibrium.
The box comes in two variants. The exchange version has dimensions
determined by total available stocks of the two goods (see Figure 1). It incorporates traders’ indifference maps, one with origin sited in the southwest
corner and the other in the northeast corner. The box depicts opportunities for
mutually beneficial trade. Thus a movement from initial endowment point E
to point Z on the contract curve—a movement accomplished through a trade
of ER units of the second good for RZ units of the first—benefits both traders
simultaneously by putting them on higher indifference curves. In general, so
long as the straight trading line EZ, whose slope measures the price of the first
good in terms of the second, cuts the indifference curves of both parties at
point E, it pays each to move along that line to the contract curve. Once on
the contract curve, however, the potential for further mutually advantageous
trades is at an end. Since the contract curve is the locus of indifference-curve
tangency points, it follows that movements along the contract curve improve
the welfare of one trader only by reducing that of the other.
For valuable comments on earlier drafts of this article, the author is indebted to D. P. O’Brien,
J. Patrick Raines, Tibor Scitovsky, and especially to his Richmond Fed colleagues Peter
Ireland, Jeff Lacker, Ned Prescott, and John Weinberg.

Federal Reserve Bank of Richmond Economic Quarterly Volume 82/1 Winter 1996

37

38

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Exchange Box Diagram
Second
Trader’s Origin

02
E

Price Line

Endowment
Point

Autarky
Indifference
Curves

Z

R
Contract
Curve

0

1

First
Trader’s Origin

Good 1

Both traders prefer allocations in the shaded area to initial endowment E.
Movement down the price vector connecting endowment point E to efficiency
point Z on the contract curve puts both traders on higher indifference curves
a n d t h u s m a k e s t h e m b e t t e r o f f . A t p o i n t Z, h o w e v e r , a l l p o t e n t i a l m u t u a l l y
beneficial trades are at an end. Movements along the contract curve benefit
one trader only at the cost of hurting the other.

+

The alternative production variant of the box depicts the fabrication of two
goods from two factor inputs. It replaces indifference maps representing preference functions with isoquant maps representing production functions. It lets
available factor quantities determine the dimensions of the box. Efficient factor
allocations occur along the contract-curve efficiency locus. There, isoquants are
tangent to each other such that the output of one good is maximized given the
output of the other.
The chief appeal of the box diagram is its ability to explain much with
little. A simple plane diagram, the box can, in Kelvin Lancaster’s words, “show
the interrelationships between no less than twelve economic variables” ([1957]
1969, p. 52). Moreover, it can do so without resort to algebra and calculus,
techniques inaccessible to the mathematically untrained. Small wonder that
economists extol the analytical and pedagogical properties of the box or that
textbooks feature it as an expository device.
Where some textbooks go astray, however, is in their ahistoric presentation
of the diagram. Typically, they say little or nothing about its origins and evolu-

T. M. Humphrey: Early History of the Box Diagram

39

tion. They simply present it as an accomplished fact without inquiring into its
genealogy. A leading international trade theory textbook authored by Richard
Caves and Ronald Jones (1981) provides a prime example. It attributes the box
to no progenitor, not even to Francis Edgeworth or Arthur Bowley. The result
is that the student is unaware of the circumstances prompting the diagram’s
development. He knows not who invented it, why it was invented, what problems it originally was designed to solve, or how it evolved under the impact of
attempts to perfect it and extend its range of application. Nor can he appreciate
the intellectual effort involved in its creation and refinement. Unaware of such
matters, he may surmise that the diagram sprang fully developed from the brain
of the latest theorist. Ahistoric textbooks indeed foster that very impression.
Such are the hazards of disassociating an idea from its historical context and
presenting it as a timeless truth.
Far from being timeless, the box diagram possesses a definite chronology. That chronology features some of the leading names in neoclassical and
modern economics. Francis Edgeworth, Vilfredo Pareto, A.W. Bowley, Tibor
Scitovsky, Wassily Leontief, Kenneth Arrow, Abba Lerner, Wolfgang Stolper,
Paul Samuelson, T. M. Rybczynski, and Kelvin Lancaster all contributed to the
diagram’s development.
Edgeworth invented the exchange box in 1881. He used it to demonstrate
the indeterminacy of isolated barter and the determinacy of competitive equilibrium. He showed that all final settlements are on the contract curve, that
the competitive equilibrium is one such settlement, and that the contract curve
shrinks to the competitive equilibrium as the number of traders increases. Pareto
in 1906 demonstrated his celebrated optimality criterion with the aid of the
box. Bowley in 1924 generalized Edgeworth’s work with his notion of the
bargaining locus. Scitovsky in 1941 employed the box to formulate his famous
double-bribe test of increased efficiency. Leontief coordinated, consolidated,
and clarified the earlier accomplishments in his 1946 rehabilitation of the exchange box. In so doing, he paved the way for the post-war popularity of the
diagram. Following hard on Leontief’s heels, Arrow in 1951 employed the
concepts of convex sets and supporting hyperplanes to analyze the problem of
corner solutions on the boundary of the box. And Samuelson in 1952 employed
the box to investigate how international transfers affect the terms of trade.
When the foregoing contributions threatened to exhaust the analytical potential of the exchange box, economists turned to the alternative production
version. Already, Lerner had presented the first production box in a pioneering
1933 paper whose publication unfortunately was delayed for nineteen years.
In the meantime, Stolper and Samuelson published the first production box
diagram to appear in print. As employed by them in 1941, by Rybczynski in
1954, and by Lancaster in 1957, the production diagram proved indispensable
to the derivation and illumination of certain core propositions of the emerging
Heckscher-Ohlin theory of international trade.

40

Federal Reserve Bank of Richmond Economic Quarterly

The paragraphs below attempt to trace this evolution and to identify specific contributions to it. Besides unearthing lost or forgotten insights, such an
exercise may serve as a partial antidote to the textbooks’ ahistorical treatment
of the diagram. One conclusion emerges: namely that the box hardly developed
autonomously. Rather it evolved in a two-way interaction with its applications.
Thus an unsolved puzzle in microeconomics prompted the invention of the
box—a prime example of a seemingly intractable problem inducing the very
tool required for its solution. The resulting availability of the diagram then
spurred economists to find new applications for it. These new uses in turn triggered modifications of the diagram. Applications were both cause and effect
of the diagram’s development.

1. FRANCIS Y. EDGEWORTH
The box diagram makes its first appearance on pages 28 and 113 of Francis
Edgeworth’s 1881 Mathematical Psychics. Motivated by a problem in microeconomic theory, Edgeworth invented the diagram and its constituent
indifference-map and contract-curve components to solve the problem.
Edgeworth’s predecessors had long known that equilibrium price in isolated, two-party exchange is indeterminate. They also understood that equilibrium between numerous buyers and sellers operating in competitive markets is
determinate. But they had been unable to reconcile the two results. They could
not show rigorously how increasing numbers lead to price determinacy.
This task Edgeworth sought to accomplish. Using the box diagram, he
established (1) that final outcomes must be on the contract curve, (2) that the
contract curve shrinks as the number of competitors increases, (3) that competitive equilibrium is one point on the contract curve, and therefore (4) that
as the number of competitors increases without limit the contract curve shrinks
to a single point, namely the competitive equilibrium.1 Here was his rationale
for inventing the diagram.
Edgeworth’s Invention and its Components
Edgeworth’s diagram depicts two isolated individuals, A and B, trading fixed
stocks of two goods, x and y, whose quantities determine the dimensions of the
box (see Figure 2). Individual A initially holds the entire stock of good x and
individual B the entire stock of good y. Superimposing indifference maps on
the box, Edgeworth sites the origin of A’s map in the lower right corner and
the origin of B’s map in the upper left corner. This arrangement fixes point 0
1 In the case of multiple competitive equilibria, the contract curve shrinks not to one but to
several points. Edgeworth recognized such a possibility. But he tended to focus on the case of
singular rather than multiple equilibrium. See Newman (1990, p. 261).

T. M. Humphrey: Early History of the Box Diagram

41

Figure 2 Edgeworth’s Version of the Box
B’s Origin
Contract
Curve

C

I
B

N

OA

Offer
Curves

0
Autarky
Indifference
Curves

OB

I
A0

M
Price
Ray

C
Good x

0

A’s Origin

Endowment
Point
Edgeworth’s original diagram depicts three components: (1) autarky
indifference curves going through the endowment point and defining the cigarshaped area of mutually advantageous trades; (2) offer curves or sets of
points of tangency of indifference curves and the price ray as it swings about
the endowment point; and (3) the contract curve, whose relevant segment lies
between the autarky indifference curves.

+

in the lower left corner as the endowment point of both individuals. That is,
the length of the lower horizontal axis measured from right to left indicates the
amount of good x held by A just as the length of the left vertical axis measured
from top to bottom indicates the amount of good y held by B. Since A holds
no y nor B any x, these axes establish the endowment point.
From the indifference curves radiating outward from their respective origins, Edgeworth selects one particular curve for each trader, namely the curves
passing through the endowment point. These curves show alternative combinations of goods that yield the same satisfaction as the endowment bundle.
They indicate the level of utility each person would enjoy if he consumed his

42

Federal Reserve Bank of Richmond Economic Quarterly

endowment bundle and refrained from exchange. They also trace out the zone
of mutually beneficial exchanges that make both traders better off than they
would be under autarky.
Next, Edgeworth draws in the contract curve CC along which indifference
curves are tangent such that one trader cannot occupy a higher indifference
curve unless the other is forced to occupy a lower one. Especially significant is
the portion of the contract curve bounded by the autarky indifference curves.
Since traders require that potential exchanges make them at least as well off
as they would be under autarky, they will never voluntarily agree to trades
outside those bounds. It follows that the relevant segment of the contract curve
lies in the lens-shaped area between the indifference curves going through the
endowment point.
Finally, Edgeworth sketches traders’ reciprocal demand schedules or offer
curves. These curves apply to the special case where the two traders act as
representative price-takers operating on opposite sides of a competitive market. Offer curves show how much each trader is willing to exchange at all
possible prices. Edgeworth of course did not invent such curves. That honor
goes to Alfred Marshall. But he was the first to derive them as the locus of
points of tangency of indifference curves and the price ray as it pivots about
the endowment point. He likewise was the first to explain that each point on
an offer curve represents an outcome of constrained utility maximization in
which the commodity price ratio, or slope of the price ray, equals the ratio of
marginal utilities, or slope of the indifference curves.
Exploiting Potential Mutual Gains from Exchange
Having derived the exchange box and its constituent components, Edgeworth
employed it to illuminate five basic propositions. His first proposition states that
final settlements must be on the contract curve. At any other point, both parties
could make themselves better off by renegotiation. Consider any point lying off
the contract curve. Going through that point are intersecting indifference curves
enclosing a cigar-shaped area that spells unexploited potential mutual gains
from exchange. Traders will not let such opportunities go unrealized. Instead,
they will exploit them until they reach the contract curve where indifference
curves are tangent and further mutual gains are at an end.
Efficiency of Competitive Equilibrium
Edgeworth’s second proposition refers to the efficiency of competitive equilibrium. It states that such equilibrium is always on the contract curve. The
reason? Competition establishes a common, market-clearing price ratio.
Competitive price-takers independently respond to that ratio by trading at the
point where each supplies the quantity the other demands and vice versa.
That is, price-takers operate at the point where their offer curves intersect (see

T. M. Humphrey: Early History of the Box Diagram

43

Figure 3 Competitive Equilibrium versus the Indeterminacy
of Two-Party Barter

B’s Origin
Best
for A

C

I
B0
N

OA

Competitive
Equilibrium

Offer
Curves

I
OB

Price
Ray

M

A0

Best
for B

C
0

Good x

A’s Origin

Bargaining between two isolated traders can lead to an outcome anywhere on
the segment of the contract curve between the autarky indifference curves. By
contrast, the competitive equilibrium is uniquely determined at the intersection
of the price ray and offer curves. There the traders’ indifference curves are
tangent to the common price ray and thus to each other.

+

Figure 3). At this point, indifference curves are tangent to the common price
ray emanating from the endowment point and thus are tangent to each other.
Since such tangencies occur only on the contract curve, it follows that competitive equilibrium is on that same curve.
Indeterminacy of Isolated Two-Party Barter
Edgeworth’s third proposition refers to the indeterminacy of isolated two-party
exchange. In such bilateral monopoly situations, it is impossible to determine,
from indifference maps and endowments alone, the precise price-quantity equilibrium that will emerge. All one can say is that equilibrium must lie on the

44

Federal Reserve Bank of Richmond Economic Quarterly

segment of the contract curve between the autarky indifference curves. But
which one of the infinity of possible equilibria will prevail will depend upon
considerations external to Edgeworth’s model, namely the relative bargaining
skills and strengths of the traders as well as the strategies and tactics they
employ. Economists traditionally have had little to say about such matters.2
They cannot confidently predict any unique outcome. The precise ingredients of shrewd, effective bargaining remain subtle, elusive, and obscure. Still,
economists can note that the gain one bargainer gets from exchange is limited
only by the other’s effort to get the best for himself. Final settlement will be
near point N if A is the superior bargainer. It will be nearer to point M if B has
the bargaining advantage.
Neither outcome, Edgeworth noted, necessarily coincides with the point
of maximum aggregate welfare on the contract curve. There the sum of the
traders’ satisfactions is at its peak. Identifying this unique maximum point of
course requires that utility be cardinally measurable and comparable across
individuals—properties Edgeworth thought utility possessed. It was on these
grounds that he advanced his famous principle of arbitration. Compulsory
arbitration, he argued, could do what unrestricted bargaining could not do.
By imposing the utilitarian sum-of-satisfactions solution on the bilateral monopolists, arbitration would yield a determinate, socially optimum outcome.
Conversely, in the absence of such arbitration indeterminacy would continue
to characterize the isolated two-party case.
Recontracting and the Role of Numbers
Edgeworth’s fourth proposition, his recontracting theorem, refers to the role of
numbers in reducing indeterminacy. It states that as the number of traders gets
large, the contract curve shrinks to a single point, the competitive equilibrium.
Edgeworth sketches a proof on pages 35–37 of his Mathematical Psychics
(see Creedy [1992], pp. 158–65, for a particularly clear and insightful interpretation). He starts with the two-person case in which party A provisionally
contracts with party B to reach point C on A’s indifference curve I A 0 (see Figure
4a). He then introduces a new pair of traders identical to the first pair. This
2 They have said something, however. John Nash (1950) thought the bargainers might agree
to maximize the multiplicative product of their respective utility gains from trade (the excess
of post-trade over autarky levels of satisfaction). John Harsanyi (1956) showed that the Danish
economist Frederick Zeuthen (1930) had proposed essentially the same solution two decades
before Nash. Ariel Rubinstein (1982) showed that the Nash solution is the outcome of a noncooperative, offer-counteroffer game. John Creedy (1992, pp. 193–99) suggested a variant of
the Nash solution, namely the maximization of a geometrical weighted average of the traders’
utility gains, with the weights measuring the relative bargaining powers of the two parties. These
solutions establish unique potential agreement points on the contract curve. Since it is unlikely
that the bargainers would always agree to go to such proposed points, however, indeterminacy
remains.

T. M. Humphrey: Early History of the Box Diagram

45

Figure 4 Edgeworth’s Recontracting Process

a

b
I

C

I

B0
IA

C

B0
IA

0

IA

IA

1

0

2

C1
C

C

P

P1
0

0

c

d
I 0
B

I
B0

C

IA

C

I

0

IA

A0

P

3

C*
*
P

0

C

C

0

Recontracting plus convexity of indifference curves implies that the contract
curve shrinks as traders become more numerous. With a single pair of traders,
the contract curve is CC' as shown in panel a. Adding another pair shrinks the
c u r v e b y t h e a m o u n t CC * a t b o t h e n d s ( p a n e l c ) . W h e n t h e n u m b e r o f p a i r s
gets very large, the curve shrinks to the competitive equilibrium (panel d).

+

maneuver allows him to use the same box diagram to deal with four parties. It
permits him to represent the preferences of each of the As (and the Bs) with a
single indifference map.
It also means that the agreement reached at point C cannot be final. For
the two As can now ignore one of the Bs and deal with the other at point C.
When they split the resulting bundle equally among themselves, they will each
reach the half-way point P on the trade vector 0C. That point, because of the
convexity of indifference curves, is on a higher such curve than before. Thus
the As are better off, their trading partner B is just as well off as initially, and
the excluded B is at his endowment point.

46

Federal Reserve Bank of Richmond Economic Quarterly

In retaliation, the excluded B then underbids his competitor by offering the
As a trade on better terms at point C1 (see Figure 4b). The As, by accepting, can
share the resulting bundle among themselves to each attain point P1 on a still
higher indifference curve than before. Repeated recontracting brings the parties
to point C∗ (see Figure 4c). There the As are indifferent between (1) trading with
both Bs at C∗ and (2) dealing with just one B at C∗ and splitting the resulting
bundle at point P∗ . Either option puts them on the same indifference curve.
There being no advantage to choosing option 2 over option 1, the As will
trade with the two Bs at point C∗ . The result is that recontracting, which began
at point C, ends at point C∗ . Adding a trader to each side of the market shrinks
the contract curve by the amount CC∗ . The same logic of course applies to
point C at the other end of the contract curve. Recontracting initiated there
shrinks the contract curve inward as each of the As continually underbids the
other to attract the business of the Bs. In other words, the contract curve shrinks
at both ends.
Although two pairs of traders shrink the range of indeterminacy, they hardly
eliminate it. To reduce it further, Edgeworth adds a third pair. Doing so gives
room for two Bs to underbid the third for the patronage of the As. Dealing with
the two Bs, the three As each can reach a point P two-thirds the distance from
the origin to any point C on the contract curve. Final settlement occurs when
point C shrinks inward sufficiently to lie on the same A-indifference curve as
point P. The same reasoning holds for the other end of the contract curve,
which of course shrinks too.
Let the number of pairs of traders N grow without limit. Then point P,
which according to Edgeworth is (N − 1)/N times the distance from the origin
to point C, converges on that latter point. Expressed geometrically, final settlement in the large-numbers case occurs where an A-indifference curve is tangent
to a ray from the origin (see Figure 4d). The same holds true for an indifference
curve of the Bs. The result is that both indifference curves are tangent to the
same ray and thus to each other just as in the competitive equilibrium. Large
numbers shrink the contract curve to the point of competitive equilibrium.
Monopoly Pricing—An Exception to Edgeworth’s Rule?
Finally, Edgeworth considered a case that apparently violated his postulate that
final settlements lie on the contract curve. That case has two bargainers agreeing
on price but making no agreement on the quantities to be traded. An extreme
example confronts a representative competitive price-taker with a monopolistic
price-maker. The monopolist is of the simple, or non-price-discriminating,
variety. He sets a single price for all units exchanged and leaves the competitor
free to determine how much he (the competitor) wants to trade at that price
along his offer curve.
Let A be the representative competitor and B the monopolist. If B’s monopoly power is absolute, he will set the single price that puts him on his highest

T. M. Humphrey: Early History of the Box Diagram

47

attainable indifference curve given A’s offer curve (see Figure 5). That is, he
chooses the price that takes him to point Q, where his indifference curve just
touches A’s offer curve. Of course, if his monopoly power is less than absolute,
his fear of losing A’s patronage to potential rival traders may induce him to
charge the slightly lower price shown by the slope of ray 0q. In any case, the
result is that trade takes place at a point like Q (or q) on A’s offer curve rather
than on the contract curve. Here is an apparent exception to the rule that final
settlements tend to be efficient.
Edgeworth was quick to point out, however, that the exception stems from
the assumption that the parties contract over price alone. Were they to contract
over quantity as well, they both could move advantageously to the contract
curve. Thus Edgeworth questioned the validity of the assumption. To him,
rational behavior required that parties bargain over both price and quantity
dimensions of a deal, especially when it was to their mutual advantage to
do so.
In illustration, Edgeworth referred again to monopoly point Q reached
through a price-only contract. From that point, superior outcomes are possible
in the sense that both parties can move to higher indifference curves than
those crossing through Q. Edgeworth realized, however, that such improved
positions would never be attained by new price settings alone. For, given that
the monopolist is constrained by the competitor’s offer curve, any change in the
slope of the price line 0Q would make him (the monopolist) worse off than he
is at Q and for that reason would be resisted. But mutually beneficial positions
could be reached if the competitor somehow could be induced to leave his offer
curve. Such an inducement could take the form of a new contract specifying
quantity as well as price.
For example, monopolist B might dictate terms corresponding to point Z,
thus improving his own welfare. He would lower the price against himself in
exchange for a more-than-compensating rise in quantity traded. And he would
do so confident that A would gladly agree to the larger trade volume in return
for the guarantee of a lower price. In other words, A would concur with any
price-quantity package moving him to an indifference curve higher than the
one he would otherwise occupy at point Q. And if such a negotiated package
fell short of the contract curve, the parties could renegotiate other packages
until they finally arrived there.3

3 Tibor Scitovsky, in his classic 1942 article “A Reconsideration of the Theory of Tariffs,”
showed that the parties could reach point Z by an alternative route. Competitor A could bribe
monopolist B to act as a competitor operating on his own offer curve. The bribe, paid in A’s own
good, would result in a rightward shift of the endowment point and its attendant offer curves
by the amount of the payment. So shifted, the offer curves would intersect at point Z. The
monopolist would gain from the bribe and the price-taker would gain from the lower, competitive price. Edgeworth, however, said nothing of this scheme.

48

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Monopoly Outcomes

A’s Indifference
Curve
at the
Monopoly
Point

Contract
Curve

OA
B’s Offer
Curve (were he
not a Monopolist)

OB

q
Monopoly
Price
Line

Q
B’s
Monopoly
Point

A’s
Offer
Curve

0

Better
Point

Z

B’s
Monoply
Indifference
Curve

Good x

Monopolist B sets the price that puts him on his highest attainable indifference curve given the offer curve of the competitive price-taker A . The
monopolist goes to tangency point Q. T h e r e , h o w e v e r , A’s indifference curve
i s t a n g e n t t o t h e p r i c e r a y a n d n o t t o B’ s i n d i f f e r e n c e c u r v e . T h e r e s u l t i n g
intersecting indifference curves create a lens-shaped area of unexploited
mutually beneficial exchanges. If the parties could agree to let monopolist B
set both price and quantity, they could move to efficient point Z where both
are better off than at point Q.

+

In short, Edgeworth showed that agreements fixing both price and quantity
inevitably lead to the contract curve. By contrast, agreements limited to establishing price alone may, under certain circumstances, lead only to the offer
curve. But he insisted that rational agents have an incentive to choose the
former agreements over the latter. Thus all final settlements tend to be on the
contract curve.
Appraisal
Edgeworth’s contribution must be judged one of the greatest virtuoso performances in the history of economics. Going beyond the mere creation of the

T. M. Humphrey: Early History of the Box Diagram

49

box diagram itself, he invented its principal components, the indifference map
and the contract curve. True, he did not invent offer curves. But he did give
the earliest demonstration of their derivation from the underlying indifference
contours and price ray. Moreover, in showing that offer curves intersect at the
contract curve, he was the first to use them to demonstrate the efficiency of
competitive equilibrium.
Edgeworth’s work is remarkable in another respect. His five propositions
essentially point the way to all of modern economics. One finds in them both a
treatment of competitive equilibrium and its efficiency in exhausting the gains
from trade and a framing of the problems that arise when perfect competition
ceases to prevail. These problems arguably constitute the fundamental motivation for the development of game theory.
Indeed, Edgeworth himself contributed to this development by anticipating key game-theoretic ideas. He demonstrated that final allocations must lie
on the segment of the contract curve spanning the indifference curves going
through the endowment point. In so doing, he identified what game theorists
some seventy-five years later were to call the core of the economy. And, in
illustrating that the contract curve shrinks to a single point, he showed how the
core behaves as its agents increase in number. Finally, his recontracting theory
foreshadowed the game-theoretic notion that no coalition of traders can block
the emergence of competitive equilibrium. Mark Blaug (1986, p. 70) said it
all when he described Edgeworth’s theory of the core as “his most beautiful
contribution.”

2. VILFREDO PARETO
The next to present the box diagram was Vilfredo Pareto, who did so in his
1906 Manuale d’economie politica. Pareto’s work obviously owes much to
Edgeworth. Indeed, commentators including Maffeo Pantaleoni (1923, p. 584)
and John Creedy (1980, p. 272) have stressed that very point. But Pareto also
modified Edgeworth’s work in at least two key respects.
For one thing, he presented the box in its now-conventional form. That is,
he located the origins of the indifference maps in the southwest and northeast
corners, respectively, rather than in the other two corners as Edgeworth had
done (see Figure 6). The result was that the succession of indifference-curve
tangency points—Pareto did not draw the efficiency locus—sloped upward
from left to right rather than downward as in Edgeworth’s version.
Second and more important was Pareto’s interpretation of the welfare
implications of the box. Unlike Edgeworth, who believed that interpersonal
comparisons of utility make it possible in principle to identify a unique point
of maximum aggregate welfare on the contract curve, Pareto denied that such
comparisons could be made and indeed refused to make them.

50

Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 Pareto’s Diagram
Second
Trader’s Origin
0

1

B
2
E

3

4

4
3

A
2

D
1

0

First
Trader’s Origin

Good 1

Movements from point D to any point within the lens-shaped area
are unambiguously welfare-improving since both parties gain and
n e i t h e r l o s e . B u t m o v e m e n t s a c r o s s t a n g e n c y p o i n t s A, E , a n d B
are welfare-ambiguous and defy comparison since one party gains
while the other loses.
+

Pareto Optimality
Accordingly, he held that only outcomes involving gains for some and losses for
none are unambiguously welfare-improving just as outcomes involving gains
for none and losses for some are unambiguously welfare-decreasing. By contrast, outcomes involving gains for some and losses for others are ambiguous.
They cannot be judged in terms of quantitative utility comparisons. The inadmissibility of interpersonal comparisons of utility (or “ophelimity” as Pareto
termed the utility concept) foils their evaluation.
It follows that movements from points like D, where indifference curves
cross, to points like A, E, and B, where the curves are tangent, constitute

T. M. Humphrey: Early History of the Box Diagram

51

Pareto-superior moves. They put at least one party on a higher indifference
curve and none on a lower one. But movements across successive tangency
points like A, E, and B, involving as they do higher curves for one person and
lower curves for the other, defy comparison. An infinity of such Pareto-optimal
points exists, none of which can be judged superior to the others.
In short, there is no single point of maximum welfare, Edgeworth’s claim
to the contrary notwithstanding. All one can say is that points off the tangency
locus are economically inefficient since everyone could gain by moving to a
point at which no mutually advantageous reallocations are possible. Likewise,
points on the locus are economically efficient in the sense that no reallocation
could improve the position of both parties. Edgeworth’s notion of a unique
welfare optimum gave way to Pareto’s notion of an infinity of noncomparable
optima.

3. ARTHUR W. BOWLEY
After Pareto’s Manuale, fully eighteen years elapsed before the box diagram
made its next appearance in A.W. Bowley’s famous 1924 Mathematical
Groundwork of Economics. Inspired by Edgeworth and Pareto, Bowley generalized and extended their work in three ways. First, he replaced their assumption
that each hypothetical trader initially holds the entire stock of one good and
none of the other. He replaced it with the alternative assumption that each
trader initially holds some of both goods. The result was to fix the endowment
point in the interior of the box rather that at one of its corners (see Figure 7).
Bowley’s innovation is conventional practice today.
Bargaining Locus
Second, he supplemented Edgeworth’s analysis of bilateral monopoly with his
concept of the bargaining locus. In defining that locus, which consists of the
offer-curve segments Q1 QQ2 , Bowley argued as follows. If the two parties
contract over price alone, equilibrium may well be on the offer curves rather
than on the contract curve. The party possessing the superior bargaining power
will set the price and leave the other free to determine the trade volume at that
price along his offer curve. Accordingly, the outcome will be somewhere on
the price-taker’s offer curve.
Suppose B is the price-maker whose bargaining superiority is absolute. He
will set the price to reach point Q2 where his highest attainable indifference
curve just touches A’s offer curve. But if his bargaining superiority is somewhat weakened by the countervailing bargaining skills of A, he will be forced
to shade his price downward and occupy a position on A’s offer curve in the
direction of competitive point Q. These considerations trace out the lower Q2 Q
segment of the bargaining locus.

52

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 Bowley’s Version of the Box

B’s Origin

C

Contract
Curve

R
A’s Offer
Curve

Q

B’s Offer
Curve

Q
1

Q
2

T

C
Endowment
Point

0

Good x

A’s Origin

Bargaining between price-making, price-taking traders establishes the
curve Q1QQ 2 a s t h e l o c u s o f f i n a l o u t c o m e s . F i n a l s e t t l e m e n t o c c u r s o n
the upper or lower segment depending upon whether A o r B i s t h e
dominant bargainer. If both parties possess equal and offsetting
monopoly power, final settlement occurs at Q.

+

Similarly, if A is the price-maker, trade will occur at the point of intersection of the price ray he sets and B’s offer curve. Trader A will aim at reaching
point Q1, where the offer curve is tangent to his highest attainable indifference
curve. But if A’s bargaining power is less than absolute, he may be forced to
lower the price against himself and thus move to a point on B’s offer curve to
the right of point Q1 . These considerations establish the upper Q1 Q segment
of the bargaining locus.
The upshot is that if either one trader or the other sets the price, trade
occurs at some point on the combined upper and lower segments of the offer

T. M. Humphrey: Early History of the Box Diagram

53

curves between points Q1 and Q2 .4 With the single exception of point Q, where
equal and offsetting bargaining power yields the competitive equilibrium, all
these points are off the contract curve. Thus Bowley confirms Edgeworth’s
contention that when price-maker confronts price-taker over price alone the
outcome is rarely efficient.
Trading at Disequilibrium Prices
Finally, Bowley advanced an alternative to Edgeworth’s treatment of how the
economy converges to its core. As mentioned above, Edgeworth, in considering
such convergence, ruled out trading at disequilibrium prices. For him, contracts
become binding and exchanges occur only at final equilibrium prices corresponding to points on the contract curve. Disequilibrium contracts he treated as
tentative, provisional, non-binding, and subject to revision until the equilibrium
contract emerged.
By contrast, Bowley permitted exchanges to take place at disequilibrium
prices. He envisioned traders moving across a succession of intermediate positions in the lens-shaped area enclosed by indifference curves emanating from
the endowment point. From each such intermediate trading position, they would
move to a subsequent, Pareto-improving one changing the price as they went.
They would continue in this fashion until they reached the core. The resulting
path to equilibrium is described by a broken, or segmented, price line and final
settlement can occur anywhere on the section RT of the contract curve.
For all its apparent realism, however, Bowley’s analysis comes at a high
cost. It greatly complicates the diagram. Each disequilibrium trade means a new
allocation of goods such that the endowment point shifts continually. Since offer
curves emanate from such endowment points, a new set of offer curves has to be
drawn at each stage of the process. The result is to clutter the diagram unduly.
For this reason, Edgeworth’s simplification seems superior pedagogically to
Bowley’s treatment.

4. TIBOR SCITOVSKY
In the twenty-two years following the publication of Bowley’s Mathematical
Groundwork, the exchange box virtually disappeared from the literature. It
surfaced briefly in 1941 when Tibor Scitovsky employed it to expose a flaw
4

This result holds even when both bargainers, after agreeing on a noncompetitive price,
treat it as given and act as price-takers operating on their respective offer curves. In this special
case, the price ray will cut the respective offer curves at different points. One party, in other
words, will wish to trade a larger quantity at the bargained price than will the other. Here, the
smaller quantity will be the one actually traded. The outcome will be exactly the same as if one
party unilaterally set the price (see Scitovsky 1951, p. 418).

54

Federal Reserve Bank of Richmond Economic Quarterly

in compensation tests of increased efficiency. Nicholas Kaldor and John R.
Hicks had proposed such tests to circumvent Pareto’s prohibition banning the
evaluation of changes favoring some people while hurting others. Applied to
such situations, the compensation test was supposed to reveal whether a change
from one non-optimal state to another was, on balance, welfare-improving if
some gained and some lost. The change was said to pass the test if the gainers
could fully compensate the losers and still be better off.
But Scitovsky noted a paradox. The test might reveal both states to be superior to each other. Observe a change-induced reallocation from goods-bundle
A to bundle B (see Figure 8). Let points A and A have the same vertical
height with the same being true of points B and B . Then compensation can
be represented as a quantity of the horizontally measured good alone. Gainer
J (whose indifference map originates at the lower left) could fully compensate
loser I (whose indifference map originates in the upper right) by an amount B B
and still be better off. The hypothetical transfer would leave him occupying
a higher indifference curve than the one going through his initial position A.
Similarly, in a reverse transition from B to A, individual I could bribe individual
J by an amount AA and still be better off than at B. The test would reveal
allocation B as preferred to allocation A. Once at B, however, the same test
would reveal A as the superior allocation.
Double-Bribe Criterion
To avoid such contradictions, Scitovsky proposed a double test. Situation B is
preferred to situation A if the gainers from the change can profitably compensate
the losers, or bribe them to accept it, while the potential losers cannot profitably
bribe the gainers to oppose the change.
Scitovsky’s double-bribe criterion impressed economists far more than did
the box diagram he used to exposit it. For that reason, his paper served merely to
interrupt rather than to halt the diagram’s pre-World War II lapse into obscurity.
That lapse persisted for five more years.

5. WASSILY LEONTIEF
Then came Wassily Leontief’s 1946 Journal of Political Economy article on
“The Pure Theory of the Guaranteed Annual Wage Contract.” Employing
perhaps the most elaborate version of the exchange box to be found in the
scholarly literature of the time, Leontief summarized, consolidated, and clarified all earlier work. He spelled out such notions as the lens-shaped zone of
mutually advantageous trades, the contract curve, offer curves, the competitive
and simple monopoly (price-maker, price-taker) outcomes and their welfare
implications with a lucidity and elegance unmatched in earlier work. In so
doing, he reawakened economists to the power and subtlety of the diagram and
thus initiated its post-war revival.

T. M. Humphrey: Early History of the Box Diagram

55

Figure 8 Scitovsky’s Paradox
I’s Origin

J
2
J
1

I2

I1

A

A

B

B

J2
I

1

J
1
I2

J’s Origin

Good 1

Paradoxically, the Kaldor-Hicks compensation test may justify both a
move from situation A to situation B and a reverse move from B back
to A. I n t h e m o v e f r o m A t o B, a g e n t J c o u l d c o m p e n s a t e a g e n t I by
the amount BB ' and still be better off. He would still occupy a higher
indifference curve than at A. Contrariwise, in the reverse move from B
to A, agent I could compensate agent J by the amount AA ' a n d s t i l l b e
better off than at B.

+

Perfectly Discriminating Monopoly
Leontief’s main contribution, however, was to specify exactly how a dominant bargainer might extract for himself all the potential gains from trade. Let
that bargainer present his passive counterpart with an all-or-nothing, take-it-orleave-it option to trade the entire fixed bundle C at a fixed price equal to the
slope of ray 0C (see Figure 9). The passive party either accepts the option or
rejects it and remains at his endowment point. Since the option leaves him no
worse off than does the autarky outcome, he accepts it. The resulting settlement
is at one end of the core, namely at the extreme that yields the dominant party
all the gains from exchange.

56

Federal Reserve Bank of Richmond Economic Quarterly

Figure 9 Perfectly Discriminating Monopolist

0

Endowment
Point

B’s Origin

C

M
A

A2
M
B

A
1

E
C

B
1
B
2

A’s Origin

B’s
Offer
Curve

A’s
Offer
Curve

B
0

A
0

Good 1

T r a d e r B, a p e r f e c t l y d i s c r i m i n a t i n g m o n o p o l i s t , c a p t u r e s a l l t h e p o t e n t i a l
g a i n s f r o m t r a d e f o r h i m s e l f b y g o i n g t o p o i n t C o n h i s t r a d i n g p a r t n e r A ’s
autarky indifference curve A0 . H e p r e s e n t s A with an all-or-nothing, take-itor-leave-it option to trade the fixed bundle C at the fixed price denoted by
t h e s l o p e o f r a y 0 C. A l t e r n a t i v e l y , B a c h i e v e s t h e s a m e r e s u l t b y m o v i n g
d o w n A’s autarky indifference curve, charging the highest price he can get
f o r e a c h s u c c e s s i v e u n i t o f t r a d e u n t i l h e a r r i v e s a t p o i n t C. U n l i k e t h e
s i m p l e m o n o p o l y o u t c o m e MB , t h e d i s c r i m i n a t i n g m o n o p o l y o u t c o m e i s o n
the contract curve and therefore is efficient.

+

Leontief further noted that all-or-nothing option contracts are equivalent
to perfect price discrimination. With price discrimination, the dominant trader
moves along the autarky indifference curve of the passive trader. He does so
by charging the highest price he can get for each successive unit of trade—that
is, the highest price his partner is willing to pay rather than do without the
unit—until he (the dominant trader) reaches the core at a point most favorable
to himself. The result, in terms of the distribution of the gains from trade, is
clearly the same as that achieved by the take-it-or-leave-it option.
In stressing this point, Leontief also emphasized that price discrimination,
because it leads to the contract curve, is economically efficient. Like perfect

T. M. Humphrey: Early History of the Box Diagram

57

competition, it wastes no resources. In this respect, the discriminating monopoly
outcome is preferable to the simple monopoly one.
The significance of Leontief’s contribution was this. Edgeworth and Bowley had stated that final settlement might occur at either extreme of the core.
But they had failed to identify such outcomes with all-or-nothing options and
discriminatory pricing. Leontief did so and established once and for all the
exact price-quantity agreements that produce such outcomes.

6. OTHER POST-WAR CONTRIBUTIONS:
KENNETH ARROW AND PAUL SAMUELSON
Leontief’s rehabilitation of the exchange box contributed greatly to its popularity in the late 1940s and early 1950s. Extensions and generalizations followed
when Kenneth Arrow and Paul Samuelson found imaginative new uses for
the box.
Arrow, in his 1951 essay “An Extension of the Basic Theorems of Classical Welfare Economics,” did at least three things. First, he introduced modern
set-theoretic concepts into the box. He interpreted the relevant regions of indifference maps as convex consumption sets and price or budget lines as their
supporting hyperplanes. Doing so allowed him to replace local or first-order
optimality criteria—the familiar marginal conditions—with global criteria.
Second, he employed the foregoing concepts to establish the two fundamental theorems of welfare economics. Theorem one states that every competitive
equilibrium, because it occurs at a point where each agent maximizes his satisfaction given the level of satisfaction of the other, is a Pareto optimum. Theorem
two states that every Pareto optimum, because it can be supported by a price
vector that equates supply and demand, is a competitive equilibrium. Arrow
demonstrated that both theorems hold for the standard case where indifferencecurve tangencies occur in the interior of the box.
Third, he analyzed boundary optima in which interior tangencies give way
to corner solutions on the edges of the box. His analysis yielded a positive
and a negative result. The positive result was that competitive equilibria retain
their optimality properties even when they occur on the borders of the box.
His negative result was that, without extra assumptions, there may be Pareto
optimal points on the boundaries that cannot possibly be equilibrium allocations
(see Figure 10).
Consider point X. There agent A’s downward-sloping indifference curve
meets the corresponding curve of agent B at its peak. Clearly this is a Paretoefficient allocation since each agent is on his highest attainable indifference
curve given the curve of the other. Nevertheless, this optimum cannot sustain
an equilibrium. For given the flatness of B’s curve at its peak, the tangent
price vector that separates the two indifference curves at X is necessarily a

58

Federal Reserve Bank of Richmond Economic Quarterly

Figure 10 Arrow’s Exceptional Case
Good 1
A
2

0B
A
4

A
3

A1

B1

0A

B2

X
B
3

Pareto-efficient solution X contradicts the notion that optimality guarantees a
competitive equilibrium. For the horizontal, tangent price line separating the
indifference curves at X induces agent B to maximize his satisfaction by
remaining at that point. Contrariwise, it induces agent A to maximize his
satisfaction by moving as far to the right as possible. The upshot is that A
and B seek incompatible allocations and the market fails to clear.

+

horizontal line coinciding with the lower edge of the box. Its slope implies
a zero relative price that induces the agents to register incompatible claims.
Given the zero price, B maximizes his utility by remaining at point X. By
contrast, A maximizes his utility by moving rightward as far as possible along
the price line, reaching ever-higher indifference curves as he goes.
The upshot is that A and B seek inconsistent allocations at the prices implied by corner-solution X and so the market fails to clear. Students refer to
this curiosum as Arrow’s Exceptional Case. It violates the theorem that every
Pareto optimum guarantees a competitive equilibrium.5
5 The theorem holds, however, when both indifference curves possess negative slopes at
boundary optima. In such cases, a downward-sloping, tangent price line can always be fitted
between the curves. Its slope represents the market-clearing price ratio that induces both parties
to go to the optimum point.

T. M. Humphrey: Early History of the Box Diagram

59

Samuelson, in his classic 1952 Economic Journal article on “The Transfer
Problem and Transport Costs,” used the exchange box to determine if a transfer payment made by Europe to America would worsen or improve Europe’s
terms of trade. According to him, the transfer shifts the endowment point to the
left and with it the offer curves and terms-of-trade ray that intersect at world
trade equilibrium (see Figure 11). But whether the new ray is less or more
steeply sloped than the old depends on the relative marginal propensities to consume Europe’s export good, clothing, in both countries. If the transfer reduces
Europe’s clothing consumption more than it expands America’s, the result is
an excess world supply of clothing whose relative price must therefore fall.
The terms of trade will turn against Europe. On the other hand, if the transferinduced fall in Europe’s demand for its exportable good, clothing, is less than
the rise in America’s demand for that same good, the resulting excess world
demand for clothing will bid up its relative price. Europe’s terms of trade
will improve. The slope of the terms-of-trade ray can become either flatter or
steeper. It all depends on the relative propensities to consume.
These extensions, however, brought the evolution of the exchange box to
a halt. For the combined contributions of Leontief, Arrow, and Samuelson had
virtually exhausted the analytical potential of the diagram and left it with little
new to do. True, it maintained its popularity in the textbooks. But it was clear
to all that the exchange box had seen its heyday. By the mid-1950s, its main
use was to illustrate established ideas rather than to generate new ones.6 Not
so the alternative production variant, however. Economists were increasingly
finding new applications for that version of the box.

7. ABBA LERNER
Already, in December 1933, Abba Lerner had drawn perhaps the earliest version
of the production box. He presented it in a term paper on factor-price equalization which he wrote for Lionel Robbins’s seminar at the London School of
Economics.
Lerner’s diagram superimposes isoquant, or production indifference, maps
of two industries fully employing two factor inputs whose fixed quantities
determine the dimensions of the box (see Figure 12). Each isoquant shows alternative factor combinations capable of producing a given level of output. Any
point in the box represents a particular allocation of the two factors between
6

This situation, however, proved to be temporary. Unforeseen at the time was the post-1970
resurrection of the box to depict Kenneth Arrow’s notion of insurance as trade in state-contingent
commodities (see Duffie and Sonnenschein [1989], pp. 584–86, and Niehans [1990], pp. 493–95).
By exchanging such commodities, agents could in principle profitably insure themselves against
the risks of unfavorable states of the world. In so doing, they could reach the contract curve
where the allocation of risk-bearing is optimal.

60

Federal Reserve Bank of Richmond Economic Quarterly

Figure 11 The Transfer Problem

American Clothing

America’s
Origin 0
A

C

B
B

C
0

E

Europe’s
Origin

A

A

European Clothing

Before transfer, Europe and America trade along the terms-of-trade
v e c t o r AB l i n k i n g t h e a u t a r k y e n d o w m e n t p o i n t A t o t h e f r e e - t r a d e
e q u i l i b r i u m p o i n t B. W h e n E u r o p e p a y s t o A m e r i c a a t r i b u t e o f AA '
units of cloth, the endowment point shifts horizontally leftward from A
to A' . T h e c o r r e s p o n d i n g n e w t r a d e e q u i l i b r i u m p o i n t i s B'. Whether
t h e p o s t - t r a n s f e r t e r m s - o f - t r a d e v e c t o r A' B' i s s t e e p e r o r f l a t t e r t h a n
the old one depends on the countries’ marginal propensities to spend
the proceeds of the tribute on cloth versus food. Either result is
possible. Thus a transfer can cause the paying country's terms of
trade to deteriorate, improve, or remain unchanged.

+ the production of the two goods. Isoquants going through such points show

quantities of both goods produced with this factor allocation.
Regarding factor allocations off the locus of isoquant tangency points,
Lerner notes that they are technologically inefficient. They squander scarce
resources. They leave room for at least one industry, via mere reallocation of
existing inputs, to increase its output with no loss of output of the other. Thus,
starting from point Z, one can, by moving along isoquant B until it reaches
tangency with isoquant A, increase the output of good A with no decrease in
the output of good B. No such feat is possible on the efficiency locus itself,
however. There, one industry’s expansion spells the other’s contraction. There,
factor allocations are technologically efficient in the sense that they maximize
the output of one good given the output of the other.
Efficiency, then, requires producers to operate on the contract curve. And,
according to Lerner, competition and factor mobility together ensure that they

T. M. Humphrey: Early History of the Box Diagram

61

Figure 12 Lerner’s Production Box Diagram

A

B

0B

Z

P

A

0A

Factor 1

B

A t i n e f f i c i e n t p o i n t Z , g o o d A’ s o u t p u t c a n b e i n c r e a s e d w i t h n o l o s s i n
g o o d B’ s o u t p u t . P r o d u c e r s s i m p l y r e a l l o c a t e f a c t o r i n p u t s t o Aproduction so as to move along the given B isoquant cutting successively
higher A isoquants in the process. At efficient point P on the contract
curve, however, the output of one good can be increased only by
decreasing the output of the other.

+

will do so. Competition forces producers to hire both factors until the ratio
of their marginal products, represented by the slopes of isoquants, equals the
ratio of their prices, represented by the slope of a relative factor-price line. And
factor mobility dictates that resource prices, and so their ratio, are the same
in both industries. Consequently, both industries operate at a point where their
isoquants are tangent to a common factor-price-ratio line and thus are tangent
to each other. Such points lie on the contract curve.

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Federal Reserve Bank of Richmond Economic Quarterly

Delayed Publication
Unfortunately, economists had to wait for nineteen years to see Lerner’s pioneering diagram. Tibor Scitovsky, in his essay on “Lerner’s Contributions to
Economics,” tells why. In 1948 and 1949, Paul Samuelson published his celebrated proof of the factor-price-equalization theorem. Robbins, upon reading
Samuelson’s papers, recalled Lerner’s 1933 term paper on the same subject.
Robbins still had a copy of the paper in his files. Upon his urging, Lerner published the manuscript without alteration as the 1952 Economica piece “Factor
Prices and International Trade.”
As to why Lerner neglected to publish the paper in 1934, Tibor Scitovsky
recounted a story he heard in 1935 when he was one of Lerner’s students.
Evidently, Lerner had given his only corrected copy to another student to be
typewritten for submission to a scholarly journal. But the student lost the paper
on a London bus and was unable to retrieve it. Lerner, who was busy writing
other papers at the time, could not find the time to reproduce the lost manuscript.
The resulting delay made Lerner’s pathbreaking work and its innovative diagram seem less-than-novel when they finally appeared. In any case, it was not
from Lerner but from Wolfgang Stolper and Paul Samuelson that the economics
profession first learned of the production box.

8. WOLFGANG STOLPER AND PAUL SAMUELSON
Wolfgang Stolper and Paul Samuelson published the first production box diagram to appear in print. It features prominently in their 1941 Review of
Economic Studies article on “Protection and Real Wages.” Of the two authors,
Stolper (1994, p. 339) credits Samuelson with the idea of using the box. In any
case, they applied it to derive their famous theorem according to which free
trade benefits the relatively plentiful factor and hurts the relatively scarce one
while protective tariffs do the opposite.
Stolper-Samuelson Theorem
The Stolper-Samuelson theorem rests on two propositions. First, compared with
autarky, free trade raises the price of the relatively abundant factor and lowers
the price of the relatively scarce one. Conversely, trade restriction raises the
scarce factor’s price and lowers the plentiful factor’s. Second, it follows that a
tariff-induced restriction of trade may benefit labor in countries where labor is
the scarcer factor. In such countries, a tariff may raise real wages and increase
labor’s real income both absolutely and relatively as a percent of the national
income.
Stolper and Samuelson reached these conclusions via the following route.
Suppose in the absence of trade a country produces wheat and watches with a

T. M. Humphrey: Early History of the Box Diagram

63

fixed factor endowment consisting of much capital and little labor (see Figure
13). Measure capital on the horizontal axes of the box and labor on the vertical
ones. The box, being wider than it is tall, indicates a high ratio of capital to
labor and thus identifies capital as the relatively plentiful factor and labor as
the relatively scarce one.
Next assume that, at any given factor-price ratio, wheat production requires
a higher ratio of capital to labor than does watch production. Wheat, in other
words, is capital intensive and watches are labor intensive. The slopes of laborto-capital factor-proportion rays going through any point on the contract curve
show as much. Those rays are steeper for watches than for wheat. Moreover,
the contract curve lies everywhere below the diagonal of the box. Were factor
intensities the same in both industries, the contract curve would coincide with
the diagonal. And were factor-intensity reversals to occur, the contract curve
would cross the diagonal. Neither possibility is allowed. Both are ruled out by
assumption.
Initially, in the absence of trade, the country produces and consumes at
point M on the contract curve. Wheat, embodying relatively large amounts of
relatively cheap and plentiful capital, is the low-cost good. Conversely, watches,
embodying much scarce and hence relatively dear labor, constitute the highcost good. Given that the opposite conditions prevail in the rest of the world,
the result is that wheat is cheaper in terms of watches at home than abroad.
Free Trade Helps the Plentiful Factor
When trade opens up, foreigners will import the home country’s cheap wheat
and home residents will import foreigners’ cheap watches. The consequent
increased demand for the home country’s wheat and the decreased demand for
its watches bids up the domestic price of wheat relative to the price of watches.
The resulting price rise induces wheat producers to expand by hiring capital and
labor from watch producers so as to move to free-trade point N. But the contracting watch industry, being labor-intensive, releases relatively little capital
and relatively much labor compared to the ratio in which the capital-intensive
wheat industry wants to absorb those factors. The ensuing labor surplus and
capital shortage bids wages down and capital rentals up. The lower wages
and higher rentals in turn induce both industries to substitute cheaper labor
for dearer capital. The upshot is that the labor-to-capital ratio rises in both
industries. And it does so even as the overall economy-wide endowment ratio
shown by the slope of the diagonal stays unchanged. In terms of the diagram,
both factor-proportion rays through point N are steeper than those through
point M.
With less capital working with each unit of labor in both industries, the
marginal product of labor falls and the marginal product of capital rises. Under
competitive conditions, those marginal products constitute factor real rewards

64

Federal Reserve Bank of Richmond Economic Quarterly

Figure 13 Stolper-Samuelson Theorem

Watch
Origin

Capital

0

N

M

0
Wheat
Origin

Capital

Free trade moves the capital-rich economy from autarky point M to free-trade
p o i n t N. T h e d a s h e d r a y s s h o w t h a t t h e l a b o r - t o - c a p i t a l r a t i o r i s e s i n b o t h
industries. With more labor working with each unit of capital in each industry, the
marginal productivity of capital and hence its real return rises while the marginal
productivity of labor and hence its real wage falls. Free trade helps the plentiful
factor and hurts the scarce one. Conversely, a protectionist move from point N to
point M helps the scarce factor and hurts the plentiful one.

+

which factor mobility equalizes across industries. It therefore follows that real
wages fall and real rentals rise expressed in terms of either good. Indeed, the
flatter common slope of the isoquants at point N than at point M signifies as
much. Those slopes indicate the rise in capital’s and the fall in labor’s real
return. They show that free trade benefits the country’s abundant capital factor
and hurts its scarce labor one.
Protection Helps the Scarce Factor
Conversely, protection does the opposite. It raises the relative price and thus
stimulates the output of import-competing watches at the expense of wheat

T. M. Humphrey: Early History of the Box Diagram

65

production. In so doing, protection moves the domestic product-mix and its
associated interindustry factor allocation from free-trade point N toward
autarky point M. To induce the expanding watch industry to absorb factors
in the proportion released by the contracting wheat industry, rentals must fall
relative to wages. The consequent fall in capital’s relative price encourages
both sectors to adopt more capital-intensive techniques. The result is a rise
in the capital-to-labor ratio in both industries as shown by the flatter slope of
the rays going through M than through N. With more capital working with
each unit of labor in both sectors, the marginal product of labor rises and the
marginal product of capital falls. With factor real rewards equal to marginal
products, the real wage of scarce labor rises while the real rental of abundant
capital falls, as shown by the steeper common slope of the isoquants at M
than at N. In short, import tariffs raise real wages and lower real rentals when
the import-competing sector is more labor-intensive than the export sector.
Protection benefits the scarce factor and hurts the plentiful one.
Evaluation
The Stolper-Samuelson paper is a milestone in the history of the box diagram
and the evolution of trade theory. It crystallized certain components of the
emerging Heckscher-Ohlin theory of international trade into a two-good, twofactor general equilibrium model. It then condensed that model into a simple
box diagram capable of showing how commercial policy affects distributive
shares. In so doing, it demonstrated the box’s power in handling a large number of interrelated variables and thus established it as the standard tool of trade
theory. Once established, the box proved indispensable in the derivation of
such key trade propositions as the factor-price-equalization, Heckscher-Ohlin,
and Rybczynski theorems.
Most important, the box diagram, in Stolper’s and Samuelson’s hands,
taught that informal intuition on trade issues could be misleading. Before
Stolper and Samuelson, most economists believed instinctively that free trade
benefits all factor inputs. In demonstrating rigorously that such was not necessarily the case, Stolper and Samuelson made economists more cautious in
discussing the benefits of trade. Thereafter, economists would acknowledge
possible losses to the scarce factor in movements to free trade. But they would
insist, on the grounds that trade benefits the country as a whole, that the gains
of the abundant factor exceed the scarce factor’s losses. Citing Scitovsky, they
would argue that the abundant factor could in principle compensate the scarce
factor for its losses and still be better off whereas the scarce factor would be
unable to profitably bribe the abundant factor to oppose free trade.

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Federal Reserve Bank of Richmond Economic Quarterly

9. T. M. RYBCZYNSKI
Stolper and Samuelson had used the box to link trade- or tariff-induced changes
in commodity prices to changes in factor prices. They had shown how a product
price increase causes a more-than-proportional rise in one factor’s real reward
while lowering the reward of the other. By contrast, T. M. Rybczynski in 1955
used the box to link changes in factor endowments to changes in commodity
outputs. He showed that when one factor increases in quantity (product prices
held constant), it causes a more-than-proportional increase in the output of
one good and an absolute fall in the output of the other. Here was a startling
revelation. Before Rybczynski, most economists felt that an increase in the
endowment of one non-specific factor would lead to a rise in the output of all
goods.
Rybczynski’s demonstration goes as follows (see Figure 14). Let the country’s initial factor endowment be that indicated by the dimensions of box ABCD.
The economy initially produces at point P on the contract curve. The slope of
the factor-intensity ray emanating from the wheat origin A, being flatter than
its counterpart originating from the watch origin C, identifies wheat as the
capital-intensive good and watches as the labor-intensive one.
Rybczynski Theorem
Now assume that the economy’s capital endowment expands by the amount
BE while its labor endowment remains unchanged. The result is that the box
annexes the new rectangle BEFC. How does the capital accumulation and
the corresponding expansion of the box affect the output-mix of wheat and
watches? Rybczynski’s assumption of constant commodity prices provides the
answer. Such constancy holds for small open economies taking their prices as
given exogenously from the closed world economy.
Constant commodity prices imply constant factor prices. And with linear
homogeneous production functions, constant factor prices imply unchanged
factor proportions in both industries. Point Q in the new box satisfies that latter
criterion. Only at that point are the capital-to-labor ratios (as shown by the
slopes of the factor-intensity rays) the same as they were at point P in the old
box. Thus the new equilibrium factor allocation must be Q. This new allocation,
however, sees more labor and capital devoted to wheat production and less of
both to watch production. The result is that wheat production expands and
watch production contracts. Here is the famous Rybczynski theorem: Let one
factor increase while the other stays constant. Then output of the good intensive
in the increased factor will, at constant commodity prices, increase in absolute
amount. Conversely, output of the other decreases absolutely.
The reasoning is straightforward. The expanding factor must be absorbed
in producing the good using it intensively. To keep factor proportions fixed,
as implied by the assumption of constant commodity prices, the expanding

T. M. Humphrey: Early History of the Box Diagram

67

Figure 14 Rybczynski Theorem

D

Capital

C

Watches

F

Q
P

A

Wheat

B

Capital

E

At constant commodity prices, a rise in the country’s capital stock with no
growth in its labor force raises the output of capital-intensive wheat and lowers
the output of labor-intensive watches. Why? Because fixed commodity prices
imply fixed factor prices which imply unchanged factor proportions in both
industries. Thus as wheat output expands to absorb the extra capital, it requires
extra labor to keep factor proportions unchanged. The only source of this extra
labor is the watch industry, which therefore must contract. We go from point P
to point Q with the slopes of the factor proportion rays remaining unchanged
throughout.
+

industry must hire the non-increasing factor too. The only source of this factor is
the other industry, which therefore must contract. Once again, the box diagram
had rendered a seemingly counterintuitive proposition transparent.

10. KELVIN LANCASTER
The box diagrams of Lerner, Stolper-Samuelson, and Rybczynski referred to a
single country only. As such, they were hardly equipped to accommodate twocountry models of international trade. The emerging Heckscher-Ohlin model
was a prime example of such a model. True, the above-mentioned writers had
introduced some Heckscher-Ohlin components into their single-country diagrams. But the list of included components was incomplete. Exposition of the
full model required boxes referring to at least two countries.

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Federal Reserve Bank of Richmond Economic Quarterly

Credit for developing the two-country box in its Heckscher-Ohlin form
goes to Kelvin Lancaster.7 His diagram, as presented in his 1957 Economica
article on “The Heckscher-Ohlin Trade Model: A Geometric Treatment,” embodies the standard features of that two-by-two-by-two model. Two countries
produce two goods from two factor inputs. The countries are incompletely
specialized. They produce both goods before and after trade. One good is
always more capital-intensive than the other. Factor endowments differ across
countries. Full employment prevails as does perfect competition in product and
factor markets. Both countries share the same linear homogeneous production
technology exhibiting constant returns to scale. Such technology ensures that
factor marginal productivities are determined by factor-input ratios and not by
scale of output.
The diagram incorporating these assumptions superimposes production
boxes representing the countries’ different factor endowments (see Figure 15).
Capital is measured horizontally, labor vertically. The wide box ABCD identifies
country I as the relatively capital-abundant nation. Similarly, the tall box AEFG
specifies country II as the relatively labor-plentiful nation. Lancaster assumes
that wheat production is always capital-intensive and watch production laborintensive in both countries. The contract curves indicate as much. They lie
below the diagonals of the boxes. Thus as one moves along a contract curve
from left to right, the capital-to-labor ratio declines in response to a rising
rental-to-wage ratio. But, at any given factor- price ratio, the capital-to-labor
ratio is always higher in wheat than in watches. Were such not the case, the
contract curves would either coincide with the diagonal or cross it.

7

Even before Lancaster, Jan Tinbergen (1954, p. 137) had presented an alternative version
of the two-country box. But his diagram, unlike Lancaster’s, maps production possibility curves
into commodity space. Depicting global trade equilibrium, he fits an equilibrium world price
line between the production transformation curves and consumption indifference maps of the two
countries. Both countries produce at the common point of tangency of their respective transformation curves and the price line. Then they trade along that line, each exporting its comparative
advantage good and importing its comparative disadvantage one, until they reach the point of
maximum satisfaction on their highest attainable indifference curves. In this way, trade enables
both to consume beyond their transformation curves.

T. M. Humphrey: Early History of the Box Diagram

69

Figure 15 Heckscher-Ohlin Theorem and Factor-Price Equalization

Watches

G

F

Country ll

Watches

D

C

Country l

H

K

J

A Wheat

L
E

Capital

B

Free trade moves countries I a n d II from autarky production points L a n d H to
post-trade points K a n d J. Capital-rich country I produces more capitali n t e n s i v e w h e a t a n d f e w e r l a b o r - i n t e n s i v e w a t c h e s . L a b o r - r i c h c o u n t r y II d o e s
the opposite. The slopes of the isoquants are the same at both post-trade
points. This implies equal labor marginal productivities and equal capital
marginal productivities in both countries. Since trade equalizes product prices
worldwide and factor prices equal product prices times factor marginal
productivities, it follows that trade equalizes factor prices too.

+

Heckscher-Ohlin Theorem
Having constructed the diagram, Lancaster used it to demonstrate the celebrated Heckscher-Ohlin and factor-price-equalization theorems. These theorems, together with their companion Stolper-Samuelson and Rybczynski postulates, constitute the core propositions of Heckscher-Ohlin trade theory. The
Heckscher-Ohlin theorem predicts that each country will export the good intensive in its abundant factor and import the good intensive in its scarce factor.
And the factor-price-equalization theorem says that free trade in commodities
equalizes factor prices worldwide just as unrestricted factor mobility would do.
The box diagram clarifies the underlying logic.

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Federal Reserve Bank of Richmond Economic Quarterly

Initially, in the absence of trade, the countries operate in isolation at points
L and H on their respective contract curves. At those autarky points, factor
prices and factor combinations used to produce each good differ across the two
countries as do product prices. Wheat, the capital-intensive good, is cheapest
in terms of watches in capital-rich country I. Conversely, watches, the laborintensive good, are cheapest in terms of wheat in labor-abundant country II.
When trade opens up, country I produces more of its export good, wheat,
and fewer import-competing watches. The country moves along its contract
curve to the free-trade point K. There, I’s relative commodity prices, or terms
of trade, are the same as those abroad such that no incentive remains for further
expansion of trade. At point K the rays AK and CK, whose slopes represent
the factor proportions employed in I’s wheat and watch industries, respectively,
intersect as required by the full-employment assumption.
Lancaster proves that the corresponding free-trade point for country II is J.
The reason is simple. Free trade equalizes the ratio of commodity prices worldwide. In equilibrium, that ratio equals the marginal rate of factor substitution
which equals the ratio of factor prices. Relative factor prices in turn uniquely
determine factor input ratios in production functions exhibiting constant returns
to scale. With both countries facing the same relative factor prices and sharing
the same production functions, it follows that both must use the same factor
input ratios too. Geometrically, the capital-to-labor factor-proportion rays going through country II’s free-trade point must have the same slopes as those
intersecting at country I’s free-trade point. Such indeed is the case. Ray AJ is
identical to ray AK. And ray FJ is parallel to ray CK. So country II’s free-trade
point J corresponds to country I’s free-trade point K. Point J is the only point
on II’s contract curve cut by factor-intensity rays of the same slope as those
going through point K on I’s contract curve.
Having established the corresponding free-trade points, Lancaster required
one final step to complete his demonstration. He took advantage of the property that linear homogeneous production functions allow output to be measured
as the distance along any ray from the origin. He employed this property to
compare the post-trade product mixes in the two countries. Country I produces
more wheat (AK > AJ) and fewer watches (CK < FJ) than does country II.
Thus country I’s product mix is heavily weighted toward wheat and country
II’s toward watches. Here is Lancaster’s demonstration of the Heckscher-Ohlin
theorem: each country produces (and exports) relatively more of the good intensive in its abundant factor.
Factor-Price-Equalization Theorem
As for absolute factor-price equalization, Lancaster offered the following
demonstration. Observe the tangent isoquants at the free-trade equilibrium
points J and K. Constant-returns-to-scale considerations dictate that these

T. M. Humphrey: Early History of the Box Diagram

71

isoquants, lying as they do on identical or parallel factor-proportion rays,
possess the same slopes at one equilibrium point as they do at the other.
But these slopes represent the ratios of factor marginal productivities which,
as noted above, free trade equalizes across countries. Indeed, Lancaster shows
that a stronger condition holds. When the two countries share the same linear
production technology, free trade equalizes absolute as well as relative marginal productivities. Each factor’s individual marginal productivity is the same
in both nations.
Two additional steps complete the argument. The first cites the law-ofone-price notion that free trade renders the price of any traded commodity
everywhere the same. The second refers to the competitive equilibrium condition that the price of any factor equals its marginal productivity multiplied by
commodity price. Since trade equalizes commodity prices and marginal productivities worldwide, it equalizes their multiplicative product, factor prices, as
well.
Lancaster’s demonstration appeared at a time when other scholars were
contributing to production-box analysis. Complementing his work were Kurt
Savosnick’s 1958 derivation of production possibility curves within the confines
of the box and Ronald Jones’s 1956 use of the diagram to examine the effects of
factor-intensity reversals. These applications would have delighted Edgeworth.
Apparently there was no end to what his invention could accomplish.

11.

CONCLUSION: HOW THE BOX EVOLVED

The history of the box diagram reveals how analytical tools evolve in a complex interaction with their uses (see Koopmans [1957] 1991, pp. 169–71, for the
definitive statement of this thesis). In this interaction, tools play a double role
of servant and guide. As servants, they help solve problems motivating their
invention. As guides, they alert their users to other problems solvable with their
aid. Certainly Edgeworth regarded the box as servant when he invented it to
demonstrate gains from exchange and to resolve the puzzle of how increasing
numbers lead to the competitive equilibrium.
Once invented, however, the exchange box took on the status of guide. Its
very existence made economists aware of other phenomena potentially seeking
its application. In short order, it was employed to explain the rationale of
such things as Pareto optimality, simple and discriminatory monopoly pricing,
compensation tests, the transfer problem, and corner solutions. All were manifestations of the drive to generalize the exchange box and extend its range of
application.
This same drive produced the alternative production box. Here a simple
analogy sufficed. Economists saw how the exchange box depicted the allocation
of fixed stocks of goods between the utilities of two individuals. They quickly

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realized that an analogous version could depict the allocation of a fixed stock
of factor inputs between the production of two goods. Thus was born the production box, whose initial use was to devise rules for optimal factor allocation.
Once available, however, the production box spurred economists to find new
applications for it. Chief among these applications was the Heckscher-Ohlin
theory of international trade. Accordingly, the box was deployed to derive,
prove, or illustrate the key propositions of that theory.
The above experience contradicts Tjalling Koopmans’s ([1957] 1991, p.
175) contention that diagrams, though a powerful aid to intuition and exposition, are nevertheless no match for higher mathematics in rigorous economic
analysis. For the box diagram is surely an exception to that rule. In support of
his allegation, Koopmans cites (1) the unreliability of diagrams as guides to
reasoning, (2) their confining effect on the choice of problems studied, and (3)
their inability to handle problems of more than two dimensions.
While these charges may stand up in a general comparison of diagrammatic
and mathematical techniques, the box diagram itself pleads innocent to them.
Far from being unreliable, it proved to be a highly accurate tool in the hands
of economists ranging from Edgeworth to Samuelson. So accurate was it, in
fact, that successive users found little need to modify it substantially. Far from
being confining, it freed its users to attack long-unsolved problems such as
how free trade affects the absolute and relative income shares of factor inputs.
Its limited dimensionality likewise proved to be no handicap. On the contrary,
Lancaster noted that the diagram could show the interrelationships between no
less than twelve economic variables. Edgeworth likewise found the diagram’s
two-dimensionality no bar to analyzing what happens when unlimited pairs
of traders are introduced into the model. In short, the history shows that this
simple geometrical diagram, in terms of its ability to yield penetrating insights
into problems of economic theory, has been a powerful mathematical tool in
its own right.
Of course the box, like any diagram, cannot handle all problems. Far from
it. Nobody would deny, for example, the diagram’s insufficiency to represent
infinite-horizon models involving infinite-dimensional commodity space. Such
complex models are beyond the capacity of the box. Rather the box’s strength
lies in depicting simple general equilibrium models. As these models are extremely useful, so too is the diagram that embodies them.
In any case, it was the box diagram itself, more than any accompanying mathematics, that captured the attention of the economics profession. The
result was that the box became a fixture of trade and welfare theory and a
commonplace of textbooks. The survival of the concept testifies to its continued usefulness. Even today, if one wishes to understand the sources of and
gains from exchange as well as the optimality of competitive equilibrium and
the logic of efficient resource allocation, one can do no better than study the
diagram.

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73

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