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Staggered Price Setting
and the Zero Bound on
Nominal Interest Rates
Alexander L. Wolman

T

he nominal interest rate cannot be less than zero: no one would choose
to hold assets bearing a guaranteed negative nominal return when they
could instead hold money, which bears a guaranteed zero nominal return. Does the zero bound have normative implications for monetary policy?
The nominal interest rate tends to be low when expected inflation is low, so the
lower expected inflation is, the more likely it is that zero nominal interest rates
would be encountered. Some have argued that the zero bound’s proximity at
low inflation constitutes an argument against policy that results in low inflation
or deflation.1 Here we compare moderately deflationary and moderately inflationary regimes using a macroeconomic model to evaluate whether the zero
bound introduces distortions that make low inflation undesirable.
The model and the method distinguish our analysis from other recent research on the same topic.2 The model has optimizing behavior by individuals
and firms, with the qualification that firms’ price setting is staggered. Other
analyses of the zero bound have also used sticky-price models; the zero bound
is more likely to be important if nominal disturbances have real effects, as
they do with sticky prices. Individuals in the model choose to hold money
because it decreases the time they must spend shopping. Other analyses have
not modeled money demand. The method we employ involves solving the entire
model nonlinearly, which means directly imposing the zero bound on nominal
Alexander.Wolman@rich.frb.org. I thank Mike Dotsey, Marvin Goodfriend, Bob Hetzel,
Andreas Hornstein, Tom Humphrey, Bob King, Jeff Lacker, Bennett McCallum, Yash Mehra,
Athanasios Orphanides, David Small, and Alex Tabarrok for helpful discussions and comments. The views expressed herein do not represent the views of the aforementioned individuals, the Federal Reserve Bank of Richmond, or the Federal Reserve System.
1 Notable

examples are Vickrey (1954), Okun (1981), and Summers (1991).
examples include Fuhrer and Madigan (1997), Rotemberg and Woodford (1997),
and Orphanides and Wieland (1998). Section 1 contains a discussion of these articles.
2 Notable

Federal Reserve Bank of Richmond Economic Quarterly Volume 84/4 Fall 1998

1

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Federal Reserve Bank of Richmond Economic Quarterly

interest rates. We then compare the two inflation regimes in several ways, one
of which involves using an explicit welfare metric, the representative agent’s
expected utility.
In the model, a deflationary regime where nominal interest rates are occasionally zero generates higher welfare than a moderate inflation regime where
nominal interest rates are always positive. This striking result—which conflicts
with the spirit if not the letter of previous work—can be attributed to two factors mentioned above. First, the fact that money demand is explicitly modeled
means that there is a distortion associated with positive nominal interest rates:
individuals waste resources economizing on real money balances. Second, while
the two-period staggered price-setting requirement makes prices sticky, it does
not make inflation sticky. When inflation is sticky, as in the models used by
Fuhrer and Madigan (1997) and Orphanides and Wieland (1998), for example,
the zero bound on nominal interest rates effectively means that real interest
rates are constrained in low-inflation regimes. In contrast, in the sticky-price
model used here, real interest rates are not constrained at low inflation. The
monetary authority can create temporary expected inflation when nominal rates
are zero, thereby pushing real rates down, as described by Mishkin (1996).

1.

BACKGROUND AND RELATED WORK

Nominal interest rates are interest rates on marketable securities or loans denominated in an economy’s unit of account. In contrast, real interest rates
apply to assets denominated in a market basket of goods and services. Irving
Fisher, who used the terms “money interest” and “real interest,” is traditionally
credited with being the first to distinguish between nominal and real interest
rates. Fisher himself acknowledged, however, that he had many predecessors
who understood the distinction between nominal and real interest rates to some
degree.3
In Fisher’s original analysis, the relationship between the nominal (money)
interest rate and the real interest rate is but a special case of the relationships between interest rates denominated in any two standards of value. The celebrated
Fisher equation first appears in “Appreciation and Interest,” in an example
where the two standards are gold and wheat. But, when Fisher introduces that
analysis, he poses the general question, “If a debt is contracted in either of two
standards and one of them is expected to change with reference to the other,
will the rate of interest be the same in both? Most certainly not” (Fisher 1896,
p. 6). The Fisher equation follows three pages later: 1 + j = (1 + a)(1 + i),
3 Fisher’s important works on this subject are “Appreciation and Interest,” The Rate of Interest, and The Theory of Interest. See Humphrey ([1983] 1986), and Laidler (1991) for a discussion
of Fisher’s predecessors.

A. L. Wolman: Zero Bound on Nominal Interest Rates

3

where j and i are the rates of interest in wheat and gold, respectively, and a
is the (certain) expected rate of appreciation of gold in terms of wheat. This
generality on Fisher’s part is important, because it provides him with a principle
for understanding why the money interest rate is bounded by zero. Fisher states
that the interest rate cannot be negative in any standard that can be hoarded
without loss. The argument is straightforward: individuals would choose to
hoard the standard itself rather than hold securities or loans denominated in
that standard and yielding negative interest. For perishable standards, however,
the situation is different: “One can imagine a loan based on strawberries or
peaches contracted in summer and payable in winter with negative interest”
(Fisher 1896, p. 32). Since fiat money is storable at near zero cost, it follows
that the nominal interest rate in a modern, fiat-money economy is approximately
bounded by zero.
The zero bound is clearly a constraint on monetary policy, but is it an
important constraint? In order to answer this question, one needs a macroeconomic model and a criterion for measuring importance. To understand the
contribution made by this article, one should first know something about the
models and criteria used in recent analyses by Fuhrer and Madigan (1997),
Rotemberg and Woodford (1997), and Orphanides and Wieland (1998).4
Fuhrer and Madigan (1997) and Orphanides and Wieland (1998) use similar models, so we will consider them together. As with our analysis below,
they assess the zero bound’s importance by comparing their models’ performance at a moderate inflation target to that at an inflation target low enough
to make the nominal interest rate occasionally zero. Fuhrer and Madigan use a
small model that contains (i) a backward-looking IS curve, (ii) an overlapping
price-contracting specification, and (iii) a monetary policy reaction function.5
Orphanides and Wieland’s model shares the same contracting specification but
disaggregates the IS curve into separate spending equations for consumption,
fixed investment, inventory investment, net exports, and government spending. Neither model includes money. Monetary policy operates by changing the
short-term nominal interest rate. Long-term real interest rates enter the spending equations, but because the contracting specification makes inflation sticky,
persistent changes in the short-term nominal rate generate changes in the longterm real rate. Thus, monetary policy can affect real spending and hence output.
In both models, the equations representing private sector behavior are posited
rather than derived from explicit optimization problems.
Fuhrer and Madigan evaluate the zero bound’s importance by comparing
their model’s responses to IS curve shocks at inflation targets of zero and 4
4 The “liquidity trap” literature associated with Keynes ([1936] 1964) and Patinkin (1965)
concerned the possibility of a positive lower bound on nominal interest rates. Relating that literature to recent work would be an article by itself.
5 It is the same model used in Fuhrer and Moore (1995).

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Federal Reserve Bank of Richmond Economic Quarterly

percent. In contrast, Orphanides and Wieland simulate their model using estimated shock processes and compare the variance of output at different inflation
targets. The general conclusion of these papers is that at a zero inflation target,
monetary policy is significantly constrained by the zero bound, in the sense
that the zero bound is encountered regularly, and output is consequently more
variable than at a moderate inflation target. The easiest explanation for this
result comes from the first example in Fuhrer and Madigan, a permanent shock
to the IS curve. The monetary authority responds to this shock by lowering
short-term nominal interest rates. When the inflation target is zero, the monetary authority cannot lower the nominal rate by as much as it would choose if
the inflation target were 4 percent. With sticky inflation, the decline in the real
interest rate is also smaller, and therefore—because of the interest rate effect
on spending—there is a larger fall in output at the zero inflation target. This
fall in output is presumed to be bad, although that presumption is not implied
by the model.
The principal virtue of the analysis conducted by Fuhrer and Madigan
(1997) and Orphanides and Wieland (1998) is that it is performed using models that fit a particular sample of data quite well. However, their low inflation
experiments are conducted in an economic environment quite different from
the data sample. Therefore, the fact that the models’ equations are not derived
from explicit objective functions makes it doubtful that those equations would
be stable in the face of the contemplated policy experiments. Although the
model we use has not been shown to fit recent data well, it is valuable because
it is set up with explicit objective functions for individuals and firms. This
means that the model can legitimately be used for policy and welfare analysis.
Rotemberg and Woodford (1997) come to a slightly different conclusion
about the importance of the zero bound as a constraint on monetary policy,
using a different model and approach from those of Fuhrer and Madigan and
Orphanides and Wieland. As we will also, Rotemberg and Woodford use a
sticky-price model whose equations are derived from explicit optimization problems, and they use the utility function of agents in the model to measure the
welfare associated with different monetary policy rules. However, Rotemberg
and Woodford linearize their model to simplify the analysis, and this precludes
them from directly imposing the zero bound. They account for the zero bound
indirectly by assuming that the variability of the monetary authority’s interest
rate instrument is constrained by the average level of interest rates, that is,
by the inflation target. Specifically, they assume that the ratio of the standard
deviation of the nominal interest rate to the average level of the nominal interest rate can be no greater than the ratio that describes their 1980 –1995 U.S.
sample. Thus, policy rules that generate high variability of nominal rates are
incompatible with low inflation targets. Since a generic implication of models
such as theirs is that stable inflation requires volatile nominal interest rates,
their assumption implies a sharp tradeoff between the level of inflation and

A. L. Wolman: Zero Bound on Nominal Interest Rates

5

its variability. While this assumption has the effect of increasing the optimal
inflation target from zero in their model, the optimum does not move far from
zero.
All three papers discussed above exclude money from the models. Rotemberg and Woodford correctly state that the behavior of their model would
be unchanged if they used a money-in-the-utility function specification where
money was additively separable in the period utility function. However, ignoring money demand also means ignoring the welfare costs of positive nominal
interest rates. That is, while the behavior of real and nominal variables may be
invariant to incorporating money in an additively separable way, the welfare
implications of different monetary policies are not invariant to this modification. Since concern about the zero bound on nominal interest rates boils down
to concern about the welfare level associated with very low inflation targets,
leaving money out of the model may be an important omission.

2.

A MODEL WITH STAGGERED PRICE SETTING

Our analysis of the zero bound’s importance for monetary policy is based on
an explicit optimizing sticky-price model similar to, but simpler than, the one
in Rotemberg and Woodford (1997). As in Fuhrer and Madigan (1997) and
Orphanides and Wieland (1998), we impose the zero bound directly, rather
than measuring its importance indirectly.6 However, we take our analysis two
steps further. First, we explicitly model money demand (using a shopping-time
technology), so there is a force working in favor of zero nominal interest rates.
Second, no linear approximations are employed to solve the model, which is
fundamentally nonlinear.
The model follows the tradition of Taylor (1980), in that price setting is
staggered: each firm sets its price for two periods, with one half of the firms
adjusting each period.7 As in Taylor’s model, monetary policy is nonneutral
in the model because stickiness in individual prices gives rise to stickiness in
the price level. There are a continuum of firms, and they produce differentiated
consumption goods using labor provided by consumers at a competitive wage
as the sole input. Consumers are infinitely lived and use income from their
labor, which is supplied elastically, to purchase consumption goods. Consumers
hold money in order to economize on transactions time, as in McCallum and
Goodfriend ([1987] 1988).

6 Fuhrer and Madigan use three different approaches, one of which involves directly imposing
the zero bound.
7 The remainder of this section is loosely based on Section 2 in King and Wolman (forthcoming 1999). The model analyzed here differs in that it explicitly motivates money demand with
a shopping-time technology.

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Federal Reserve Bank of Richmond Economic Quarterly

Consumers
Consumers have preferences over a consumption aggregate (ct ) and leisure (lt )
given by
Et

∞
X

β t · [ln (ct ) + χt · lt ] .

(1)

t=0

The discount factor β is set to 0.985, and the variable χt is a random preference
shock.8 The consumer’s budget constraint is
ct +

Mt
Bt /Pt
Mt−1 Bt−1
St
+
=
+
+ wt nt + dt + ,
Pt
1 + Rt
Pt
Pt
Pt

and the time constraint is
nt + lt + h[Mt /(Pt ct )] = E,

(2)

where Pt is the price level, Mt is nominal money balances chosen in period t
to carry over to t + 1, Bt is holdings of one-period nominal zero-coupon bonds
maturing at t + 1, Rt is the interest rate on nominal bonds, wt is the real wage,
nt is time spent working, dt is real dividend payments from firms, St is a lumpsum transfer of money from the monetary authority, h[Mt /(Pt ct )] is time spent
transacting, and E is the time endowment. Defining real balances to be mt ≡
Mt /Pt , the function h(·) is parameterized as in Wolman (1997):
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 ,

(3)

with φ = 1.4 × 10−3 , A = 1.7 × 10−2 , and ν = −0.7695. Transactions time is
thus decreasing in real balances and increasing in consumption, up to a satiation
level of the ratio of real balances to consumption.
Goods Market Structure
As in Blanchard and Kiyotaki (1987), we assume that every producer faces a
downward-sloping demand curve with constant elasticity ε. 9 The consumption
R
ε
ε−1
aggregate is an integral of the differentiated products ct = [ c(ω) ε dω] ε−1 ,
as in Dixit and Stiglitz (1977).
8 This value of β implies a steady-state real interest rate of 6.5 percent per annum and hence
a steady-state nominal interest rate of about 11.5 percent when there is 5 percent annual inflation.
While the number assigned to β has quantitative implications for the results reported below, it
does not have qualitative implications.
9 We assume ε = 10.

A. L. Wolman: Zero Bound on Nominal Interest Rates

7

Since all producers that adjust their prices in a given period choose the
same price, it is easier to write the consumption aggregate as
µ
¶ ε
1 ε−1
1 ε−1 ε−1
· c0,tε + · c1,tε
,
(4)
ct = c(c0,t , c1,t ) =
2
2
where cj,t is the quantity consumed in period t of a good whose price was set
in period t − j. The constant elasticity demands for each of the goods take the
form
µ ∗ ¶−ε
Pt−j
· ct ,
(5)
cj,t =
Pt
where P∗t−j is the nominal price at time t of any good whose price was set j
periods ago, and Pt is the price index at time t, given by
·
¸ 1
1 ¡ ∗ ¢1−ε 1 ¡ ∗ ¢1−ε 1−ε
· Pt
+ · Pt−1
.
(6)
Pt =
2
2
Optimization
If we attach Lagrange multipliers λt and µt to the budget and time constraints,
respectively, so that λt is the marginal value of real wealth and µt is the marginal value of time, the first-order conditions for the individual’s maximization
problem, with respect to ct , lt , nt , Bt , and Mt , are
mt
1
= λt − µt · h0 (·)( 2 ),
(7)
ct
ct
χt = µt ,

(8)

µt = wt · λt ,

(9)

λt
λt+1
= β · (1 + Rt ) · Et
,
Pt
Pt+1

(10)

λt+1
µt 0
1
λt
+
· h (·)( ) = βEt
.
Pt
Pt
ct
Pt+1

(11)

and

In choosing consumption optimally (as in [7]), the individual weighs the benefit of consuming a marginal unit, which is the left-hand side of (7), against
the cost, which consists of both forfeited real wealth (the first term on the
right-hand side) and time spent transacting (the second term on the right-hand
side). In choosing leisure and labor supply optimally (as in [8] and [9]), the
individual weighs the marginal value of time against both the marginal utility
of leisure and the wage earnings that the time would yield. The choice of bond
holdings (equation [10]) equates the marginal value of nominal wealth today
to (1 + Rt ) times the marginal value of nominal wealth tomorrow. And finally,

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Federal Reserve Bank of Richmond Economic Quarterly

optimal money holdings (equation [11]) imply that the individual equates the
transactions-facilitating benefit to the foregone interest cost of holding money.10
Firms
Each firm produces with an identical technology:
cj,t = nj,t , j = 0, 1,

(12)

where nj,t is the labor input employed in period t by a firm whose price was
set in period t − j. Given the price a firm charges, it hires enough labor to meet
the demand for its product at that price. Firms that do not adjust their price
in a given period can thus be thought of as passive, whereas firms that adjust
their price do so optimally, that is, in order to maximize the present discounted
value of their profits. Given that it has set a relative price
for a firm of type j are

P∗
t−j
Pt

P∗t−j
· cjt − wt · njt ,
Pt

, real profits

(13)

that is, revenue minus cost.
Optimal Price Setting
Maximization of present value implies that a firm chooses its current relative
price, taking into account the effect on current and expected future profits. Substituting into (13) the demand curve (5) and the technology (12), the present
discounted value of expected profits is given by
"µ ¶
µ ∗ ¶−ε #
P∗t 1−ε
Pt
− wt ·
+
ct ·
Pt
Pt
λt+1
· ct+1 ·
βEt
λt

"µ

P∗t
Pt+1

¶1−ε

µ

− wt+1 ·

P∗t
Pt+1

¶−ε #

(14)

for the two periods over which a price will be in effect. Differentiating (14)
with respect to P∗t and setting the resulting expression equal to zero, one sees
that the optimal relative price satisfies
P1
j
ε
ε
P∗t
j=0 β Et [λt+j · wt+j · (Pt+j /Pt ) · ct+j ]
· PJ−1
=
.
(15)
j
ε−1 · c
Pt
ε−1
t+j ]
j=0 β Et [λt+j · (Pt+j /Pt )
10 The
λt
Pt

transactions-facilitating benefit is given by
λ
βEt P t+1
t+1

µt
Pt

· h0 (·)( c1 ), and the foregone interest
t

−
(see [10]). A conventional money demand equation can be derived by
cost is
combining (9)–(11): mt /ct = A · {[Rt / (1 + Rt )] · (ct /wt ) + φ}ν .

A. L. Wolman: Zero Bound on Nominal Interest Rates

9

Essentially, the optimal relative price equates discounted marginal revenue with
discounted marginal cost; the numerator of (15) represents marginal cost and
the denominator marginal revenue.11 In a noninflationary steady state, the firm
ε
would choose a markup over marginal cost of ε−1 . In an inflationary or deε
flationary steady state, the markup would differ from ε−1 , as adjusting firms
would take into account the future erosion (or inflation) of their relative price
(see King and Wolman [forthcoming 1999] for details). With uncertainty, the
markup becomes time varying: it depends on the current and expected future
marginal utility of wealth, price level, aggregate demand, and real wage.
Driving Process
The only exogenous variable in the model is the preference shock χt , and it is
assumed to follow a two-state Markov process:
¡
¢
Pr χt = χ | χt−1 = χ = 0.8, and
³
´
Pr χt = χ | χt−1 = χ = 0.8, χ < χ.

(16)

Thus, χt varies between high and low values, and on average each value persists for five periods before switching. This process is not meant to replicate
actual features of the U.S. economy. Rather, it is chosen to make the economy
alternate between periods of high and low output in a way that makes the real
interest rate vary over time. It is by no means the only process that would
yield such behavior. The equilibrium behavior of the real interest rate will be
affected by monetary policy as well as by the shock process.
Monetary Policy
As described below, we assume that policy is characterized by a feedback rule
for the nominal interest rate. One component of the feedback rule is a “target”
inflation rate, an inflation rate that the rule would deliver in the absence of
shocks. In general, the feedback rule makes the nominal rate a differentiable
function of observable variables. In certain states of the world, however, that
differentiable function would make the nominal rate negative. In those states
of the world, we assume that the policy rule sets Rt = 0. Given the nominal
interest rate implied by the policy rule, the monetary transfer (St ) is determined
by money demand. Note that money demand is an integral part of the model. It
is sometimes asserted that when the monetary authority follows an interest rate
rule, money demand can be left out of the model, as it only serves to determine
the value of the money supply. Here that is not the case, because the quantity
11 Note

that in this sentence, marginal revenue and cost are with respect to price, not quantity.

10

Federal Reserve Bank of Richmond Economic Quarterly

of money enters other equations of the model in addition to the money demand
equation (specifically [7] and [2]).
The nominal interest rate is the rate on one-period bonds, which are assumed to be in zero net supply. This is somewhat problematic from the standpoint of justifying the zero bound. That is, the zero bound is a necessary
characteristic of nominal bonds that are willingly held, but nominal bonds are
not actually held in the model (they are priced). This inconsistency can be
rectified by assuming that there is a fixed real quantity of outstanding government bonds, and the government pays the interest on those bonds by levying
lump-sum taxes as necessary.
Solving the Model
The standard method used for solving dynamic stochastic models such as this
one is to calculate the steady state for a given inflation rate, and then linearize
the model’s equations around that steady state. Linearization would be inappropriate here, because it would rule out imposing the zero bound on nominal
interest rates. Instead of linearizing, then, we solve the model using a crude
version of the finite element method (see McGrattan [1996]). This method
involves picking a grid of points for the model’s state variable, P∗t−1 , and then
finding values of the “control” variables numerically for each grid point and for
each value of the preference shock such that the model’s equations are approximately satisfied. The solution consists of mappings from the state variable to
each of the other variables. Those mappings can be used in conjunction with the
stochastic process for the preference shock to simulate the model. Because this
solution method involves a finite number of grid points, it necessarily yields
only an approximate solution. However, to the extent that the true mappings
from the state variables to the other variables are smooth functions, the grid
method can yield an extremely accurate solution. Furthermore, the extent that
the mappings appear nonlinear gives an indication of the error that would be
associated with linearization methods.

3.

IMPLICATIONS OF THE ZERO BOUND IN THE MODEL

Using the model described above, one can determine whether the zero bound
means that a very low inflation target (here it will be deflation) significantly
modifies economic performance relative to a moderate inflation target. For a
particular specification of monetary policy, we will simulate the model at moderate inflation and then at moderate deflation, and compare the results along
three dimensions. The first involves simulating the model for 30 periods with
the same shocks at high and low inflation, and informally comparing the results.
The second involves the variances of inflation and output, which has been the
conventional metric in the literature on monetary policy rules (see the papers in

A. L. Wolman: Zero Bound on Nominal Interest Rates

11

Taylor [forthcoming 1999]). Given that the model yields an obvious choice for
a welfare function (the representative agent’s expected utility), we also compare
the two regimes in terms of welfare.
Model Simulations
Recent research on monetary policy has emphasized “Taylor rules,” that is,
specifications of policy where the monetary authority sets a short-term interest
rate as a linear combination of deviations of inflation from a target and deviations of output from some trend or potential level. These rules, popularized by
John Taylor (1993), have been shown to be parsimonious approximations of
the behavior of actual central banks and to have reasonable properties in certain
theoretical models. The rule used below is similar to a Taylor rule, except that
instead of inflation on the right-hand side it uses the price level. Concretely,12
½ ∗
¾
R + 1.5 · [ln(Pt ) − ln(P̄t )] + 1.0 · [ln(ct ) − ln(c̄)],
Rt = max
,
(17)
0
where R∗ is the steady-state nominal interest rate consistent with the chosen
inflation target, P̄t is a target price-level path that grows at the targeted inflation
rate, and c̄ is the steady-state level of consumption associated with the inflation
target.13 This rule implies that the price level will always be expected to return
to the same trend path. In contrast, the standard Taylor rules imply that inflation
will always be expected to eventually return to target, but the price level will
be expected to drift away from any previous trend path.14
Introduction to the Functions Describing General Equilibrium
As background to the simulation results, Figure 1 displays the relationships
between key endogenous variables and the state variable, which is the price set
last period by adjusting firms. Figure 1 is generated with an inflation target of 5
percent. The solid lines show the relationship between P∗t−1 (detrended by the
targeted inflation rate) and each endogenous variable when the preference shock
takes on a high value, and the dashed lines show the relationships when the
preference shock takes on a low value. Using panel b, and with knowledge of
P∗0 , one can trace out a path for P∗t by drawing values of χt from the stochastic
12 The interest rate in (17) is a quarterly interest rate, whereas the rates plotted in Figures
1–4 are annual rates.
13 The inflation target affects steady-state consumption for two reasons. First, the markup
chosen by adjusting firms varies with the inflation target in a way that does not exactly offset
the inflation erosion of nonadjusting firms’ markups. Second, by lowering real balances, higher
inflation effectively makes consumption more expensive.
14 The original motivation for using a price-level target here instead of an inflation target
was computational ease. It turns out, however, that analyzing an inflation-targeting policy is no
more demanding than analyzing price-level targeting. We are studying inflation-targeting policies
in ongoing research.

12

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Functions Mapping State Variable (P∗t−1 ) to Other Variables at
5 Percent Inflation Target

a. Ct

1.25
1.20

x
y

1.15

χ

y

1.10

x

1.05
1.00
1.25

1.30

1.35
1.40
LN(P*t - 1 )

χ
1.45

1.50

b. LN(detrended P*t )

1.50
1.45

y

x

χ

1.40
χ

1.35

x

y

1.30
1.25
1.25

1.30

1.35
1.40
LN(P*t - 1 )

1.45

1.50

1.45

1.50

c. Nominal Ratet
0.16
0.12
0.08
χ
0.04

χ

0.00
1.25

1.30

1.35
1.40
LN(P*t - 1 )

d. LN(detrended Mt )
0.80

χ
χ

0.40

0.00

- 0.40
1.25

1.30

1.35
1.40
LN(P*t - 1 )

1.45

1.50

A. L. Wolman: Zero Bound on Nominal Interest Rates

13

process governing it. Then, with the path for P∗t in hand, the relationships in
panels a, c, and d can be used to generate paths for the other variables for the
given sequence of χt . What follows is a discussion of the model’s principal
mechanisms in light of the relationships shown in Figure 1.
There are essentially two determinants of current-period variables in the
model. One is the value of the stochastic preference parameter (χt ), and the
other is the value of the price that adjusting firms set last period. When χt
takes on a high value, the marginal utility of leisure is high. Agents react by
supplying less labor to the market, and this reaction brings with it a decrease in
consumption. Thus, in panel a, the level of consumption is low when χt = χ.
For low values of P∗t−1 , the lower level of consumption causes the monetary
authority to set a lower value for the nominal interest rate, as in the left-hand
part of panel c, and the lower nominal interest rate in turn drives up money
demand (panel d). However, when P∗t−1 is especially high, the nominal rate is
lower in the χ (high-consumption) state. Why is this the case? The feedback
rule for monetary policy sets the nominal rate as an increasing function of both
consumption and the price level, so it must be that in the high-P∗t−1 region the
price-level effect dominates in the feedback rule. The policy functions for the
price level (not shown) indeed reflect this fact. The price level is higher in the
χ state than in the χ state, and the gap between the price levels in the two
states is increasing in P∗t−1 .
Another perspective on the nominal interest rate functions in panel c comes
from thinking about two relationships emphasized by Irving Fisher. We have
already seen the “Fisher Equation: I,” which states that the nominal interest
rate is approximately equal to the sum of the real interest rate and expected
inflation.15 But Fisher also provided the seminal discussion of the relationship
between real interest rates and current and future marginal utilities of consumption. Since the real interest rate is the price at which agents can trade
current consumption for future consumption, it follows that agents will choose
an expected consumption path to equate the real interest rate to the ratio of
marginal utilities of current and future consumption. When utility is logarithmic
in consumption, as it is here, this “Fisher Equation: II” implies that the real
interest rate is approximately equal to expected consumption growth.16
From panel a, we know that consumption and the preference parameter
move in opposite directions. Further, the stochastic process for the preference
parameter is mean reverting, so that when χt is low it is expected to increase,
and, therefore, consumption is expected to fall. From Fisher’s second equation,

15 For an explanation of why the “Fisher Equation: I” is only approximately correct, see
Sarte (1998).
16 The relationship is only approximate here because the shopping time requirement means
that the marginal utility of consumption is greater than the marginal value of a unit of real wealth.
To derive this approximate relationship, combine (7) and (10) above.

14

Federal Reserve Bank of Richmond Economic Quarterly

real interest rates are then low when the preference parameter is low. Note,
however, that the policy rule typically makes nominal rates high in those cases
when we have just argued that real rates are low. From Fisher’s first equation,
it must then be that high nominal rates correspond to high enough expected
inflation to counteract the low real rates. From panel d we can see that monetary policy does in fact deliver high expected inflation when the preference
parameter is low. The money supply is low when the preference parameter is
low, and mean reversion implies that the money supply is expected to increase
in those periods, generating high expected inflation.
Note that the behavior of real interest rates conflicts with the behavior
displayed in the other articles discussed above. There the monetary authority
lowers nominal interest rates when output is low, and real rates fall as well.
Here, for the most part, the monetary authority also decreases nominal interest
rates when output is low. However, real interest rates are to a great extent
determined by the shock process in conjunction with Fisher’s second equation.
For a large class of such processes that includes the one used here, real interest
rates are low when output is high. More generally, it has proven difficult to
produce models where the cyclical behavior of real rates matches the data
without resorting to the type of reduced form modeling employed by Fuhrer
and Madigan (1997) and Orphanides and Wieland (1998).
Simulated Time Paths
Figure 2 displays the time paths of the variables from Figure 1 other than P∗t ,
as well as the price level, the real interest rate, and expected inflation, for a
sequence of 30 χt drawn from the stochastic process described above. This
sequence will be a benchmark for comparison with the low inflation target
case below. Focusing first on consumption (panel a), note that there are essentially three regions: low, high, and intermediate. The high-consumption region
is attained with any sequence of at least two consecutive low values for χt
(the realizations of χt are plotted in panel b). Likewise, the low-consumption
region is attained with any sequence of at least two consecutive high values for
χt . These regions correspond to the points marked x in Figure 1a and b. The
intermediate-consumption region corresponds to the transition from one value
of the preference shock to the other; these are the points marked y in Figure
1a and b. The fact that it takes two periods to transit between the high- and
low-consumption regions is an implication of two-period price stickiness. To
see this, suppose the economy had been in the low preference parameter/highconsumption state for several periods. If χt then took on a high value, in the
initial period the state variable (P∗t−1 ) would be at the level associated with χ,
so that the economy could not immediately transit to low consumption. If χt
remained high in the next period, consumption would settle at a lower level,
because the state variable had changed; by the period after the shift in χt , all

A. L. Wolman: Zero Bound on Nominal Interest Rates

15

Figure 2 Time Paths from 30-Period Simulation
(5 Percent Inflation Target)
b. LN(detrended P) and Level of χ

a. Consumption

1.18

1.40

1.16

1.30

1.14

1.20

LN(Pt )

1.10

1.12

1.00
1.10

0.80
0.70

1.06
5

0.20

10

15
t

20

25

30

c. Expected Inflation

5

0.13

0.10

0.12

0.05

0.11

0.00

0.10

- 0.05

0.09
5

10

15
t

20

25

30

e. LN(detrended money supply)

5

0.25

0.15

0.20

0.10

0.15

0.05

0.10

0.00

0.05

- 0.05

0.00

- 0.10

- 0.05
5

10

15
t

20

25

30

15
t

20

25

30

25

30

0.08

0.20

- 0.15

10

d. Nominal Rate

0.14

0.15

- 0.10

χt

0.90

1.08

- 0.10

10

15
t

20

f. Ex Ante Real Rate

5

10

15
t

20

25

30

firms would have had a chance to adjust their price. If prices were flexible,
the transition would be immediate, whereas with prices set for more than two
periods the transition would be correspondingly longer.
Note that in some of the periods when consumption takes on an intermediate value, the real rate is negative (Figure 2f). Specifically, this occurs in
periods when χt = χ and χt−1 = χ (periods 12, 17, and 20). Referring back

16

Federal Reserve Bank of Richmond Economic Quarterly

to Figure 1, one can see that in this situation consumption is expected to fall
towards the low level associated with χ. With consumption expected to fall
significantly, the real rate must be negative. Because the inflation target is 5
percent, the zero bound does not inhibit the real rate from going negative. However, one might expect that with a very low inflation target, the real rate would
be inhibited from going negative, and thus the zero bound would interfere with
the economy’s “natural” behavior.
Figures 3 and 4 correspond to Figures 1 and 2, with an inflation target of −5
percent. From Figure 3a–c, we see that for a wide range of values of the state
variable, including the region corresponding to high consumption, the nominal
rate is zero. This drastically different behavior of the nominal rate, however,
does not correspond to significantly different functions for consumption (Figure
3a). The simulation in Figure 4 confirms these results. Whereas we surmised
that the nominal rate might hit the zero bound when χt = χ and χt−1 = χ,
in fact it hits the bound whenever χt = χ. However, consumption behavior
is almost indistinguishable from Figure 2, the 5 percent inflation target. From
Fisher’s second equation, we know that similar consumption behavior must
correspond to similar real rate behavior, as confirmed in Figure 4f. How is a
zero nominal rate consistent with a negative real rate in periods 12, 17, and
20? From Fisher’s first equation, the real rate is the difference between the
nominal rate and expected inflation, so in those periods the monetary authority
is making expected inflation positive (panel c). The targeted rate of deflation
is consistent with periods of high expected inflation, because the policy rule
unambiguously makes the expected inflation temporary, and there is no uncertainty about whether the monetary authority will adhere to the policy rule.
Simulations such as those in Figures 2 and 4 are an informal means of
evaluating whether the zero bound is important. However, those simulations
provide clear evidence—at least in the model used here—that monetary policy
can offset the zero bound by generating temporary expected inflation. With
real rates thus unconstrained, the existence of the zero bound does not appear
to constitute an argument against a low inflation target. Figure 4 illustrates an
additional feature of the model that favors a very low inflation target. In panels
a and f, the series for consumption and real rates from Figure 2, corresponding to a 5 percent inflation target, are reproduced along with the new series
corresponding to 5 percent deflation. In panel a, we see that consumption is
actually higher in every period with the 5 percent deflation target than it is with
the 5 percent inflation target. The lower inflation target corresponds to lower
nominal interest rates on average, as is shown clearly in panel d of Figures
2 and 4. Lower nominal interest rates in turn correspond to a smaller money
demand distortion, as in Bailey (1956) and Friedman (1969). Individuals hold
higher real balances because the opportunity cost of real balances has fallen,
and higher real balances effectively make consumption cheaper, because they
decrease the time that an individual must spend transacting.

A. L. Wolman: Zero Bound on Nominal Interest Rates

17

Figure 3 Functions Mapping State Variable (P∗t−1 ) to Other Variables at
5 Percent Deflation Target

a. C t

1.24

1.16
χ

1.08
χ

1.00
-0.45

-0.40

-0.35

-0.45
-0.45

-0.30 -0.25
LN(P*t - 1 )

-0.20

-0.15

-0.20

-0.15

b. LN(detrended P*t )

-0.15

-0.25

-0.35

χ
χ

-0.40

-0.35

-0.30 -0.25
LN(P*t - 1 )

c. Nominal Ratet

0.050

χ

0.040
0.030
0.020

χ

0.010
0.000
-0.45

-0.40

-0.35

-0.30 -0.25
LN(P*t - 1 )

-0.20

-0.15

d. LN(detrended Mt)

1.20
0.80
0.40
0.00

χ

-0.40
-0.80
-1.20
-0.45

χ

-0.40

-0.35

-0.30 -0.25
LN(P*t - 1 )

-0.20

-0.15

18

Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Time Paths from 30-Period Simulation
(5 Percent Deflation Target)

a. Consumption

b. LN(detrended price level)

1.18

-0.25

1.16

-0.27

1.14
-0.29
1.12
-0.31

1.10

-0.33

1.08
1.06
5

10

15 20 25
t
c. Expected Inflation

30

-0.35

0.10

0.025

0.05

0.020

0.00

0.015

-0.05

0.010

-0.10

0.005

-0.15

5

10

15 20 25 30
t
e. LN(detrended money supply)

0.000

1.00

0.25

0.80

0.20

0.60

0.15

0.40

5

5

10

15 20 25
t
d. Nominal Rate

30

10

30

15 20 25
t
f. Ex Ante Real Rate

0.10

0.20
0.05

0.00

0.00

-0.20

-0.05

-0.40
-0.60

5

10

15
t

20

25

30

-0.10

5

10

15
t

20

25

30

Note: Dashed series are from Figure 2 (5 percent inflation target).

Variances
The simulations in Figures 2 and 4 provide strong evidence on the importance
of the zero bound, and the welfare results below give the bottom line. To enhance comparability with the articles by Rotemberg and Woodford (1997) and
Orphanides and Wieland (1998), we also provide information on variability at

A. L. Wolman: Zero Bound on Nominal Interest Rates

19

high and low inflation targets. Table 1 shows the standard deviations of some of
the main variables in the model for both regimes, based on simulations of 5,000
periods. As suggested by Figures 1– 4, the variability of consumption is barely
affected by the inflation target. On the other hand, the nominal interest rate is
much less variable when the inflation target makes zero occasionally binding.
There is a tradeoff in the model between the average level of inflation and the
minimum feasible variability of inflation, just as described in Rotemberg and
Woodford (1997). Also as in that paper, the large difference in nominal interest
rate variability in the two regimes translates into only a small difference in
inflation variability. A striking feature of Table 1 is the tremendous increase
in money supply variability in the deflation regime. This can be traced to the
fact that the money demand function exhibits increasing sensitivity to nominal
interest rates as the nominal interest rate falls.

Table 1 Standard Deviations in the Two Policy Regimes
Consumption

Inflation

Nominal rates

Money

5 percent inflation

0.0427

0.0706

0.0145

0.0910

5 percent deflation

0.0435

0.0786

0.0093

0.7562

Welfare
The motivation for this article came from the idea that low inflation targets
might be bad because of distortions introduced by the zero bound on nominal
interest rates. It is clear from the simulations presented thus far that in fact
the real (as opposed to nominal) distortions associated with the zero bound are
small. Nevertheless, it is interesting to know whether the inflation or deflation
regime is preferred on welfare grounds. When the zero bound is not a factor,
a welfare comparison will hinge on the other distortions present in the model.
Those other distortions involve the inflation tax and the interaction between
sticky prices and monopolistic competition. The inflation tax distortion makes
deflation preferable to inflation. Sticky prices and monopolistic competition
make the optimal inflation target near zero, so neither 5 percent inflation nor
deflation targets would obviously be preferred to the other on that basis. It
therefore seems likely that the unambiguous effect of the inflation tax will
dictate that the lower inflation regime is preferred. However, to resolve the
issue definitively, we must compare the representative individual’s expected
utility in the inflation and deflation regimes.
We calculate expected utility by performing 1,000 simulations of 1,000
periods each, with each simulation beginning from a random value for the
state variable. The initial condition is chosen by simulating the model for 50

20

Federal Reserve Bank of Richmond Economic Quarterly

periods, starting from the steady state, and then setting P0 = P50 . Each simP
t
ulation (k = 1 to 1, 000) yields a value for Uk ≡ 1,000
t=0Pβ · [ln (ct ) + χt · lt ] ,
1,000
−1
and then expected utility is given by E (U) = 1, 000 · k=1 Uk . With values
for expected utility in both regimes, we compare the regimes by pretending
that they were generated in a steady state. We calculate the average per-period
utility in the two regimes and then the percentage increase in consumption that
would make an agent living in the lower utility regime just as well-off as an
agent in the higher utility regime. The results of this exercise are that an agent
living in the inflationary regime would be indifferent between receiving a 2.6
percent increase in per-period consumption and switching to the deflationary
regime.
To illustrate the importance of the inflation tax in these results, we can
repeat the comparison of the two inflation regimes with a slight modification.
That modification is to eliminate the money demand distortion; we modify (7)
to λt = 1/ct and replace (11) with Mt = Pt ·ct . With the inflation tax eliminated,
the 5 percent inflation target regime is marginally preferred to the 5 percent
deflation target regime, although the difference in welfare is minuscule compared to the difference found (with opposite sign) when the inflation tax played
a role. The results from eliminating the money demand distortion mean that
money demand is crucial in making the deflationary regime welfare-superior to
the inflationary regime. However, even without the money demand distortion,
the fact that the nominal interest rate is occasionally zero in the deflationary
regime does not significantly affect the behavior of real variables. In particular,
the policy rule is still able to generate temporarily high expected inflation when
real rates need to be negative.
Open Questions
With respect to the specific model used here, at least three modifications would
be interesting to analyze. The first modification deals with the specification
of price stickiness. Structural models of sticky inflation are ad hoc, but they
have been shown to fit recent data well. It should be possible to modify the
price block of the current model to make inflation sticky. The resulting specification would not simply repeat the work of Orphanides and Wieland (1998)
and Fuhrer and Madigan (1997), because it would incorporate money demand.
Solving such a model would be more computationally intensive than solving
the model in this article, because it would include additional state variables
associated with the pricing specification.
The second modification is related to the first; it involves changing the policy rule from the price-level form to the more common inflation form. Possibly
with such a rule and a low inflation target the monetary authority would be less
able to generate the temporary expected inflation necessary to drive real rates
negative. More generally, it would be interesting to study the properties of a

A. L. Wolman: Zero Bound on Nominal Interest Rates

21

wide range of rules and to find out what the optimal rule is. Experiments with
a rule that specifies the money supply instead of the nominal interest rate as the
policy instrument yield similar results to those above, in that the deflationary
regime is preferred to the inflationary regime. The interest rate rule generates
higher welfare than the money rule, but that comparison is limited, focusing
on two specific rules as opposed to classes of rules. In terms of optimal rules,
King and Wolman (forthcoming 1999) find that it is optimal to stabilize the
price level if the money demand distortion is nonexistent. With that distortion
present, optimal policy will undoubtedly involve some deflation, but it is not
clear exactly what the optimal policy rule is.17
The third modification is one that takes more seriously the fiscal aspect
of monetary policy. Work by Woodford (1996) and Sims (1994) emphasizes
the joint behavior of fiscal and monetary policy. This joint behavior might be
especially relevant when interest rates are near zero, because at zero nominal
interest rates, fiscal and monetary policy effectively become unified; money
and government bonds are perfect substitutes.
Apart from the specifics of the model, the assumption that agents in the
model have perfect information about the policy rule is crucial. We found that
zero nominal interest rates did not prevent the real rate from falling, because
the monetary authority could generate expected inflation when the nominal rate
was zero. Agents know that any inflation that ensues will be temporary, and
that the monetary authority remains committed to its stated inflation target,
so these occasional periods of high expected inflation do not trigger inflation
scares. In practice, central banks might have concerns about being able to
generate occasional episodes of high expected inflation without endangering
the credibility of their low inflation target. In principle it would be possible to
analyze this sort of issue in an extension of the current framework.
A fundamental assumption underlying all recent work on the zero bound is
that negative ex ante real interest rates are occasionally a natural characteristic
of the U.S. economy. It is a trivial matter to look at data on ex post real rates
and see that at the short end of the yield curve they have been negative on
many occasions. It is less clear that ex ante real rates have been negative. From
Irving Fisher, we know that real rates defined by the CPI can be negative only
to the extent that the market basket that makes up the CPI is not storable at
zero cost. Undoubtedly the inclusion of various services and perishable goods
means that in principle the ex ante real rate can be negative. Nonetheless, lack
of consensus about how to estimate inflation expectations means that widely
accepted series for ex ante real rates do not exist.
17 The

approach taken in this article would suggest defining the optimal policy rule as the rule
that generates the highest level of unconditional expected utility. King and Wolman (forthcoming
1999) use a different criterion; they ask what policy rule is implied by assuming that the monetary
authority maximizes agents’ expected utility given some arbitrary initial conditions.

22

4.

Federal Reserve Bank of Richmond Economic Quarterly

CONCLUSIONS

Two general conclusions are supported by the theoretical analysis in this article.
First, the way money demand is modeled is important for how one evaluates
the zero bound on nominal interest rates. Existing work presumes that the
zero bound makes low inflation bad, because it prevents monetary policy from
optimally responding to shocks. But monetary theory supports a strong benefit
to zero nominal interest rates, namely, eliminating inefficiencies associated with
holding “too little” money. The existence of those inefficiencies contributed to
the result in this article that, taking into account the zero bound, a regime with
moderate deflation yields higher welfare than a regime with moderate inflation.
The second conclusion is that stickiness of inflation is crucial in generating
costs of low inflation associated with the zero bound. If prices are sticky but
inflation is not, then real rates can fall even if nominal interest rates are very
low: the monetary authority simply creates some expected inflation if it wants
to drive real rates down.

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Blanchard, O. J., and N. Kiyotaki. “Monopolistic Competition and the Effects
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Dixit, Avinash, and Joseph Stiglitz. “Monopolistic Competition and Optimum
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297–308.
Dotsey, Michael, Robert G. King, and Alexander L. Wolman. “State Dependent
Pricing and the General Equilibrium Dynamics of Money and Output.”
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1996.
Federal Reserve Bank of Kansas City. Achieving Price Stability. Kansas City:
Federal Reserve Bank of Kansas City, 1996.
Fisher, Irving. The Theory of Interest. New York: Macmillan, 1930, reprinted,
New York: Kelley and Millman, Inc., 1954.
. The Rate of Interest. New York: Macmillan, 1907.
. “Appreciation and Interest,” in Publications of the American
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23

Friedman, Milton. “The Optimum Quantity of Money,” in The Optimum
Quantity of Money, and Other Essays. Chicago: Aldine Publishing
Company, 1969.
Fuhrer, Jeffrey C., and Brian F. Madigan. “Monetary Policy When Interest
Rates Are Bounded at Zero,” Review of Economics and Statistics, vol. 79
(November 1997), pp. 573–85.
Fuhrer, Jeffrey C., and George Moore. “Inflation Persistence,” Quarterly
Journal of Economics, vol. 110 (February 1995), pp. 127–59.
Humphrey, Thomas M. “The Early History of the Real/Nominal Interest Rate
Relationship,” Federal Reserve Bank of Richmond Economic Review, vol.
69 (May/June 1983), pp. 2–10, reprinted in Thomas M. Humphrey, Essays
on Inflation, 5th ed., Federal Reserve Bank of Richmond, 1986.
Keynes, John Maynard. The General Theory of Employment, Interest and
Money. London: Macmillan and Company, 1936, reprinted, San Diego:
Harcourt Brace Jovanovich, 1964.
King, Robert G., and Alexander L. Wolman. “What Should the Monetary
Authority do When Prices are Sticky?” in John B. Taylor, ed., Monetary
Policy Rules. Chicago: University of Chicago Press for NBER, forthcoming
1999.
. “Inflation Targeting in a St. Louis Model of the 21st Century,”
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Laidler, David E. W. The Golden Age of the Quantity Theory. Princeton, N.J.:
Princeton University Press, 1991.
Lucas, Robert E., Jr. “On the Welfare Cost of Inflation,” Working Paper 394.
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McCallum, Bennett T., and Marvin S. Goodfriend. “Demand for Money:
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McGrattan, Ellen R. “Solving the Stochastic Growth Model with a Finite
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(January–March 1996), pp. 19– 42.
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Orphanides, Athanasios, and Volker Wieland. “Monetary Policy Effectiveness
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Patinkin, Don. Money, Interest, and Prices: An Integration of Monetary and
Value Theory, 2d ed. New York: Harper and Row, 1965.
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Government Loan,
Guarantee, and
Grant Programs:
An Evaluation
Wenli Li

R

ecently, there has been a trend toward loan guarantee programs over
other programs that support the credit market. From 1970 to 1998, the
real value of outstanding federal loan guarantees rose at an accelerated pace, while the real value of direct loans, the other major government
loan program, has remained about the same (see Figure 1). In particular, the
Small Business Administration (SBA), which has provided government loan
guarantees to small businesses since 1953, has experienced an unprecedented
increase in its loan volume over the past three years. In December 1997, with
the growing popularity of SBA loans, Congress passed an SBA funding bill
that set aside $39.5 billion and $11 billion, respectively, for the SBA’s 7(a) and
504 business loan programs over the next three years. This more than tripled
the current 7(a) level which was $10.3 billion in fiscal year 1997.1
The surge in loan guarantee programs prompts the question: Are loan guarantees the best way to provide benefits to targeted borrowers or to channel
additional resources to targeted sectors? As the following paragraphs show, not
in all cases. This article explains that conclusion by examining the economic
consequences of three distinct methods of channeling resources to targeted
borrowers: direct government lending, loan guarantees, and outright grants.
While the logic applies to any credit market segment, the article particularly
focuses on the small business sector. The analysis studies the changes in firm
investment, bankruptcy cost, and business entry under each loan program in a

I would like to thank Tom Humphrey, Jeff Lacker, Pierre Sarte, and John Weinberg for
many important suggestions. Jeff Walker provided excellent research assistance. The views
expressed are the author’s and not necessarily those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
1 Bureau

of National Affairs, Inc. (1997).

Federal Reserve Bank of Richmond Economic Quarterly Volume 84/4 Fall 1998

25

26

Federal Reserve Bank of Richmond Economic Quarterly

theoretical model economy designed to capture the essential features of small
business borrowing.
One thing is sure. These credit policies cannot make the private economy
any more efficient, the reason being that the government does not have information or technology advantage over private agents. Therefore, there will
not be any efficiency gain associated with credit policies. (In other words, the
absence of efficiency gains means that policies cannot make any agent better
off without hurting other agents.) In this article, we take as given a political
desire to assist a particular group of borrowers and look at how the different
alternate credit programs redistribute resources.
Perhaps it is most appropriate to explore the effects of government credit
programs within a model of financial frictions. It is natural to do so because
many economists contend that such frictions have a greater effect on certain
kinds of borrowers, such as small businesses and students, than on others.
Accordingly, the environment studied here is one in which financial frictions
are caused by private information: in particular, moral hazard.2 Moral hazard
occurs when the very act of insuring a borrower against risk induces him to take
on additional risk. Such frictions drive a wedge between the cost of internal
funds and that of external funds as in Townsend (1979) and Gale and Hellwig
(1985). The central notion is that wealth affects people’s decisions, creating
liquidity constraints.
The relevance of such a model is supported by empirical evidence. HoltzEakin, Joulfaian, and Rosen (1994), Evans and Leighton (1989), Blanchflower
and Oswald (1998), and Evans and Jovanovic (1989) among many others, find
that a lack of wealth affects people’s ability to become self-employed, even
after accounting for the possible correlation between entrepreneurial ability
and wealth. In a more recent study, Bond and Townsend (1996) reported on
the results of a survey of financial activity in a low-income, primarily Mexican neighborhood in Chicago and found that borrowing is 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, while 50 percent of
the respondents financed their start-up entirely out of their own funds.

1.

AN OVERVIEW OF GOVERNMENT
CREDIT PROGRAMS

In the United States, the federal government regularly proposes and endorses
programs that are designed to direct and encourage the flow of funds to
2 Adverse

selection—namely, situations in which borrowers have unverifiable hidden knowledge about their likelihood of repayment—is another form of private information that gives rise to
financial frictions. See de Meza and Webb (1987), Gale (1991), Innes (1991), and Lacker (1994)
for discussion.

W. Li: Government Loan, Guarantee, and Grant Programs

27

selected consumers and businesses. For instance, the Community Reinvestment
Act (CRA) attempts to increase the flow of funds to disadvantaged communities or persons by requiring depository institutions to make a minimum effort
to fund these groups. Similarly, the SBA’s section 7(a) loan program and its
Small Business Investment Company program encourage the flow of funds
to small businesses through government guarantees of debt issued by the financial intermediaries providing the funds to the small business. Numerous
other government-sponsored enterprises (GSEs) such as Fannie Mae, Sally
Mae, Freddie Mac, etc., operate on secondary markets and provide credits for
targeted groups in exchange for preferential treatment from the government.
Government intervention in the financial market has occurred mainly via
direct loans, grants, and indirect loan guarantees. In the case of direct loans, a
government agency acts as an intermediary in place of banks; it issues loans
directly to the targeted group, obtaining funds from the capital markets by
issuing Treasury securities and/or imposing taxes. Direct loans typically offer
large subsidies, usually to the agricultural and rural sectors. Unlike direct loans,
grants and loan guarantees do not involve any repayment from the recipients.
Grants, provided by the government directly to the targeted recipients, are often
received at the end of the period when they are added to business profits to help
defray costs. Loan guarantees provide investors with assurance that the government will make up any difference between a given guaranteed loan payment and
an agent’s actual loan payment. A loan guarantee requires the participation of
three parties: the government agency, the borrower, and the private lender. The
government agency deals indirectly with the borrower through a private lender.
Typically, the acquisition of an SBA loan proceeds as follows. The borrower
first presents the appropriate financial data for the lender to review. Based on
the lender’s evaluation, three courses of action are possible: the lender (1) may
decide to finance the loan without an SBA loan guarantee; (2) may provide
financing conditional upon obtaining an SBA loan guarantee; or (3) may reject
the loan. If the lender approves the loan based on the SBA’s willingness to
provide a guarantee, then the lender must help the borrower prepare the SBA
loan application. Upon completion of the application, the SBA reviews the
loan. Over 90 percent of all loan guarantee applications are approved by the
SBA (Haynes 1996). Of course loan guarantee programs assist a wide range
of borrowers besides small businesses, including homeowners, students, and
exporters.3
Figure 1 depicts the recent trend in government direct loan and loan guarantee programs (GSEs included). As shown here, federal credit outstanding in
3 In

addition to direct loans and loan guarantees, GSEs aid borrowers in housing, agricultural,
and student loan markets, primarily through the operation of secondary markets. The tax-exempt
status of state and local governments allows them to borrow at reduced cost and to direct the
interest savings to preferred borrowers.

28

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Real Value of Federal Credit and Guarantees Outstanding
(1992 dollars)

900

800
Loan Guarantees

Dollars in Billions

700

600

500

400
Direct Loans

300

200

100
1970

1975

1980

1985
Year

1990

1995

2000

+

the form of loan guarantees has experienced an explosive growth relative to
that of direct loans.4
Tables 1 and 2 present the various direct loan and guaranteed loan programs
that existed in the 1996 fiscal year. As the tables show, virtually every sector
of the economy is covered by some type of program, and assistance to some
sectors takes the form of both direct loans and guaranteed loans. In this article,
we focus on the kinds of programs associated with investment behavior. Examples of such programs include those targeted to the entrepreneurial community
and students.

2.

THE THEORETICAL MODEL

A sensible model for our purpose must have two key features. First, the model
should display asymmetric information that gives rise to financial frictions so
4 Grants

are not used as much as direct loans and loan guarantees. We do not have timeseries data on the spending of government grants in the United States.

W. Li: Government Loan, Guarantee, and Grant Programs

29

Table 1 Direct Loan Transactions of the Federal Government:
1996 Fiscal Year (Millions of Dollars)
Net Outlays
National defense
Internal affairs
Energy
Natural resources and environment
Agriculture
Commerce and housing credit
Transportation
Community and regional development
Education, training, employment
and social services
Health
Income security
Veteran benefits and services
General government direct loans
Total

Outstandings

1,674
1,036
34
6,183
1,570
47
1,963
9,120

1,384
38,983
34,125
294
15,580
40,897
314
17,739
12,431

25
93
1,442
379

834
2,303
1,188
462

23,566

166,534

Source: The Budget of the United States Government, 1996.

that agents’ wealth affects their investment demand. Second, the model should
also demonstrate that the amount of desired investment (not simply whether to
invest) varies with the cost of borrowing.
Here we describe an economic environment that contains the above features. It is a simple environment with borrowing and lending occurring under
the condition of moral hazard. The main characteristic of this environment is
that some information regarding the return to investment projects is concealed
and is observable to project owners but not to lenders. Because lenders do not
have full information, they cannot determine the state of the projects so they
have to spend real resources to verify borrowers’ reports. The economy studied
here also includes another important characteristic: agents decide whether to
start a new business or remain an employee. Since imperfect information limits
risk-sharing, this self-selection turns out to be correlated with the amount of
assets that agents hold, as well as the quality of their business projects. Therefore, both margins of business activity are captured in the model, namely, the
intensive margin of business investment and the extensive margin of entry.
To introduce some notation, we refer to a two-period economy with a
continuum of agents of measure one. Consumption takes place in both periods,
and we denote them by ci , i = 1, 2. The utility function is assumed to take
the form U(c1 ) + c2 . In the first period, each agent receives some wealth w
and a project that can be operated in the second period. Wealth w has a cumulative distribution function G(w) on the interval [ w, w], where 0 < w < w. The

30

Federal Reserve Bank of Richmond Economic Quarterly

Table 2 Guaranteed Loan Transactions of the Federal Government:
1996 Fiscal Year (Millions of Dollars)
Net Outlays
National defense
Internal affairs
Energy
Natural resources and environment
Agriculture
Commerce and housing credit
Transportation
Community and regional development
Education, training, employment
and social services
Health
Income security
Veteran benefits and services
General government direct loans
Subtotal
less secondary guaranteed loans
Total

Outstandings

276
8,418

441
34,341
691

5,082
181,277
826
839
19,816

12,309
987,420
2,154
2,565
101,874

210
5
28,676
379

3,113
3,867
154,762
462

245,425

1,303,537

−101,540

−497,433

143,885

806,104

Source: The Budget of the United States Government, 1996.

project is indexed by its probability of success p: if a project succeeds, it
produces output f (k), where k is total investment; if it fails, no output will be
produced. Function f (k) is assumed to be increasing in k and concave, i.e.,
f 0 (.) > 0 and f 00 (.) < 0. The project success probability p is characterized
by a cumulative distribution function denoted by Γ(p) with support [ p, p]. The
probability of success p is a measure of business quality.
In the first period, after receiving his endowment of assets and a project, an
agent determines his consumption for this period and his saving for the second
period. He also decides whether he wants to carry out his project. In period
2, the agent, if he is an entrepreneur, decides how much to invest. If the total
amount of investment exceeds his saving, then he needs to borrow. If the agent
is a worker, he draws his income from lending and a fixed income q from
working an outside option in period 2.5 The following timeline describes the
sequence of actions.

5 We could assume that production takes both capital and labor as inputs and that q corresponds to the wage that is endogenously determined. This assumption would further complicate
the analysis here without much gain.

W. Li: Government Loan, Guarantee, and Grant Programs

31

period 1: all agents receive w and learn p
⇓
formulate period 1 consumption
make occupational decision that will affect period 2
⇓
period 2:
z
}|
{
entrepreneurs:
workers:
borrow or lend
lend
⇓
⇓
manage the project
work the outside option
⇓
⇓
repay loan
receive interest payment from deposit
⇓
⇓
consume
consume
The information structure of the economy is as follows. Everything in
the first period is public information: the level of assets, the quality of the
project, and the decision about whether or not to be an entrepreneur. In period
2, however, when production takes place, only those carrying out the project
observe the outcome of the project. An outsider can learn the outcome only
after bearing a verification (auditing) cost. Given that financing a project may
require loans from more than one lender, the optimal financial structure is one
where all lending is transacted by a large financial intermediary who lends to
a large number of borrowers and borrows from a large number of depositors.
Because it has a comparative advantage in doing so, the financial intermediary
monitors the borrowers to economize on verification costs; if there were direct
lending, each of the lenders who lent to an entrepreneur would have to verify
the investment project’s return in the event of default.
Those wishing to borrow attempt to do so by announcing loan contract
terms: the amount of loans borrowed, repayment after production conditional
on borrowers’ report, and when monitoring occurs. If the financial intermediary accepts the terms, it then takes deposits, makes loans, and monitors project
returns as required by the contracts it accepts. We assume perfect competition
in the financial sector. Then, in equilibrium, the financial intermediary will
be perfectly diversified, will earn zero profits, and will have a nonstochastic
return on its portfolio. Therefore, the intermediary need not be monitored by
the depositors.
The two-outcome distribution of returns is a special case of the more general distributions discussed in Townsend (1979) and Gale and Hellwig (1985).
We rule out randomized verification strategies, that is, the financial intermediary
cannot verify the return of an agent’s project with some probability. The optimal
contract in this setting is a debt contract where entrepreneurs pay a fixed amount

32

Federal Reserve Bank of Richmond Economic Quarterly

if the project succeeds and default if the project fails, in which case verification
takes place. We can interpret the act of verification as implying bankruptcy for
two reasons. First, in the more general setup, the optimal contract turns out to
be the standard debt contract under which the return is observed if and only if
the firm is insolvent. Second, real-world bankruptcy does appear to involve a
transfer of information. The cost of bankruptcy can be substantial and is likely
to be a function of the level of the firms’ debt. For simplicity, we assume that
bankruptcy cost takes the form of β + γb, where β corresponds to the fixed
cost, and γ is the per-unit variable cost. The amount of borrowing is denoted
by b. Firms’ total investment k is then the sum of its own internal fund or
savings from first period s and loan borrowing b.
Let x denote the payment by the entrepreneur to the financial intermediary,
and let r be the interest rate the financial intermediary pays to investors. It
follows that the financial intermediary is willing to accept loan contract offers
yielding an expected rate of return of at least r. Borrowers differ in the amount
s of their initial wealth that they save, and their project’s probability of success
p. A loan contract with a borrower (s, p) must satisfy the following constraint,

px = rb + (1 − p)(β + γb),

(1)

if the intermediary is willing to accept it. Investment k is the total of saving s
and loan borrowed b. The loan contract also has to be feasible for the borrower

x ≤ f (k).

(2)

This expression says that the borrower has enough to repay the loan in the
good state.
Borrowers will then maximize their own expected utility by setting investment level k, subject to the constraints just described. Therefore, announced
loan contracts will be selected so that they solve

π(s, p) = max{pf (b + s) − px} = max{pf (b + s) − rb − (1 − p)(β + γb)}, (3)
b

b

where b = k − s, subject to conditions (1) and (2). The function π(s, p) is the
expected second-period consumption of a borrower with saving s and business
project p.
The return v to a representative worker (s, p) is equal to

v(s) = q + rs,

(4)

W. Li: Government Loan, Guarantee, and Grant Programs

33

consisting of the income q plus the gross return rs on savings. In period 1, an
agent chooses his period 1 consumption c1 , saving s, period 2 consumption c2 ,
and occupational decision δ to solve the following problem:6
max U(c1 ) + Ec2 ,

(5)

Ec2 = δπ(s, p) + (1 − δ)v(s),

(6)

s = w − c1 ,

(7)

δ ∈ {0, 1}.

(8)

subject to

Condition (6) says the second-period consumption depends on the agent’s occupation, π(s, p) for an entrepreneur and v(s) for a worker. Condition (7) indicates
that saving is the difference between an agent’s asset endowment and his firstperiod consumption. Condition (8) restricts δ to be a binary variable that takes
a value 1 when the agent chooses to be an entrepreneur in the second period
and 0 when he chooses to be a worker.
Saving in period 1 is a solution to the following first-order condition:
½
π1 (s, p) if π(s, p) > v(s),
(9)
U0 (w − s) =
r
otherwise.
Figures 2 and 3 describe the determination of occupational choice for a
given project and a given endowment of asset. The asset level is measured
on the horizontal axis in Figure 2, the project success probability is measured
6 Another

way of writing an agent’s problem is as follows:
max{Uw , Ue },
δ

where Uw is the utility of being a worker in the second period, and Ue is the utility of being an
entrepreneur in the second period. The occupational decision is denoted by δ; it takes a value of
1 when Uw < Ue and 0 otherwise. Moreover,
Uw = max U(c1 ) + E(c2 ),
c1 ,s,c2

subject to
Ec2 = v(s),
s = w − c1 .
Ue = max U(c1 ) + E(c2 ),
subject to

c1 ,s,c2

Ec2 = π(s, p),
s = w − c1 .

34

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Determination of Occupational Choice - I

0.9

0.8

0.7

Utility

Being an Entrepreneur
0.6

0.5
Being a Worker
0.4

0.3

0.2
1.0

1.1

1.2

1.3

1.4

1.5
1.6
Wealth

1.7

1.8

1.9

2.0

+

on the horizontal axis in Figure 3. The utility of being either an entrepreneur
or a worker is measured in the vertical axis of both figures. Note first that
all entrepreneurs equate the marginal product of investment to the marginal
cost of funds, which includes the monitoring cost associated with lending, i.e.,
pf 0 (k) = r + (1 − p)γ. Workers save additional wealth, so utility rises with
wealth at rate r for workers. Entrepreneurs also save any additional wealth,
and additional saving for this group reduces future borrowing needs, saving
r + (1 − p)γ. This holds as long as saving is less than desired capital stock. If
saving is greater than that, investment is self-financing, and extra wealth will
first increase utility at rate pf 0 (s) (which is less than r + (1 − p)γ), then r.
Thus, there is a cutoff level of wealth, as shown in Figure 2, such that agents
with wealth higher than the cutoff level will become entrepreneurs. It is clear
from the profit function that entrepreneurs’ utility increases with the quality of
their business, while the utility of workers does not vary with their endowed
project. Hence, as shown in Figure 3, there exists a cutoff level of business
quality for each wealth level so that agents with projects above the cutoff level
will become entrepreneurs. Results 1 and 2 summarize the analysis.

W. Li: Government Loan, Guarantee, and Grant Programs

35

Figure 3 Determination of Occupational Choice - II

0.8
0.7
Being an Entrepreneur

0.6

Utility

0.5
0.4
Being a Worker

0.3
0.2
0.1
0.0
1.0

1.1

1.2

1.3

1.4
1.5
1.6
1.7
Success Probability p

1.8

1.9

2.0

+

Result 1. Given a project, there is a threshold asset level such that agents
with assets higher than the threshold will choose to undertake their projects.
Result 2. Given the asset endowment, there is a threshold probability of
success such that agents whose projects have a higher probability of success
become entrepreneurs.
The competitive equilibrium of this economy is defined as a resource allocation for workers and entrepreneurs together with an interest rate for which
two conditions hold. First, agents maximize expected utility by choosing several
decision variables, including their consumption in both periods, their saving in
period 1, their occupational decisions and, in the case of entrepreneurs, their
investment and loan size in period 2. Second, the market for capital clears, i.e.,
Z Z

Z Z

max{s(w, p) − k, 0}δ(w, p)dG(w)dΓ(p)

b(w, p)δ(w, p)dG(w)dΓ(p) =
p

w

Z pZ w

s(w, p)[1 − δ(w, p)]dG(w)dΓ(p),

+
p

w

(10)

36

Federal Reserve Bank of Richmond Economic Quarterly

where δ denotes the occupational choice. The left-hand side of (10) is demand
for loans by entrepreneurs; the first term on the right-hand side is saving by
entrepreneurs, and the second term is saving by workers. Agents’ saving s,
investment k, and occupational decision δ are all functions of their assets w
and their project quality p.
The Case of the First Best without Information Asymmetry
We now briefly analyze the economy without information asymmetry in order
to draw comparisons. Starting with period 2, in the absence of information
asymmetry, the interest rate charged by intermediaries is equal to their cost of
funds. Hence direct lending performs equally as well as financial intermediation, and there will also be no need for financial intermediaries. Agents face
the same interest rate regardless of their asset holdings. The entrepreneurial
decision will be determined solely by the quality of the business project. To
see this, note that the profit function for an entrepreneur with saving s and
success probability p is
π(s, p) = max{pf (s + b) − rb}
b

= pf (k∗ ) − r(k∗ − s),

(11)

where k∗ is the solution to the following first-order condition
pf 0 (k∗ ) = r.
The income for a worker with saving s and project p is
v(s) = rs + q.
It is clear that the difference between π(s, p) and v(s) is independent of s.
Additional saving has the benefit of reducing required borrowing for the entrepreneur, which is worth r per unit in period 2. Rate r is the same as the rate
of return that workers obtain on their savings. Therefore, greater initial wealth
does not make entrepreneurship any more attractive than working.
The key difference between the economy with imperfect information and
the economy examined here is that wealth enters into the decision rules of
agents in the information-constrained economy. Private information reduces
aggregate output in two ways. First, as Result 2 demonstrates, it is not always
true that the most efficient projects are chosen. Some inefficient projects are
carried out simply because the owners have higher internal funds, and some
efficient projects are not activated because the owners have insufficient funds.
Second, there is a social cost associated with monitoring. This cost does not
accrue to any member of the economy and hence is viewed as a deadweight
loss. The discussion of government policies in the credit market in the next
section will be centered around these two dimensions. The first relates to the

W. Li: Government Loan, Guarantee, and Grant Programs

37

extent of business activity in the economy, while the second is a measure of
the transaction costs associated with financial intermediation.

3.

GOVERNMENT CREDIT PROGRAMS

The government finances loans by borrowing from lenders at a competitive,
risk-free interest rate. Correspondingly, it finances subsidies through imposition
of an income tax, which we assume is a lump-sum levy.7 The government has
access to the same information and verification technology as the private financial intermediary, therefore, as shown earlier, government subsidies cannot
be Pareto-improving. However, government subsidies and taxation do have
distributive effects. We will focus on the use of government credit programs
for redistributive purposes and will ask which programs are most efficient in
channeling resources to the desired groups.
Direct Loans
Suppose the government institutes a direct loan program that is available to
a subset of the population, identified by race or location. The targeted group
otherwise has the same characteristics as the population as a whole and is a
fraction µ of the general population. We assume that direct government loans
will bear a below-market interest rate, and we denote the difference between
this interest rate and that of the market rate by ε.8 A lump-sum income tax τ
is levied on all agents in order to finance the subsidy.
We examine the subsidized entrepreneurs first. It is convenient to consider
the situation where the private financial intermediary administers all the loans
and is compensated by the government for the amount of the loan subsidy.
Using the same notation as before, in period 2 the break-even condition for the
financial intermediary becomes
px = (r − ε)b + (1 − p)(β + γb),

(12)

7 There is another potential avenue for the government to finance its loans that is not captured
by the model: the government can issue securities and require private financial intermediaries and
households to hold a certain proportion of these securities. An increase in the number of government subsidies will then increase the amount of government securities that must be held by banks
or by households. This increase in private agents’ holding of government securities can in turn
affect the behavior of households and private intermediaries. For example, the U.S. Farm Credit
System has at least the implicit support of the U.S. government, permitting it to issue bonds at
an interest rate only very slightly above Treasury security yields. Effectively, this support lowers
the opportunity cost of funds to the lender. Interested readers can find related discussion in Fried
(1983).
8 In our setup, it does not matter whether entrepreneurs receive all their loans from the
government at a below-market interest rate or only receive a fraction of their loans at a belowmarket interest rate. That is, the two cases are the same as long as the net subsidy is the same in
both cases.

38

Federal Reserve Bank of Richmond Economic Quarterly

where εb is the direct loan subsidy. The profit function for a subsidized entrepreneur (s, p) is
π s (s, p) = max{pf (b + s) − px}
b

= max{pf (b + s) − (r − ε)b − (1 − p)(β + γb)}.
b

(13)
(14)

An entrepreneur decides loan borrowing b according to
pf 0 (b + s) = r − ε + (1 − p)γ.

(15)

Consider first the partial equilibrium effects of the direct loan program
where the effect of the change of interest rate is not taken into account. Agents
now borrow more and have a lower marginal productivity of capital.9 Given
our monitoring technology, this increases social cost in the sense that additional
resources will be allocated to monitoring. The decrease in marginal productivity
of capital is independent of the success probability of the project p. ³
¶π s
=
Since profits are strictly increasing in the loan subsidy rate ε
¶ε
b > 0), subsidized entrepreneurs will benefit. Moreover, it is the cash-poor
entrepreneurs with good projects who benefit the most. The intuition is clear.
The direct loan subsidy studied here is proportional to the amount of loans
borrowed, and it is precisely those who are either poor or have a good business
who need to borrow the most.
An unsubsidized entrepreneur’s profit function remains the same as equation (3). We denote it by π u (s, p), where the superscript u stands for unsubsidized. A worker’s income also remains the same as equation (4).
The agent’s problem is now
max U(c1 ) + Ec2 ,

(16)

Ec2 = δ[ξπ s (s, p) + (1 − ξ)π u (s, p)] + (1 − δ)v(s),

(17)

s = w − c1 − τ ,

(18)

δ ∈ {0, 1},

(19)

subject to

where ξ is 1 if the agent belongs to the targeted group and 0 if not.

9 We limit our attention to cases where (1−p)γ ≥ ε. If the inequality is not satisfied, external
funds will be more attractive than internal funds, and entrepreneurs will choose to deposit all their
savings with the financial intermediary—an unrealistic situation.

W. Li: Government Loan, Guarantee, and Grant Programs

39

The corresponding first-order condition that solves for saving is as follows:
 s
¶π (s,p)

, subsidized entrepreneurs;


 ¶us
(20)
U0 (w − s − τ ) = ¶π (s,p) , unsubsidized entrepreneurs;

¶s



r,
worker.
The imposition of a lump-sum tax reduces the incentive to save for all
agents in the economy, while the reduction in the marginal productivity of saving (and therefore of capital) further discourages subsidized entrepreneurs from
saving. Taxation and public provision of the subsidy thus crowd out private
saving. This reduction in private saving would further increase the demand for
external funding and hence increase the monitoring cost associated with external finance in the event of failure. Moreover, loan subsidies give the targeted
group an advantage over the nontargeted group: holding everything else the
same, an agent belonging to the targeted group is more likely to become an
entrepreneur. Therefore, some agents in the nontargeted group will be crowded
out of entrepreneurship.
To summarize, the partial equilibrium analysis above indicates that on one
hand a direct loan encourages cash-poor agents with good projects to carry
out their projects. On the other hand, it creates an incentive for subsidized
entrepreneurs to overinvest beyond the desired investment level; a disincentive for all agents, particularly entrepreneurs, to save; and a disincentive for
unsubsidized agents to become entrepreneurs.
The competitive general equilibrium of this economy with government subsidy rate ε is easily defined. It is a resource allocation of workers, entrepreneurs,
an interest rate, and a lump-sum tax rate τ that satisfies three conditions. First,
agents choose their consumption in both period 1 and period 2; their savings
in period 1; their occupational decisions and, in the case of entrepreneurs,
their borrowing in period 2 to maximize the expected discounted utility from
consumption. Second, the market for capital clears. Third, government balances
its budget, i.e.,
Z Z
Z Z
µεb(s, p)δ(s, p)dG(w)dΓ(p) =
τ dG(w)dΓ(p),
(21)
p

w

p

w

the left-hand side represents government expenditure on direct loan subsidies,
and the right-hand side represents government revenue from lump-sum tax.
The general equilibrium effect of direct loans from the government is more
involved. The increase in loan demand and the decrease in private saving will
drive the interest rate up, the increase in interest rate will have offsetting effects on savings and the demand for loans. Therefore, in equilibrium, the above
partial equilibrium results will be lessened. Moreover, fewer unsubsidized entrepreneurs will choose to become entrepreneurs, and those that do will invest
less in response to the increased interest rate, i.e., the government subsidy will

40

Federal Reserve Bank of Richmond Economic Quarterly

crowd out unsubsidized entrepreneurs and their investment. We summarize
these findings in Result 3 and plot them in Figure 4. This figure shows how the
population is divided into workers and entrepreneurs for the benchmark case
and for the case of direct subsidies. In the benchmark model, the cutoff line
for being an entrepreneur is downward sloping. Any agents above the cutoff
line will become entrepreneurs, any below will be workers. Under direct loans,
the cutoff line for the targeted group shifts downward and becomes steeper,
reflecting that cash-poor entrepreneurs with good business prospects benefit
the most from direct loans. For the nontargeted group, the cutoff line shifts
upward, reflecting the crowding effect caused by the advantage that subsidized
entrepreneurs have over the unsubsidized, along with the effect of taxation.
Result 3. Under direct loans from the government, subsidized entrepreneurs will for a given interest rate invest more in their projects, reducing their
marginal return on capital. Entrepreneurs in the targeted group with few assets
and good projects (low w and high p) benefit most from a direct loan subsidy.
Savings for all agents decline, but savings for subsidized entrepreneurs decline
even more. Unsubsidized entrepreneurs have less incentive to carry out their
projects, hence some of them will be crowded out of entrepreneurship. These
results are likely to be weakened in general equilibrium because the interest
rate is higher.
Loan Guarantees
Now consider a government loan guarantee program. Motivated by SBA practices, we assume that the government guarantees a proportion η of each private
loan made by targeted entrepreneurs. In other words, the private lender, in case
of default, is guaranteed η percent of the loan payment. Again to facilitate
comparison, we assume that only a fraction µ of the population are members
of the targeted group.
We consider first the entrepreneurs who receive loan guarantees. Let x
denote loan payment in the event of success. Then the break-even condition
for the financial intermediary is
px + (1 − p)ηx = rb + (1 − p)(β + γb).

(22)

The corresponding profit function for a subsidized entrepreneur becomes
π s (s, p) = max{pf (b + s) − px}
b

= max{pf (b + s) − p[rb + (1 − p)(β + γb)]/[p + (1 − p)η]},
b

where x =

rb+(1−p)(β+γb)
p+(1−p)η

by equation (22).

(23)

Ÿ

W. Li: Government Loan, Guarantee, and Grant Programs

41

Figure 4 Determination of Occupational Choices under Direct Loans

1.1
1.0
0.9

Asset

0.8
Benchmark

0.7

Nontargeted Group

0.6
Targeted Group

0.5
0.4
0.3
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Business Success Probability p
+

Loan borrowing b is determined by the following equation, which requires
the marginal productivity of capital to be equal to the marginal cost
pf 0 (b + s) = r + (1 − p)γ − (1 − p)η
=

p[r + (1 − p)γ]
.
p + (1 − p)η

r + (1 − p)γ
p + (1 − p)η
(24)

Again, we will only study the case where agents weakly prefer internal
funds to external funds even under loan guarantees. As in the case of direct loan
programs, the marginal productivity of capital is smaller than the benchmark
r+(1−p)γ
case. However, unlike direct loan programs, the difference (1 − p)η p+(1−p)η is
a function of both the loan guarantee percentage and the success probability of
the project. In fact, the difference decreases with p, implying that the investment
behavior of agents with riskier projects is more distorted; that is, there is more
overinvestment, compared with the benchmark economy.
To find out how loan guarantees affect entrepreneurs, we can examine the
profit function of a typical subsidized entrepreneur π s (s, p),

42

Federal Reserve Bank of Richmond Economic Quarterly

rb + (1 − p)(β + γb)
¶π s
rb + (1 − p)(β + γb)
= (1 − p)
− (1 − p)2 η
¶η
p + (1 − p)η
[p + (1 − p)η]2
= (1 − p)

rb + (1 − p)(β + γb)
p > 0.
[p + (1 − p)η]2

(25)

The derivative of expected utility with respect to the subsidy rate η is positive, indicating that all subsidized entrepreneurs benefit from the loan guarantee.
To see which subsidized entrepreneurs benefit most, we can examine how the
effect of the subsidy rate varies with saving and project quality.
³ s´
³ s´
¶π
¶π
¶
¶
¶η
¶η ¶b
=
¶s
¶b ¶s
= −(1 − p)
¶

³

¶π s
¶η
¶p

r + (1 − p)γ
p < 0,
[p + (1 − p)η]2

(26)

´

= −{b[r + (1 − p)γ] + (1 − p)β}
−(1 − p)(bγ + β)

p
[p + (1 − p)η]2

p
[p + (1 − p)η]2

+(1 − p){b[r + (1 − p)γ] + (1 − p)β}

η − p(1 − η)
.
[p + (1 − p)η]3

(27)

Intuitively, given that a fixed proportion of a loan is guaranteed in the
event of failure, those who borrow more and/or have a higher probability of
failure will benefit more from loan guarantees. This explains why those with
low savings enjoy relatively more benefits. The effect of loan guarantees on
an agent with a good project is determined by two forces. On the one hand,
having a good project means borrowing more and hence being able to enjoy
the benefits of large loan guarantees in the event of failure; on the other hand, a
good project means a lower probability of failure and therefore less need for a
loan guarantee. The first two terms on the right-hand side of equation
³
´ (27) are
s
¶
π
¶
¶η |
negative, while the sign of the third one is ambiguous. Since
p=0 > 0
¶p
in the neighborhood of p = 0, agents
³ will
´ benefit more if they have a higher
s
¶
π
¶
¶η |
probability of success. Conversely,
p=1 < 0 indicates that, in the neigh¶p
borhood of p = 1, agents with a lower probability of success will benefit more.
These results suggest that a middle range of entrepreneurs benefits the most
from the loan guarantees.

W. Li: Government Loan, Guarantee, and Grant Programs

43

An unsubsidized entrepreneur has the same profit function and investment
behavior as in the benchmark economy. We denote an unsubsidized entrepreneur’s profit function by π u (s, p). The income of workers remains the same.
An agent’s problem in period 1 is defined as follows:
max U(c1 ) + Ec2 ,

(28)

Ec2 = δ[ξπ s (s, p) + (1 − ξ)π u (s, p)] + (1 − δ)v(s),

(29)

subject to

s = w − c1 − τ ,

(30)

δ ∈ {0, 1},

(31)

where ξ takes a value of 1 if the agent belongs to the targeted group and 0
otherwise.
Agents in period 1 will determine their saving for period 2 so that the
marginal gains from saving in the latter period equal the marginal cost of
reduced consumption in the former one. Under loan guarantees, the marginal
gains from saving are lower than in the benchmark economy, thus inducing subsidized entrepreneurs to reduce their savings. For unsubsidized entrepreneurs
and workers, the lump-sum income tax acts to increase their marginal benefits
of consumption in period 1; accordingly, under loan guarantees, unsubsidized
entrepreneurs and workers will increase their consumption and reduce their savings. Moreover, unsubsidized entrepreneurs will receive less from their projects
than their subsidized counterparts and as a result are likely to be crowded out
of entrepreneurship.
The competitive equilibrium can be defined similarly to that of an economy
with direct loans with the government budget constraint being
Z Z
Z Z
µ(1 − p)ηx(s, p)δ(s, p)dG(w)dΓ(p) =
τ dG(w)dΓ(p),
(32)
p

w

p

w

where δ(s, p) is the indicator for occupational decision; it has a value of 1
for entrepreneurs and 0 for workers. The left-hand side is the government’s
expense to guarantee a fraction µ of entrepreneurs a portion η of their loans in
the event of default, and the right-hand side is government revenue from the
lump-sum tax.
In general equilibrium, the increased loan demand and decreased loan
supply raise the equilibrium interest rate, in which case borrowing is more
expensive for entrepreneurs and saving is more attractive for all agents. Consequently, the partial equilibrium results will be lessened. Moreover, unsubsidized
entrepreneurs will reduce their investment in response to the higher interest rate.
We summarize these findings in Result 4. Figure 5 describes the determination
of occupational choices under loan guarantees. Agents above the cutoff lines

44

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Determination of Occupational Choices under Loan Guarantees

1.2

1.0
Nontargeted Group

Asset

0.8

Benchmark

0.6

0.4

0.2
0.0

Targeted Group

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Business Success Probability p
+

become entrepreneurs, and those below become workers. Under loan guarantees, the cutoff line for targeted entrepreneurs shifts downward and becomes
more convex, indicating that entrepreneurs with businesses of mediocre quality benefit the most from the loan guarantees; the cutoff line for nontargeted
entrepreneurs shifts upward, reflecting the crowding out of unsubsidized entrepreneurs.
Result 4. With government loan guarantees, investment by subsidized
entrepreneurs for a given interest rate is higher, and marginal returns to capital
are lower, than in the benchmark economy by an amount that decreases with
p. Poor entrepreneurs with mediocre projects (low w and medium p) benefit
more than others from the loan guarantees. Private savings are lower, especially for entrepreneurs. The increase in the equilibrium interest rate in general
equilibrium will lessen these results.
Grants
Instead of lending directly to entrepreneurs or providing investors with a guarantee on entrepreneurial loans, the government can offer targeted entrepreneurs

W. Li: Government Loan, Guarantee, and Grant Programs

45

a grant of φ, payable at the end of the period and financed by lump-sum income
tax τ . Added to firm profits, the grant would be available for investors. Again,
we assume that the targeted group is a fraction µ of the population and that
they share the same wealth and business quality characteristics as the general
population.
For subsidized entrepreneurs in period 2, using the same notation as before,
the loan payment x for an entrepreneur with saving s, project success probability
p, and borrowing b satisfies the break-even condition
px = rb + (1 − p)(β + γb).

(33)

A subsidized entrepreneur (s, p) chooses b to maximize his profit function
in the second period,
π s (s, p) = max{pf (s + b) + φ − px}
b

= max{pf (s + b) + φ − rb − (1 − p)(β + γb)}.
b

(34)

It is easy to see that the first-order condition that determines firms’ investment is unchanged so that a grant does not alter an entrepreneur’s investment
choices. Additionally, from the first-order condition (9), a grant does not change
an entrepreneur’s saving decision in period 1 either.10 However, it does increase
an agent’s incentive to become an entrepreneur since carrying out a risky ac¶π(s,p)
= 1,
tivity is associated with a higher payoff now. Obviously, since
¶φ
the benefit is fixed for all entrepreneurs regardless of their assets and business
projects.
The problem of an unsubsidized entrepreneur remains the same as in the
benchmark economy. An agent’s problem at period 1 is now
max U(c1 ) + Ec2 ,

(35)

Ec2 = δ(ξπ s (s, p) + (1 − ξ)π u (s, p)) + (1 − δ)v(s),

(36)

subject to

s = w − c1 − τ ,

(37)

δ ∈ {0, 1},

(38)

where δ is 1 if the agent chooses to be an entrepreneur in period 2 and 0
otherwise; ξ takes a value of 1 if the agent belongs to the targeted group and
0 otherwise.
The marginal gain from saving is unaffected by the grant. However, the
marginal cost of saving at period 1 is increased by the imposition of a lump-sum
10 As

with direct loans and loan guarantees, the associated lump-sum income tax has a
distortionary effect on agents’ saving in period 1.

46

Federal Reserve Bank of Richmond Economic Quarterly

tax. Therefore, all agents will reduce their saving. The incentive to consume
more in period 1 is smaller for grants than for those of direct loans and loan
guarantees.
The definition of general equilibrium under grants is similar to the cases of
direct loans and loan guarantees except for the government’s budget constraint
Z Z
Z Z
µφδ(w, p)dG(w)dΓ(p) =
τ dG(w)dΓ(p).
(39)
p

w

p

w

Here the left-hand side is the government’s expense from giving out a fixed
grant φ to targeted entrepreneurs, and the right-hand side is lump-sum tax
revenue.
As with direct loans and loan guarantees, in general equilibrium increased
loan demand and decreased loan supply drive up the interest rate, loan borrowing becomes more expensive and saving more attractive. The partial equilibrium
results discussed will be lessened. These findings are summarized in Result 5.
Figure 6, which depicts how grants affect agents’ occupational choices, shows
the asset-project success probability cutoff line shifting downward for the targeted group and upward for the nontargeted group.
Result 5. With grants, the investment behavior of subsidized entrepreneurs for a given interest rate is unaffected. All entrepreneurs benefit equally
from the subsidies regardless of their asset holdings and project quality. Agents
in period 1 will reduce their saving in response to the imposition of the lumpsum tax. These effects are reduced in general equilibrium due to the increase
in the equilibrium interest rate.
Our analysis, despite its partial equilibrium nature, provides some evidence
on the direction and magnitude of the many channels through which agents are
affected under different loan programs. First, along the investment margin,
both direct loans and loan guarantees create incentives for entrepreneurs to
overinvest (compared with the benchmark economy). The incentive is stronger
for owners of poor projects under loan guarantees. Grants, on the contrary, do
not alter investment behavior.
Second, with respect to risk-shifting, owners of good projects who are less
wealthy benefit the most from direct loans. While poor agents do benefit more
from loan guarantees, those with medium-quality projects benefit the most.
Grants are nondiscriminatory; a fixed amount is assigned to all entrepreneurs.
Third, government subsidies in the form of direct loans and loan guarantees
crowd out the saving of all agents in the economy, especially those of the entrepreneurs. Lump-sum income taxation reduces consumption in period 1 and
hence increases the marginal utility of consumption in period 1. Moreover, it
reduces savings for all agents under all loan programs.
To summarize, grants have the least distortionary effect, direct loans are
capable of targeting efficient projects, and loan guarantees are more likely to

W. Li: Government Loan, Guarantee, and Grant Programs

47

Figure 6 Determination of Occupational Choices under Grants

1.1

1.0

Asset

0.9
Benchmark

0.8

Nontargeted Group

0.7
Targeted Group
0.6

0.5

0.4
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Business Success Probability p
+

attract relatively riskier entrepreneurs. Since direct comparison of the general
equilibrium impact of different government credit programs is not as transparent
as that of partial equilibrium analysis, we now turn to numerical analysis for
some insights.

4.

A NUMERICAL EXAMPLE

This section reiterates the lessons of the previous analysis in general equilibrium
by incorporating the effect of loan programs on the interest rate. These lessons
are conducted by applying a hypothetical numerical example.
Before we launch our numerical analysis, note that all these forms of government subsidies shift loan demand outward, while lump-sum taxation shifts
private loan supply inward so that in the new equilibrium, the interest rate will
go up. This rise in the equilibrium interest rate offsets some of the benefits
created by government subsidies for entrepreneurs, since loans are more expensive now. In contrast, the rise in the interest rate benefits workers who are
disadvantaged by taxation.

48

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 A Numerical Example
Benchmark

Direct Loan

Entrepreneurs
total
targeted
nontargeted

0.2204
0.0441
0.1763

0.2205
0.0463
0.1742

0.2207
0.0467
0.1740

0.22271
0.0472
0.1753

Monitoring cost

0.02066

0.020733

0.02081

0.020733

Total output

0.64543

0.646114

0.64542

0.64548

Cutoff w level
targeted
nontargeted

1.2574
1.2574

1.2433
1.2610

1.2344
1.2613

1.2336
1.2592

Cutoff p level
targeted
nontargeted

0.6379
0.6379

0.6341
0.6392

0.6316
0.6392

0.6307
0.6385

Average w for entrepreneurs

0.9715

0.96635

0.9664

0.9668

Average p for entrepreneurs

0.7895

0.78986

0.7896

0.7876

1
2
3
4
5
6

Guarantee

Grant

most overall entrepreneurial activity
most entrepreneurial activity within targeted group
least monitoring cost
most total output
least average wealth for entrepreneurs
highest average business quality for entrepreneurs

In our numerical example, the utility function is chosen to be of log form
c1−2.5

1
+ 3c2 .
in the first period, and linear in the second period, i.e., U(w, p) = 1−2.5
The wealth variable w is a random draw from a uniform distribution over
the interval [0.2, 1.6], in which the richest person with wealth 1.6 is 8 times
richer than the poorest person having wealth 0.2. The success probability p
of an agent’s endowed project follows a uniform distribution over the domain
[0.3, 0.85]. The production function takes the form 1.7k0.67 . The fixed monitoring cost β is set to be 0.1, and the unit cost γ is 0.4. The wage that workers
get from the outside option q is 0.4.
We fix the lump-sum tax to be 0.001 per person; the fraction of agents who
are eligible for subsidies µ is 0.2. Then we study the different loan programs
whose rates—ε for direct loans, η for loan guarantees, and φ for grants—are
chosen so that the government balances its budget in equilibrium. Table 3
reports the results.
The results are consistent with our analysis in the previous section. One
thing common with all three loan programs is that agents in the targeted
group are helped at the cost of the agents in the nontargeted group. Though

W. Li: Government Loan, Guarantee, and Grant Programs

49

entrepreneurial activity increases under all loan programs in the targeted group,
it declines in the nontargeted group. The threshold levels of both wealth and
project quality increase for the nontargeted agents.
When comparing direct loans and loan guarantee programs, we find that
loan guarantees are better at promoting entrepreneurship at the cost of lower
average business quality and higher bankruptcy cost. The reason is straightforward. As shown in Section 3, direct loan programs benefit poor agents with
good projects the most. So these agents tend to borrow more and therefore
require most of the subsidies. Under loan guarantees, however, entrepreneurs
with few assets and mediocre projects benefit the most. The resulting benefits
are somewhat more evenly distributed. For the same reason, under direct loans
the average wealth of entrepreneurs is lower and the average quality of their
projects is higher than under loan guarantees. Since entrepreneurs with low
quality projects are more likely to bankrupt, the bankruptcy cost is higher
under loan guarantees. These results survive different parameter specifications
in our experiments.
Another interesting result is that grants seem to outperform loan guarantees
in promoting entrepreneurship at lower monitoring cost. However, grants induce
the lowest average business quality among all the programs and do not seem
to help the poor. This has to do with the nondiscriminatory nature of grants.

5.

CONCLUSION

Are loan guarantees the best way to channel assistance to targeted classes
of borrowers? Our analysis of a credit market with asymmetric information
indicates that grants are most effective at promoting entrepreneurship. Loan
guarantees attract relatively riskier businesses with few assets. Direct loans
do best at targeting cash-poor borrowers with good projects. Subsidized entrepreneurs overinvest under direct loans and loan guarantees.
All of the programs, especially direct loan and loan guarantee programs,
discourage private saving. So why are loan guarantees so popular? Although
there is no clear answer, it may be that differences in government budgetary
accounting allow guarantees to be passed easily since loan guarantees often
do not appear in the budget until a payment is made. Webb (1991) provides
an excellent review and an estimate of the unfunded liabilities of the U.S.
government budget. Another possibility, as suggested by the model, is that the
benefits of guarantees spread more evenly over a broad set of agents than do
the benefits of direct loan subsidization. This more equitable distribution of
benefits perhaps appeals to the public’s conception of fairness and therefore
can help generate more political support for guarantees.

50

Federal Reserve Bank of Richmond Economic Quarterly

REFERENCES
Blanchflower, David G., and Andrew J. Oswald. “What Makes an Entrepreneur?” Journal of Labor Economics, vol. 16 (January 1998), pp.
26–60.
Bond, Philip, and Robert Townsend. “Formal and Informal Financing in
a Chicago Ethnic Neighborhood,” Federal Reserve Bank of Chicago
Economic Perspectives, vol. 20 (July–August 1996), pp. 3–27.
Bureau of National Affairs, Inc. “Senate Small Business Panel Approves Hikes
in Popular Small Business 7(a) Loan Program,” BNA’s Banking Report,
vol. 69 (July 7, 1997), p. 17.
de Meza, David, and David C. Webb. “Too Much Investment: A Problem
of Asymmetric Information,” Quarterly Journal of Economics, vol. 102
(May 1987), pp. 281–92.
Evans, David S., and Boyan Jovanovic. “An Estimated Model of Entrepreneurial Choice under Liquidity Constraints,” Journal of Political Economy,
vol. 97 (August 1989), pp. 808–27.
Evans, David S., and Linda S. Leighton. “Some Empirical Aspects of
Entrepreneurship,” American Economic Review, vol. 79 (June 1989), pp.
519–35.
Fried, Joel. “Government Loan and Guarantee Programs,” Federal Reserve
Bank of St. Louis Review, vol. 65 (December 1983), pp. 22–30.
Gale, Douglas, and Martin Hellwig. “Incentive-Compatible Debt Contracts:
The One-Period Problem,” The Review of Economic Studies, vol. 52
(October 1985), pp. 647–64.
Gale, William G. “Economic Effects of Federal Credit Programs,” American
Economic Review, vol. 81 (March 1991), pp. 133–52.
Haynes, George W. “Credit Access for High-Risk Borrowers in Financially
Concentrated Markets: Do SBA Loan Guarantees Help?” Small Business
Economics, vol. 8 (December 1996), pp. 449–61.
Holtz-Eakin, David, Douglas Joulfaian, and Harvey S. Rosen. “Entrepreneurial
Decisions and Liquidity Constraints,” Rand Journal of Economics, vol. 25
(Summer 1994), pp. 334– 47.
Innes, Robert. “Investment and Governmental Intervention in Credit Markets
When There Is Asymmetric Information,” Journal of Public Economics,
vol. 46 (December 1991), pp. 347–81.
Lacker, Jeffrey M. “Does Adverse Selection Justify Government Intervention in
Loan Markets?” Federal Reserve Bank of Richmond Economic Quarterly,
vol. 80 (Winter 1994), pp. 61–95.

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51

Office of Management and Budget. “Analytical Perspectives,” Budget of The
United States Government, Fiscal Year 1996. Washington: Government
Printing Office, 1996.
Townsend, Robert M. “Optimal Contracts and Competitive Markets with
Costly State Verification,” Journal of Economic Theory, vol. 21 (October
1979), pp. 265–93.
Webb, Roy H. “The Stealth Budget: Unfunded Liabilities of the Federal
Government,” Federal Reserve Bank of Richmond Economic Review, vol.
77 (May/June 1991), pp. 23–33.

Fisher’s Equation
and the Inflation Risk
Premium in a Simple
Endowment Economy
Pierre-Daniel G. Sarte

O

ne of the more important challenges facing policymakers is that of
assessing inflation expectations. Goodfriend (1997) points out that
one can interpret the meaning of a given interest rate policy action
primarily in terms of its impact on the real rate of interest. However, evaluating
this impact requires not only that one understands the various links between
the nominal rate and expected inflation but also that one can quantify these
relationships.
To find an approximate measure of expected inflation, one often turns to the
behavior of long bond rates. Two key ideas explain why this approach might be
appropriate. First, Fisher’s theory holds that the real rate of interest is just the
difference between the nominal rate of interest and the public’s expected rate
of inflation. Second, the long-term real rate is generally thought to exhibit very
little variation.1 Alternatively, and still based on Fisher’s theory, one might use
the yield spread between the ten-year Treasury note and its inflation-indexed
counterpart as an estimate of expected inflation. In January 1997, the U.S.
Treasury indeed began issuing ten-year inflation-indexed bonds.
While economic analysts typically attempt to capture inflation expectations
using Fisher’s equation, this method has its flaws. When inflation is stochastic,
Fisher’s relation may not actually hold. Barro (1976), Benninga and Protopapadakis (1983), as well as Cox, Ingersoll, and Ross (1985), show that the
decomposition of the nominal rate into a real rate and expected inflation should
The opinions expressed herein are the author’s and do not represent those of the Federal
Reserve Bank of Richmond or the Federal Reserve System. For helpful suggestions and
comments, I would like to thank Mike Dotsey, Tom Humphrey, Yash Mehra, and Alex
Wolman. Any remaining errors are, of course, my own.
1 See

Simon (1990).

Federal Reserve Bank of Richmond Economic Quarterly Volume 84/4 Fall 1998

53

54

Federal Reserve Bank of Richmond Economic Quarterly

include an additional component excluded from Fisher’s equation: the inflation
risk premium. This premium reflects the outcome of random movements in
inflation that effectively cause nominal bonds to be risky assets relative to
inflation-indexed bonds. As we shall see in this article, the sign of the premium may be positive or negative, depending on how unexpected movements
in inflation co-vary with surprises in consumption growth.
Another reason the Fisher equation may not hold is that when one links
the nominal rate to the real rate and expected inflation, one must consider the
nonlinearity inherent in inflation when calculating expectations. Specifically,
inflation is a ratio of prices. We shall see that this nonlinearity works through
the variance of inflation surprises.
Since it is evident that Fisher’s equation does not work in all situations,
why should one consider the equation useful? (Note that if both the inflation
risk premium and the variance of inflation surprises are negligible, then Fisher’s
equation holds precisely.) This article answers the question by building on earlier work by Labadie (1989, 1994). In particular, the analysis below relies upon
three key building blocks. First, to study the effect of inflation risk on nominal
rates, we formally incorporate uncertainty as part of the environment surrounding households’ optimal bond purchasing decisions. Second, we assume that a
bivariate vector autoregression (VAR) in the logs of consumption growth and
inflation drives the model. This assumption makes it possible to work out exact
analytical solutions for bond yields and expected inflation. Finally, we estimate
the driving process empirically by using U.S. consumption-growth data to calibrate the model’s analytical solutions. In contrast to Labadie (1989), we are
able to derive solutions consistent with a general-order VAR process instead of
a VAR(1). This allows us to better capture the joint time-series properties of
consumption growth and inflation. Moreover, whereas Labadie’s work focuses
on the equity premium, we will concentrate mainly on the model’s quantitative
implications for the inflation risk premium.
Two important conclusions emerge from the analysis. One is that the
model’s quantitative estimates of the inflation risk premium are insignificant.
This result occurs primarily because little covariation exists between shocks to
consumption growth and unexpected movements in inflation in U.S. data. In
other words, since inflation surprises are as likely to occur whether consumption growth is high or low, there is no reason why the inflation risk premium
should be substantially positive or negative. This notion is unrelated to the fact
that the equity premium tends to be very small in consumption-based asset
pricing models. We will show that adopting a pricing kernel that helps explain
the equity premium does not necessarily change the size of the inflation risk
premium in any meaningful way. The implication is that, in practice, Fisher’s
equation may be a reasonable approximation even when inflation is stochastic.
The other important conclusion (for the sample period covering 1955
to 1996) is that the model’s historical estimates of the yield on a one-year

P.-D. G. Sarte: Inflation Risk Premium

55

nominal bond match the actual yield on one-year Treasury notes relatively
well. However, the model’s estimates of the one-year nominal rate perform
very poorly during the late 1970s. The model’s inability to track the nominal
rate during that period may reflect the unusual tightening by the Federal Reserve
(the Fed) in an effort to bring down inflation at that time. Our benchmark model
suggests a consumption-based real rate whose standard deviation is around 1
percent. Surprisingly, this is more than half the standard deviation of the ex
post real rate despite the fact that consumption growth is relatively smooth.
Using a different methodology, we find additional supporting evidence in favor
of Fama (1990), who suggests that expected inflation and the real rate move in
opposite directions. Finally, our model indicates that it is difficult to determine
whether expected inflation is more or less volatile than the real rate at short
horizons. Although conventional wisdom suggests that the real rate varies more
than expected inflation in the short run, we find that the choice of preference
specification is crucial for this result.
This article is organized as follows. Section 1 presents the basic framework used to price nominal and inflation-indexed bonds. Section 2 describes
the joint driving process linking consumption growth and inflation. Sections 3
and 4 present the results which obtain under different preference specifications.
Finally, Section 5 offers some concluding remarks.

1.

PRICING NOMINAL AND
INFLATION-INDEXED BONDS

The economy is populated by a continuum of infinitely lived households. These
households are identical in terms of their preferences and endowments. The
per capita endowment is nonstorable, exogenous, and stochastic. The typical
household’s wealth consists of currency, one-period inflation-indexed and oneperiod nominal discount bonds. Thus, an indexed bond purchased at time t pays
one unit of the endowment good with certainty at time t + 1. As in Labadie
(1989, 1994), this instrument provides a benchmark that helps isolate real from
inflationary effects. Contrary to the indexed bond, the nominal bond is subject
to inflation risk. That is, a nominal bond purchased at date t pays one unit of
currency, say dollars, at date t + 1.
Each household maximizes its lifetime utility over an infinite horizon. The
timing of trade follows that of the cash-in-advance economy described in Lucas
(1982). Specifically, at the beginning of each period and before any trading takes
place, a stochastic monetary transfer, νt Mt−1 , and a real endowment shock, yt ,
are realized and observed publicly. After receiving the money transfer, as well
as any payoffs on maturing bonds, the representative household decides on how
to allocate its nominal wealth between money balances, Mtd , indexed bonds,
zt , and nominal discount bonds, zNt . Once the asset market has closed, the

56

Federal Reserve Bank of Richmond Economic Quarterly

household uses its money balances acquired at the beginning of the period
Mtd to finance its consumption purchases pt ct , where pt is the price level at
date t. The household then receives its nominal endowment income pt yt , which
it cannot spend until the subsequent period. To summarize, the representative
household solves
max U = Et

∞
X

β s−t u(cs ), 0 < β < 1,

(1)

s=t

subject to the constraints
xt
Md
pt−1
Mt−1 + νt Mt−1
zt−1
pt−1
ct−1 + qt zt + zNt + t =
yt−1 +
+ zt−1 +
, (2)
pt
pt
pt
pt
pt
pt
and
ct ≤

Mtd
.
pt

(3)

We denote by qt and xt the real price of a one-period indexed bond and the price
of a one-period nominal bond, respectively. Et is the conditional expectations
operator where the time t information set includes all variables dated t and
earlier.
Appendix A contains the first-order conditions associated with the above
problem. These optimality conditions yield the following Euler equation,
u0 (ct ) = βEt (1 + rt )u0 (ct+1 ),

(4)

where we have defined 1/qt as (1 + rt ). Equation (4) states that in choosing how
much to consume versus how much to save in the form of an indexed bond,
the representative household explicitly compares marginal benefit and marginal
cost. The marginal benefit, in utility terms, of consuming one additional unit of
the endowment good today is given by u0 (ct ). Alternatively, the household could
save that additional unit and use it to purchase an indexed bond that would
yield (1 + rt ) with certainty in the following period. Therefore the right-hand
side of equation (4) captures the marginal cost of consuming one additional
unit of the endowment good today in utility terms. As equation (4) indicates,
the optimal consumption/savings allocation naturally equates marginal benefit
and marginal cost.
Now, in this setup, the representative household also has the option of
saving through a nominal pure discount bond. Optimality implies that
u0 (ct+1 )
u0 (ct )
= βEt (1 + rtN )
,
pt
pt+1

(5)

where (1 + rtN ) is defined as 1/xt . Analogous to the situation we have just described, the marginal benefit of consuming one additional dollar’s worth of the
endowment good today, where one dollar is worth 1/pt units of the endowment

P.-D. G. Sarte: Inflation Risk Premium

57

good, is u0 (ct )/pt . By instead saving this additional dollar in a nominal bond,
the representative household would reap (1 + rtN )/pt+1 units of the endowment
good next period. The right-hand side of equation (5), therefore, represents
the marginal cost of consuming an additional dollar’s worth of the endowment
good in the current period. As in equation (4), the optimal consumption/savings
allocation still dictates equating marginal benefit to marginal cost.
Since equations (4) and (5) simply show different methods of how to best
allocate income towards consumption and savings, one might naturally expect
a precise link to emerge between the real rate and the nominal rate. Using
equation (5) yields
u0 (ct+1 ) pt
1
=
βE
,
t
u0 (ct ) pt+1
1 + rtN
which may be rewritten2 as
µ
¶ µ
¶
µ 0
¶
1
pt
u (ct+1 ) pt
1
,
=
E
+
βcov
.
t
t
1 + rt
pt+1
u0 (ct ) pt+1
1 + rtN

(6)

Note that if inflation is deterministic, then the covariance term on the righthand side of equation (6) disappears and the above equation reduces to Fisher’s
relation,
µ
¶
pt+1
.
(7)
(1 + rtN ) = (1 + rt )
pt
To understand the nature of the differences between the modified Fisher equation and equation (7), let us first examine the covariance term in (6). This term
is known as the inflation risk premium and already emerges in Benninga and
Protopapadakis (1983) or Cox, Ingersoll, and Ross (1985). Recall that saving
one additional dollar in period t yields (1 + rtN )/pt+1 units of the endowment
good in period t + 1. However, the price level next period, pt+1 , is unknown at
date t. Inflation, therefore, makes the nominal discount bond a risky asset; the
premium in effect alters the nominal rate to account for this additional risk.
To make matters more concrete, let us temporarily suppose that momentary utility is given by the Constant Relative Risk Aversion (CRRA) function
u(c) = c1−γ − 1/1 − γ, γ > 0. Consequently, the ratio of marginal utilities
in equation (6) is decreasing in consumption growth and given by (ct+1 /ct )−γ .
Therefore, when the conditional covariance term is negative, inflation is likely
to be high when consumption growth is low. In other words, the return on the
nominal bond is adversely affected by inflation precisely when the household
suffers from low consumption growth. Now observe that relative to a world
without inflation uncertainty, a negative conditional covariance raises the nominal rate. We may, therefore, interpret this higher nominal yield as compensating
2 Here

E(x)E(y).

we use the fact that for any two random variables x and y, cov(x, y) = E(xy) −

58

Federal Reserve Bank of Richmond Economic Quarterly

the household for the additional inflation risk associated with the nominal bond.
The reverse is true when the conditional covariance term is positive.
Another reason equation (6) does not correspond to Fisher’s relation when
inflation is stochastic, even if the conditional covariance term were zero, has
to do with Jensen’s Inequality. In particular, Et (pt+1 /pt ) is generally not equal
to 1/Et (pt /pt+1 ). As one might expect, we shall see below that the difference
between Et (pt+1 /pt ) and 1/Et (pt /pt+1 ) rises with the volatility of inflation surprises. In a world without such surprises, the conditional expectations operator
is irrelevant, so this difference would vanish.
To close the model, we simply note that in equilibrium, ct = yt , while
Mtd = Mt . In addition, since households are identical, indexed and nominal
bonds are in zero net supply so that zt = zNt = 0. In what follows, we assume
for simplicity’s sake that νt ≥ β so that the cash-in-advance constraint always
binds.

2.

THE ENDOWMENT AND INFLATION PROCESSES

We now define a driving process for this economy. Let endowment growth and
the inflation rate be denoted by yt+1 /yt = ζt+1 and pt+1 /pt = φt+1, respectively.
We assume that the joint time-series behavior of ln ζt+1 and ln φt+1 can be
described by a covariance stationary bivariate VAR (p).3 The law of motion
for the endowment process is
ln ζt+1 = δζ0 +

p
X

δζζ, j ln ζt−j +

j=0

p
X

δζφ, j ln φt−j + εζ,t+1 .

(8)

j=0

Similarly, the inflation rate follows a process that can be described by
ln φt+1 = δφ0 +

p
X
j=0

δφζ, j ln ζt−j +

p
X

δφφ, j ln φt−j + εφ,t+1 .

(9)

j=0

Shocks to endowment growth and inflation, (εζ,t , εφ,t ), are assumed to be
jointly distributed normal random variables such that E(εζ,t ) = E(εφ,t ) = 0,
var(εζ,t ) = σζ2 , var(εφ,t ) = σφ2 , and cov(εζ,t , εφ,t ) = σζφ . Moreover, as in
Labadie (1989), the shocks satisfy E(εζ,t , εφ,s ) = E(εζ,s , εφ,t ) = 0, for s 6= t.

3.

RESULTS WITH CRRA UTILITY

Analytical Solutions
In this section, we assume that momentary utility is of the CRRA form.
Our main focus will be to derive and interpret solutions for bond prices or,
3 Since

the cash-in-advance constraint is assumed to bind, this bivariate system implicitly
dictates the behavior of money growth.

P.-D. G. Sarte: Inflation Risk Premium

59

alternatively, rates of return on the indexed and nominal discount bonds. The
goal is to assess to what degree Fisher’s equation approximates its generalized
version in (6) in a calibrated consumption-based asset pricing model. With
CRRA utility, equation (4) becomes
µ
¶
ct+1 −γ
,
(10)
qt = βEt
ct
which may also be written as
ln qt = ln β + ln Et ζt+1 −γ .
Using the properties of log-normal random variables described in Appendix B,
as well as those of the driving process in Section 2, it immediately follows that
ln qt = ln β − γδζ0 +

p
p
X
X
γ 2 σζ2
−γ
δζζ, j ln ζt−j − γ
δζφ, j ln φt−j .
2
j=0
j=0

The real price of the one-period inflation-indexed bond can therefore be expressed as
"
#
γ 2 σζ2
) Qt ,
(11)
qt = β exp(−γδζ0 +
2
−γδ

−γδ

where Qt = Πpj=0 ζt−j ζζ, j Πpj=0 φt−j ζφ, j . Equation (11) suggests that the real
rate, 1/qt , is not only a function of past endowment growth but also of past
inflation rates. This result arises since, by equation (8), past inflation rates help
forecast endowment growth next period, ζt+1 . In addition, observe that greater
volatility in unexpected endowment growth movements, as captured by σζ2 ,
raises qt and, therefore, lowers the real rate. Put another way, a more risky
endowment growth process serves to lower the real rate of return. This latter
effect, however, is only present to the degree that households care about risk
so that γ > 0. When households are risk-neutral and γ = 0, qt is independent
of σζ2 .
Turning our attention to the behavior of the nominal rate, equation (5) can
be rewritten as
µ
¶
ct+1 −γ pt
(12)
xt = βEt
ct
pt+1
so that ln xt = ln β + ln Et (ct+1 /ct )−γ (pt /pt+1 ). Again, using the properties of
log-normal random variables yields
µ
¶
ct+1 −γ pt
−γ
= Et ln ζt+1
+ Et ln φ−1
(13)
ln Et
t+1 +
ct
pt+1
¡
¢
1
γ2
vart ln ζt+1 + vart ln φt+1 + γcovt ln ζt+1 , ln φt+1 .
2
2

60

Federal Reserve Bank of Richmond Economic Quarterly

As before, we can use the properties of the driving process to obtain
"
#
σφ2
γ 2 σζ2
+
+ γσζφ ) Xt ,
xt = β exp(−γδζ0 − δφ0 +
2
2
−(γδ

+δ

)

−(γδ

+δ

(14)

)

where Xt = Πpj=0 ζt−j ζζ, j φζ, j Πpj=0 φt−j ζφ, j φφ, j . As expected, the behavior of the nominal rate depends on the time-series characteristics of both
endowment growth and the inflation rate. In particular, observe that the greater
the unconditional variance of inflation surprises, the lower the nominal rate,
since 1 + rtN = 1/xt . Furthermore, a larger negative covariance between unexpected movements in endowment growth and inflation surprises raises the
nominal rate (so long as γ > 0). As mentioned earlier, this result reflects that
when σζφ < 0, high inflation shocks tend to occur when endowment growth is
unexpectedly low. In this case, the household, therefore, requires a higher yield
on nominal bonds to account for the inflation risk. Alternatively, we can see
this notion by tracing the effect of the covariance between endowment growth
shocks and inflation shocks on the inflation risk premium directly. By using
equation (6) and solving for ln 1/(1 + rt )Et (pt /pt+1 ) as we have done above, one
sees that
õ

covt

ct+1
ct
"

¶−γ
Ã

pt
,
pt+1

!

=

!#
σφ2
γ 2 σζ2
¤
£
+
exp −γδζ0 − δφ0 +
exp(γσζφ ) − 1 Xt . (15)
2
2
¡
¢
Hence, it is now clear that covt (ct+1 /ct )−γ , pt /pt+1 ≤ 0 whenever σζφ ≤ 0,
regardless of the other terms in equation (15). As suggested by equation (6),
this effect raises the nominal rate over and above that implied by movements
in the real rate and expected inflation alone. Finally, it should be intuitive that
when households are risk-neutral and γ = 0, the inflation risk premium is
identically zero irrespective of σζφ .
Earlier in the analysis, we hinted that even if the inflation risk premium
were zero at all dates, equation (6) would not necessarily reduce to the Fisher
equation when inflation is stochastic. We argued that, generally, Et ( pt+1 /pt ) 6=
1/Et ( pt /pt+1 ) and that this difference would rise with the volatility of inflation
surprises. This result is shown formally in Appendix C and, in particular,
"
Ã
!
à 2 !#
−σφ2
σφ
1
pt+1
−
− exp
P−1
(16)
= exp(δφ0 ) exp
Et
t ,
pt
Et ( pt /pt+1 )
2
2
δ

δ

φζ, j p
φφ, j
Πj=0 φt−j
. Figure 1 illustrates how the right-hand side
where Pt = Πpj=0 ζt−j
of this last equation varies as a function of φ2φ . Since the result in equation (16)
is essentially driven by Jensen’s Inequality, the greater the variance of inflation

P.-D. G. Sarte: Inflation Risk Premium

61

Figure 1 Effect of Jensen’s Inequality on the Simple Fisher Equation
4.5
4.0
3.5

exp(σ 2φ /2)

3.0
2.5
2.0
1.5

E t (p t+1 /p t )-1/E t (p t /p t+1 )

1.0
exp(-σ 2φ /2)

0.5
0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

σ 2φ

+

shocks, the more the convexity inherent in the price ratio matters. In a world
without inflation surprises, σφ2 = 0, and the right-hand side of (16) vanishes.
Note that in the latter case, inflation is not necessarily constant but is deterministic and described by equation (9), without the εφ,t+1 shock. Therefore the
conditional expectations operator in (16) becomes, in some sense, irrelevant.
Thus far, we have been able to show that the discrepancy between the
modified Fisher equation in (6) and the Fisher equation in (7) ultimately boils
down to two crucial aspects of the environment; namely, the covariance between unexpected movements in endowment growth and inflation surprises, as
well as the unconditional volatility of inflation surprises. However, whether this
difference is quantitatively significant remains to be seen.
Quantitative Implications
To address the quantitative features of the model just presented, we must first
tackle the issue of calibration. As a benchmark case we first fix the discount
rate, β, to 0.996 and set the risk-aversion parameter, γ, to 0.75. The value of
the discount rate is chosen so that, in the benchmark scenario, the mean of
the model-implied ex post real rate matches its counterpart in the data at 2.32
percent. Note that since U.S. real consumption has generally been growing at
about 2 percent over the sample, our discount factor is scaled up by a factor of

62

Federal Reserve Bank of Richmond Economic Quarterly

(1.02)γ relative to one that would be appropriate for stationary data. We then
examine how the results vary with changes in the risk-aversion parameter. The
only other necessary parameters of the model relate to the exogenous driving
process. To this end, we estimate the bivariate VAR described by equations (8)
and (9) using the following data:
• Consumption refers to per capita annual U.S. consumption of nondurable goods, durable goods, and services, spanning 1955 to 1996
and expressed in 1992 dollars.
• Price level refers to the ratio of nominal consumption to real
consumption.
Note that we are using annual data in order to avoid estimating equations
(8) and (9) with variables averaged over extended periods. Using data averaged over a ten-year period, for instance, would result in a substantial loss of
information. A VAR of order 4 is estimated with resulting R2 s of 0.75 and
0.90 for equations (8) and (9), respectively. The point estimates for σζφ and
σφ2 are 3.20 × 10−5 and 2.43 × 10−4 . It directly follows that the inflation
risk premium generated by this model is all but negligible. Observe that this
result has little to do with the notion that the equity premium is typically
small in this type of framework. Instead, it is driven almost exclusively by the
fact that inflation surprises move in a way unrelated to unexpected changes
in consumption growth. (We return to this point more fully in the next section.) Moreover, consistent with the high R2 associated with the estimation of
equation (9), the volatility of shocks to inflation also appears to be very small.
Therefore, by equation (16), we may think of Et (pt+1 /pt ) as essentially equal to
1/Et (pt /pt+1 ).
Figure 2 presents the historical estimates generated from the model for the
period 1955 to 1996 using the benchmark parameters. We chose this time span
so that we could directly compare the model-implied nominal rate with the
actual yield on one-year Treasury notes.
As we can see from Figure 2, panel c, except for the late 1970s and early
1980s, the model performs relatively well in matching the actual nominal rate.
The model’s inability to capture the sharp rise in nominal rates in the late 1970s
can perhaps best be explained by the unusually aggressive disinflationary policy
adopted by the Fed at that time. In response to strong inflationary pressures in
the fall of 1980, Goodfriend (1993, pp. 11–12) notes that “the Fed began an
unprecedented aggressive tightening. . . . Thus, the run-up of the funds rate
to its 19 percent peak in January 1981 marked a deliberate return to the high
interest rate policy.” It may be, therefore, that the assumptions concerning the
driving process described by equations (8) and (9) are not entirely justified. In
particular, a specification for the driving process that included the possibility
of a regime switch around 1980 might have been more appropriate.

P.-D. G. Sarte: Inflation Risk Premium

63

Figure 2 Simulated Results with CRRA Utility

a. Ex Ante Real Rate

Percent

0.040

0.025

0.010

-0.005

1955

1965

1975
Year

1985

1995

b. Expected Inflation

Percent

0.08
0.06
0.04
0.02
0.00

1955

1965

1975
Year

1985

1995

c. Nominal Interest Rate

Percent

0.14

Model
Data

0.10

0.06

0.02

1955

1965

1975
Year

1985

1995

d. Inflation Risk Premium

Percent

0.0000268

0.0000256

0.0000244

+

1955

1965

1975
Year

1985

1995

64

Federal Reserve Bank of Richmond Economic Quarterly

As shown in panel d, the inflation risk premium is insignificant over the
entire period and since the variance of inflation surprises is small, the modified
Fisher equation collapses almost exactly to the Fisher equation. To be specific,
the gap that separates equation (6) from equation (7) is never more than 3 basis
points over the entire period. Thus, while the Fisher equation does not hold
in theory when inflation is stochastic, it may very well serve as a reasonable
approximation in practice.
Panel a of Figure 2 also shows that the ex ante real rate can be quite
volatile. Observe in particular the severe real rate drops that occur in 1975
and 1980. In the context of this model, recall that the real rate in equation
(11) is in part a function of recent consumption growth. According to the
driving process described in Section 2, past consumption growth helps predict
future consumption growth in equation (10). Consequently, the sharp fall in real
rates in 1975 and 1980 correspond respectively to the two recessions typically
associated with the severe rise in oil prices and the credit controls imposed
by the Carter Administration. Over the period under consideration, the oneyear real rate fluctuates between 0.25 percent and 3.7 percent. This range is
substantially greater than the 75-basis-point range found by Ireland (1996) for
the ten-year real rate. Our findings therefore lend support to the stylized view
that as maturity increases, variations in the nominal rate are more likely due to
variations in expected inflation than variations in the real rate. Table 1 presents
some key sample statistics concerning the time-series properties of the historical
estimates generated by the model as we vary the risk-aversion parameter.
As suggested by the estimates in Table 1, the standard deviation of the real
rate is about 1 percent in the benchmark case. This rate is more than half the
standard deviation of the ex post real rate of 1.80 percent over the same period.
Therefore, in spite of relatively smooth consumption growth, this framework
generates a real rate with considerable volatility.
In Table 1, we also note that both the mean and the standard deviation of the
real rate increase sharply with the risk-aversion parameter. This result emerges
because a rise in the degree of risk aversion implies a fall in the elasticity of
intertemporal substitution in consumption. Since the representative household
is less willing to smooth consumption across periods, it generally requires a
higher return on bonds in order to save. More importantly, this feature of the
model is precisely that which makes it difficult to match the equity premium.
As observed in Abel (1990), although the return on stocks typically rises with
γ, the fact that the return on Treasury notes also rises with γ essentially leaves
the difference between the stock return and the bond return unchanged, even
for large increases in risk aversion. Ideally, to have a better chance of matching
the equity premium without requiring extreme values of γ, one would like a
framework in which increases in the degree of risk aversion do not necessarily
yield increases in the real rate.

P.-D. G. Sarte: Inflation Risk Premium

65

Table 1

γ = 0.75

γ = 1.75

γ=6

Ex Ante Real Rate:
rt

Expected Inflation:
Et (pt+1 /pt )

Nominal Rate:
rtN

mean: 2.14

mean: 4.03

mean: 6.23

std: 1.00

std: 2.43

std: 2.39

corr(rt , Et (pt+1 /pt )): −0.29

E (p
/p )
var( t t+1 t ):
rt

mean: 4.51

mean: 4.03

mean: 9.90

std: 2.37

std: 2.43

std: 3.01

corr(rt , Et (pt+1 /pt )): −0.29

E (p
/p )
var( t t+1 t ):
rt

mean: 15.10

mean: 4.03

mean: 19.64

std: 9.02

std: 2.43

std: 8.98

corr(rt , Et (pt+1 /pt )): −0.28

E (p
/p )
var( t t+1 t ):
rt

5.99

1.03

0.07

Finally, because the volatility of the real rate depends so crucially on γ in
the above experiment, it is difficult to say whether the volatility of the real rate
relative to that of expected inflation is greater or less than one. In addition, the
model consistently generates a negative correlation between the real rate and
expected inflation across all values of the risk-aversion parameter. The latter
result supports earlier evidence to that effect by Fama (1990).

4.

RESULTS WITH “KEEPING-UP-WITH-THEJONESES” UTILITY

Analytical Solutions
Thus far, estimates of the inflation risk premium based on the above framework as well as U.S. consumption data appear to be quantitatively small. We
have also suggested that this result is unrelated to the fact that the equity
premium tends to be small in consumption-based asset pricing models. To see
why this is true, we now adopt an alternative preference specification that we
refer to as the “keeping-up-with-the-Joneses” (KUPJ) specification. Under this
alternative way of modeling preferences, which defines utility as a function
of relative consumption, Abel (1990) shows that while the return on stocks
typically increases with the risk-aversion parameter, the real return on bonds
generally remains constant. Therefore, when the degree of relative risk aversion
is sufficiently high (γ = 6 in Abel [1990]), the author is able to generate an

66

Federal Reserve Bank of Richmond Economic Quarterly

equity premium that is within the range of that observed in the data. We now
formally show that even when utility is of the KUPJ form, the inflation risk
premium remains small irrespective of the degree of risk aversion.
Following Abel (1990) and Gali (1994), momentary KUPJ utility is given
by
u(ct ) =

(ct /Ct−1 )1−γ − 1
,
1−γ

(17)

where Ct−1 denotes average consumption in the previous period. Thus, the
specification in (17) captures the idea that it is not consumption per se but
qt as the price of a onerelative consumption that matters to households. Using e
period inflation-indexed bond under this alternative functional form for utility,
equation (4) now reads as
µ
¶
µ
¶
Ct γ−1
ct+1 −γ
e
qt = β
Et
(18)
Ct−1
ct
µ

=

Ct
Ct−1

¶γ−1

qt ,

where qt , given by equation (10), is the price of an inflation-indexed bond when
utility is CRRA. Similarly, the inverse of the nominal rate in equation (12) is
now given by
µ
¶
µ
¶
Ct γ−1
ct+1 −γ pt
e
xt = β
Et
(19)
Ct−1
ct
pt+1
µ

=

Ct
Ct−1

¶γ−1

xt .

In equilibrium, Ct = ct when households are identical. Therefore, the solutions
qt and ext can simply be obtained by scaling up equations (11) and (14),
for e
respectively, by a power function of current consumption growth, ζtγ−1 . More
importantly, these results also indicate that the new inflation risk premium is
now given by equation (15) multiplied by ζtγ−1 . Since the inflation risk premium under KUPJ utility is simply the premium that emerges under CRRA
utility scaled up by current consumption growth (to the power γ − 1), a value
of σζφ = 0 still implies that the inflation risk premium is identically zero irrespective of γ. In other words, it is still true in this case that when unexpected
movements in consumption growth and shocks to inflation are uncorrelated, the
inflation risk premium is zero regardless of the degree of risk aversion. Given
our estimate in the previous section of σζφ = 3.20 × 10−5 , it follows that even
when preferences follow the KUPJ specification, the simple Fisher equation in
(7) remains a good approximation to the generalized Fisher equation in (6).

P.-D. G. Sarte: Inflation Risk Premium

67

Quantitative Implications
Figure 3 presents the historical estimates from the benchmark case where utility
is of the KUPJ form. The parameter values for the bivariate driving process are
the same as those used in the previous section. A direct comparison with Figure
2 reveals little difference between the two sets of figures. In particular, observe
that, as expected, the inflation risk premium continues to be negligible over the
entire period under consideration. As in the earlier experiment, the model still
fails to capture the behavior of the nominal rate at the end of the 1970s and
beginning of the 1980s. However, it is interesting that both the ex ante real rate
and the model-implied nominal rate seem to exhibit more variation relative to
Figure 2. This result is consistent with the earlier work of Abel (1990) who
finds that, while the mean return on bonds remains relatively constant as the
degree of risk aversion rises with KUPJ preferences, the volatility of bond
returns tends to exceed that which emerges with CRRA utility. The following
table makes the last point more concretely.
When one compares Table 2 with Table 1, it is clear that under the alternative preference specification, the real rate is largely invariant with respect to the
degree of risk aversion. This invariance property is precisely the mechanism
that, for a high enough value of γ, allows Abel (1990) to generate an equity risk
premium close to the one found in the data. Table 2 also clearly suggests that
in all cases, the volatility of both the real rate and nominal rate is greater than
its corresponding value in Table 1. As in the previous section, it remains that
the volatility of the real rate increases sharply with the degree of risk aversion.
Accordingly, whether the real rate varies more or less than expected inflation
at short horizons still depends heavily on the particular preference specification
adopted. In addition, as in Fama (1990), the model continues to suggest a consistent negative correlation between the real rate and expected inflation across
different values of γ. Therefore, although we find that Fisher’s equation holds
relatively well in this framework, the nominal yield moves generally less than
one-for-one with expected inflation at the one-year horizon.

5.

CONCLUDING REMARKS

This article investigates the extent to which the simple Fisher equation can be
interpreted as a reasonable approximation to its more complete counterpart in
a dynamic endowment economy. The expanded Fisher equation, in addition to
capturing movements in real rates and expected inflation, differs from its simpler version along two dimensions. First, it accounts for random movements
in inflation through an inflation risk premium. Second, it acknowledges the
inherent nonlinearity of inflation in drawing a link between the nominal rate
and expected inflation.
Given U.S. consumption data, we find that the quantitative historical
estimates of the inflation risk premium for the period 1955 to 1996 are small.

68

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Simulated Results with KUPJ Utility

a. Ex Ante Real Rate

Percent

0.05

0.03

0.01

-0.01
1955

1965

1975
Year

1985

1995

b. Expected Inflation

Percent

0.08
0.06
0.04
0.02
0.00
1955

1965

1975
Year

1985

1995

c. Nominal Interest Rate

Percent

0.14
Model
Data
0.10
0.06
0.02
1955

1965

1975
Year

1985

1995

Percent

d. Inflation Risk Premium

0.0000226

0.0000214

+

1955

1965

1975
Year

1985

1995

P.-D. G. Sarte: Inflation Risk Premium

69

Table 2

γ = 0.75

γ = 1.75

γ=6

Ex Ante Real Rate:
rt

Expected Inflation:
Et (pt+1 /pt )

Nominal Rate:
rtN

mean: 2.75

mean: 4.03

mean: 6.86

std: 1.24

std: 2.43

std: 2.62

corr(rt , Et (pt+1 /pt )): −0.16

E (p
/p )
var( t t+1 t ):
rt

mean: 2.69

mean: 4.03

mean: 6.78

std: 2.43

std: 2.43

std: 3.75

corr(rt , Et (pt+1 /pt )): −0.39

E (p
/p )
var( t t+1 t ):
rt

mean: 2.65

mean: 4.03

mean: 6.65

std: 11.06

std: 2.43

std: 10.75

corr(rt , Et (pt+1 /pt )): −0.37

E (p
/p )
var( t t+1 t ):
rt

3.80

1.00

0.04

This result emerges primarily because unexpected movements in consumption
and inflation surprises appear to have little covariation in U.S. data. In other
words, since inflation surprises are largely unrelated to consumption growth,
there is no reason why the inflation risk premium should be either positive
or negative. Moreover, the latter notion was shown to have little to do with
the equity premium being typically small in consumption-based asset pricing
models. Therefore, although the Fisher equation does not theoretically apply in
an environment with stochastic inflation, it may serve as an adequate approximation in practice.
Using two different preference structures, we also find that the modelimplied nominal yield on one-year bonds matches the actual one-year yield
on Treasury notes relatively well for most of the sample period. However,
the model fails to track the nominal rate adequately in the late 1970s. We
suspect that this latter result is partly driven by the singularly aggressive stance
adopted by the Federal Reserve at that time in order to bring down very high
inflation rates. In interpreting our results concerning the inflation risk premium, one needs to be cognizant of the model’s failure along this dimension.
Our benchmark cases also suggest a real rate whose volatility is more than
half that of its U.S. ex post counterpart. Further, our framework in all cases
provides additional evidence to support Fama’s (1990) view that expected inflation and the real rate tend to move in opposite directions. Finally, we find
that under both preference specifications, whether the real rate is more or less
volatile than expected inflation depends heavily on households’ degree of risk

70

Federal Reserve Bank of Richmond Economic Quarterly

aversion. Taken together, these last two points suggest one should proceed with
caution when interpreting movements in short-term nominal yields in terms of
movements in expected inflation.

APPENDIX A:

HOUSEHOLD OPTIMALITY
CONDITIONS

Let λt and µt represent the Lagrange multipliers associated with constraints
(2) and (3), respectively. Then, the first-order conditions associated with the
household’s problem are given by
u0 (ct ) = µt + βEt λt+1

pt
,
pt+1

(20)

qt λt = βEt λt+1 ,

(21)

λt
λt+1
= βEt
,
pt
pt+1

(22)

xt
and

λt = µt + βEt λt+1

APPENDIX B:

pt
.
pt+1

(23)

USEFUL PROPERTIES OF LOG-NORMAL
RANDOM VARIABLES

This appendix describes properties of log-normal random variables that are
useful in deriving the solution for bond prices described in Section 3. Let x be
a log-normal random variable, then
•

ln E(x) = E(ln x) + (1/2)var(ln x) and

•

ln E(xa ) = aE(ln x) + (a2 /2)var(ln x) for a ∈ R.

Furthermore, if y is a log-normal random variable, then so is z = xy. To
see this, note that ln z = ln x + ln y, which is the sum of two normal random
variables and thus itself normally distributed. It directly follows from the first
of the above properties that
•

ln E(xy) = E(ln x) + E(ln y) + (1/2)var(ln x) + (1/2)var(ln y)
+ cov(ln x, ln y).

P.-D. G. Sarte: Inflation Risk Premium

APPENDIX C:

71

JENSEN’S INEQUALITY AND THE
VARIANCE OF INFLATION SHOCKS

Since
ln Et

pt+1
pt+1 1
pt+1
= Et ln
+ vart ln
pt
pt
2
pt
1
= Et ln φt+1 + vart ln φt+1 ,
2

equation (9) directly implies that
Et
δ

σφ2
pt+1
)Pt ,
= exp(δφ0 +
pt
2

(24)

δ

φζ, j p
φφ, j
Πj=0 φt−j
. Furthermore, since ln Et pt /pt+1 can simply
where Pt = Πpj=0 ζt−j
−1
be expressed as ln Et (pt+1 /pt ) , we also have that

Et

σφ2 −1
pt
= exp(−δφ0 +
)P .
pt+1
2 t

(25)

It then follows that

"
#
σφ2
−σφ2
pt+1
1
= exp(δφ0 ) exp(
) − exp( ) P−1
−
Et
t .
pt
Et (pt /pt+1 )
2
2

(26)

Hence, the difference on the left-hand side of equation (26) rises with σφ2 as
conjectured. This is shown in Figure 1.

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Federal Reserve Bank of Richmond Economic Quarterly

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