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Two Perspectives on
Growth and Taxes
Peter N. Ireland
R
obert Solow’s (1956) neoclassical model reigns as the standard theory
of economic growth. The Solow model begins with the assumption
that capital accumulation is subject to diminishing marginal returns.
It attributes sustained growth in national income per capita to technological
progress that proceeds at a constant, exogenously given rate. Thus, it is an
exogenous growth model.
Recently, economists have developed alternatives to the Solow model that
build on Frank Knight’s (1944) earlier theory of economic growth. These
economists follow Knight by adopting an all-encompassing definition of capital
that accounts for improvements in land, human capital, and scientific knowledge as well as for physical capital. Again along with Knight, they argue that
under this broad definition, capital accumulation should be subject to constant,
rather than diminishing, marginal returns. In their models, sustained growth
occurs even in the absence of exogenous technological change. Hence, these
are endogenous growth models.
This article presents versions of both the Solow model of exogenous growth
and the Knightian model of endogenous growth. In doing so, it illustrates that
the differences between these two models are more than purely technical ones;
indeed, the differences are of great relevance to contemporary policy debate
in the United States. Specifically, Section 1 shows that in the Solow model,
a change in the rate of income taxation affects the level, but not the growth
rate, of per-capita output. Section 2 demonstrates that in the Knightian model,
in contrast, changes in tax rates do influence long-run growth.
The author is an economist at the Federal Reserve Bank of Richmond. He would like to
thank Mike Dotsey, Tom Humphrey, Tony Kuprianov, and especially Bob Hetzel for helpful
comments and suggestions. He would also like to thank Chris Otrok for assistance in obtaining
the tax rate data. The opinions expressed herein are his own and do not necessarily reflect
those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
Federal Reserve Bank of Richmond Economic Quarterly Volume 80/1 Winter 1994
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Federal Reserve Bank of Richmond Economic Quarterly
Since the two models have such very different policy implications, determining which more accurately describes growth in the U.S. economy remains
an important task. Thus, Section 3 concludes the article with a review of the
empirical work on income taxation and economic growth.
1.
THE SOLOW MODEL WITH TAXES
In the Solow model, time periods are indexed by t = 0, 1, 2, . . . . Like many
other contemporary macroeconomic models, the Solow model considers the
behavior of a single, infinitely lived representative agent. This representative
agent’s individual quantities translate into per-capita quantities when the results
are applied to understand actual economies.
The representative agent produces output Yt with capital Kt during each
period t according to a production function of the form
Yt = AKtα .
(1)
Since 0 < α < 1, equation (1) indicates the presence of diminishing marginal
returns to capital accumulation: successive incremental additions to the capital
stock yield progressively smaller increases in total output. Figure 1 illustrates
this property of equation (1). It shows that the marginal return on capital, equal
to R = αAKtα−1 , is a decreasing function of the capital stock Kt .
The representative agent saves amount St during period t in order to add
to his capital stock in period t + 1. The Solow model assumes that saving St is
governed by
St = S(R − R∗ ),
(2)
where S is an increasing function of the marginal return on capital R. Thus,
the representative agent saves more when the return on capital is higher. R∗
represents the rate of return on capital that is so low that the agent no longer
finds it worthwhile to save; when R = R∗ , St = S(0) = 0.
Figure 1 traces out the dynamics generated by the interaction between the
production function (1) and the saving function (2). With the initial capital
stock given by K0 , the marginal rate of return R0 exceeds R∗ , so that saving
is positive. The representative agent continues to save and accumulate capital
until the marginal return has fallen to R∗ . At this point, Kt = K ∗ , and saving
stops.
When the government imposes an income tax τ , the representative agent’s
after-tax income becomes (1 − τ )AKtα and his after-tax marginal return on
capital becomes Rτ = (1 − τ )αAKtα−1 . Thus, the income tax τ induces the
parallel downward shift in the marginal return schedule that is also shown in
Figure 1. Since the representative agent cares only about his after-tax return,
his saving now stops when Rτ reaches R∗ . Starting from the initial capital stock
K0 , capital accumulation continues only until Kt = K τ .
P. N. Ireland: Two Perspectives on Growth and Taxes
3
Thus, Figure 1 reveals how the income tax affects aggregate output in the
Solow model. Since it lowers the effective marginal return on capital, the tax
weakens the representative agent’s incentives to save. Lower saving translates
into a smaller capital stock. Consequently, the level of output ultimately attained
with the tax, A(K τ )α , is lower than the level of output A(K ∗ )α achieved without
the tax.
Figure 1 also shows that with or without the tax, the Solow model implies
that the marginal return on capital eventually falls to R∗ , so that capital accumulation and growth ultimately cease. Historically, many economists have used
this implication of the diminishing marginal returns assumption to argue that
the growth of the U.S. economy, or indeed any capitalist economy, cannot be
sustained. Alvin Hansen (1939), for example, interprets the Great Depression
of the 1930s as a symptom of a low rate of return on capital and warns that
the U.S. economy might stagnate permanently.
In light of the U.S. economy’s recovery from the Depression and its continuing expansion since then, however, Solow augments the production function
Figure 1 Taxes in the Solow Model
Marginal Return
R =
␣ AK t␣- 1
␣- 1
R = ( 1 - ) ␣ AK t
R0
R*
K0
+
K
Capital Stock
K*
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Federal Reserve Bank of Richmond Economic Quarterly
(1) so that his model can account for sustained growth. Specifically, Solow
assumes that the parameter A increases steadily over time at the rate µ. Output
at time t is then described by
Yt = At Ktα ,
(3)
At+1 = µAt .
(4)
where
When At increases, the representative agent produces more output with the
same capital stock. Thus, equations (3) and (4) capture the effects of constant
technological progress.
Figure 2 illustrates the effect of an increase in the parameter A. The government continues to impose the tax τ . At the end of time t, the capital stock Ktτ
has reached the level consistent with the minimum rate of return R∗ ; without
technological change, the economy would grow no further. When A increases
from At at time t to At+1 at time t + 1, however, the marginal return schedule
shifts upward from Rτt to Rτt+1 . The marginal return rises above R∗ and capital
Figure 2 Technological Change in the Solow Model
Marginal Return
␣- 1
R t +1 = ( 1 - ) ␣ A t +1 K t +1
␣- 1
Rt = (1- )␣ AtKt
R*
Kt
Capital Stock
+
Kt+1
P. N. Ireland: Two Perspectives on Growth and Taxes
5
τ
accumulation begins again. The capital stock increases to Kt+1
. Thus, by
constantly offsetting the effects of diminishing marginal returns, the kind of
continual technological progress described by equation (4) generates sustained
growth in the Solow model.
A key assumption behind equation (4) is that µ is completely exogenous.
This assumption makes the Solow model an exogenous growth model. Although
the tax τ creates adverse effects on incentives that lead to a lower level of output, it has no influence on the process of technological change that determines
the economy’s long-run rate of growth. Thus, taxes have level effects but not
growth effects in the Solow model.
The Solow model’s key implication that tax policies have level effects but
not growth effects can also be derived more rigorously with a mathematical
treatment of the model. As above, the representative agent produces output
according to the production function with exogenous technological change described by equations (3) and (4). The government levies the flat-rate income
tax τ .
The government uses its tax revenue to provide the representative agent
with a lump-sum transfer of Gt units of output at each date t. The distinction
between the flat-rate tax and the lump-sum transfer must be emphasized. The
flat-rate tax reduces the agent’s effective return on capital and hence weakens
his incentives to save. The agent receives the lump-sum transfer no matter how
much he saves; the payment Gt has no effect on incentives.
At each date t, the representative agent’s total income consists of his aftertax output (1 − τ )Yt and government transfer Gt . The agent divides this income
between consumption Ct and investment It ; he faces the budget constraint
(1 − τ )Yt + Gt = Ct + It .
(5)
From consuming Ct during period t, the representative agent derives utility
measured by ln(Ct ), where ln denotes the natural logarithm. His lifetime utility
is then
∞
β t ln(Ct ),
(6)
t=0
where 0 < β < 1 is a factor that discounts utility in future periods relative
to utility in the current period. By investing It at time t, the agent adds to his
capital stock at time t + 1, so that
Kt+1 = (1 − δ)Kt + It ,
(7)
where δ is capital’s depreciation rate.
The representative agent maximizes the utility function (6) subject to the
constraints (3)–(5) and (7). Cass (1965) demonstrates that the solution to this
maximization problem dictates that consumption and capital eventually grow
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Federal Reserve Bank of Richmond Economic Quarterly
at the same rate γ and the consumption-capital ratio converges to the constant
ξ. Formally,
limt→∞ Ct+1 /Ct = limt→∞ Kt+1 /Kt = γ
(8)
limt→∞ Ct /Kt = ξ.
(9)
and
Like the representative agent, the government faces a budget constraint
that requires that its receipts τ Yt equal its expenditures Gt in every period t :
τ Yt = Gt .
(10)
Together, equations (3), (5), (7), and (10) imply that the economy’s aggregate
resource constraint is
At Ktα = Yt = Ct + It = Ct + Kt+1 − (1 − δ)Kt .
(11)
That is, output equals consumption plus investment.
In light of equations (8) and (9), equation (11) determines the long-run
growth rate that is predicted by the Solow model. Dividing (11) by Kt and
taking the limit of both sides yields
limt→∞ At /Kt1−α = ξ + γ − (1 − δ).
(12)
Since the right-hand side of equation (12) is constant, this condition implies
that At and Kt1−α must eventually grow at the same rate. Equation (4) indicates
that the growth rate of At is µ; equation (8) implies that the long-run growth
rate of Kt1−α is γ 1−α . Hence, it must be that µ = γ 1−α or, equivalently, that
γ = µ1/(1−α) .
(13)
Equations (3) and (4) then imply that the long-run growth rate of output is
limt→∞ Yt+1 /Yt = limt→∞ (At+1 /At )(Kt+1 /Kt )α = µµα/(1−α) = µ1/(1−α) . (14)
Equation (14) reveals again the Solow model’s key implication: the longrun growth rate is ultimately determined by the rate of technological progress
µ. The tax rate τ appears nowhere in equation (14); changes in the tax rate
have no effect on long-run growth.
A numerical example illustrates the effects of taxes on growth in more
detail. With α = 0.333, µ = 1.0133, β = 0.988, and δ = 0.1, King and Rebelo
(1990) show that if each period in the Solow model represents one year, then the
model economy’s long-run annual growth rate is 2 percent, about the average
growth rate of output per capita in the twentieth-century United States.
With these parameter values, Figure 3 plots the growth rates of two Solow
economies. Both have constant tax rates, but τ = 0.20 in the first and τ = 0.25
in the second. The economies both start with the same capital stock, set so that
P. N. Ireland: Two Perspectives on Growth and Taxes
7
Figure 3 Growth in the Solow and Knightian Models
2.1
20% Tax - Both Models
2.0
1.9
25% Tax - Solow Model
1.8
Percent
1.7
1.6
1.5
1.4
1.3
25% Tax - Knightian Model
1.2
1.1
0
10
20
30
40
Year
+
the economy with τ = 0.20 always grows at its long-run rate of 2 percent.
Thus, this numerical exercise isolates the effects of an increase in taxes from
20 to 25 percent.
The figure shows that the higher tax rate does slow the economy’s growth
in the short run. Initially, the growth rate is less than 1.54 percent under the
25 percent tax, compared to 2 percent under the 20 percent tax. Eventually,
however, the growth rates of the two economies converge, exactly as required
by equation (14). Thus, the results once again illustrate that changes in tax
policy have level effects but not growth effects in the Solow model.
2.
A KNIGHTIAN MODEL WITH TAXES
Maddison (1987, Table 1, p. 650) reports that the growth rate of the U.S. economy averaged 4.2 percent annually during 1870–1913, 2.8 percent annually
during 1913–1950, 3.7 percent annually during 1950–1973, and 2.3 percent
annually during 1973–1984. As the results of the previous section demonstrate,
the Solow model attributes all changes in an economy’s long-run growth rate
to changes in its rate of exogenous technological progress. Thus, according
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Federal Reserve Bank of Richmond Economic Quarterly
to Solow’s model, the variation in long-run growth documented by Maddison
must be due to fluctuations over time in µ. Similarly, among the 92 countries
for which complete data are recorded by Summers and Heston (1991, Table
III, pp. 356–58), the average growth rate of real per-capita GDP from 1980
through 1988 ranged from 7.8 percent for China to −8.2 percent for Trinidad
and Tobago; growth in the United States during this period averaged 2.3 percent. The Solow model also implies that this international variation in growth
rates must be due to cross-country differences in µ.
Solow’s model does not suggest how µ is ultimately determined, however.
Thus, the model essentially leaves unexplained the enormous variation in longrun growth rates that is observed both within countries over time and across
countries at any given point in time. This shortcoming of the Solow model
has led economists to search for alternative models that do identify sources of
variation in long-run growth. One new line of inquiry draws on a theory of
economic growth, due to Frank Knight (1944), that predates the formulation
of the Solow model.
Knight challenges the conventional view that it is useful to organize factors
of production into the three categories of land, labor, and physical capital. According to this conventional view, land is permanently fixed in supply. Labor
supply is slightly more variable, but is ultimately limited in the short run by
the size of the working population. Only the stock of physical capital can be
quickly and easily increased over time.
Knight points out that while land may be fixed in quantity, there is no limit
to improvements that can be made in its quality. As a matter of fact, landowners
continually develop and improve their property. The productive capacity of an
economy’s land thereby increases over time, much as the productive capacity
of its physical capital stock continually expands as a result of new investment.
Similarly, argues Knight, the quantity of labor may be fixed in the short run,
but the quality of the workforce is easily augmented. Just as an entrepreneur
invests today to obtain a more productive capital stock tomorrow, a worker
allocates time to education and training today so as to become more productive
tomorrow. In other words, an economy accumulates human capital as well as
physical capital.
In fact, Knight goes on to suggest that the process of technological change
is itself just the fruit of another kind of investment. Entrepreneurs and workers
search continually for new, more efficient methods of production. Their research
and development efforts require resources in the present, but yield a return in
the form of increased productivity in the future.
Thus Knight, like Solow, assigns an important role to technological progress in his theory of economic growth. But while Solow assumes that technological progress occurs at an exogenously given rate, Knight views the process
as endogenous: the same incentives that induce agents to accumulate physical
P. N. Ireland: Two Perspectives on Growth and Taxes
9
capital drive them to search for technological and scientific advances. In light
of this distinction, Knight’s is an endogenous growth model.
Knight replaces the traditional categories of land, labor, and physical capital
with an all-encompassing definition of capital that accounts for improvements
in the quality of land, the accumulation of human capital, and the endogenous
process of technological change as well as for physical capital. Knight then
argues that the various forms of capital he identifies are complements in production. There may be diminishing returns to accumulating one type of capital
alone, but there is no tendency for their marginal product to fall as all types
are increased together. Under Knight’s broad definition of capital, therefore,
production features constant, rather than diminishing, returns.
Recent papers by Jones and Manuelli (1990), Barro (1990), King and Rebelo (1990), and Rebelo (1991) incorporate Knight’s ideas into contemporary
models of economic growth. The simplest of these models differs from Solow’s
only in its specification of the production function. Here, the representative
agent produces output with capital in each period t according to the linear
production function
Yt = AKt .
(15)
The parameter A is once again constant in equation (15); there is no exogenous technological change. Instead, this model adopts Knight’s idea that
technological advances occur endogenously and should be accounted for in
a comprehensive definition of the capital stock Kt . Equation (15) also drops
the exponent α on capital, reflecting Knight’s assumption of no diminishing
returns.
The implications of Knight’s theory can be derived by considering the
properties of the production function (15) along with the saving function St =
S(R − R∗ ). Consider first the case where there are no taxes. Equation (15) then
implies that the marginal return to capital is the constant A; because of the
no-diminishing-returns assumption, this return does not depend on the size of
the capital stock Kt . Since the return on capital R = A never falls to the critical
level R∗ , capital accumulation continues forever.
Unlike the Solow model, therefore, this Knightian model accounts for
sustained growth in output per capita even in the absence of exogenous technological change. Recalling that Knight’s all-encompassing definition of capital
accounts for endogenous technological progress, the return R = A > R∗ provides the representative agent with an incentive to add continually to the stock
of technological knowledge. Economic growth continues as long as this incentive is preserved. The Knightian model therefore contradicts Hansen’s (1939)
view that an economy will stagnate without exogenous technological change.
The flat-rate income tax τ in the Knightian model shifts the marginal return
schedule down from R = A to Rτ = (1 −τ )A. Sustained growth still occurs if it
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Federal Reserve Bank of Richmond Economic Quarterly
is the case that Rτ > A. But since the tax permanently lowers the effective rate
of return, it also permanently weakens the agent’s incentive to accumulate all
types of capital. Hence, unlike the Solow model, this Knightian model predicts
that taxes affect the growth rate as well as the level of aggregate output. That
is, tax policies have both level and growth effects.
The policy implications of the Solow and Knightian models can be traced
back to the different ways in which these two models account for the process
of technological change. The Solow model depicts technological change as
a completely exogenous process; hence, tax rates do not influence long-run
growth. The Knightian model, on the other hand, treats technological change as
part of the endogenous process of capital accumulation. Just as higher tax rates
weaken the representative agent’s incentive to accumulate physical capital, they
induce him to slow down his search for more efficient methods of production.
Consequently, tax rates do help determine the rate of long-run growth.
In fact, the Knightian model predicts that any economic policy that changes
incentives for the accumulation of broadly defined capital will also influence
the rate of long-run growth. Since such policies differ widely over time and
across countries, this model identifies potential sources of variation in long-run
growth rates that the Solow model does not.
As suggested by the analysis above, the mathematical formulation of the
Knightian model differs from that of the Solow model only in terms of the production function. Now the representative agent maximizes the utility function
(6) subject to the budget constraint (5) and the capital accumulation equation
(7) as well as the linear production function (15). King and Rebelo (1990)
show that the solution to this maximization problem is such that consumption,
capital, and output always grow at the constant rate ω:
Ct+1 /Ct = Kt+1 /Kt = Yt+1 /Yt = ω,
(16)
ω = β[(1 − τ )A + (1 − δ)].
(17)
where
Equation (17) indicates that the economy’s growth rate depends negatively
on the tax rate τ , so that taxes have growth effects as well as level effects
in the Knightian model. Figure 3 illustrates these effects in more detail by
repeating its numerical exercise for the Knightian model. As before, β = 0.988
and δ = 0.1. With A = 0.165, output grows at the annual rate of 2 percent
under a constant 20 percent tax rate (King and Rebelo 1990). The figure uses
these parameter values to plot the growth rates of two Knightian economies,
one with τ = 0.20 and the other with τ = 0.25. Thus, as before, the exercise
illustrates the effects of a tax increase from 20 to 25 percent.
The figure shows that the growth rate decreases from 2 percent under the
20 percent tax rate to 1.19 percent under the 25 percent tax rate. Moreover,
P. N. Ireland: Two Perspectives on Growth and Taxes
11
as equations (16) and (17) imply, there is no tendency for the growth rates of
the two economies to converge; the growth rate falls permanently in response
to higher taxes. Once again, therefore, the results demonstrate that tax policies
have both level and growth effects in the Knightian model.
3.
EMPIRICAL STUDIES OF TAXATION AND GROWTH
Although Figure 3 illustrates that the Solow and Knightian models have different implications for the effects of taxation on long-run growth, this difference
may seem to be just a technical matter at first. After all, the tax increase in
the Solow model does not permanently decrease growth as in the Knightian
model, but it does result in slower growth for more than two decades. When
expressed in terms of the level rather than the growth rate of output, however,
the difference is enormous. In the Solow model, the increase in the tax rate
from 20 to 25 percent decreases the level of output by 3.17 percent over 40
years. In the Knightian model, the same tax increase reduces the level of output
by 27.5 percent over 40 years.
Policymakers in the United States have recently called for increases in marginal tax rates, which they argue will help to close the federal budget deficit
without significant losses in output. Others disagree, claiming that higher taxes
inevitably lead to slower growth. As the results of the previous sections show,
competing economic theories lend support to both sides in this debate. On the
one hand, the Solow model describes an environment in which tax rates do not
affect long-run growth; on the other, the Knightian model confirms the view
that higher taxes do hinder growth.
Thus, the next step in applying the theories to understand the U.S. economy
is to determine empirically whether or not changes in tax rates actually translate
into changes in long-run growth. If taxes do not influence long-run growth, then
the Solow model and its policy implications should be taken seriously. If taxes
do help determine long-run growth, however, then the Knightian model and its
implications are to be preferred.
Figure 4 plots the growth rate of real per-capita GDP (taken from the
Economic Report of the President 1993) along with Barro and Sahasakul’s
(1986) tax rate series (updated to run through 1989) for the United States. The
graph suggests that there has been a negative relationship between growth and
taxes over the postwar period. In fact, a simple regression of the growth rate
on the tax rate yields a negative coefficient that is statistically significant at
the 10 percent level. Cebula and Scott (1992) use quarterly series from 1957
through 1984 to regress growth in real per-capita GDP on various measures of
fiscal policy, including the top personal income tax rate. They also find that
changes in taxes have a negative and statistically significant effect on growth.
A problem with using these results to discriminate between the Solow and
Knightian growth models arises because even in the Solow model, changes
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Growth and Tax Rates in the U.S. Economy
40
35
Tax Rate
30
Percent
25
20
15
10
Growth Rate
5
0
-5
1959
1963
1967
1971
1975
1979
1983
1987
Year
+
in tax rates affect growth in the short run. As Figure 3 reveals, it is only in
the long run that taxes influence growth in the Knightian model but not in
the Solow model. Thus, the negative short-run correlation between taxes and
growth that appears in Figure 4 is consistent with the implications of both
models. Likewise, since Cebula and Scott do not distinguish between short-run
and long-run changes in growth, their results cannot be interpreted as decisive
evidence against the Solow model either.
Kocherlakota and Yi (1993) recognize the problem of distinguishing between short-run and long-run changes in growth and sidestep this problem
by taking a slightly different approach to test the Solow model against the
Knightian theory. They note that in addition to having distinct implications for
the effects of taxes on the growth rate of output, the two models have different
predictions for the effects of taxes on the level of output. Specifically, the
Solow model predicts that temporary changes in tax rates have only temporary
effects on the level of output. The Knightian model, in contrast, predicts that
temporary changes in taxes permanently affect the level of output.
P. N. Ireland: Two Perspectives on Growth and Taxes
13
Kocherlakota and Yi assume that all changes in U.S. tax rates, 1917–1983,
are temporary ones, and they use a statistical model that distinguishes between
temporary and permanent changes in the level of real GNP. Their estimate
indicates that temporary increases in tax rates have translated into permanent
decreases in the level of output; this result supports the Knightian model. On
the other hand, the estimate is not statistically significant, which suggests that
the Solow model may be more realistic. Overall, Kocherlakota and Yi’s results
may simply indicate that even with 65 years of data and with the most powerful
statistical techniques, it is very difficult to extract much information about the
determinants of long-run growth from the U.S. time series.
Other researchers circumvent the problem of distinguishing between shortrun and long-run changes in growth by using international cross-sectional data
rather than time series data. With cross-sectional data, growth rates within
each country can be averaged over extended periods of time in order to smooth
out short-run fluctuations and thereby identify long-run trends. In addition, by
drawing on the experiences of many different countries, cross-sectional data
bring more information to bear on the question of whether tax rates affect
long-run growth. On the other hand, compared to time series studies, those that
use cross-sectional data must make the additional assumption that the same
mechanisms through which taxes influence aggregate activity in the United
States operate in the other countries as well, so that conclusions that apply
internationally also hold for the United States.
Existing cross-sectional studies differ in that some use the average tax
rate, the ratio of total tax receipts to national income, while others use the
marginal tax rate, the additional taxes paid when income rises incrementally,
to estimate the effects of taxes on growth. In both of the theoretical models
presented above, the simple flat-rate tax is such that the average and marginal
tax rates coincide. In reality, however, tax rates differ with the source and level
of income, so that average and marginal tax rates diverge. Since economic
decisions depend on the marginal tax rate, this measure is more appropriate
for investigating the effects of taxes on growth. Data on marginal tax rates are
often unavailable, however; average tax rates must then serve as a proxy.
Marsden (1983) takes data from 20 countries, 1970–1979. He organizes
these countries into ten pairs; each pair consists of countries with similar levels
of per-capita income but different average tax rates. In each pair, he finds that
the country with the lower tax rate has a higher rate of real GDP growth. As
a matter of fact, all of the ten low-tax countries have higher growth rates than
any of the high-tax countries. This pattern also appears in Marsden’s regression
results, which show that average tax rates have a significantly negative effect
on growth across countries.
In Reynolds’ (1985) sample, industrial countries with high average tax
rates, including Sweden, Belgium, and the Netherlands, grew at an average
rate of 1.7 percent between 1976 and 1983. In contrast, those with low average
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Federal Reserve Bank of Richmond Economic Quarterly
tax rates, such as the United States, Portugal, and Japan, averaged 4.1 percent
growth. About the effects of marginal tax rates, Reynolds notes:
Supply-side tax theory would predict that economic performance in Ontario,
Canada, with a top tax rate of 51 percent, would be superior to that of Quebec,
with its 60 percent rate. It would predict that development in Puerto Rico,
with a top tax rate of 68 percent, would fall behind that of any U.S. state.
It would predict that Australia would outperform New Zealand, that Cyprus
would outperform Greece, that the state of New Jersey would grow faster than
New York, and so on. All of these predictions are correct.
(P. 557)
Among developing countries, Reynolds finds that those with the highest marginal tax rates have economies that contracted by 1.4 percent annually, 1979–
1983. Those with the lowest marginal tax rates, on the other hand, have
economies that grew by 4.9 percent annually.
Skinner (1987) uses cross-sectional data from 31 sub-Saharan African
countries, 1965–1982. His regression equation shows that the average tax rate
has a negative and statistically significant effect on growth.
Average tax rates and growth turn out to be positively correlated in
Rabushka’s (1987) sample of 49 developing economies, 1960–1982. Rabushka
interprets this finding as evidence that governments in more prosperous countries are able to levy more taxes than those in slower-growing nations, rather
than as evidence that higher taxes lead to faster growth. Unlike average tax
rates, he notes, marginal tax rates are negatively correlated with growth. The
country with the lowest marginal tax rate, Hong Kong, has one of the highest
growth rates in the sample, averaging 7 percent annually. A group of countries
with the highest marginal tax rates, in contrast, grew at the average annual rate
of only 1.9 percent.
Most recently, Martin and Fardmanesh (1990) and Engen and Skinner
(1992) estimate cross-sectional regression equations and find that average tax
rates are significantly and negatively related to growth in real income per capita.
Martin and Fardmanesh’s sample includes data from 76 countries, 1972–1981;
Engen and Skinner’s consists of data from 107 countries, 1970–1985.
Thus, many cross-sectional studies appear to support the hypothesis that
tax rates influence long-run growth and thereby point to the Knightian model
as the more accurate description of the U.S. economy. Only three of these
studies, however, use more than ten years’ worth of data. The others suffer
from the same problem as Cebula and Scott’s: by averaging data over a brief
time interval, they may not adequately distinguish between short-run and longrun variation in growth. In addition, there are still other cross-country studies
that lend support to the Solow model by indicating that changes in tax rates do
not affect the long-run growth rate of output.
Using a sample of 63 countries, 1970–1979, Koester and Kormendi (1989)
begin by noting that both average and marginal tax rates appear to be negatively
P. N. Ireland: Two Perspectives on Growth and Taxes
15
associated with growth. They go on to point out, however, that the level of GDP
is negatively related to growth, suggesting that smaller countries grow faster
than more developed ones. They note, in addition, that average tax rates are
positively correlated with the level of GDP; like Rabushka, they suggest that
this correlation indicates that more affluent countries have governments that
levy more taxes. Together, these last two correlations raise the possibility that
earlier studies may have mistakenly concluded that changes in tax rates have
long-run effects on growth, since the negative correlation between taxes and
growth may simply reflect the fact that for independent reasons, both tax rates
and growth rates are related to the level of income.
To allow for this possibility, Koester and Kormendi add the level of GDP
to their growth regressions; by holding the level of income constant, they focus
on the direct link between taxes and growth. While the coefficients on both
average and marginal tax rates are still negative in the expanded regressions,
neither is statistically significant. Garrison and Lee (1992) find that these results
continue to hold when the data set is extended through 1984. Thus, these studies
suggest that changes in tax rates do not have important growth effects.
Easterly and Rebelo (1993) calculate marginal tax rates for 32 countries
in 1984. They regress the growth rate of per-capita consumption from 1980
through 1988 on the marginal tax rate as well as the level of per-capita GDP
(following Koester and Kormendi) and measures of two other variables, educational attainment and political instability, that may explain cross-country
differences in growth rates. Like Koester and Kormendi, Easterly and Rebelo
obtain a negative but statistically insignificant coefficient on the marginal tax
rate. Thus, their results also appear to support the Solow model.
Easterly and Rebelo note, however, that they cannot reject the hypothesis
that all of their regressors are jointly insignificant. Like Kocherlakota and Yi’s
time series results, therefore, Easterly and Rebelo’s cross-country results may
simply indicate that there is insufficient information about the determinants of
growth in their sample.
Thus, a review of the literature reveals that no strong conclusions can yet be
reached as to which model, Solow’s exogenous growth model or the Knightian
endogenous growth model, is more appropriate for studying the effects of taxation on growth in the U.S. economy. A number of papers present evidence that
tax rates do affect long-run growth, but others find no significant relationship.
The literature points to several problems that need to be overcome in future empirical work. Time series studies must effectively discriminate between
short-run and long-run changes in growth, for it is only long-run changes
that distinguish the competing models. Cross-sectional studies must distinguish between average and marginal tax rates, since marginal tax rates most
directly affect economic decisions but are frequently difficult to measure. Crosssectional studies must also address the possibility, first raised by Koester and
Kormendi, that simple correlations may not reflect the direct effects of taxation
16
Federal Reserve Bank of Richmond Economic Quarterly
on growth. Finally, Easterly and Rebelo’s results suggest that efforts to collect
tax and growth rate data from a wider sample of countries than has been
previously considered might prove useful in sharpening the statistical results.
The massive federal budget deficit in the United States makes it likely that
policymakers will continue to advocate significant tax increases in the years
ahead. Since the Solow and Knightian models offer such different predictions
for the effects of these tax increases on output, the empirical relationship between taxation and growth remains an important unsettled issue for future
research.
REFERENCES
Barro, Robert J. “Government Spending in a Simple Model of Endogenous
Growth,” Journal of Political Economy, vol. 98 (October 1990, Part 2),
pp. S103–25.
, and Chaipat Sahasakul. “Average Marginal Tax Rates from Social
Security and the Individual Income Tax,” Journal of Business, vol. 59
(October 1986, Part 1), pp. 555–66.
Cass, David. “Optimum Growth in an Aggregative Model of Capital Accumulation,” Review of Economic Studies, vol. 32 (July 1965), pp. 233–40.
Cebula, Richard J., and Gerald E. Scott. “Fiscal Policies and Growth: An
Extension,” Rivista Internazionale di Scienze Economiche e Commerciali,
vol. 39 (January 1992), pp. 91–94.
Easterly, William, and Sergio Rebelo. “Marginal Income Tax Rates and
Economic Growth in Developing Countries,” European Economic Review,
vol. 37 (April 1993), pp. 409–17.
Economic Report of the President. Washington: United States Government
Printing Office, 1993.
Engen, Eric M., and Jonathan Skinner. “Fiscal Policy and Economic Growth,”
Working Paper 4223. Cambridge: National Bureau of Economic Research,
December 1992.
Garrison, Charles B., and Feng-Yao Lee. “Taxation, Aggregate Activity and
Economic Growth: Further Cross-Country Evidence on Some Supply-Side
Hypotheses,” Economic Inquiry, vol. 30 (January 1992), pp. 172–76.
Hansen, Alvin H. “Economic Progress and Declining Population Growth,”
American Economic Review, vol. 29 (March 1939), pp. 1–15.
Jones, Larry E., and Rodolfo Manuelli. “A Convex Model of Equilibrium
Growth: Theory and Policy Implications,” Journal of Political Economy,
vol. 98 (October 1990, Part 1), pp. 1008–38.
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King, Robert G., and Sergio Rebelo. “Public Policy and Economic Growth:
Developing Neoclassical Implications,” Journal of Political Economy, vol.
98 (October 1990, Part 2), pp. S126–50.
Knight, Frank H. “Diminishing Returns from Investment,” Journal of Political
Economy, vol. 52 (March 1944), pp. 26–47.
Kocherlakota, Narayana R., and Kei-Mu Yi. “A Simple Time Series Test of
Endogenous vs. Exogenous Growth Models: An Application to the United
States.” Manuscript. University of Iowa, Department of Economics, July
1993.
Koester, Reinhard B., and Roger C. Kormendi. “Taxation, Aggregate Activity
and Economic Growth: Cross-Country Evidence on Some Supply-Side
Hypotheses,” Economic Inquiry, vol. 27 (July 1989), pp. 367–86.
Maddison, Angus. “Growth and Slowdown in Advanced Capitalist Economies:
Techniques of Quantitative Assessment,” Journal of Economic Literature,
vol. 25 (June 1987), pp. 649–98.
Marsden, Keith. “Taxes and Growth,” Finance and Development, vol. 20
(September 1983), pp. 40–43.
Martin, Ricardo, and Mohsen Fardmanesh. “Fiscal Variables and Growth:
A Cross-Sectional Analysis,” Public Choice, vol. 64 (March 1990), pp.
239–51.
Rabushka, Alvin. “Taxation, Economic Growth, and Liberty,” Cato Journal,
vol. 7 (Spring/Summer 1987), pp. 121–48.
Rebelo, Sergio. “Long-Run Policy Analysis and Long-Run Growth,” Journal
of Political Economy, vol. 99 (June 1991), pp. 500–521.
Reynolds, Alan. “Some International Comparisons of Supply-Side Tax Policy,”
Cato Journal, vol. 5 (Fall 1985), pp. 543–69.
Skinner, Jonathan. “Taxation and Output Growth: Evidence from African Countries,” Working Paper 2335. Cambridge: National Bureau of Economic
Research, August 1987.
Solow, Robert M. “A Contribution to the Theory of Economic Growth,”
Quarterly Journal of Economics, vol. 70 (February 1956), pp. 65–94.
Summers, Robert, and Alan Heston. “The Penn World Table (Mark 5):
An Expanded Set of International Comparisons, 1950–1988,” Quarterly
Journal of Economics, vol. 106 (May 1991), pp. 327–68.
Firm Size, Finance,
and Investment
John A. Weinberg
T
here is something in our national consciousness that looks fondly upon
the small firm. This affection for small business is not entirely unwarranted. Small firms account for an important part of economic activity.1
A vast majority of all businesses in the United States are small; over 90 percent
have fewer than 20 employees. Small firms accounted for between 40 and 50
percent of GNP and over 60 percent of net job growth in the 1980s.2 Such
figures have drawn considerable attention in recent discussions of attempts to
ease securities and bank regulation or to promote other policies concerning the
financing of small firms.
This long-standing affection has at times generated significant public policy. Much of our antitrust policy was arguably generated more by a general
mistrust of bigness and desire to protect small business than by a concern for the
inefficiencies of monopoly pricing. Recently, much attention has been paid to
the plight of the small firm in raising capital in the face of recently strengthened
bank regulation. Indeed, it seems that a necessary part of the debate over any
proposed public policy action, from health care to tax policy, is the question
of how it will affect small firms.
A related working paper, “Learning, Firm Size and Investment,” was presented at the
University of Kentucky, and the author thanks the seminar participants, and Dan Black
in particular, for their comments. This article has also benefited from discussions with
Jeff Lacker and with Gordon Phillips on this and ongoing joint work. The views expressed
herein are the author’s and do not necessarily represent the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
1 Small firms can be defined as those with fewer than 500 employees or those with revenues or assets below some standard. The Small Business Administration uses the employment
definition, while the Securities and Exchange Commission uses a revenue standard ($15 million
annually) for exemption from some registration requirements.
2 These figures are drawn from the Small Business Administration (1992).
Federal Reserve Bank of Richmond Economic Quarterly Volume 80/1 Winter 1994
19
20
Federal Reserve Bank of Richmond Economic Quarterly
This article examines some aspects of the financial behavior of small firms
as compared to larger firms. A particular focus is the question of whether there
is a failure in financial markets that limits the activities of small firms. Both
theoretical and empirical analyses of financial behavior have suggested that such
a market failure might exist. The theoretical arguments center on problems of
asymmetric information; when lenders are less well informed than borrowers
about borrowers’ conditions and activities, credit markets may not clear in the
conventional fashion.3 Such a failure of markets to function efficiently might
suggest a role for government intervention to improve the allocation of financial
capital.
Recent evidence on the investment behavior of large and small firms suggests the possibility that informational problems weigh more heavily on small
firms. In particular, there is evidence that investment by smaller firms is more
sensitive to factors that, in a world of perfect capital markets, are not expected to
affect investment. The first section of this article surveys some of the evidence
on differences in financial behavior across firm sizes, including the evidence
on investment behavior.
The second section turns to theoretical interpretations of the evidence. The
first of these interpretations is the theory of market failures due to asymmetric
information, building on the idea that a firm’s insiders will often know more
than outsiders about the firm’s prospects. This asymmetry can increase the cost
of raising funds from outside investors. The asymmetric information perspective
has led some to conclude that the market typically fails to provide sufficient
financial capital.
While the likely effects of informational constraints may well vary with
firm size, the interpretation of differences in behavior in terms of asymmetric
information implicitly treats firm size as exogenous. The central point of this
article is that an attempt to provide a theoretical explanation of differences in
behavior across firm sizes should begin with a theory of firm size. The next
subsection describes such a theory, drawing from the industrial organization
literature; it is a life cycle theory in which firms, when they are young, learn
about their productive capabilities. This learning drives the differences in behavior between large and small firms. It turns out that this theory, without
informational market failures, is consistent with much of the evidence on firm
size and financial behavior. Hence, movement toward a theory that jointly
determines size and financial behavior weakens the case for a market failure
interpretation of the evidence.
Section 3 discusses some implications for public policy toward the financing of small firms. Under the theory of market failure due to adverse selection,
3 Greenwald
and Stiglitz (1986) present a general theory of market failures in the presence
of asymmetric information.
J. A. Weinberg: Firm Size, Finance, and Investment
21
investment undertaken by small firms is inefficiently low compared to a world
of perfect information. Some have argued that government intervention can
move financial markets in the direction of greater efficiency by giving favorable treatment to small firms. Under the alternative theory, there is no market
failure and no role for government intervention.
1.
FINANCIAL BEHAVIOR OF SMALL AND
LARGE FIRMS
An image of small business that has appeared in the popular media in recent
years is one of entrepreneurs starved for capital. According to this image,
recent banking legislation reduced the flow of bank loans to small firms. At the
same time, venture capital provision of equity financing fell from the peaks it
achieved in the middle of the last decade. Without access to external financing, small firms have been limited in their ability to grow and contribute to
employment.
Parts of the above image are no doubt accurate. Most measures of the flow
of external finance to small firms show a decline in recent years. Such numbers,
however, must be understood in the proper context. Has the recent experience
of small firms been qualitatively different from that of larger firms? By at least
some measures, the answer is no. For instance, commercial and industrial bank
loans to all firms, large and small, fell in 1991.4
To gain greater perspective on the recent experience of small and large
firms, one might ask whether there are any systematic differences in the financial behavior of firms of different sizes. One approach to such a question
is to examine the balance sheet characteristics of small firms. The Census
Bureau’s Quarterly Financial Report provides aggregate balance sheet data for
all manufacturing firms and for small manufacturing firms (firms with less than
$25 million in assets). These data give rise to a few observations. Most notably,
small manufacturing firms use more bank debt, as a percent of assets, than do
larger firms. From 1986 to the first quarter of 1993, small firms’ loans from
banks have averaged about 20 percent of total assets, while the corresponding
figure for all firms has been less than 10 percent. The difference in the reliance
on bank loans is particularly pronounced in long-term debt (with a maturity
of greater than one year). While smaller firms have fewer long-term liabilities
(about 40 percent of total liabilities compared to almost 60 percent for all firms),
more than half of all long-term debt of small firms is in the form of bank loans.
For all firms, bank loans constitute less than one-third of all long-term debt.
The observations above on the reliance of small firms on banks are consistent with findings from earlier periods. Andrews and Eiseman (1981) find
4 The
Federal Reserve Bulletin provides figures on commercial and industrial loans.
22
Federal Reserve Bank of Richmond Economic Quarterly
the same pattern in data from the 1970s and from 1958. The importance of
banks for small firms is also apparent in survey evidence, such as the Federal
Reserve’s National Survey of Small Business Finance. In an analysis of that
survey’s data, Elliehausen and Wolken (1990) uncover the additional result that
the smaller the firm, the greater the importance of local rather than distant banks.
This result suggests the importance to small firms of having a close relationship
with suppliers of funds. Correspondingly, small firms are less likely to raise
funds in public securities markets.
Since the set of firms that have not issued public securities tends to consist
of firms smaller than those in the set of public corporations, it should not be
surprising that those firms issuing securities for the first time are often small
relative to those already public. Most often, a firm’s first public issue is of common stock equity (an initial public offering, or IPO). While the size distribution
of firms undertaking IPOs varies from year to year, it typically includes many
small firms (assets less than $10 million). In 1984, virtually all IPOs were by
small firms, while in 1985 and 1986, small firms conducted about half of all
offerings.5
Even within the population of only public corporations, there are differences across firm size categories. In addition to the same tendencies cited
above, it is worth noting the covariation of firm size and dividend behavior
among public firms. Fazzari, Hubbard, and Petersen (1988), in their study of
the investment behavior of a panel of firms, divide their sample into three
classes based on dividend behavior: firms with a dividend to income ratio
persistently less than 0.1; those with a dividend to income ratio between 0.1
and 0.2; and those who persistently paid out at least 20 percent of their income
in dividends. The average size (measured by 1970 capital stock) of the highest
dividend-paying group was more than four times that of the middle group and
more than ten times that of the lowest group.
Another way in which smaller firms seem to differ systematically from
larger firms is in the relationships between financial variables and real economic
decisions. Most notably, there appear to be differences in the determinants of
investment. A useful benchmark for thinking about investment and its relation
to financial conditions is the irrelevance result of Modigliani and Miller (1958).
The “Modigliani-Miller Theorem” states that a firm’s financial policy (capital
structure, payment of dividends, etc.) has no effect on its real decisions, including investment. Technological and product market opportunities determine
investment and other real decisions. The firm’s financial choices, for instance,
of debt versus equity financing, should have no bearing on its real opportunities.
The Modigliani-Miller result applies to a frictionless world of perfect markets in which all market participants are always fully informed about firms’
5 Small
Business Administration (1992).
J. A. Weinberg: Firm Size, Finance, and Investment
23
opportunities. Empirically, the result seems to fail frequently. Financial characteristics are correlated with firm behavior, and such relationships are most
apparent for smaller firms.
One focus in the literature on financial characteristics and real behavior has
been on the relationship between cash flow and investment. In the frictionless
world of the Modigliani-Miller theorem, the two should be unrelated. A firm
with good investment opportunities should be able to fund its investment either
out of its own cash flow or by raising external funds. In a world of perfect
information, a firm with good opportunities will face no barrier in raising funds
from outside investors or financial institutions. Hence, unless the size and quality of the firm’s investment opportunity set is correlated with current cash flow
performance, there is no reason to expect a correlation between cash flow
and investment. Contrary to this theoretical perspective, there is considerable
evidence that for at least some firms, cash flow does help determine investment.
The evidence on investment and cash flow comes in two forms, corresponding to two standard approaches to the empirical study of investment behavior.
The first of these is based on the Tobin’s q theory of investment.6 Under this
theory, the ratio of a firm’s market value to the replacement cost of its assets
(Tobin’s q) serves as a measure of the firm’s investment opportunities. The
theory suggests a regression equation of the following nature:
Iit = β0i + β0t + β1 qit + β2 CFit + it ,
(1)
where Iit is firm i’s investment in fixed capital in time period t (as a fraction of
current fixed capital input), qit is the Tobin’s q ratio and CFit is cash flow (as
a fraction of current fixed capital input). The null hypothesis is that β2 = 0.
This approach is followed by Fazzari, Hubbard, and Petersen (1988). They
reject the null hypothesis, estimating positive values of β2 on a sample of
public corporations. In particular, when the sample is divided into subsamples
according to dividend behavior, cash flow is most strongly related to investment
for the subsample of firms paying the lowest dividends.
A central focus in interpreting the results on cash flow and investment is
the extent to which, conditional on other variables included in the analysis, cash
flow provides information on the firm’s investment opportunities. If cash flow
does provide such information, then the empirical findings are not necessarily
contrary to the Modigliani-Miller results. This issue is the concern of much
of the next section. With this concern in mind, there have been some recent
studies that have supplemented the evidence on cash flow and investment. One
such study is by Fazzari and Petersen (1993). They augment equation (1) to
estimate
Iit = β0i + β0t + β1 qit + β2 CFit + β3 ∆Wit + it ,
6 Tobin
(1969).
(2)
24
Federal Reserve Bank of Richmond Economic Quarterly
where ∆Wit is the change in firm i’s working capital in period t (as a fraction
of current fixed capital input).7 Like cash flow, working capital can serve as
an internal source of funds for fixed investment. For a sample of firms paying
low dividends, Fazzari and Petersen estimate a statistically significant negative
value for β3 . They interpret this result as further suggesting the importance
of internal finance to these firms; holding cash flow constant, a firm finances
increasing investment by drawing down its holdings of liquid assets. A similar
finding was obtained by Whited (1991) who examined the tendency of firms
to accumulate liquid financial assets before undertaking a program of fixed
investment.
The second approach used in studying investment behavior involves the
direct estimation of the “first-order condition” in a firm’s value-maximizing
choice of investment. A simplified version of such a condition for a typical
firm can be expressed as
Et [mpkt+1 ] = ρt ,
(3)
where Et denotes expectation conditional on information available at time t,
mpkt+1 is the marginal product of the capital input, and ρt is the “user cost of
capital,” which, in its simplest form, includes the rates of interest and capital
depreciation between times t and t + 1. Equation (2) determines the desired
amount of capital in the next period (period t + 1), and (net) investment is
simply the change in capital input from the current to the next period. Gilchrist
(1990) and Whited (1992) are among the authors using this approach, the socalled Euler equation approach. The findings tend to parallel that of Fazzari,
Hubbard, and Petersen. Equation (2) fits the data well for a sample of firms that
regularly pay dividends but not for firms with low, irregular, or no dividend
payment histories. While both approaches outlined above divide the samples
of firms according to dividend policy, it should be noted that this procedure
also tends to divide firms by size. As mentioned above, Fazzari, Hubbard, and
Petersen provide evidence on the correlation between size and dividend policy.
The evidence suggests distinct differences in financial behavior across firms
in different size classes. Smaller firms tend to make considerably less use of
public securities markets for raising external funds. Accordingly, when they do
raise external funds, they are more likely to borrow from a bank or other financial institution. Lastly, smaller firms seem to rely more on internally generated
funds to finance their investment activities.
7 Working
capital is current assets (primarily inventories, cash, and accounts receivable) less
current liabilities (short-term debt and accounts payable).
J. A. Weinberg: Firm Size, Finance, and Investment
2.
25
TWO THEORETICAL PERSPECTIVES ON
FIRM SIZE AND FINANCE
While the empirical studies reviewed above provide a picture of how small and
large firms differ, they give little insight into why they differ. Providing such
insight is the role of economic theory. This section provides two theoretical perspectives that might be used to interpret the empirical picture painted above.
The focus of the first is on imperfections in financial markets. The second
focuses on the causes of variations in firm sizes in a dynamic, competitive
economy.
Informational Market Failures
The apparent rejections of Modigliani-Miller results have led many economists
to seek out the market imperfections, or sources of market failure, that cause financial behavior to differ from the idealized model. One imperfection on which
much attention has been focused is the problem of incomplete or asymmetric
information. A transaction is made under incomplete information when one
party to the transaction has information that is relevant to the other party’s
decision. For instance, a seller may know details about the quality of the product or service being sold. It may be difficult for the buyer to perfectly discern
all these details on inspection, or even upon receiving the product or service.
In such a situation, a seller of a truly high-quality product may be unable to
receive a price which fully reflects the product’s quality. If high quality is
more costly to provide, then the inability to extract a higher price may serve to
drive high-quality providers out of the market. This problem, first analyzed in
some detail by Akerlof (1970), is known as the “lemons” problem or “adverse
selection” problem.
Another variety of asymmetric information problem is the “moral hazard”
problem. The term “moral hazard” refers to the fact that the observable performance of one party to a transaction often depends partly on that party’s
unobservable actions and partly on random events. Hence, the contract governing the transaction cannot directly prescribe the “morally hazardous” action.
Desired actions must be indirectly induced through the payment incentives in
the contract.
An extensive theoretical literature has examined the implications of privateinformation problems for financial markets. The focus, here, will be on that part
of the literature which finds that asymmetric information can raise a firm’s cost
of obtaining external finance. If the cost is raised enough, the firm may be forced
to rely entirely on internal funds to finance its investment projects. One line
of this research examines the implications of adverse selection for the ability
of firms to raise funds through the issuance of debt to a competitive market of
26
Federal Reserve Bank of Richmond Economic Quarterly
investors or institutions.8 The key insight in this line of work is that, as in the
lemons problem, a high-quality borrower (that is, one with a low probability
of default) may have difficulty credibly conveying credit quality information
to lenders. Hence, even a good borrower will have to pay an interest rate that
compensates for the probability that any borrower might be a bad borrower
(with a high probability of default). In some cases examined in this work, the
problem becomes so severe that some (high-quality) borrowers are unable to
obtain funds at any interest rate. Stiglitz and Weiss (1981), among others, have
argued that such a credit rationing result is to be expected in financial markets
subject to incomplete information.
There also has been work that has argued that moral hazard can impair a
firm’s access to external funds. A notable example is Gertler (1992). In such
models, outside investors are unable to directly monitor all of the resource
allocation decisions made inside the firm. An insider (manager) may have an
incentive to misallocate resources for personal benefit. This incentive is reduced
when the manager’s own resources are put at risk in the enterprise.
When viewing financial markets through the lens of asymmetric information theory, financial intermediaries often emerge as institutions that can
partially resolve the problems of adverse selection and moral hazard by spending resources on information production. A bank or other intermediary might,
for instance, invest resources in evaluating a borrower prior to lending, as in
Boyd and Prescott (1986). Alternatively, such an institution might engage in
costly monitoring of the borrower’s performance after a loan has been made,
as in Diamond (1984). This perspective is consistent with the popular view of
banks and other intermediaries as institutions that specialize in informationintensive financial arrangements.
When asymmetric information affects the availability or cost to a firm of
securing external funds, then the Modigliani-Miller results on the independence
of financial behavior and real investment may not hold. A simple example may
be useful.9 Consider a firm that initially has no assets, either in the form of fixed
capital or in the form of more liquid assets. The firm chooses its investment
in fixed capital, k (in nominal value), and funds its purchase in a competitive
credit market. If it is successful, the firm will produce output according to
a production function, f (k) (giving output in nominal value). Corresponding
to this production function is a downward-sloping marginal product curve,
as in Figure 1. If unsuccessful, the firm produces nothing and defaults on its
8 Lacker provides a thorough and critical review of this line of research in this issue of the
Economic Quarterly.
9 The analysis of this example is admittedly incomplete, for a number of reasons. The
purpose is not to give a complete model of financial markets but to give a sense of the directions
in which asymmetric information can move financial behavior away from the benchmark case of
full information.
J. A. Weinberg: Firm Size, Finance, and Investment
27
Figure 1 Demand for Capital by a High-Quality Firm
R⬘
R*
mpk
k⬘
k*
k
R* = /P H
R⬘= /[ P H +( 1 - )P L ]
+
loan. There are two possible types of firm. High-quality firms succeed with
probability PH and low-quality firms succeed with probability PL < PH .
Consider the problem facing a high-quality firm. If quality is known to
all participants in the credit market, then the high-quality firm can borrow at
an interest rate R per unit borrowed such that PH R = ρ, where ρ is the rate
of return available to lenders from an alternative risk-free investment. In this
case, the firm’s choice of k would be determined by equating the (expected)
marginal product of capital to its (expected) marginal cost, as in equation (2).
This choice is given by k∗ in Figure 1.
Suppose now that only a firm’s insiders know the firm’s true quality.
Lenders know only that some fraction, π, of all firms are high-quality. If the
financial market cannot discriminate and must lend to all on equal terms, then
the interest rate on loans, R, must be such that [πPH + (1 − π)PL ]R = ρ. Facing
such a rate, a high-quality firm chooses an amount of capital given in Figure
1 by k < k∗ .
28
Federal Reserve Bank of Richmond Economic Quarterly
The presence of low-quality borrowers who cannot be screened out might
be said to impose an externality on the high-quality borrowers. Note, however,
that this externality is only relevant to a high-quality borrower without internal resources. If, before making its investment decision, the firm received a
windfall of cash, it would make the higher investment k∗ . Hence, this simple
example suggests how, in the presence of asymmetric information, a firm’s
investment decision can be sensitive to random shocks to cash flow. It is also
worth noting that the example suggests conditions under which a firm might
find it worthwhile to utilize the type of costly information production provided
by a bank. If we think of this information production as providing a “stamp
of approval” or certification of true quality, then the value of obtaining such
certification depends on the premium resulting from asymmetric information.
This premium, the difference between R and R∗ in Figure 1, is decreasing in
π, the fraction of high-quality borrowers in the population. Hence, if the role
of financial intermediaries is to produce information that counteracts problems
of adverse selection, then the services of intermediaries will have greater value
the more severe the adverse selection problem faced by high-quality borrowers.
Under the asymmetric information view of financial markets, some firms
will undoubtedly be more subject to the problems of adverse selection than
others. Some firms will have a track record of past performance that will
make it difficult to hide flaws and overstate virtues. Others, particularly young
firms, will come to financial markets as relatively unknown entities. Hence,
if one looked at a cross section of firms, one might expect deviations from
the benchmark of frictionless finance to be inversely related to a firm’s age
and experience. The empirical evidence summarized above suggests an inverse
relationship between such deviations and firm size. Therefore, the results of
the asymmetric information approach will best conform to observed behavior
if firm size and age are correlated. It is probably not surprising that age and
size are, in fact, positively correlated in large cross sections of firms. One might
imagine, then, a life cycle theory of the firm: as firms grow, they acquire publicly observed experience that enables them to loosen the bounds of financial
constraints. Occasionally, as a result of changes in technology, preferences, or
personnel, a firm’s past experience becomes irrelevant for its future performance. At this stage, a firm either ceases to exist or returns to an earlier stage
in the life cycle.
Life cycle models like that suggested above have, in fact, been used in
analyzing the distribution of firm sizes in markets and economies. As will be
discussed in the next subsection, the examination of such a model reveals that
many of the empirical facts outlined above can be explained simply by the
life cycle features of the model, without the additional feature of asymmetric
information. This finding should prompt caution in considering the possible
public policy implications of analyses based on informational market failures.
J. A. Weinberg: Firm Size, Finance, and Investment
29
A Life Cycle Approach to Firm Size and Behavior
Analyzing differences in financial behavior among firms of different sizes is a
bit like reading a book from the middle onward. You find characters reacting
to a situation, but you do not know how they got into that situation. Similarly,
in understanding differences between large and small firms, it may be useful
to have a notion of what determines firm size. In other words, it may be useful
to have a theory of the size distribution of firms in a market or an economy.
Such a theory should be broadly consistent with empirical facts about size
distributions.
The industrial organization literature has established a number of facts
about size distributions. Simon and Bonini (1958) observed many of these
facts, and more recent studies have provided some confirmation and some
revision.10 The first such fact is that there are, indeed, persistent differences
in firm size within industries as well as across industries. Size distributions,
either at the industry or aggregate level, tend to be skewed, with relatively
small numbers of the largest firms and a large mass of firms in the smaller size
ranges. Earlier studies concluded that rates of growth were independent of firm
size, but more recent work, such as Evans (1987) and Hall (1987), has found
this to be true only among larger firms. Overall, there is a negative correlation
between size and growth. In addition, firm size is positively correlated with
firm age, as found, for instance, by Evans (1987) and Dunne, Roberts, and
Samuelson (1988). This last fact strongly suggests that life cycle effects may
be important for understanding differences between the average behavior of
small and large firms.
What the facts outlined above suggest is that there is a considerable amount
of heterogeneity among firms. A model of a competitive economy that recognizes these facts of industrial organization should incorporate some form of
heterogeneity into the fundamentals of the model economy. One such model
has been provided by Lucas (1978). In a simplified version of that model,
there is a generic technology available for using capital input to produce an
output. Productivity, however, also depends on the ability of the entrepreneur
or manager using the input.11 Hence, the manager-specific technology can be
represented by y = θf (k), where y is output, k is capital input, and θ is the
ability of the manager. Choice of inputs is like that represented in Figure 1, in
which marginal product of capital is set equal to ρ, the market cost of capital.
The curve mpk is higher the greater is the parameter θ. Accordingly, for any
market cost of capital ρ, firms managed by managers with higher θs will be
bigger than those with lower θs.
10 For
instance, see Evans (1987) and Hall (1987).
model also considers labor input and the division of the economy’s population
between workers and manager-entrepreneurs.
11 Lucas’
30
Federal Reserve Bank of Richmond Economic Quarterly
In the Lucas model, the underlying distribution of ability in the population
determines the size distribution of firms. As a static model, however, it cannot
directly address facts concerning the growth of firms. A related model, first
studied by Jovanovic (1982), adds a dynamic learning process to an environment similar to that of Lucas. A firm begins its lifetime uncertain of the value
of θ, its firm-specific productivity parameter. Output has a stochastic component, so that experience provides imperfect information about ability. As a firm
accumulates more experience over time, uncertainty about the parameter, θ,
declines.
In both the Lucas and Jovanovic models, there is an opportunity cost to
the manager of continuing to produce. This might be, for instance, the value
of working for the market wage or the value of starting a new productive
endeavor. In the static model of Lucas, the existence of such an opportunity
cost simply means that there is a “marginal” level of ability θ0 . Anyone with
θ > θ0 becomes a manager and hires inputs, while anyone with θ < θ0 pursues
the alternative activity. More precisely, if the value of not being a manager is
C (independent of ability), then the marginal ability level is determined by
θ0 f (k(θ0 )) − ρk(θ0 ) = C.
(4)
In equation (4), the notation k(θ) indicates that the optimally chosen level of
capital input is a function of managerial ability. The determination of θ0 is
depicted in Figure 2. In that figure, v(θ) is the return to being a manager
with ability θ. Figure 3 shows how the determination of θ0 serves to truncate
the distribution of abilities in the population. Hence, even if the underlying
distribution is symmetric, as in the figure, the distribution of ability among
those who operate firms will be skewed. This skewness carries over to the
distribution of sizes, because size rises with ability.
In the Jovanovic model, the marginal ability level would be determined
exactly as in equation (4) if managers were fully aware of their abilities from
the outset. With initial uncertainty and learning through experience, a manager
with expected ability less than θ0 may find it worthwhile to continue to produce
on the chance that, through favorable experience, he will learn that he is able
enough to remain a manager in the long run. In other words, a firm may be
willing to take an operating loss, because production has informational value.
A manager’s willingness to incur a short-term loss in exchange for information depends on two things: the current expected value of θ and the age of
the firm. The lower the manager’s expected ability, the greater the expected
operating loss from continuing to operate and the smaller the probability that
the next observation will be good enough to raise expected θ above θ0 . The
older the firm, the more experience it has accumulated. This experience serves
to reduce the remaining uncertainty about θ. Less uncertainty about θ, in turn,
implies a lower probability of experiencing output much greater than expected.
J. A. Weinberg: Firm Size, Finance, and Investment
31
Figure 2 The “Marginal” Firm
Return
v ( ) ⬅f ( k ( ) ) -k ( )
C
0
+
Figure 3 The Distribution of Abilities Is Truncated
+
0
32
Federal Reserve Bank of Richmond Economic Quarterly
Consequently, the probability of a substantial shift in expectation is reduced.
In the learning model, then, a manager will continue to produce as long as
expected ability is greater than some marginal value, θ0n , where n is the age of
the firm (the number of periods for which it has been operating). Hence, there
is a sequence of thresholds for firms to continue operating. This sequence has
two notable properties. First, θ0n < θ0 , indicating that firms will be willing to
take losses in the short run. Second, θ0n+1 > θ0n , stating that older firms will
be less willing to take such losses.
Since expected ability determines size, the learning model predicts a positive correlation between size and age; the further below θ0 , the smaller the
firm and the less experienced it is likely to be. A very stark version of this
model appears in Weinberg (1993). In that version, a firm starts with a prior
expectation of its productivity. This prior might come from the manager’s past
experiences in other activities or from pre-production research and development
work. Hence, there are a variety of prior expectations in the population. By
producing for a fixed amount of time (one period), the firm learns its true ability
with certainty. In this way, the population of firms can be separated into two
classes: young firms, who are in the process of learning, and mature firms, who
have already learned their types. If each firm, young or old, faces an exogenous probability of disappearing (due to exogenous shocks to its technology or
personnel), there will tend to be a steady-state mixture of young and mature
firms in the economy. Young firms will be smaller, on average, than mature
firms. They will also face a higher probability of exit, since they can exit either
because of exogenous shocks or because they learn that their ability is not great
enough to merit continued operation.
The simple, two-class version of the learning model makes it quite easy to
examine differences in investment behavior. The investment of mature firms is
very simple. Since they have learned their firm-specific abilities, their investment (acquisition of capital for the next production period) will not respond
to current output. In fact, their investment will, on average, merely replace
depreciation (unless there are other sources of firm growth). Young firms, on
the other hand, learn about their abilities from their current output. Hence,
conditional on initial size, better performance implies a higher realized ability
level, which, in turn, implies greater investment.
Notice that the relationship between investment and current performance
is very similar to the empirical relationship discussed in Section 1, where cash
flow was used as the measure of current performance. In the models discussed
in this section, there are no imperfections in the capital markets; firms face no
purely financial constraints. The authors of studies that found an effect of cash
flow on investment certainly recognize that cash flow could be serving as an
indicator of investment opportunities. What they overlooked, perhaps, was that
economic theory should give us a strong a priori reason to believe that, in such
regressions, cash flow is playing that role, for small firms in particular.
J. A. Weinberg: Firm Size, Finance, and Investment
33
Comparing the Theoretical Approaches
The two perspectives sketched above represent the two most common explanations of findings of a cash flow effect in investment behavior of small, growing
firms; an unexpected boost to cash flow might loosen the financial constraints
arising from asymmetric information, or it might provide a signal of enhanced
profitability and thereby shift investment demand. Notice that information plays
a central role in both of these stories. In one, problems arise from the inability
of some market participants to credibly convey private information to other
participants. In the other story, information accumulates over time, but in a
public way. While either one of these approaches can potentially explain the
relationship between investment and cash flow, how do they compare in addressing some of the other facts outlined in Section 1? This section examines
that question.
In the asymmetric information approach, cash flow affects investment, because firms subject to adverse selection pay a premium for external funds.
Some of the evidence seems to support this notion. The firms for whom the
cash flow effect is the greatest are firms that pay very little in dividends to
shareholders. For these firms, working capital, which consists of short-term,
liquid assets, can serve as an additional source of internal investment funds.
In a full information, Modigliani-Miller world, a firm would be indifferent
between the use of internal and external funds. If, as in the learning model,
current income served as a signal of profitable investment opportunities, then
paying the income out as dividends and raising investment funds externally
would be equivalent to using the income to fund investment internally. Hence,
the Modigliani-Miller framework makes no prediction about the choice between internal and external funds. Suppose that, in an otherwise frictionless
environment, there were a small transactions cost associated with raising external funds.12 Firms would then have sufficient reason to prefer internal funds.
That is, rather than paying dividends and raising funds externally as needed, a
young firm with good growth prospects will retain earnings to fund its likely
investment needs. Hence, problems of asymmetric information are sufficient
but not necessary for a preference for internal funding.
The learning model, then, is consistent with the observations on investment
behavior and the use of internal funds. Small firms are more likely to be young
firms and engaged in learning. For these firms, the presence of favorable investment opportunities is correlated with the presence of ample internal funds,
generated from current and recent favorable performance. Larger firms are more
likely to be mature. For these firms, investment opportunities are less tied to
firm-specific learning from experience. They are correspondingly more likely
12 The
type of cost considered here might be the cost of negotiating with an individual
investor or the cost of making the public aware of an issue of public securities.
34
Federal Reserve Bank of Richmond Economic Quarterly
to have opportunities arise in times of low internal resources, requiring them
to go to external sources for funds.
Other than the observations on investment behavior, the key facts discussed
in Section 1 concerned where firms go for external funds. Most significantly,
small firms go to banks for more of their financing than do large firms. Under the asymmetric information approach, one might suppose that asymmetric
information problems are more severe for small firms, so that the value of
using bank evaluation and monitoring services is greater for small firms than
for large firms. Diamond (1991) develops a model in which such monitoring
is provided to firms with mid-level reputations. If such a firm enjoys good
performance, it can improve its reputation and raise public funds. While firm
size is not directly incorporated in that model, it is not difficult to imagine
a direct link between reputation and size. A similar line of reasoning can be
followed under the life cycle approach. Small, young firms are likely to face the
greatest uncertainty about their own long-run productivities. Again, the value
of the information production services of banks will be greatest for these firms.
In both approaches, banks are seen as producers of information. In the former
case, they produce information in an attempt to undo the effects of asymmetric
information, while in the latter, they produce new information that is useful to
the firm in making its resource allocation decisions. In either case, once a firm
has accumulated enough information to know (or to convince others) that it is
profitable enough to continue producing, it enters the class of more mature firms
that utilize public debt and equity markets for their external financing needs.
In summary, a theoretical perspective based on asymmetric information
that produces financial constraints is capable of explaining observed deviations
from the type of behavior predicted by the frictionless framework of Modigliani
and Miller. By itself, however, this perspective cannot fully explain how those
deviations tend to be more apparent for smaller than for larger firms. Some
explanation of why the asymmetric information problems weigh more heavily
on some firms is needed. Such an explanation can be found in a life cycle
perspective. As firms age and grow, they acquire a public reputation that can
partially undo the constraints imposed by informational frictions. One finds,
however, that the life cycle perspective is capable of explaining a great deal of
the observed behavior by itself.
Clearly the two theoretical approaches discussed herein are not mutually
exclusive. Firms that are young and still accumulating knowledge about themselves are likely to be firms about which insiders are better informed than
outsiders. Knowledge of self precedes public reputation. However, the presence
of financial constraints seems not to be necessary for explaining the empirical
facts discussed above. Since the magnitude of asymmetric information problems
is inherently difficult to measure, it would be discomforting to rely on a theory
that draws its explanatory power from informational frictions. The life cycle
approach provides an attractive alternative.
J. A. Weinberg: Firm Size, Finance, and Investment
3.
35
SOME PUBLIC POLICY IMPLICATIONS
There has been a great deal of concern in recent years about the difficulties that
small firms face in securing funds from financial markets and institutions. A
number of regulatory and legislative initiatives have been put forward to address
the financial needs of small firms.13 Some of these proposals seek to expand
credit to small firms by easing regulations. The federal agencies with regulatory responsibilities for depository institutions have jointly developed a plan to
allow well-capitalized banks to make some small business loans with reduced
documentation requirements. Similarly, the Small Business Incentive Act would
exempt small issuers from some of the registration and disclosure requirements
for issuing public debt and equity securities. A third regulatory approach is
represented by the Small Business Loan Securitization and Secondary Market
Enhancement Act. This measure would ease banking and securities regulations
to facilitate the establishment of a secondary market in small business loans,
similar to that which exists for home mortgage loans. The establishment of
such a secondary market is also the aim of the Small Business Credit Act,
which would create a government-sponsored enterprise to buy and securitize
small business loans. Hence, under this proposal, the federal government would
play a more direct role in intermediating between loan originating banks and
the secondary market. Finally, there have been proposals to provide direct
government subsidies to small business lending. The Small Business Capital
Enhancement Act would create a loan loss reserve fund with contributions from
the government as well as from lenders and borrowers.
While the proposed approaches vary in how they would expand credit to
small firms, they all share a fundamental premise: if faced with the same terms
and rules as other firms, small firms would be underserved by the financial
markets. Such a premise is consistent with the conclusions that have been
drawn by some from the asymmetric information perspective. In this view,
financial constraints impose inefficient limitations on the operations of small
firms. Some have argued that such inefficiency can be countered by government intervention in financial markets. Even within the asymmetric information
framework, however, the case for efficiency-enhancing intervention is weak.14
Briefly, there is no reason to suppose that the practices we observe in financial markets and institutions are not efficient responses to the informational
frictions present in the economic environment. Since government intervention
cannot remove those frictions, there is no reason to suspect that the government
can improve on the responses developed by market participants. Under the life
cycle approach, there is no market failure and, therefore, no reason to suspect
13 Humes
and Samolyk (1993) describe the proposals mentioned here.
critique of the case for intervention in the presence of asymmetric information is given
by Lacker in this issue of the Economic Quarterly.
14 A
36
Federal Reserve Bank of Richmond Economic Quarterly
that the allocation of financial capital can be improved upon by government
intervention.
As noted above, a number of the recent proposals have taken the form of
easing regulatory requirements as opposed to directly or indirectly subsidizing the financing of small firms. These proposals might be based less on the
notion of informational market failure than on the idea that financial markets
and institutions face an excessive regulatory burden. If regulation is excessive,
however, why should its easing be targeted to small firms? Again, there must be
some reason why small firms are underserved. One possibility is the presence
of informational market imperfections. Another is that the regulatory burden
may be excessive for the financing of small firms but not for larger firms. This
possibility could arise if, for instance, the costs of complying with regulations
had a sizeable fixed component. This line of thinking probably lies behind
proposals to allow small-firm exemptions from documentation and disclosure
requirements for bank lending and issues of public securities.
Government intervention in favor of small firms, then, can be viewed as
partially offsetting the effects of existing government intervention. Desirability
of such a move depends on the reasons for the original intervention and on
the judgment of how much regulation is excessive. Consider the case of easing
bank regulations for small-firm lending. Suppose that financial behavior follows
a version of the life cycle model in which banks provide information production
services that aid firms (younger firms in particular) in their productive decisions.
In this model, the population of potential firms is divided into three groups,
depending on their priors: those that raise funds in public securities markets;
those that receive bank funding and information services; and those that do not
receive funding. Regulations on bank lending can be interpreted as increases in
the costs of producing these information services. This increase in costs does
two things. On the “high end,” firms with sufficiently favorable priors will be
induced to forgo bank services and raise more of their funds in public markets.
These firms will be larger than the average bank client but smaller than the
average public firm. On the “low end,” firms with marginal priors will find
themselves priced out of the market for bank lending. They will be among the
smallest firms. Recent years have seen just such a coincidence of reduced bank
lending with increasing numbers of initial public offerings.
A small-firm exemption to some bank regulatory requirements will reverse
the low-end effect and, depending on the cut-off size, possibly the high-end
effect. Is such a reversal desirable? Suppose that the original regulations were
put in place to counter the perceived incentives for excessive risk taking induced by (implicit or explicit) government guarantees to bank depositors. The
effects of such guarantees are similar to the effects of reducing the cost of bank
information services. Hence, the various policies and counter policies serve to
shift the margin between those firms that rely on bank financing and those that
use public markets as well as the margin between those that are able to obtain
J. A. Weinberg: Firm Size, Finance, and Investment
37
bank financing and those that are priced out of the market. Choosing the “best”
setting for those margins is a difficult judgment.
The question of choosing the best setting of bank regulations has arisen in
discussions of recent legislative and administrative changes in bank regulatory
policy. Some have argued that stricter examination standards in the Federal
Deposit Insurance Corporation Improvement Act (FDICIA) of 1991 and riskbased capital requirements in the Basel Accord have driven the cost of bank
lending so high as to contribute to a “credit crunch.” Such arguments have
been made, for instance, by Bizer (1993) and others, on the editorial pages
of The Wall Street Journal. On the other hand, with only a limited time since
the Act’s implementation, it is difficult to disentangle the effects of FDICIA
from other influences on aggregate credit market behavior.15 While it does
seem to be the case that financing of small firms has been particularly slow
since the implementation of FDICIA, it is also true that small-firm financing
and productive activity is generally more volatile and responsive to business
cycle fluctuations than that of larger firms.16 This greater volatility is consistent
with the life cycle approach; while the responses of large firms to a change in
market conditions are likely to be mostly “movements along demand curves,”
the responses of small firms are more likely to include changes in decisions to
enter or exit from markets.
In summary, neither the asymmetric information approach nor the life
cycle approach provides a definitive justification for a tilt toward small firms
in financial market policy. One might argue for a policy favoring small firms
on other grounds. Since small firms account for a large share of employment
growth and since many small firms engage in highly innovative activities, one
might argue that small-firm activity generates external benefits that contribute
to the long-run growth of our economy. If such an argument is used to justify
policies favoring small firms, it is not clear why such policies should work
through financial market manipulation. A simpler approach might come in the
form of targeted tax breaks.
4.
CONCLUDING REMARKS
Financial behavior should not be viewed in a vacuum. If we observe systematic
financial differences across firm sizes, or across some other firm characteristic,
we should seek to understand those differences in the proper context. This
article has asked the question, “What does economic theory have to say about
15 For
a discussion of the problem of identifying credit crunches, see Owens and Schreft
(1993).
16 The
generally greater volatility of small-firm behavior is found, for instance, by Gertler
and Gilchrist (1991).
38
Federal Reserve Bank of Richmond Economic Quarterly
the joint determination of firm size and financial behavior?” By contrast, the
interpretations of financial behavior leading to conclusions of market failure
have been conducted out of context, lacking an explicit theory of the determination of firm size. While the market failure interpretation might suggest a
positive role for government intervention in financial markets, this conclusion
is less tenable when the empirical facts are viewed in the context of a theory
of the size distribution of firms.
By taking size differences as given in interpreting financial differences,
the market failure approach amounts to partial equilibrium analysis; one market (the financial market) is examined in isolation from other markets in the
economy. Attempting to understand the joint determination of size differences
and financial behavior is a step toward general equilibrium analysis. Hence,
the arguments presented in this article might be viewed as contributions to the
case for the benefits of conducting applied economic analysis within a general
equilibrium framework.
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M2 and Monetary Policy:
A Critical Review of the
Recent Debate
Michael Dotsey and Christopher Otrok
R
ecently the question of whether a monetary aggregate, and in particular
M2, is a useful intermediate target for monetary policy has been the
subject of intense debate. The most striking feature of this debate,
which has been largely empirical, is that the central issues that are relevant
for analyzing M2’s usefulness in the conduct of monetary policy have been
neglected. Issues involving the controllability of M2 and the structural relationship between M2 and economic activity have not been adequately addressed.
Rather, much of the debate has focused on the notion of predictive content.
The argument expressed in much of the literature is that if money lacks
predictive content, then it has no useful role as either an information variable,
an intermediate target, or, when possible, an instrument of monetary policy.1
This argument basically misses the point. In this article we argue that Grangercausality tests generally are not a proper test of the usefulness of money as
an intermediate target. We also argue that evaluating the usefulness of any
monetary aggregate in the conduct of monetary policy requires a structural
model.
The authors would like to thank Tim Cook and Peter Ireland for many helpful suggestions.
The views expressed in this article are those of the authors and do not necessarily reflect
those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
1 This view is expressed rather strongly in Friedman and Kuttner (1992) and seems at least
to be implicit in most of the literature cited in this article. In what follows we use the terms information variable, intermediate target, and instrument in standard ways. An information variable
is one that provides information about future economic activity and in particular about variables
in the Fed’s objective function. An intermediate target is a variable that the Fed explicitly tries
to hit by altering its monetary instrument, which is a variable under direct control. That is, an
instrument is either the federal funds rate or an element of the Fed’s balance sheet.
Federal Reserve Bank of Richmond Economic Quarterly Volume 80/1 Winter 1994
41
42
Federal Reserve Bank of Richmond Economic Quarterly
The first section of this article reviews the empirical debate concerning
M2’s usefulness as an intermediate target. In particular we look at the result
of Friedman and Kuttner (1992, 1993) which indicates that in the presence
of financial market variables, M2 contains no predictive content for real income. We find that this result is fragile and that M2 does have significant
predictive content for both real and nominal GDP when the statistical tests
are properly specified. This agrees with similar evidence presented in Feldstein
and Stock (1993), Hess and Porter (1992), Becketti and Morris (1992), and
Konishi, Ramey, and Granger (1992). Thus under current operating procedures
M2 provides useful information about the economy.
In Section 2 we take a deeper look at the notion of predictive content
and its limitations in designing monetary policy. In particular, we show that
a failure to find predictive content says very little about the potential usefulness of a monetary aggregate as an intermediate target. The lack of predictive
content can arise as a result of operating procedures. It is entirely possible
that the same monetary aggregate could serve as a reliable intermediate target
or information variable under alternative operating procedures. Thus merely
analyzing reduced-form relationships for the purpose of making theoretical
arguments about the potential usefulness of M2, or any other measure of money,
in formulating policy can be misleading.
Also, the presence of predictive content does not necessarily imply that
M2 would make a good intermediate target. For that to be the case, M2 must
be controllable and must have an effect on economic variables that the Fed
ultimately wishes to influence. Feldstein and Stock (1993) realize that M2’s
usefulness as an intermediate target hinges on its controllability and assume
that the Fed can perfectly control M2. In Section 3 we analyze the effects of
relaxing this assumption and show that allowing for imperfect control weakens
their argument substantially.
In Section 4 we look at another assumption that is crucial to the Feldstein
and Stock analysis and to the entire recent empirical debate. That assumption
involves the invariance of the estimated reduced-form structures to changes
in Federal Reserve operating procedures. Here we show that the changes in
operating procedures advocated by Feldstein and Stock should have substantial effects on the economy’s reduced form. Thus, issues related to the Lucas
critique cannot easily be dismissed and there is reason to question the validity
of their policy experiments.
The combined analysis presented in Sections 2 and 4 indicates that the lines
of research evaluated in this article are not likely to be productive from the
standpoint of understanding and designing monetary policy. Granger-causality
tests say very little about the usefulness of a variable as a monetary instrument or intermediate target. Further, the assumption that the Lucas critique
is not measurably important in discussions concerning alternative operating
procedures is rather heroic.
M. Dotsey and C. Otrok: M2 and Monetary Policy
1.
43
A REVIEW OF THE STATISTICAL EVIDENCE
In this section we look at the statistical evidence regarding the ability of M2 to
help predict future movements in either real or nominal income. The key issue
here is the treatment of cointegration. Engle and Granger (1987) show that
proper estimation of a nonstationary system must explicitly account for any
cointegrating relationships. Merely differencing nonstationary data and then
performing statistical analysis does not properly account for long-run relationships, while leaving the data in levels omits relevant parameter restrictions.
To highlight the differences that can occur with alternative specifications, we
present Granger-causality tests results using differenced data with and without
the inclusion of a cointegrating vector. The finding of Friedman and Kuttner
that M2 has no predictive content is shown to be in part a result of an improper
statistical representation.
Specifically we analyze equations of the following form:
∆yt =
n
αyi ∆yt−i +
i=1
+
n
αsi st−i +
i=1
∆Yt =
n
i=1
n
αmi ∆Mt−i +
n
αpi ∆pt−i +
n
i=1
αri ∆rt−i
i=1
ατ i τt−i + βzt−1 + εt
(1)
i=1
ayi ∆Yt−i +
i=1
+
n
n
n
ami ∆Mt−i +
i=1
ati τt−i + bzt−1 + et ,
n
i=1
ari ∆rt−i +
n
asi st−i
i=1
(2)
i=1
where the symbol “∆” indicates first differencing, y (Y) is the log of real
(nominal) GDP, M is the log of nominal M2, p is the log of deflator on GDP, r
is either the three-month federal funds rate or the six-month commercial paper
rate,2 s is the spread between the six-month commercial paper rate and the
three-month Treasury bill rate, τ is a term structure variable measuring the
yield difference between the ten-year Treasury bond and the three-month funds
rate, and z is the cointegrated vector between M2, the price level, income, and
nominal interest rates implied by a stable money demand function.
The spread is included to take account of the financial effects emphasized
by Friedman and Kuttner (1992), namely, that this spread reflects a risk premium that can vary cyclically. It is also influenced by changing liquidity needs
since Treasury bills are more liquid than commercial paper. Thus the spread
variable is a stand-in both for changes in the demand for financial liquidity
2 We also experimented with the Treasury bill rate. The major difference is that the spread
and the term structure are less significant (sometimes very much so) when the Treasury bill rate
is employed.
44
Federal Reserve Bank of Richmond Economic Quarterly
and for changes in risk. Increases in risk or liquidity demands should have a
negative effect on economic activity.
The term structure variable is included to portray the stance of monetary
policy (see Bernanke and Blinder [1992]). An upward slope in the term structure indicates expectations of rising inflation and loose monetary policy. Thus
the coefficients on this variable should be positive in the nominal GDP regressions and positive in the real GDP regressions if there are significant nominal
rigidities in the economy. Alternatively, an upward-sloping yield curve could be
associated with an upward-sloping term structure of real interest rates signaling
expected consumption growth and thus has a positive association with future
real output. If the term structure variable is largely reflecting the expected behavior of future real rates, then the effect on nominal GDP could be ambiguous
since real output growth could result in lower inflation.
The regressions depicted in equations (1) and (2) are run in two different
ways. One way is that of Friedman and Kuttner in which the cointegrating relationship is ignored (i.e., the constraints β = b = 0 are incorrectly imposed).
The alternative methodology includes the cointegrating relationship among M2,
the price level, real income, and nominal interest rates.3
The importance of including cointegration can be examined within the
confines of a simple linear rational expectation model. Suppose the real part of
the economy was exogenous and the nominal side could be depicted as
rt = Et pt+1 − pt + wt
(3)
Mt = pt − crt + vt
(4)
Mt = µ + Mt−1 + xt ,
(5)
where wt , vt , and xt are white-noise disturbances to the nominal interest rate,
money demand, and money supply, respectively. Equation (3) is the Fisher relationship relating nominal interest rates to expected inflation and a stochastic
real rate of interest. The money demand disturbance, vt , incorporates changes
in real income and transactions costs, while xt is a money control error. The
model displays a cointegrating relationship between money and prices, with
Mt − pt being stationary.
The reduced form for money and prices is
Mt = µ + Mt−1 + xt
(6)
use a cointegrating vector similar to Feldstein and Stock (1993) in which yt +Pt −Mt −
αrt is stationary. With interest rates multiplied by 100, α, the interest semi-elasticity of money
demand, is equal to .0052 for commercial paper regressions and .0041 for regressions using the
federal funds rate.
3 We
M. Dotsey and C. Otrok: M2 and Monetary Policy
pt = (1 + c)µ + Mt−1 + xt +
c
1
wt −
vt .
1+c
1+c
45
(7)
Using equations (6) and (7) we can examine the importance of including cointegrating terms when testing if money growth helps predict future inflation.
From the structure, it is obvious that money does help predict the future price
level, but statistical tests will generally fail to confirm this feature of the model
if the cointegrating relationship is ignored. To illustrate this point, we generated 2,000 samples of 100 observations each and tested for Granger-causality.4
Without cointegration the lagged money growth was only significant at the
5 percent significance level 4.5 percent of the time, while with cointegration
money Granger-caused prices 95 percent of the time.
Having illustrated the potential importance of including cointegrating vectors, we reinvestigate the Friedman-Kuttner results that M2 does not Grangercause real output. The results are depicted in Table 1, where we report p-values
or significance levels on Granger-causality tests. The sample period is 1960:1
through 1993:1 and four lags of each variable are included (i.e., n = 4 in
equation [1]). Column 1 of Panel (a) basically replicates Friedman and Kuttner’s result that the spread has significant predictive content while M2 does
not. Replacing the commercial paper rate by the funds rate implies that M2 is
significant at the 10 percent level, while adding a term structure term implies
that M2 is significant at the 5 percent level. Like Feldstein and Stock (1993)
we find that including a cointegrating term yields the result that M2 is highly
significant in all specifications, even the one favored by Friedman and Kuttner,
as is the spread and the term structure.5
Since the monetary authority may also be interested in the forecasts of
nominal magnitudes when making policy decisions, we also look at the predictive content of M2 growth and the spread in regressions where nominal GDP
is the dependent variable. The results are depicted in Table 2. Here both the
spread and M2 are found to be highly significant predictors of nominal output.
Granger-causality tests are not the only way of examining predictive content. One may also wish to know if the effects of certain variables are long-lived
or if they die out quickly over time. To address these issues, we look at impulse
4 In the simulations both w and v are independently drawn from an N (0,1) distribution while
xt ∼ N (0, .1). The parameter µ = .05 and c = 2 reflect an interest elasticity of approximately
.10. The VARs were run with five lags of money growth and inflation.
5 Friedman and Kuttner’s results have been attacked on other grounds. Becketti and Morris
(1992) find that eliminating the period October 1979–October 1982 when the Fed altered its
operating procedures implies that M2 Granger-causes real output, while Konishi, Ramey, and
Granger (1992) attribute most of the spread’s predictive power to the inclusion of the 1971–1975
period. We were able to replicate the Becketti and Morris result but did not find the Konishi,
Ramey, and Granger result to be robust to alternative specifications. Using equation (1) with
β = 0, we still find the spread has predictive content for real GDP when using the funds rate,
the commercial paper rate, or both the funds rate and the term structure.
46
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 p-Values for Variables in Real Income Equations
1960:1–1993:1
Panel (a): No Cointegration
Independent
Variable
∆y
∆m
∆p
∆rff
∆rcp
s
τ
.5273
.1520
.0217
na
.0593
.0155
na
.6551
.0912
.0301
.0264
na
.0147
na
.5429
.0348
.0196
na
.0816
.0063
.0650
.6942
.0364
.0096
.0095
na
.0054
.0167
.0355
.0037
.0012
.0010
na
.0026
na
.0022
.0246
.0012
.0007
na
.0029
.0012
.0476
.0019
.0122
.0004
.0001
.0001
na
.0008
.0044
.0005
Panel (b): With Cointegration
Independent
Variable
y
m
p
rff
rcp
s
τ
zmd
.0291
.0070
.0009
na
.0026
.0031
na
.0025
Definition of variables: y = ln(real GDP), m = ln(nominal M2), p = ln(implicit price deflator
for GDP), rff = quarterly average federal funds rate, rcp = six-month commercial paper rate,
s = six-month commercial paper rate minus the three-month Treasury bill rate, τ = ten-year
Treasury bond rate minus the federal funds rate.
response functions. These results are displayed in Figures 1 and 2. Here we
only use the specification that includes cointegration. The impulse response
functions and the dashed lines that depict their 95 percent confidence band
are displayed in Figure 1 and indicate that M2 has a short-lived effect on real
output growth when ordered second in the orthogonalization procedure (panel
A) but not much effect when ordered third (panel B). The spread, however, has
a significant negative effect on real growth independent of ordering (panels C
and D). The results for nominal income are depicted in Figure 2. Here a shock
to M2 has a significant effect on nominal GDP that is quite long-lived (panels
A and B), while the spread’s effect is significant and surprisingly positive at
business cycle frequencies.6
6 We
also tested if the sum of coefficients on the spread variable were significant and found
that we could not reject this sum being equal to zero in either the real or nominal GDP regressions.
M. Dotsey and C. Otrok: M2 and Monetary Policy
47
Table 2 p-Values for Variables in Nominal Income Equations
1960:1–1993:1
Panel (a): No Cointegration
Independent
Variable
∆y
∆m
∆rff
∆rcp
s
τ
.7074
.0261
na
.0313
.0803
na
.6172
.0204
.0109
na
.1046
na
.8817
.0065
na
.0756
.0390
.2649
.8975
.0053
.0067
na
.0309
.0731
.0520
.0013
.0007
na
.0490
na
.0042
.1048
.0005
na
.0059
.0246
.2411
.0052
.0416
.0001
.0002
na
.0152
.0277
.0013
Panel (b): With Cointegration
Independent
Variable
y
m
rff
rcp
s
τ
zmd
.0739
.0020
na
.0024
.0402
na
.0053
Definition of variables: y = ln(nominal GDP), m = ln(nominal M2), rff = quarterly average
federal funds rate, rcp = six-month commercial paper rate, s = six-month commercial paper rate
minus the three-month Treasury bill rate, τ = ten-year Treasury bond rate minus the federal funds
rate.
2.
A DEEPER LOOK AT GRANGER-CAUSALITY
The analysis conducted in the previous section supports the results of a number
of other studies that M2 has significant predictive content for future movements
in both real and nominal GDP. Given the tremendous amount of effort exerted
in analyzing this issue, it is important to ask whether Granger-causality is a
relevant and essential property of a variable if that variable is to be useful
in conducting monetary policy. In particular, does the absence of Grangercausality imply that a variable cannot be used as an intermediate target or
instrument? The somewhat counterintuitive answer is no. Thus, for example,
the fact that some studies show that the monetary base does not Granger-cause
real economic activity provides little guidance concerning the potential role of
the base in conducting monetary policy.7
7 Examples
depicting the usefulness of the base as an instrument of policy can be found in
McCallum (1988), Judd and Motley (1991), and Hess, Small, and Brayton (1993).
48
Federal Reserve Bank of Richmond Economic Quarterly
Figure 1 Impulse Response Functions for Real Income
A. Shock to DM2
Response of Dy (Order = y, M, s, r, p)
B. Shock to DM2
Response of Dy (Order = y, s, M, r, p)
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
-0.10
-0.10
0
4
8
12
16
0
C. Shock to Spread
Response of Dy (Order = y, M, s, r, p)
4
8
12
16
D. Shock to Spread
Response of Dy (Order = y, s, M, r, p)
0.3
0.24
0.2
0.16
0.08
0.1
0.00
0.0
-0.08
-0.1
-0.16
-0.2
-0.24
-0.3
-0.32
-0.4
-0.40
0
4
8
12
16
0
4
8
12
16
+
Symmetrically the observation of Granger-causality does not necessarily
imply that a variable will be useful as an intermediate target or instrument.
Issues of controllability and the ability of a variable to causally influence
economic variables of primary concern must be addressed as well. Grangercausality, therefore, merely indicates that under existing policy a variable provides useful information about future economic activity.
In this section we examine by way of an illustration why the lack of
Granger-causality may not be particularly relevant. In the next two sections we
look at the other side of the coin and show that Granger-causality does not
necessarily imply that a variable will be useful as an intermediate target.
To illustrate our point, we use the simple economic framework in the
preceding example. Instead of using money as an instrument, the Fed uses an
interest rate instrument whose behavior is given by
rt = µr rt−1 + µp pt−1 .
(5 )
M. Dotsey and C. Otrok: M2 and Monetary Policy
49
Figure 2 Impulse Response Functions for Nominal Income
A. Shock to DM2
Response of DY (Order = Y, M, s, r)
B. Shock to DM2
Response of DY (Order = Y, s, M, r)
0.40
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
-0.10
-0.10
0
4
8
12
16
0
8
12
16
D. Shock to Spread
Response of DY (Order = Y, s, M, r)
C. Shock to Spread
Response of DY (Order = Y, M, s, r)
0.32
0.3
0.24
0.2
0.16
0.1
0.08
0.0
0.00
-0.1
-0.08
-0.2
-0.16
-0.3
-0.24
4
-0.4
0
4
8
12
16
0
4
8
12
16
+
Thus the economy is depicted by equations (3), (4), and (5 ). The reduced form
for this economy has the following ARMA (1, 1) representation:8
pt
Mt
= δ1
+
8 As
1
0
1 + c − cδ1 0
pt−1
Mt−1
µr /δ2
0
(1 + c − cδ1 )µr /δ2 0
+
1/δ2 0
1/δ2 1
rt
vt
rt−1
.
vt−1
(8)
shown in Boyd and Dotsey (1993), equation (8) is the unique nonexplosive solution
to this economic system. δ1 and δ2 are the eigenvalues of the matrix
| δ1 | ≤ 1 and | δ2 | ≥ 1 for appropriate choices of µr and µp .
0
−µr +µp
1
1+µr
with
50
Federal Reserve Bank of Richmond Economic Quarterly
The reduced form of equation (8) can be written as an infinite order AR process
∞
pt
pt−i
1/δ2 0
rt
=
Πi
+
,
(9)
Mt
M
1/δ
v
1
t−i
2
t
i=1
where each upper right-hand element of each Πi matrix is zero. Thus money
will fail to Granger-cause prices. This failure says nothing about money’s usefulness as a monetary instrument or intermediate target, since it is obvious that
in this simple model economy controlling money will control prices.9,10
3.
M2 AS AN INTERMEDIATE TARGET AND
THE FELDSTEIN-STOCK ANALYSIS
As mentioned, the other side of this last result, that a lack of predictive content does not rule out the usefulness of a monetary aggregate in formulating
monetary policy, is that predictive content does not necessarily imply that an
aggregate should be an instrument or an intermediate target. Predictive content
merely indicates that under current operating procedures a variable provides
some useful information for forecasting future economic activity. In order to
make the case that a variable would be a good instrument or intermediate
target, one must show that a policy that incorporates the variable in either role
improves economic performance in a welfare-enhancing way. Feldstein and
Stock (1993) undertake such an exercise for M2.
They perform this exercise by estimating equations of the form (2), with a
cointegrating vector included showing that M2 has significant predictive content for future nominal GDP. They also perform very sophisticated tests of the
stability of the M2-nominal GDP relationship and find that it is indeed stable.
The conclusion drawn from these results is that the reduced-form relationship
they have estimated is likely to be invariant to changes in operating procedures,
since the Fed changed operating procedures over their sample. The Fed, therefore, can profitably exploit the relationship between M2 and nominal GDP to
reduce the variability of nominal GDP growth.
To see to what extent this variability can be reduced, they calculate the optimal M2 supply function, treating their reduced form as a structural relationship
and assuming that M2 is perfectly controllable. They use the non-cointegrated
system for this purpose. They find that optimally controlling M2 would result
9 In particular, studies that show that the monetary base has little predictive content for
future economic activity do not imply that the base would be an ineffective monetary instrument
or intermediate target. McCallum (1993b) makes a similar argument with respect to stability of
the base nominal GDP relationship.
10 We also looked at some alternative policies. It appears that money’s failure to Grangercause output occurs whenever the Fed insulates the economy from money demand disturbances.
M. Dotsey and C. Otrok: M2 and Monetary Policy
51
in quarterly growth rates of nominal GDP that are 88 percent as variable as
they are now.11 They then show that a simple feedback rule
Mt = −λYt−1 + (1 − λ)Mt−1
(10)
performs almost as well.
One of the critical assumptions in their analysis is that M2 is perfectly
controllable. Taking for the moment the assumption that their reduced form is
invariant to the change in operating procedures that they propose, we wish to
see how their simple rule (for optimally chosen λ) would perform if M2 were
not perfectly controllable. We use the cointegrated specification of equation (2)
since our earlier results indicate that this is the preferred specification. Like
Feldstein and Stock we drop the term structure and spread variable.
To see what the effect of using rule (10) has on the variance of nominal
GDP, we conduct the same Monte-Carlo experiment that they do. First we
generate 2,000 simulations of 40 quarters each using a four-variable vector
autoregression that includes a cointegrating vector. The variables are nominal
GDP, the price deflator on GDP, the three-month Treasury bill rate, and M2.
More precisely, the first equation of this system looks like (2) with the term
structure and spread terms omitted. In performing the simulations the random
disturbances and coefficients are drawn from the appropriate distributions. We
then replace the estimated M2 equation with (10) and perform the same exercise. Using the simulated data, ratios of the variances of nominal GDP growth
can be constructed and analyzed. The results of the analysis are presented in
Table 3.
For the case λ = 0.3 and perfect controllability, the mean of these ratios
is .948, indicating that under the rule (10) nominal GDP’s variance could be
reduced to roughly 95 percent of its current value. Also 68 percent of the ratios
are less than one, indicating that following (10) reduced variability most of the
time. As the assumption of controllability is relaxed by adding to (10) a control
error scaled by the percentage of M2’s actual variance, the performance of the
rule deteriorates. For example, if an attempt to control M2 as an intermediate
target resulted in half the quarterly variability we now see, nominal output
variance would only be reduced to .985 of its value under current procedures.
Variability also only declines in 55 percent of the simulations. Thus the strength
of the argument for using M2 as an intermediate target is intimately related
to the issue of controllability. But the potential controllability of M2 cannot
be answered by this exercise. In order to answer that question, one needs a
11 They report a good deal more information. For example, they find that in 90 percent
of their simulated decades, simulated GDP is less variable than actual GDP. Had they used
the cointegrated system for this exercise, they would have found even greater improvement since
including the cointegrating vector improves the R2 of the model. (We thank Jim Stock for pointing
this out to us.)
52
Federal Reserve Bank of Richmond Economic Quarterly
Table 3 The Predicted Reduction in the Variance of Nominal
GDP Growth from Following a Simple Monetary Rule,
−λ
λYt−1 + (1 − λ)Mt−1
Variance
of M2
Value of
λ
Mean
Standard
Deviation
Median
Fraction
<1
0
0
0
25% of actual
25% of actual
25% of actual
50% of actual
75% of actual
100% of actual
0
.1
.3
0
.1
.3
.3
.3
.3
.913
.941
.948
1.153
.998
.965
.985
.999
1.015
.115
.121
.120
.268
.119
.103
.098
.098
.101
.914
.938
.954
1.085
.992
.972
.989
1.004
1.015
.81
.72
.68
.31
.51
.64
.55
.48
.42
Note: These results were constructed by a Monte Carlo procedure that produced 2,000 draws of
40 quarters of predicted nominal GDP growth using the proposed money rule and random draws
from the distribution of the reduced-form parameters and the reduced-form disturbances.
structural model since there is no period in which the Fed actually tried to
control M2.
4.
A DEEPER LOOK AT REDUCED-FORM INVARIANCE
Of equal if not greater importance than the issue of controllability is the assumption of reduced-form invariance to the change in operating procedures
proposed by Feldstein and Stock. In this section we investigate the likely effects
on reduced-form parameters if the Fed were to change its operating procedures
from an interest rate feedback rule that responds to economic performance to a
rule that targets M2. We do this for both an interest rate instrument and a total
reserves instrument.
We use a simple log-linear rational expectations model for our investigation. Since monetary economics lacks an acceptable model, we choose to
examine the issue of reduced-form invariance by examining a calibrated linear
rational expectations model. While this model falls short of representing reality,
it contains a number of key features found in many macroeconomic models
and is useful for broadly illustrating the points we wish to make. The model is
given by
yst = yt−1 + as ( pt − Et−1 pt ) + ut
(11)
ydt = a0 + yt−1 − ad ( pt + rt − Et pt+1 ) + wt
(12)
M. Dotsey and C. Otrok: M2 and Monetary Policy
53
mdt = pt − cr rt + cy yt + vt
(13)
rt = b0 + bp ∆pt + by ∆yt + br rt−1 + bm mt−1 + xt .
(14)
All variables with the exception of the nominal interest rate are in logs. Equation (11) is the standard Lucas supply curve relating real output to unexpected
price-level movements, while (12) is an IS curve in which aggregate demand
responds negatively to increases in the real rate of interest. Equation (13) is
the demand function for M2 and (14) is the Fed’s interest rate rule.12 The Fed
is modeled as responding to inflation, ∆Pt , and real output growth, ∆yt , while
maintaining concern for some degree of interest rate smoothing. The Fed also
responds to past M2 behavior using M2 as an information variable as opposed
to using it as an intermediate target. (For a more complete discussion of models
of this kind, see McCallum [1980].)
We can illustrate the extent to which the reduced form of this hypothetical
economy is invariant to changes in operating procedures by examining how
endogenous variables fluctuate around their expected value. A more complete
analysis would present the entire reduced form, but the anticipated parts of the
solution do not yield a simple analytical representation. We therefore present
only the unanticipated portion of the reduced-form solution. For the system
(11)–(14) these fluctuations are
ỹt = (1/D){ad (1 + bp )ut + as wt + xt }
(15)
p̃t = (1/D){−(1 + ad by )ut + wt + xt }
(16)
r̃t = (1/D){(ad by − bp )ut + (as by + bp )wt + axt }
(17)
m̃t = (1/D){[−(1 + ad by ) − cr (ad by − bp ) + cy (ad (1 + bp ))]ut + vt
+ [1 − cr (as by + bp ) + cy as ]wt + [1 − cr (as + ad ) + cy ]xt },
(18)
where the “˜” notation indicates unexpected deviations (e.g., ỹt = yt − Et−1 yt )
and D = as + ad + ad (bp + as by ).
If the Fed were to alter its policy rule (14) and use a noisy interest rate
instrument, such as borrowed reserves, instead of directly controlling the funds
rate, then the basic change in the economic system would be captured by an
increased variance in xt (the unexplainable part of policy). The solutions for
the reduced-form parameters in terms of the structural parameters would be
largely unchanged.13
12 Fuhrer and Moore (1993) find a similar rule helps fit the data quite well when included
in their contracting model.
13 Dotsey (1989) shows that allowing banks to have private information does affect the
reduced-form coefficients. For reasonable parameters, however, this effect is very small.
54
Federal Reserve Bank of Richmond Economic Quarterly
The changes in operating procedures over the period of the Feldstein and
Stock analysis—the announced move to nonborrowed reserve targeting under
lagged reserve requirements and the gradual emphasis placed on a borrowed
reserve target later on—amounted to a noisy interest rate instrument. Also,
as long as the Fed is using an interest rate instrument changes in reserve
requirements, or the move from lagged reserve to contemporaneous reserve
requirements, are largely inconsequential. Finally, the removal of Regulation
Q ceilings should have very little impact on the interest elasticity of money
demand if banks face fairly constant marginal costs of providing transactions
services and if they price deposits competitively.14 The stability that Feldstein
and Stock find in their reduced-form estimates is, therefore, not surprising.
The change in operating procedures they contemplate, namely, directly
targeting M2, may be an entirely different matter. To investigate the effect
on the economies reduced form, we replace equation (14) by an equation that
describes the Fed as targeting M2 with an interest rate instrument. This equation
is given by
rt = (1/cr )[Et−1 pt + cy Et−1 yt − m∗t ],
(14 )
where m∗t is the target level of M2. Equations (15)–(18) would become
ỹt = (ad /a)ut + (as /a)wt
(15 )
p̃t = (−1/a)ut + (1/a)wt
(16 )
r̃t = 0
(17 )
m̃t = [(ad − 1)/a]wt + [(1 + as )/a]wt + vt ,
(18 )
where a = as + ad . This reduced-form system is quite different from the
one shown previously, implying that the assumption of structural invariance is
somewhat tenuous.
Alternatively, the Fed could attempt to control M2 by placing a uniform
reserve requirement, η , on M2 balances and controlling M2 through the supply
of total reserves. Under this policy, required reserves, RR, would then be equal
to η(M2−C), where C is currency, and total reserve demand would equal RR
14 If
one thinks of the demand for money as responding to opportunity costs, then prior to the
removal of Regulation Q money demand was influenced by the nominal rate, r. After removal of
rate Q it was influenced by r − rM , where rM is the own rate. If marginal costs are fairly constant
and the banking system is competitive, then rM = (1 − λ)r and the opportunity cost of holding
money will be λr. One sees that d log M/d log r is invariant to this regulatory change when money
demand is of the constant elasticity form. If the demand for money is semi-logarithmic, then its
interest elasticity would be scaled by λ. Since the elasticity is very small to begin with, stability
tests may not be very sensitive to the removal of Regulation Q.
M. Dotsey and C. Otrok: M2 and Monetary Policy
55
plus excess reserve demand, ER. A log-linear representation of total reserve
demand would be
tr dt = log η + mt − ζt + εt ,
(13 )
where tr is the log of total reserves, mt is the log of M2 balances, ζt = (ct /M2t )
and εt = ERt /[η(M2t − Ct )].15 If the Fed supplied total reserves in an attempt
to hit an M2 target, it could do so by setting tr st = log η + m∗t . If total reserve
control was exact then mt = m∗t + ζt − εt and M2 will vary with movements in
currency and excess reserves. If the Fed instead controlled the monetary base,
then movements between currency and deposits would generally change M2.
Only if η = 1 and excess reserves were unimportant would strict M2 control
be achievable. In this case M2 would equal the monetary base. Note, however,
that although M2 is controllable, there is no longer a banking system since
without fractional reserves banks have no assets with which to make loans.
One suspects that such a policy would lead to financial changes that would
considerably affect the correlation of nominal output and M2.
Again if we replace (13) with equation (13 ) and (14) with
tr st = trt∗ + xt ,
(14 )
where trt∗ = log η + m∗t and xt represents reserve control errors, then the
reduced-form representation for unexpected changes in economic activity are
ỹt = (1/DD){−(ad cy + cr )ut + cr wt − ad (vt − xt − ζt + εt )}
(15 )
p̃t = (1/DD){−(ad /a)(ad cy − 1)ut − (ad /a)(1 + as cy )wt
− ad (vt − xt − ζt + εt )}
r̃t = (1/DD){(ad cy − 1)ut + (1 + as cy )wt + a(vt − xt − ζt + εt )}
m̃t = xt + ζt − εt ,
(16 )
(17 )
(18 )
where DD = (as + ad )cr + ad (1 + as cy ). Again, the reduced form is not
invariant to this proposed change in operating procedures.
To investigate the extent of the reduced-form invariance, we calibrate the
three distinct structural models and examine the implied variance-covariance
matrices. In doing so, we set as = 1.0 and ad = .7. The first value is taken
from King and Plosser (1986) while the second comes from Fuhrer and Moore
(1993).16 We assume M2 has a unitary income elasticity, cy = 1, and cr = 2
= log(RR + ER) = log[RR(1 + ER
)] = log RR + log(1 + ER
) ≈ log η + log(M2−C)
RR
RR
1−C
+ εt = log η + log[M( M )] + εt ≈ log η + m − ζ + ε.
16 Fuhrer and Moore’s coefficient is actually the impact effect of aggregate demand to a
change in lagged value of the long-term real interest rate.
15 tr
56
Federal Reserve Bank of Richmond Economic Quarterly
implies an interest elasticity of approximately −.10. Again following Fuhrer
and Moore, the coefficients bp and by are assumed to be .1 and we set br =
bm = .1 as well.
To produce numerical results, we also need to say something about the variances of the structural disturbances. We assume that the variance of the shocks
to aggregate supply, aggregate demand, money demand, and Fed behavior are
all of the same magnitude σ 2 . The variances for the currency/M2 ratio, ζ, and
the excess reserve/M2 ratio, ε, are assumed to be somewhat smaller. These are
(1/16)σ 2 and (1/1000)σ 2 , respectively. For example, a value of σ 2 = .0001
would imply that 95 percent of the shocks to quarterly output supply growth
or output demand growth are ±2 percent. The (1/16)σ 2 value for the variance
of ±C/M2 then implies that fluctuations in this ratio are generally no larger
than .005 and the (1/1000)σ 2 value implies that fluctuations in ER/M2 seldom
exceed .0006.
Under this parameterization the variance-covariance matrices for the three
models are
.77
.52
.35 .93
2 .10 .69
2
σ ,
σ , and
.55 .57 .87
0.0 0.0 0.0
.005 .13 −.60 1.54
.61 .80 0.0 2.42
.51
−.04
.10
.15
.07
−.18
.15
.43
−.35
2
σ ,
1.06
respectively. The differences are noticeable, indicating that it is inappropriate
to use a reduced-form structure from one type of policy rule to make inferences
about the effects of an alternative policy rule. Of interest, however, is the fact
that using a total reserves instrument to target M2 produces the lowest variance
in prices and real output. Whether this result would carry over to more detailed
structural models is unknown. Also, other types of rules, for example, those
advocated by McCallum (1988, 1993a), may be better still. It is obvious that
before jumping on an M2 bandwagon a lot of work remains to be done.
4.
CONCLUSION
In this article we have critically examined the debate over using M2 as an
intermediate target of monetary policy. We have done this by focusing on two
largely empirical studies. We found that Friedman and Kuttner’s central result
that money does not Granger-cause real or nominal output is due to a model
misspecification. When the cointegrating relationship among money, income,
prices, and interest rates is accounted for, money does indeed Granger-cause
M. Dotsey and C. Otrok: M2 and Monetary Policy
57
output. We also found that Feldstein and Stock’s main result, that a monetary
policy that uses M2 as an intermediate target can substantially reduce the variance of nominal GDP growth, depends critically on their assumption that M2 is
perfectly controllable. When this assumption is realistically relaxed, the ability
of their policy to reduce the variance of nominal income growth is seriously
diminished.
More generally, our results cast doubt on the idea that either empirical
exercise is useful in analyzing alternative monetary policies. Studies, such
as Friedman and Kuttner’s, that base their conclusions solely on the use of
Granger-causality and reduced-form models do not provide a firm basis for
making a decision about the usefulness of a monetary aggregate under some
alternative operating procedure. The absence of Granger-causality may provide
little guidance for evaluating the usefulness of M2 or any other aggregate
for monetary policy. All that the absence of Granger-causality tells us is that
under current operating procedures some variable does not help forecast future
economic activity.
Further, the presence of Granger-causality does not in and of itself imply
that targeting the aggregate in question would be good monetary policy. In order
to undertake that exercise, one needs a theory and the corresponding structural
model. Simply using a reduced-form model as Feldstein and Stock do is inappropriate when that reduced form will not remain invariant to the contemplated
changes in policy. There does not seem to us any shortcuts that will substitute
for the hard and necessary work of building and analyzing structural models.
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Becketti, Sean, and Charles Morris. “Does Money Still Forecast Economic
Activity?” Federal Reserve Bank of Kansas City Economic Review, vol.
77 (4th Quarter 1992), pp. 65–78.
Bernanke, Ben S., and Alan S. Blinder. “The Federal Funds Rate and the
Channels of Monetary Transmission,” American Economic Review, vol.
82 (September 1992), pp. 901–21.
Boyd, John H., III, and Michael Dotsey. “Interest Rate Rules and Nominal
Determinancy.” Manuscript. April 1993.
Dotsey, Michael. “Monetary Control Under Alternative Operating Procedures,”
Journal of Money, Credit, and Banking, vol. 21 (August 1989), pp. 273–
90.
Engle, Robert F., and C. W. J. Granger. “Cointegration and Error Correction:
Representation, Estimation, and Testing,” Econometrica, vol. 55 (March
1987), pp. 251–76.
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Federal Reserve Bank of Richmond Economic Quarterly
Feldstein, Martin. “The Recent Failure of U.S. Monetary Policy,” Working
Paper 4236. Cambridge, Mass.: National Bureau of Economic Research,
December 1992.
, and James H. Stock. “The Use of Monetary Aggregate to Target
Nominal GDP,” Working Paper 4304. Cambridge, Mass.: National Bureau
of Economic Research, March 1993.
Friedman, Benjamin M., and Kenneth K. Kuttner. “Another Look at the
Evidence on Money-Income Causality,” Journal of Econometrics, vol. 57
(June 1993), pp. 189–203.
. “Money, Income, Prices, and Interest Rates,” American Economic
Review, vol. 82 (June 1992), pp. 472–92.
Fuhrer, Jeff, and George Moore. “Monetary Policy and the Behavior of LongTerm Real Interest Rates,” Finance and Economic Discussion Series No.
93-16. Washington: Board of Governors of the Federal Reserve System,
May 1993.
Hess, Gregory D., and Richard D. Porter. “Comparing Interest-Rate Spreads
and Money Growth as Predictors of Output Growth: Granger Causality
in the Sense Granger Intended.” Manuscript. Board of Governors of the
Federal Reserve System, September 1992.
Hess, Gregory D., David H. Small, and Flint Brayton. “Nominal Income
Targeting with the Monetary Base as Instrument: An Evaluation of
McCallum’s Rule,” in Marvin Goodfriend and David H. Small, special
issue eds., Operating Procedures and the Conduct of Monetary Policy:
Conference Proceedings. Finance and Economics Discussion Series,
Working Studies 1: Part 2. Washington: Board of Governors of the Federal
Reserve System, 1993.
Judd, John P., and Brian Motley. “Nominal Feedback Rules for Monetary
Policy,” Federal Reserve Bank of San Francisco Economic Review,
Summer 1991, pp. 3–17.
King, Robert G., and Charles I. Plosser. “Nominal Surprises, Real Factors and
Propagation Mechanisms,” Working Paper 50. University of Rochester,
Center for Economic Research, July 1986.
Konishi, Toru, Valerie A. Ramey, and Clive W. J. Granger. “Stochastic
Trends and Short-Run Relationships Between Financial Variables and Real
Activity.” Manuscript. University of California, San Diego, December
1992.
McCallum, Bennett T. “Monetary Policy Rules and Financial Stability.”
Manuscript. August 1993a.
. “Comment on Feldstein and Stock’s ‘The Use of a Monetary
Aggregate to Target Nominal GDP.’ ” Manuscript. April 1993b.
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. “Robustness Properties of a Rule for Monetary Policy,” CarnegieRochester Conference Series on Public Policy, vol. 29 (Autumn 1988),
pp. 173–203.
. “Rational Expectations and Macroeconomic Stabilization Policy,”
Journal of Money, Credit, and Banking, vol. 12 (November 1980, Part 2),
pp. 716–46.
Stock, James, and Mark W. Watson. “A Simple Estimator of Cointegrating
Vectors in Higher Order Integrated Systems,” Working Paper WP-91-3.
Federal Reserve Bank of Chicago, February 1991.
. “New Indexes of Coincident and Leading Indicators,” in O.
Blanchard and S. Fischer, eds., NBER Macroeconomic Annual. Cambridge,
Mass.: MIT Press, 1989, pp. 351–93.
Does Adverse Selection
Justify Government
Intervention in
Loan Markets?
Jeffrey M. Lacker
G
overnment involvement in loan markets in the United States is substantial. For example, federal government direct and assisted lending
for 1980 through 1987 amounted to $1208.1 billion, or 25.3 percent of total net lending in nonfinancial credit markets over the period (Gale
1991). Only $115.5 billion of this amount represents direct federal lending;
the rest is accounted for by guaranteed loans ($251.3 billion), lending through
government-sponsored enterprises ($441.8 billion), and lending subsidized by
tax-exemptions ($399.5 billion). Can such interventions be justified as welfareimproving corrections of market failures?
Some economists have argued that market failure is particularly likely in
credit markets because of “adverse selection”—borrowers have unverifiable
hidden knowledge about their likelihood of repayment.1 There is a type of
externality in loan offers, and this can sharply constrain loan market outcomes.
The author thanks Tom Humphrey, John Weinberg, Tony Kuprianov, and Tim Cook for
helpful comments on earlier drafts. Some of the material in this article resulted from joint
work with John Weinberg. The author is solely responsible for the contents of the article,
and the views expressed do not necessarily reflect those of the Federal Reserve Bank of
Richmond or the Federal Reserve System.
1 “The Federal government has played a central role in the allocation of credit among competing uses. This paper illustrates that this sort of government program can under plausible conditions
improve on the unfettered market allocation. A necessary condition for efficient government intervention is unobservable heterogeneity among would-be borrowers regarding the probability of
default. The greater is such heterogeneity, the greater is the potential for default” (Mankiw [1986],
p. 469). See also de Meza and Webb (1987), Gale (1990), Greenwald and Stiglitz (1986), Innes
(1991), and Smith and Stutzer (1989).
Federal Reserve Bank of Richmond Economic Quarterly Volume 80/1 Winter 1994
61
62
Federal Reserve Bank of Richmond Economic Quarterly
For example, a good borrower may not receive ideal loan terms, since an
otherwise indistinguishable bad borrower would have an incentive to apply for
that loan as well, making it unprofitable. The market failure literature argues
that government credit guarantees can ease these constraints by improving loan
terms for less creditworthy borrowers. The resulting improvement in loan terms
for good borrowers leaves them willing to fund the subsidy to bad borrowers.
In a separate literature, some economists have argued that adverse selection
can help explain the role of financial intermediaries.2 In their models, financial
intermediaries often emerge endogenously as part of equilibrium arrangements,
attaining allocations that cannot be attained through direct lending alone. One
notable result from these models is that the resulting financial arrangement
cannot be improved upon by government intervention; private financial arrangements do as well as any government scheme.
How can we reconcile these two contrasting approaches? As I show in this
article, both are based on virtually identical economic environments. They differ, however, in how they predict outcomes for given economic environments;
each adopts a different definition of equilibrium. In the models rationalizing
government intervention, equilibrium is defined by the way agents play a specific multi-stage game. The rules of the game have strong implications for
how agents rationally play and what outcome emerges. In contrast, models
of financial intermediation are careful not to impose any institutional arrangement on the agents in the economy, so that institutional structure can emerge
endogenously.
I argue in this article that the different definitions of equilibrium yield
contrasting policy conclusions because the market failure approach imposes
ad hoc restrictions that prevent mutually beneficial contractual arrangements.
In the models of market failure, a seemingly reasonable implication of the
game agents play is that each type of loan contract must break even. This
condition prevents lenders from offering a menu of contracts that breaks even
on average but involves cross-subsidies across contracts. In this case, government tax and subsidy schemes, such as credit guarantees, can bring about the
cross-subsidization that private agents cannot. Thus government intervention
can make all people better off, even though the government is subject to the
same informational constraints as private agents.
The same equilibrium condition rules out endogenous financial intermediaries in the market failure models. Intermediaries arise in adverse selection
environments to reap the benefits of cross-subsidy; by subsidizing one borrower
the incentive constraints impeding a better borrower can be relaxed, making
both types better off. Since financial intermediaries are a prominent feature
2 See
Miyazaki (1977), Boyd and Prescott (1986), Boyd, Prescott, and Smith (1988), and
Lacker and Weinberg (1993).
J. M. Lacker: Adverse Selection in Loan Markets
63
of loan markets, it seems desirable to adopt a model that allows financial
intermediaries to emerge when they have a role to play. This suggests that the
adverse selection justification for government intervention in loan markets is
based on an overly restrictive definition of equilibrium. I conclude that based
on the models now available, adverse selection does not justify government
intervention in loan markets.
In this article I focus solely on situations of adverse selection, in which
agents have relevant private information ex ante, that is, before they first meet.
Situations of ex post private information, in which agents obtain hidden information after they first meet, do not present the same possibilities for market
failure. It is well known that the standard theorems on the optimality of competitive equilibria continue to hold under ex post private information (Prescott
and Townsend 1984). As a consequence, adverse selection has received far
more attention as a potential source of market failure. There remain, of course,
possible justifications on redistributive grounds, but these are beyond the scope
of this article.3
1.
AN ADVERSE SELECTION CREDIT ECONOMY
In this section I describe a simple economic environment with borrowing and
lending under adverse selection. The central feature of the environment is that
borrowers have private information about the risk and return on their investment
projects. Lenders do not know as much as borrowers, but try to infer as much
as they can from the repayment promises borrowers issue. Lenders’ beliefs and
borrowers’ actions are linked in a delicate interdependence that is the hallmark
of adverse selection environments. In order to make this interdependence manageable and understandable, I will work with a drastically simplified economy.
Various versions of adverse selection credit economies have been studied by
economists. However, there is a basic structure shared by virtually all adverse
selection environments, and my argument carries over to more general settings.
The economy I examine contains one feature that is not standard in adverse
selection credit market models. I assume that borrowers are able to costlessly
hide the return to their project. This feature, along with the properties of the
collateral good, implies that borrowers’ repayment promises must take the form
of collateralized debt, as I showed in an earlier paper (Lacker 1991). In most
of the literature on adverse selection in credit markets, either debt contracts are
3 Some government credit programs might be justified to ameliorate the effects of other government regulations that inhibit diversification by private financial intermediaries, such as legal
restrictions on bank branching (Williamson 1993). The best solution, however, is to eliminate
the legal restrictions themselves. Lang and Nakamura (1993) argue that lending can generate an
informational externality via publicly disclosed appraisals.
64
Federal Reserve Bank of Richmond Economic Quarterly
imposed by the theorist or the equilibrium contracts are not debt at all.4 This
feature does not alter my argument in any way, but it makes the predicted financial arrangements somewhat more realistic and demonstrates that the argument
does not depend on the ad hoc imposition of debt contracts.
To begin then, there are two periods and all consumption takes place in
the second period. There are a large number of borrowers, each with a single
investment project that requires exactly one unit of input in the first period and
yields a random return in the second period. The return can take on one of two
values: either R units of output (the “good state”) or zero (the “bad state”).
Borrowers’ returns are independent of one another. In addition, each borrower
has an amount K of a collateral good in the second period. This good is more
valuable to the borrower than it is to any other agent. One interpretation of
the collateral good is chattels—portable personal property such as clothing or
furniture.
Borrowers are risk-averse. They have identical utility functions over
second-period consumption, given by u(c1 + c2 ), where c1 is second-period
consumption of output and c2 is consumption of the collateral good. I assume
that the function u is strictly concave.
There are two types of borrowers—good and bad. The good borrowers
have a high probability of a good return, pg , and the bad borrowers have a low
probability of a good return, pb . I assume 0 < pb < pg < 1. A borrower’s type
is known only to that borrower; borrowers are observationally indistinguishable
to all other agents. The number of good borrowers is Ng , and the number of
bad borrowers is Nb , both of which should be thought of as large.
There are a large number of lenders, more than the number of borrowers.
Each lender has one unit of input good in the first period. Like borrowers,
lenders only desire to consume in the second period. Unlike borrowers, however, lenders have linear, risk-neutral utility functions. Their utility is given by
c1 + βc2 , where c1 is second-period consumption of output and c2 is consumption of a borrower’s collateral good. I assume that the preference parameter
β is positive but strictly less than one. This reflects the assumption that a
collateral good is more valuable to a borrower, relative to the payment good,
than it is to any lender. The difference in valuations could represent a special
match between the borrower and the collateral good or the resource costs of
transferring the good.
Lenders have an alternative investment opportunity available to them that
yields ρ units of output in the second period for every unit of input good
invested in the first period. Because they only want to consume in the second
period, lenders will want to lend or invest all of their first-period input goods.
Because of their alternative investment opportunity, a loan to a borrower will
4 An
exception is Boyd and Smith (1993).
J. M. Lacker: Adverse Selection in Loan Markets
65
have to provide at least as much expected utility as ρ. Because there are more
lenders than there are borrowers, there will always be an elastic supply of loans
on terms that provide lenders with at least as much expected utility as ρ.
A loan contract, or contract for short, could in general specify payments of
output and collateral by the borrower for each possible return. In other words,
a contract could consist of four numbers: payments of output and collateral
when the return is high and payments of output and collateral when the return
is low. We can restrict attention to simpler contracts without doing any harm,
however. First, note that the low return is zero, so the payment of output when
the return is low will always be zero. Second, consider the collateral payment
when the return is high. It is easy to show that as long as R is large enough,
it is never desirable to have a positive collateral payment in the good state
since the collateral is more valuable to the borrower than to the lender. More
precisely, any contract with a positive collateral payment in the good state
is dominated by one with no collateral payment and commensurately larger
payment of output in the good state.5 Therefore, we can restrict attention to
contracts that specify two numbers: r, a borrower’s payment of output when
the return is good, and C, a borrower’s payment of collateral when the return
is bad.
Because there are two types of borrowers and they might receive different
contracts, we need a notation for each. Let (rg , Cg ) be the contract for a good
borrower, and let (rb , Cb ) be the contract for a bad borrower. To be physically
feasible, the output payment must not be greater than the return in the good
state, R, and the collateral payment must be nonnegative and no greater than
the borrower’s collateral, K. I assume that R and K are large enough that these
feasibility constraints never bind. Since they play no role in the analysis, from
here on I will ignore them.
In the second period a borrower can hide the return R, making it appear
that the return is zero. The hidden return can be consumed secretly by the
borrower. This possibility constrains the contracts to which the borrower can
credibly agree. For example, if a contract calls for no collateral payment when
the return is low, but some positive payment r when the return is high, the
borrower will always hide the return, pay nothing, and consume the entire
return R in private; the alternative is to make the payment r and consume
R − r. A collateral payment when the return is low can provide an incentive to
make the required payment when the return is high. In this case the borrower
compares paying r to hiding the return and transferring collateral with value
C. If C ≥ r, then the borrower has no incentive to hide the return. Therefore,
contracts must satisfy the following.
5 This
is true under either of the definitions of equilibrium that appear below.
66
Federal Reserve Bank of Richmond Economic Quarterly
Incentive constraints:
rh ≤ Ch
for
h = g, b.
(1)
Condition (1) ensures that the borrower has an incentive to repay the loan
as agreed when the return is high, rather than hand over the collateral. The
incentive constraints imply that loans must be “fully collateralized,” meaning
that the value of the collateral (to the borrower) is at least as large as the value
of the promised repayment. If C > r, the loan is “overcollateralized.” The
only contracts that satisfy the incentive constraint (1) are collateralized debt
contracts—noncontingent except when the return is insufficient to make the
payment r.6
I can now describe the most crucial condition that contracts must satisfy
in this environment. Suppose that the end result of the interactions between
agents in this economy is that good borrowers receive contracts (rg , Cg ) and
bad borrowers receive contracts (rb , Cb ). Since a borrower’s type is private
information, one type of borrower could conceivably participate pretending
to be the other type of borrower, receiving the contract meant for the other
type. All participants might be expected to be aware of this possibility. If the
designation of contract (rg , Cg ) for good borrowers and contract (rb , Cb ) for
bad borrowers is to correspond to reality, then it must not be in any borrower’s
interest to masquerade as the other type of borrower. This condition, which
I call the self-selection constraints, must be satisfied by the outcome of any
mechanism agents adopt. Stated formally, we have
Self-selection constraints:
pg u[(R − rg ) + K] + (1 − pg )u(K − Cg )
≥ pg u[(R − rb ) + K] + (1 − pg )u(K − Cb )
(2)
pb u[(R − rb ) + K] + (1 − pb )u(K − Cb )
≥ pb u[(R − rg ) + K] + (1 − pb )u(K − Cg ).
(3)
Constraint (2) states that the expected utility of a good borrower is at least
as high under contract (rg , Cg ) as under contract (rb , Cb ). Similarly, constraint
(3) states that the expected utility of a bad borrower is at least as high under
contract (rb , Cb ) as under the contract (rg , Cg ). Neither type of borrower has an
incentive to pose as the other.
6 The distinction between a risky debt contract and a more general contingent claim is blurred
when there are only two returns. For example, one might just as well call a contract satisfying (1)
a collateralized profit-sharing plan; the borrower promises to pay a fraction r/R of the return or
else hand over collateral when the return is zero. When there are many possible returns, however,
the distinction is quite meaningful.
J. M. Lacker: Adverse Selection in Loan Markets
67
The self-selection constraints are illustrated in Figure 1. The good state
output payment r is measured on the vertical axis, while the bad state collateral
payment C is measured on the horizontal axis. Any arbitrary contract (r, C) can
be represented by a point on the graph. Borrower preferences over contracts
are shown by means of indifference curves. The curve labeled Vg is the set
of contracts that leaves a good borrower indifferent to the contract (rg , Cg ).
Similarly, the curve labeled Vb is the set of contracts that leaves a bad borrower
indifferent to the contract (rb , Cb ).
Borrowers would like smaller payments in either state, so indifference
curves slope down and borrower utility is increasing toward the lower left
corner of the graph. A contract like (r̂, Ĉ ) is preferred over (rg , Cg ) by good
borrowers, since it lies below Vg , and is preferred over (rb , Cb ) by bad borrowers, since it lies below Vb . The indifference curves of a bad borrower are
everywhere steeper than the indifference curves of a good borrower. Because
the probability of having to surrender collateral is larger for bad borrowers,
Figure 1 Self-Selection and Incentive Constraints
r
45º
Vb
V
g
(r b ,C b )
ˆ ˆ
(r,C)
+
(r g ,C g )
C
Notes: r is the loan repayment amount, and C is the collateral that is transferred in the event
of default. Vb is a bad-borrower indifference curve through the contract (rb , Cb ). Vg is a goodborrower indifference curve through the contract (rg , Cg ). Utility is increasing to the lower left.
Incentive constraints for voluntary repayment of the loan require that contracts lie below the 45◦
line.
68
Federal Reserve Bank of Richmond Economic Quarterly
they are more averse to collateral requirements than are good borrowers. As a
result, it is always possible to find a pair of contracts that “separate” the two
types of borrowers, as shown in Figure 1. Bad borrowers prefer their contract
(rb , Cb ) to the good borrowers’ contract (rg , Cg ), because the latter would place
them on an inferior indifference curve. Similarly, good borrowers prefer their
contract to the bad borrowers’ contract.
The incentive constraints imply that contracts must lie below the 45◦ line;
the collateral payment must be at least as large as the good state payment. If
(1) is not satisfied (r > C), then the borrower would hide the return in the good
state and transfer collateral rather than pay more.
Figure 2 shows “break-even lines” for each type of borrower. The line
labeled πg is the set of loans to good borrowers that earn no excess profits
for lenders. (Throughout this article “profits” refers to the expected profits of
lenders.) In other words, πg is the set of contracts that satisfy
pg r + (1 − pg )βC = ρ.
(4)
Contracts above πg earn positive profits for lenders and contracts below πg earn
negative profits. The line labeled πb is similarly defined as the set of loans to
bad borrowers that earn no excess profits for lenders.
pb r + (1 − pb )βC = ρ.
(5)
The break-even line for bad borrowers is steeper than the break-even line for
good borrowers because the probability of default and collateral transfer is
larger for bad borrowers. Contracts above πg but below πb (to the left of their
intersection) earn excess profits on good borrowers but negative profits on bad
borrowers. Figure 2 also shows the “pooling break-even line,” πgb . This is the
set of contracts that earn no excess profits when all borrowers apply, satisfying
[ pg r + (1 − pg )C ] Ng + [ pb r + (1 − pb )C ] Nb ≥ ρ(Ng + Nb ).
(6)
Overcollateralization is inefficient in this environment, ceteris paribus. As
Figure 2 shows, a borrower’s indifference curve is always steeper than the
break-even line for that borrower. In the absence of the incentive and selfselection constraints, a borrower choosing among all of the contracts on the
appropriate break-even line would prefer the one on the vertical axis, where
collateral transfer is zero. There are two reasons for this. First, such a contract
minimizes the risk borne by the borrower. Second, collateral is more valuable in the hands of the borrower, so better loan terms are available if the
collateral transfer is minimized. The inefficiency of collateral transfer implies
that contracts will attempt to minimize the collateral component, subject to the
incentive and self-selection constraints.
J. M. Lacker: Adverse Selection in Loan Markets
69
Figure 2 Collateral Transfers Are Undesirable, Ceteris Paribus
r
45º
πb
π gb
πg
Vb
Vg
C
+
Notes: πg (πb ) is the set of contracts earning zero expected profits on good (bad) borrowers. πgb
is the set of contracts earning zero expected profits on the average borrower.
2.
AN ARGUMENT FOR GOVERNMENT INTERVENTION
IN LOAN MARKETS
A Definition of Equilibrium: The Wilson Equilibrium
I have described an economic environment, in other words, the preferences, endowments and technologies (including information technologies) of an artificial
economy. An economic model also requires a means of selecting a predicted
outcome from among the many possible outcomes that are feasible for any
given environment—in other words, a definition of “equilibrium.” The usual
70
Federal Reserve Bank of Richmond Economic Quarterly
candidate is the competitive equilibrium in which agents take prices as given
and select utility- or profit-maximizing quantities.
The standard notion of a competitive equilibrium is problematic in adverse
selection environments, however. In frictionless environments, the value of a
commodity is known to the buyer and so the perceived desirability of a transaction depends only on the preferences of the buyer and the price. In an adverse
selection environment, buyers’ beliefs about the value of the item—a financial
claim, in our case—hinge on the actions of sellers, which in turn may depend
on the entire array of options available to sellers. Thus the desirability of a
given transaction may depend on all of the other transactions taking place. In
our environment, for example, a lender’s beliefs about which borrowers have
applied for a loan depend on what other loan contracts are available. Another
lender could offer a contract that takes away the best borrowers, leaving behind
high-risk borrowers.
A number of definitions of equilibrium have been proposed for adverse
selection economies. Many have defined equilibrium as the outcome of some
game agents are assumed to play. Formally, a game consists of a sequence
of moves and countermoves available to agents, along with a specification of
the payoffs they receive for any particular sequence of chosen moves. Players
adopt strategies, functions determining their choice of move in various circumstances. The outcome of the game is presumed to be a Nash equilibrium, in
which agents take other agents’ strategies as given and choose a strategy that
maximizes their expected payoff. In a Nash equilibrium, each player’s strategy
is a “best response” to other players’ strategies.
The simplest version of this approach to adverse selection economies is a
two-stage game. In the first stage lenders simultaneously offer loan contracts,
and in the second stage borrowers choose which contracts to accept. Accepted
contracts are then executed, determining payoffs. A lender decides on a loan
offer, taking as given the loans offered by other lenders and the way in which
borrowers select from the available loans. Unfortunately, as Rothschild and
Stiglitz (1976) showed in a closely related environment, there often is no equilibrium for this game. The problem is that “pooling” contracts, in which all
borrowers receive the same contract, are always vulnerable to contracts that
“cream-skim” the best borrowers away, while separating contracts can be vulnerable to pooling contracts that make both types of borrowers better off. Thus
in some cases this notion of equilibrium makes no prediction at all!
One alternative that has been proposed is a particular four-stage game. The
first two stages are as before, with lenders making offers and borrowers accepting. In the third stage lenders can withdraw any loan offer made in the first
stage, but no contracts can be added. Lenders cannot precommit to not withdraw
a contract in the third stage. In the final stage borrowers choose contracts again,
the game ends, and contracts that have been accepted are executed. This game
always has an equilibrium, so it avoids the serious existence problem of the
J. M. Lacker: Adverse Selection in Loan Markets
71
two-stage game. This definition of equilibrium, which I describe more explicitly
below, was first proposed by Charles Wilson in 1977 and is widely known as
the “Wilson equilibrium.” As I will show, there is a rationale for government
loan market intervention in models adopting the Wilson equilibrium.7
Many other definitions of equilibrium have been proposed for adverse selection environments, and models adopting some of them have been used to
justify government intervention in loan markets. The rationale for government
intervention under these other equilibria is similar to that of the Wilson equilibrium, and I will briefly comment on them at the end of this section. One
advantage of the Wilson equilibrium is that it always exists.
To formally define a Wilson equilibrium, let S be a set of contracts. The
set S could be a pair of separating contracts or a single pooling contract.
Definition (defeats): Given a set of contracts S and another set of contracts S ,
suppose borrowers self-select among both sets of contracts. If any contracts in
S earn negative profits, delete the smallest number of contracts in S such that
the remaining contracts are all profitable after borrowers again self-select. If all
of the contracts in S now earn nonnegative profits and at least one earns strictly
positive profits, then the set of contracts S defeats the set of contracts S.8
Definition: A Wilson Equilibrium is a set of contracts S satisfying the following
conditions:
(i)
(ii)
(iii)
(iv)
the incentive constraints (1);
the self-selection constraints (2) and (3);
each contract earns nonnegative profits for lenders; and
no other set of contracts S exists that defeats the set of contracts S.
The first two conditions require that contracts be consistent with the informational imperfections of the environment. Condition (iii) states that each
individual contract must at least break even. The essential idea in condition (iv)
is that a set of equilibrium contracts cannot be trumped by some other contracts
earning excess profits. The critical component of the definition concerns the
conjectures of lenders contemplating introducing the deviating contracts. A new
contract might attract good borrowers from other lenders and might earn excess
profits, but some of the original contracts may then earn negative profits. If the
7 Innes (1991) describes the case for government intervention in loan markets based on
the Wilson equilibrium. Crocker and Snow (1985a) prove the existence of Pareto-improving
government tax schemes in the Rothschild-Stiglitz insurance model.
8 This definition, following Wilson, does not rely on an explicit formal definition of a game.
Wilson viewed these conditions as defining a competitive equilibrium. As is typical, the game has
multiple equilibria. One could view the definition as selecting a particularly plausible equilibrium.
Hellwig (1987) conjectures that recently proposed equilibrium “refinements” select the Wilson
equilibrium.
72
Federal Reserve Bank of Richmond Economic Quarterly
original contracts now earning negative profits are withdrawn, borrowers will
reallocate themselves among the remaining contracts and some new contracts
may now earn negative profits. Such contracts do not defeat the equilibrium.
If the new contracts remain profitable, then they have defeated the original set
of contracts, which could not have been an equilibrium.
Condition (iii) is crucial. It is sometimes called the “type-wise break-even”
condition, since it requires that each contract earn nonnegative profits on its
own. This condition is derived from the ability of lenders to withdraw any
unprofitable contracts at the third stage of the game. Furthermore, lenders cannot precommit to not withdraw unprofitable contracts. As we will see below,
condition (iii) implies a welfare-enhancing role for government intervention.
What Does the Wilson Equilibrium Look Like?
There is a unique set of contracts that constitutes a Wilson equilibrium. Depending on parameter values, the Wilson equilibrium is one of two types.9
One type, the separating equilibrium, is shown in Figure 3. The bad borrower
receives the contract (rb∗ , C∗b ), where the bad-type break-even line, πb , intersects
the 45◦ line. Of all the contracts that break even on bad borrowers and satisfy
the incentive constraint (1), the contract (rb∗ , C∗b ) is the one most preferred by
bad borrowers. The good borrower’s contract has to lie on or above Vb∗ , the
bad borrower’s indifference curve through (rb∗ , C∗b ), in order to satisfy the selfselection constraint (3). Since it must at least break even, it must also lie on
or above πg . Of all the contracts satisfying (3) and the good-type break-even
condition, the contract (rg∗ , C∗g ) is the one most preferred by good borrowers.
It is easy to see why this is an equilibrium. First, imagine trying to attract
good borrowers without attracting the bad borrowers. To do so the new contract
would have to lie below the good-type indifference curve Vg∗ but above Vb∗ , to
the southeast of (rg∗ , C∗g ) in Figure 3. But such contracts earn negative profits
since they lie below πg . Similarly, there is no contract that attracts only the
bad borrowers, satisfies the incentive constraint, and earns nonnegative profits.
Finally, imagine introducing a pooling contract that attracts both good and
bad borrowers. Such a contract would have to lie on or above the pooling
break-even line, πgb . No such contract would succeed in attracting the good
borrowers, since πgb lies everywhere above Vg∗ .10
The other type of Wilson equilibrium is a pooling equilibrium. Both types
of borrowers receive the same loan contract, (r∗∗ , C∗∗ ) in Figure 4. This contract
lies at the tangency of the pooling break-even line, πgb , and a good borrower’s
9 There is a knife-edge case of a single set of parameter values for which both types of
equilibria coexist, which I will ignore.
10 In this case the Wilson equilibrium is the same as the equilibrium of the two-stage game
analyzed by Rothschild and Stiglitz (1976).
J. M. Lacker: Adverse Selection in Loan Markets
73
Figure 3 The Wilson Equilibrium When Nb /Ng Is Large: Type-Wise
Break-Even Separating Contracts
r
45º
π gb
πb
V*
b
,C *
)
(r *
b
b
V*
g
πg
*
(r *
g ,C g )
V*
b
V*
g
C
+
Notes: Bad borrowers receive the contract (rb∗ , C∗b ), where the bad-borrower break-even line, πb ,
intersects the 45◦ line. Good borrowers receive the contract (rg∗ , C∗g ), where the good-borrower
break-even line intersects the bad borrower’s indifference curve through (rb∗ , C∗b ); the bad borrower
is indifferent between (rb∗ , C∗b ) and (rg∗ , C∗g ). Good borrowers prefer (rg∗ , C∗g ) to any contract on
the pooling break-even line, πgb . This type of equilibrium occurs for high and very high levels
of Nb /Ng in Table 1.
indifference curve, Vg∗ . This contract provides higher expected utility for a good
borrower than the separating equilibrium, since it lies below the indifference
curve through the separating allocation, Vg∗ . Of all of the pooling contracts,
(r∗∗ , C∗∗ ) is most preferred by the good borrowers.11
To see why this is an equilibrium, consider how a lender might try to attract good borrowers by offering a contract like (r̂g , Ĉg ) in Figure 4. This would
indeed attract good borrowers and it would make positive profits as well, since
it lies above the good-type break-even line, πg . But now the contract (r∗∗ , C∗∗ )
would lose money, since it would only be selected by bad borrowers. It would
that the pooling equilibrium might lie on the 45◦ line. This occurs if there is no
point below the 45◦ line on πgb tangent to a good borrower’s indifference curve.
11 Note
74
Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 The Wilson Equilibrium When Nb /Ng Is Small:
A Pooling Contract
r
45º
πb
π gb
V*
b
V*
g
**
Vg
πg
(r **,C** )
ˆ )
(rˆg ,C
g
+
V **
b
C
Notes: The Wilson equilibrium contract (r∗∗ , C∗∗ ) is the pooling contract that maximizes the
expected utility of the good borrower. The rival contract (r̂g , Ĉg ) would attract all of the good
borrowers, but fails to defeat (r∗∗ , C∗∗ ) because the latter would then lose money on just bad
borrowers and be withdrawn, forcing all borrowers to take (r̂g , Ĉg ), which would then lose money.
This type of equilibrium occurs for intermediate, low, and very low levels of Nb /Ng in Table 1.
be withdrawn in the third stage of the game, leaving only the new contract
(r̂g , Ĉg ). The new contract would now attract both types of borrowers, and since
it lies below πgb it would earn negative profits. Such a contract thus fails to
defeat (r∗∗ , C∗∗ ). It should be clear that no other pooling contract is able to
defeat (r∗∗ , C∗∗ ) either, since none would break even and attract just the good
borrowers after (r∗∗ , C∗∗ ) is withdrawn.12
Which equilibrium occurs depends on whether πgb , the pooling breakeven line, intersects Vg∗ , the good borrower’s indifference curve that passes
through the good-type separating contract. If it does, the equilibrium is a pooling
12 In this case the Rothschild and Stiglitz equilibrium does not exist because the contract
(r̂g , Ĉg ) would defeat the candidate equilibrium (r∗∗ , C∗∗ ); their equilibrium does not allow subsequent withdrawal of contracts earning negative profits.
J. M. Lacker: Adverse Selection in Loan Markets
75
contract like (r∗∗ , C∗∗ ) in Figure 4. If it does not, the equilibrium is the set of
break-even separating contracts shown in Figure 3. This depends on whether
πgb lies closer to πg or πb , which in turn depends on the ratio Nb /Ng . If there
are many bad borrowers relative to good borrowers, pooling allocations are
unattractive to the good borrower, so the type-wise break-even separating allocation is the equilibrium. If there are few bad borrowers relative to good
borrowers, then the good borrower does well in a pooling allocation. As Nb /Ng
approaches zero and the bad borrowers become a negligible portion of the
market, the equilibrium allocation approaches the intersection of πg and the
45◦ line, the loan that the good borrower would receive if there were no bad
borrowers.
Is the Wilson Equilibrium Pareto-Optimal?
Are there any alternative allocations that make no agents worse off and make
at least one agent strictly better off? If the answer is no, the given allocation
is Pareto-optimal. The only relevant alternative allocations to check, of course,
are those that are attainable—allocations that respect the resource, incentive,
and self-selection constraints of the environment.
Is the Wilson equilibrium Pareto-optimal? Often the answer is no. Figure 5
shows why. Suppose the set of contracts {(rg∗ , C∗g ), (rb∗ , C∗b )} is a separating Wilson equilibrium, as before. (This occurs when the ratio of bad to good borrowers
is above a certain threshold.) Now replace the bad borrowers’ contract with the
contract (r̂b , Ĉb ), down and to the left along the 45◦ line. This new contract
makes bad borrowers better off, but earns negative profits since it lies below πb .
In order to maintain resource feasibility, the new contract for good borrowers
must earn excess profits. As a result, the good borrowers’ new contract must lie
on or above a line parallel to (but above) πg , shown as a dashed line in Figure 5.
The new contract for bad borrowers relaxes the self-selection constraint, which
now requires that (r̂g , Ĉg ) lie on or above V̂b . Among the contracts that satisfy
the two constraints, the contract (r̂g , Ĉg ), at the intersection of the dashed line
and V̂b , is the one most preferred by good borrowers.
As shown in Figure 5, the good borrowers’ new contract lies on an indifference curve that is superior to Vg∗ , the indifference curve through the Wilson
equilibrium contract.13 The new set of contracts makes both types of borrowers
better off, and by construction, lenders receive just as much expected consumption (and therefore expected utility) as before. In addition, the new contracts
have been constructed to satisfy incentive and self-selection constraints. Therefore, the set of contracts {(r̂g , Ĉg ), (r̂b , Ĉb )} Pareto-dominates the separating
Wilson equilibrium contracts.
13 The
new contract gives the good borrower a consumption pattern that is less risky than
under the original contract. In addition, the good borrower benefits from reduced collateralization.
76
Federal Reserve Bank of Richmond Economic Quarterly
Figure 5 The Wilson Equilibrium Is Pareto-Dominated:
The Separating Case
r
45º
πb
ˆ
V
b
V*
g
ˆ )
(rˆb ,C
b
V*
b
(r * ,C * )
b
b
ˆ
V
g
π ⬘g
*
(r *
g ,C g )
πg
ˆ )
(rˆg ,C
g
+
C
Notes: Bad borrowers prefer the contract (r̂b , Ĉb ) to the equilibrium contract (rb∗ , C∗b ). Because
(r̂b , Ĉb ) lies below πb , the resource feasible contracts for good borrowers now lie along πg . The
good borrower can now obtain the contract (r̂g , Ĉg ), where the bad borrower’s indifference curve,
V̂b , intersects πg . As shown, (r̂g , Ĉg ) yields greater expected utility than (rg∗ , C∗g ).
The Wilson equilibrium is Pareto-optimal when Nb /Ng is very large. Whenever Nb /Ng is above some critical threshold, the dashed good-borrower resource
feasibility line lies so far above πg that no improvement for good borrowers is
possible. This might even be true for any possible choice of alternative badborrower contract along the 45◦ line. If Nb /Ng is below the critical threshold
the separating equilibrium is Pareto-dominated.
When the Wilson equilibrium is a pooling contract and does not lie on
the 45◦ line, it is easy to show that it is not Pareto-optimal. The equilibrium
allocation is dominated by a pair of separating contracts lying along Vb∗∗ ; the
alternative contract for bad borrowers is above πgb and the contract for good
borrowers is below. Thus bad borrowers are indifferent and good borrowers
are made strictly better off. If the Wilson equilibrium is the 45◦ line pooling
contract, then it is Pareto-optimal. This occurs for values of Nb /Ng below some
threshold. To summarize then, for a range of intermediate values of Nb /Ng the
J. M. Lacker: Adverse Selection in Loan Markets
77
Wilson equilibrium is not Pareto-optimal. For values of Nb /Ng above or below
this range, the Wilson equilibrium is Pareto-optimal.14
Government Intervention Can Be Pareto-Improving
A crucial feature of the alternative allocations that Pareto-dominate the Wilson
equilibrium is that they involve cross-subsidy. A pair of feasible contracts involve cross-subsidy if they do not lie on the individual break-even lines πg and
πb ; in other words, one earns positive expected profits while the other earns
negative expected profits. In the allocations that Pareto-dominate the Wilson
equilibrium, the good borrowers subsidize the bad borrowers, loosening the
self-selection constraint and allowing good borrowers a less risky consumption pattern and reduced collateral transfer. Such allocations cannot be Wilson
equilibria because they violate the type-wise break-even condition. When the
Wilson equilibrium is not Pareto-optimal, government intervention can help by
performing the cross-subsidization that is ruled out in equilibrium. Tax and
subsidy schemes can provide bad borrowers with better loan terms, relaxing
the bad borrower’s self-selection constraint and allowing more desirable loan
terms for good borrowers. Good borrowers are better off, even though they
bear the tax burden.
Government intervention in this credit market can take many forms. One
method is a subsidy for high-interest (bad-type) loans. The government could
fund the subsidy through taxes levied on lenders’ returns. This would relax the
bad-borrower break-even line faced by lenders, making them willing to offer
subsidized loan terms. The net tax on loans to good borrowers would shift
upward the good-type break-even line. Tax and subsidy rates can be selected
so that the resulting Wilson equilibrium Pareto-dominates the no-intervention
equilibrium.15
One difficulty with a subsidy scheme of this sort is that it must be applied
only to the loan contracts selected in equilibrium by the bad borrowers. A simpler alternative is a government loan guarantee applicable to all loans, funded
by a tax on lenders’ interest income. The government would guarantee a fraction
δ of the stipulated loan repayment r, where β < δ < 1. If the collateral transfer
yielded βC < δr, the government would pay the lender δr − βC. This could
be funded by a tax, τ , on lenders’ net interest earnings, (r − ρ). The parameter
β can be set so that only bad borrowers are subsidized in equilibrium. The
break-even lines for loans to type h borrowers (h = g, b) is now
ph [r − τ (r − ρ)] + (1 − ph )MAX[βC, δr] = ρ.
(7)
14 For a complete welfare analysis of the closely related Rothschild-Stiglitz insurance environment, see Crocker and Snow (1985b).
15 Crocker and Snow (1985a) consider such tax/subsidy schemes.
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Federal Reserve Bank of Richmond Economic Quarterly
The government budget constraint is
τ pg (rg − ρ)Ng + τ pb (rb − ρ)Nb ≥ (1 − pg )Ng MAX[0, δrg − βCg ]
+ (1 − pb )Nb MAX[0, δrb − βCb ].
(8)
The effect of the tax is to rotate both break-even lines in a clockwise direction
around the point at which they intersect. The effect of the guarantee is to make
the lines flat to the left of where δr = βC. The combined effect is illustrated
in Figure 6.
The allocation {(r̂g , Ĉg ), (r̂b , Ĉb )} can be attained by setting τ so that
(r̂g , Ĉg ) satisfies (7), and then setting δ so that (r̂b , Ĉb ) satisfies (7). The feasibility of the new set of contracts implies that the government budget constraint
is satisfied.16 In the case of the pooling Wilson equilibrium, parameters could
similarly be set to achieve a Pareto-dominating separating allocation.
Other welfare-enhancing schemes are easy to imagine, but all share the
same principle. The government is able to cross-subsidize loan contracts in a
way that is ruled out in the Wilson equilibrium. Cross-subsidies are inconsistent
with rational strategies in the multi-stage game that agents are assumed to play.
A natural question that arises is: Why would agents play this particular game?
Other Definitions of Equilibrium
As I mentioned earlier, other definitions of equilibrium in adverse selection
models have been used to justify government intervention in loan markets.
Some authors select the Rothschild-Stiglitz equilibrium (type-wise break-even
separating contracts) and restrict attention to cases in which it exists (Smith
and Stutzer 1989; Gale 1990). Some authors adopt pooling allocations and note
that such allocations are sometimes Pareto-dominated (Greenwald and Stiglitz
1986; de Meza and Webb 1987; Mankiw 1986). John Riley (1979) proposed an
equilibrium very similar to Wilson’s, in which lenders cannot withdraw contracts in the third stage (as they can under the Wilson setup) but can propose
new contracts if they wish.17
16 This scheme would be affected by the possibility that the new pooling break-even line
may now intersect the good borrower’s indifference curve through (r̂g , Ĉg ). If it did, the Wilson
equilibrium in the presence of this government guarantee is a pooling contract on the 45◦ line. If
the tax and subsidy parameters are set to balance the budget at the separating contracts, they may
violate the budget constraint at the pooling equilibrium. This problem might limit the magnitude
of the Pareto-improvement. Adding a fixed lump-sum component to the tax schedule can get
around the problem. See Crocker and Snow (1985a).
17 A vast literature studies adverse selection environments as “signaling games,” in which
the informed agents (our borrowers) move first by taking some irrevocable action or making
a contract offer; see Cho and Kreps (1987). This approach has not yet been applied to policy
analysis.
J. M. Lacker: Adverse Selection in Loan Markets
79
Figure 6 A Pareto-Improving Government Credit Guarantee
r
45º
ˆ
V
b
V*
g
πb
V*
b
␦r = C
ˆ )
(rˆb ,C
b
πˆ b
ˆ
V
g
πg
πˆ g
ˆ )
(rˆg ,C
g
+
C
Notes: π̂g (π̂b ) is the set of contracts that break even after taxes when accepted by good (bad)
borrowers. The tax on net interest income rotates the break-even lines, while the guarantee flattens
them out to the left of the line δr = βC, where the guarantee just pays off.
All of these other definitions of equilibrium impose a particular structure on the way agents interact, some through explicit games, some in an
ad hoc fashion. All share the feature that “equilibrium” allocations can fail
to be Pareto-optimal, providing a role for government intervention. Under all
definitions, equilibrium is Pareto-dominated in all the cases in which the Wilson equilibrium is Pareto-dominated. Under some definitions, equilibrium is
Pareto-dominated in other cases as well. In a sense, the Wilson equilibrium
provides the strongest case for government intervention because, of the equilibria that have been proposed, the laissez-faire Wilson equilibrium is least likely
to be Pareto-dominated; if government intervention is warranted for the Wilson
equilibrium, it is warranted under other definitions as well. In any event, the
Wilson equilibrium is representative of definitions that give rise to market
failure in adverse selection environments, and my remarks apply with equal
force to all.
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Federal Reserve Bank of Richmond Economic Quarterly
What About Credit Rationing?
Adverse selection models of credit markets are often associated with the notion of “credit rationing” (Stiglitz and Weiss 1981). The parameter values I
have assumed for my economy imply that credit rationing never occurs.18 It
should be clear that the adverse selection justification of government intervention in loan markets does not depend on the existence of credit rationing
(Smith and Stutzer 1989; Gale 1990). The justification relies on the effects of
self-selection constraints, and these perturb equilibrium whether or not credit
rationing occurs.
3.
ADVERSE SELECTION MODELS OF FINANCIAL
INTERMEDIARIES19
Financial intermediaries such as banks, pension funds, and insurance companies
surely play an important role in loan markets. And yet standard frictionless
models have little to say about financial intermediaries. Either organizations
such as banks or firms are taken as primitive elements, or they are economically inessential because equilibrium allocations can be achieved without them.
Financial intermediaries can be viewed as large multilateral arrangements that
arise to overcome the problems of asymmetric information that are absent in the
standard frictionless models. Much recent effort has gone into the search for environments in which “realistic” multilateral arrangements, or at least aspects of
them, are in some sense endogenous outcomes rather than imposed constraints.
Much of this effort has focused on environments in which information is limited
in some way, either being asymmetrically distributed or costly to obtain.
Adverse selection environments have been the basis for a number of prominent recent models of financial intermediation.20 In this section I will describe
a simple model of financial intermediaries using the economic environment
laid out in Section 1. In Section 2 I took the same environment, adopted a
particular definition of equilibrium—the Wilson equilibrium—and showed that
government loan market intervention could be Pareto-improving. In this section
I adopt a different definition of equilibrium; this is the only difference between
the two models. Under the definition adopted here, financial intermediaries
can emerge endogenously in equilibrium. Furthermore, there is no welfareenhancing role for government intervention, since equilibrium allocations turn
out to be Pareto-optimal.
18 Insufficient
collateral—a low value of K—often gives rise to borrowing constraints.
results presented in this section are due to joint ongoing work with John Weinberg.
20 See Boyd and Prescott (1986), Boyd, Prescott, and Smith (1988), and Lacker and Weinberg (1993). One should add Hajime Miyazaki (1977), who interprets cross-subsidizing wageemployment contracts as an “internal labor market,” in other words, a firm. Adverse selection
is not the only possible approach; Diamond (1984) and Williamson (1986) present models of
endogenous financial intermediaries based on costly verification and delegated monitoring.
19 The
J. M. Lacker: Adverse Selection in Loan Markets
81
Financial Intermediaries Are Inhibited Under the Wilson Equilibrium
One hallmark of financial intermediaries is that almost all of their assets and
liabilities are financial claims, as opposed to physical assets. Because financial
intermediaries hold large portfolios, they do not necessarily need to break even
on each individual claim. In contrast, when individual claims are sold directly
by borrowers to ultimate lenders, equilibrium requires that each claim at least
break even. Cross-subsidization thus appears to be inconsistent with nonintermediated lending. Therefore, financial intermediaries should be expected to
arise whenever allocations require cross-subsidization. Adverse selection models of financial intermediaries are based on just such reasoning.21
If we want to allow for the possibility of financial intermediaries, the
equilibrium notion adopted in the previous section is clearly inadequate. The
Wilson equilibrium assumes that lenders and borrowers can only enter into bilateral financial contracts. Multilateral financial arrangements are precluded by
assumption. For simplicity, lenders in our environment have only one unit each
to lend, exactly the amount each borrower wants to borrow. Thus no lender
offers more than one contract. It would make no difference for the models,
however, if each lender was large relative to borrowers and made many loans.
More to the point, the Wilson equilibrium imposes a particular game on
the agents in the economy. Agents are assumed to interact through a specific
sequence of moves and countermoves governed by a specific set of rules. In
particular, the game underlying the Wilson equilibrium specifies that in the third
stage lenders are able to withdraw individual loan contracts. This prevents a
lender from offering a menu of contracts as a whole in the first stage and
precommitting not to drop any single contract. This feature gives rise to the
break-even constraint, which implies a welfare-enhancing role for government
intervention. The same feature rules out the cross-subsidizing allocations associated with financial intermediaries.
In many instances, participants in real world economies interact within
highly structured institutions, governed by rules, laws, customs, and so forth.
A wide variety of market institutions come to mind, from decentralized search
markets, to trading fairs, to auctions, to highly centralized (and organized)
open-outcry markets. Many of these institutional arrangements can easily be
cast as games since they impose binding restrictions on the interaction of participants. Game theory is obviously quite useful for analyzing the implications
of alternative institutional and market structures.
However, when we are interested in predicting institutional arrangements,
when we want a model of which game agents will play, we need a different
21 The situation is analogous to a multiproduct firm with economies of scope across products.
In a sustainable equilibrium one product might be subsidized in the sense that the price is less
than the stand-alone marginal cost.
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Federal Reserve Bank of Richmond Economic Quarterly
approach. Indeed, outcomes in adverse selection environments are known to be
particularly sensitive to the assumed “market convention,” as Wilson’s subsequent (1979) research demonstrated. He showed that equilibrium can be very
different depending on whether the informed agents (the borrowers in our environment) or the uninformed agents (the lenders) propose contracts. Wilson’s
1979 results stand as a strong warning about the reliability of predictions from
adverse selection models in which equilibrium is identified as the outcome of
one particular game.
A Different Definition of Equilibrium: The Sustainable Equilibrium22
A different definition of equilibrium is needed then, a different way of selecting
a predicted outcome for this environment, one that allows for the possibility of
financial intermediaries. Three ideas guide the definition described below. First,
I want to be agnostic about the game agents might play to implement the resulting allocation, since imposing a particular game could arbitrarily restrict the
allocations agents can achieve. Second, there should be some notion of competition between rival financial intermediaries. If a financial intermediary is part of
an equilibrium, there must be no other rival financial intermediary that “beats” it
by doing better for the agents involved. Third, for a rival financial intermediary
to beat a candidate intermediary, the rival must be “credible” in the sense that it
cannot be beaten by any other (credible) rival intermediary. If a rival intermediary is itself vulnerable to another rival, it cannot be taken seriously as a threat
to overturn the candidate allocation. It is important to note that the credibility
requirement is imposed on any subsequent proposed rival intermediary.
While the definition of the Wilson equilibrium was stated in terms of contracts, the definition of the sustainable equilibrium can be stated more clearly in
terms of coalitions and allocations. A coalition is simply a collection of some
or all of the agents in the economy. Let n designate a typical coalition; n is a
list of the names of each agent in the coalition. Let N designate the coalition
consisting of every agent in the economy, that is, the coalition of the whole.
An allocation for a given coalition is a list of the consumption plans of all
agents in a coalition, together with their investment decisions. An allocation is
equivalent to specifying all of the contracts among agents in a coalition. Let a
designate a typical allocation.
The central ingredient in the definition of a sustainable equilibrium is
the idea of blocking, which captures the notion of competition between rival
22 The
equilibrium described here was introduced in Lacker and Weinberg (1993) and is
related to the idea of “coalition-proof Nash equilibrium” formulated by Bernheim, Peleg, and
Whinston (1987) and Greenberg (1989). Kahn and Mookherjee (1991) independently developed
a closely related equilibrium notion for games in adverse selection environments also based on
the coalition-proof idea, but their approach differs enough in certain details that their results have
a very different flavor.
J. M. Lacker: Adverse Selection in Loan Markets
83
coalitions. Intuitively, an allocation for a given coalition can be blocked by a
subcoalition if the subcoalition can feasibly make all its members at least as
well off and some strictly better off. In an adverse selection environment this
idea must be specified with some care. A key consideration is that allocations
for a subcoalition are limited by the incentive and self-selection constraints, as
are all allocations. An additional consideration arises, however. If a subcoalition
is proposed, will any of the agents left behind in the original coalition wish to
misrepresent themselves in order to gain entry into the deviating subcoalition?
If so, the self-selection constraints for the subcoalition will be undermined,
making the proposed deviation infeasible. The following definition of blocking
takes these considerations into account.
Definition: An allocation a for coalition n is blocked by a subcoalition n ,
together with an allocation a , if:
(i) the blocking allocation a satisfies the incentive and self-selection constraints and is resource feasible for n ;
(ii) all agents in n are at least as well off under a as they would be under a,
and at least one agent is made strictly better off; and
(iii) no agents that the subcoalition leave behind in the original coalition
could make themselves better off by joining the subcoalition, including
by claiming to be a different type.
Conditions (i) and (ii) are standard. Condition (iii) implies that if one type
of agent is made strictly better off, the coalition attracts all of that type of
agent. Condition (iii) also implies that if a deviating subcoalition wants to
attract some, but not all, of a given type of agent, they must make that type
of agent indifferent between joining the subcoalition (truthfully) and receiving
the original allocation. Also, that type of agent must have no incentive to join
the subcoalition by claiming to be another type of agent in the subcoalition.
Condition (iii) merely extends the self-selection constraints to cover potential
blocking subcoalitions; it recognizes a subcoalition’s vulnerability to strategic
behavior.
I am now ready to define a sustainable equilibrium. In order to do so
I must define the sustainable allocations for each possible subcoalition, as
well as for the coalition of the whole. The sustainable equilibrium is then just
the sustainable allocation for the coalition of the whole. Let s(n) denote the
set of sustainable allocations for coalition n. The definition is then simple:
an allocation is sustainable if it is not blocked by any subcoalition together
with a sustainable allocation for that subcoalition. The mapping s(n) is defined
formally as follows.
Definition: The mapping s(n) is the set of sustainable allocations for each
coalition n if it satisfies the following properties:
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Federal Reserve Bank of Richmond Economic Quarterly
(i) allocations a in s(n) satisfy the incentive and self-selection constraints
and are resource feasible for the coalition n and
(ii) an allocation a is in s(n) if and only if there does not exist a subcoalition
n together with an allocation a in s(n ) such that (a , n ) blocks a.
Condition (i) merely states that sustainable allocations must satisfy resource
and informational constraints. Condition (ii) captures the notion of credibility.
An allocation is sustainable if it is not blocked by any subcoalition together
with an allocation that is sustainable for that subcoalition. If an allocation is
blocked by such a subcoalition and allocation, then it is not sustainable.
A sustainable equilibrium, then, is any allocation that is sustainable for the
population as a whole.23
What Does the Sustainable Equilibrium Look Like?
It turns out that there is a simple way to find the sustainable equilibrium for
our economy. Solutions to a particular maximization problem, shown below,
are sustainable allocations.
The Miyazaki Problem:
MAX
pg u(R − rg + K) + (1 − pg )u(K − Cg )
s. t.
pb u(R − rb + K) + (1 − pb )u(K − Cb )
≥ pb u(R − rg + K) + (1 − pb )u(K − Cg )
[ pg rg + (1 − pg )βCg ]Ng + [ pb rb + (1 − pb )βCb ]Nb ≥ ρ(Ng + Nb )
rh ≤ Ch
h = g, b
pb u(R − rb + K) + (1 − pb )u(K − Cb ) ≥ V 0b ,
(10)
(11)
(12)
(13)
23 The sustainable equilibrium is closely related to the core—the set of allocations that are
simply unblocked. The core is empty in the cases in which the Rothschild-Stiglitz equilibrium
does not exist—that is, the cases in which the Wilson equilibrium is a pooling allocation. It
should be clear that the set of sustainable equilibria always contains the set of core allocations,
when they exist, because the latter allows “easier” blocking. Townsend (1978) studies the core in
a perfect information economy with fixed costs of bilateral exchange. There, intermediaries are
required to overcome the nonconvexity. Interestingly, he describes a noncooperative game that
allows contract proposals to include multilateral financial arrangements. The equilibrium of the
noncooperative game attains the core allocation, thus bridging the gap between the game-theoretic
and cooperative approaches. Boyd and Prescott (1986), Boyd, Prescott, and Smith (1988), and
Marimon (1988) also study core-like equilibria in adverse selection environments. Given the
definition of blocking, the core is the set of unblocked allocations. Unfortunately, the core is
often empty in our economy, as it is in many adverse selection economies.
J. M. Lacker: Adverse Selection in Loan Markets
85
where
V 0b ≡ MAX
s. t.
pb u(R − rb + K) + (1 − pb )u(K − Cb )
pb rb + (1 − pb )βCb ≥ ρ
rb ≤ Cb .
The Miyazaki Problem maximizes the expected utility of the good borrowers.
The first constraint (10) states that bad borrowers have no incentive to pretend
to be good borrowers. The second constraint (11) is just resource feasibility.
The third constraint (12) ensures repayment incentives. The fourth constraint
(13) states that the bad borrowers receive no less expected utility than V 0b ,
the expected utility they would receive if they were on their own. V 0b is the
maximum expected utility for bad borrowers under a contract that breaks even
and respects the incentive constraint. It should be apparent that V 0b is equal
to Vb∗ , the expected utility under the Wilson equilibrium separating contract,
(rb∗ , C∗b ) in Figure 3.
Hajime Miyazaki, in a 1977 paper in the Bell Journal of Economics, proposed that equilibrium be defined as solutions to an analogous problem in an
adverse selection labor market economy. He argued that employers (analogous
to lenders in our economy) are able to offer cross-subsidized wage-employment
schedules, a situation he identified as an “internal labor market”—in other
words, a multilateral financial arrangement. The “Miyazaki equilibrium,” as it
has come to be called, has been neglected in the adverse selection literature
because cross-subsidization seemed hard to reconcile with a narrow conception
of competitive behavior.
Using a few key properties of the sustainable equilibrium, there is a simple
procedure that finds it. One important property is that bad borrowers receive
contracts on the 45◦ line, minimizing the risk they bear and the collateral they
transfer. Any other contract providing the same expected utility for the bad
borrowers would use more resources and would thus make good borrowers
worse off.
The first step in the procedure to find a sustainable equilibrium is to trace
out the set of contracts that are feasible for the good borrower, shown as a
dashed line in Figure 7. This set is constructed by varying the bad borrower’s
contract along the 45◦ degree line between (rb∗ , C∗b ), the separating Wilson
equilibrium contract, and (r̄, C̄), the pooling contract on the 45◦ line. Start by
taking the contract (rb∗ , C∗b ) for the bad borrower as given. Then the best possible
contract for the good borrower is (rg∗ , C∗g ), where both the self-selection and
the resource constraints bind. Now consider the contract (rb1 , C1b ) for the bad
borrower, a short distance down along the 45◦ line from (rb∗ , C∗b ). Since (rb1 , C1b )
lies below πb , the overall resource constraint is tightened; now contracts along
πg1 are feasible for the good borrower. Because the bad borrower’s self-selection
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 The Feasible Allocations
r
45º
(r 1 ,C 1 )
b
b
πb
V*
b
2
(r b2 ,C b
)
,C *
)
(r *
b
b
π gb
1
Vb
¯¯
(r,C)
V b2
π g2
π g1
πg
(r g2 ,C g2 )
1)
(r g1 ,C g
+
*
(r *
g ,C g )
C
Notes: This figure shows the construction of the dashed line—the set of the best feasible contracts
for the good borrower. Each point on the dashed line corresponds to a particular bad-borrower
contract along the 45◦ line. For the bad-borrower contract (rb1 , C1b ), the best contract for the good
borrower is (rg1 , C1g ), where the bad-borrower indifference curve Vb1 intersects the break-even line
πg1 . Similarly, for bad-borrower contract (rb2 , C2b ), the best feasible contract for the good borrower
is (rg2 , C2g ).
constraint is relaxed, the best possible contract for the good borrower is now
(rg1 , C1g ), up and to the left of (rg∗ , C∗g ). Continuing this procedure for every badborrower contract between (rb∗ , C∗b ) and (r̄, C̄) traces out the dashed line, the set
of the best possible good-borrower contracts for various levels of bad-borrower
utility.
The second step is to select the contract along the dashed locus that maximizes the good borrower’s expected utility; the associated allocation is the
sustainable equilibrium. The best contract for the good borrower is shown as
(rgs , Csg ) in Figure 8, where the dashed locus is tangent to a good-borrower
indifference curve. The bad borrower receives (rbs , Csb ). Depending on the ratio
of bad borrowers to good borrowers, the sustainable equilibrium could instead
be at either of the endpoints of the dashed line. If the ratio of bad borrowers
to good borrowers is relatively large (the range labeled “very high” in Table
1), the dashed line is very steep and the sustainable equilibrium is the set of
J. M. Lacker: Adverse Selection in Loan Markets
87
Figure 8 The Sustainable Equilibrium
r
45º
πb
s
Vb
V*
b
(r *
,C *
)
b
b
π gb
s
s
Vg
¯ ¯
(r,C)
s
πg
s
(r b ,C b )
s
(r g ,C g )
¯ )
(r¯g ,C
g
ˆ )
(rˆg ,C
g
s
Vb
s
Vg
*
(r *
g ,C g )
C
+
Notes: The contract along the dashed line that maximizes the expected utility of the good borrower
is the sustainable equilibrium: (rgs , Csg ). The associated contract for the bad borrower is (rbs , Csb ).
The contract (r̂g , Ĉg ) for the good borrowers fails to credibly block the sustainable equilibrium
because it in turn is credibly blocked by (r̄g , C̄g ), the sustainable allocation for the coalition of
just good borrowers.
break-even separating contracts, the same as the Wilson separating equilibrium.
This case is shown in Figure 9. If there are few bad borrowers (the range labeled
“very low” in Table 1), the dashed line is relatively flat and the pooling contract
on the 45◦ line is the sustainable equilibrium, the contract (r̄, C̄) in Figure 10.
Table 1 summarizes the different types of sustainable equilibria for various
values of the ratio of bad borrowers to good.
What prevents lenders from skimming off the good borrowers, offering
a contract they prefer and which earns excess profits (a contract like [r̂g , Ĉg ]
in Figure 8)? Such a deviation lacks credibility because it does not meet the
sustainability requirement defined above. If such a coalition were to form, it
would consist entirely of good borrowers, but it would be vulnerable to the
sustainable allocation for that coalition—the contract (r̄g , C̄g ) at the intersection of the good-type break-even line, πg , and the 45◦ line. In other words,
agents would anticipate that if the proposed deviation (r̂g , Ĉg ) were to occur,
it would itself be blocked by a subcoalition proposing (r̄g , C̄g ). Since the latter is
88
Federal Reserve Bank of Richmond Economic Quarterly
Figure 9 The Sustainable Equilibrium for Nb /Ng “Very High”
r
45º
π gb
πb
V*
b
s
s
(r b ,C b )
V*
g
¯ ¯
(r,C)
πg
s
s
(r g ,C g )
C
+
Notes: When Nb /Ng is very high, the dashed line is steep and the best contract for the good borrower lies at the lower endpoint. In this case, sustainable equilibrium is identical to the separating
Wilson equilibrium and is Pareto-optimal, and financial intermediaries are unnecessary.
sustainable, that threat is credible and succeeds in blocking the cream-skimming
deviation. There is no sustainable allocation that attracts the good borrowers
away from the equilibrium contract (rgs , Csg ). Attracting just the bad borrowers
is unsuccessful, since any contract that they alone prefer earns negative profits. Finally, there is no pooling contract that would succeed in attracting the
good borrowers, since every feasible pooling allocation gives them utility lower
than Vgs .
Government Intervention Is Never Pareto-Improving
The sustainable equilibrium is always Pareto-optimal. The Miyazaki Problem
maximizes the expected utility of the good borrowers subject to resource, incentive, and self-selection constraints, and a participation constraint for the bad
borrowers. No other feasible allocation yields higher expected utility for good
borrowers without violating the bad borrower’s participation constraint. The
sustainable equilibrium is not necessarily the best possible allocation for the
bad borrowers, but any other feasible allocation that provides higher expected
Ratio of Bad to Good Borrowers, Nb /Ng
Very Low
Low
Intermediate
High
Very High
pooling,
45◦ line
pooling,
45◦ line
pooling,
below 45◦
line, Fig. 4
separating,
break-even,
Fig. 3
separating,
break-even,
Fig. 3
Is the Wilson equilibrium Pareto-optimal?
yes
yes
no
no
yes
Can government intervention
be Pareto-improving?
no
no
yes
yes
no
Wilson equilibrium
Sustainable equilibrium
pooling,
45◦ line,
Fig. 10
separating, cross-subsidizing,
Fig. 8
separating,
break-even,
Fig. 9
Is the sustainable equilibrium Pareto-optimal?
yes
yes
yes
Are financial intermediaries necessary?
no
yes
no
J. M. Lacker: Adverse Selection in Loan Markets
Table 1 Properties of Equilibria as Nb /Ng Varies
89
90
Federal Reserve Bank of Richmond Economic Quarterly
utility for the bad borrower must provide lower expected utility for the good
borrower.24
Government intervention in whatever form it takes must respect the resource, incentive, and self-selection constraints of the environment. Since the
sustainable equilibrium is Pareto-optimal with respect to those constraints, there
are no allocations that the government can feasibly achieve that Pareto-dominate
the sustainable equilibrium. In contrast, sometimes Wilson equilibrium allocations are not Pareto-optimal with respect to resource, incentive, and selfselection constraints; in exactly these cases government intervention can make
all agents better off (see Table 1).
When Do Financial Intermediaries Arise?
In some cases the sustainable equilibrium involves cross-subsidization across
contracts. This occurs whenever the sustainable equilibrium is a pair of distinct
contracts that do not lie on the individual break-even lines, πg and πb . In
these cases the sustainable equilibrium is somewhere along the dashed line,
as in Figure 8. When the ratio of bad to good borrowers is very high, the
sustainable equilibrium is a pair of contracts that each break even, as in Figure
9; no cross-subsidization occurs in this case. When the ratio of bad to good
borrowers is very low, as in Figure 10, the sustainable equilibrium is a single
pooling contract and so cross-subsidization occurs.
Financial intermediaries are required whenever the sustainable equilibrium
involves cross-subsidy across contracts. In this case a financial intermediary can
break even on a portfolio of loans to both good and bad borrowers even though
individual contracts do not break even; the bad contract earns positive expected
profits while the good contract earns negative expected profits. Direct lending,
with each investor making a single loan, is inconsistent with cross-subsidized
contracts, since no lender would make a single loan earning negative profits.
The allocation that can be achieved by direct bilateral lending, the Wilson
equilibrium, cannot be a sustainable equilibrium in this case because it can
be blocked by a financial intermediary offering contracts preferred by both
borrowers.
For extreme values of the ratio of bad to good borrowers, financial intermediaries are not necessary to achieve the sustainable equilibrium. When the ratio
is very high, each contract breaks even in the sustainable allocation. Individual
lenders know they will break even on the borrowers that request the loans they
offer. When the ratio of bad to good borrowers is very low, lenders offering the
24 All of the allocations corresponding to contracts along the dashed line between the sustainable equilibrium and the pooling contract on the 45◦ line are Pareto-optimal. For a very high
ratio of bad borrowers to good borrowers, as in Figure 9, all of the dashed line corresponds to
Pareto-optimal allocations. For a very low ratio of bad borrowers to good borrowers, as in Figure
10, only the pooling contract on the 45◦ line is Pareto-optimal.
J. M. Lacker: Adverse Selection in Loan Markets
91
Figure 10 The Sustainable Equilibrium for Nb /Ng “Very Low”
r
45º
V*
b
πb
V*
g
V
s
g
π gb
V
s
b
¯ ¯
(r,C)
πg
s
Vb
s
Vg
C
+
Notes: When Nb /Ng is very low, the dashed line is relatively flat and the best contract for the good
borrower is at the upper endpoint; both borrowers receive the contract (r̄, C̄). In this case, the
sustainable equilibrium is identical to the Wilson equilibrium and is Pareto-optimal, and financial
intermediaries are unnecessary.
pooling contract make excess profits on good borrowers and negative profits
on bad borrowers. Lenders do not know which type of borrower accepts their
loan, but if they believe that the probability that a given borrower is of a given
type is the same as that type’s representation in the population, then ex ante
expected profits are zero.25
Financial intermediaries arise in all cases in which the Wilson equilibrium
is not Pareto-optimal. This occurs for ranges of the ratio of bad to good borrowers labeled “intermediate” and “high” in Table 1. In these situations the Wilson
equilibrium can be improved upon by government intervention. For the same
reason government intervention is Pareto-improving, the Wilson equilibrium
allocation is unsustainable because it is vulnerable to a financial intermediary
offering a Pareto-improving set of contracts. Thus whenever the Wilson equilibrium suggests a role for government intervention, the sustainable equilibrium
25 Note
required.
that financial intermediaries could be operative in this equilibrium, but they are not
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Federal Reserve Bank of Richmond Economic Quarterly
suggests a role for financial intermediaries and no role for government intervention. The welfare-enhancing role of government intervention in the Wilson
equilibrium is the direct result of restrictions that prevent the emergence of
financial intermediaries.
4.
CONCLUDING REMARKS
In this article I examined a single economy under two different definitions of
equilibrium. The Wilson equilibrium assumes that agents interact by playing a
specific four-stage game. Equilibrium allocations are often not Pareto-optimal
under this definition of equilibrium, and a government tax/subsidy scheme can
be Pareto-improving. Under the other equilibrium, agents are free to communicate and propose alternative arrangements, and outcomes are required to be
sustainable in a certain sense. The sustainable equilibrium is Pareto-optimal and
implies no welfare-enhancing role for government intervention. The sustainable equilibrium also gives rise to financial intermediaries, a widely observed
phenomenon in loan markets. By contrast, conditions implicit in the Wilson
equilibrium prevent intermediaries from playing any role. This observation
suggests that the Wilson equilibrium, and others like it, are too restrictive and
that models based on them are unreliable guides to policy. Thus, on the basis of
these considerations, I conclude that adverse selection does not justify government intervention in loan markets. Intervention could, of course, be desirable
on redistributive grounds.
One might wonder if the approach advocated here is somehow rigged
to minimize the potential efficiency role of the government. The notion of
sustainability places only minimal restrictions on agents’ interactions. Does
this approach give private agents an unrealistic capacity to coordinate their
activities to achieve the best of all possible allocations? Are these assumptions
Panglossian?
This is a legitimate question to raise. To put the question another way,
Under what conditions would such an approach ever predict that government
intervention is welfare-improving? One response is to give the models a normative rather than positive interpretation. In other words, treat the model as if
it were telling us the best allocation. If we are confident the primitive assumptions on preferences, endowments, and technologies match well with the actual
economy and we observe that the recommended allocation is not being attained,
then government intervention to achieve the optimal allocation is warranted.
The difficulty with this approach, however, is that without a positive model
of why the observed allocation falls short of the one recommended by the
economist, we can have little confidence that we have accurately identified all
of the relevant impediments to trade. Without such confidence, we are forced
to rely on ignorance or irrationality to explain out-of-equilibrium observations.
J. M. Lacker: Adverse Selection in Loan Markets
93
An alternative response is to view government intervention as a potential
outcome, an endogenous component of the equilibrium multilateral arrangement. Indeed, in many models it is hard to distinguish between endogenous
financial intermediaries and government-mandated reallocations. A rationale
for government intervention would require a model in which government actions and private contracts are clearly distinguishable, perhaps in the methods of
enforcing contracts. A case for government intervention could then be made if a
plausible model predicted allocations that could not be achieved through private
arrangements alone, but instead required identifiably governmental arrangements. I know of no such model at the present time that justifies government
intervention in loan markets.
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