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Zero Inflation and
the Friedman Rule:
A Welfare Comparison
Alexander L. Wolman

A

distinct trend in recent years has been for central banks to emphasize low and stable inflation as a primary goal. In many cases zero
inflation—or price stability—is promoted as the ultimate long-run goal
(Federal Reserve Bank of Kansas City 1996). Economic theory also stresses
the benefits of low inflation. However, in contrast to the current fashion among
central banks, one of the most famous—and robust—results in monetary theory
is that the optimal rate of inflation is negative: in many economic models in
which money plays a role, welfare is maximized when the inflation rate is low
enough so that the nominal interest rate is zero. Central bankers are certainly
aware of this result, yet they never seriously advocate a long-run policy of
deflation (negative inflation).
How much welfare is lost from a zero inflation policy as opposed to an
optimal deflation policy? As shown below, the shape of the economy’s money
demand function with respect to nominal interest rates holds the key to answering the question. Lucas (1994) argues for a specification where real balances
increase toward infinity as the nominal interest rate approaches zero. He finds
that zero inflation is not much of an improvement over moderate inflation
but that optimal deflation offers sizable benefits. The analysis in this article
supports a different conclusion: reducing inflation from a moderate level to
zero entails substantial welfare benefits, and the additional benefit achieved
by optimal deflation is small. My analysis is based on estimating a general

This article is based on the third essay in my 1996 doctoral dissertation at the University of
Virginia. I would like to thank Robert King, my dissertation advisor, for his support. Thanks
also to Michael Dotsey, Robert Hetzel, Andreas Hornstein, and Thomas Humphrey for their
comments. The views expressed here are the author’s and not necessarily those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 83/4 Fall 1997

1

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Federal Reserve Bank of Richmond Economic Quarterly

money demand function that nests the one preferred by Lucas. The estimates
imply a satiation level of real balances, which proves to be important for the
comparison of zero inflation and optimal deflation.1
The original analysis of the relationship between money demand and the
welfare cost of inflation is credited to Bailey (1956). I review both Bailey’s
analysis and that of Friedman (1969), whose “Friedman rule” is the famous
result previously mentioned. I then describe informally Lucas’s (1994) recent
work on quantifying the costs of deviating from the Friedman rule. Whereas
Lucas’s work is guided by inventory theory, my own estimates follow from a
broader interpretation of the transactions-time approach to money demand. I use
these estimates for welfare analysis similar to Lucas’s. Although the analysis
suggests that the Friedman rule may not offer much of a benefit in comparison
to zero inflation, it does not explain why central banks do not choose to pursue
deflation. I thus point out several channels absent from my analysis through
which inflation may have welfare effects. These additional channels may help
to explain why central banks seem content to shoot for zero inflation.

1.

MONEY DEMAND AND THE WELFARE
COST OF INFLATION

Bailey (1956) showed how a money demand relationship could be used to
derive estimates of the welfare cost of inflation. He assumed a money demand
function that gave real balances (M/P, where M is the nominal quantity of
money and P is the price level) as a function of the nominal interest rate (R)
and made a consumer surplus argument: just as the area under the demand curve
for any good measures the total private benefits of consuming that good, so
the area under a money demand curve represents the private benefit of holding
money. At a nominal interest rate of 5 percent, since people are willingly giving
up 5 cents per year per dollar of money held, the marginal benefit of holding
the last dollar must be 5 cents per year. Similarly, at a nominal interest rate
of zero, people are not giving up any interest payments to hold money, so the
marginal benefit of holding the last dollar must be zero. At a social optimum,
the marginal benefit to society of holding money should equal the marginal cost
to society of producing money. With the reasonable simplifying assumption that
the cost to society of producing money is zero, the optimal nominal interest rate
is zero.2 In a steady state the nominal interest rate is approximately equal to
the real interest rate plus the inflation rate, so optimal policy, commonly known
as the Friedman rule, involves deflation at a rate equal to the real interest rate.
1 Chadha, Haldane, and Janssen (1997) have performed an analysis similar to this article
using U.K. data. They emphasize a distinction between short-run and long-run money demand.
2 Lacker (1996) reports manufacturing and operating costs for coin and currency of approximately 0.2 percent of face value.

A. L. Wolman: Zero Inflation and the Friedman Rule

3

With a nominal interest rate of zero as the optimal policy, it is possible to
measure the cost of any inflation rate for a particular money demand function.
Simply measure the area under the inverse money demand curve between the
real balances corresponding to the Friedman rule and the real balances corresponding to the nominal interest rate in question.3 That is, add up all of the
marginal benefits that are foregone by following a suboptimal policy; those
marginal benefits are measured by the nominal interest rate (the inverse money
demand function) at each level of real balances.4 At this point the term “cost
of inflation” may seem misleading; according to the theory sketched above, it
would be more appropriate to use the term “cost of positive nominal interest
rates.” Since the former term is so widely used, however, I will stick with it.
Particular theories of money may imply more complicated money demand
relationships than the one assumed by Bailey; for example, the analysis in
Section 3 will involve consumption and the real wage as arguments in the
money demand function. However, it is still the case that the Friedman rule is
optimal, and holding consumption and the real wage constant, the area under
the inverse money demand curve still provides an approximate measure of the
direct cost of inflation.5
While the optimality of the Friedman rule holds as long as real balances
are a decreasing function of the nominal interest rate (subject to the caveats
in Section 5), the welfare costs of inflation can vary with the money demand
function in two ways. First, the overall benefit of reducing inflation from, say,
10 percent to the Friedman rule can vary. Second, the apportionment of that
benefit may vary, in the following sense. According to one money demand
function, reducing inflation from 10 percent to zero may generate 99 percent
of the total welfare benefit, with the remaining reduction to the Friedman rule
adding essentially nothing. Another function could reverse this; reducing inflation from 10 percent to zero might generate only 1 percent of the total welfare
benefit, with the remaining reduction to the Friedman rule being crucial for
generating any significant benefits. This article is concerned mainly with the
latter issue.
Lucas (1994) contrasts the welfare implications of two particular money
demand functions, both of which specify the ratio of real balances to real

3 The standard money demand curve expresses real balances as a function of nominal interest
rates, whereas the inverse money demand curve inverts this relationship to express nominal interest
rates as a function of real balances.
4 This measure of the cost of inflation does not take into account indirect effects of inflation,
as will be explained in Section 4. I thus refer to the area under the money demand curve as a
measure of the direct cost of inflation.
5 If the money demand relationship involves variables other than the nominal interest rate,
the area under the inverse money demand curve (R(m)) only approximates the direct cost of
inflation, because these other variables will generally vary across different values of the nominal
interest rate.

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Federal Reserve Bank of Richmond Economic Quarterly

consumption as a function of the nominal interest rate. The ratio of real balances to consumption is used because the money demand functions discussed
here are assumed to apply to long-run data, and in the long run real balances
move roughly one for one with consumption.6 In the first specification, semilog, there is a fixed relationship between the change in the nominal interest rate
and the percentage change in the real balances to consumption ratio. That is,
if the nominal interest rate rises from zero to 1 percent, the percent decrease
in real balances/consumption is the same as if the nominal interest rate rises
from 5 percent to 6 percent. In the second specification, log-log, there is a fixed
relationship between the percentage change in the nominal interest rate and the
percentage change in the real balances to consumption ratio. Thus an increase
in the nominal interest rate from zero to 1 percent will cause a much larger
percentage drop in real balances/consumption than an increase in the nominal
interest rate from 5 percent to 6 percent. Note that if the log-log relationship is
taken literally, the ratio of real balances to consumption must be infinite when
the nominal interest rate is zero.
How do the two specifications compare in terms of welfare? With the loglog function, a slight increase in the nominal interest rate near zero generates
a tremendous decline in the ratio of real balances to consumption. Using Bailey’s (1956) reasoning, there must be a significant welfare cost of deviating just
slightly from the Friedman rule. The semi-log specification generates smaller
costs of slight deviations from the Friedman rule but roughly the same benefits of reducing inflation from, say, 5 percent to zero. Lucas argues that for
the United States, the log-log specification fits the data more closely than the
semi-log specification.7 Most of the benefits to reducing inflation would then
accrue only if the inflation rate were made negative, as it would need to be in
order to achieve the Friedman rule. In his own words, “log-log demand implies
a substantial gain in moving from zero inflation to the Friedman optimal deflation rate needed to bring nominal interest rates to zero, while under semi-log
demand this gain is trivial” (Lucas 1994, p. 5).
Is log-log demand an accurate characterization of the data? Lucas argues
that it is more accurate than semi-log demand, but is it reasonable to restrict
the search to those two alternatives? Answering these questions requires one
to be explicit about a model of money demand.

6 Lucas refers to the ratio of real balances to income. In his model there is no investment,
so consumption equals income. The model I will use does have investment, and the appropriate
ratio will be real balances to consumption rather than real balances to income.
7 Lucas (1994, p. 3) plots semi-log and log-log functions for various interest semi-elasticities
and elasticities and concludes that “the semi-log function . . . provides a description of the data
that is much inferior to the log-log curve.”

A. L. Wolman: Zero Inflation and the Friedman Rule

2.

5

THE TRANSACTIONS-TIME APPROACH TO
MONEY DEMAND

Economists have developed a wide range of models of money, none of them
entirely satisfactory. The models that are most appealing in terms of their
microfoundations—that is, their descriptions of the obstacles that individuals
overcome by holding money—tend to be ill-suited to quantification (e.g., estimating the welfare cost of inflation in the United States). An example is the
search-theoretic class of models developed by Kiyotaki and Wright (1989).8 On
the other hand, those models that are easiest to quantify do not convincingly describe the obstacles that cause individuals to hold money. Examples include the
money-in-the-utility function and cash-in-advance approaches (Sidrauski 1967
and Lucas and Stokey 1983, respectively). A middle ground is the transactionstime approach, developed by McCallum (1983) and McCallum and Goodfriend
(1987). Their fundamental assumption is that consumption requires time spent
shopping (or transacting), and transactions time may be decreased by holding a
greater quantity of real balances. The analysis in this article will be conducted
in the transactions-time framework.
Denoting transactions time in period t by ht , and the transactions-time
function by h(c, m), the assumptions that transactions time is increasing in consumption and decreasing in real balances mean that ∂h/∂c > 0 and ∂h/∂m < 0.
I make the further assumption that the function is homogeneous of degree zero
in c and m: if c and m increase or decrease by the same percentage, then
transactions time is unchanged. It follows that only the ratio of m to c matters:
ht = h(mt /ct ). Lucas (1994) shows that the transactions-time approach can be
explicitly linked to earlier inventory-theoretic models of money demand developed by Baumol (1952) and Tobin (1956). The simplest inventory-theoretic
model corresponds to the transactions-time technology,
h(mt /ct ) = κ · (mt /ct )−1 ,

(1)

where κ can be interpreted as a fixed cost of replenishing money holdings.9
More complicated inventory-theoretic approaches can be shown to imply similar h(.) functions, with the difference being that m/c would be raised to some
power less than −1:
h(mt , ct ) = κ · (mt /ct )−1/γ , γ ∈ (0, 1).

(2)

See Lucas (1994).

8 This is not to rule out the possibility that in the future, search-based models will be useful
for quantitative exercises.
9 While McCallum and Goodfriend interpreted h() in terms of shopping time, Lucas interpreted it as going-to-the-bank time.

6

Federal Reserve Bank of Richmond Economic Quarterly

The inventory-theoretic interpretation imposes strong restrictions on the
form of the transactions-time technology and hence, as I will describe below,
on the form of the money demand function. Specifically, for the transactionstime technology, it implies that no matter how high the ratio of real balances
to consumption, there is still some additional benefit to increasing that ratio
further. Lucas (1994, p. 16) defends this implication as follows: “Managing an
inventory always requires some time, and a larger average stock must always
reduce this time requirement, no matter how small it is.” One cannot argue
with this statement, according to a narrow interpretation of what it means to
manage an inventory. However, holding a higher inventory of real balances also
requires increased resources to protect the inventory, a point made by Friedman
(1969, p. 17), who described a shopkeeper hiring guards to “protect his cash
hoard.”
Given an arbitrary transactions-time technology, the associated money demand function can be derived by specifying some additional features of the
economic environment. Assume that individuals face a budget constraint,
Pt ct + Mt +

Bt
= Mt−1 + Bt−1 + Pt wt nt + Dt ,
1 + Rt

(3)

and a time constraint,
nt + lt + ht = 1,

(4)

where Pt is the price level, Mt is nominal money balances (mt pt ), Bt is holdings
of one-period nominal zero-coupon bonds maturing at t + 1, Rt is the interest
rate on bonds, wt is the real wage, nt is the fraction of time spent working,
Dt is dividend payments from firms, lt is the fraction of time spent as leisure,
and ht is the fraction of time spent carrying out transactions. In a given period,
individuals’ sources of funds are the money balances with which they enter the
period, the bonds they redeem, the wage income they earn, and the dividends
they receive from firms. These sources fund current consumption and money
balances and bonds to carry over into the next period.
Deriving the money demand function requires knowing what it means for
an individual to hold an optimal quantity of real balances. Optimal behavior
involves balancing marginal benefit and marginal cost. What are the marginal
benefit and marginal cost of holding money? From Section 1, the marginal cost
of an additional dollar is the interest foregone in the next period (Rt ); the marginal benefit of an additional dollar is the decrease in transactions time that it
1
brings about. This decrease in transactions time is −h (mt /ct )· P ·c , and the extra
t t
time can be spent in the labor market earning the nominal wage (Pt · wt ). Since
marginal cost is measured as of the subsequent period, marginal benefit needs to
be adjusted correspondingly: current period labor earnings can be invested in the
1
bond market, so their value tomorrow is −Pt · wt · (1 + Rt ) · P ·c · h (mt /ct ) .
t t

A. L. Wolman: Zero Inflation and the Friedman Rule

7

Equating marginal cost and marginal benefit implies

−h (mt /ct ) =

Rt
ct
· ,
1 + Rt wt

(5)

which can be used to confirm the Friedman rule result: at a nominal interest
rate of zero, money holdings are chosen so that the marginal benefit of an
additional unit of money is zero.
Under the inventory-theoretic interpretation, as mentioned earlier, the marginal benefit of an additional unit of money is never zero. Combining (5) with
the specification in (2), the strictly positive marginal benefit of additional real
balances corresponds to infinite real balances at the Friedman rule (R = 0):
κ
Rt
ct
· (mt /ct )−1−1/γ =
· , γ ∈ (0, 1].
γ
1 + Rt wt

(6)

The inventory-theoretic approach has appeal, but the implication that real
balances would be infinite at the Friedman rule is extreme and argues for
considering transactions-time technologies that do not share that implication.
If real balances are finite at the Friedman rule, there is some quantity of real
balances at which the marginal benefit of holding an additional unit of real
balances is zero. That level of real balances—if it exists—will be referred to
as the satiation level. A key proposition, namely that the welfare gains from
low nominal interest rates are concentrated near the Friedman rule, depends
crucially on the assumption of no satiation level; the log-log money demand
function does not have satiation, whereas the semi-log function does.
The log-log function is roughly consistent with inventory theory: assuming that c and w are constant, and noting that Rt /(1 + Rt ) ≈ Rt , (6) yields a
nearly linear relationship between the log of real balances and the log of the
nominal interest rate. In contrast, the semi-log function is inconsistent with
inventory theory, as it posits a linear relationship between the log of real balances and the level of the nominal interest rate. Thus Lucas’s purely empirical
argument favoring the log-log specification over semi-log is strengthened by
his theoretical argument favoring the inventory approach. However, inventorytheoretic models do not offer the only alternative to semi-log money demand.
And the fact that the inventory approach implies infinite real balances at a zero
nominal interest rate suggests searching across a wider class of models. In the
next section, I present estimates of a money demand function that allows for
satiation and is consistent with the basic assumptions of the transactions-time
model. This function nests nonsatiation (log-log) as a special case, but for many
parameter values it is not consistent with inventory theory.

8

3.

Federal Reserve Bank of Richmond Economic Quarterly

ESTIMATES OF A GENERAL MONEY
DEMAND FUNCTION

From (5), in order for a transactions-time function to be consistent with satiation, it must be that for some positive value of m/c, further increases in that
ratio do not decrease transactions time. In (6), under the inventory approach,
transactions time is always decreasing in m/c, so subtracting a constant from the
left-hand side of (6) will yield a technology consistent with satiation. That is, if
h (mt /ct ) = φ − (κ/γ) · (mt /ct )−1−1/γ , with φ ≥ 0, then the implied transactionstime technology allows for satiation. Since it will be convenient below to
specify the parameters in a slightly different way, I define ν ≡ −γ/(1 + γ), and
A ≡ (κ/γ)−γ/(1+γ) , so that h (mt /ct ) = φ−A−1/ν ·(mt /ct )1/ν , with ν < 0, A > 0.
The technology can be found by integrating the previous expression:
h(mt /ct ) = φ · (mt /ct ) −

1+ν
ν
A−1/ν · (mt /ct ) ν + Ω, for mt /ct < A · φν ,
1+ν

h(mt /ct ) = Ω, for mt /ct ≥ A · φν ,

(7)

where Ω is a nonnegative constant that represents the minimum possible transactions time. This function is decreasing in mt /ct as long as mt /ct is less than
A · φν , and the satiation level of real balances is given by (m/c)s = A · φν .
If φ = 0, then there is no satiation level, and the function is consistent with
inventory theory. The implicit money demand function is given by
A−1/ν · (mt /ct )1/ν − φ =

Rt
ct
· ,
1 + Rt wt

(8)

which can be rewritten to yield an explicit money demand function:
mt /ct = A ·

ct
Rt
·
+φ
1 + Rt wt

ν

.

(9)

My strategy now is to estimate A, φ, and ν using (9) and to test the hypothesis
that there is no satiation level of real balances (φ = 0). The theory as presented
thus far suggests that (9) should hold exactly. Of course it does not; I choose to
model the error term as additive, but the estimation results do not change significantly if the error is assumed to be multiplicative. The data, which are from
the United States for the period 1915 to 1992, are described in the appendix.
Although four separate variables enter (9), for estimation purposes it is simplest to define the two composite variables, yt ≡ mt /ct and xt ≡ [Rt / (1 + Rt )] ×
[ct /wt ] . Then the estimation equation is
yt = A · (xt + φ)ν + εt .

(10)

A. L. Wolman: Zero Inflation and the Friedman Rule

9

Figure 1 displays a plot of yt versus xt . Estimates of A, ν, and φ are found by
solving the following nonlinear least squares (NLS) problem:10
T

min

yt − A · (xt + φ)ν

2

.

(11)

A,ν,φ t=1

In general, the NLS estimates are consistent and asymptotically normal, as
shown by Amemiya (1985, pp. 127–35); here I do not make any distributional
assumptions about εt , the residual. Confidence intervals for the parameters
were generated by bootstrapping, which allows one to construct a sampling
distribution without any distributional assumptions and without relying on the
accuracy of linear approximations.11
Table 1 contains estimates for A, ν, and φ, along with centered 95 percent
confidence intervals. Although the estimated value of φ is close to zero, the
implied satiation level of m/c is fairly low, 2.674. Following Amemiya (1985,
p. 136), I construct a t-test of the nonsatiation hypothesis (φ = 0). The test
statistic is 26.99, meaning that nonsatiation is overwhelmingly rejected. Using
the sampling distribution for the parameters A, φ, and ν, Figure 2 plots the
implied sampling distribution for the satiation level of m/c. According to the
sampling distribution, 90 percent of the probability mass for the satiation ratio
lies below a value of 5. However, the right-hand tail of the distribution is fat;
the x-axis would need to go all the way to 46,000 to encompass 97.5 percent of
the probability mass, meaning that the satiation level is imprecisely estimated.
This imprecision follows from the properties of the data: the lowest nominal
interest rate in the sample is 0.7 percent, and for the observations with the
lowest nominal interest rates, there is substantial variation in the ratio of real
balances to consumption.12 The solid line in Figure 1 shows the fitted values.
10 The

presence of consumption in the numerator of x and the denominator of y can cause
the NLS estimator to be biased, as it may induce a correlation between the residual (εt ) and x.
More generally, if the residual represents a shock to the transactions-time technology, then in
general equilibrium such a correlation would arise even without consumption on both sides of the
estimation equation. I have investigated these problems by estimating with instrumental variables
using the generalized method of moments (GMM). The GMM estimates are highly sensitive to
the choice of instruments, so I report only the NLS results.
11 The bootstrapping approach involves three steps. The first step is to produce the NLS
estimates. The second step is to fit an AR model to the NLS residuals, producing a new set
of disturbances, et , that are approximately white noise (an AR(2) was fit to εt to produce et ).
The final step is to draw randomly with replacement from the et , producing N new vectors, yt ,
each of size T. For each of those new samples the parameters are estimated by nonlinear least
squares. The yt are generated by combining the xt data and the random draws of et with the initial
parameter estimates.
12 Working in a different money demand framework, Mulligan and Sala-i-Martin (1996)
have developed a method of estimating the behavior of money demand near zero nominal interest
rates. Their fundamental insight is that if there is a fixed cost of holding nonmonetary assets, the
behavior of individuals who hold only monetary assets at positive nominal interest rates can yield
information about aggregate money demand at a nominal interest rate of zero.

10

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Data and Predicted Values

3.5
3.0
2.5

m/c

2.0

●
●
●●
●
● ●
●
● ●
● ●
●
●
●
●
● ●

1.5
1.0

●
●
●
● ●
● ●
●
● ● ●●
● ● ●● ● ●
● ●●
● ●● ● ● ● ●
●● ● ● ●
●
●
●
●●
●
●
● ●

nonsatiation imposed

●
●
● ●
●

●
●
●
● ●

●●● ●

●

●

unrestricted

0.5
0
0

0.001

0.002

+

0.003
0.004
R x c
1+R
w

0.005

0.006

0.007

Table 1 Unrestricted Estimates and 95 Percent Confidence Intervals
A

ν

φ

0.01702

−0.7695

0.001399

(6.1 × 10−7 , 0.421)

(−3.31, −0.20335)

(2.5 × 10−19 , 0.0131)

For comparison purposes, I also estimated A and ν under the nonsatiation
restriction. Table 2 contains the estimates, and the dashed line in Figure 1
shows the fitted values when nonsatiation is imposed. With money demand
estimates in hand, we can now look at their implications for the welfare cost
of inflation.

Table 2 Restricted Estimates: Nonsatiation
A

ν

0.2526

−0.2699

A. L. Wolman: Zero Inflation and the Friedman Rule

11

Figure 2 Distribution of Estimated Satiation Level
1.0
Cumulative Sampling Distribution

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1

2

3

4
5
Satiation Level of m/c

+

4.

6

7

WELFARE ANALYSIS

By specifying a general equilibrium model, I can use the above estimates of
the transactions-time technology to compute the exact welfare cost of inflation. I use a standard real business cycle model, as in Prescott (1986) or King,
Plosser, and Rebelo (1988), augmented by the transactions-time money demand
specification to answer the following question: in a world of constant 5 percent
inflation, how much income would an individual willingly forfeit (or require)
in order to live in a world with some lower (or higher) constant inflation rate?13
The economy consists of a representative individual who chooses consumption and money balances and leisure to maximize lifetime utility:
∞

β j u(ct+j , lt+j ),

Et
j=0

13 For the purpose of computing this welfare measure, I define full income as the sum of
consumption and w · l, where w is the real wage and l is leisure. The computation holds the
real wage constant at its benchmark level. That is, what amount of additional full income at
the old real wage would give the individual the same utility as the decrease in inflation under
consideration?

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Federal Reserve Bank of Richmond Economic Quarterly

where u(c, l) = ln(c) + ψ ln(l). This maximization is subject to the budget
constraint (3), the transactions-time technology, and the time constraint (4).
Optimal choices of consumption, leisure, bond holdings, and money holdings
imply
uc (ct , lt ) = λt · Pt · 1 + wt h (

ct 1
)( ) ,
mt mt

ul (ct , lt ) = λt · wt · Pt ,

(12)
(13)

and
1 + Rt = Et

λt
,
βλt+1

(14)

as well as the money demand relationship (5). In these expressions λt is
the shadow price of nominal wealth—the multiplier on (3). Since consumption requires a time expenditure, there is a wedge between the marginal utility of consumption and the marginal utility of wealth in (12). That wedge,
c
1
λt · wt · Pt · h ( mt ) · ( m ), is the value in utility terms of the marginal transactions
t
t
time associated with an additional unit of consumption. The efficiency condition for leisure, (13), sets the marginal utility of leisure equal to the marginal
utility of foregone earnings, and the efficiency condition for bond holding, (14),
describes the equivalence between having $1 of wealth today and $(1 + R) of
wealth tomorrow. An additional equation defines transactions time as (7).
Firms produce the economy’s single good using capital, which they own,
and labor, which they hire on a period-by-period basis, according to a constant
returns to scale production function,
yt = at f (kt , gt nt ),

(15)

where yt is output, at is a random productivity factor, kt is the capital stock,
and g is the exogenous growth rate of labor-augmenting technical progress. In
a steady state, the exogenous technical progress will mean that output, consumption, real balances, the capital stock, investment, and the real wage will
also grow at rate g. Capital accumulates according to
kt+1 = kt · (1 − δ) + it ,

(16)

where it is investment and δ is the depreciation rate. Since firms own the capital
stock, they earn rents in equilibrium; those rents are paid out as dividends to
individuals, who own the firms. Firms maximize the expected discounted stream
of future profits—all of which are paid out as dividends—where the discount
rate for period t + j is the consumer’s marginal rate of substitution between a
dollar of wealth in periods t and t + j:
∞

βj·

Vt = Max Et
j=0

λt+j
· Pt · at · f (kt , gt nt ) − wt · Pt · nt − Pt · it .
λt

A. L. Wolman: Zero Inflation and the Friedman Rule

13

This maximization is subject to (15) and (16). Thus the firm’s first-order condition with respect to next period’s capital stock is
Pt = β · Et

λt+1
· Pt+1 · at+1 · fk (kt+1 , gt+1 nt+1 ) + Pt+1 · (1 − δ) .
λt

(17)

According to (17), the decrease in current-period profit that results from a marginal increase in investment should be exactly offset by the increase in future
profits associated with a higher capital stock next period. Optimal choice of
labor input implies that the real wage equals the marginal product of labor:
at

∂f (kt , gt nt )
= wt .
∂nt

(18)

This completes the description of the model economy.
The benchmark economy has 5 percent annual inflation, and the other parameters are chosen as follows. The real growth rate (g) is 3 percent annually.
The production function is Cobb-Douglas with labor’s share equal to 2/3, and
the depreciation rate, δ, is 0.025. The preference parameters ψ and β are set
so that the real interest rate is 6.5 percent annually and steady-state hours
worked are 20 percent of the time endowment.14 All of the above values are
within the normal range chosen in the real business cycle literature. Given
the estimated parameters of the transactions-time technology, the constant of
integration (Ω) is chosen so that steady-state transactions time is 2 percent
of the time endowment, consistent with the data presented by Andreyenkov,
Patrushev, and Robinson (1989). The values of the parameters ψ, β, and Ω,
as well as the remaining endogenous variables, are found by solving for a
deterministic steady state of the system of equations given by (12)–(18), (4),
(7), and (9). To compute the welfare measure, the inflation rate alone is varied,
and the new steady state is computed at each desired inflation rate.
Figure 3 plots the quantity of full income (defined in footnote 13), as a
percentage of its benchmark level, that individuals would be willing to forego
(would require) to live in a lower (higher) inflation world. The solid line represents the unrestricted estimated money demand specification, and the dashed
line represents the restricted estimates that impose nonsatiation. Both specifications imply that if inflation were reduced to the Friedman rule from a 5 percent
annual rate, for individuals in the model economy it would be as if their full

14 The real interest rate, r, is equal to (1 + R)/(1 + π), where π is the inflation rate. As in
King and Wolman (1996), I assume that the risk-free real interest rate relevant for calculating
the opportunity cost of money holding is 1 percent annually, whereas the real rate that implicitly
enters (15) and (18) is 6.5 percent. The risk premium is not modeled explicitly. In practice this
means that there are two real interest rates in the model, but since both of them are “known,” no
equations or unknowns are added to the steady-state computation. See below for a discussion of
the implications of this ad hoc approach.

14

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Welfare Compared to Baseline of 5 Percent Inflation

0.8

Percent of Baseline Income

0.6
unrestricted

0.4
0.2
0

nonsatiation imposed

-0.2
-0.4
-0.6
-1

+

0

1

2

6
3
4
5
Percent Annual Inflation

7

8

9

10

income had increased by about 0.6 percent.15 However, the apportionment of
these benefits differs in the two cases. Under nonsatiation, less than 3/4 of this
benefit can be achieved with zero inflation, whereas the unrestricted money
demand specification implies that almost 9/10 of the benefit can be achieved
with zero inflation. While I have argued that nonsatiation is an implausible
assumption, one should keep in mind that the satiation level was imprecisely
estimated. To the extent that one believes the actual satiation level is higher
than it was estimated in Section 3, there would be higher costs associated with
zero inflation relative to the Friedman rule than are indicated by the solid line.
While the results are not sensitive to small changes in most of the model’s
other parameters, they are sensitive to the underlying real interest rate. I have
assumed that the real return on capital is 6.5 percent annually and that the riskfree real rate of return relevant for measuring the opportunity cost of holding
money is 1 percent. Since uncertainty is not explicitly incorporated into the
model, this assumption is ad hoc. The assumption is made because in the
United States these have been the average real returns on equity and Treasury
bills, respectively. Ideally, one would explicitly model the banking system and
15 The

(1994).

magnitude of these welfare benefits is similar to the magnitudes reported in Lucas

A. L. Wolman: Zero Inflation and the Friedman Rule

15

thus endogenize the spread between risky and (nominally) riskless returns. The
assumption of a 1 percent riskless real rate is important for the magnitude of
the welfare cost of inflation. If that rate were instead assumed to be 6.5 percent,
then the Friedman rule would involve a 6.5 percent deflation rate instead of
a 1 percent deflation rate. From Figure 3, this would imply a significantly
larger benefit to achieving the Friedman rule. However, the behavior of money
demand at the Friedman rule—that is, whether or not there is satiation—would
remain important regardless of the assumed value for the real rate.
There is a long tradition, discussed earlier, of measuring the cost of inflation
by the area under a money demand curve. Here that calculation would have
yielded curves almost identical in shape to those in Figure 3. However, the
area calculation would describe time saving only, without accounting for the
effect on welfare of changes in consumption. In the case of the estimates with
satiation, there is roughly a 1 percent difference in the level of consumption
between the Friedman rule and 5 percent inflation. I take this difference in
consumption into account in Figure 3. In general, the area under the money
demand curve may misstate the welfare cost of inflation because it measures
only the direct effect of increases in real balances; here the direct effect is the
decrease in transactions time and the indirect effects are summarized by the
increase in consumption. This distinction is especially important in Dotsey and
Ireland (1996), where the (endogenous) growth rate of the economy is indirectly
affected by the inflation rate. The direct effect on money demand is dwarfed by
the indirect effect on growth in their model. An additional reason for preferring
exact welfare calculations is that the area under the money demand curve does
not take into account agents’ preferences and thus cannot actually be interpreted
in terms of welfare.

5.

OTHER EFFECTS OF INFLATION

The above analysis compares different rates of steady inflation in a model where
the only welfare effects of inflation work through the demand for money. This
narrow focus was chosen to highlight the importance of assumptions about the
behavior of money demand at low nominal interest rates. However, in more
general models, the quantitative results involving money demand may vary.
Furthermore, there may be welfare effects of inflation unrelated to the demand
for money. In this section, I briefly discuss some ways in which analysis of
the welfare effects of inflation differs in more general models. The references
I provide are meant to serve as entry points to what in each case are extensive
literatures.
Much of the literature on macroeconomic models with money has involved
nominal rigidities, such as sticky prices. In contrast, the model in this article
has flexible prices. Sticky prices lead to effects of steady inflation that work

16

Federal Reserve Bank of Richmond Economic Quarterly

through other channels in addition to money demand. Models with sticky prices
usually involve imperfect competition, and inflation can affect the magnitude
of the distortion from imperfect competition. In King and Wolman (1996), for
example, the markup of price over marginal cost—which is a distortion—varies
with inflation because firms incorporate into their pricing decisions the possibility that the price they choose will remain fixed for several periods. While
some have suggested that inflation can have beneficial effects on the markup
(Rotemberg 1996; Benabou 1992), King and Wolman (1996) find the opposite
effect, as firms choose a high markup when they set price to compensate for the
deterioration that will be caused by inflation. Whether that result generalizes
to a wider class of models is an open question.
A literature beginning with Phelps (1973) extends the type of analysis
performed in this article by incorporating distortionary taxes. Inflation, or more
properly, money creation, is a source of revenue (seigniorage) for the government. Implicitly, my analysis has assumed that this revenue can be replaced
by a lump sum tax, which does not distort individual decisions. If lump sum
taxes are unavailable, so that seigniorage must be replaced by a distortionary
tax such as an income tax, then the optimal rate of inflation in principle could
be higher than that corresponding to the Friedman rule; there would be a welfare benefit to inflation counteracting the welfare cost associated with money
demand. Recent work by Chari, Christiano, and Kehoe (1996) and Correia
and Teles (1997), among others, suggests that this benefit is small enough that
the Friedman rule remains optimal with distortionary taxes for a wide range
of money demand specifications. With satiation, however, distortionary taxes
would probably make the optimal nominal interest rate positive, because with
satiation the marginal welfare cost of inflation is zero at the Friedman rule.
Feldstein (1997) has emphasized another way in which inflation interacts
with public finance, namely the costs of inflation that result from a nonindexed tax code. With a nonindexed tax code, inflation raises the effective tax
rate on both individuals and businesses. Feldstein argues that these tax-related
distortions alone cost the U.S. economy about 0.8 percent of GDP per year.
Finally, the steady-state analysis in this article leaves open the question of
transitional effects of a significant decrease in the inflation rate. These transitional effects would be small in the model used here. However, models with
sticky prices or other nominal rigidities may imply significant welfare costs
of a transition to lower inflation, with the costs depending on such factors as
how credible the disinflation is. Friedman himself stressed transitional issues:
“Any decided change in the trend of prices would involve significant frictional
distortion in employment and production” (1969, p. 45). This topic is currently
being studied intensively; see Ball (1994a,b) and Ireland (1995) for examples
of recent work. It is important to note, however, that in contrast to a one-time
cost of lowering the inflation rate, the benefit of low inflation emphasized in
this article accrues year after year.

A. L. Wolman: Zero Inflation and the Friedman Rule

6.

17

CONCLUSIONS

At positive nominal interest rates, individuals incur an opportunity cost by
holding money instead of interest-bearing securities. Since the social cost of
producing money is nearly zero, there is a divergence between the private
and social costs of holding money when nominal interest rates are positive.
Individuals choose to equate the marginal benefit of holding money with the
private cost, so positive nominal interest rates generate an inefficiency. Policymakers, by setting the nominal interest rate at zero, and so equating private
and social costs, can eliminate this inefficiency. In models where there are no
other distortions, it follows that this same monetary policy is optimal from
a welfare perspective. Lucas (1994) has argued that the form of the money
demand function implies significant welfare losses at even very low nominal
interest rates. His conclusion results from his assumption that individuals do
not become satiated with real balances as the nominal rate declines toward
zero. Equivalently, the marginal benefit of holding real balances is positive no
matter how high are individuals’ money holdings.
I have estimated the money demand function implied by a general
transactions-time technology and found evidence that the marginal benefit of
holding real balances declines to zero at a nominal interest rate of zero. In
other words, individuals can become satiated with real balances. My conclusions regarding satiation, however, are vulnerable to the criticism that zero
nominal interest rates have never occurred. Nonetheless, my results imply that
the welfare cost of low nominal interest rates is small. Most of the benefits
from reducing inflation below, say, 5 percent can be achieved with price stability
(zero inflation), and those benefits are significant. In my model a reduction in
inflation from 5 percent to zero is equivalent to an increase in consumption of
0.6 percent of output. This result helps reconcile the optimality of zero nominal
interest rates with the tendency of central banks to emphasize zero inflation.
Still, one wonders why central banks do not simply advocate the optimal policy.
Probably the explanation involves factors such as transitional costs of disinflation. This point aside, it is easier to understand why central banks would
advocate sub-optimal policy if that policy is close to being optimal.

18

Federal Reserve Bank of Richmond Economic Quarterly

APPENDIX
The data used to estimate (9) are annual, from 1915 to 1992. The nominal
interest rate is the yield on commercial paper from the National Bureau of
Economic Research (NBER) database (1915–1946) and Citibase (1947–1992).
I use nominal data for consumption, the wage rate, and the money supply;
taking ratios causes the price indexes to cancel. The consumption series consists
of three spliced series. From 1915 to 1929, I combine personal consumption expenditures per capita in 1929 dollars, with the deflator for the same. The former
is series A25 from Kendrick, reproduced in the U.S. Commerce Department’s
Long-Term Economic Growth (LTEG). The latter is series B64 from LTEG.
Both are annual series. From 1930 to 1945, I combine personal consumption
expenditures per capita in 1958 dollars, with the deflator for the same. The
former is series A26 and the latter is series B65, both from LTEG, and both
annual. Finally, from 1946 to 1992, I use personal consumption expenditures in
current dollars, divided by population. The former is series GC, from Citibase;
it is in billions of dollars and is seasonally adjusted quarterly data, which I
average to create annual data. The latter is PAN (Citibase 1946 –1991), with
data for 1992 estimated by extrapolating the average rates of change from 1990
to 1991; population is in thousands.
As mentioned above, I use nominal wage data. Also, since the raw wage
data is hourly, I multiply by the number of hours in a quarter (2,184) to get
a quarterly wage. From 1915 to 1946, I “reflate” total compensation per hour
at work for manufacturing production workers, using the CPI. The former
is series B70 from LTEG; it is in 1957 dollars. The latter is m04045 from
the NBER database. From 1947 to 1992, I use average hourly earnings of
production workers in manufacturing, in current dollars. This is series LEHM
from Citibase. Finally, since the relevant wage variables from a theoretical
perspective are after-tax wages, I multiply wages by the average marginal tax
rates provided by Barro and Sahasakul (1983) and updated through 1992 in the
manner they describe.16
For money, from 1915 to 1970 I use the M1 series from Friedman and
Schwartz (1963) and the Federal Reserve, which is reproduced as series B109
and B110 in LTEG. From 1970 to 1992 I use FM1 from Citibase. Both series
are in billions of dollars and are deflated by the POPM population measure
mentioned above. Prior to 1946, that population measure is the annual series
in the Bureau of the Census’s Historical Statistics (Series A–6–8, p. 8).

16 The

conclusions reached above are unchanged if before-tax wage rates are used.

A. L. Wolman: Zero Inflation and the Friedman Rule

19

REFERENCES
Amemiya, Takeshi. Advanced Econometrics. Cambridge, Mass.: Harvard
University Press, 1985.
Andreyenkov, Vladimir G., Vasily D. Patrushev, and John P. Robinson. Rhythm
of Everyday Life: How Soviet and American Citizens use Time. Boulder,
Colo.: Westview Press, 1989.
Bailey, Martin J. “The Welfare Cost of Inflationary Finance,” Journal of
Political Economy, vol. 64 (April 1956), pp. 93–110.
Ball, Laurence. “Credible Disinflation with Staggered Price-Setting,” American
Economic Review, vol. 84 (March 1994a), pp. 282–89.
. “What Determines the Sacrifice Ratio,” in N. Gregory Mankiw,
ed., Monetary Policy. Chicago: The University of Chicago Press, 1994b.
Barro, Robert J., and Chaipat Sahasakul. “Measuring the Average Marginal
Tax Rate from the Individual Income Tax,” Journal of Business, vol. 56
(October 1983), pp. 419–52.
Baumol, William J. “The Transactions Demand for Cash: An InventoryTheoretic Approach,” Quarterly Journal of Economics, vol. 66 (November
1952), pp. 545–66.
Benabou, Roland. “Inflation and Markups: Theories and Evidence from the
Retail Trade Sector,” European Economic Review, vol. 36 (April 1992),
pp. 566–74.
Bureau of the Census. Historical Statistics of the United States, Colonial Times
to 1970. Washington: U.S. Department of Commerce, 1975.
Chadha, Jagjit S., Andrew G. Haldane, and Norbert G. J. Janssen. “ShoeLeather Costs Reconsidered.” Manuscript. Bank of England, January
1997.
Chari, V. V., Lawrence J. Christiano, and Patrick J. Kehoe. “Optimality of the
Friedman Rule in Economies with Distorting Taxes,” Journal of Monetary
Economics, vol. 37 (April 1996), pp. 203–23.
Correia, Isabel, and Pedro Teles. “The Optimal Inflation Tax,” Discussion
Paper 123. Institute for Empirical Macroeconomics, Federal Reserve Bank
of Minneapolis, August 1997.
Dotsey, Michael, and Peter N. Ireland. “The Welfare Cost of Inflation in
General Equilibrium,” Journal of Monetary Economics, vol. 37 (February
1996), pp. 29– 47.
Federal Reserve Bank of Kansas City. Achieving Price Stability. Kansas City:
Federal Reserve Bank of Kansas City, 1996.

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Feldstein, Martin. “The Costs and Benefits of Going from Low Inflation
to Price Stability,” in Christina D. Romer and David H. Romer, eds.,
Monetary Policy. Chicago: University of Chicago Press, 1997.
Friedman, Milton. “The Optimum Quantity of Money,” in The Optimum
Quantity of Money, and Other Essays. Chicago: Aldine Publishing
Company, 1969.
, and Anna J. Schwartz. A Monetary History of the United States
1867–1960. Princeton: Princeton University Press, 1963.
Ireland, Peter N. “Optimal Disinflationary Paths,” Journal of Economic
Dynamics and Control, vol. 19 (November 1995), pp. 1429–48.
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Growth and Business Cycles: I. The Basic Neoclassical Model,” Journal
of Monetary Economics, vol. 21 (March/May 1988), pp. 195–232.
King, Robert G., and Alexander L. Wolman. “Inflation Targeting in a St. Louis
Model of the 21st Century,” Federal Reserve Bank of St. Louis Review,
vol. 78 (May/June 1996), pp. 83–107.
Kiyotaki, Nobuhiro, and Randall Wright. “On Money as a Medium of
Exchange,” Journal of Political Economy, vol. 97 (August 1989), pp.
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Lacker, Jeffrey M. “Stored Value Cards: Costly Private Substitutes for
Government Currency,” Federal Reserve Bank of Richmond Economic
Quarterly, vol. 82 (Summer 1996), pp. 1–25.
Lucas, Robert E., Jr. “On the Welfare Cost of Inflation,” Working Paper 394.
Stanford University: Center for Economic Policy Research, 1994.
, and Nancy L. Stokey. “Optimal Fiscal and Monetary Policy in an
Economy without Capital,” Journal of Monetary Economics, vol. 12 (July
1983), pp. 55–93.
McCallum, Bennett T. “The Role of Overlapping-Generations Models in
Monetary Economics,” Carnegie-Rochester Conference Series on Public
Policy, vol. 18 (Spring 1983), pp. 9–44.
, and Marvin S. Goodfriend. “Demand for Money: Theoretical
Studies,” in The New Palgrave: A Dictionary of Economics. London:
Macmillan Press, 1987, reprinted in Federal Reserve Bank of Richmond
Economic Review, vol. 74 (January/February 1988), pp. 16–24.
Mulligan, Casey B., and Xavier X. Sala-i-Martin. “Adoption of Financial
Technologies: Implications for Money Demand and Monetary Policy,”
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Prescott, Edward C. “Theory Ahead of Business Cycle Measurement,” Federal
Reserve Bank of Minneapolis Quarterly Review, vol. 10 (Fall 1986), pp.
9–22.
Rotemberg, Julio J. “Commentary,” Federal Reserve Bank of St. Louis Review,
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Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary
Economy,” American Economic Review, vol. 57 (May 1967), pp. 534 – 44.
Tobin, James. “The Interest-Elasticity of Transactions Demand for Cash,”
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Washington: Bureau of Economic Analysis, 1973.

Group Lending and
Financial Intermediation:
An Example
Edward S. Prescott

I

magine a small group of people, each of whom borrows money from a financial intermediary. The intermediary does not require collateral because
the borrowers are relatively poor and do not own much property. Instead,
the intermediary requires group members to be jointly liable for each other’s
loans. That is, if a member defaults on a loan, the rest of the group is liable
for the remainder of the loan. If the group does not honor this joint obligation,
then the entire group is cut off from future access to credit.
The lending arrangement I just described is not fictitious. Two million
villagers, most of whom are female and poor, borrowed in this way from the
Grameen Bank in Bangladesh. In Bolivia, 75,000 urban entrepreneurs, roughly
one-third of the banking system clientele, borrowed money via group loans
from BancoSol. Even in nineteenth-century Ireland, many rural residents took
out loans similar to group loans.
Motivated in part by group lenders in less-developed countries, organizations in the United States have developed similar programs. The 1996 Directory
of U.S. Microenterprise Programs lists 51 organizations that issue group loans.
The programs operate in both rural and urban areas. Often they are run by
nonprofit organizations.
The underlying idea of group lending is to delegate monitoring and enforcement activities to borrowers themselves. Borrowers who know a lot about each
other, such as those who live in close proximity or socialize in the same circles,
are the most promising candidates for group lending. For example, the rural
villages that Grameen lends in would seem ideally suited for group lending,
I would like to thank Hiroshi Fujiki, Tim Hannan, John Walter, Roy Webb, and John
Weinberg for helpful comments. The views expressed in this paper do not necessarily
represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 83/4 Fall 1997

23

24

Federal Reserve Bank of Richmond Economic Quarterly

because they are relatively self-contained communities, and people live close
to each other and interact regularly. In such an environment, residents should
be better than outsiders at assessing and monitoring the creditworthiness of
fellow residents. They should also be better able to apply social pressure on
potential defaulters.
The first goal of this paper is to analyze group lending, particularly as a
potential method for lending to the poor in the United States. Studying alternatives to traditional lending is important because there is economic evidence
that the poor in the United States have an unmet demand for finance. Zeldes
(1989) finds that the poor are borrowing-constrained; that is, they would like
to borrow more money at existing rates than they can. Evans and Jovanovic
(1989), even after accounting for possible correlation between entrepreneurial
ability and wealth, find that the lack of wealth affected the poor’s ability to
engage in self-employment activities. Bond and Townsend (1996), reporting on
the results of a survey of financial activity in a low-income, primarily Mexican
neighborhood in Chicago, find that bank loans are not an important source of
finance for business start-ups. In their sample, only 11.5 percent of business
owners financed their start-up with a bank loan. Furthermore, 50 percent of the
respondents financed their start-up entirely out of their own funds.
Two services provided by financial intermediaries are delegated monitoring and asset transformation. Banks provide both of these services and, maybe
surprisingly, groups do too. Group members monitor each other and through
joint liability, transform the state-contingent returns of its members’ loans into
a security with a different state-contingent payoff. Consequently, groups can
be interpreted as financial intermediaries, albeit small ones.
Interpreting groups as financial intermediaries is an important part of my
second goal: to place group lending in the context of the rest of the financial
intermediation sector. In this paper, groups have a comparative advantage at
some types of financial intermediation. Understanding comparative advantage
and specialization in financial intermediation to the poor is important because
it can help answer questions such as: Which financial intermediary is best at
what activity? How are different intermediaries financially linked? Do legal
and regulatory restrictions, through their effect on the organization of financial
intermediation sector, change the services they offer? These are the questions
that underly the assessment of legislative acts aimed towards lending to the
poor, like the Community Development Financial Institutions Act (CDFIA)
and the Community Reinvestment Act (CRA).1

1 The two acts take different views on the importance of the structure of the financial intermediation sector for lending to the poor. The recently enacted CDFIA seems to take the view
that alternatives to traditional financial institutions are needed to provide financial services to
low-income communities. (See Townsend [1994] for a critical discussion of the act.) It funds
institutions that specialize in providing financial services to low-income communities. In contrast,

E. S. Prescott: Group Lending and Financial Intermediation

25

Theoretical Framework
The theoretical framework I use is the delegated monitoring model developed
in Diamond (1984, 1996). In his work, there are lots of small lenders and a
smaller number of borrowers. Lenders lend to borrowers through a financial
intermediary in order to economize on monitoring costs.
My model makes two additions to this framework; the major one is to
allow some borrowers to monitor each other at a lower cost than outsiders.
The heterogeneity in monitoring costs drives the coexistence of two types of
financial intermediaries, large ones like banks and smaller ones like groups.
Both types transform assets and provide monitoring services. In my model,
just like in Diamond’s model, lenders’ funds flow through a large financial intermediary. But in my model, the large financial intermediary does not directly
lend to all borrowers. Instead, for those borrowers who can monitor each other
at a low cost, it lends to groups that in turn lend to their members.
I use Diamond’s framework for three reasons. First, delegated monitoring
is an important feature of group lending. Second, it allows for the embedding
of groups into the financial intermediation sector. Finally, it demonstrates the
similarities between groups and other financial intermediaries.
There is a small economic literature on group lending. This literature examines group versus individual lending but not in a model designed to study
the existence of financial intermediaries. Stiglitz (1990) examines a problem
where group members can assess whether other members are shirking. Varian
(1990) examines the important screening role groups may provide, that is, their
use of their prior knowledge about others to form groups. He also examines
learning from fellow members as a potential advantage of groups. Besley and
Coate (1995) examine the potential enforcement advantages groups may have.
For example, social ostracism of defaulters is an option available to groups
but not to outsiders. These penalties can reduce incentives to default but not
in all cases. Sometimes, they increase the chance of default. While all of these
features of group lending are important, I abstract from them.
In the following section, I provide background on group lending in practice,
after which my model is developed and analyzed. Then I analyze the portability
of group lending to the United States in the context of the model.

the CRA seems based on the premise that lending to low-income people is best done by existing
financial institutions but that these institutions underserve low-income communities because of
neglect, or even discrimination in the most egregious cases. The CRA works by requiring regulators to evaluate banks on criteria such as financial services provided to low-income communities.
Banks that score poorly are subject to sanctions such as limits on merger activities.

26

1.

Federal Reserve Bank of Richmond Economic Quarterly

GROUP LENDING IN PRACTICE

Microfinance is the provision of financial services to the poor. The prefix micro
is used because the amounts involved in transactions are small. Often, microfinance is provided by nonprofit organizations; their targets are people who have
not participated in the formal financial sector. The financial services that their
clients do use tend to be supplied by relatives, or in some parts of the world, by
moneylenders. Formal financial institutions have avoided this market because
the loan sizes are small, administrative costs per dollar lent are high, and they
perceive the risk of default to be significant. It is the absence of the formal
sector from these markets that has led nonprofit organizations, often with the
goal of poverty alleviation instead of profit maximization, to supply financial
services. It is also the inappropriateness of traditional financial products that
has led to the introduction of financial products such as group lending.
Group lending is not the only tool used to provide microfinance. Many
microfinance organizations make loans only to individuals while others make
loans to both individuals and groups. Others provide savings and insurance services. Much microfinance is provided informally, by rotating savings and credit
associations, or between friends and family. While these issues are important,
I do not discuss them because this paper is a study of the narrower question
of what conditions favor group lending.
Group Lending in Less-Developed Countries
The most famous group lender is the Grameen Bank, which was founded in
Bangladesh in the mid-1970s. This bank makes loans to groups of five unrelated individuals who are poor. Most groups consist of landless women from
the same village. Loans are made sequentially with remaining members not receiving their loans until other members repay their loans. Loan size is increased
after the group has successfully repaid earlier small loans.2
The bank has grown tremendously. In 1992, it lent to 2 million people at
real interest rates of around 12 to 16 percent. Their repayment rate is high,
around 97 to 98 percent. The bank even shows a profit, though it would not do
so without the low-interest loans and grants it has received (Morduch 1997).
The Grameen Bank is far from the only institution to make group loans.
Even in Bangladesh, there are at least two other organizations, Bangladesh
Rural Advancement Committee (BRAC) and Thana Resource Development
and Employment Programme (TRDEP), that make group loans (Montgomery,
2 There are several other interesting features of the bank’s organization. For example, collections of six groups are formed into Centres. All payments are made at Centre meetings in public
view of other Centre members. Savings funds are also developed to provide for contingencies
like death or disability. See Rashid and Townsend (1993) and Fuglesang and Chandler (1987) for
more details.

E. S. Prescott: Group Lending and Financial Intermediation

27

Bhattacharya, and Hulme 1996). Like Grameen, BRAC is a sizable institution,
lending to over 600,000 borrowers in 1992. Other countries with lending institutions that make group loans include Kenya (Mutua 1994), Malawi (Buckley
1996), Costa Rica (Wenner 1995), Columbia, and Peru, just to name a few.
One of the most successful group lenders is BancoSol, located in Bolivia.
It is a chartered bank, subject to the supervision of SIB, the Bolivian bank regulatory agency. It makes uncollateralized loans for periods of 12 to 24 weeks.
Repayments are made frequently, every week or two. Loans are made to what
they call solidarity groups, each of which can have four to ten members. The
group takes a loan from the bank and apportions it among its members. Like
Grameen’s groups, group members are jointly liable for each other’s debts.
Loans are usually made to provide working capital for small-scale commercial
activities. Also like Grameen, the majority (77 percent) of clients are women.
But unlike the Bangladesh bank, most of the borrowers are located in urban
areas. Nonetheless, borrowers still have good information about each other
because BancoSol requires all members of a solidarity group to work within a
few blocks of each other. Most borrowers are market vendors, though half of
the portfolio is lent to small-scale producers like shoemakers, bakers, and tailors
(Glosser 1994). Lending is not the only financial service provided by BancoSol.
It also offers deposit services in both boliviano and U.S. dollar-denominated
accounts.3
BancoSol’s growth has been extraordinary. In 1996, it lent to about 75,000
people, roughly one-third of the people who use the Bolivian banking sector.
In 1996, BancoSol had a loan portfolio of $47.5 million. It also earned $1.1
million on revenues of $13 million (Friedland 1997). Two important reasons
for this success is that the bank charges real interest rates of 34 percent and
has a default rate of less than 1 percent (Agafonoff 1994). The high interest
rates are no doubt required to cover the high administrative costs required by
its lending strategy. As a basis of comparison, 80 percent of BancoSol’s costs
are administrative, while the comparable number for the rest of the Bolivian
banking industry is only 20 percent (Glosser 1994).

3 BancoSol is a chartered bank because Bolivian law requires deposit-taking institutions to
be chartered banks subject to governmental supervision. BancoSol was created by PRODEM, a
nonprofit organization that specialized in making loans. Its operations were financed mainly by
grants, usually from foundations and USAID. The organization felt that grants were an insufficient
source of capital, so it decided to create a regulated bank in order to have the legal right to collect
deposits. Interestingly, the bank’s nontraditional activities complicated the granting of the charter.
For example, existing Bolivian banking law required that uncollateralized credit be less than twice
paid-up capital. Unfortunately, for BancoSol, uncollateralized credit is all they supply! The bank
negotiated a compromise in which loans under $2,000 do not count towards this total. The costs
of the conversion were not trivial. They exceeded $500,000, according to one estimate (Glosser
1994).

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Federal Reserve Bank of Richmond Economic Quarterly

Group Lending in the United States
Recently, several lenders have tried group lending in the United States. These
lenders are nonprofit organizations whose main goal is to assist the poor—in
particular, women and minorities—by financing self-employment. Since these
efforts have started relatively recently, published information is still limited.
One source of information is a study by Edgcomb, Klein, and Clark (1996),
who examine seven microenterprise programs. Of the seven, four make group
loans.4 Each program provides services other than group lending. Several lend
to individuals, others provide training, and some provide all three services.
All four programs followed Grameen’s example but with modifications.
Each agency started with groups of five members. However, the agencies found
that if an individual dropped out of the group, the rest of the group would
disband. Currently, three of the agencies allow more flexibility in group size.
One program allows four to ten members, while another allows four to six
businesses per group.
The scale of the agencies’ operations are still small. For example, the
number of loans made by the programs in 1994 ranged from 27 to 103, and
average loan sizes ranged from about $2,100 to $4,900. Making these loans
is expensive. The average cost per loan varied from $4,500 to $15,300, so
these programs are far from self-sufficient. However, when compared with job
training and other assistance programs, their costs seem more reasonable. I
discuss possible reasons for the high costs after I describe the model.
Historical Group Lending
Group lending is often considered a recent innovation, and its recent popularity certainly is connected with the success of the Grameen Bank. There are,
however, at least two types of institutions that existed long before the Grameen
Bank and that used variants on group lending.
To the best of the author’s knowledge the earliest institutions that used
a form of group lending were the Irish Loan Funds (Hollis and Sweetman
1997a, b).5 The funds developed in the early 1700s, peaked in size in the early
1800s, and then slowly declined throughout the rest of the nineteenth century.
Interestingly, Hollis and Sweetman trace their development to Jonathan Swift,
the Anglican priest best known for writing Gulliver’s Travels.
4 The four that made group loans were the Coalition for Women’s Economic Development
(CWED), based in Los Angeles; the Good Faith Fund (GFF), located in Pine Bluff, Arkansas;
the Rural Economic Development Center (REDC), which lends throughout North Carolina; and
the Women’s Self-Employment Project (WSEP), based in Chicago.
5 All reported information about the Irish Loan Funds is taken from Hollis and Sweetman
(1997a, b).

E. S. Prescott: Group Lending and Financial Intermediation

29

The Irish Loan Funds were usually located in rural areas, took deposits,
and made small loans. The institutions generally made uncollateralized loans
to finance a small investment project, such as the purchase of an animal. As a
rule, the loans were repaid on a weekly basis. These loans most resembled
present-day group loans in that all borrowers were required to obtain two
cosigners for each loan, and both cosigners were liable for repayment.6 While
each fund was independent, the funds were regulated by a Central Loan Fund
Board.
Another historical example of European group lenders was that of the
German credit cooperatives that developed in the late nineteenth century
(Guinnane 1993; Banerjee, Besley, and Guinnane 1994). They were often located in rural areas where individuals knew each other well. These cooperatives
provided credit services, and importantly, many had a policy of unlimited liability. That is, if the cooperative failed, any member could be sued for the
entire amount owed by the cooperative. Interestingly, these credit cooperatives
were the inspiration for the credit union movement in the United States.

2.

THE MODEL

The model in this paper is designed to study the following three features of
group lending and the financial intermediation sector:
• the existence of joint liability groups
• the existence of more traditional financial intermediaries
• large financial intermediaries lending to the groups
Analysis of these issues requires a model in which it is possible to lend funds
either directly to an individual or indirectly through a financial intermediary.
With two additions, the framework in Diamond (1984, 1996) provides an environment that satisfies these conditions.
Diamond considered an economy where there are borrowers and lenders,
and funding each borrower’s project requires the resources of several lenders.
Borrowers’ returns are unobserved by a lender unless he spends resources to
monitor the borrower. Lenders face the choice of whether to lend directly to
borrowers or to lend to them indirectly through the financial intermediary. In
equilibrium, lenders lend to the financial intermediary and the intermediary in
turn lends to the borrowers. The reason that lenders lend through the financial
intermediary is that it avoids costly duplicative monitoring.
This paper operates in the same framework but with two additions, heterogeneous monitoring costs and screening costs. The important addition is the
former. In particular, some borrowers are given the ability to form small groups,
6 The

loans are much like the ones Swift made. Using his own money, he made small
uncollateralized loans, required cosigners on loans, and required frequent repayments.

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Federal Reserve Bank of Richmond Economic Quarterly

and in these groups they can monitor their fellow members. This ability is
potentially valuable because group members monitor each other at a lower cost
than a more traditional financial intermediary. People who live close to each
other, those who work near each other, or those who socialize together would
be most likely to satisfy these conditions. As in Diamond’s model, it is optimal
for lenders to lend to a traditional financial intermediary, but in this paper the
financial intermediary lends to groups that in turn lend to their members. As we
will see, the incentive problem underlying the contract between lenders and the
large financial intermediary is the same as the incentive problem underlying the
contract between the large financial intermediary and the groups. It is in this
sense that groups and institutions, such as banks, are financial intermediaries
for the same reason.
Environment
The model in this section is really a numerical example that closely follows
Diamond (1996). In this economy, there are two main types of people, lenders
and borrowers. Both types are risk-neutral, and consumption cannot go below
zero. Each lender is endowed with 1/m, m > 1, units of the investment good.
The investment good cannot be consumed, but it can be used to create the
consumption good. Lenders have access to a safe but low-return investment
technology. Their investment technology takes x units of the investment good
and turns it into 1.05x units of the good, receiving an interest rate of 5 percent.
The borrowers are better at producing the consumption good, but they start
without any units of the investment good. Each borrower’s investment technology requires an input of exactly 1.0 unit of the investment good. An investment
of less than 1.0 produces an output of zero and any investment over 1.0 unit is
wasted. The former assumption means that for each borrower it takes the funds
of at least m lenders to finance his investment. Their investment technology
is also riskier than that of lenders. In this example, an investment of 1.0 unit
produces an output of 1.0 with a probability of 0.2 and an output of 1.4 with a
probability of 0.8. Expected output for a borrower is (0.2)1.0+(0.8)1.4 = 1.32,
which is greater than 1.05, the return on the safe investment. However, 20 percent of the time output is less than what it would have been if the lender’s
investment technology had been used. Finally, I assume that each borrower’s
return is independent of other borrowers’ returns.
In this model, the owners and the productive users of the investment good
are different people. As the problem presently stands, the initial mismatch
between owners and users is easily rectified through simple loan contracts.
Lenders would lend to borrowers as long as their expected repayment was
equal to 1.05. There is no role for intermediaries.
To introduce intermediaries requires the addition of complications to writing and enforcing contracts, complications that intermediaries are better able

E. S. Prescott: Group Lending and Financial Intermediation

31

to overcome than lenders. I now describe four features to the model that affect
the feasibility and desirability of various contracts and ultimately lead to a
role for financial intermediaries, both large ones like banks and smaller ones
like groups. The four features are private information on borrowers’ returns,
liquidation costs, costly monitoring, and costly screening.
Private Information
It is assumed that borrowers’ returns are private information. That is, a borrower
is the only person who knows the success of his project; lenders do not observe
it, nor do other borrowers. Private information makes some contracts infeasible.
For example, consider a contract where lenders receive 1.0 if the low output
is produced and 1.0625 if the high output is produced. If lenders knew that
the contractual terms would be honored by the borrower, they would make
the loan because their expected return is 0.2(1.0) + 0.8(1.0625) = 1.05. Under
private information, however, they cannot be sure that this contract would be
honored. The reason is that lenders do not know the true value of the output
so the borrower could always claim that he received a low output. That is, if
the lender received the high output the borrower could claim he received the
low output, pay 1.0 to the lender, and keep the difference. Lenders would be
powerless to stop this deception; they cannot find out if he is telling the truth,
and as things are presently specified, they cannot punish him. All they can do
is refuse to lend, despite the acknowledged quality of his project.
Liquidation Costs
A contract with the option of liquidation is one way out of this dilemma. In this
model, a liquidation cost serves as an ex post penalty imposed by the lender
on the borrower. If the borrower does not meet the terms of his agreement, the
lender can liquidate the borrower’s assets. In this model, I interpret liquidating
as meaning that the borrower and the lender receive zero. This means, among
other things, that there are no assets that the lender can seize and sell. (In
microfinance, projects are so small that one would gain very little from seizing
and selling physical assets.)
The penalty imposed on the borrower by liquidation is important because
it prevents him from always claiming he received the low output, as in the contract described above. For example, consider a debt contract with a face value
F of 1.3125. If the borrower does not repay 1.3125, he has defaulted. When the
output is 1.4, the borrower pays 1.3125. When the output is 1.0, the borrower
cannot pay the full amount, so the lender liquidates, giving the borrower (and
the lender) zero. The expected return to the lender is (0.8)(1.3125) = 1.05, so
the loan is made and the borrower receives zero in the low-return state and
0.0875 in the high-return state. The threat of liquidation is enough to force
repayment in the high-return state. The cost of liquidation is that output, which

32

Federal Reserve Bank of Richmond Economic Quarterly

is 0.2 in expected value terms, is destroyed. But in this example, the benefits
of financing the loan outweigh the liquidation costs.7
Costly Monitoring
Costly monitoring is the other way to make lending feasible. In this paper
there are two types of monitoring: costly monitoring by a lender and mutual
monitoring within a group. Monitoring by a lender is identical to monitoring in
Diamond’s model; the lender pays an ex ante cost that allows him to observe a
borrower’s output. In essence, the lender uses resources to observe the private
information. The resource costs could be as simple as spending time with the
borrower or as complex as receiving regular reports on the project’s financial
status.
Observing output is valuable because then repayment can be made dependent on output, which avoids the need for liquidation. For example, consider
the following contract: the lender monitors and the borrower pays 1.0 if the low
output is realized, and 1.2 if the high output is realized. The expected return
for the lender is 0.2+0.96−K, where K is the cost of monitoring. If the cost of
monitoring is K ≤ 0.11, then a lender’s expected return (assume for the moment
there is only one lender) is greater than 1.05, making monitoring worthwhile.
Furthermore, this contract with monitoring is better for the borrower than the
liquidation contract. (In both cases the borrower keeps zero in the low state,
but under the monitoring contract, he keeps more in the high state.)
The second type of monitoring, mutual monitoring within a group, is the
main departure from Diamond’s model. I assume that within a subset of borrowers there are pairs of borrowers who know each other well, maybe because
they live near each other or maybe because they are in the same social or
ethnic circles.8 Each one of these pairs may form a group at a per-person cost
of Kg . Membership in a group allows a group member to observe the other
group member’s output. Furthermore, because of the close social ties within a
group, or maybe even because their time is less valuable than a loan officer’s,
I assume that the cost of being in a group is lower than the cost of anyone else
monitoring them, that is, Kg < K.
At this point I should say more about what it means to be a group and how
that affects the group’s interaction with nongroup members. I am assuming that
group members observe each other’s outputs and act cooperatively or collude.
In many models where people can collude, their interaction is complicated
7 There is no advantage from a contract that liquidates for the high output but not for the
low output. Under such a contract, the borrower would always claim the low output, avoiding
liquidation and keeping the difference between the high-output and the low-output payment. More
generally, if the technology allows for more than two realizations of the output, even a continuum,
then the optimal contract will still be a debt contract. The optimal contract will require a constant
payment and liquidation if that payment is not made.
8 For simplicity, I assume that groups consist of only two people.

E. S. Prescott: Group Lending and Financial Intermediation

33

and even disadvantageous.9 In this model, there are no such disadvantageous
effects. Furthermore, the analysis is simple because the borrowers are riskneutral and thus utility is transferable. In this model, transferable utility eases
the analysis because it means that the division of output between the group
members does not affect the group’s decisions. That is, regardless of how the
group shares their returns, the group acts as if it is maximizing total expected
output. In this paper, I assume that they share the returns equally. Besley and
Coate (1995) examine a group-lending arrangement where there is an element
of strategic play between group members, and they show that this can be a
problem in some cases. I abstract from this consideration.
Screening Costs
The last element, and the remaining addition to Diamond’s setup, is the addition
of a screening cost. What I have in mind is a preliminary form of monitoring.
A lender needs to meet with the borrower, discuss his project, and record and
verify information about the borrower. In contrast with the previously discussed
monitoring costs, screening costs do not reveal the final output. They only represent the effort that goes into ensuring that the project has a chance of success.
To model these ideas, I assume that there is a fixed cost of Ks per lender to
screen a borrower. I do not model what happens if the lender or lenders do not
screen a borrower; I simply assume that they must screen a borrower before
they make a loan.
I also assume that screening is only necessary for lending to borrowers.
By borrowers I mean the second type of people, those who have access to the
high return and risky technology, and not any entity that receives funds for
investment. In particular, there is no need to screen a financial intermediary,
though the financial intermediary still needs to screen any borrowers to whom
it lends. This assumption is admittedly strong but not without merit. It seems
reasonable to assume that it is harder to do a preliminary evaluation on small,
idiosyncratic investment projects than on a large, well-known institution such
as a bank. The only role of this assumption is to ensure that lending to groups
is done by the financial intermediary and not directly by lenders.10
Where I am going . . .
In this economy there are lenders who have funds and borrowers who do
not. The productivity of borrowers’ investment projects creates a demand for
9 See, for example, Holmstr¨ m and Milgrom (1990), Itoh (1993), Ramakrishnan and Thakor
o
(1991), or Prescott and Townsend (1996).
10 There are other ways to ensure that lending to groups goes through the large financial
intermediary, though they add additional issues that complicate the analysis. For example, making
lenders risk-averse would be sufficient, since then each lender would want to lend directly to more
than one group. Consequently, each lender would screen several groups, raising screening costs.

34

Federal Reserve Bank of Richmond Economic Quarterly

finance. Private information, however, precludes lending unless there is monitoring or the penalty of liquidation. Before describing how these elements
create a demand for financial intermediation, it is helpful to show what the
lending flows will be and where each type of financial intermediary fits into
the flow pattern.
Figure 1 describes the direction of lending flows in the model. Arrows
indicate the direction of lending and an M indicates whether or not there is
monitoring. The lenders, who start with the investment good, make unmonitored
loans to the large financial intermediary.11 This financial intermediary makes
two types of loans, monitored loans to individuals and unmonitored loans to
groups. Groups, the smaller financial intermediary, in turn make monitored
loans to its members.
My strategy for analyzing the model is to split the analysis into two sections. In the first section, I take as given that there is one large financial
intermediary and analyze its decision of whether to make a loan to an individual or to a group. To do this analysis, I consider each type of loan the
financial intermediary may make to the borrowers and enumerate the trade-offs
of lending to a group versus lending to individuals and also whether or not it
is beneficial to monitor the loans. Next, I consider the lending decisions for
lenders and show that it is indeed optimal for them to lend to borrowers through
the financial intermediary rather than to lend to them directly.
Lending by the Financial Intermediary
The large financial intermediary has three options for lending funds:
• It can lend to borrowers, not monitor them, and use the threat of
liquidation;
• It can lend to borrowers and monitor them; or
• It can lend to borrowers through groups.
For this last case, we need only concern ourselves with unmonitored loans
to the groups, since if the bank monitored them, it might as well bypass the
groups altogether.
Recall that for each borrower who invests 1.0 unit of capital, he produces
the low output of 1.0 with a probability of 0.2 and the high output of 1.4 with
a probability of 0.8. Also, borrowers need 1.0 unit of the good to invest and
for reasons explained later, the large intermediary requires an expected return
of 1.05.
The expected returns to a project can be broken into five components: the
expected payment to the financial intermediary R, the expected utility (return)
11 There

can be more than one large financial intermediary as long as each one has a sufficiently large portfolio. For our purposes, it is simplest to assume there is only one.

E. S. Prescott: Group Lending and Financial Intermediation

35

Figure 1 Lending Flows in the Model
M
M
M

Lenders

Borrowers
(who are
not group
members)

LARGE
FINANCIAL
INTERMEDIARY

M
Group

M

+

M
Group

M

Borrowers
(who are
group
members)

Notes: M indicates that the loan is monitored. Arrows indicate direction of lending flows.

of the borrower U, the liquidation costs L, the monitoring costs M, and the
screening costs S. These will sum to 1.32, the project’s expected output. In the
following sections, when each contract is analyzed, I will list the values of the
five components for each contract. Also, I assume that the financial intermediary
receives 1.05, the opportunity cost of the lenders’ funds. Thus, any excess accrues to the borrower. Under this (unimportant) assumption, maximizing social
welfare is equivalent to maximizing the utility to the borrower.
Individual Lending with Liquidation but No Monitoring
The enforcement device used for this contract is liquidation. Since there is no
monitoring, a state-contingent contract without liquidation cannot be offered.
Instead, a debt contract with a face value of F is written. Under this contract
the borrower must pay F, or his project is liquidated. To make the problem
interesting I assume that the parameters are such that 1.0 < F < 1.4. This
means that if the borrower receives the high return he pays F, but if he receives

36

Federal Reserve Bank of Richmond Economic Quarterly

the lower return, his project is liquidated, and both he and the intermediary
receive zero. An F guaranteeing that the intermediary receives 1.05 in expected
return is the solution to the following equation:
1.05 = (0.2)0 + (0.8)F − Ks .
The intermediary receives a zero payment 20 percent of the time, it receives a
payment of F 80 percent of the time, and this return has to be high enough to
cover the screening costs Ks and the opportunity cost of the funds 1.05. The
solution to the equation is F = 1.3125 + Ks /(0.8). The borrower’s expected
utility is U = (0.2)0 + (0.8)(1.4 − F). Calculations for utility and the other
variables of interest are as follows:
U = 0.07 − Ks ,
R = 1.05,
L = 0.20,
M = 0, and
S = Ks .
Notice that these values sum to 1.32, the project’s expected return.
Individual Lending with Monitoring
There is no need to liquidate when monitoring because output is observed by
the financial intermediary. For simplicity, I assume that 1.0 is paid out if the
low return occurs, and F is paid out if the high return occurs. A face value
of debt F that gives the intermediary a return of 1.05 is the solution to the
following equation:
1.05 = (0.2)(1.0) + (0.8)F − K − Ks .
Compared with the previous contract, the intermediary now receives 1.0 if the
low output is observed but must also bear the monitoring cost K. The solution
to this equation is F = 1.0625 + (K + Ks )/(0.8). The borrower’s expected utility
is again U = (0.2)0 + (0.8)(1.4 − F). Carrying out the calculations for the
variables produces the following numbers:
U = 0.27 − K − Ks ,
R = 1.05,
L = 0,
M = K, and
S = Ks .
Comparing the utilities from a loan with monitoring and a loan using liquidation shows that the former is preferred when 0.27 − K − Ks > 0.07 − Ks , or
equivalently, K < 0.20.

E. S. Prescott: Group Lending and Financial Intermediation

37

Group Lending
The group-lending contract includes elements of monitoring and liquidation.
The group members monitor each other, but since the large financial intermediary does not know the results of their monitoring, it needs to include a
liquidation provision in the contract. As I mentioned earlier, the two members
of a group pool their resources so the group’s distribution of returns is
return

probability

2.0
2.4
2.8

0.04
0.32
0.64

The assumptions made concerning group membership are that group members observe each other’s output and act cooperatively. In this context, acting
cooperatively means they maximize the expected value of the group’s return.
Thus, the contract needs to be written in terms of the total returns to the group,
since the group can always move funds around to pay off a debt. Therefore, the
optimal contract will again be a debt contract, with liquidation if the face value
of the debt is not repaid. To facilitate comparison with the other contracts, we
put the face value of the debt in per-group-member terms, that is, the face value
of the group’s debt is 2F.
For the intermediary to receive an expected payment of 2.10 (1.05 per
group member), F needs to solve the following equation:
2.10 = (0.04)0 + (0.32)(2F) + (0.64)(2F) − 2Ks .
I assume that the large intermediary rather than the group bears the screening
cost. This assumption is not important.
At this point, it is necessary to make one more assumption. I assume that
2.4 units of output is enough to pay off the face value of the group’s debt, 2F.
The value of 2F will depend on the other parameters, so I am assuming their
values are such that this condition holds. Under these assumptions, the solution
to the equation is F = 1.09375 + Ks /(0.96). Each borrower’s utility, assuming
equal division of returns, is calculated from U = (0.04)0 + (0.32)(2.4 − 2F −
2Kg )/2+(0.64)(2.8−2F−2Kg )/2. I include the monitoring cost in this equation
because the group pays it themselves. The values of the variables in per-groupmember terms are
U = 0.23 − (0.96)Kg − Ks ,
R = 1.05,
L = 0.04,
M = (0.96)Kg , and
S = Ks .

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Federal Reserve Bank of Richmond Economic Quarterly

Special attention should be paid to the liquidation cost, L. Under group
lending, L = 0.04, which is dramatically lower than the case where the intermediary lends but does not monitor. (Recall that the liquidation cost in that
case was 0.20.) The reason for the dramatic reduction is that the distribution of
the group’s output is different from the distribution of the individual’s output.
In particular, the group’s distribution has less variance. The decreased dispersion of group returns reduces the incentive problem caused by the private
information. In turn, a weakened incentive problem means that liquidation is
invoked less often than a liquidation contract between the intermediary and an
individual.
The argument is easier to understand if we compare two borrowers borrowing F each as a group with the same borrowers borrowing F each as individuals
under the unmonitored liquidation contract. Also, assume that 1.0 < F < 1.2.
When the funds are lent to the individuals, each borrower’s project is liquidated 20 percent of the time. This means that 4 percent of the time both are
liquidated, 32 percent of the time one is liquidated, and 64 percent of the
time neither is liquidated. Now compare these liquidation probabilities with
those of the group. Under the group contract, 4 percent of the time both are
liquidated, but 96 percent of the time neither is liquidated. The reason is that if
one borrower gets a bad return and the other gets a good return, then the latter
bails out the former. The transfers between the group members, in effect, alter
their distribution of returns. This change reduces the probability of liquidation,
which is beneficial.
One more way to view this problem, and an argument I will return to
when discussing the large intermediary, is to consider a group consisting of
a very large number of borrowers. (More formally, assume there is a continuum of them.) Because there are so many group members, the law of large
numbers means that the group’s total return is 1.32 − Kg with probability
1.0. All idiosyncratic risk averages out. In this case, there is never a need
to liquidate since any claim that total output was less than 1.32 − Kg would
not be credible.
To reiterate, groups greatly reduce the probability of being liquidated.
Still, they have to pay a monitoring cost, and the relative size of these two
costs (along with the intermediary’s monitoring cost) determine whether group
lending is better than the other types of lending. In this example, group
monitoring is better than individual lending with monitoring if 0.23−(0.96)Kg −
Ks > 0.27 − K − Ks ; that is, the utility accruing to a borrower from group
monitoring is greater than the utility accruing to a borrower from an individual
lending with monitoring contract. Rearranging terms, the condition is
(0.96)Kg + 0.04 < K.

(1)

Equation (1) says that group monitoring is better if the sum of the group monitoring cost Kg and the liquidation cost of 0.04 is less than the intermediaries

E. S. Prescott: Group Lending and Financial Intermediation

39

monitoring cost K. This is not strictly true because Kg is multiplied by 0.96.
That number, however, is only in the equation because groups bear the cost of
monitoring; if their projects are liquidated, they receive zero and do not have
to bear the monitoring cost.
I can now provide conditions under which the large financial intermediary
will lend according to the pattern described by Figure 1. First, I assume that
monitoring by the intermediary satisfies K < 0.20 (so individual lending with
monitoring is better than individual lending without monitoring). Second, I
assume that for some pairs of borrowers Kg is small enough to satisfy equation
(1) and for other pairs of borrowers it is not. The former borrowers could
be those who live near each other like Grameen’s clients or work near each
other like BancoSol’s clients. For parameter values satisfying these conditions,
borrowers who cannot form a group borrow as individuals with a monitored
loan, while other borrowers who can form a group do so and borrow from the
intermediary as a group, using the liquidation contract.
Lending to the Large Financial Intermediary
Now return to the lenders’ lending decision. In equilibrium, as indicated by
Figure 1, lenders lend to the large financial intermediary rather than directly to
individuals or groups. Most of the pieces are already in place to demonstrate
why this is the case. Lenders can either transform the asset themselves by using
the low return but riskless technology, or they can choose one of the following
four lending options:
• Lend directly to borrowers and use a liquidation contract;
• Lend directly to borrowers and monitor them;
• Lend directly to the group and use a liquidation contract; or
• Lend to the large financial intermediary.
The last option, lending to the large financial intermediary, is the optimal
arrangement. I will demonstrate this by first showing that the costs to lenders
of lending directly is greater than the same costs faced by the large financial
intermediary making the same loans. Then, I will show that the lenders can
lend to the large financial intermediary at no cost. This will mean that lending
through the large financial intermediary is better than direct lending. Finally, if
the large intermediary receives a return of 1.05, as was assumed in the previous
analysis, and the intermediary adds no costs to lending, then it is optimal for
lenders to lend to the large intermediary.12
The first three cases listed above are the direct-lending options available
to lenders. Each one of these options corresponds to one of the cases worked
12 Technically, lenders are indifferent between this option and using the safe investment
technology. Among these two choices, I assume that the lenders choose the socially optimal one,
which is to lend to the large financial intermediary.

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Federal Reserve Bank of Richmond Economic Quarterly

through earlier in the section. The difference is that now monitoring and screening costs have to be borne by m > 1 lenders rather than just the large financial
intermediary. The algebra is easy enough to work through but it is simpler
to use the following observations. The incentives faced by a borrower do not
depend on whether his funds are obtained from lenders or via the large financial
intermediary. Consequently, the problem is unchanged from the earlier analysis
except that screening costs (in all three cases) and monitoring costs (in the
second case) are m > 1 times as much under direct lending. Therefore, it is
cheaper for lenders to lend through the large financial intermediary rather than
directly.
However, there still remains the issue of whether or not lenders need to
monitor and screen the financial intermediary. If they do not, they can lend
to the intermediary, which in turn lends to borrowers (either directly or indirectly through groups). This flow of funds will economize on monitoring and
screening costs relative to direct lending.
By assumption, there is no need to screen the intermediary. However, some
work is needed to demonstrate that lenders do not need to monitor the large
financial intermediary. How do lenders know that the intermediary actually
monitors the borrowers? How do they know the return of the intermediary?
(At this point, it is helpful to think of the large intermediary as a person,
possibly a lender, who if he did not monitor would save himself monitoring
costs.)
In the previous section’s analysis of lending to the group, the increased
size of the group made the liquidation contract more effective. The larger the
group, the more effective a liquidation contract was. If the group consisted of a
continuum of members, then there was no need to monitor because the group’s
return is certain.
The same logic applies to the problem facing the lenders lending to the
intermediary. If the intermediary lends to a continuum of borrowers, then the
intermediary’s return is certain. Thus, the optimal contract between lenders and
the large financial intermediary is an unmonitored debt contract of face value
F = 1.05. As part of the debt contract, the lenders liquidate the intermediary’s
assets if it claims its return is less than 1.05. But in equilibrium, the intermediary’s portfolio is so diversified that its assets are never liquidated. Thus,
there is no liquidation cost to lending through the financial intermediary, and
there is no need to monitor it. The entire return of 1.05 that the intermediary
receives from borrowers can be passed to the lenders. Lending through the
large financial intermediary is better than direct lending.
To summarize, the large financial intermediary economizes on monitoring
and screening costs while the groups economize only on monitoring costs. Relative to direct lending, both types of intermediaries economize on monitoring
costs in the same way. Lending through the intermediaries avoids the duplicative monitoring of borrowers by lenders while the intermediary’s diversification

E. S. Prescott: Group Lending and Financial Intermediation

41

reduces the need for lenders to monitor it. Thus, total monitoring is lowered
in the economy. The reduction of these costs is the financial intermediary’s
special role in transforming assets.
There is, however, one way in which the two types of intermediaries differ
in how they economize on monitoring costs. Compared with monitoring by the
large financial intermediary, the groups save on monitoring costs because they
have a cost advantage. It is efficient for the large financial intermediary to lend
through groups if this cost saving outweighs the liquidation cost from using
the group. The remaining observation—that lenders lend to groups through the
large financial intermediary—occurs to economize on screening costs.13

3.

ANALYSIS

Ideally, the model would be used in the following way. We would start with
measurements of parameters in the model, such as distribution of returns, costs
of monitoring, etc. These measurements would come from economies, like
villages in Bangladesh or urban areas in Bolivia, where group lending is successfully used. Using these measurements we would evaluate the model on the
criterion of whether or not it predicts there will be groups. If it does predict
groups, the experiment proceeds by solving the model using parameter values
taken from low-income U.S. communities. Then, the model could be used to
evaluate the potential of group lending in the United States.
Precise measurement of many of these values is beyond the scope of this
paper. Indeed, measurement of a concept like monitoring is a research project
in and of itself. Consequently, the following discussion is necessarily sketchy,
guided by what little information is available. Still, it is valuable, and one can
gain some broad ideas about the role group lending and financial structure may
play in channeling credit to the poor. The discussion should be considered a
starting point, particularly for researchers and practitioners who are looking for
guidance as to what variables to measure.
Business Opportunities
The model analyzes the problem of financing investment projects. It takes
as given that potentially profitable investment projects exist. The financing
13 One difference from the Diamond (1984, 1996) setup is worth mentioning. In his paper,
financial intermediaries exist only to economize on monitoring costs. In this paper, the large financial intermediary economizes on monitoring costs, but it also economizes on screening costs.
The latter costs, in fact, are sufficient in this model for the large intermediary to exist. In this
paper, monitoring costs serve the role of obtaining a nontrivial trade-off between individual and
group lending. They are necessary to generate the existence of the small financial intermediaries,
that is, the groups.

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problem, however, is irrelevant if there are no profitable microenterprise
projects to finance.
The evidence presented in the introduction suggests that there are profitable
investment projects in the United States that would be financed in the absence
of information constraints. There are, however, reasons to think that there may
be less of these opportunities in the United States than in Bangladesh or Bolivia. For example, in less-developed countries 60 to 80 percent of the labor
force is engaged in self-employment (Edgcomb, Klein, and Clark 1996), while
in the United States only about 12 percent of the labor force is self-employed
(Segal 1995). Ultimately, of course, the existence of profitable self-employment
opportunities must be determined by empirical investigation.
A related issue, applicable to most microfinance programs, is what type
of investments can be financed with group lending or any other microfinance
program. For example, one key feature of the studied lending programs is the
required frequency of repayments. Frequent repayment requires that an investment produce cash flow for the entire course of the loan. If it does not, then
the borrower will default. This time path would seem to preclude loans for
investments that pay off sometime in the future. For example, a planting loan
to a farmer is poorly suited for frequent repayment because planting does not
generate income until harvest.
A cursory examination of the type of loans made by Grameen, BancoSol,
or the Irish Loan Funds bears out this observation. Despite their rural location,
planting loans are not made by Grameen nor were they frequently made by
the Irish Loan Funds. Many loans tend to be for investments that produce a
flow of income. The purchases of a cow that produces milk or a chicken that
lays eggs are examples of such an investment. BancoSol’s loans, while in a
different context, serve a similar purpose. They tend to be made for working
capital.
Conceivably, there are many valuable investments that do not produce the
steady cash flow demanded by group and other microfinance lending schemes.
The important question here is why are the loans made with these terms? Are
frequent repayments an important part of monitoring? The answers to these
questions are important not just to the evaluation of group lending in the United
States but also for the evaluation of lending in less-developed countries.
Source of Funding and Comparative Advantage in Lending
The source of funding is important because it can limit the activities of a financial intermediary, and it can influence the optimal structure of the financial
intermediation structure. In the model, there were many lenders per borrower.
This ratio was responsible for the existence of the large financial intermediary
since the number of lenders needed to finance a borrower determines the costs
of direct lending, and consequently the savings in monitoring and screening

E. S. Prescott: Group Lending and Financial Intermediation

43

costs from intermediation. For microfinance programs it is reasonable to ask if
there are lots of lenders per borrower. First, the loans are for small amounts,
and second, many lenders are donors with large amounts to lend.
BancoSol receives some of its funding from deposits. Agafonoff (1994)
reports that in 1994 BancoSol’s average loan was $499 and its average deposit
was $225. (The majority of the bank’s loans and deposits are denominated
in U.S. dollars rather than Bolivian bolivianos.) These numbers are consistent
with the model’s assumption.
Still, many investors are large organizations whose investments are much
higher than the amount any single individual borrows. In terms of the model
some modifications would need to be made to ensure that donors lend through
an intermediary rather than directly. The simplest, and most obvious, would
be to assume that donors do not have the expertise to lend themselves. Consequently, K and Ks are much higher if they lend themselves rather than through
an intermediary. Another possibility is that donors, particularly those overseas,
find it expensive to monitor because of physical, linguistic, and even cultural
distance from the borrowers. (See Boyd and Smith [1992] for a model in which
people at different locations have a comparative advantage in lending in their
home location.)
A comparison of the United States and Bolivia suggests that a group lender
may desire different sources of funds in the two countries. In Bolivia, BancoSol
raises some of its funds from deposits, but it is a country where a large fraction
of the population does not use the banking sector. The banking sector, and more
generally the financial structure, is much more extensive in the United States.
Consequently, raising deposits might not be a group lender’s comparative advantage. Instead, debt or equity might be a better source of capital for a group
lender in the United States.
In the United States, group lenders’ comparative advantage should be in
lending rather than in collecting deposits. Lending to the poor likely requires
a different set of skills than other types of lending. BancoSol’s high administrative costs relative to the rest of the Bolivian banking sector is supportive of
the latter conjecture.14
Indeed, it is not difficult to imagine a highly specialized financial system
where traditional financial intermediaries collect deposits and then direct funds
to specialists in microfinance, who in turn lend to groups (or individuals).
There is no reason to think that traditional financial intermediaries are the best
institutional vehicles for delivering credit to the poor.

14 In the model, groups save on monitoring costs, yet in the data, group lenders spend a lot
of resources on monitoring. This is not a contradiction. The issue is how much more resources
would have to be used to monitor in the absence of groups. That is what the model captures.

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Federal Reserve Bank of Richmond Economic Quarterly

Costs to the Large Intermediary
In the model, for some parameter values the large intermediary saved costs
relative to direct lending. In practice, monitoring and screening costs may be so
high as to make any form of financing unprofitable. The problem is particularly
acute for microfinance because loans are for small amounts, and they require
frequent repayments. In the context of the model’s parameters, K and Ks might
be much higher in the United States than in less-developed countries.
The data bears out the importance of these costs. Eighty percent of BancoSol’s costs are administrative while the cost figures for the U.S. agencies
exceed the face value of the loans. BancoSol has surmounted these problems
through a combination of a low default rate and a high interest rate (about 34
percent per year). In 1994, their average cost per dollar lent was 0.16; their
borrower-to-loan-officer ratio was about 320.
Any microfinance program in the United States that desires to even approach self-sufficiency will need a similar strategy and results. None of the
four agencies have reached BancoSol’s scale. No agency made more than 107
loans in 1994. Their loan-loss ratios vary from about 2 to 17 percent, and their
costs per dollar lent uniformly exceed one. These programs are far from selfsufficient. Of course, these programs are relatively new and any activity takes
time to learn, not to mention the time needed to obtain economies of scale.
It would be interesting to compare these agencies’ default rates with those of
Grameen or BancoSol in their early years of operation.15
Still, self-sufficiency may be too strong an evaluative criterion. Many services and transfers are distributed through the social welfare system and these
programs are the right basis for comparison. Under this interpretation, microfinance is unusual in that it directs aid to specific people in the population; those
who are willing to start businesses. Furthermore, unlike most social welfare
programs, the recipients face the explicit incentive to perform or lose their aid.
Under this criterion, group lending may very well be an effective method for
targeting aid to the poor, particularly since these agencies’ costs are comparable
with those of job-training programs.
Monitoring within Groups
One of the most critical issues concerning group lending is how high is Kg ,
the cost of group monitoring?16 There are reasons to think that Kg is higher in
15 A potential problem for any program with the goal of self-sufficiency is that the interest
rates necessary to cover costs may be illegal, violating usury laws in many states of the union.
16 In the model, monitoring was an either–or proposition. The only options available were to
pay the monitoring costs and observe fellow members’ output or to not pay the cost and not see
the output. In practice, there are degrees of monitoring. Still, for the purposes of our discussion,
Kg provides a useful way to summarize these degrees.

E. S. Prescott: Group Lending and Financial Intermediation

45

the United States than in developing countries. There is more anonymity, the
costs of being excluded from a group are smaller in a rich country, and people
do not necessarily work in such close quarters.
Edgcomb, Klein, and Clark (1996) provide some indirect evidence in support of this view. They conclude that the group-lending programs have had
the most trouble in rural areas. The programs found that rural residents do not
tend to know each other well enough to be able to support groups, in part
because of the low density of the population and in part because of the low
number of self-employed people in rural areas. One agency has even resorted
to purchasing credit reports on fellow members for potential groups.
Another complication is that self-employment opportunities are more diverse in the United States than in less-developed countries (Edgcomb, Klein,
and Clark 1996).17 Group members engaged in similar activities can learn from
each other and can evaluate the borrowing proposals of fellow group members.
It probably also makes monitoring easier. This is another reason Kg may be
higher in the United States. Some of the resources used on training by the U.S.
programs may be designed to compensate for this.

4.

CONCLUSION

Lending groups are financial intermediaries, albeit small ones. The model shows
how groups, as well as larger financial intermediaries, economize on monitoring costs and transform assets. Through diversification, financial intermediaries
alleviate incentive problems and reduce the costs of monitoring and screening.
Throughout the paper, I provide extensive description of existing grouplending programs to demonstrate that group lending is a type of intermediation
that is viable in at least several environments, including some of older origin
than many probably realized. Whether it is viable in the United States is an
open question, though the conditions here appear to be less favorable for it
than in less-developed countries. Still, while the narrow focus of this paper is
on the relative merits of group lending, the broader goal is to study financial
structure. Understanding financial structure is a necessary prerequisite to the
proper formulation of policy involving financial intermediation and low-income
communities.

17 However,

different activities may have less-correlated returns. In my model, group lending
is more valuable when returns are less correlated.

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Federal Reserve Bank of Richmond Economic Quarterly

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Investing in Equities:
Can it Help Social Security?
Michael Dotsey

S

ocial Security is in trouble. A recent report by the U.S. General Accounting Office (1997) indicates that absent any changes to the current system,
payments to beneficiaries will exceed revenues from payroll taxes in
2012, and by 2029 the Social Security Trust Fund will be depleted. That Social
Security is in trouble is not really news. The system has a long history of being
underfinanced, and the current difficulties are not historically large. Recently,
the 1994–1996 Advisory Council on Social Security issued its report with various recommendations for putting the system on firm financial footing. From
an economic perspective, making the Social Security System sound is not a
difficult task. There exist a multitude of ways for doing so, but most involve
either increases in taxes, reductions in benefits, or both. Thus, any plan inherently involves difficult political decisions. However, one part of the solution
that is included in each of the three separate plans that were presented to the
Commissioner of Social Security was the recommendation that some portion
of the current trust fund be invested in the stock market. By taking advantage
of the higher returns earned by equities, this recommendation seemingly would
reduce the increases in taxes or the reduction in benefits that would be needed
to return the Social Security System to financial viability.
In this article I address the economic merits of this recommendation. My
analysis suggests that the ownership of the capital stock has very few consequences for the government’s budget. The economic opportunities available to
society are not increased by a transfer of capital from the private sector to the
government. In short, there is no free lunch.

I wish to thank Douglas Diamond, Andreas Hornstein, Thomas Humphrey, Kent Smetters,
and Alex Wolman for many useful suggestions and comments. The views expressed herein
are the author’s and do not represent the views of the Federal Reserve Bank of Richmond or
the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 83/4 Fall 1997

49

50

1.

Federal Reserve Bank of Richmond Economic Quarterly

A BRIEF HISTORY

The Inception of Social Security
Social Security was created in 1935 as an intergenerational transfer program
from workers to retirees. Its design also provided for income redistribution
among the elderly, because replacement rates (the ratio of the benefit paid in
the first year of retirement to taxable earnings in the preceding year) are higher
for low-income workers than for high-income workers. Social Security is a
pay-as-you-go system. In years when the revenue from Social Security taxes
exceeds outlays, the U.S. Treasury uses the proceeds to finance other expenditures, thereby reducing the level of government debt from what it otherwise
would have been. There exists the accounting fiction of a trust fund, but from
an economic perspective no such fund exists. The lower level of government
debt makes it more likely that future claims will be honored, but there is no
dedicated set of securities belonging to the Social Security Administration that
has the same legal standing as a government bond issued to a private citizen.
Because it is a pay-as-you-go system, there is the potential that, for a variety
of reasons, promised payments could become increasingly difficult to honor.
This is what has happened repeatedly to the Social Security System.
Despite its problems, the Social Security System has been remarkably successful in terms of its growth and its economic importance. At the time of its
creation, the old age and survivors insurance (OASI) part of the program was
fairly small, with benefits equaling 0.03 percent of GDP in 1940. By 1950 that
percentage had risen to only 0.33 percent of GDP, but by 1996 it had risen to
over 4 percent of GDP. In terms of taxable payrolls, benefits were 10.7 percent
in 1994, which is very close to the 10 percent envisioned in the 1939 Act
(see Miron and Weil [1997]) and represented roughly 19 percent of all federal
outlays. Over time, the fraction of the labor force covered by Social Security
has risen from 63.7 percent in 1940 to 97.6 percent in 1993.
Social Security has also played a major role in reducing the percent of
those over 65 who live below the poverty line. In 1959, 35.2 percent of the
elderly were characterized as poor. By 1994, that figure had dropped to 11.7
percent. The increase in Social Security benefits is in large part responsible for
this decline. Expressed in terms of 1995 dollars, the average monthly benefit in
1950 was $269.30, in 1960 it was $381.38, and in 1995 it was $719.80. Also,
the number of beneficiaries has risen substantially from 222,488 in 1940 to
37.5 million in 1995. In terms of percentages of the population over 65, only
7 percent received benefits in 1940, whereas 91.3 percent received benefits in
1995. More importantly from the standpoint of helping the poor, Social Security currently provides over 90 percent of income for half the seniors below
the poverty line and 50 percent of income for two-thirds of all beneficiaries.
As the preceding figures show, the scope and amount of coverage has
increased greatly since the inception of Social Security. The original act

M. Dotsey: Investing in Equities: Can it Help Social Security?

51

promised benefits only to those who contributed, but in 1939 benefits were
extended to spouses and surviving widows. Over time, various changes have
expanded the scope of Social Security, with perhaps the most important extension resulting from the 1950 Act that brought 10 million new workers into
the system. Also, various changes in computing benefits, coupled with high
inflation and growth in wages, served to increase benefits, which consequently
grew much faster than the economy.
Initially a 2 percent tax rate, equally divided between employer and employee, was levied on income up to $3,000. The first benefits to contributing
retirees were not to be paid until 1942, but the 1939 Act moved that date
forward to 1940. Further, no benefits were to be paid in any month that a
retiree earned more than $15. To put that figure in perspective, the average
annual wage in 1937 was $979. This feature of the system indicates that Social
Security was in part envisioned as insurance against destitution. However, under
the assumption of no inflation and no wage growth, the replacement rate for
a worker earning $1,000 for 45 years, and retiring at age 65 in 2002, would
have been 0.60 under the initial act. That means that this hypothetical worker
would have received $600 a year in perpetuity, implying that the initial act also
possessed features that went far beyond mere insurance. A 60-year-old worker
earning the same salary ($1,000) and retiring in 1942 would have received
benefits of $200 a year. With the extension of benefit eligibility, the 1939 Act
also reduced the replacement rate to 0.43. Thus, our hypothetical worker would
receive only $430 upon retirement, while his spouse would receive $215.
Under the 1939 Act, the combined tax rates on employer and employee
were 2 percent and were scheduled to rise to 6 percent by 1949 and remain
fixed thereafter. Full benefits would not begin until 1991, when workers with a
full history of contributions would be retiring. According to projections at the
time, the internal real rate of return for those retirees would be 3.9 percent, not
much above the 3 percent rate of return that was projected on the accumulated
trust fund. Or, in more relevant terms, the internal rate of return would not
be too far above the economy’s growth rate and benefits could be paid by
issuing government debt without increasing the debt-to-GDP ratio. Thus the
initial planning attempted to create a sustainable system.
A History of Problems
Over its history the Social Security System probably has never been sound.
The chief reason is that Congress tended to make benefits more generous than
originally intended and refused to raise tax rates as fast as the 1939 Act prescribed. Tax rates did not reach 6 percent until 1960. Also, economic factors
such as usage growth interacted with the methodology for calculating benefits,
increased the level of benefits in unintended ways during the 1970s, and placed
the system under tremendous strain. Corrections to the methodology were not

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Federal Reserve Bank of Richmond Economic Quarterly

made quickly enough, and tax rates were not raised sufficiently, so that the system almost defaulted in the early 1980s. Demographic changes also conspired
to make the system less sound than it would have been under stable population
growth. Thus, under current law someone just entering the labor force will earn
a rate of return on Social Security contributions that is probably negative, while
the rate of return for those that have already retired is significantly higher than
was intended.
The 1950 Act, which brought in 10 million new workers, also calculated
their benefits in a way that provided them with large transfers. Expansion in
scope need not have been detrimental to the soundness of the system, but
these workers received benefits that were based on their wage history after
1950 rather than on their entire wage history. Thus, individuals from this group
who retired soon after 1950 received full benefits and a large transfer from
the existing system. Basically for this group the link between the replacement
rate and the number of years of paying into the system was cut, and these
new retirees received the same benefits as those who had been in the system
since its inception. To accommodate this change, average benefits were slightly
reduced.
Perhaps the most severe problem for the system was created by the 1972
Act, which for the first time included automatic price adjustments. Previously,
such adjustments were made on an ad hoc basis. However, the adjustment
procedure ended up overcompensating workers and made replacement rates
unstable (for an excellent discussion see Munnell [1977]). The cost-of-living
adjustment for retirees did not present a problem. Rather, the calculated replacement rates for newly retired workers were overstated. In essence these
workers received an increase in their benefits that accounted not only for inflation but for wage growth as well. Because wages tend to rise with inflation,
new retirees received a double counting. The amount of initial benefits also
increased with the disparity between real wage growth and inflation. In this
manner, the economic climate at the time, along with the unsound method of
computing initial benefits, placed great stress on the system, with replacement
rates rising from 47.9 percent in 1970 to 66.7 percent in 1980. As a result,
the individual that retired in the 1970s received the largest net transfer of any
cohort under Social Security.
The mistakes in the 1972 Act led to the rescue package of 1977, which constituted the largest peace-time tax increase in U.S. history. The rescue package
also stopped initial benefits from rising faster than wage growth. The system was pronounced sound for the rest of the century and well into the next
one. Unfortunately, the pronouncement was wrong. By 1981, there was a high
probability that the system would not be able to meet its promised benefits.
A commission was appointed to deal with the problem. Its lack of complete
success is in part why Social Security restructuring is currently receiving so
much attention. The 1983 Act did raise the schedule of tax rates and the annual

M. Dotsey: Investing in Equities: Can it Help Social Security?

53

maximum on taxable earnings. It also effectively reduced benefits by taxing
some portion of Social Security payments. Finally, it gradually raised the age
to 67 at which full benefits were paid for the cohort born in 1960. Combined,
these changes averted a problem of failing to honor legislated benefits but failed
to solve the problem of long-term insolvency.
The Current Problem
The Social Security System as currently constituted is not actuarially sound.
In this regard, the important date is 2012, because that is when expenditures
will exceed receipts. At that point the federal government will have to raise
taxes, reduce government spending, or increase its borrowing in order to make
the promised payments to retirees. Beyond that date, the revenue shortfall will
increase and the necessary adjustments will be more dramatic. It is estimated
that the revenue shortfall will be $57 billion in 2015 and grow to $232 billion
in 2020. Put in perspective, current total OASI payments are approximately
$308 billion dollars. This deficit will occur in part because there will be an
estimated 50.4 million beneficiaries in 2015, up from 37.5 million in 1995.
As mentioned earlier, the system’s current troubles are a consequence of
increasing benefits, due both to the increased number of retirees and the more
generous benefits that each retiree receives. One way to gauge the increase in
the level of benefits is to compare them with average wages. For example, in
1953 the maximum benefit was equivalent to 30.5 percent of the average wage.
By 1981 the corresponding figure was greater than 50 percent, and in 1995 it
equaled 60.5 percent (Marcks 1997). Unquestionably, retirees’ benefits have
been rising relative to the tax base that can support those benefits.
The problem is also one of demographics. In 1945 there were 42 workers
per retiree. In 1995 that number had shrunk to 3.3, and it is projected that in
2030 there will only be 2 workers per retiree. Furthermore, the life expectancy
of individuals has increased since the inception of the system, meaning that
a greater fraction of contributors have become beneficiaries. Also, the length
of retirement has increased. In 1940 a 65-year-old male and female had a life
expectancy of 12 and 13 years, respectively. By 2015 the comparable numbers
will be 16 and 20 years.
These demographic features imply that maintaining the current level of
benefits requires a significant increase in taxes. The Report of the 1994–1996
Advisory Council on Social Security (Department of Health and Human Services 1997) indicates that taxes would have to be raised immediately by 2.13
percent to attain 75-year balance.1 These calculations explicitly take into ac1 It should be noted that 75-year balance and actuarial soundness are not the same thing,
because the problems of the system tend to worsen in the future. Thus 75-year soundness today
implies 75-year unsoundness tomorrow.

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Federal Reserve Bank of Richmond Economic Quarterly

count interest payments and payments on principal from the fictitious trust
fund. To make these payments, the government would have to increase the
level of the debt, reduce spending, or increase tax revenue from other sources.2
Thus, total tax payments could be substantially higher if all forms of taxes are
considered. Waiting to adjust tax rates will only make the problem worse.

2.

THE STOCK MARKET TO THE RESCUE?

Over the period 1926 to 1993, the real return on the Standard & Poor’s 500
averaged 9 percent, while the real yield on intermediate-term U.S. government
bonds averaged only 2.2 percent. This difference in yields is large. For example, earning 9 percent implies that your investment doubles approximately
every eight years as compared to every 35 years with a 2.2 percent return.
Furthermore, in every 22-year period since 1926, equities have outperformed
bonds. These considerations have spurred many observers to argue that investing at least some portion of the Social Security Trust Fund in equities can avert
the financial difficulties facing the system.
In one sense the proposition is true. Increasing the yield on the trust fund
can make the Social Security System more viable in isolation. However, it
can only do so by making the rest of the government worse off. On net, an
individual taxpayer will be little affected by this investment policy. In order for
the government, or some part of it, to take an equity position, the government
as a whole must issue more bonds. This swap of paper claims with the public
affects the allocations and the risk characteristics of the respective portfolios
but quantitatively does not have any appreciable effect on the government’s
overall budget. The economy cannot produce any more goods, and although
the consumption profile of the representative household may be somewhat altered, the effects of this alteration are small. Since taxpayers are the ultimate
receivers of any government earnings or losses, it matters little who owns the
capital stock.
A number of economists recognize this fairly simple notion. Federal Reserve Board Chairman Alan Greenspan expressed the idea cogently in his recent
Remarks at the Abraham Lincoln Award Ceremony of the Union League of
Philadelphia (1996, p. 8), “Bonds and equities are merely the paper claims to
income earning assets, and the value of the income stream is not determined by
short-run changes in the supply and demand for securities. Rather, equity prices
2 If

the payments promised by Social Security are equivalent to payments promised on
government bonds, then increasing the level of the measured debt to pay off these claims does
not affect the overall indebtedness of the U.S. government. This action just transfers a promise
into an explicit security. Treating the promised Social Security benefits in a similar way to any
other government IOU implies that the true level of the government debt is closer to $17 trillion
instead of the $5 trillion currently calculated.

M. Dotsey: Investing in Equities: Can it Help Social Security?

55

must, in the long run, reflect the underlying earnings of the corporations on
which the equities are a claim, as well as society’s need to be compensated for
postponing consumption into the future and its perception and attitudes toward
risk as a consequence of uncertainty about the future. Indeed, the total market
value of debt plus equities is, to a first approximation, likely to be unaffected
by a shift in the balance of paper claims.” These sentiments are also reflected in
the views of Herbert Stein (1997, p. A18), “. . . privatizing the Social Security
funds would not add to national saving, private investment, or the national
income and would not allow the system to earn more income without anyone
earning less.”
Others, however, have argued to the contrary and have made the purchase
of equities by the trust fund seem like a free lunch. For example, editorial
commentary in Barron’s Online by Thomas G. Donlan (1997) states that “Unless the system invests in private enterprise and those investments continue
to earn historically high returns the Baby Boom generation will pay for its
own retirement.” Investing in equities is a major component of all three plans
presented by the 1994 –1996 Advisory Council on Social Security (Department
of Health and Human Services 1997).
The Trust Fund
The Social Security System is but one part of the government. It is the largest
part, with transfers amounting to 22 percent of government expenditures in
1995. The system’s trust fund is really a myth. Social Security receives contributions or taxes from workers and their employers and pays out benefits
to retirees, their dependents, and those on disability. Excesses in receipts over
expenditures are handed over to the U.S. Treasury to be used in financing other
governmental activities. Employing an accounting fiction, the Social Security
System treats these transfers as investment in government securities and adds
them to an imaginary portfolio that also collects fictitious interest payments.
From the perspective of the government’s total budget, this practice implies
that the Treasury issues fewer bonds to the public than it would if there were
no surplus received from Social Security. Unlike Treasury bills issued to the
public, however, the IOUs from the Treasury to Social Security are not counted
as government debt.
What would happen if the Social Security System invested in equities? The
system would currently turn over less surplus to the Treasury, and the Treasury
would have to issue more bonds to the public. Again, from the perspective of
the government as a whole, this transaction amounts to a trade of bonds for
equities with the public. Can such a trade benefit the public? Since equities
are a claim on firms, government ownership of stock amounts to government
ownership of some portion of the country’s capital stock. So the preceding
question can be rephrased. Does it matter who owns the capital stock? The
analysis presented below attempts to shed light on that question. It turns out

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Federal Reserve Bank of Richmond Economic Quarterly

that the policy of government investment in equities has either only minor or
no effects on the government’s budget and the saving rate of the economy.
Whether the government’s financing decisions have any economic effect depends on its ability to transfer risk across individuals. In the models considered
in Section 3, that ability is absent, and hence government portfolio decisions
are irrelevant. The overlapping generations model of Section 4 allows some
scope for more efficient risk-sharing and the government’s portfolio decision
does affect economic behavior. Quantitatively, this effect turns out to be small.

3.

A MODEL WITH INFINITELY LIVED AGENTS

In this section I use a model populated by infinitely lived agents (or more generally, the dynastic families possessing bequest motives as in Barro [1974]) to
explore whether government investment in private capital affects the amount of
tax revenue needed to support a given stream of transfer payments. Answering
this question is analogous to answering the question of whether investment by
the Social Security Administration in the stock market would have any impact
on the financing of a given stream of Social Security payments. I analyze this
question in a sequence of models that highlight the key issue, namely that equity
premium considerations are unimportant and it is only the transferring of risk
across generations that has any effect on economic outcomes.3 The model with
infinitely lived agents clearly makes the point that when there is no possibility
of transferring risk among agents, because all agents are essentially the same,
the existence of an equity premium does not in any way allow government
ownership of capital to influence economic outcomes.
To begin, I shall consider a world in which all transfers and taxes are lump
sum. Private agents own some portion of the capital stock and the government
owns the rest. The government may also issue debt. It finances transfer payments and the interest payments on debt through its earnings on capital and
through taxes. I will show that in such a world the behavior of individuals
is unaffected by the portion of the capital stock owned by the government.
Essentially, any distribution of ownership of the capital stock is consistent with
the initial path of transfers and taxes and has no effect on the consumption
or saving decisions of individuals. In other words, the government’s portfolio
decision is irrelevant. I shall then extend this model to include distortionary
taxes and show that the results are unchanged.
The Model with Lump Sum Taxes
This model economy is populated by people who live forever or, more generally, by the dynastic families in Barro (1974). Output is stochastic and is
3 For

a detailed analysis of these issues, see Bohn (1997a, b).

M. Dotsey: Investing in Equities: Can it Help Social Security?

57

produced via a standard neoclassical production function using capital and
labor. The government finances lump sum transfers through lump sum taxes,
the issuance of debt, and the return from its ownership of capital.
Individual Decisions
To start the analysis, consider the problem of the individual agent who wishes
to maximize lifetime well-being or utility subject to a budget constraint. The
individual owns some capital that earns ρ(st ) in state s at time t. That is,
the return to capital is stochastic and, while one observes the actual return in
any given period, future returns are uncertain and depend on the state of the
economy in that period. Individuals also receive transfer payments from the
government Tr(st ) and pay taxes T(st ). These transfers and taxes may, but need
not, depend on the state of the economy. Individuals also own government
bonds, b(st ), that pay r(st ) units of consumption in all states in period t + 1.
Finally, given a capital stock at the beginning of period t, agents choose how
much capital to bring into the next period, k(st ), and how much to consume
this period, c(st ).
Formally, the representative agent maximizes discounted expected lifetime
utility
β t u[c(st )]π(st )

max
t,St

subject to per-period budget constraints in each possible state st .
c(st )+ bd (st )+ k(st ) ≤ w(st )n+ρ(st )k(st−1 )+[1+r(st−1 )]b(st−1 )+Tr(st )−T(st ),
where w is the real wage rate, n is exogenous labor supply, and ρ is the rate of
return on capital. For simplicity, I assume that capital fully depreciates each period. Thus, agents are maximizing their utility, taking into account expectations
of all possible future events. In the notation above, st is the realization of one of
finitely many states of the economy at time t. st represents a particular history
of realizations up to time t. That is, st = (s0 , s1, . . . st ) is a particular history
of events up to time t. The set St represents all the possible histories that can
occur. Each event occurs with probability π(st ) and each history occurs with
probability π(st ). Each agent rents out labor and capital to firms in competitive
rental markets and earns the appropriate marginal product of each factor.
The first-order conditions for optimal bond and capital accumulation are
u [c(st )] = β

π(st+1 )u [c(st+1 )][1 + r(st )]

(1a)

π(st+1 )u [c(st+1 )]ρ(st+1 ).

(1b)

st+1

and
u [c(st )] = β
st+1

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Federal Reserve Bank of Richmond Economic Quarterly

These conditions imply that agents accumulate assets so that they are just indifferent between consuming an extra unit of consumption in any particular state
st or investing in either another bond or an extra unit of capital and consuming
the proceeds of that investment next period. Also, since a government bond
returns the same amount in each state at t + 1, it is less risky than holding
capital whose return is uncertain. The interest on a government bond will,
therefore, generally be less than the expected return on capital. That is, capital
will on average earn a premium over bonds with the amount of the premium
depending on the agent’s aversion to risk and the underlying riskiness of the
return on capital. It is this feature of bonds and capital that initially seems to
suggest that the government, by issuing bonds and owning some more capital,
can reduce the tax burden associated with any stream of transfer payments.
However, as the first-order conditions make clear, both of these choices have
the same value when adjusted for risk, namely the current marginal utility of
consumption. Thus, there is no free lunch.
The Government
Each period the government makes some transfers, collects some taxes, and
adjusts its portfolio by either issuing or repurchasing some government bonds
or buying or selling some capital, x (or claims to the capital, which amount to
the same thing). In each state, the government’s net holding of assets obeys
bs (st ) − x(st ) = b(st−1 )[1 + r(st−1 )] + Tr(st ) − T(st ) − ρ(st )x(st−1 ).

(2)

It is clear from this expression that, all other things equal, an increase in the
capital stock held by the government at time t − 1 reduces the taxes that are
necessary to maintain the same net asset position. The experiment we are interested in, however, is not what happens if someone donates an extra unit of
capital to the government but what happens when the government increases its
holdings of capital by issuing additional debt.
Market Clearing
For any allocation of consumption, bonds, and capital to be an equilibrium,
it must be consistent with the resource constraints of the economy and with
supply equaling demand. In particular, for each state the following equations
hold:
c(st ) + k(st ) + x(st ) = A(st )[k(st−1 ) + x(st−1 )]α n1−α

(3)

bs (st ) = bd (st ).

(4)

and

Equation (3) indicates that the amount consumed and invested must equal the
output produced in the current period, and equation (4) requires that the supply

M. Dotsey: Investing in Equities: Can it Help Social Security?

59

of bonds issued by the government must be equal to the demand for these
bonds by the public.
Solution
The consumption decision of agents will now be shown to be independent of
portfolio decisions of the government. Alternatively, agents do not care who
owns the capital stock since they are indifferent between holding an extra unit
of government debt or an extra unit of capital. In particular, consumption in
any state is given by
c(st ) = (1 − β)A(st )K(st−1 ),

(5)

where K is the aggregate capital stock equal to k + x. The accumulation of
private capital is then expressed as
k(st ) = βA(st )K(st−1 ) − x(st−1 ).

(6)

As long as government capital does not exceed βA(st )K(st−1 ), the above solutions satisfy the first-order conditions of agents and do not violate the economy’s
overall resource constraint. Thus, for any supportable path of taxes and transfer
payments, individuals are indifferent as to who owns the capital stock.
The Model with Distortionary Taxes
Next consider the case where the government raises revenue through distortionary taxation. In this setting it is not so easy to represent analytically the
solution to the decision problem of agents. However, by looking at the individual’s first-order conditions and budget constraints along with the budget
constraint and transversality condition of the government, one sees that the
proportion of the capital stock owned by the government is irrelevant.
Individual Decisions
With distortionary taxes on both labor and interest income, the representative
agent maximizes lifetime utility subject to the following per-period budget
constraint,
c(st ) + bd (st ) + k(st ) ≤ ρ(st )[1 − τ (st )]k(st−1 ) + w(st )n[1 − τ (st )]
{1 + r(s

t−1

)[1 − τ (s )]}b(s
t

t−1

(7)

t

) + Tr(s ).

Unlike the previous budget constraint, the government now taxes wages and the
return on capital and bonds at the rate τ . The first-order necessary conditions
for optimal bond holdings and investment are
π(st+1 )u [c(st+1 )]{1 + r(st )[1 − τ (st+1 )]}

u [c(st )] = β
st+1

(8a)

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Federal Reserve Bank of Richmond Economic Quarterly

and
π(st+1 )u [c(st+1 )]{ρ(st+1 )[1 − τ (st+1 )]}.

u [c(st )] = β

(8b)

st+1

The consumer’s accumulation of assets must also satisfy the transversality
condition,
lim
p(st+j )[k(st+j ) + b(st+j )] = 0,
(9)
t
t
t+1
j→∞

t+j

t+j

j,st+1 ∈ St+1

where p(st+j ) = β j π(st+j ){u [c(st+j )]/u [c(st )]} is the price of a contingent
t
t
t
claim. In the above expression st+j indicates a particular history of states from
t
t to t + j and St+j is the set of all possible histories.
t
Government
The government’s budget constraint is given by
bs (st ) − x(st ) = b(st−1 ){1 + r(st−1 )[1 − τ (st )]} + Tr(st )
−ρ(st )τ (st )k(st−1 ) − τ (st )w(st )n − ρ(st )x(st )

(10)

and indicates that the government’s net liability position depends on its debt,
the net interest paid on that debt, its revenues from taxing income earned from
capital and labor, as well as the revenue it earns on its own capital stock.4
The budget constraint implies that in states where capital has a relatively high
rate of return, some debt is retired, while in states where capital’s return is
low, debt is issued. The government’s net asset position must also satisfy the
transversality condition
p(st+j )[b(st+j ) − x(st+j )] = 0.
t
t
t+1

lim

j→∞

(11)

t+j
t+j
j,st+1 ∈ St+1

Equilibrium
Formally, the definition of an equilibrium is given by
4 Using the above budget constraint and the first-order conditions of the representative agent,
the government’s lifetime budget constraint as of period t can be expressed as
t+j

ρ(st )τ (st )k(st−1 ) + ρ(st )x(st−1 ) − b(st−1 ) =

p(st

t+j

)Tr(st

)+

t+j
t+j
j,st ∈ St

t+j

t+j

p(st+1 )ρ(st

t+j

)[k(st

t+j

) + x(st

)].

t+j
t+j
j,st+1 ∈ St+1

Notice that only the sum of private and government-owned capital stock enters the right-hand
side of equation (11). Therefore, for any sequence of state-contingent prices, only the total capital
stock and not its distribution affects the tax policies that are necessary to support a given stream
of transfer payments.

M. Dotsey: Investing in Equities: Can it Help Social Security?

61

Equilibrium: Given the initial conditions b(st−1 ), x(st−1 ), and k(st−1 ), an equilibrium is a sequence of quantities and prices {b(s), k(s), x(s), K(s), c(s), w(s),
r(s), ρ(s), τ (s), Tr(s)} for all histories s ∈ S∞ satisfying the individual’s firstt
order conditions (8a) and (8b), the individual’s budget constraint (7), the
government’s budget constraint (10), the economy’s resource constraint (3),
and the transversality conditions of both the individual and the government (9)
and (11).
Irrelevance Proposition:5 Suppose that {b(s), k(s), x(s), K(s), c(s), w(s), r(s),
¯
¯
ρ(s), τ (s), Tr(s)} is an equilibrium, then any {b(s), k(s), x(s), K(s), c(s), w(s),
¯
¯
¯
r(s), ρ(s), τ (s), Tr(s)} is an equilibrium if b(s), k(s), x(s) satisfy (a) k(s), x(s) ≥ 0
¯
¯
¯
¯
and k(s) + x(s) = K(s), and (b) b(s) is defined recursively by (10).
¯
Proof: The individual’s first-order conditions and the economy’s resource constraint are satisfied because the real allocations are identical in the two equilibriums. The individual’s transversality condition is, therefore, also satisfied.
Equilibrium in the goods market and condition (b) imply that the household’s
budget constraint is also satisfied. Examining the lifetime budget constraint of
the government from date t onward, one derives that
T

p(st+j )[Tr(st+j ) − w(st+j )τ (st+j )n
t
t
t
t

b(st−1 ) − x(st−1 ) =
j=0 st+j ∈ St+j
t
t

−ρ(st+j )τ (st+j )K(st+j )]
t
t
t

(12)

p(st+T+1 )[b(st+T+1 ) − x(st+T+1 )].
t
t
t

+
st+T+1 ∈ St+T+1
t
t

Because the first two terms are the same for both equilibriums, the last term
must be the same for both equilibriums. Therefore, the transversality condition
must hold for the second equilibrium. Hence, different distributions of the
capital stock do not affect the aggregate capital stock, consumption, rates of
return, tax rates, wages, or transfer payments.
As demonstrated in these models, the ownership of the capital stock has no
effect on economic outcomes and is not an avenue that can be used to rescue
the Social Security System in an economy where agents are altruistically linked
to future generations and, hence, behave as if they were infinitely lived.

4.

A MODEL WITH FINITE LIVED AGENTS

The previous two cases demonstrate that a premium in the return to capital relative to bonds is not sufficient for government portfolio decisions to have any
5I

would like to thank Andreas Hornstein for suggesting and helping me with this particular
form of the argument.

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Federal Reserve Bank of Richmond Economic Quarterly

real effect on consumer decisions. Changes in portfolio allocations do not affect
the lifetime opportunities of the average individual, so they do not have any real
consequence. In a model with finite lived agents, however, portfolio decisions
generally will affect the economic behavior of consumers—not because capital
earns a higher return than bonds but because a change in the portfolio of old
agents must affect their consumption decisions. In the last period of life it is
the only decision they have left to make. Thus, the government ownership of
capital means that the current old agents hold more bonds. This consideration
implies that their consumption stream has different risk characteristics than
if the government owned no capital. Because the government’s ownership of
capital can transfer risk between current and future generations, it can change
behavior.6 In the setting of infinitely lived agents, there is no one to whom they
can transfer risk. But because we are now considering different generations,
there is the potential for risk transfer. How big an effect policies involving
portfolio composition may have is an open question. In this section, some
rough estimates are formed in a simple two-period overlapping generations
model. The results suggest that government ownership of capital may not be
the boon that its proponents suggest.
The Individual
In the first period of life, a young individual works a fixed number of hours, n.
With his earnings, he pays taxes, saves to finance consumption when old, and
purchases goods for current consumption. Saving takes the form of ownership
of the capital stock and government bonds. When the individual reaches old
age, he receives transfers from the government, rental on the capital stock that
then fully depreciates, and after-tax interest plus principal on his government
bonds. With this income he purchases consumption goods. In this model economy production is stochastic and transfer payments are fixed. Formally, the
individual’s problem is
max {u[cy (st )] + β

u[co (st+1 )]π(st+1 )}
st+1

subject to the budget constraints
(13a)

co (st+1 ) ≤ ρ(st+1 )[1 − τ k (st+1 )]k(st ) +
{1 + r(st )[1 − τ k (st+1 )]}b(st ) + T o ,

and

cy (st ) + k(st ) + b(st ) ≤ w(st )n[1 − τ (st )] + T y

(13b)

6 This idea is discussed in Volume II of the Report of the 1994 –1996 Advisory Council on
Social Security (Department of Health and Human Services 1997). The effects of government
financing decisions on intergenerational risk-sharing are formally derived in Bohn (1997a, b),
Smetters (1997), and Mariger (1997). Smetters shows that the risk-sharing engendered by the
government’s purchase of equities is equivalent to options contracts between generations.

M. Dotsey: Investing in Equities: Can it Help Social Security?

63

where the last constraint must hold for each possible state st+1 drawn from the
set St+1 . The superscripts y and o refer to young and old, respectively.
Here, as in the previous examples, s indexes the various possible states that
can occur. The above specification assumes that agents know what state they
are currently in but are unsure about next period’s state. All they know is the
probability, π, of any particular state occurring. Specifically, at time t, agents
know how productive the economy is, the transfers that are given to both the
current old and current young, the current tax rates on labor income, τ , and
interest income, τ k , the current wage rates, and the promised rate of interest on
government bonds. They do not, however, know what these variables will be
in the future. Thus, they attempt to maximize not only the utility from current
consumption but expected utility from future consumption.
The first-order conditions for the problem are
u [cy (st )] = λy (st ),

(14a)

βπ(st+1 )u [co (st+1 )] = λo (st+1 ) for each st+1 ∈ St+1 ,

(14b)

u [co (st+1 )]π(st+1 )ρ(st+1 )[1 − τ k (st+1 )],

u [cy (st )] = β

(14c)

st+1

and
{1 + r(st )[1 − τ k (st+1 )]}u [co (st+1 )]π(st+1 ),

u [cy (st )] = β

(14d)

st+1

where a prime indicates the first derivative and λy (st ) and λo (st+1 ) are the multipliers associated with the constraints (13a) and (13b). The last two constraints
give the efficient consumption-saving decisions of the current young. These
conditions state that at an optimum the marginal utility of forgoing one unit
of consumption today must be equal to expected marginal utility of additional
consumption tomorrow earned from the proceeds of investing in another unit
of either capital or bonds. Notice that the last two equations also imply that
the certain yield on a bond and the expected after-tax yield on capital must be
such that the agent is indifferent between holding a bond or capital. As before,
because the return on capital is uncertain, the premium that capital earns over
bonds depends on the agent’s degree of risk aversion.
Firms
Firms produce output by employing the labor of the young and renting capital
from the old and from the government. The production function is constant
returns to scale and is given by
Y(st ) = A(st )K(st−1 )α n1−α ,

(15)

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Federal Reserve Bank of Richmond Economic Quarterly

where Y is aggregate per capita output, and K is the aggregate per capita capital
stock. The maximization of profits implies that each factor receives its marginal
product, which will depend on the productivity shock A(st ).
Government
The government issues bonds and purchases capital. It also supplies transfers
to the young, T y , and the old, T o . These latter transfers may be thought of as
Social Security although in reality the old receive more than just OASI payments alone. The government also raises revenue by taxing wage and capital
income as well as the interest earned on bonds. Specifically, the government’s
budget constraint is
B(st ) − x(st ) = {1 + r(st−1 )[1 − τ k (st )]}B(st−1 ) − ρ(st )x(st−1 )
+ T + T − τ (s )w(s )n − τ (s )ρ(s )k(s
y

o

t

t

k

t

t

t−1

(16)

),

where B(s) is the per capita aggregate supply of government bonds and x(s) is
the per capita capital stock owned by the government. The government’s net
indebtedness B − x is positively influenced by its repayment of existing debt,
the interest on that debt, and transfer payments. The government’s earnings on
its capital stock, as well as the revenue from the taxation of labor, bonds, and
the private sector’s return on capital, all reduce the government’s indebtedness.
Equilibrium
Equilibrium in this model is defined as a sequence of quantities (consumption,
capital, and bond allocations), factor prices (wages, interest rates, and rental
rates), and taxes and transfers that are consistent with each agent’s maximization
of expected utility, and the firms’ maximization of profits. Equilibrium satisfies
the individual’s budget constraints (13a) and (13b), the government’s budget
constraint (16), and the government’s transversality condition and results in the
clearing of both the bond and goods markets. In particular for each possible
history,
Y(st ) = co (st ) + cy (st ) + K(st )

(17a)

B(st ) = b(st ).

(17b)

and

Also, the per capita capital stock must equal its individual components, i.e.,
K(s) = k(s) + x(s).
Unlike the case where agents are in effect infinitely lived, a similar irrelevance proposition does not apply. In the overlapping generations model,
two separate budget constraints, one for the current old, (13b), and one for
the current young, (13a), must hold simultaneously. Notice that the sum of
these two budget constraints is the same as the budget constraint for the

M. Dotsey: Investing in Equities: Can it Help Social Security?

65

infinitely lived agent. Thus any allocation that satisfies the economy’s resource
constraints and the government’s budget constraint will satisfy the sum of the
two agents’ budget constraints; hence, total consumption will be unchanged.
However, this allocation will not generally satisfy each budget constraint separately, and individual consumption will vary with changes in the distribution
of capital. The variation in individual consumption implies that rates of return
will have to change as well and that the same sequence of tax rates cannot
support an identical path of transfer payments.
Analyzing the Effects of Government Ownership of Capital
To analyze the effects of government ownership of capital, I analyze the effect
on average tax rates of changes in the proportion of the capital stock owned
by the government. In doing so, the pattern of transfer payments and the government’s net asset position, B − x, are fixed. As a result the experiment does
not create any additional government indebtedness and maintains the level of
benefits received by the elderly. The results of this experiment are suggestive
but not definitive. The model I use is admittedly stylized. Moreover, I do not
investigate plausible alternative fiscal policies, including those fixing the net
present discounted value of government liabilities rather than fixing them in
each and every period. The latter policy would produce a smoother stream of
taxes than the one analyzed here but would be computationally much harder
to implement. Also, because of the assumption that people live for two periods only, the benefits of risk-sharing are likely to be overemphasized in this
framework. Old agents are required to hold all of the capital stock; thus any
ownership of capital by the government reduces their exposure to rate-of-return
risk. If the model included more periods, old agents could shift some of this
burden to agents in their middle ages and thus reduce the risk-sharing benefits
that ensue from the government’s ownership of capital. The model also excludes
other forms of risk-sharing arrangements, such as capital-gains loss-offsets and
progressive taxation. Adding these features to the model would further reduce
the gains to intergenerational risk-sharing.7
The equations used to solve the model include one that specifies the policy
of fixing the government’s net indebtedness and an equation that specifies the
taxation of labor income relative to interest income. Equations 13(a,b), 14(c,d),
15, 16, and the two first-order conditions that determine the marginal product
of capital and labor are also employed. Together with a behavioral relationship
that specifies the government’s purchase of capital, the solution to the model involves solving 11 independent equations in 11 unknowns. The variables solved
for are the privately held capital stock, the publicly held capital stock, govern7I

wish to thank Douglas Diamond and Kent Smetters for bringing these points to my
attention.

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Federal Reserve Bank of Richmond Economic Quarterly

ment bond issue, consumption by the young, consumption by the old, output,
the interest rate paid on bonds, the rental rate on capital, wages, and the tax
rates on labor and interest income, respectively. This system can be reduced
to three equations that determine the interest rate, the aggregate capital stock,
and the tax rate. In deriving these equations, I assume that the government
maintains ownership of a fixed percentage of the capital stock, µ. It is also
assumed that utility displays constant relative risk aversion and takes the form
c1−σ −1

u(c) = 1−σ . Thus, the solution to this three-equation system yields the
policy function for K(st ) = hk [K(st−1 ), A(st ), A(st−1 )], the functions τ (st ) =
hτ [K(st−1 ), A(st ), A(st−1 )], and r(st ) = hr [K(st−1 ), A(st ), A(st−1 )].
To analyze the effects of government investment in capital, two slightly
different models are simulated, one in which only labor is taxed, τ k = 0, and
one in which all income is taxed at the same rate, τ = τ k . For given values
of transfers and net government indebtedness, I then compare tax rates and
the aggregate capital stock in model economies in which the government owns
0, 2.5, 5, and 10 percent of the capital stock. The proposal of investing up
to 40 percent of the Social Security Trust Fund in equities would result in a
much smaller proportion of government ownership of the capital stock than
any of the percentages considered. In 1995 the value of the Social Security
Trust Fund was approximately $458 billion, while the value of traded equity
was greater than $7.7 trillion. Thus, the experiments will, on this dimension,
overstate the effects of the current proposal. In essence, I am comparing the
equilibrium outcomes of four different economies. Transitional questions are,
therefore, not addressed by this experiment.
Calibration
In calibrating the model, I envision a period as corresponding to 25 years.
β is set at 0.5, which corresponds to an annual discount factor of roughly
0.973. Labor’s share of output, α, is 2/3, and the coefficient of relative risk
aversion, σ, is set at 10, implying an average equity premium between 5.7 percent and 7.1 percent. Transfers to the old generation are set to equal 4 percent
of steady-state output in the model. When only labor is taxed, such transfers are
equal to the actual percentage of output distributed by OASI. The government’s
indebtedness is 1 percent of output and transfers to the young are roughly 2.5
percent of output, implying a steady-state tax rate on labor of 10.67 percent.
This tax rate is close to the current tax rate of 10.52 percent on the OASI
portion of the Social Security tax. Thus, the labor-tax-only model is calibrated
to approximate the tax rate and the transfers that actually occur. Allowing the
government also to tax capital increases the tax base and results in a lower
steady-state tax rate and a somewhat higher level of capital and more output.
The fraction of output transferred to the old is, therefore, also somewhat lower
at 3.65 percent, although the old are receiving the same transfer in both models.

M. Dotsey: Investing in Equities: Can it Help Social Security?

67

To analyze the effect on the average tax rate of government ownership
of capital, I simulate both model economies over four generations or periods
1,000 times and take averages of the tax rates and capital stock that are produced by the simulations. Each simulation is started at capital’s nonstochastic
steady state, which is invariant to the government’s portfolio allocation, and
each succeeding capital stock is solved for based on the preceding realized
value of capital and the past technology shocks. The tax rates and interest rate
that are consistent with this solution are also obtained. The stochastic process
for technology is identically and independently distributed with mean 1 and
standard deviation of 0.08. The standard deviation was chosen to match the
standard deviation of 25-year cumulative deviations from trend over the post–
World War II period. This figure would represent the standard deviation of any
generation’s income from trend income. The standard deviation of this cumulative deviation from trend output was 0.13. I then used a standard deviation
that was as close to 0.13 as possible and that still allowed for well-behaved
policy functions of the capital stock.8 Because of the positive comovement of
inputs with the technology shock, 0.13 is an upper bound on the variation in
the technology shock. For example, Christiano and Eichenbaum (1992) obtain
estimates of the relative variability of the technology shock to output anywhere
from 48 to 90 percent. Therefore, 0.08 may not be an unreasonable number.

Results
The results of this experiment are reported in Tables 1 and 2. Table 1 includes the results when only labor is taxed, and Table 2 contains the results
when both labor and interest income are taxed. For the case when only labor
is taxed, one sees that average tax rates fall from 0.1059 to 0.1041 as the
government increases its ownership of capital from zero to 10 percent of the
aggregate capital stock. At 2.5 percent ownership, the decline in the average tax
rate needed to support the level of transfer payments is negligible. It follows
that ownership of equities by the Social Security Trust Fund would have little
effect on the viability of the Social Security System. Because the decline in
tax rates is so small, the capital stock is only marginally higher under the
policy of government ownership of capital. In short, this proposed policy has
little economic effect. The case where all income is taxed at the same rate is
qualitatively similar. Basically, each economy’s performance is not influenced
by government portfolio decisions.

8 The models investigated above possess two steady states. One steady state, which is unstable, occurs at relatively low values of the capital stock. If the technology shock is too large,
the capital stock potentially can enter this unstable region and the policy functions diverge.

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Table 1 Effects of Government Ownership of Capital
(only labor is taxed)
Fraction of capital owned

0

2.5

5

10

Average tax rate
Standard deviation of tax rate
Average capital stock
Standard deviation of capital stock

0.1059
0.0074
0.1059
0.0139

0.1054
0.0082
0.1061
0.0141

0.1049
0.0089
0.1063
0.0143

0.1041
0.0105
0.1066
0.0147

Table 2 Effects of Government Ownership of Capital
(all income is taxed)
Fraction of capital owned

0

2.5

5

10

Average tax rate
Standard deviation of the tax rate
Average capital stock
Standard deviation of the capital stock

0.0610
0.0042
0.1420
0.0163

0.0606
0.0048
0.1421
0.0165

0.0603
0.0053
0.1422
0.0166

0.0596
0.0064
0.1425
0.0169

5.

CONCLUSIONS

Current proposals for modifying Social Security have one key feature in common: namely, investing part of the trust fund in equities. Advocates believe
that such a reallocation of the trust fund’s portfolio will make the system more
viable, and maintain the level of benefits without resorting to large increases in
taxes. After analyzing the effects of such reallocation in some basic economic
models, the results are not encouraging. Even though capital on average earns
a higher rate of return than bonds, the government is not able to take much
advantage of this differential, because only the ability to shift risk matters. The
results in this regard are similar to those found in Bohn (1997a, b), Mariger
(1997), and Smetters (1997). Quantitatively, this risk shifting from old to young
does not significantly affect the government’s budget or the economic behavior
of individuals. In short, under the fiscal policies studied above, there is not
much to be gained by government ownership of the capital stock. Actuarial
soundness of the Social Security System will have to be achieved through
other means.

M. Dotsey: Investing in Equities: Can it Help Social Security?

69

REFERENCES
Barro, Robert J. “Are Government Bonds Net Wealth?” Journal of Political
Economy, vol. 82 (November/December 1974), pp. 1095–1117.
Bohn, Henning. “Risk Sharing in a Stochastic Overlapping Generations
Economy.” Manuscript. June 1997a.
. “Social Security Reform and Financial Markets.” Manuscript.
June 1997b.
Christiano, Lawrence J., and Martin Eichenbaum. “Current Real-BusinessCycle Theories and Aggregate Labor-Market Fluctuations,” American
Economic Review, vol. 82 (June 1992), pp. 430–50.
Department of Health and Human Services. Report of the 1994–1996 Advisory
Council on Social Security. Washington: Government Printing Office,
January 1997.
Donlan, Thomas G. “Social Investment: Reform of Social Security Requires
Private Investment and More,” Barron’s Online (http://www.barrons.com),
January 31, 1997.
Greenspan, Alan. Remarks at the Abraham Lincoln Award Ceremony of the
Union League of Philadelphia. December 6, 1996.
Marcks, Ronald H. “Social Security’s Most Basic Infirmity,” Wall Street
Journal, January 16, 1997.
Mariger, Randall P. “Social Security Privatization: What It Can and Cannot
Accomplish,” Finance and Economics Discussion Series, No. 1997–32.
Washington: Board of Governors of the Federal Reserve System, Divisions
of Research & Statistics and Monetary Affairs, June 1997.
Miron, Jeffrey A., and David N. Weil. “The Genesis and Evolution of Social
Security,” NBER Working Paper 5949. March 1997.
Munnell, Alicia H. The Future of Social Security. Washington: Brookings
Institution, 1977.
Smetters, Kent. “Investing the Social Security Trust Fund into Equity: An
Options Pricing Approach.” Washington: Congressional Budget Office,
July 1997.
Stein, Herbert. “Social Security and the Single Investor,” Wall Street Journal,
February 5, 1997.
U.S. General Accounting Office. Report to the U.S. Senate Committee on
Finance and the House of Representatives Committee on Ways and
Means, Social Security Administration: Significant Challenges Await New
Commissioners. Washington: Government Printing Office, February 1997.

Fisher and Wicksell on
the Quantity Theory
Thomas M. Humphrey

T

he quantity theory of money, dating back at least to the mid-sixteenthcentury Spanish Scholastic writers of the Salamanca School, is one of
the oldest theories in economics. Modern students know it as the proposition stating that an exogenously given one-time change in the stock of money
has no lasting effect on real variables but leads ultimately to a proportionate
change in the money price of goods. More simply, it declares that, all else being
equal, money’s value or purchasing power varies inversely with its quantity.
There is nothing mysterious about the quantity theory. Classical and neoclassical economists never tired of stressing that it is but an application of the
ordinary theory of demand and supply to money. Demand-and-supply theory, of
course, predicts that a good’s equilibrium value, or market price, will fall as the
good becomes more abundant relative to the demand for it. In the same way,
the quantity theory predicts that an increase in the nominal supply of money
will, given the real demand for it, lower the value of each unit of money in
terms of the goods it commands. Since the inverse of the general price level
measures money’s value in terms of goods, general prices must rise.
In the late nineteenth and early twentieth centuries, two versions of the
theory competed. One, advanced by the American economist Irving Fisher
(1867–1947), treated the theory as a complete and self-contained explanation
of the price level. The other, propounded by the Swedish economist Knut Wicksell (1851–1926), saw it as part of a broader model in which the difference,
or spread, between market and natural rates of interest jointly determine bank
money and price level changes.

For helpful comments, thanks go to Mike Dotsey, Alice Felmlee, Bob Hetzel, Rowena
Johnson, Elaine Mandaleris, Ben McCallum, Ned Prescott, and Alex Wolman. The views
expressed are those of the author and not necessarily those of the Federal Reserve Bank of
Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 83/4 Fall 1997

71

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Federal Reserve Bank of Richmond Economic Quarterly

The contrasts between the two approaches could hardly have been more
pronounced. Fisher’s version was consistently quantity theoretic throughout
and indeed focused explicitly on the received classical propositions of neutrality, equiproportionality, money-to-price causality, and independence of money
supply and demand. By contrast, Wicksell’s version contained certain elements
seemingly at odds with the theory. These included (1) a real shock explanation
of monetary and price movements, (2) the complete absence of money (currency) in the hypothetical extreme case of a pure credit economy, and (3) the
identity between deposit supply and demand at all price levels in that same
pure credit case rendering prices indeterminate.
Despite these anomalies, Wicksell was able to derive from his analysis essentially the same conclusion Fisher reached. Both concluded that the monetary
authority bears the ultimate responsibility for price level stability, a responsibility it fulfills either by determining some nominal variable—such as dollar
price of gold, monetary base, bank reserves—under its control or by adjusting
its lending rate in response to price level deviations from target.
The story of how Fisher and Wicksell reached identical policy conclusions
from seemingly distinct models is instructive. It reveals that models appearing
to be substantially different may be only superficially so. In the case of Fisher
and Wicksell, it reveals that their models may not have been as dissimilar as
often thought. Indeed, the alleged non-quantity-theory elements in Wicksell’s
work prove, upon careful examination, to be entirely consistent with the theory.
In an effort to document these assertions and to establish Wicksell’s position
in the front rank of neoclassical quantity theorists with Fisher, the paragraphs
below identify the two men’s contributions to the theory and show how their
policy conclusions derived from it.

1.

FISHER’S VERSION OF THE QUANTITY THEORY

In his 1911 book The Purchasing Power of Money, Fisher gave the quantity
theory, as inherited from his classical and pre-classical predecessors, its definitive modern formulation. In so doing, he accomplished two tasks. First, he
expressed the theory rigorously in a form amenable to empirical measurement
and verification. Indeed, he himself fitted the theory with statistical data series,
many of them of his own construction, to demonstrate its predictive accuracy.
Second, he spelled out explicitly what was often merely implicit in the
work of John Locke, David Hume, Richard Cantillon, David Ricardo, John
Wheatley, and other early quantity theorists, namely the five interrelated propositions absolutely central to the theory. These referred to (1) equiproportionality
of money and prices, (2) money-to-price causality, (3) short-run nonneutrality
and long-run neutrality of money, (4) independence of money supply and demand, and (5) relative-price/absolute-price dichotomy attributing relative price

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

73

movements to real causes and absolute price movements to monetary causes
in a stationary fully employed economy.1
Fisher enunciated these propositions with the aid of the equation of exchange P = (MV + M V )/T, which he attributed to Simon Newcomb even
though Joseph Lang, Karl Rau, John Lubbock, and E. Levasseur had formulated it even earlier. Here P is the price level, M is the stock of hard or metallic
money consisting of gold coin and convertible bank notes, V is the turnover
velocity of circulation of that stock, M is the stock of bank money consisting
of demand deposits transferable by check, V is its turnover velocity, and T is
the physical volume of trade. Fisher’s assumption that metallic money divides
in fixed proportions between currency and bank reserves and that reserves are a
fixed fraction of deposits allowed him to treat checkbook money as a constant
multiple c of hard money. His assumption allows one to simplify his expression
to P = MV ∗ /T, where V ∗ = V + cV .
Of the equation’s components, Fisher ([1911] 1963, p. 155) assumed that,
in long-run equilibrium, the volume of trade is determined at its full-capacity
level by real forces including the quantity and quality of the labor force, the size
of the capital stock, and the level of technology. Save for transition adjustment
periods in which the variables interact, these real forces and so the level of
trade itself are independent of the other variables in the equation. Likewise,
institutions and habits determine aggregate velocity, whose magnitude is fixed
by the underlying velocity turnover rates of individual cashholders, each of
whom has adjusted his turnover to suit his convenience (Fisher [1911] 1963, p.
152). Like the volume of trade, velocity is independent of the other variables
in the equation of exchange. And with trade and velocity independent of each
other and of everything else in the equation, it follows that equilibrium changes
in the price level must be due to changes in the money stock.
Classical Propositions
All the fundamental classical quantity theory propositions follow from Fisher’s
demonstration. Regarding proportionality, he writes that “a change in the quantity of money must normally cause a proportional change in the price level”
([1911] 1963, p. 157). For, with trade and velocity independent of the money
stock and fixed at their long-run equilibrium levels, it follows that a doubling
of the money stock will double the price level.
Fisher realized, of course, that proportionality holds only for the ceteris
paribus thought experiment in which trade and velocity are provisionally held
fixed. In actual historical time, however, trade and velocity undergo secular
changes of their own independent of the money stock. In that case, proportionality refers to the partial effect of money on prices. To this partial effect must
1 For

a discussion of these classical propositions, see Blaug (1995) and Patinkin (1995).

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Federal Reserve Bank of Richmond Economic Quarterly

be added the parallel effects of coincidental changes in velocity and trade (see
Niehans [1990], p. 277). The sum of these separate effects shows the influence
of all on the price level. Thus if M, V ∗ , and T evolve secularly at the percentage
rates of change denoted by the lowercase letters m, v∗ , and t, respectively, then
the price level P evolves at the percentage rate p = m + v∗ − t. Fisher ([1911]
1963, pp. 246–47) himself expressed the matter precisely when he declared
that the history of the price level is a history of the race between increases in
the money stock and increases in the volume of trade.
Fisher was equally adamant on the neutrality of money other than during
transition adjustment periods. Regarding long-run neutrality, he says that “An
inflation of the currency cannot increase the product of . . . business” since the
latter “depends on natural resources and technical conditions, not on the quantity of money” ([1911] 1963, p. 155). In short, trade’s long-run independence
of money in the equation of exchange means that money cannot permanently
influence real activity.
Money can, however, influence real activity temporarily. Indeed, the classical proposition regarding the short-run nonneutrality of money posits that very
point. Fisher ([1911] 1963, pp. 58–72), in his theory of the cycle, attributes such
nonneutrality to delays in the revision of lenders’ inflation expectations and the
resulting sluggish adjustment of nominal interest rates. A monetary shock sets
prices rising. Rising prices generate inflation expectations among business borrowers whose perceptions of current and likely future price changes are superior
to those of lenders. These inflationary expectations engender corresponding expectations of higher business profits. Sluggish nominal loan rates, however, fail
to rise enough to offset these rising expectations. Consequently, real loan rates
fall. Spurred by the fall in real rates, business borrowers increase their real
expenditure on factor inputs. Employment and output rise. Eventually, nominal
loan rates catch up with and surpass business profit (and inflation) expectations.
Real rates rise thereby precipitating a downturn.2
As for the proposition of unidirectional money-to-price causality, Fisher
established it two ways. First, he denied that causation, under the gold standard then prevailing, could possibly run in the reverse direction from prices to
money ([1911] 1963, pp. 169–71). To demonstrate as much, he supposed prices
miraculously to double, the other variables in the exchange equation initially
remaining unchanged. Far from inducing an accommodating expansion in the
money stock, the price increase would, in an open trading economy, actually
prompt that stock to contract. The stock would contract as the price increase,
by rendering domestic goods expensive in relation to foreign ones, engendered
2 As

we will see, such nonneutralities are absent from Wicksell’s work. Adhering as he did
to a real theory of the cycle, he denied that business fluctuations stem from monetary shocks (see
Leijonhufvud [1997]). Such shocks in his view leave the economy always at full employment.
Consequently, he held that neutrality of money prevailed in the short run as well as the long.

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

75

a trade balance deficit and a resulting external drain of monetary gold. The upshot is that the price increase would not cause a supporting rise in the money
stock as reverse causation implies. Nor for that matter could the price increase
spawn validating changes in the other variables of the exchange equation. The
independence of those variables with respect to the price level rules out this
possibility. In short, the price level is “the one absolutely passive element in
the equation” (Fisher [1911] 1963, p. 172). Its movements are the result, not
the cause, of prior changes in the quantity of money per unit of trade.
Alternatively, Fisher demonstrates M–to–P causality by showing that no
variables in the exchange equation can intervene to absorb permanently the
impact of a change in M and thus prevent the force of that impact from being
transmitted to P. No compensating changes in trade will occur to blunt M’s
impact since the two variables are independent in long-run equilibrium. Nor
will M exhaust its effect in reducing velocity permanently. For cashholders
have already established velocity at its desired level, a level independent of M.
Instead, Fisher ([1911] 1963, pp. 153–54) argued that money will transmit
its full effect to prices through the following cash-balance adjustment mechanism. Let the money stock double from M to 2M, the price level initially
remaining unchanged. With prices and trade given, actual velocity V ∗ = PT/2M
falls to one-half the level cashholders desire it to be, or PT/M. In an effort to
restore actual velocity to its desired level, cashholders will increase their rate of
spending. The increased spending will, because trade is fixed at its full-capacity
level, put upward pressure on prices. Prices will continue to rise until actual
and desired velocities are the same (V ∗ = 2PT/2M = PT/M). At this point,
prices will have doubled equiproportionally with money.
The remaining classical propositions follow directly from Fisher’s analysis. Regarding the relative-price/absolute-price dichotomy, he denied that real
factors change the absolute price level in a stationary, fully-employed economy.
In particular, he insisted that price level changes cannot be caused by cost-push
forces emanating from trade-union militancy, business-firm monopoly power,
commodity shortages, and the like ([1911] 1963, pp. 179–80).3 Such forces,
he says, drive relative prices, not absolute ones. In other words, given the
money stock, velocity, and trade, real shock-induced changes in some relative
prices produce compensating changes in others, leaving the absolute price level
unchanged. Real shocks, if they are to affect absolute prices as well as relative
ones, must somehow also cause changes in M, V ∗ , or T. Fisher saw little reason
to expect them to do so. And even if they did, their effect would always be so

3 In his 1920 book Stabilizing the Dollar, Fisher listed 41 frequently cited nonmonetary
causes of inflation and noted that “while some of them are important factors in raising particular
prices, none of them . . . has been important in raising the general scale of prices” (p. 11). In his
view “no explanation of a general rise in prices is sufficient which merely explains one price in
terms of another price” (p. 14).

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small as to be swamped by exogenous changes in money.
Finally, with respect to independence of money supply and demand, Fisher
attempts to establish it by arguing that the money stock owes its determination
to “influences outside the equation of exchange,” that is, to influences other
than the trade-to-velocity ratio T/V ∗ (= M/P) which constitutes the public’s real
demand for money ([1911] 1963, p. 90). For a closed gold-standard economy,
these outside influences include the rate of gold production as influenced by
new gold discoveries and technological innovations, both of which temporarily
lower the metal’s production cost below its market value and so give a profit
boost to mining. For open economies operating on the gold standard, additional external influences include foreign price levels. These, when high or low
relative to the domestic price level, induce specie flows through the balance
of payments. Such specie flows in turn raise or lower the domestic money
stock and through it the domestic price level. From the viewpoint of the open
domestic economy, money-stock changes are predetermined exogenously by
the height of the foreign price level. These money-stock changes then endogenously affect domestic prices. As Fisher put it, “the price level outside of New
York City . . . affects the price level in New York City only via changes in
the money in New York City. Within New York City it is the money which
influences the price level, and not the price level which influences the money”
([1911] 1963, p. 172).
Today, of course, we would say that an open economy’s money stock is
endogenously determined by the requirement that domestic price levels move
in step with foreign ones to maintain equilibrium in the balance of payments
(see Friedman and Schwartz [1991], p. 42). But Fisher, by contrast, argued
that the open economy’s money stock is determined exogenously by the given
state of the balance of payments resulting from the given foreign (relative to
domestic) price level.
We will see in Section 4 below, however, that he did correctly apply the
exogeneity, or independence, proposition to so-called compensated dollar and
inconvertible paper standard regimes. He recognized that, in such regimes, the
policy authority governs money exogenously either through control of the gold
weight of the dollar or through the high-powered monetary base consisting of
the authority’s own liabilities. Through these instruments, the authority renders
the money stock independent of money demand.

2.

WICKSELL’S INTERPRETATION OF THE
QUANTITY THEORY

Knut Wicksell’s perception of the classical quantity theory, as expounded in
his 1898 Interest and Prices and Volume 2 of his 1906 Lectures on Political
Economy, was less comprehensive than Fisher’s. Wicksell understood the

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

77

theory to mean only the proposition that prices are proportional to hard money,
or metallic currency, in long-run equilibrium. This proportional relationship
was, he believed, established through the operation of a real balance effect. In
his view, cashholders had a well-developed demand for a constant stock of real
cash balances. This demand together with the given nominal money supply
ensured price level determinacy.
Thus a random shock to the price level that temporarily raised it above its
equilibrium level would, by making actual real balances smaller than desired,
induce cashholders either to cut their expenditure on or to increase their sales
of goods in an effort to restore the desired level of real balances. The resulting
excess supply of goods on the market would put downward pressure on prices
until they reestablished their initial proportional relationship to the unchanged
money stock, thus restoring real balances to equilibrium. In Wicksell’s own
words:
suppose that for some reason or other commodity prices rise while the
stock of money remains unchanged . . . . The cash balances will gradually
appear to be too small in relation to the new level of prices . . . . I therefore
seek to enlarge my balance. This can only be done . . . through a reduction
in my demand for goods and services, or through an increase in the supply
of my own commodity . . . or through both together. The same is true of all
other owners and consumers of commodities. But in fact nobody will succeed
in realizing the object at which each is aiming—to increase his cash balance;
for the sum of individual cash balances is limited by the amount of the available stock of money, or rather is identical with it. On the other hand, the
universal reduction in demand and increase in supply of commodities will
necessarily bring about a continuous fall in all prices. This can only cease
when prices have fallen to the level at which the cash balances are regarded
as adequate, [that is, when] prices . . . have fallen to their original level.
([1898] 1965, pp. 39– 40)

This same stability condition, Wicksell noted, ensured that a decrease in
the money stock would, by rendering real balances smaller than desired, induce
a proportional fall in spending, and therefore prices, to restore real balances
to their desired level. For Wicksell, then, the classical quantity theory implied
money stock and price level proportionality achieved through real balance effects.
Pure Cash Economy
Wicksell found the theory to be perfectly valid for hypothetical pure cash
economies in which no banks exist to issue checkable deposits, all transactions
being mediated entirely by gold currency. In such economies, a demand for
a fixed quantity of real gold balances ensures that prices move proportionally
to money in long-run equilibrium. Thus newly discovered gold in a closed
economy will, at initially unchanged prices, make real balances larger than

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desired. Cashholders will spend the excess, thereby putting upward pressure on
prices which rise proportionally to the increased monetary gold stock.
In an open trading economy, cashholders’ adjustments will induce equilibrating real balance effects abroad as well as at home. For let all goods
worldwide be tradeables—exportables and importables—whose prices are, by
the law of one price, kept everywhere the same by the operation of commodity
arbitrage. Then the increased home expenditure on these goods, induced by the
gold discovery and resulting excess cash balance, will raise prices abroad thus
eroding real balances there. In an effort to rebuild their balances, foreigners
cut their spending on and increase their offer of tradeables. The resulting trade
surplus is financed by a specie inflow that restores foreign real balances to their
desired level. Real balance effects operate to establish proportionality between
money and prices throughout the world (see Myhrman [1991], pp. 269–70).
Mixed Cash-Credit Economy
To Wicksell, however, the classical quantity theory, applying as it did to pure
cash economies, seemed much too narrow and antiquated. It omitted banks
and the deposit liabilities they issue by way of loan. It therefore could account
neither for the influence of checking deposits on the price level, nor for how
both variables move from one equilibrium level to another. Nor for that matter
could it account for the forces inducing their movement. To overcome these deficiencies, Wicksell sought to supplement the quantity theory with a description
of the mechanism through which monetary equilibrium is disturbed and subsequently restored in mixed cash-credit, or currency-deposit, economies. Thus
was born his celebrated analysis of the cumulative process (see Jonung [1979],
pp. 166–67, Laidler [1991], pp. 135–39, Leijonhufvud [1981], pp. 151–60, and
Patinkin [1965], pp. 587–97).
That analysis attributes deposit and price level movements to discrepancies
between two interest rates. One, the market or money rate, is the rate banks
charge on loans and pay on deposits. The other is the natural or equilibrium
rate that equates desired saving with intended investment at full employment
and that also corresponds to the expected marginal yield or internal rate of
return on newly created units of physical capital. Or, equivalently, it is the rate
that equates aggregate demand for real output with the available supply.
When the loan rate lies below the natural rate such that the cost of capital
is less than capital’s expected rate of return, planned investment will exceed
planned saving. Entrepreneur investors seeking to finance new capital projects
will wish to borrow more from banks than savers deposit there. Since banks
accommodate these extra loan demands by creating checking deposits, a deposit
expansion occurs. This expansion, by underwriting the excess desired aggregate
demand implicit in the investment-saving gap, transforms it into excess effective
aggregate demand that spills over into the commodity market to put upward

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

79

pressure on prices. In so doing, the deposit expansion produces a persistent and
cumulative rise in prices for as long as the interest differential lasts.
Now Wicksell argued that, in mixed cash-credit economies using currency
and bank deposits convertible into currency, the rate differential would quickly
vanish. The public’s demand for real cash balances ensures as much. For let
cashholders transact a certain portion of their real payments in currency. Then
a rise in prices stemming from the rate differential necessitates additional currency to satisfy that real transaction demand. The ensuing public conversion of
deposits into currency and the resulting drain on bank reserves induces banks
to raise their loan rates until they (loan rates) equal the natural rate. This last
step stems the reserve drain and also brings the price rise to a halt. If banks,
because they initially possessed excess reserves, were willing to let reserves
run down a bit, then prices would stabilize at the new, higher level. But if banks
possessed no excess reserves and so had to restore reserves to their initial level
following the price rise, then they (banks) would continue to raise the market
rate above the natural rate until prices returned to their pre-existing level. Either
way, a quantity theory element in the form of the public’s demand for currency
works to anchor the price level in the mixed cash-credit economy. Nominal
determinacy prevails in that economy as it did in the pure cash economy.
Cumulative Process Model
Expressed symbolically and condensed into a simple algebraic model, Wicksell’s cumulative process can be put through its paces to reveal the exact
workings of its constituent quantity theory elements. Since these elements have
provoked so much controversy in the Wicksell literature, it is important to specify precisely how Wicksell used them.4 Assume with Wicksell that all saving
is deposited with banks, that all investment is bank-financed, that banks lend
solely to finance investment, and that full employment prevails such that shifts
in aggregate demand affect prices but not real output. Then his model reduces
to the following equations linking the variables investment I, saving S (both
planned, or ex ante, magnitudes), loan rate i, natural rate r, loan demand LD ,
loan supply LS , excess aggregate demand E, change in the stock of checkable
deposits dD/dt, price level change dP/dt, and market-rate change di/dt.
The first equation says that planned investment exceeds saving when the
loan rate of interest falls below its natural equilibrium level (the level that
equilibrates saving and investment):
I − S = a(r − i),

4 For

(1)

similar attempts to model algebraically the cumulative process see Brems (1986), Eagly
(1974), Frisch (1952), Laidler (1975), Niehans (1990), and Uhr (1960).

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where the coefficient a relates the investment-saving gap to the interest differential that creates it.
The second equation states that the excess of investment over saving equals
the additional checkable deposits newly created to finance it,
dD/dt = I − S.

(2)

In other words, since banks create new checkable deposits by way of loan,
deposit expansion occurs when banks lend to investors more than they (banks)
receive from savers. Thus equation (2) admits of the following derivation.
Denote the investment demand for loans as LD = I(i), where I(i) is the schedule relating desired investment spending to the loan rate of interest. Similarly,
denote loan supply as the sum of saving plus new deposits created by banks
in accommodating loan demands. In short, LS = S(i) + dD/dt. Equating loan
demand and supply and solving for the resulting gap between investment and
saving yields equation (2).
The third equation says that the new deposits, being spent immediately, spill
over into the commodity market to underwrite the excess aggregate demand
for goods E implied by the gap between investment and saving:
dD/dt = E.

(3)

The fourth equation says that this excess aggregate demand bids up prices,
which rise in proportion to the excess demand:
dP/dt = bE,

(4)

where the coefficient b is the factor of proportionality between price level
changes and excess demand.
Substituting equations (1), (2), and (3) into (4), and (1) into (2), one obtains
dP/dt = ab(r − i)

(5)

dD/dt = a(r − i),

(6)

and

which together state that price inflation and the deposit growth that underlies
it stem from the discrepancy between the natural and market rates of interest.
Finally, since bankers must at some point raise their loan rates to protect their gold reserves from inflation-induced cash drains into hand-to-hand
circulation, one last equation,
di/dt = gdP/dt,

(7)

closes the model. This equation says that bankers, having worked off excess
reserves, now raise their rates in proportion to the rate of price change (g
being the factor of proportionality). The equation ensures that the loan rate

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

81

eventually converges to its natural equilibrium level, as can be seen by substituting equation (5) into the above formula to obtain
di/dt = gab(r − i).

(8)

Solving this equation for the time path of the loan rate i yields
i(t) = (i0 − r)e−gabt + r,

(9)

where t is time, e is the base of the natural logarithm system, i0 is the initial
disequilibrium level of the loan rate, and r is the given natural rate. With the
passage of time, the first term on the right-hand side vanishes and the loan rate
converges to the natural rate. At this point, monetary equilibrium is restored.
Saving equals investment, excess demand disappears, deposit expansion ceases,
and prices stabilize at their new, higher level.5

3.

WAS WICKSELL A QUANTITY THEORIST?

At first glance the preceding model, especially equation (5), appears to attribute
price level changes directly to the interest rate differential rather than to monetary causes. This point is sometimes cited as evidence that Wicksell was not
a quantity theorist (see Greidanus [1932], p. 83, and Adarkar [1935], p. 27, as
cited in Marget [1938], pp. 183, 187). But it is patently obvious that the model
is perfectly consistent with the quantity theory when monetary shocks generate
the rate differential. Under these conditions the differential and the resulting
price movements clearly have a monetary origin.
Indeed, Wicksell himself described how a monetary impulse would trigger
the cumulative process consistent with the classical quantity theory. Assuming
the monetary impulse took the form of a gold inflow from abroad, he noted
that the new gold ordinarily would be deposited in banks. So deposited, the
gold would augment bank reserves beyond the level banks desired to hold. The
resulting pressure of excess reserves would, he argued, induce banks to lower
their loan rate below the natural rate, thus precipitating the cumulative rise in
the volume of bank money (deposits) and prices. Under these conditions, one
could confidently attribute changes in both the stock of deposits and the price
level to preceding changes in the monetary gold stock.
Having recognized potential monetary origins of the cumulative process as
a theoretical possibility, however, Wicksell rejected this possibility on empirical
grounds. His study of nineteenth-century British prices and interest rates had
5 Of

course if there were no excess reserves to begin with, prices would have to stabilize at
their pre-existing level. Bankers, having no excess reserves to lose, would adjust their loan rates
either to forestall all reserve drains or to reverse (annul) drains that had already occurred. Either
way, prices would stabilize at their initial level.

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convinced him that the cumulative process typically originated not in monetary shocks to the loan rate but rather in real shocks to the natural rate. His
consequent stress on real shocks in the form of wars, technological progress,
innovations, and the like has spurred some scholars to ask: if real shocks predominate over monetary shocks in generating the rate differential, doesn’t it
follow that the resulting price level movements are real rather than monetary
phenomena, contrary to the quantity theory?
In answering this question in the affirmative, these scholars imply that
Wicksell may have done more to subvert the theory than to support it. Thus
Lars Jonung states:
Wicksell’s approach emphasizes nonmonetary developments, that is “real” factors, as the principal sources of price changes. Although the monetary sector
has a central position in the transmission mechanism from “real” developments to changes in prices, there is a tendency to ignore monetary factors in
a theory that assumes that movements in the real rate are the driving force
behind deflations and inflations. It is thus easy to end up with a theory of
the price level that relates the behavior of prices directly to variables that
influence the real rate, such as changes in the flow of innovations and technological improvements. Here Wicksell’s theory has much in common with the
Schumpeterian “longwave explanation,” which associates price level changes
with the introduction of new production techniques, which implies that nonmonetary factors are the causes behind long-run changes in prices. (Jonung
[1979], p. 179; see also Cagan [1965], p. 253, and Laidler [1997], p. 5)

What such interpretations overlook, however, is that Wicksell himself always saw his cumulative process model as embodying the quantity theory and
being entirely consistent with it. His model was to him nothing less than a
full-scale extension of the theory to account for the influence of bank deposits
on the price level. In particular, his equations (3) and (4) upon substitution
reduce to dP/dt = b(dD/dt). In so doing, they reveal that a price level change
could never occur without the accompanying change in the supply of deposits
to support it.
In short, real shocks and the resulting rate differential alone could never
sustain price level changes. Instead, something else is required to translate
shocks into commodity price inflation. Something, in other words, must finance
the excess demand for goods that keeps prices rising. That something is deposit
expansion. Without it, excess demand and price increases could never occur
and the cumulative process would be abortive. The upshot is that Wicksell
thought the key factor underlying and permitting price movements was deposit
expansion, not real shocks and rate differentials.
Of the few commentators who underscore this point, none are more emphatic than Charles Rist and Arthur Marget. Rist ([1938] 1966, p. 300) likens
Wicksell to Voltaire’s sorcerer, whose incantations could kill a herd of cattle if
accompanied by a lethal dose of arsenic. In Wicksell’s case, the arsenic—the

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

83

true cause—was an elastic supply of deposits. The incantations took the form
of rate differentials. Similarly, Marget (1938, p. 183) cites “abundant passages
in Wicksell’s writings which show that he did think of the ‘plentiful creation of
money’ (that is, bank-credit, or the M of our equation) as being the crucial link
in the [cumulative] process.” In short, changes in the stock of deposits were to
Wicksell the one absolutely necessary and sufficient condition for price level
movements.
Critique of Tooke’s Interest Cost-Push Theory
Nowhere did Wicksell express this view more forcefully than in his famous
critique of Thomas Tooke (Wicksell [1898] 1965, pp. 99–100, and [1906] 1978,
pp. 180–87). Tooke, author of the celebrated History of Prices and leader of the
English Banking School, had disputed, indeed scorned, the quantity theoretic
doctrines of the rival Currency School. In opposition to those doctrines, Tooke,
in his 1844 volume An Inquiry into the Currency Principle (Tooke [1844]
1959), argued that price level changes stem from cost-push forces originating
in the real economy rather than from disturbances originating in the monetary
sector. In particular, he argued that interest rate increases, by raising the cost of
doing business, would raise general prices as the increased costs were passed
on to buyers. The resulting price inflation, Tooke implied, would occur even
in the face of a constant money stock.
Wicksell, however, maintained that such price level increases could never
occur unless underwritten by expansion of that stock. According to him, it is
deposit growth stemming from a two-rate differential, and not interest cost-push
per se, that constitutes the necessary condition for general prices to rise. Without
the accommodating monetary growth, the interest cost-push forces would, he
insisted, exhaust themselves in changing relative, not absolute, prices ([1906]
1978, p. 180). The prices of interest-intensive goods would rise relative to
the prices of non-interest-intensive ones. But the general price level would
remain unchanged. For if the money stock were constant and banks possessed
no excess reserves, any rise in the natural rate would force bankers to engineer
a matching rise in the loan rate to protect their reserves from cash drains into
hand-to-hand circulation. The two rates would remain equal and prices would
stay constant. Only if banks initially possessed excess reserves could a positive
shock to the natural rate permanently raise the equilibrium price level. And
even here the price increase is attributable to the monetary factor—the excess
reserve—that permits it to occur. All of which is consistent with the quantity
theory and confirms Wicksell’s adherence to it.
Pure Credit Economy
To summarize, Wicksell had shown that the quantity theory applies perfectly to
the pure cash economy. He had then shown that, when augmented to account

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for the influence of deposit-financed demand on prices, it applies to mixed cashcredit economies as well. In both cases, he had established that a real currency
demand together with an independent nominal currency supply are sufficient
to pin down the price level. Seeking to extend the theory to its logical limit,
he next applied it to the hypothetical extreme case of a pure credit economy
in which no currency exists and all transactions are settled by transfers of
deposits on the books of banks. Here he showed that the theory fails to hold
in the absence of central bank intervention.
According to him, it fails to hold in the first place because the pure credit
economy employs no currency to which the theory can apply. With currency
absent, no demand for and supply of it exists to determine the price level. Nor
can deposit demand and supply be relied upon to determine the price level.
For, in the pure credit economy, the two deposit variables are identical to each
other at all price levels. Being identical, they cannot exhibit demand-supply
independence as price determinacy requires. Wicksell explains:
in our ideal [pure credit] state every payment . . . is accomplished by means
of cheques or giro facilities. It is then no longer possible to refer to the supply
of money as an independent magnitude, differing from the demand for money.
No matter what amount of money may be demanded from the banks, that is
the amount which they are in a position to lend . . . . The banks have merely
to enter a figure in the borrower’s account to represent a credit granted or a
deposit created. When a cheque is then drawn and subsequently presented to
the banks, they credit the amount of the owner of the cheque with a deposit
of the appropriate amount (or reduce his debit by that amount). The “supply
of money” is thus furnished by the demand itself. . . . It follows that . . .
the banks can raise the general level of prices to any desired height. ([1898]
1965, pp. 110–11)

With deposit supply identical to demand at all prices, there is no unique
equilibrium price level or deposit quantity. Rather, there is an infinity of
price-quantity equilibria. The price level, in other words, is indeterminate.
Wicksell’s cumulative process model applied to the pure credit economy cannot
determine it.
Instead, his credit economy model specifies the rate of rise of the price level
dP/dt (see Leijonhufvud [1997], p. 8). Starting from some historically given
position, this rise can continue indefinitely as long as a natural-rate/market-rate
disparity persists, that is, as long as banks are under no reserve pressure to raise
their rates. Since no currency demand exists to drain reserves in the pure credit
economy, banks need hold no reserves other than central bank credit. And even
this form of reserve is unnecessary in a banking system—Wicksell’s “ideal”
system—composed of a single central bank with branches in every town and
hamlet (see Uhr [1960], p. 222). As a central bank, the ideal bank need hold no
credit reserves with itself. Moreover, as a monopoly institution, the ideal bank
can lose no reserves through the clearing house to other banks (of which there

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

85

are none) and so need hold no reserves whatsoever. The result is a system
totally devoid of reserve constraints to anchor nominal variables. In such a
system, deposit supply possesses potentially unlimited elasticity. Consequently
prices, in addition to being indeterminate, theoretically can rise (or fall) forever.
Wicksell insisted, however, that it was up to the central bank to impose
nominal determinacy in this case. The central bank could do so through control
of the market rate. By adjusting the rate when prices threaten to rise or fall,
the bank could close and reverse the rate differential. In so doing, the bank
could maintain prices and the supporting volume of deposits at fixed, determinate levels. Here the central bank’s obligation to impose price determinacy
replaces the missing reserve constraint to force equilibrating rate adjustment.
Nominal determinacy is preserved, consistent with the quantity theory. In this
way, Wicksell ensures that at least one element of the theory survives even in
the pure credit case.

4.

POLICY REFORM PROPOSALS

The preceding remarks contend that Wicksell was, commentators’ views to the
contrary notwithstanding, every bit as much a quantity theorist as Fisher. Evidence reveals that he, like Fisher, understood and indeed enriched the theory’s
postulates.
But there is a simpler way to prove he and Fisher saw things much the same
as far as the quantity theory was concerned. That way is to compare the policy
views of the two. One can employ a simple litmus test: a person essentially
is a quantity theorist if he believes the monetary authority can stabilize the
price level through control, direct or indirect, of the stock of money or nominal
purchasing power. Both Fisher and Wicksell pass this test with flying colors.
Both advocated price level stability, albeit for different reasons. Fisher
thought such stability would smooth, if not eliminate completely, the business
cycle. In so doing, it would alleviate the overuse (stress, strain, exhaustion) of
labor and capital resources endured in business booms and the loss of output
and employment suffered during depressions. By contrast, Wicksell thought
price stability would stop the arbitrary and unjust redistribution of income and
wealth that unanticipated inflation and deflation produce. In this way, it would
prevent the loss in aggregate social welfare that occurs, because of diminishing
marginal utility of income, when unanticipated price movements transfer real
income from losers to gainers.
Both also advocated that price stability be achieved through feedback policy
rules. In this connection, both devoted their best efforts to devising effective
rules. Each writer proposed rules directing the monetary authority to adjust its
policy instrument in corrective response to price level deviations from target.
Such instrument adjustment would in turn produce a corresponding adjustment

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in the money stock. This latter adjustment would act to stabilize prices. The
money stock was of key importance here. Only by operating through it could
instrument adjustment stabilize prices.
In Fisher’s famous compensated dollar plan, the policy instrument is the
gold content of the dollar, or official dollar price of gold (see Patinkin [1993]).
The monetary authority adjusts this price in response to price level deviations
from target. Since the price level, or dollar price of goods, is by definition the
dollar price of gold times the world gold price of goods, the authority must
offset movements in the gold price of goods with compensating adjustments in
the dollar price of gold so as to keep the general price level constant.
Fisher made it clear, however, that his compensated dollar plan would operate on the price level through the money stock. It would do so by changing
both the physical amount and the nominal valuation of the nation’s stock of
monetary gold. Thus when world gold inflation was raising the dollar price
of goods, the American policy authority would lower the official buying and
selling price of gold. Industry and the arts, finding gold less expensive, would
therefore demand more of it. Consequently, part of the nation’s gold stock
would be diverted from monetary to nonmonetary uses (see Lawrence [1928],
p. 432). The resulting shrinkage in the stock of monetary gold would lower the
price level. In addition, the reduced official price of gold, by producing a corresponding reduction in the nominal value of physical gold reserves, would lessen
the nominal volume of paper money issuable against such backing (see Patinkin
[1993], p. 16). This reduced nominal issue too would put downward pressure
on prices. In sum, whether through physical reduction or nominal revaluation,
the monetary gold stock would shrink and so too would the quantity of money
and level of prices it could support.
Later on, in the mid-1930s, Fisher (1935, p. 97) proposed another policy
rule. It had the central bank adjusting, via open market operations, the monetary
base in response to price deviations from target. In this case, the price level
was the goal variable, the monetary base was the instrument, and the money
stock was the intermediate variable. To minimize slippage between the base
instrument and the money stock, Fisher advocated a system of 100 percent
required reserves behind deposit money.
Although Wicksell’s preferred policy instrument differed from Fisher’s, his
activist feedback rule followed exactly the same pattern as Fisher’s. The authority would adjust its policy instrument, namely its lending rate, in response
to price deviations from target. In Wicksell’s own words (1919, p. 183, cited
in Jonung [1979], p. 168), “the Riksbank’s tool to keep the price level . . .
constant is to be found exclusively in its interest rate policy, such that the
Riksbank has to increase its rates as soon as the price level shows a tendency
to rise and lower them, as soon as it shows a tendency to fall.” Such rate adjustments would in turn produce corresponding corrective movements in the money

T. M. Humphrey: Fisher and Wicksell on the Quantity Theory

87

stock. These latter movements then would stabilize prices.6 Together, these
propositions constitute what Howard S. Ellis, in his classic German Monetary
Theory: 1905–1933, called Wicksell’s “central theorem,” namely the theorem
“that bank rate controls the price-level through its effect on the amount of
available purchasing power” (Ellis 1934, p. 304).
Thus if prices were rising, the central bank would raise the bank rate.
The rise in the bank rate would close the gap between it and the natural rate.
The closing of the gap would eliminate the differential between the investment demand for and saving supply of loanable funds. The elimination of
that differential would arrest growth in the stock of deposits and bring price
rises to a halt. Further raising of the bank rate would cause deflationary monetary contraction, thereby reversing the preceding inflationary price movement
and restoring prices to target. Here is a classic quantity theoretic prescription
for achieving price stability through monetary means. It is proof positive that
Wicksell, like Fisher, was a bona fide quantity theorist.

5. CONCLUSION
What then remains of the alleged difference between Fisher’s and Wicksell’s
interpretation of the quantity theory? Not much, in this observer’s opinion. Any
existing difference seems superficial rather than substantive, more semantic than
real. And it virtually vanishes once their policy reform proposals are taken into
account.
Commentators typically claim that interest rates are the key to Wicksell’s
analysis, whereas for Fisher the money stock is pivotal. They likewise claim
that real shocks initiate the inflationary process in Wicksell’s model, whereas
monetary shocks do so in Fisher’s. True enough. But these distinctions largely
lose force when one realizes that both men saw changes in the stock of monetary purchasing power consisting of bank deposits and currency as the one
absolutely indispensable and potentially controllable factor responsible for price
level changes. Moreover, both regarded this stock as the crucial intermediate
variable connecting policy instruments to price targets. Finally, both concluded
that the monetary authority bears the ultimate responsibility for monetary and
price level stability, a responsibility it discharges by giving some nominal variable under its control a stable, determinate value. In so doing, both enunciated
the principle of nominal determinacy, the sine qua non of the quantity theory.
These similarities would seem to outweigh any differences.
One reads Fisher and Wicksell today not so much to note the contrasts in
their analytical models as to appreciate the brilliant, prescient, and imaginative
6 Uhr (1991, p. 94) notes that Wicksell believed that the application of his rule would
prevent the price level from varying more than three percentage points above or below its target
or base-year level.

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ways they applied the quantity theory. In arguing for price stability achievable
through monetary means, both were adherents of monetary policy in the classical quantity theory tradition. Their two treatments are complementary rather
than competitive.

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Cagan, Philip. Determinants and Effects of Changes in the Stock of Money,
1875–1960. New York: Columbia University Press, 1965.
Eagly, Robert. The Structure of Classical Economic Theory. New York: Oxford
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Ellis, Howard S. German Monetary Theory: 1905–1933. Cambridge, Mass.:
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Greidanus, Tjardus. The Value of Money. London: P. S. King & Son, 1932.
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. The Golden Age of the Quantity Theory: The Development
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