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Federal Reserve Bank of Chicago

The Cost of Business Cycles under
Endogenous Growth
Gadi Barlevy

WP 2003-13

The Cost of Business Cycles under Endogenous Growth∗
Gadi Barlevy
Federal Reserve Bank of Chicago and NBER
Economic Research Department
Federal Reserve Bank of Chicago
230 South LaSalle Street
Chicago, IL 60604
e-mail: gbarlevy@frbchi.org

Abstract
In his famous monograph, Lucas (1987) put forth an argument that the welfare gains
from reducing the volatility of aggregate consumption are negligible. Subsequent work that
revisited Lucas’ calculation continued to ﬁnd only small beneﬁts from reducing the volatility
of consumption, further reinforcing the perception that business cycles don’t matter. This
paper argues instead that ﬂuctuations can aﬀect welfare by aﬀecting the growth rate of
consumption. I present an argument for why ﬂuctuations can reduce growth starting from
a given initial consumption, which could imply substantial welfare eﬀects as Lucas (1987)
already observed in his calculation. Empirical evidence and calibration exercises suggest
that the welfare eﬀects are likely to be substantial, about two orders of magnitude greater
than Lucas’ original estimates.

First Version: February, 2000
Current Version: August, 2003

∗
I have beneﬁtted from comments on this and an earlier version of the paper by Fernando Alvarez, Marco
Bassetto, Larry Christiano, Elhanan Helpman, Zvi Hercowitz, Sam Kortum, Robert Lucas, Alex Monge, Assaf Razin, Helene Rey, and seminar participants at Northwestern, Wisconsin, Harvard, Columbia GSB, NYU,
Princeton, Maryland, Ohio State, the Canadian Macro Study Group, Society of Economic Dynamics, and the
National Bureau of Economic Research. I am also grateful to Valerie Ramey for supplying me with the data and
the original code from her paper. The views expressed here do not necessarily reﬂect the position of the Federal
Reserve Bank of Chicago or the Federal Reserve System.

1. Introduction
In his famous monograph, Lucas (1987) argued that business cycles in the post-War U.S. involved at most negligible welfare losses, thereby challenging the presumption that stabilizing
the cyclical ﬂuctuations that persisted during this period would have been highly desirable. His
argument can be stated as follows. Consider a representative consumer with a conventional
constant-relative risk aversion (CRRA) utility function over consumption streams {Ct }∞ , i.e.
t=0
U ({Ct }) =

∞
X
t=0

βt

1−γ
Ct − 1
1−γ

where γ ≥ 0 measures relative risk-aversion and β < 1 denotes the rate at which the agent
discounts future utility. Suppose this consumer is given a consumption stream
Ct = λt (1 + εt ) C0

(1.1)

where λ is a constant greater than one and εt is an i.i.d. random variable with mean zero. That
is, average consumption starts out at C0 and grows at rate λ, so that average consumption
after t periods will equal λt C0 . Actual consumption will deviate from this average by a factor
of 1 + εt . Using per-capita consumption growth from the post-War era, we can estimate λ and
the variance of εt . To determine the costs of ﬂuctuations, Lucas asked what constant fraction
of each year’s consumption the consumer would be willing to give up to avoid ﬂuctuations, i.e.
to replace εt with zero in each and every period. For reasonable values of γ, the answer turns
out to be astonishingly small, less than one-tenth of one percent. By contrast, Lucas calculated
that a consumer would sacriﬁce as much as 20% of his consumption each year when γ = 1
to increase the average growth rate λ by one percentage point. Thus, Lucas concluded that
growth matters, but business cycles, at least of the magnitude that occurred in the U.S. over
the post-War period, do not.
Although the calculation above abstracts from several important considerations, its implication that consumption volatility at business cycle frequencies does not matter much for welfare
has proven to be quite robust. For example, Imrohoroglu (1989), Atkeson and Phelan (1994),
Krusell and Smith (1999), Storesletten, Telmer, and Yaron (2001), and Beaudry and Pages
(2001) depart from the representative agent setting and calibrate income streams to individuallevel data, assuming only imperfect insurance arrangements across agents. Since agents can
still save against future income shocks, the implied costs of business cycles in these papers are
small, typically less than 1% of consumption per year, and in some cases are even negative.
Obstfeld (1994) maintains a representative agent framework but allows shocks to be persistent
1

rather than i.i.d., and again ﬁnds relatively small costs. Lastly, Obstfeld (1994), Campbell
and Cochrane (1995), Pemberton (1996), Dolmas (1998), Alvarez and Jermann (1999), Tallarini (2000), and Otrok (2001) examine departures from CRRA utility. With the exception
of Campbell and Cochrane and Tallarini, whose ﬁndings are challenged in some of the other
papers, these experiments again tend to ﬁnd small costs of ﬂuctuations at business cycle frequencies for reasonable parameter conﬁgurations. Thus, it appears that there is limited scope
for generating large costs of business cycles from aversion to consumption risk.1
The results above would seem to deny that business cycles can ever matter much for welfare.
Nevertheless, this paper argues that cycles can be associated with considerable welfare costs,
about two orders of magnitude greater than those Lucas originally computed. These costs
are not due to consumption volatility per se, as maintained in previous work, but to the fact
that cyclical ﬂuctuations can aﬀect the economy’s long-run growth rate. The motivation for
examining growth eﬀects comes from Lucas’ original insight that changes in growth rates can
have large welfare eﬀects: even if aggregate ﬂuctuations have a modest eﬀect on the rate at
which an economy grows, they could in principle have a signiﬁcant eﬀect on welfare.
The notion that business cycles might lead to large welfare costs because of their eﬀect on
growth has been raised in previous work, including Mendoza (1997), Jones, Manuelli, and
Stacchetti (1999), Epaulard and Pommeret (2000), and Matheron and Maury (2000). However,
these models fail to generate large welfare costs for ﬂuctuations of the magnitude of the post-War
U.S. experience. The reason, as I argue below, is that these models do not produce growth eﬀects
of the type implicit in Lucas’ calculation on the beneﬁts from faster growth. Lucas computed
how much an agent would sacriﬁce to obtain a consumption stream that starts at the same
initial level of consumption but grows more rapidly, as illustrated by the shift from the solid to
the dashed line in the ﬁrst panel of Figure 1. But in the models cited above, ﬂuctuations aﬀect
long-run growth by changing the average amount of resources that are allocated to growthenhancing activities in equilibrium. If eliminating ﬂuctuations increases average investment,
consumption will still grow at a faster rate, but since resources must be diverted to investment,
average initial consumption will fall, as illustrated in the bottom panel of Figure 1. Clearly,
the welfare implications of a given change in the growth rate will be very diﬀerent for these
two experiments. Previous work has therefore focused on the beneﬁts of allowing agents to
reoptimize between present and future consumption, not the beneﬁts of more rapid growth in
the sense that Lucas described.
1

Lucas (2003) surveys the literature on the costs of consumption risk over the business cycle that emerged
since his original monograph and reaches a similar conclusion.

This paper reformulates the endogenous growth models above in a way that does give rise to
the growth eﬀect illustrated in the top panel of Figure 1. It does this by introducing diminishing
returns to investment, which implies that the growth rate will be a concave function of the level
of investment. As a result, even if eliminating ﬂuctuations had no eﬀect on average investment,
and thus no eﬀect on average initial consumption, it would still lead to more rapid growth
by making investment less volatile. Intuitively, eliminating ﬂuctuations reallocates investment
from periods of high investment to periods of low investment. Since the marginal return to
investment is higher when there is less investment, this allows agents to achieve more growth
from the same resources. The ﬁrst part of the paper formalizes this argument using a familiar
model of endogenous growth.
In the remainder of the paper, I assemble various pieces of evidence on the extent of diminishing returns to investment. These consistently suggest that eliminating ﬂuctuations will
increase the growth rate of per-capita consumption from 2.0% to about 2.4 − 2.5% if we held
average investment ﬁxed. For γ = 1, this implies a cost of cycles of 8 − 10% of consumption

per year, and the cost only rises when I turn to an alternative set of preferences that can better
replicate the volatility of aggregate investment over the cycle. While this suggests business
cycles are quite costly, it need not follow from this result that stabilization is desirable. This
largely depends on the source of aggregate shocks. For example, if cycles are due to productivity shocks, policymakers will only end up making agents worse oﬀ by attempting to oﬀset
these shocks with countercyclical policy (although agents could potentially be made better oﬀ
by coordinating to an equally productive but less volatile technology). Thus, this paper does

not directly advocate for stabilization; rather, its purpose is to oﬀer a rationale for why business
cycles matter. Documenting a large cost of cycles is an important result in its own right given
the grave importance people often seem to attach to economic ﬂuctuations despite previous
results that suggest they shouldn’t, and it opens the door to a role for stabilization, at least in
some scenarios, that previous calculations would deny. The basic insights developed here can
and should be used to explore environments where policy can play a beneﬁcial role in order to
determine whether stabilization policy is ultimately desirable.
The paper is organized as follows. Section 2 develops a model of endogenous growth that
allows for diminishing returns to investment. Section 3 attempts to quantify the welfare cost
of ﬂuctuations using data on growth. Section 4 examines whether calibrating the endogenous
growth model to produce a signiﬁcant growth eﬀect can accord with other time series data, e.g.
investment. Section 5 calibrates the model to data on Tobin’s q. Section 6 concludes.

2. Endogenous Growth with Diminishing Returns to Investment
To study the eﬀects of economic ﬂuctuations on growth and welfare, I need a model in which the
growth rate is determined endogenously. Towards this end, I use a stochastic AK growth model.
This speciﬁcation has become a staple for modeling endogenous growth under uncertainty, and
using it facilitates comparison with the previous literature. The ﬁrst to analyze this model were
Levhari and Srinivasan (1969), who used it to study savings decisions under uncertainty. They
in turn solved an inﬁnite-horizon version of a problem that was originally studied by Phelps
(1962). Leland (1974) subsequently reinterpreted this model in terms of long-run economic
growth. Many authors have since used variations of this basic model to study endogenous
growth in the face of aggregate uncertainty.
The economy consists of a representative agent who derives utility only from consumption. To
simplify the exposition, I focus on the planning problem for this economy. One can show that
the allocation of resources that solves this problem coincides with the equilibrium allocation in
a decentralized market economy. Time is discrete, and the agent discounts the future at rate β.
For now, I assume per-period utility is isoelastic in consumption, i.e. for a given consumption
stream {Ct }∞ , the utility of the agent is equal to
t=0
∞
X
t=0

βt

1−γ
Ct − 1
1−γ

(2.1)

In evaluating the beneﬁts of growth, Lucas (1987) considered the case where γ = 1, i.e. log
utility, a value that falls within the range of estimates for intertemporal elasticity of substitution
found in the literature, e.g. Epstein and Zinn (1991). I similarly use this case as my benchmark
for calculating the cost of ﬂuctuations. However, below I argue that this speciﬁcation for utility
does a poor job of matching certain empirical regularities, which motivates me to eventually
turn to a more general formulation that includes (2.1) as a special case.
The agent has access to a production technology that converts inputs into consumption goods.
The only input for producing consumption goods is capital. Speciﬁcally, production is linear
in capital, but the number of units that can be produced from a unit of capital ﬂuctuates
stochastically over time, i.e.
Yt = At Kt
where At follows a Markov process. Thus, the source of ﬂuctuations in this economy are shocks
to productivity. However, as noted by Eaton (1981), we can always reinterpret these shocks as
government policy shocks. That is, suppose aggregate productivity is constant over time, but
4

there is a government sector that collects a random fraction τ t of income in taxes which it uses
to ﬁnance contemporaneous purchases that do not enter into the agent’s utility. This leaves the
agent with an income of Yt = (1 − τ t ) AKt ≡ At Kt .
In what follows, it will prove necessary to impose some regularity conditions on the Markov
process At . First, I require that for any x, Prob(At+1 ≤ x | At ) is weakly decreasing in At .
Thus, a higher realization of productivity today is associated with a weakly higher expected
productivity next period. This assumption ensures the agent will be better oﬀ at higher levels
of aggregate productivity. This includes the case where At is i.i.d. I further assume that the
values of At are such that expected discounted utility is well-deﬁned, i.e. the growth rate cannot
exceed the discount rate, and the planning problem has an interior solution. Finally, I assume
the initial distribution over At corresponds to the invariant distribution of the Markov chain
(which implicitly assumes such a distribution exists and is unique). I evaluate welfare using
this distribution throughout the paper, i.e. the expected utility of the agent is calculated prior
to the resolution of uncertainty over the initial level of productivity A0 .
Since output is proportional to capital, the time path of output (as well as consumption)
depends on the evolution of the capital stock Kt . At date 0, the agent is endowed with some
initial amount of capital K0 . Beyond this date, the level of capital depends on the endogenous
decisions of the agent. If the agent begins the period with Kt units, a fraction δ of the capital
is assumed to depreciate over the period, so that at the beginning of the next period only
(1 − δ) Kt units remain. The agent can add to this stock by setting aside some of the output
from the current period and converting it, together with his existing capital, into capital for
use in the subsequent period. The technology for producing new capital is characterized by a
function Φ (It , Kt ) that depends on the amount of output set aside for investment It and the
existing stock of capital Kt . The function Φ (·, ·) is assumed to be homogenous of degree 1 and
increasing in its ﬁrst argument. Hence, we can rewrite this production function as
µ ¶
It
Φ (It , Kt ) = φ
Kt
Kt
where φ0 (·) > 0. The stock of capital available for production in period t + 1 is thus
¶
µ
µ ¶
It
Kt+1 = 1 + φ
− δ Kt
Kt

Repeated substitution of the above equation yields the capital stock at date t as a function of
the initial capital stock K0 :
" t µ
µ ¶
¶#
Y
Is
1+φ
−δ
K0
(2.2)
Kt =
Ks
s=0
5

The original Levhari and Srinivasan (1969) model is a special case of this model where the
function φ (·) is equal to the identity function and δ = 1. In this case, equation (2.2) reduces
to Kt+1 = It , i.e. the wealth Kt+1 the agent holds at the beginning of each period is just his
savings from the previous period It = Yt −Ct . While several authors have by now departed from

the assumption of full depreciation, in line with the interpretation of Kt as physical capital,
the assumption that φ (·) is linear is still commonplace. But following Uzawa (1969), it is just

as natural to allow φ (·) to be concave, implying diminishing returns to investment. Formally,
concavity in φ (·) implies that for a ﬁxed amount of capital, each additional unit of investment
will contribute less to the stock of capital. We can alternatively interpret this concavity as
adjustment costs denominated in units of capital, so that increasingly more capital is required
to merge new investment goods with the existing capital stock. As I discuss in more detail
below, empirical evidence suggests there is indeed some curvature in φ (·) in practice.2
To summarize, output in period t is produced using all of the capital available at the beginning
of the period. Out of this output, the agent chooses to consume an amount Ct , and uses the
remainder It = Yt − Ct to invest in capital for the next period. It will prove convenient to deﬁne
ct = Ct /Yt as the fraction of output the agent consumes and it = It /Yt = 1−ct as the fraction he
sets aside for investment. Using this notation, we can rewrite the agent’s consumption stream
in a form reminiscent of Lucas’ original speciﬁcation:
Ct = ct At Kt
#
" t
Y
= ct At
(1 + φ (is As ) − δ) K0
≡

s=0

t
Y

s=0

λs (1 + εt ) C0

(2.3)

ct At
− 1 is the
c0 A0
deviation of consumption from its trend, and C0 = c0 A0 K0 is the initial level of consumption.
Note that if the growth rate of capital λs were constant, this consumption stream would simplify

where λs ≡ 1 + φ (is As ) − δ is the growth rate of the capital stock, εt ≡

to λt (1 + εt ) C0 , precisely the form Lucas posited. But this consumption stream now emerges
endogenously as an optimal response to the underlying economic environment rather than as
an exogenous speciﬁcation.
2

As an aside, diminishing returns also serve to dampen the response of investment to changes in A. This helps
to address a common critique of AK growth models, namely that policy changes that ought to aﬀect investment
have little eﬀect on growth empirically. While some have argued that policy changes have no eﬀect at all on
long-run growth, McGrattan (1998) argues that these eﬀects are apparent in longer time series.

The cost of aggregate ﬂuctuations is deﬁned as the ex-ante utility gain for the agent (i.e.
prior to the realization of the sequence {At }∞ ) if we were to move him from the stochastic
t=0
environment above to one with the same initial capital stock but where productivity is constant
and equal to the unconditional average productivity in the stochastic environment. That is,
both environments are equally productive on average, but in one environment productivity
ﬂuctuates around its mean. For notational convenience, let an asterisk denote the value of a
variable in the counterfactually stable environment. Thus, productivity A∗ is constant for all
t
dates t and is equal to E [At ] ≡ A∗ . Following Lucas, I measure the cost of cyclical ﬂuctuations
in terms of compensating variation, i.e. by the additional fraction of consumption per period
the agent would require in the stochastic environment to attain the same level of expected
utility as in the stable environment.
A few remarks about the interpretation of this cost are in order. If At reﬂects exogenous
technology shocks, the above cost is purely hypothetical in the sense that it cannot be avoided
by stabilization policy. Although the government can replicate the eﬀects of productivity shocks
on the agent through taxes and subsidies, the original allocation is Pareto optimal, and any selfﬁnancing scheme that fools him into diﬀerent consumption and savings choices will only lower
his welfare. Hence, the cost of cycles does not represent the gains from any feasible stabilization
policy. Still, establishing that this cost is large can explain why individuals often cite business
cycles as an important concern. By contrast, if ﬂuctuations in At reﬂect spurious ﬁscal policy
shocks, the cost of cycles can be avoided simply by eliminating arbitrary policy variability, in
which case the cost of cycles is also the value to stabilization. However, for this logic to go
through, it is important that ﬂuctuations be spurious rather than an optimal response to some
other underlying shock.3 For example, we would not want local governments to even out their
expenditures on snow removal over the year. Although smoothing these expenditures over the
year would eliminate one source of volatility, it would prevent resources from being allocated
to address underlying weather shocks that are seasonal in and of themselves.
Since the utility of the agent depends on the consumption stream he faces, we ﬁrst need
to solve the optimal consumption for a given economic environment. We therefore set up the
3
Arbitrary ﬂuctuations are not limited to capricious policymaking. Sunspots and coordination failures can
also lead to spurious volatility in measured productivity At . For example, Benhabib and Farmer (1994) generate
sunspots in models with increasing returns to scale, while Shleifer (1986) generates volatile productivity through
coordination on the implementation of exogenous technological improvements.

problem of the representative agent:
V (K0 , A0 ) ≡ max E0
Ct

subject to

"∞
X
t=0

1−γ
t Ct
β
1−γ

(2.4)

¶
¸
· µ
At Kt − Ct
+ 1 − δ Kt
1. Kt+1 = φ
Kt
2. the law of motion for At

We can rewrite this problem recursively as
(
)
1−γ
Ct
+ βE [V (Kt+1 , At+1 ) | At ]
V (Kt , At ) = max
Ct
1−γ
(
)
(ct At Kt )1−γ
= max
+ βE [V ((1 + φ (it At ) − δ) Kt , At+1 ) | At ]
1−γ
ct ∈[0,1]
where recall it = 1 − ct .
Using the maximization problem above, I next argue that in the face of aggregate ﬂuctuations,
the agent will choose to vary his investment rate it At over time. Consequently, the growth rate
λt = 1 + φ (it At ) − δ will ﬂuctuate with At . Establishing this result requires additional technical
assumptions, namely that φ (·) is strictly concave and that it satisﬁes the boundary conditions
limx→0 φ0 (x) = ∞ and limx→∞ φ0 (x) = 0. With these assumptions, we have the following
proposition, whose proof is contained in an Appendix:
Proposition: Suppose Prob(At+1 ≤ x | At ) is weakly decreasing in At . Then it At =

increasing in At .

It
is
Kt

An implication of this proposition is that if we decompose investment or output into a stochastic trend and deviations from trend, we would ﬁnd that trend growth for both investment
and output is higher when these variables are above their trend. Formally, rewrite Yt and It in
the same way as Ct in (2.3), i.e.
"t−1 #
Y
¡
¢
Yt =
λs 1 + ε0 Y0
(2.5)
t
s=0

and
It =

" t−1
Y

s=0

λs

¡
¢
1 + ε00 I0
t

(2.6)

At
it At
− 1, and ε00 =
− 1. Then ε0 and ε00
t
t
t
A0
i0 A0
will be positively correlated with λt . This will not necessarily be true for consumption, since
ct At = At − it At could either increase or decrease with At , and hence εt in (2.3) could be
where λs = 1 + φ (is As ) − δ as before, ε0 =
t

negatively correlated with λt . However, since trend consumption growth and trend output
growth are both equal to λs , consumption growth will generally be positively correlated with
output growth. This observation will be important for interpreting some of the empirical results
below.
To understand the welfare implications of cyclical ﬂuctuations, it will help to work through
the diﬀerent ways in which shifting from stochastic productivity {At } to a constant productivity

A∗ induces the agent to change his consumption. First, eliminating ﬂuctuations induces the
agent to choose a consumption path that does not ﬂuctuate around its trend. Formally, when
A∗ is constant for all t, the agent will choose to consume a constant fraction c∗ of output. But
t
this implies the deviation from trend consumption in (2.3) will equal
ε∗ =
t

c∗ A∗
c∗ A∗
t t
−1= ∗ ∗ −1=0
c∗ A∗
c A
0 0

Since the agent is risk-averse, eliminating deviations from trend makes him better oﬀ. But
when Lucas (1987) computes the implied welfare gain assuming that all observed ﬂuctuations
in consumption growth reﬂect deviations from trend, the welfare gains from eliminating such
deviations turn out to be negligible for reasonable utility speciﬁcations.4 Simply put, deviations
from trend per-capita consumption over the post-War period are not large enough to generate
signiﬁcant costs of cyclical ﬂuctuations for reasonable degrees of risk aversion.
Second, since the agent will choose to set aside a constant fraction i∗ of his income for investment when productivity is stable, eliminating ﬂuctuations in aggregate productivity induces
the agent to choose a consumption path with a deterministic trend rather than a stochastic
trend. Thus, stabilization eliminates both ﬂuctuations around trend consumption as well as
ﬂuctuations in trend consumption. Once again, this will make the agent better oﬀ given his
aversion to risk. But since ﬂuctuations in trend consumption are permanent, they reduce welfare by more than stationary ﬂuctuations around trend. Still, when Obstfeld (1994) calculates
the implied welfare gain under the assumption that all observed ﬂuctuations in consumption
growth represent i.i.d. permanent shocks to trend, he again ﬁnds relatively small welfare costs.
4
Lucas’ calculation assumes deviations from trend are i.i.d., while the model allows deviations to be serially
correlated. More persistent shocks would generate larger costs. But as I discuss below, allowing for more
persistent shocks to the level of consumption does not lead to substantial costs of ﬂuctuations.

Somewhat larger costs can arise if we assumed persistent rather than i.i.d. shocks to trend, as is
allowed under the more general Markov structure of At . But empirically, consumption growth
exhibits fairly low persistence, as documented among others by Christiano, Eichenbaum, and
Marshall (1991).5 This reaﬃrms Lucas’ original observation, namely that even though eliminating ﬂuctuations makes it easier for the agent to smooth consumption, the volatility of per-capita
consumption over the cycle is so small that this welfare gain is negligible.
Lastly, eliminating ﬂuctuations in productivity in this environment can induce the agent to
maintain a diﬀerently-sloped consumption proﬁle from the one in the stochastic environment.
Here, we need to distinguish between changes in the consumption proﬁle that are due to changes
in the volatility of investment and those that are due to changes in the level of investment. To
separate the two, suppose we were to eliminate ﬂuctuations in At but force the agent to save
a constant fraction i∗ such that the average investment-to-capital ratio is the same as in the
stochastic environment, i.e. i∗ solves
i∗ A∗ = E [it At ]
Given the concavity in φ (·) and the results in the proposition, this level of investment will be
associated with a higher average growth rate, since for any non-degenerate distribution of it At ,
1 + φ (i∗ A∗ ) − δ = 1 + φ (E [it At ]) − δ
> 1 + E [φ (it At )] − δ
Thus, merely eliminating volatility in investment will lead to more rapid growth, even if
average amount of resources set aside for investment remains unchanged. If we then allow
agent to freely choose his investment optimally and he chooses some i∗ 6= E [it At ] /A∗ ,
long-run growth rate will change further. The direction of the change depends on whether

the
the
the
the

agent desires higher or lower investment in the absence of shocks. Both scenarios are possible
for diﬀerent parameter values. The eﬀect of eliminating ﬂuctuations in At on growth can thus
be decomposed into two parts: the part due to eliminating the volatility of investment around
its original average, and the part due to changes in average investment in response to moving
to a more stable environment.
5
Bansal and Yaron (2001) argue instead that consumption growth is persistent. They assume consumption
growth follows an ARMA process gt+1 = ρgt + η t − ωη t−1 where η t is i.i.d. If ω is close to ρ, it will be hard to
distinguish this process from white-noise even when ρ is large, which could lead us to erroneously infer growth
is not highly autocorrelated if we fail to take into account the moving-average term. Bansal and Yaron estimate
ρ = 0.95 and ω = 0.85 from dividend growth (which is the same as consumption growth in their model), and
use this to argue the costs of business cycles are large. However, ARMA models for consusmption growth data,
such as Lewbel (1994), fail to ﬁnd high values of ρ.

To see why it is important to distinguish between these two eﬀects, note that by forcing
the agent to keep the average investment-capital ratio unchanged, we leave his expected initial
consumption unchanged, since
∗
∗
c∗ A∗ K0 = (A∗ − i∗ A∗ ) K0

= [E (At ) − E (it At )] K0
= E (ct At ) K0
Thus, eliminating ﬂuctuations and forcing the agent to maintain the same average investment
allows him to attain a consumption path that starts at the same level on average but grows more
rapidly. Lucas (1987) provides some calculations on just how much the agent would be willing
to sacriﬁce to achieve this scenario. For γ = 1, he reports the agent would be willing to sacriﬁce
20% of consumption each year to increase the growth rate of consumption by one percentage
point. By contrast, if the growth rate is higher because the average investment rate i∗ A∗ is
∗
∗
higher, initial consumption will be lower, since C0 = (A∗ − i∗ A∗ ) K0 is decreasing in i∗ A∗ . The
two channels are therefore associated with welfare changes of very diﬀerent magnitudes (and
possibly of diﬀerent signs) for a given change in the average growth rate. For example, the agent
might optimally choose a lower average investment rate, i∗ A∗ < E (it At ), even though it leads
to lower growth. Increasing growth by undertaking more investment does not automatically
raise welfare as it does in Lucas’ thought experiment.6
Since previous authors assume φ (·) is linear, they allow ﬂuctuations to aﬀect growth only
through changes in the level of investment. Consequently, those who previously allowed for
growth eﬀects have found only small costs. This is not surprising: if eliminating ﬂuctuations
increases growth by increasing investment, the gain the agent reaps from more rapid consumption growth will be largely oﬀset by the loss he incurs from a lower initial consumption. To
generate more substantial welfare gains of the magnitude suggested by Lucas’ calculations on
the value of growth, it is necessary that cycles retard growth for a ﬁxed average investment,
which is precisely what happens under diminishing returns.
In sum, the key to generating large welfare costs of cyclical ﬂuctuations appears to be the
presence of diminishing returns to investment, i.e. a concave function φ (·). In the next several
6

A similar distinction arises in models where growth is driven by learning-by-doing rather than factor accumulation, e.g. Ramey and Ramey (1991). In these models, eliminating ﬂuctuations can simultaneously raise
both current output and future output growth. The inherent tradeoﬀ in these models is not between present and
future consumption, but between present leisure and present (and future) consumption. Once again, we would
want to distinguish between changes in the growth rate due to changes in the level of output, which require
sacriﬁcing initial leisure, and those due to changes in the volatility of output, which do not.

sections, I argue that empirical evidence supports the notion of diminishing returns and thus
a large cost of business cycles. At ﬁrst glance, this result might seem puzzling: if investment
volatility has such an adverse eﬀect on welfare, why wouldn’t the agent smooth investment at
the expense of consumption volatility, especially if consumption volatility is so inconsequential?
But this intuition is misleading. Previous literature has only established that the consumption
volatility we observe over the cycle is inconsequential. But the consumption volatility that would
be required to counterfactually stabilize investment over the cycle may be substantially larger
than the volatility of consumption we actually observe over the cycle. As the proposition above
demonstrates, it is in fact optimal to vary investment when productivity ﬂuctuates. Ultimately,
the costs of aggregate ﬂuctuations come not from the fact that investment is volatile per se, but
from the fact that the underlying stochastic environment forces the agent to choose a volatile
path for investment. As such, the agent might be signiﬁcantly better oﬀ if the shocks that
caused him to behave this way were eliminated.

3. Quantitative Analysis: Evidence from Growth Data
Since the cost of ﬂuctuations above stems from the eﬀects of ﬂuctuations on growth, it seems
natural to begin with evidence on growth rates themselves to quantify this cost. In this section,
I pursue two diﬀerent approaches to exploiting such data to gauge the eﬀects of ﬂuctuations on
growth. First, I use a reduced-form approach that exploits variation in volatility and growth
across observations to predict the counterfactual growth rate in the U.S. if aggregate volatility
were set to zero. This approach has the virtue that it does not require estimating returns to
investment. Second, I make use of estimates of diminishing returns together with the distribution of consumption growth over the post-War period as an alternative approach to estimating
the same counterfactual growth rate.

3.1. Reduced-Form Estimates
In computing the cost of ﬂuctuations, recall that cycles aﬀect growth in two ways: they make
investment ﬂuctuate around its average level, and they can change the average level of investment itself. For the model above, eliminating the volatility of investment while holding average
investment ﬁxed provides a lower bound on the cost of business cycles. This is because investment is Pareto optimal, so allowing agents to also change the level investment can only
make them better oﬀ. In what follows, then, I abstract from the eﬀects of ﬂuctuations on the
average level of investment. This provides me with a lower bound on the cost of cycles, at least
12

when investment is socially eﬃcient. If investment is ineﬃcient, my approach could potentially
overstate (or understate) the true cost of aggregate ﬂuctuations. However, the welfare eﬀects
of changing average investment are typically much smaller than the eﬀects of reducing growth
for a given initial level of consumption, so it seems unlikely that abstracting from them will
signiﬁcantly overstate the true cost of cycles. Moreover, there is some evidence (which I cite
below) that macroeconomic volatility does not appear to be related to the level of investment,
suggesting these eﬀects are likely to be of minor signiﬁcance in any event.
Suppose we had cross-section or time-series data on growth rates, investment, and macroeconomic volatility. We could use this data to estimate average growth as a function of volatility
and other variables, i.e. to estimate
E [λ] = f (X, σ)

(3.1)

where E [λ] denotes average growth over a given time period, σ is a measure of aggregate
volatility over this same period, and X denotes other variables that aﬀect average growth.
If X includes the average level of investment over the relevant time period, we can use the
function f (·, ·) to project average growth to the case where σ = 0 but average investment
is held ﬁxed. Conveniently, previous authors have already carried out such an exercise. For
example, Ramey and Ramey (1995) estimate a linear speciﬁcation of (3.1) using cross-country
data. They estimate a model of the form
∆ ln yit = µσi + θXi + εit
¢
¡
εit ∼ N 0, σ 2
i

(3.2)

where ∆ ln yit denotes the growth rate of real GDP per capita in country i and year t, the
vector Xi is a set of country-speciﬁc explanatory variables, and the error term εit is distributed
normally with a variance σ2 speciﬁc to each country.7 This speciﬁcation assumes that in any
i
given year, the growth rate in each country varies from its historical mean by a normally
distributed error term εit , where the historical mean µσ i + θXi depends on the size of the
standard deviation of the error term for the country in question.
The model in (3.2) can be estimated by maximum likelihood. Ramey and Ramey estimate
(3.2) for a sample of 92 countries using observations between 1962 and 1985, as well as a
7

Although the relevant growth rate for calculating the welfare cost of ﬂuctuations involves the growth rate of
consumption, the two series grow at the same rate in the long-run in the model constructed above. Thus, the
fact that Ramey and Ramey use the growth rate of output per capita does not pose a problem for my analysis.

subset of 24 OECD countries using observations between 1952 and 1988. They ﬁnd that after
conditioning on average investment, growth and volatility are negatively related. The coeﬃcient
µ on the volatility term σ i is signiﬁcantly diﬀerent from zero at conventional signiﬁcance levels.
Their point estimate for µ varies across samples and speciﬁcations between −0.1 and −0.9,

although estimates appear clustered around −0.2. Since the standard deviation of per-capita
output growth in the U.S. is 2.5%, using the point estimate of −0.2 suggests that eliminating
aggregate shocks altogether should increase the growth rate by half a percentage point, from
2.0 to 2.5% per year. Applying Lucas’ estimate that an agent would sacriﬁce approximately

20% of consumption for a 1 point increase in growth when γ = 1, we obtain a cost of aggregate
ﬂuctuations of at least 10% of consumption per year, two orders of magnitude greater than
Lucas’ original estimate. It is also noteworthy that Ramey and Ramey ﬁnd that controlling for
the investment to output ratio has no eﬀect on the point estimate for µ; the negative correlation
between growth and volatility does not operate through diﬀerences in average investment shares.
This accords with additional evidence they present that the investment share of output does not
vary systematically with the volatility of GDP growth across countries. Thus, macroeconomic
volatility appears to be correlated with growth directly and not through its eﬀect on investment.
Ramey and Ramey’s results have been subsequently extended. For example, Martin and
Rogers (2000) consider the relationship between growth and volatility across European regions.
They estimate µ at −0.274, compared with Ramey and Ramey’s estimates of −0.211 and −0.384
for the full-country sample and the OECD sample, respectively. This point estimate suggests
a larger growth eﬀect of almost seven-tenths of a percentage point.8
As a further check, I examine the same relationship using time-series variation for the U.S.
Previous authors, including Zarnowitz and Moore (1986) and Ramey and Ramey (1991), already
demonstrated that there is a signiﬁcant negative correlation between the level and the volatility
of output growth in the U.S. over time. However, these studies only report raw correlations
rather than point estimates for µ, and neither controls for average investment. I therefore reexamine the time-series data to derive estimates that are comparable to the above speciﬁcation.
I follow Ramey and Ramey (1991) in dividing the data into distinct four-year episodes between
presidential elections. This identiﬁcation assumes that elections play an important role in gen8
Martin and Rogers’ methodology is not identical to Ramey and Ramey’s. For example, they regress average
growth on the unconditional standard deviation of output growth rather than use (3.2). But the explanatory
variables account for such a small share of the variation in growth that this is insigniﬁcant. Martin and Rogers
also do not control for average investment. But as noted above, average investment is uncorrelated with volatility
across countries, suggesting that omitting this variable may not be of much consequence either.

erating changes in macroeconomic policy, which in turn aﬀects the level of aggregate volatility.9
I apply this time-series data to estimate the same model as in (3.2). That is, I assume that the
growth rate each quarter is a deviation from the mean growth rate over the relevant four year
period, where the deviation is drawn from a normal distribution with standard deviation σi
speciﬁc to that period. Using quarterly data from 1953:1 - 2000:4 on output and gross domestic
investment from publicly available Bureau of Economic Analysis (BEA) sources (and converted
to per-capita levels using Census Bureau population ﬁgures), I constructed a panel dataset of
12 four-year periods, with each panel consisting of 16 quarterly observations. In contrast with
the cross-country data, there is now no need to introduce additional control variables such as
measures of human capital, which are unlikely to have much explanatory power for a single
country within a relatively short time horizon.
The results for the 12 four-year panels are reported in Table 1. The parameter of interest
µ is the coeﬃcient on the standard deviation of output growth. The point estimate of µ is
negative, although we cannot reject the null hypothesis that µ ≥ 0. Its point value is −0.348,

which again falls within the range from cross-country data in Ramey and Ramey (1995). I
experimented with dropping observations to gauge the robustness of the point estimate of µ.
For all of the subsamples I considered, the point estimate for µ remained consistently on the
order of −0.2 and −0.3. Thus, various sources of variation that can be used to estimate a

reduced-form relationship — countries, regions, and time periods — all suggest that eliminating
volatility in the U.S. should increase growth by between half and three quarters of a percentage
point when holding average investment ﬁxed.

3.2. Estimates Based on Diminishing Returns
As an alternative approach, we can use data on growth together with actual estimates of
diminishing returns to predict the same counterfactual rate at which the economy would grow
if we stabilized investment at its mean that the reduced-form estimates above are meant to
capture. To be more precise, suppose φ (·) is given by
µ

I
φ
K

I
K

¶ψ

(3.3)

I also experimented with accounting for the two presidential successions that occurred between elections during this period, but this change proved unimportant. I likewise considered grouping observations by presidential
administrations rather than by terms. However, a likelihood ratio test overwhelmingly rejected this speciﬁcation
in favor of one that allows volatility to diﬀer across terms.

where ψ ∈ (0, 1] governs the extent of diminishing returns to investment. Suppose further that
we know the distribution of trend per-capita consumption growth, λt = 1 + φ (it At ) − δ. By
inverting the function φ (·) in (3.3), we can compute the implied growth rate if we were to
stabilize investment at its mean value. Speciﬁcally,
i´ψ
³ h
1
1 + φ (E [it At ]) − δ = 1 + E (λt − 1 + δ) ψ
−δ

(3.4)

Thus, if we estimate ψ together with the distribution of the permanent part of consumption
growth, we can directly compute what the growth rate would be in the absence of ﬂuctuations
if we were to hold average investment ﬁxed.
Turning ﬁrst to ψ, note that the ﬁrst order condition of the agent’s maximization problem,
which is derived in the Appendix, can be rewritten as
·

E [βVK (Kt+1 , At+1 )]
φ (it At ) =
U 0 (Ct )
0

¸−1

(3.5)

The expression inside the brackets is the ratio of the marginal value of a unit of capital relative
to the value of an additional unit of investment. This ratio is commonly referred to as marginal
q.10 For the isoelastic function, we obtain the following relationship between the investmentto-capital ratio and q:
ln

I
K

= constant +

1
ln q
1−ψ

(3.6)

Hence, we can recover the curvature parameter ψ from the elasticity of the investment-to-capital
ratio with respect to q. Estimates of this elasticity abound in the literature. One of the few
papers that explicitly estimates an isoelastic speciﬁcation in line with (3.6) is Eberly (1997),
who estimates investment equations using micro data for various OECD countries. For the
U.S., her point estimate is equal to 1.22, with a 95% conﬁdence interval of [1.08, 1. 36], which
implies ψ ∈ [0.07, 0.26]. If the investment technology is identical in the 11 OECD countries in
her sample, we can obtain a more precise point estimate by averaging the estimated elasticity
across countries. Weighting by the number of observations in each country, the average elasticity
of investment with respect to q is equal to 1.36. This estimate lies within the 95% conﬁdence
interval for 10 out of the 11 OECD countries in the sample (including the U.S.). Since the crosscountry comparison in Ramey and Ramey (1995) is only valid if the investment technology φ (·)
is the same across countries, the consistency of this elasticity across countries in Eberly’s sample
is reassuring.
10
Since the production technologies are all homogeneous of degree 1, marginal q in this economy is equal to
average q, as shown in Hayashi (1982). The latter measure is the one frequently used in empirical work.

Other studies that look at the relationship between investment and q in the U.S. yield estimates that are broadly consistent with those reported above. For example, various papers
estimate a version of (3.6) that uses levels as opposed to logs. These can still be used to compute
a point elasticity at the sample mean to gauge the magnitude of ψ. Few of the estimates that
are reported in the literature exceed those reported above, but a signiﬁcant number purport
to ﬁnd lower elasticities. For instance, in an early contribution to this literature, Abel (1980)
reports estimates of this elasticity that range between 0.50 and 1.14, which captures the range
of most subsequent estimates. Estimates below 1 are problematic, since they imply a negative
value of ψ. However, such low values have been dismissed on the grounds that measurement
error in q would bias the estimated elasticity towards 0. For example, Cummins, Hassett, and
Oliner (1999) argue that instrumenting for q routinely yields estimates of elasticity above unity.
This claim is conﬁrmed in Eberly’s dataset: when she estimates (3.6) in levels using the same
data, her point elasticity for U.S. ﬁrms evaluated at sample means is equal to 0.56 for ordinary
least squares, and 1.06 when she instruments for q. An alternative approach to estimating the
curvature parameter ψ is pursued by Christiano and Fisher (1998). They use a method of moments approach to estimate various parameter values for their model, including the elasticity of
investment with respect to q. Their estimate of elasticity is 1.31, within the range of Eberly’s
estimates.
Next, to recover the distribution of λt from consumption data, recall that consumption growth
in the model is given by
∆ ln Ct = ln λt + ∆ ln (1 + εt )
Although we cannot simply plug in the distribution of consumption growth ∆ ln Ct into our
calculation, we can use it to recover the distribution of trend consumption growth λt . That
is, assuming productivity At follows a Markov process, we can estimate the distribution of λt
using maximum likelihood. Let P = [pij ] be an N × N matrix that denotes the transition

probabilities between the N values At can assume. For the model to accord with observed
consumption growth, I need to introduce an additional noise term; otherwise, consumption
growth can only assume at most N 2 values. I therefore assume that measured consumption
growth is given by
∆ ln Ct = ln λt + ∆ ln (1 + εt ) + η t
¢
¡
where ηt ∼ N 0, σ 2 is assumed to reﬂect measurement error. The parameters of the model
that need to be estimated are a set of trend growth rates {ln λj }N−1 , a set of ﬁrst diﬀerences in
j=0

consumption levels {∆ ln (1 + εj,j+1 )}N−2 , the transition matrix P , and the variance term σ2 .
j=0

Using the estimated transition matrix P , we can solve for the invariant distribution over the

N possible states of the world. With this distribution and the estimates for ln λt , I can then
compute expression (3.4) for a given ψ.
For my estimation, I use annual consumption data from the Bureau of Economic Analysis
(BEA), divided by population estimates obtained from the Census Bureau. The data spans the
years 1951-1998. I constructed the likelihood function recursively following Hamilton (1994).
In estimating the likelihood function, I restrict the unconditional expected growth rate E [λt ],
evaluated at the invariant distribution of the estimated transition matrix P , to equal 2.0% per
year. This accords with conventional estimates for the growth rate of per-capita consumption
based on even longer horizons than my sample period. This restriction is not too stringent,
since consumption grows at roughly this rate over my sample period. However, because of
the ﬁnite length of the sample, there is potential for the maximum likelihood estimate to force
transient phenomena into estimates of long-run trends. Table 2 reports the maximum likelihood
estimates for all the parameters for the case of N = 2 and N = 3, respectively. The estimates
reveal substantial variation in the growth of trend consumption over time, ranging between
−0.7% and 4.3% per year. There is no conﬂict between the ﬁnding of a negative growth rate

in the data and the fact that the isoelastic speciﬁcation implies φ (x) ≥ 0, since growth can be
negative if the depreciation rate δ exceeds φ (iA). The fact that estimates for ∆ ln (1 + ε) are
negative imply that trend consumption grows more rapidly when consumption is below trend,
i.e. the level of consumption falls when aggregate productivity is high. As noted earlier, this
could simply reﬂect a disproportionate shift towards investment when aggregate productivity
rises, and is not inconsistent with the fact that consumption growth is procyclical over the long
run. By contrast, the model does imply that output and investment grow more rapidly when
they are above trend. Estimating a similar decomposition for these two series conﬁrms this
prediction, but I omit these results for the sake of brevity.
Using the invariant distribution for consumption growth reported in Table 2, and assuming
δ = 0.09 as is standard in the literature, the implied growth rate (3.4) in the two-regime model
is given by

´
³
1
1 ψ
1 + 0.37 (0.0857) ψ + 0.63 (0.1241) ψ
− 0.09

(3.7)

´
³
1
1
1 ψ
− 0.09
1 + 0.29 (0.0830) ψ + 0.42 (0.1127) ψ + 0.29 (0.1329) ψ

(3.8)

and in the three-regime model it is given by

Figure 2 plots these two values against the investment elasticity (1 − ψ)−1 . Abel’s largest
estimate for this elasticity, 1.14, implies that stabilizing investment at its average value will
18

raise the growth rate from 2% to 2.78% and 2.91%, respectively. Eberly’s point estimate for
this elasticity, 1.22, yields more conservative estimates of 2.58% and 2.64% per year, respectively.
Christiano and Fisher’s estimate of 1.31, implies growth rates of 2.46% and 2.49% per year,
respectively. Finally, an elasticity of 1.36, which is Eberly’s average estimate across OECD
countries, implies growth rates of 2.39% and 2.42% per year, respectively. Thus, for the range
of estimates of the investment elasticity reported in the literature, the two- and three-regime
models produce similar growth eﬀects, although the three-regime model suggests a slightly larger
eﬀect. In either case, cyclical ﬂuctuations appear to lower growth by about 0.4 − 0.9 percentage
points, which is the same range as the one we obtain using the reduced-form approach above.
Thus, the two approaches to using growth data are consistent, and both translate into a welfare
cost of at least 8% of consumption per year.

4. Quantitative Analysis: Calibration
The previous section provided estimates of the cost of cyclical ﬂuctuations based only on growth
data. In this section, I examine whether a growth eﬀect of the magnitude implied in the
previous section can be reconciled with other macro data. More precisely, since the growth rate
ultimately depends on investment, I examine whether a large cost of cycles can be reconciled
with empirical evidence on aggregate investment. That is, I calibrate the process for {At } in
the model from Section 2 to match the empirical volatility of trend growth reported in Table 2,
and then check whether the path for investment given this process is consistent with the actual
behavior of investment over the cycle.
I begin my analysis using the standard CRRA utility in Section 2. This function involves
two parameters: the discount rate β and the inverse intertemporal elasticity of substitution γ.
Following Lucas, I set β to 0.95 and γ to 1. On the production side, I need to specify the Markov
process for {At } that governs the production of consumption goods. To simplify matters,
I assume At follows a two-state Markov process, where the transition probabilities between
the two states are taken from the two-regime model in Table 2. Since the previous section
suggests that the economy would likely grow at a rate of 2.5% in the absence of ﬂuctuations, I
calibrate the mean of At , denoted by A, to deliver a growth rate of 2.5%. Given the transition
¡
¢
probabilities, I then set A1 = (1 + x) A and A0 = 1 − 0.633 x A to ensure that the average of
0.367

{At } is A, and I choose x to match the standard deviation of trend consumption growth λt in

Table 2 of 1.85 percentage points.

Lastly, I need to parameterize the technology for producing investment goods. I maintain
the isoelastic form for φ (·) in (3.3). To set ψ, I could have used estimates of the elasticity of
investment with respect to q as before. However, the model suggests another way of selecting
this parameter. Consider the deterministic steady state of the model, i.e. where x = 0. As
previously observed by Jones, Manuelli, and Stachetti (1999), if we set ψ = 1 and choose A to
match the empirical average growth rate, the consumption share of output will only equal 0.32.
Yet empirically, consumption accounts for about two thirds of output. Jones, Manuelli, and
Stacchetti argue that this discrepancy can be resolved by counting a fraction of investment in the
model as consumption, under the pretext that investments in human capital such as education
and health care expenditures are counted as private consumption in national income accounts.
However, between 1951 and 1998, total expenditures for these two categories account for at most
20% of consumption, and in earlier years account for only 6% of total consumption, far less
than the nearly 50% share needed to reconcile with the data.11 An alternative interpretation
is that constant returns to investment provide overly powerful incentives for agents to invest.
This suggests we can use the fact that consumption accounts for about two thirds of income to
determine the extent of diminishing returns. Formally, I calibrate ψ and A to match a growth
rate of 2.5% and a consumption share of output of 0.67. This yields a value of ψ = 0.234, or
an elasticity of investment with respect to q of 1.31, on target with estimates of this elasticity
described in the previous section.
Once all of the parameter values are assigned, we can solve the model for diﬀerent values of
x and choose the one for which the standard deviation of trend consumption growth is 1.85
percentage points. A brief discussion of how I solve the model is contained in the Appendix.
The relevant value of x turns out to be 0.374. At this value, the average growth rate along
the equilibrium path is 2.13% per year, despite the fact that the deterministic steady state
growth rate is calibrated to 2.5% per year. The lower bound on the cost of ﬂuctuations is
actually a little larger than the amount an individual would pay to increase the growth rate
from 2.13% to 2.5%, since the increase in growth from eliminating volatility in investment is
oﬀset by the fact that average investment E [it At ] falls slightly once we eliminate ﬂuctuations
(i.e. the absence of ﬂuctuations will lead the agent to re-optimize between present and future
consumption towards slightly higher present consumption). But the calibration conﬁrms my
previous results that suggest cyclical ﬂuctuations will lower the long-run average growth rate
11

Alternatively, some human capital expenditures might not be counted in either GDP or investment, in which
case the measured investment share would be smaller than its true value. Jones and Manuelli explore this
possibility in subsequent work, but they ﬁnd that only a negligible fraction of such expenditures is likely to be
missing from GDP when returns to investment are assumed constant.

by about 0.4 percentage points relative to the non-stochastic environment.
That said, the calibration also reveals that the investment path implied by the model is
much less volatile than what we actually observe over the cycle. Along the optimal path, the
investment share of output it varies between 0.280 and 0.350. This implies a standard deviation
of 10%, only two-thirds of the empirical counterpart of 14%. A more dramatic illustration of
this discrepancy involves the volatility of investment relative to the volatility output, a measure
often emphasized in the real business cycle literature. If we compare the standard deviation of
it At to the standard deviation of At , the fact that it is not very volatile implies the standard
deviation of detrended investment is only 11% larger than the standard deviation of detrended
output. But the actual standard deviation of investment is 2.5 − 3 times as large as that of
output. Thus, if we introduce empirically plausible degrees of curvature in the investment
sector, investment in the model winds up being too smooth along the equilibrium path when
we calibrate the model to match U.S. data. To put it another way, the model suggests that if
investment volatility were really so costly, aggregate investment should be much less volatile
than its empirical counterpart. Agents in the model try to avoid varying investment over time
given the large cost of investment volatility, and the only reason growth rates ﬂuctuate so much
in the calibration is that output is assumed to be so volatile that it is hard to ﬁnd resources to
smooth investment over time without making consumption overly volatile.
At ﬁrst glance, the above analysis suggests there is contradiction inherent within the idea
that growth eﬀects lead to a large cost of business cycles: if cycles are so costly, agents will
act to undo that cost. However, one has to be careful about using the above model to draw
this conclusion. To appreciate why the model might be problematic, recall that the ﬁrst-order
condition for (2.4) is given by
µ ¶
It
0
φ
= q −1
Kt
where q is the ratio of the market value of equity to the replacement value of physical capital.
As such, the volatility of investment in the model is intimately related to the volatility of asset
prices. But it is well known that in the standard real business cycle model, asset prices are far
too smooth in comparison with the data. Formally, let q0 denote the value of qt when At = A0
and similarly for q1 . For the CRRA utility above with γ = 1, the ratio of q across the two
regimes is given by
q1
λ1 C0
=
(4.1)
q0
λ0 C1
where λ = 1 + φ (iA) − δ denotes the trend growth rate for each respective level of productivity.
λ1
≈ 1 and consumption is not very volatile for plausible parameter values, q will hardly
Since
λ0
21

vary over the cycle, and thus neither will investment. But empirically, q is highly volatile, as
will be discussed in more detail in the next section. It is therefore unreasonable to expect
a model which does a notoriously poor job of accounting for the volatility of asset prices to
generate plausible investment volatility when investment inherently responds to these prices.
In fact, several authors have recently argued that if we modify the real business cycle model
so that it can match empirical evidence on asset prices, investment will be overly volatile unless
we allow for some diminishing returns to investment. This was demonstrated, among others,
by Jermann (1998), Christiano and Fisher (1998), and Boldrin, Christiano, and Fisher (2001).
These papers all set out to modify the traditional real business cycle model so that it can accord
with the data on asset prices. Building on insights from the ﬁnance literature, they suggest
replacing (2.1) with a more general function
U ({Ct }) =

∞
X
t=0

βt

(Ct − bCt−1 )1−γ − 1
1−γ

(4.2)

for some b ≥ 0. Note that the CRRA utility is just a special case of this function where
b = 0. But if we allow b to be suﬃciently diﬀerent from 0, marginal utility will be quite
volatile even when consumption is relatively smooth. This translates into a relatively volatile
series for q, which provides strong incentives to vary investment over time. In fact, in the
absence of diminishing returns, investment is too volatile, and all three papers above argue
that some friction in the investment good sector such as adjustment costs must be introduced
to moderate the volatility of investment. Jermann (1998) provides a particularly nice intuition
for why business cycle models that are capable of matching asset prices require both habit
formation and adjustment costs: “With no habit formation, marginal rates of substitution are
not very volatile, since people do not care very much about consumption volatility; with no
adjustment costs, they choose consumption streams to get rid of volatility of marginal rates of
substitution. They have to both care, and be prevented from doing anything about it.”12
Since the aforementioned work on incorporating asset prices into real business cycle models
abstracts from growth considerations, we need to verify that the same intuition carries over to
endogenous growth settings. After all, agents might have more incentive to keep investment
12

An alternative modiﬁcation of the RBC model that also generates more volatile asset prices is the generalization to a multi-country setting, as in Baxter and Crucini (1993). In this case, asset prices are volatile not because
marginal rates of substitution are volatile, but because dividends are. Speciﬁcally, production will tend to be
concentrated in the country with the highest productivity, so owning equity in a country-speciﬁc technology will
yield low dividends when that country is not the leader. Given the implied volatility of asset prices, Baxter and
Crucini similarly need to introduce adjustment costs for investment not to be too volatile.

stable if it aﬀects long-run growth. I therefore re-evaluate the model using (4.2) as my utility
speciﬁcation. I continue to set γ = 1, which previous authors have argued is a reasonable
value. I calibrate the habit parameter b so that along the equilibrium path, investment will
be 2.5 times as volatile as output, in line with the estimate provided in Jermann (1998). In
choosing the parameter ψ, note that the intertemporal Euler equation — which is provided in
the Appendix — implies that the steady-state growth rate does not depend on b. Since the
CRRA utility function is just a special case of (4.2) in which b = 0, it follows that we would
still require ψ = 0.234 for the model to yield an average consumption share of two thirds.13
Solving the model is now more diﬃcult, since there is an additional state variable in the form
of lagged consumption Ct−1 . I therefore resort to a linear approximation of the intertemporal
Euler equation to solve the model. I provide a brief discussion of my approach in the Appendix.
The value of b that generates the requisite relative volatility in investment is b = 0.7. For this
value, it ranges between 0.12 and 0.42 (although these are extreme values; it no longer assumes
a two point distribution as in the CRRA case). This value of b is consistent with what previous
researchers have argued is necessary to accord with data on the equity premium. For example,
Boldrin, Christiano, and Fisher (2001) estimate b = 0.73 in calibrating their RBC model to
accord with the empirical equity premium. Christiano and Fisher (1998) estimate a somewhat
lower value of b = 0.6, while Jermann (1998) estimates a somewhat higher value of b = 0.82.
Thus, if we were to impose the same preference parameters that previous researchers have
argued can help to explain asset price volatility, the implied relative volatility of investment
would be consistent with what we observe in the data. Moreover, the average growth rate in
equilibrium is now 2.05%, even as the non-stochastic steady state remains at 2.5%, suggesting
ﬂuctuations reduce growth by as much as 0.45 percentage points. The average investment rate
E [it At ] is approximately the same as the non-stochastic steady state investment rate i∗ A∗ , so
that nearly all of the eﬀect of ﬂuctuations on growth is due to diminishing returns.
To generate a standard deviation for the growth rate λt of 1.85 percentage points, we now
need to set x = 0.165, less than half the value required under CRRA utility. Although this still
implies rather large shocks to productivity, the marginal product of capital which At is meant
to capture will in practice depend on labor and utilization rates, and will thus be more volatile
13

Carroll, Overland, and Weil (2000) examine a related model in which growth does depend on the degree of
habit. But they use a diﬀerent speciﬁcation where agents care about the ratio between consumption and habit,
not the diﬀerence between them. If utility is logarithmic in the ratio of consumption to habit, the steady state
growth rate is again independent of habit. Since Carroll et al assume greater curvature in utility than the log
case, they ﬁnd that habit aﬀects growth. If I were to use their preferences, agents would have even greater
incentive to invest in growth, requiring a smaller value of ψ to match the consumption share of output.

than measured total factor productivity. To match the relative volatility of investment using
a less volatile process for At , we would need for the average investment share of output i to
account for less than a third, e.g. by allowing for some fraction of output that is not consumed
to be allocated to government spending. Ironically, this would only be possible if we were to
lower ψ to further discourage incentives for investment. As such, we can modify the model to
replicate the volatility of growth rates with less volatile exogenous shocks, but only if the eﬀect
of ﬂuctuations on growth is assumed to be even larger. This would be consistent with some of
the larger reduced-form estimates above of three quarters of a percentage point, but as I discuss
in the next section, such low values of ψ are probably implausible.
Why does habit formation allow for such volatile investment despite the presence of diminishing returns? Consider a negative productivity shock which lowers households’ incomes.
Consumers are reluctant to decrease their consumption immediately — habit formation implies
they will want to do it gradually. Thus, on impact, households will try to sell oﬀ their assets to
ﬁnance consumption. This causes asset prices to fall, which signals to ﬁrms that they should
cut back on accumulating capital. Eventually, if the recession continues, consumers will get
habituated to low consumption, they will not be as driven to get rid of their assets, and investment will begin to recover. This overshooting explains why investment is so volatile. By
contrast, under the standard CRRA utility, households do not need to let their consumption
fall gradually, so there is no equivalent drop in asset prices and investment.
In sum, once we modify the endogenous growth model to allow for more volatile asset prices,
it is possible to reconcile the presence of diminishing returns to investment with the fact that
investment is so volatile empirically. Essentially, we need to introduce an additional motive to
smooth consumption that oﬀsets the incentive for agents to smooth investment. Such motives
are not altogether implausible given the large premium on risky equity, which suggests that
in practice agents are averse to too much consumption volatility. However, once we modify
preferences so that agents are more willing to smooth their consumption, we need to conﬁrm
such a growth eﬀect is still costly to the agent. It turns out that a one percentage point reduction
in growth is actually more costly under (4.2) than under the CRRA preferences with the same
γ. Intuitively, habit formation implies that agents care even more about growth, since what
they care about is how much more they consume relative to past consumption. To compute
the lower bound on the cost of cycles described above, consider how much an agent would
be willing to sacriﬁce to increase the growth rate by 0.45 percentage points in the absence of
ﬂuctuations, as the above calibration suggests would happen. That is, given two consumption
0
streams Ct = λt C0 and Ct = (λ + .0045)t C0 , by what constant scaling factor should we adjust
24

0
the ﬁrst consumption stream to deliver the same utility as Ct ? For λ = 1.02 and γ = 1, the
scaling factor µ solves the equation
Ã
!
µ
¶
∞
X
1.0200 t−1 1.0200 − b
β t−1 ln (1 + µ)
=0
(4.3)
1.0245
1.0245 − b
t=1

When b = 0, which corresponds to the CRRA case, µ = 0.097, i.e. an individual would
sacriﬁce 9.7% of his consumption each year for consumption to grow more rapidly. When
b = 0.7, the corresponding µ is given by 0.107, or 10.7% per year. Allowing for habit formation
therefore serves to increase the cost of business cycles in several ways: it makes agents care
more about consumption risk; it necessitates greater diminishing returns to investment, which
in turn implies ﬂuctuations will lower average growth by a larger amount; and it causes agents
to care more about the growth rate, and thus to suﬀer more from the eﬀects of ﬂuctuations on
growth.
It should be emphasized that the large cost of business cycles under habit formation does
not simply reﬂect the fact that agents are more averse to the volatility of consumption over the
cycle. In fact, as discussed in the Introduction, previous work such as Otrok (2001) has argued
that even with non-separable preferences such as (4.2), the cost of consumption volatility should
be viewed as small. This is because they compute the cost of business cycles using the actual
volatility of per-capita consumption over the cycle, which is quite small. By contrast, the cost
of cycles due to their eﬀects on growth is proportional to the cost of the larger counterfactual
consumption volatility that would be required to keep investment constant over time (since if
the cost of cycles were less than this cost, agents would never tolerate volatile investment). In
other words, the large cost of ﬂuctuations above comes from their eﬀects on growth, not just
the increased curvature of the utility function.

5. Quantitative Analysis: Estimates Based on q Data
The previous section demonstrated that investment can be volatile even with adjustment costs
as long as q is suﬃciently volatile. Since q plays such an essential role, I now propose a diﬀerent
approach to calibration that uses shocks to q as the primitive instead of deriving them from
some other shock such as ﬂuctuations in productivity. Given a process for q, we can use (3.5)
I
, from which we can derive the growth rate λ = 1 + φ (I/K) − δ. Although q is
to derive
K
determined endogenously, abstracting from the fundamentals that determine q is legitimate,
since the true source of ﬂuctuations is not directly relevant for gauging the eﬀect of ﬂuctuations
25

on growth beyond its implications for q. As such, the question of whether cyclical ﬂuctuations
have a signiﬁcant eﬀect on growth can be analyzed separately from the question of what shock
is responsible for ﬂuctuations in q. Similarly, given a particular utility function, we can directly
calculate the welfare cost associated with this growth eﬀect independently of the source of
these ﬂuctuations. However, as already noted, the source of ﬂuctuations will be important for
determining whether there is a role for stabilization policy.
I begin with a discussion of q data. One of the ﬁrst to construct a series on aggregate q was
Summers (1981). He constructed two annual series from 1931 to 1978, one that is equal to the
value of equity divided by the value of physical capital, and another which amends this ratio
to account for taxes and depreciation allowances that should aﬀect the incentives of ﬁrms to
undertake investment. Restricting attention to the post-War period, the average value of his
measure of regular q is 0.96 with a standard deviation of 24%, while the average value of the
tax-adjusted q series over the same period is 1.92 with a standard deviation of 38%. However,
the data used in constructing these series has been subsequently revised by the BEA. Blanchard,
Rhee, and Summers (1993) reconstruct the series for standard q in light of BEA revisions, and
extend the original series up to 1990. Their estimate of average of q for the post-War period
falls to 0.64, and the implied standard deviation rises slightly to 29%. Bernanke, Bohn, and
Reiss (1988) construct a quarterly series of tax-adjusted q for the period between 1947 and
1983. Their average for tax-adjusted q is equal to 1.41, with an annual standard deviation of
37%. Thus, while Summers’ original series tend to overstate the levels of q, they provide fairly
reliable measures of the volatility of aggregate q that are conﬁrmed in subsequent studies.
Assuming a two-regime model, we can translate the standard deviation of q into a ratio
q1
that we can then use to calibrate the eﬀect of ﬂuctuations on growth. Once again, let
q0
¡
¢
q1 = (1 + x) q and q0 = 1 − 0.633 x q, so that E (qt ) = q. A simple computation reveals that
0.367
q
the standard deviation of qt in percentage terms, denoted sd, is equal to 0.633 x. Hence, the
0.367
q1
ratio
is given by
q0
q
1 + 0.367 sd
q1
1+x
0.633
q
=
0.633 =
q0
1 − 0.367 x
1 − 0.633 sd
0.367

q1
= 1. 97. If the standard deviation is instead
q0
38%, as in the tax-adjusted series, the ratio will be equal to 2.56.
For a standard deviation of 29%, it follows that

Using the fact that φ (it At ) = λt − 1 + δ, and the fact that the ﬁrst order condition implies
26

−1
that φ0 (it At ) = qt , we can use the ratio of q over the cycle above together with data on trend
growth rate λ over the cycle to pin down the extent of diminishing returns ψ. In particular,
µ
¶1
µ ¶ 1
λ1 − 1 + δ ψ
i1 A1
q1 1−ψ
=
=
(5.1)
λ0 − 1 + δ
i0 A0
q0

q1
and λ0 and λ1 are enough to solve for ψ. Intuitively, the more volatile is q
q0
for a given volatility in the growth rate λ, the more diminishing returns there must be to keep

so that the ratio

growth rates in line with the data. This is because the lower is ψ, the less investment responds
to variation in q, and the less growth responds to variation in investment. Using Table 2, we can
assign λ1 = 1.034 and λ2 = 0.996. When the ratio of q is 1.97, the value of ψ which solves (5.1)
is given by 0.35, which implies an elasticity of investment with respect to q of (1 − ψ)−1 = 1.54.
Based on Figure 2, eliminating the volatility of investment around its mean should then increase

the growth rate from 2% to 2.3%. This estimate is a little lower than the estimates reported
above. However, if the ratio q1 /q0 were instead set to 2.56, the value of ψ that solves (5.1) is
equal to 0.28, which is on par with some of the more conservative estimates reported above. In
particular, this value of ψ implies an elasticity of investment with respect to q of 1.39, which by
Figure 2 suggests that eliminating ﬂuctuations will increase the growth rate from 2% to 2.4%.
Thus, calibrating the model directly to q points to a growth eﬀect that is a little smaller than
suggested in the previous two sections, although it is still quite substantial. For example, using
(4.3), we can deduce that an agent with conventional log utility would be willing to pay at least
6.4% of consumption each year to avoid the lower growth associated with cycles when ψ = 0.35
(or 7.0% when we set the habit parameter b to 0.7), and 8.5% of consumption when ψ = 0.28
(respectively, 9.2% when b = 0.7).

Note that we can use (5.1) not only to pin down ψ, but also to back out the volatility of
investment over the cycle. That is, once we estimate the curvature parameter ψ that reconciles
q and λt , we can use it to solve for the ratio i1 A1 /i0 A0 , which reﬂects the degree to which
investment at its peak exceeds its value during troughs. Figure 3 illustrates how the implied
ratio of investment rates over the cycle varies with ψ. When ψ = 0.35, the value we get
from using the standard q series above, the investment ratio i1 A1 /i0 A0 = 2.85. This value
is reasonable given that the investment share of output it ranges between 10% and 20% over
the post-War period, so i1 /i0 = 2 is plausible, and the investment share of capital it At will be
more volatile than it given that At is positively correlated with it . When we instead consider
ψ = 0.28, as suggested by the tax-adjusted q series, the ratio of investment rates over the cycle
rises to 3.69. This value is rather large, but given that investment is so small relative to the
capital stock, even small diﬀerences in investment will appear large when translated to ratios,
27

so that this ratio is not entirely unreasonable. However, Figure 3 suggests that for still lower
values of ψ, investment has to be incredibly volatile to accord with the variability of λt over
time. For example, for ψ = 0.18, which accords with Eberly’s (1997) point estimate for the
elasticity of investment with respect to q of 1.22, investment at its peak must be almost 8 times
as large as at its trough to account for the time series of λt . For ψ = 0.12, in line with Abel’s
(1980) most generous point estimate for this elasticity, investment at its peak must be over 21
times as large as at its trough. Both of these seem highly implausible. Thus, while values of
ψ that are associated with growth eﬀects of 0.3 and 0.4 percentage points can account for the
volatility of growth rates without requiring wild investment swings over the cycle, the same
cannot be said for values of ψ that are associated with a larger eﬀect on growth.
In sum, even if we are unsure about the exact source of aggregate ﬂuctuations, we can use
data on q to estimate the eﬀects of these ﬂuctuations on growth without having to take a stand
on the precise nature of the shocks that drive these ﬂuctuations. The data reveal that cycles
reduce the growth rate by between 0.3 and 0.4 percentage points, depending on how exactly q
is measured. This is a little less than the estimates of between 0.4 and 0.9 percentage points
suggested in the previous sections. But even if cycles reduce growth by only 0.3 percentage
points, they will make agents signiﬁcantly worse oﬀ.

6. Conclusion
This paper considers a cost of aggregate ﬂuctuations that is due not to consumption volatility,
but to the eﬀects of ﬂuctuations on growth. In a sense, it closes a circle that began with Lucas
(1987), who argued that growth matters for welfare while business cycles do not; if business
cycles can aﬀect the rate of economic growth, they can matter after all. My analysis suggests
that eliminating ﬂuctuations can increase the growth rate by a little under half a percentage
point without aﬀecting average initial consumption. This produces a cost of business cycles 100
times larger than what Lucas computed based on the costs of consumption risk alone.
The key to obtaining such large costs is the presence of diminishing returns to investment.
A variety of indicators point to the presence of signiﬁcant diminishing returns in the data: the
signiﬁcant negative relationship between growth and volatility across countries, regions, and
time periods for a ﬁxed average investment; the low estimated elasticity of investment with
respect to q both in the U.S. and in other OECD countries; the high share of consumption
in aggregate output, which suggests low incentives to undertake investment; and the relatively

large volatility of q compared to more modest volatility of per-capita consumption growth.
All of these consistently point to a curvature parameter ψ on the order of 0.25 − 0.35, which

suggest that cycles reduce the average growth rate by between 0.3 and 0.5 percentage points.
In a previous version of this paper, I argued that there is similar evidence in favor of substantial
diminishing returns to R&D, which serves as the engine of growth in alternative models. Thus,
large costs are likely to arise in a variety of growth models, although the fact that investment

is not always eﬃcient in some of these models requires further work to investigate the welfare
eﬀects of changes in the level of investment. More generally, the introduction of ineﬃciencies
into the growth process that are absent in the AK framework above raises various interesting
questions, such as the timing of growth-enhancing activity, which I have begun to explore in
other work. Regardless of what costs of business cycles ultimately come out of such models, the
analysis above certainly suggests that ignoring the eﬀects of ﬂuctuations on growth can lead to
incorrect conclusions regarding the welfare implications of these ﬂuctuations.
Lastly, while the arguments presented here suggest that business cycles may be quite costly,
it bears repeating that it does not immediately follow from this that stabilization policy is
inherently desirable or could avoid these underlying costs. This conclusion hinges on the precise
nature and source of aggregate ﬂuctuations, and the degree to which government policy can
truly eliminate them. This paper opens the door to large welfare eﬀects of ﬂuctuations, and
the growth channel it explores can and should be used to examine policy in the multitude of
models that have been devised to explain business cycle ﬂuctuations.

Table 1: The Relationship between Growth
and Volatility in U.S. time-series data
Dependent variable: real per-capita GDP growth

U.S.
12 Four-year Panels
Intercept

0.0065
(0.0055)

Average I/Y

0.0132
(0.0292)

-0.3484
(0.2216)

Log likelihood
# of panels
# of obs

651.99
12
192

Data source: U.S. quarterly real GDP and real gross domestic investment (in 1996 chained dollars) are taken from
BEA estimates for 1953:1 - 2000:4. Real GDP is divided by U.S. population, where population each quarter was
interpolated geometrically from annual population figures taken from the U.S. Census Bureau for July 1st of each
year. Panels correspond to presidential terms, so each panel consists of 16 quarterly observations. µ denotes the
coefficient on the standard deviation term in equation (3.2) in the text. The method of estimation is maximum
likelihood. Numbers in parentheses denote asymptotic standard errors.

Table 2: Maximum Likelihood Estimates for
Markov Model of Consumption Growth
2 regime model

coef

p01
p10

0.5369
0.3117

0.1543
0.1113

ln λ0
ln λ1

-0.0043
0.0341

∆ln(1+ε01)

σ2

3 regime model

coef

p01
p02
p10
p12
p20
p21

0.1110
0.3936
0.2308
0.1812
0.1730
0.4724

0.1356
0.1326
0.1057
0.1086
0.1317
0.1892

-0.0066

ln λ0
ln λ1
ln λ2

-0.0070
0.0227
0.0429

-0.0033
0.0039

-0.0250

0.0049

∆ln(1+ε01)
∆ln(1+ε12)

-0.0174
-0.0201

0.0045
0.0031

0.00024

0.00009

σ2

0.00010

0.00003

Implied
Invariant distribution

State

Prob

State

Prob

0
1

0.37
0.63

0
1
2

0.29
0.42
0.29

Data sources: real consumption per capita between 1950-1998 is taken from the Economic Report of the President. pij denotes the transition rate
between states i and j. lnλj corresponds to the expression φ (iA)+1- δ in the model. Since the estimation constrains the average growth rate to
equal 2.0% per year, only N-1 growth rates are estimated in the N regime model. ln∆εij denotes the change in the level of consumption for a
transition rate between states i and j. σ2 denotes the variance of the measurement error term η. The implied invariant distribution is computed
using point estimates for the transition matrix pij. Standard errors on coefficients correspond to asymptotic standard errors computed using the
expected score method.

ln Ct

(a) Increased growth from reduced volatility of investment

ln Ct

(b) Increased growth from higher average investment

Figure 1: Consumption Paths under Endogenous Growth

E(λ )
2-Regime Model

3.2

3-Regime Model

3.0
Abel (1980)

2.8

Eberly (1997)

2.6

Christiano and Fisher (1998)

2.4
2.2

(1-ψ)

-1

2.0
1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Figure 2: Predicted growth rate after stabilization
for a given investment elasticity

2.1

2.2

2.3

2.4

i1A1
i0A0

11
10
9
8
7
6
5
4
3
2
1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75

Figure 3: The volatility of investment volatility required to match
the volatility of growth rates as a function of ψ

Appendix
Proof of Proposition: We guess that the value function V (Kt , At ) assumes the form
V (Kt , At ) =

1−γ
a (At ) Kt
1−γ

It is easy to conﬁrm that this multiplicatively separable function conforms to the Bellman
equation. Ignoring the constant term in the utility function, we can rewrite the above Bellman
equation as
(ct At )1−γ
+ βE [a (At+1 ) | At ] [φ ((1 − ct ) At ) + 1 − δ]1−γ
1−γ
ct ∈[0,1]

a (At ) = max

Note that for any function V (·, ·) which is increasing in both arguments, the expression
)
(
(ct At Kt )1−γ
max
+ βE [V ((φ ((1 − ct ) At ) + 1 − δ) Kt , At+1 ) | At ]
ct
1−γ
must be increasing in At and Kt given that Prob(At+1 ≤ x | At ) is weakly decreasing in At .
The standard ﬁxed-point argument establishes that the solution to the Bellman equation V (·, ·)
a (A) K 1−γ
, it follows that a (A) is
is increasing in both arguments. Hence, if V (K, A) =
1−γ
increasing in A. This in turn implies that the conditional expectation E [a (At+1 ) | At ] is
increasing in At .

Using the ﬁrst order condition of the Bellman equation, we have
(ct At )−γ = βE [a (At+1 ) | At ] [φ ((1 − ct ) At ) + 1 − δ]−γ φ0 ((1 − ct ) At )
We can rearrange this equation to obtain
0

φ (it At ) =

1
βE [a (At+1 ) | At ]

φ (it At ) + 1 − δ
At (1 − it )

Let x = iA. Then we can rewrite the ﬁrst order condition as
µ
¶
φ (x) + 1 − δ γ
0
φ (x) = k (A)
A−x

¶γ

φ (x) + 1 − δ
where k (A) is a positive constant
< 0. Since φ (x) is decreasing in x and
At − x
is increasing in x, there exists at most one x which solves this equation. Existence then follows
and k 0 (A)

¶γ

from the limit conditions limx→0 φ0 (x) = ∞ and limx→∞ φ0 (x) = 0. If we rewrite this equilibµ
¶
φ (x) + 1 − δ γ
0
rium condition as f ≡ φ (x) − k (At )
= 0, the fact that fx < 0 and fA > 0
A−x
imply that as A rises, x must also rise to maintain f = 0, which establishes the claim. ¥
Solving the model for CRRA utility: From the proof of the proposition above, we
a (A) K 1−γ
know that the value function V (K, A) has the form
. Substituting into the Bellman
1−γ
equation yields the following equation, one for each possible realization of At :
E [a (At+1 ) | At = A]
(c (A) AK)1−γ
a (A) 1−γ
K
+β
[(φ ([1 − c (A)] A) + 1 − δ) K]1−γ
=
1−γ
1−γ
1−γ
Taking the ﬁrst order with respect to c (A) on the RHS of the above yields the following Euler
equation for c (A):
(c (A) A)−γ = βE [a (At+1 ) | At ] [φ ((1 − c (A)) A) + 1 − δ]−γ φ0 ((1 − c (A)) A)
Thus, we have two non-linear equations for each pair of variables c (A) , a (A). Solving the
model amounts to solving this system of non-linear equations,with two equations for each value
At can assume. ¥
Intertemporal Euler Equation under Habit: The intertemporal-Euler equation for a
general utility function U ({Ct }) is given by
·
¸
λt+1
Uc,t+1
Uc,t = Et β
λt
where Uc,t denotes the marginal utility of U with respect to Ct and λt denotes the Lagrange
µ
µ ¶
¶
It
multiplier on the constraint Kt+1 = 1 + φ
− δ Kt . For (4.2), we have
Kt
Uc,t = (Ct − bCt )−γ − bβ (Ct+1 − bCt+1 )−γ
while adjustment costs imply that
µ
µ
¶
µ
¶
¶
It+1
It+1 Ct+1
λt+1
φ0 (It /Kt )
0
1−δ+φ
+φ
= 0
λt
Kt+1
Kt+1 Kt+1
φ (It+1 /Kt+1 )
Deﬁne bt =
c

Ct
. Then we can rewrite the Euler equation as
Kt
c
c c
c
Et [v (bt−1 , bt , bt+1 , bt+2 , At−1 , At , At+1 )] = 0

where the function v can be obtained by substitution. The deterministic steady state is the
constant value b which solves
c
¢
¡
v b, b, b, b, A, A, A = 0
c c c c

c
c
c
and one can show that if we set bt−1 = bt = bt+1 = bt+2 , the coeﬃcient b drops out. To solve
c
for the non-steady state dynamics, I use a linear approximation of v (·) at the deterministic

steady state by evaluating the relevant derivatives of v with respect to each of its arguments,
and then use this to derive a linear approximation of the policy rule which is given by bt+1 =
c
α0 bt + α1 At + α2 At−1 for some coeﬃcients α0 , α1 , and α2 that I need to solve for. ¥
c

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Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Dynamic Monetary Equilibrium in a Random-Matching Economy
Edward J. Green and Ruilin Zhou

WP-00-1

The Effects of Health, Wealth, and Wages on Labor Supply and Retirement Behavior
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Market Discipline in the Governance of U.S. Bank Holding Companies:
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Using Market Valuation to Assess the Importance and Efficiency
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Subordinated Debt and Bank Capital Reform
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Bank Capital Regulation With and Without State-Contingent Penalties
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Puzzles in the Chinese Stock Market
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Idiosyncratic Risk and Aggregate Employment Dynamics
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Data Revisions and the Identification of Monetary Policy Shocks
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Equilibrium Lending Mechanism and Aggregate Activity
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Impact of Independent Directors and the Regulatory Environment on Bank Merger Prices:
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Productivity Growth in the 1990s: Technology, Utilization, or Adjustment?
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WP-01-06

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Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy
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and Community Banks
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Gaps and Triangles
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A Real Explanation for Heterogeneous Investment Dynamics
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Price Level Uniformity in a Random Matching Model with Perfectly Patient Traders
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Working Paper Series (continued)
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The Effect of Part-Time Work on Wages: Evidence from the Social Security Rules
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Antidumping Policy Under Imperfect Competition
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WP-01-21

Is the United States an Optimum Currency Area?
An Empirical Analysis of Regional Business Cycles
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A Note on the Estimation of Linear Regression Models with Heteroskedastic
Measurement Errors
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The Mis-Measurement of Permanent Earnings: New Evidence from Social
Security Earnings Data
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Pricing IPOs of Mutual Thrift Conversions: The Joint Effect of Regulation
and Market Discipline
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Opportunity Cost and Prudentiality: An Analysis of Collateral Decisions in
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Outsourcing Business Services and the Role of Central Administrative Offices
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Strategic Responses to Regulatory Threat in the Credit Card Market*
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The Optimal Mix of Taxes on Money, Consumption and Income
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Expectation Traps and Monetary Policy
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Monetary Policy in a Financial Crisis
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WP-02-05

Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
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Technological Progress and the Geographic Expansion of the Banking Industry
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WP-02-06

WP-02-07

Working Paper Series (continued)
Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality  Between 1980 and the Early 1990s
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The Immediacy Implications of Exchange Organization
James T. Moser

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Maternal Employment and Overweight Children
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The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
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On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
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WP-02-12

Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
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Technology Shocks Matter
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WP-02-14

Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou

WP-02-15

Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles

WP-02-16

Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
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WP-02-17

Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman

WP-02-18

Location of Headquarter Growth During the 90s
Thomas H. Klier

WP-02-19

The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
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WP-02-20

On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones

WP-02-21

The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French

WP-02-22

Working Paper Series (continued)
Inter-industry Contagion and the Competitive Effects of Financial Distress Announcements:
Evidence from Commercial Banks and Life Insurance Companies
Elijah Brewer III and William E. Jackson III

WP-02-23

State-Contingent Bank Regulation With Unobserved Action and
Unobserved Characteristics
David A. Marshall and Edward Simpson Prescott

WP-02-24

Local Market Consolidation and Bank Productive Efficiency
Douglas D. Evanoff and Evren Örs

WP-02-25

Life-Cycle Dynamics in Industrial Sectors. The Role of Banking Market Structure
Nicola Cetorelli

WP-02-26

Private School Location and Neighborhood Characteristics
Lisa Barrow

WP-02-27

Teachers and Student Achievement in the Chicago Public High Schools
Daniel Aaronson, Lisa Barrow and William Sander

WP-02-28

The Crime of 1873: Back to the Scene
François R. Velde

WP-02-29

Trade Structure, Industrial Structure, and International Business Cycles
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WP-02-30

Estimating the Returns to Community College Schooling for Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel G. Sullivan

WP-02-31

A Proposal for Efficiently Resolving Out-of-the-Money Swap Positions
at Large Insolvent Banks
George G. Kaufman

WP-03-01

Depositor Liquidity and Loss-Sharing in Bank Failure Resolutions
George G. Kaufman

WP-03-02

Subordinated Debt and Prompt Corrective Regulatory Action
Douglas D. Evanoff and Larry D. Wall

WP-03-03

When is Inter-Transaction Time Informative?
Craig Furfine

WP-03-04

Tenure Choice with Location Selection: The Case of Hispanic Neighborhoods
in Chicago
Maude Toussaint-Comeau and Sherrie L.W. Rhine

WP-03-05

Distinguishing Limited Commitment from Moral Hazard in Models of
Growth with Inequality*
Anna L. Paulson and Robert Townsend

WP-03-06

Resolving Large Complex Financial Organizations
Robert R. Bliss

WP-03-07

Working Paper Series (continued)
The Case of the Missing Productivity Growth:
Or, Does information technology explain why productivity accelerated in the United States
but not the United Kingdom?
Susanto Basu, John G. Fernald, Nicholas Oulton and Sylaja Srinivasan

WP-03-08

Inside-Outside Money Competition
Ramon Marimon, Juan Pablo Nicolini and Pedro Teles

WP-03-09

The Importance of Check-Cashing Businesses to the Unbanked: Racial/Ethnic Differences
William H. Greene, Sherrie L.W. Rhine and Maude Toussaint-Comeau

WP-03-10

A Structural Empirical Model of Firm Growth, Learning, and Survival
Jaap H. Abbring and Jeffrey R. Campbell

WP-03-11

Market Size Matters
Jeffrey R. Campbell and Hugo A. Hopenhayn

WP-03-12

The Cost of Business Cycles under Endogenous Growth
Gadi Barlevy

WP-03-13

Full text of Working Papers (Federal Reserve Bank of Chicago) : The Cost of Business Cycles Under Endogenous Growth, Working Paper 2003-13

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