The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
orKing raper series
A s s e t R e tu rn V o la tility w ith E x tre m e ly
S m a ll C o s ts o f C o n s u m p tio n A d ju s tm e n t
David A. Marshall
5
L I B R AR Y
JAN 1 3 1995
hfc.Dfc.KAL RESERVE
BANK OF CHICAGO
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December 1994 (WP-94-23)
FEDERAL R ESER V E BANK
O F CHICAGO
A S S E T R E T U R N V O L A T IL IT Y W IT H E X T R E M E L Y S M A L L
COSTS OF CONSUMPTION ADJUSTMENT
David A. Marshall
F e d e r a l R e s e r v e B a n k o f C h ic a g o
and
J .L . K e l l o g g G r a d u a t e S c h o o l o f M a n a g e m e n t
N o r th w e s te r n U n i v e r s i t y
September 19, 1994
I would like to acknowledge helpful discussions with Larry Christiano, George Constantinides,
Marty Eichenbaum, John Heaton, Ravi Jagannathan, Bob Korajczyk, Debbie Lucas, Bob
McDonald, and the members of the NBER Workshop on Asset Pricing. All remaining errors are
mine.
Asset Return Volatility With Extremely Small Costs of Consumption Adjustment
Abstract
Extremely small non-convex costs of consumption adjustment are introduced into an
equilibrium consumption capital asset pricing model (CCAPM), in an effort to reconcile the low
variance of consumption growth with the high variance of equity returns and the large mean equity
premia observed in the data. Using the approach of Hansen and Jagannathan (1991), I present
evidence that the CCAPM is consistent with observed first and second moments of consumption
growth and asset returns if changing the level of consumption involves an adjustment cost whose
maximum value is less than (l/1000)th of average within-period consumption. In practice, costs of
this magnitude are undetectable. I conclude that the CCAPM is non-robust to the introduction of
extremely small frictions, and that small costs of consumption adjustment can reconcile the model
with the Hansen-Jagannathan mean-variance restrictions.
However, this extreme non-robustness
implies that the CCAPM is unlikely to yield an empirically useful model of the short-run co
movements of aggregate consumption and asset returns.
A S S E T R E T U R N V O L A T IL IT Y W IT H E X T R E M E L Y S M A L L
COSTS OF CONSUMPTION ADJUSTMENT
Equilibrium consumption capital asset pricing models (CCAPMs) have attracted a great deal
of attention since .'their introduction in the seminal papers by Lucas (1978), Breeden (1979), and
Grossman and Shiller (1982). The CCAPM is attractive to financial economists because it implies
an observable expression for the intertemporal marginal rate of substitution (IMRS) as a function of
aggregate consumption data.
According to the theory, all asset prices are determined by the
conditional covariance between this IMRS and the asset’s payoff. In contrast, returns-based pricing
models, such as the Arbitrage Pricing Theory of Ross (1976), treat the IMRS as an unobservable
process to be proxied by the returns on one or more mimicking portfolios. Unlike the CCAPM,
returns-based models do not derive an explicit theoretical linkage between asset returns and aggregate
quantity variables.
Unfortunately, the CCAPM has not fared well in empirical tests. When preferences are
assumed to be time- and state-separable, the CCAPM cannot reconcile the low variance of
consumption growth with the high variance of equity returns without implausibly high levels of risk
aversion. 1 These difficulties are summarized by the failure of the IMRS implied by these models
to lie within the Hansen-Jagannathan mean-variance bounds. (See Hansen and Jagannathan (1991).)
Researchers have responded to the CCAPM’s empirical failures by adopting a richer
See, for example, Hansen and Singleton (1982), Eichenbaum, Hansen and Singleton
(1988), Grossman, Melino, and Shiller (1987), Hansen and Jagannathan (1991), Eichenbaum
and Hansen (1990), Mehra and Prescott (1985), Cochrane and Hansen (1992).
1
1
preference specification or by abandoning market completeness.2
In this paper, however, I argue
that the difficulties with the CCAPM run deeper than a mis-specification of preferences or market
structure.
I argue that the implications of the CCAPM are fundamentally non-robust to extremely
small perturbations in the economic environment.
In particular, undetectably small costs of
consumption adjustment are sufficient to reconcile the model with the Hansen-Jagannathan (1991)
restrictions. While this suggests that some variant of the CCAPM may be (approximately) correct,
developing an empirically useful version of the CCAPM would require accurate measurement of
these extremely small adjustment costs. I conclude that the CCAPM is unlikely to yield a useful,
robust model of short-run movements in asset returns.
Cochrane (1989) provides the intuitive reason why small adjustment costs can have a big
effect on asset returns. He shows that the utility gains from adjusting consumption in response to
most changes in investment opportunities are extremely small. If agents faced even a tiny cost of
adjusting consumption, they would forego this trivial utility gain and choose an extremely smooth
consumption path. In order to induce even the moderate consumption variability found in aggregate
data, asset prices would have to be highly variable.
Why might it be costly for an agent to change her consumption rate? There may be time
costs in implementing a new consumption/savings plan, or search costs in finding vendors for the
new, better (or worse) quality goods to be purchased.
More generally, costs of consumption
adjustment can be interpreted as a proxy for what Cochrane (1989) calls "near-rational" behavior.
Time-nonseparable preferences are studied by Gallant and Tauchen (1989),
Constantinides (1990), Ferson and Constantinides (1991), and Heaton (1991); state nonseparable preferences are studied by Epstein and Zin (1989,1991), Weil (1990). Market
incompleteness due to uninsurable idiosyncratic risk is incorporated into asset pricing models
of Lucas (1994), Heaton and Lucas (1994), Marcet and Singleton (1990), Telmer (1993),
Constantinides and Duffie (1992).
2
2
Regardless of their interpretation, the costs I introduce into the CCAPM are too small to be detected
in practice: I do not consider costs in excess of ( 1/ 1 0 0 0 )* of per capita consumption per period.
Consumption adjustment costs affect the asset price process in at least two ways. First,
adjustment costs induce time-nonseparability into an individual’s consumption/portfolio problem.
It is known, ffom.'the work of Constantinides (1990), among others, that time-nonseparability can
substantially alter the asset-pricing implications of these models. Second, in the presence of nonconvex costs of consumption adjustment, assets are not priced by the IMRS of a fictitious
"representative agent", whose consumption mimics the aggregate consumption. In general, nonconvex adjustment costs induce some agents to keep consumption at the previous period’s level, so
the entire change in aggregate supply must be absorbed by the remaining agents. Assets are priced
by the IMRS of these marginal agents, so asset prices display more variability than would be seen
in a representative agent model.
This paper is closely related to Grossman and Laroque (1990). These authors were the first
to suggest that small costs of consumption adjustment might explain the empirical failure of
equilibrium asset pricing models, although they did not pursue this conjecture in a general
equilibrium framework. This paper is also related to the growing literature documenting how costs
of consumption adjustment at the individual level can cause aggregate consumption to deviate
substantially from the predictions of frictionless models.3 This paper complements recent work in
which costs of portfolio adjustment are introduced into equilibrium asset pricing models.4
The results of this paper support my conjecture that extremely small costs of consumption
3Caballero (1993), Heston (1992), Beaulieu (1991), Marshall and Parekh (1994).
4Luttmer (1993), He and Modest (1992), Aiyagari and Gertler (1990), Heaton and Lucas
(1992).
3
adjustment may account, in part, for the failure of the CCAPM.
For example, when agents’
coefficient of relative risk aversion is set to five, I find that the Hansen-Jagannathan criterion is
satisfied with adjustment costs whose maximum value is approximately (6 / 1 0 ,0 0 0 )Ihs of average
within-period consumption.
I conclude that the implications of the CCAPM are non-robust to the
introduction of small frictions, and that such frictions can reconcile the first and second moments of
consumption growth with the Hansen-Jagannathan restrictions.
On one hand, this result is
encouraging for the CCAPM, since it suggests that the empirical failure of this model need not
represent a failure of equilibrium asset pricing theory. On the other hand, for a model to be useful
empirically, it must be robust to small perturbations. If the CCAPM is as non-robust as this paper
suggests, it is essentially useless as an empirical model of the short-run movements of asset returns.
The paper is organized as follows. Section 1 presents a simple example, which illustrates
how consumption adjustment costs affect asset returns. Section 2 presents the general model studied
in the remainder of the paper. Section 3 discusses the approach I use to solve the model. In section
4, I use the Hansen-Jagannathan methodology to evaluate the model when there are two types of
agents. In section 5 ,1 discuss whether the results of the two-agent case generalize to multiple types
of agents. Section
6
is a brief conclusion.
1. A SIMPLE E X A M P L E O F ASSET PRICING W I T H FIXED C OSTS O F
CONSUMPTION ADJUSTMENT5
In this section, I use a simple example to illustrate how small costs of consumption
adjustment can substantially increase asset price variability. In the standard frictionless CCAPM with
SI am grateful to John Heaton for suggesting this example.
4
complete markets, aggregation obtains: asset returns are determined by IMRS of a fictitious
representative agent who consumes the aggregate consumption.
When costs of consumption
adjustment are introduced into this model, aggregation breaks down.
For some agents, the
adjustment cost exceeds (in utility terms) any possible gain from adjusting consumption. These
agents keep their consumption unchanged. The remaining agents must absorb any change in the
aggregate supply of the consumption good. Since asset returns are determined by the IMRS of this
second group, which has a higher variance than the representative agent IMRS, the asset returns are
more variable than in the frictionless model.
In the illustrative model of this section, there are three time periods and two agents, who act
as price takers. The consumption of agent i in period t is denoted c{. Agents face fixed costs of
adjusting consumption: if cj ^ cj.x, agent i must forego a units of consumption in period t. The
aggregate endowment, denoted et, is known with perfect certainty, but is higher in period
2
than in
period 1 or period 3. In particular, I set e, = e, + a, e2 = eh, and e3 = e,, where eh > e, + a > e,.
The initial conditions,
c l0,
are set at the average per capita endowment:
c'0
= (e,+e2+e3)/6 , i=l,2.
Preferences are given by:
U(cI,,c 2,,c3') = log(c/) + P log(c2‘) + P2 log(c3‘), 0 < p < 1, i = 1,2.
If a = 0, the equilibrium allocation is standard. Regardless of the initial distribution of
wealth, each agent consumes a fixed proportion of the aggregate endowment in each period:
c,1 = 8 et, ct2 = (1 - 6 )et
where
8
(1 )
is a constant between zero and one. The interest rate process Rt supporting this allocation
as an equilibrium is: Rj = e ^ p e ^ , R2 = e^P e^. If a > 0, however, allocation (1) cannot be
supported as an equilibrium for all wealth distributions. If agent 1 is sufficiently poor relative to
5
agent 2, she will be unwilling to bear the cost of adjusting consumption each period. For some such
cases, however, there is an alternative equilibrium in which agent l ’s consumption is constant, and
agent 2 absorbs the entire fluctuation in aggregate supply:
c,1 = c, t = 1,2,3;
(2)
c t2 = c,2 = e4- c - a , t odd;
2
2
—
C2 ~ Ch ~ eh C
Since agent 2 is at the margin, equilibrium asset prices are determined by her intertemporal IMRS.
The interest rate process supporting allocation (2) as an equilibrium is: R, = c^(PcJ), R2 = c*/(pcj;).
Since agent 2’s consumption is more variable than the aggregate endowment, the asset returns in
equilibrium (2 ) are more variable than in the frictionless equilibrium ( 1 ).
Table 1 shows that extremely small adjustment costs are sufficient to shift the economy from
the standard equilibrium (1) to the alternative equilibrium (2) . 6
In this table, I display four
parameterizations of this example. In all these parameterizations, I normalize e( +
P = .999.
^
= 2 and I set
In Panel A of Table 1, I seek to replicate the variability we actually observe in the
growth rate of monthly aggregate consumption. When a two-state first-order Markov chain is fit to
the growth rate of U.S. monthly consumption from 1959-1986 (see equation (15) in section 5,
below), the difference between consumption growth in the two states is .00886. In Panel A of Table
1, I match this feature of the data by choosing e4 = .99779 and
= .00886.
^
= 1.00221, so that
- e/e,,
In this case, a fixed cost of consumption adjustment of 2xl0 '6 (approximately one
millionth of the average per-capita endowment per period) rules out the standard equilibrium ( 1 ).
Intuitively, the gains from adjusting consumption in response to the time-varying interest rate are so
^ h e procedure for solving this simple example is discussed in detail in the Appendix.
6
small that even this trivial adjustment cost induces at least one of the agents to choose a flat
consumption path. With this level of fixed cost, however, an equilibrium of form (2) exists for c
as high as .29. (That is, approximately 29% of the wealth in the economy is owned by agent 1.)
When c = .29, the spread between R, and R2 is 40% larger than the corresponding spread when
a
= 0. With a slightly higher adjustment cost of 4xl0'6, equilibrium (2) obtains for values of c as high
as .41, at which point the spread between R, and R2 is 70% larger than the corresponding spread in
the ffictionless equilibrium. In both cases, a trivial adjustment cost induces a shift in the nature of
the equilibrium and a substantial increase in asset-return variability.
With larger endowment fluctuations, larger fixed costs are needed to break the standard
equilibrium. Examples are presented in the remaining panels of Table 1. Yet, even in Panel C,
where the variability of aggregate endowment (as measured by e,, - e,) is twenty times higher than
in Panel A, the standard equilibrium (1) is broken by a = .0008, which implies a fixed cost of
consumption adjustment less than (2/1000) of average per capita endowment. Only in Panel D,
where real interest rates fluctuate 40 percentage points each period, does the fixed cost required to
break the standard equilibrium approach detectability.
This example illustrates that aggregation can be disrupted by small adjustment costs if the
gains from adjusting consumption differ across agents. This intuition carries over to the dynamic,
growing economy modelled in section 2, below. Unlike the example of this section, the agents in
the model of section
2
have (approximately) the same wealth. They have different propensities to
adjust consumption because they have different lagged consumptions. As the aggregate endowment
trends upward, those agents who have not adjusted recently find that their consumption level is
substantially below their permanent income. These agents derive substantial gain from adjusting, so
they are willing to bear the adjustment cost. As a result, the role of marginal consumer alternates
7
among the agents.
2.
AN ASSET PRICING MODEL WITH COSTS OF ADJUSTING CONSUMPTION
A. Basic Structure
In this section, I describe the dynamic equilibrium model studied in the remainder of this
paper. The model extends the simple exchange economy of Lucas (1978) and Hansen and Singleton
(1982) by introducing a small non-convex cost of consumption adjustment. As in these earlier
papers, there is a single non-storable consumption good, and there exists a complete market in statecontingent claims, payable in the consumption good. The only exogenous stochastic process is the
aggregate endowment, denoted {et}~=1- I assume that the endowment growth rate et/et., is a firstorder Markov chain. The state of the economy at date t, denoted st, is the history of realizations of
the aggregate endowment: s( = {e,,e2, . .. ,et}; the set of possible realizations of s, is denoted S,. Let
7tt(s,). denote the price at date zero of a claim to one unit of consumption at date t when the state is
st, and let Pr(s,) denote the probability of state s„ conditional on date 0 information.
There are N types of agents, indexed by i. Let cXsJ denote the consumption of agent i in
period t when the state is s,. The initial wealth of agent i is denoted Wq. Unlike the traditional
CCAPM, I assume that it is costly to change consumption. In particular, A(ct,ct.,) denotes the cost
(in units of consumption good) of choosing ct when consumption in the previous period was ct.t.
Define x{(s,) as consumption inclusive of adjustment costs of agent i in period t when the state is st:
xt'(st) s ct'(st) + A(ct‘(st), ctl,(st_,)).
(3)
Agent i maximizes the expected value of a time separable utility function, (assumed to be the
8
sam e for all agents) su b ject to the budget co nstraint defined by the state p ric e s {Ttt(St)}7=1:
~
cYs)1^
MAX £ ( 3 ‘ £ Pr(st) *
/ ,
A ~ t*o
s.es
1
(4)
< W0i
(5)
S.t
E
t=0
£
7c, ( s t ) x t ‘( s t)
s«€^«
B. Fixed and Quasi-Fixed Costs of Consumption A djustm ent
Suppose agents face fixed costs of adjusting consumption, with the cost a fraction a of the
per-capita endowment. The cost function A(c„ct.,) would then be given by:
0
A(ct,ct_j)
if c, = cM
(6)
e t -r
a—
if c ,* c , .
N
‘ *•'
Fixed costs are inconvenient because the cost function is discontinuous in the agents’ choice
variable. For this reason, I use the following smooth approximation to the fixed cost function (6 ),
which I call a quasi-fixed cost adjustment function:
A C V Y J = «(_!)
l-* x p [_ ^ (lo g A )2] y>0, a > 0
(7)
"t-1
Figure 1 displays a graph of A, as a function of log(ct/ct4), for two different values of y. As y—
A converges pointwise to the pure fixed cost function (6 ). If c/c^ is a stationary stochastic process,
9
then A(ct,ct.!)/et is stationary.
C. E quilibrium
For a given stochastic process {et)7=o and given initial conditions {cl,,Wo}7=1, an equilibrium
is a vector {7tt(st),.'Cj(st)}7=u =i, ste S t, satisfying:
(i) For each i, {cXs,)}^ solves individual problem (4) - (5), given prices {7tt(s,)}7=0;
(ii) The aggregate resource constraint holds, period by period:
N
E
(8)
xt'(st)
i=l
Conventional approaches to proving the existence of equilibrium do not apply to this economy, due
to the non-convex adjustment cost A. (In section 3.B, below, I address the problem of verifying that
a proposed allocation constitutes a competitive equilibrium.)
The first-order conditions for problem (4) - (5) imply:
(9)
*,(st) = P‘Pr(st)
^>(s0)
where
A ;(ct»k(St,k)* Ct«t-l(S|»k-l))
[c,*(st)]'c + E
The variable X|(s,) is the marginal value of wealth in period t
The intuitive content of (10) is
perhaps clearer if one solves equation (10) recursively, to yield the following Euler equation:
(10)
k*i 1+Aj(ct<k(st>k), ct,k_1(suk_1))
l +A I (ct1(s,),cll. 1(st. 1))
10
[C t^
= X{[1 +A 1(c t',ct!i)] +p E t[XjtlA 2(cl! I,c ti)],
For simplicity, I have suppressed the dependence of
X
i = 1 ,...,N .
(11)
and c on st. Equation ( 1 1 ) extends the
familiar envelope condition. The left-hand side of (11) gives the marginal utility of increasing
current consumption. The right-hand side gives the marginal disutility of the wealth reduction
*
needed to pay for this increase in consumption. If there were no adjustment costs, this marginal
disutility would equal X*, the marginal utility of wealth. With adjustment costs, however, a change
in consumption also changes the adjustment costs the agent must pay, both now and in the future.
The effect of these adjustment cost changes on current and future wealth are captured by including
the marginal adjustment costs, A, and A2, in equation (11).
Equation (9) implies that any asset return r, must satisfy
1 = PE,
^1*1
V i.
(12)
Agent i’s IMRS between wealth at t and wealth at t+1 is given by P(XJtlA[). Securities markets are
complete, so this IMRS is equated across agents. Dropping the superscript "i", P(Xul/Xt) corresponds
to the IMRS studied in Hansen and Jagannathan (1991).
D. Implications of the Model for Asset Returns
The non-convex adjustment costs introduce non-convexities and time non-separability into
the individual’s decision problem. To illustrate how these features affect asset returns, it is useful
to regard x, (consumption inclusive of adjustment costs, defined in equation (3 )) as the choice
variable. Figure 2 displays two indifference curves in (xt, xt+1) space under perfect certainty. The
11
upper indifference curve looks like one implied by standard concave preferences except for two
perturbations near points
a
and
b.
At point a , the agent sets ct = ct.,, so the adjustment cost at date
t is zero. Since the agent can consume the entire
rather than losing a portion of the
\
in
adjustment costs, she can achieve the given level of utility with a lower value of xt. Similarly, point
b
is where ct+, = ct: The lower indifference curve passes through the point d , where ct., = c, = ct+1.
Figure 2 illustrates the qualitative implications of the model for asset pricing. In models with
quasi-fixed costs of adjustment, agents typically make infrequent large adjustments. Between these
large adjustments, agents make very small adjustments. That is, agents tend to be near points like
a , b,
or
d,
where a maximum of one large adjustment is made during the time periods {t, t+1}. As
can be seen from Figure 2, the local curvature of the indifference curve is much greater near these
points than at other points. Even if utility defined over c, displays modest curvature, the local
curvature of the derived utility defined over x, is very large in the neighborhood of the optimally
chosen allocations.
From the work of Hansen and Jagannathan (1991), it is known that the time- and stateseparable model fails, in part, because both the mean and the variance of the IMRS implied by that
model are too low.
Figure 2 suggests why the model with adjustment costs might perform better
in this regard. The IMRS (P(X,+1At) is the slope of the indifference curve in {x„ xt+,} space at the
equilibrium allocation. Near points a ,
b,
or d , this slope is extremely sensitive to small perturbations
of the equilibrium allocation, so one would expect this model to induce greater variability in the
IMRS than in comparable models without adjustment costs. Similarly, one would expect that this
higher level of curvature would induce agents to value risk-free assets more highly. This would
imply a lower risk-free rate, and therefore a higher mean value of the IMRS, than in comparable
models without adjustment costs.
12
3. SOLVING T H E M O D E L
It is not feasible to solve the competitive equilibrium directly. This paper computes the
competitive equilibrium in three distinct steps: (i) I use a discrete state-space dynamic programming
algorithm to solve the equally-weighted optimal resource allocation (Pareto) problem corresponding
to the conjectured equilibrium, (ii) I use equations (9) and (10) to compute the state-contingent
claims prices corresponding to this Pareto optimum, and verify numerically that these prices in fact
decentralize the Pareto optimum as a competitive equilibrium, (iii) Given this equilibrium statecontingent consumption allocation, I simulate the consumption process {c{}{=0^=1, using as the initial
condition the most common state in the discrete grid. I compute the {X,} process corresponding to
{c|} by numerically solving integral equation (11), and I study the IMRS p(X,+1At) using the approach
of Hansen and Jagannathan (1991). In the following I describe each of these steps in detail.
A. Solving the Optimal Resource Allocation Problem
The equally-weighted optimal resource allocation problem corresponding to the competitive
model of section 2 , above, is:
n
MAX
rc hi-?
E£p'£±_lL_
{c,1, tO .... t=0
1=1
^
(13)
^
subject to the aggregate resource constraint (8 ), with initial conditions {c^ jflj given. The first order
conditions for this problem are identical to equation (11), where, for all i, Xj equals the Lagrange
multiplier associated with the resource constraint (8 ).
Problem (13) cannot be solved merely by finding a vector of stochastic processes {cj}7=n=i
which satisfy the first order conditions (11).
In a non-convex economy there generally are
13
suboptimal solutions to the first-order conditions.
I proceed by directly solving the dynamic
programming problem (13), using value function iteration over discretized state and control spaces.
(Details are in the appendix.) The value function associated with problem (13) is homogeneous of
degree l-£, so the problem can be transformed into one involving only the stationary state variables
{e,/et.„ c[.,/et}^=1, and stationary control variables {cj/e,} ^ =1 . The exogenous driving process is the
growth rate of the aggregate endowment, e/e,.,, which is assumed to follow a stationary two-state
Markov chain (as in Mehra and Prescott (1985)). The endogenous state vector {cj.,/e„ . . . , c^.,/e,}
is discretized as finely as needed to achieve an acceptable level of accuracy. While this method can
approximate the true solution arbitrarily well if the state and control grids are made sufficiently fine,
it suffers from the "curse of dimensionality" as N increases. For this reason, I follow the recent
literature on asset pricing with heterogeneous agents (e.g., Lucas (1994), Heaton and Lucas (1994),
Marcet and Singleton (1990), Telmer (1993)) by setting N = 2.
B. Verifying that the Pareto Optimum Can Be Decentralized
The Pareto optimal allocation need not be decentralizable as a competitive equilibrium.
Consider Figure 3, which reproduces the two-period indifference map from Figure 2. If the Pareto
optimal allocation for a particular agent was near point a, a risk-free interest rate equal to the inverse
of the IMRS between x1 and xt+1 at that point would support that allocation. Line AA, tangent to the
indifference curve at point a , gives the set of (x,, xl+1) pairs which are budget-feasible at this interest
rate. Notice that line AA is below the indifference curve except at the point of tangency. In contrast
to this well-behaved allocation, allocations near regions of nonconvexity cannot be supported as
equilibria. An example is p o in t/in Figure 4. If the interest rate were given by the inverse IMRS
at point/, the agent could trade along line FF to an allocation which is preferred to/.
14
The task at hand, then, is to rule out the stochastic equivalent of Figure 4. In the Appendix
I describe in detail how this is done. Basically, for each initial state s0 to be considered,7 *1 compute
the state-contingent claims prices, {7t(s,)}, according to equations (9) and (10), evaluated at the Pareto
optimal state-contingent allocation, denoted {c(St)}. By construction, (c(s,)} satisfies the budget
constraint implied by prices {jt(St)}. I then perturb date 0 consumption away from c(s0), and I search
for an alternative state-contingent consumption plan (c(St), t > 1 } that satisfies the budget constraint
implied by prices {7t(s,)} and that yields a higher date t expected utility than (c(st), t > 1}. If such
a consumption plan exists, the Pareto optimum cannot be decentralized as a competitive equilibrium.
If the search algorithm fails to find such a consumption plan for any candidate state s0, I interpret
this Pareto optimal allocation as a competitive equilibrium.
C. Computing the equilibrium IMRS
Once it has been verified that {c'(s,)} is indeed an equilibrium consumption allocation, it
remains to construct a {X,'(st)}7=i process which satisfies integral equation ( 1 1 ). I do so using the
parameterized expectations algorithm. (See Marcet and Marshall (1994).) A description of this
algorithm can be found in the appendix. The IMRS process, {pXj/Aj.i} can then be simulated.
An implication of the model is that, in any given state, the IMRS is equated across the two
agents. That is,
P A X , = p x X 2-i-
<14)
Nothing in the solution procedure constrains equation (14) to hold, so the correlation coefficient
In the simulations of section 4, below, I implement this algorithm for the four
(discretized) states with the highest probability.
7
15
between PX.JA-J.1 ar>d f&tAt-i can be used as a test of the accuracy of the solution method.
4. RESULTS FROM SIMULATING THE MODEL
In this section I describe the implications of the model for asset prices, using the HansenJagannathan (1991) methodology. The timing interval is one month, and the number of agent-types
is set equal to 2. The subjective discount factor, (3, equals .999 (implying an annual discount rate
of approximately 1.2%). The coefficient of relative risk aversion (CRRA), £, is varied from 2 to 10.
The parameter a (which determines the maximal adjustment cost, as a fraction of current per-capita
endowment) is varied from .0002 to .001. The parameter y determines the curvature of the adjustment
cost function: the inflection points in Figure 1 occur when Alog(Ct) =
± \J l/y .
With fixed costs of
consumption adjustment, agents choose either a large change in consumption or no change at all.
With the quasi-fixed adjustment cost (7), the choice is between a large change and a change
substantially less than the distance to the inflection point.
In the simulations I present,
y
is varied
from 200,000 to 600,000, implying that the region of small adjustment varies from Alog(c,) =
±.00224 to AlogteJ = ±.00126.
The two-state Markov chain characterizing (e/e,^) is calibrated to the monthly growth rate
of US consumption of nondurables and services from 1959:1 through 1986:12. Specifically, let g,
denote the gross growth rate of observed consumption, and let gt denote the two-state Markov
approximation. I follow Mehra and Prescott (1985) in calibrating the two possible values of gt and
the state transition probabilities so that E(g,) = E(g,), var(g,) = var(g,), cov(gt,gt.!) = cov(gt,gt.1), and
the unconditional probabilities of the two states are equal. In the data, the sample mean of gt is
1.002658, the sample standard deviation of gt is .004429, and the sample first-order serial correlation
16
of gt is -.2454. These statistics imply the following two-state discretization for g:
geflpg*} = {0.99823,1.00709}
(15)
prob(gt=g11gt_1 =g,) = prob(gt=g21gt_t =g2) = .3773.
Table 2 displays the results from the simulations. The rows labelled "correlation" give the
%
sample correlations between the IMRS of agent 1 and of agent 2. In Panels A and B, when y equals
200,000 and 400,000 respectively, these correlations are very close to unity, indicating that the
solution procedure is quite accurate for these parameterizations. It is more difficult to maintain this
degree of accuracy when the adjustment cost function has a higher degree of curvature. In Panels
C and D, when y = 500,000 and 600,000, the correlation is still above 0.9 for most parameter
combinations. The correlation does drop as low as 0.565 for one set of parameters.
In Table 2, the rows labelled "decentralize" indicate whether agents could break the proposed
equilibrium by moving away from the Pareto optimal allocation.8 For most parameter values, I
found no evidence that this was possible.
For three parameter combinations in Panel A, four
combinations in Panel B, and one in Panel C, I found that a 10% reduction in consumption allowed
each agent to trade to a consumption allocation with a slightly higher expected utility. (All other
perturbations away from the Pareto optimal allocation unambiguously reduced expected utility.) This
could be interpreted as evidence that the Pareto optimal allocations associated with these eight
parameter combinations cannot be decentralized as competitive equilibria. Alternatively, these small
irregularities may simply be artifacts of the discretization used in the solution algorithm.
Figures 5 through
8
plot the IMRS means and standard deviations from Table 2 . In these
8 1 implemented the algorithm of Appendix C for the four most probable states in each
model. The results for all four were, in every case, virtually identical, so Table 2 reports
results for only the most probable state.
17
figures, the V-shaped solid line is the Hansen-Jagannathan frontier in mean-standard deviation space,
computed using monthly asset return data from 1959:1 to 1986:12 as in Hansen and Jagannathan
(1991, Figure 5) . 9 The three contour lines moving from southwest to northeast plot the means and
standard deviations of the IMRS as the fixed cost parameter a varies from .0002 (at the extreme
southwest of eachjcontour) to .001 (at the extreme northeast of each contour). The lowest contour
corresponds to a CRRA of two. The remaining two contours correspond to a CRRA of 5 and 10
as one moves to the northwest.
The main effect of increasing risk aversion is to shift the contour lines to the left. As in
Hansen and Jagannathan (1991, Figure 5), increased risk aversion reduces the mean of the IMRS far
more than it increases the variance, and the impact of increased risk aversion within the range studied
here is not great However, the effect of increasing a from zero to .001 is dramatic. For example,
in Figure 5, the mean and variance of the IMRS satisfy the Hansen-Jagannathan criterion with
relative risk aversion of 5 when the fixed cost parameter a is set at about .0006. In Figure
6
(with
Y= 400,000), the criterion is satisfied for this level of risk aversion when a = .0004. In other words,
the asset pricing anomalies associated with the Hansen-Jagannathan diagnostic can be resolved by
introducing a non-convex adjustment cost whose maximum value is (4/10,000) of per-capita monthly
consumption. To give an idea of the magnitude of these costs, in 1992 per-capita consumption
expenditures equalled approximately $16,000 per year, so the monthly consumption expenditure of
a family of four averaged $5,333. If the adjustment cost function set a equal to .0004, the average
9The bounds were computed using two return series and six scaled returns. The returns
were the real one-month return to the value-weighted portfolio of New York Stock Exchange
equities and the real return to one-month Treasury bills. These returns were also scaled by
lagged returns and by lagged consumption. (See Hansen and Jagannathan (1991).) I would
like to thank Ravi Jagannathan for providing me with the data used to construct the meanstandard deviation frontier.
18
family would face a maximal monthly adjustment cost of $2.13.
Costs of this magnitude are
undetectable. I conclude that tiny changes in the costs of consumption transactions can have an
impact on asset pricing far greater than substantial changes in risk aversion.
Figures 7 and
8
are analogous to Figures 5 and 6 , except that y is set to 500,000 and 600,000
respectively. At these extreme levels of curvature in the adjustment cost function, the effect of small
adjustment costs on the variability of the IMRS is somewhat attenuated. The reason is that the
curvature of the adjustment cost function affects the variability of the IMRS in two different ways.
First, for a given consumption process, increasing y increases the variability of A, and A2, the first
derivatives of the adjustment cost function. This effect directly increases the variability of
\ +1/ \
through equation (11). However, increasing y also shrinks the region between the inflection points
of A(ct,ct.,), which affects the individual consumption choices. In particular, the optimal c, for those
who choose a small consumption change will be closer to ct.,.
consumption process decreases the variability of
directions.
This endogenous shift in the
These two effects work in offsetting
For moderately high values of y (such as in Figures 5 and 6 ), the direct effect of
increased y on the variability of A, and A2 dominates. For extremely high values of y (as in Figures
6
and 7), the effect of increased y on the optimal c,/ct., process dominates.
5. INCREASING T H E N U M B E R O F A G E N T TYPES
In section 4 I assumed that there were only two types of agents. Do the results of section
4 still hold when the number of agent types is increased? An equivalent way of posing this question
is to ask whether the aggregate demand-for-consumption function in the two-agent economy is a
good approximation for the aggregate demand function with multiple agents. Unfortunately, this
question cannot be answered directly. As in Lucas (1994), Heaton and Lucas (1994), Marcet and
19
Singleton (1990), and Telmer (1993), it is not feasible to solve the full model with three or more
types of agents. However, we can gain insight into this question by studying a simpler adjustmentcost model, and looking at how the shape of the aggregate demand function changes as the number
of agent types increases.
In the simple model I wish to study, there are two time periods and N agents (indexed by i).
Let cj denote agent i’s consumption and et denote the aggregate endowment in period t, and let
k
denote the price in period 1 of a claim to one unit of consumption in period 2. The wealth of the
i* agent consists of W‘ claims to period 2 consumption, where EjW' = ej. Demand for consumption
in period
1
is increasing in
it:
in order to induce agents to increase period
1
consumption, the
relative price of period 2 consumption must rise. In the appendix, I describe a version of the model
with a particularly simple individual demand function:10
c 1i
(16)
Equation (16) implies that aggregate demand is proportional to
K.
Now, let us introduce a fixed cost of adjusting period 1 consumption. The fixed cost function
has the form given in equation (6 ). Demand function (16) no longer holds. To induce an agent to
change period
1
consumption, the asset price must move enough to make the utility gain from
consumption adjustment greater than the utility loss from paying the adjustment cost. Formally, for
1 0 1 have relegated the details of this simple model and the derivation of the demand
functions to the appendix.
20
each individual there exists a region of inactivity (jr\ jc1) with
k
' <
k
\
such that
co if 2^<7t<K'
(17)
ci = 1
W '-ae j/N
7t
I shall refer to | 7? -
otherwise
| as agent i’s "adjustment aversion". The aggregate demand function sums
equations (17) over i.
In Figure 9 , 1 display an example of this aggregate demand function for the cases N = 2, N
= 3, and the limiting case" as N -> « , as well as the aggregate demand in the absence of
adjustment costs, which is implied by equation (16). In this particular example, I impose a uniform
distribution over {W1}^, by setting W‘ = 2 e2i/(N2+N), and I set
c ‘0
= W'/2 . One can verify from
demand functions (16) and (17) and the individual budget constraints (given in the appendix in
equations (25) and (28)) that {et = 50,
= 100,
n =
1, cj =
c'2 = c'0 )
constitutes an equilibrium,
regardless of the value of a. That is, in this equilibrium there is never any incentive to adjust
consumption. However, as shown in Figure 9, consumption adjustment costs induce a discontinuity
in the aggregate demand function at ej = 50, so fluctuations in aggregate supply e, in the
neighborhood of 50 induce more variation in
n
when a > 0 then when a = 0.
The magnitude of
this effect depends on N. First, the size of the discontinuity equals the minimal adjustment aversion
among agents in the economy. If there are more agent types, this minimal adjustment aversion (and
therefore the discontinuity in aggregate demand) will be smaller. However, simply adding a small
number of agents with low adjustment aversion does not necessarily change the qualitative*20
" The limiting case was computed by setting N = 2000, although all values of N above
2 0 0 imply an aggregate demand curve indistinguishable from the limiting case displayed in
the figure.
21
implications of the model. These new agents must also represent a significant fraction of aggregate
demand. In other words, the two-agent case should approximate the limiting case if, in the limiting
model, the agent types with very low adjustment aversion represent only a small fraction of aggregate
demand.
Let us apply the intuition from this simple model to the model of sections 2 through 4. In that
model, the only source of heterogeneity in adjustment aversion is the cross-sectional distribution of
lagged consumption: those agents whose lagged consumption is very far from their optimal current
consumption (in the absence of adjustment costs) display low adjustment aversion, since the utility
gain from optimally adjusting consumption is relatively high.
To argue that the two-agent
approximation is fundamentally flawed, one would have to argue that a substantial fraction of agents
each period experience a large gap between lagged consumption and (frictionless) optimal
consumption. This could be the case if the aggregate supply were highly volatile. It is unlikely to
be a major issue in the simulations of section 4, since the aggregate supply process in these
simulations is calibrated to post-war US consumption of nondurables and services, a relatively
smooth series.
6. C O N C L U S I O N S
I draw the following conclusions from the exercises in this paper:
(1)
The implications of the C C A P M for the period-by-period joint behavior of
consumption growth and asset returns are non-robust to the introduction of extremely small
frictions in the consumption-transaction process.
(2) It is not difficult to reconcile the observed means and variances of consumption
growth and asset returns in the context of an equilibrium model with rational, optimizing
22
agents. In particular, the observed low variability of consumption growth is compatible with the
observed high variability of asset returns if agents face small non-convex costs of consumption
adjustment.
(3)
The CCAPM is unlikely to yield an empirically useful model of the short-run co
movements of aggregate consumption and asset returns. If the quantitative implications of the
CCAPM are extremely sensitive to small frictions, then this model is empirically useful only if these
frictions can be accurately measured. It is unlikely that accurate measures of adjustment costs this
small can be obtained.
Furthermore, these small frictions are sufficient to render invalid the
representative-agent paradigm, so disaggregated consumption data would be required to implement
the model empirically.
More generally, this paper leads one to be pessimistic that short run asset price movements
can be explained by an equilibrium model of aggregate quantity variables:
the nonrobustness
documented in consumption-based models is likely to be found in other equilibrium models.
Retums-based models of asset pricing do not suffer from this problem. These models treat the IMRS
as an unobserved latent variable; fluctuations in the IMRS are proxied by the returns on portfolios
that are maximally correlated with the factors determining the IMRS. The effects of adjustment costs
are incorporated in these mimicking portfolios’ returns, so adjustment costs need not be treated
explicitly. The results of this paper argue that this retums-based approach is likely to provide more
reliable models of short run movements in asset prices.
23
APPENDIX
A. Characterization of Equilibrium in the Simple Example of Section 1
Each agent is endowed with the consumption allocation in the conjectured equilibrium, plus
the adjustment costs associated with that allocation. Let Jt, denote the price, in units of period 1
consumption, of the consumption good in period t under the conjectured equilibrium. Agent i’s
wealth, denoted
W ,
is the present value of agent i’s endowment under prices {7t,}. For period t, t
= 1,2,3, each agent can either set cj = cj., or let c| * cJ.j, implying eight possible strategies.
To verify whether the conjectured consumption allocation can be supported by prices {7t,},
I compute the optimal consumption under each of the eight strategies: I derive the first order
necessary conditions for optimality under each of the eight strategies, and impose the constraint that
the present value of each optimal consumption plus required adjustment costs does not exceed W*.
I then compute the value of utility under each of these eight strategies to determine which strategy
is dominant for each agent.
In the model of section 1, the conjectured equilibrium prices are
n3 =
P(c2/Ch), and
p2. The conjectured equilibrium (2) can be supported by these prices if the dominant strategy
for agent
c\
n t = 1, n 2 =
1
is {c} * cj, c{ =
c \, c \
= c3}, and the dominant strategy for agent
2
is {Cq * c2, c2
c2,
^ c2}. In Table 1 ,1 display ranges of parameters for which this condition holds. To see whether
the standard allocation ( 1 ) can be supported, I repeat this exercise, but setting {7tJ equal to the
equilibrium prices in this standard equilibrium: %=P[(e1-a)/(e 2-a)], 7t3=p2[(e1-a)/(e 3-a)].
The
standard equilibrium can be supported if, at these prices, the dominant strategy for both agents is {c0
i6 Cj, Cj 5^ Cj, c2
c3}.
24
B.
Solving the Optimal Resource Allocation Problem using Discrete Dynamic Programming
The Bellman recursion corresponding to problem (13) is:
N
V(e,cti.. = M A X
HI,
Ic V'S
E - iV
i=i t
/
v
* PE(v(e,.,,c, 1,...,c 1N) | Sl)
(18)
subject to the aggregate resource constraint (8 ). The value function V(- ;s,) is homogeneous of
degree l-£, so (18) can be rewritten as a stationary problem. Let the stationary state process and
control variables be denoted sj and cj1, as follows:
st —(e/ et-p {ct-i/eth=i)
,
c,“ ^ {CtVeJw e C *,
where S* and C* denote the stationary state and control spaces. Let V* be defined by:
V * ( 0 = V(l,ctVet,...,c tiN/et;st).
Problem (18) can now be written as follows:
Y
K
et-i
e,
V •(<
!.■) - MAX E
p
v-(0
; T *e ."cT.
i*l [C
i-C
v. j
ml
f
N
(19)
subject to
^ [ c tVet +A*(ctVet,ctVet)] <
i=l
1
where the stationary adjustment cost function A* equals A(ct,ct.!)/et, as follows:
25
(2 0 )
(
A*
\
X
—
e,’ et
J
= a l-exp[
IT
(log
)2]
ct-i/et
2
Notice that the conditioning set in (18) is the entire history of realizations of e,, while the
conditioning set in (19) is simply e/e,.,. This is a consequence of the assumption that e/e,., is a first%
order Markov process.
I
discretize S* and C \ and compute a discrete-state approximation to V* by successive
approximations.
In section 4, N = 2, and the endowment growth, e/e,.,, is allowed to take on two
values. The endogenous state vector, {c}.,/e„ c2.,/e,} is allowed to take on either 400 or 500 values.
This algorithm also delivers the state transition function,
satisfying st‘, = T(s,’,et+,/et),
T,
and the optimal control function C,
c,’ = C(s,*).
The state prices 7t,(s,) can be written as functions of the stationary state process {sj}. From
equation ( 1 0 ), one can write
\ ( s t) = e,"?V (
(22)
0
where
VW ) =
1
1+A, (s,)
(23)
t+k
+E E
' AA S'V kml 1+A, (S,,k) et+k-i
j*1
p
(In equation (23), A*(s,), j = 1,2, denotes the derivative of A*(c/e„ c,.,/e,), defined in (21), with
respect to its j *11 argument Also, I have dropped the superscript "i" for notational convenience.) It
follows from (9) and (2 2 ) that
26
1
e
j=i
eM.
7Ct(st) = p'PrCs,’, ■A Iso*)
K( O
(24)
Xo(So*)
so JCt(St) is a function of the stationary state variables {Sq, sj,..., sj}.
C. Verification that the Pareto Optimum Can Be Decentralized
The Pareto optimum, computed using the discrete dynamic programming algorithm of part
B of this appendix, is proposed as an equilibrium allocation. The proposed equilibrium is broken
if a state-contingent allocation exists that satisfies the budget constraint, but yields a higher level of
utility than Pareto optimum. I use the following algorithm to search for such an allocation:
(i) Choose a state sj and an initial endowment e0. The dynamic programming algorithm
discretizes the state space S‘ into 800 or 1000 distinct states. Most of these are frequented rarely
(if ever) in equilibrium. For each set of structural parameters used, I choose the four states most
frequently visited. Given the homotheticity of this model, the initial endowment level e0 can be set
to any arbitrary number.
(ii) Use T and the known transition function for e/eul to compute all possible state-paths
{sj,...,s|}?=i starting from
s*0,
where J > 1. If fully specified, a state-contingent consumption plan
would give consumption in all possible states for t = 1 ,
.
It is not possible to compute an infinite
number of state-contingent consumptions, so I truncate the horizon at a finite number J. This is
equivalent to constraining agents in step (vi), below, to set {cj}7=j+1 = {cJ}7=j+i- Since e,/et., is a
two-state first-order Markov chain, there are 2I+1 - 2 state-paths be computed.
(iii) Use (24) to compute the contingent claims prices 7t,(st) associated with each of the
(2J+1-2) paths computed in step (ii). The infinite sum in (23) is truncated when the next term
27
affects the cumulative sum by less than one part in 100 million. This happens rapidly, since the
cumulative product in (23) goes to zero very quickly. The expectation in (23) can be computed
exactly, since the transition function
T
and the transition matrix of {eVe11} are known.
(iv) Using C and the arbitrary initial endowment level e0, compute the Pareto-optimal
state-contingent consumption plan, which I denote {cj}^.
Consumption plan {cj}^ is the
proposed equilibrium state-contingent consumption allocation starting from initial state sj.
(v) Compute the cost (in present value terms) of the proposed equilibrium consumption
allocation, {cj}^, at the claims prices computed in step (iii), above. Formally, if S,(s0) is the set
of states that are possible at date t when the initial state is s0, the cost of {c’t }{=0 is
£
X)
«t(s,)[e!(st)+ A (ej(st),e,'Li(st.1)].
t=0 St€St(So)
28
(vi) Perturb Cq. For each perturbation, maximize the expected utility of agent i, over
budget-feasible values of {c[}^=1. The expected utility is
J
c ‘(s)1^
X) P* 5^ Pr(st) — —L—
t=0
s eS,
The budget-
1“S
feasible state-contipgent consumption plans {cjjj^ are those plans which have the same present-value
cost as {clta, computed in step (v). (In Figure 3, if {c}}^ were at point a , the set of budget-feasible
consumption plans would lie on line AA.) In the simulations of section 4 , 1 perturb Cq by setting the
initial consumption of agent i equal to Co[l+T|] for 15 values of T| ranging from -.5 to 1.
(vii) For each perturbation considered in step (vi), compare the maximized expected
utility with the expected utility provided by the proposed equilibrium. This is equivalent to
comparing If this maximized expected utility exceeds that provided by the proposed equilibrium for
any perturbation, the Pareto optimum cannot be decentralized.
D. Solving Equation (11) Using the Method of Parameterized Expectations
Equation (11) can be rewritten as :
[ctr ?
X ,
l
1 + Aj*(st*) + PE -^ ■ a 2-(s,:,) s,. Kt
(2 5 )
where, for notational convenience, I have suppressed the superscript "i". The conditional expectation
in (21) is an unknown function of s|. This unknown function is approximated by a polynomial in
the orthogonalized elements of s*, which can be denoted P(s|;0). 0 is a vector of polynomial
29
coefficients to be determined. Thus, (21) is replaced by
[cj
(2 6 )
i + a ;(S;) +PPCS,:,;©)
An approximate equilibrium will be given by a coefficient 0* such that, if the agent uses
%
P(sJ-;0*) as her predictor of (Xt+1At)A2(sJ+1) in choosing the ct vector, then P(st,0 ’) will in fact be the
best predictor ex post in the least squares sense.
The parameter vector 0* is found as follows: A sample path for the equilibrium consumption
vector, {ct}t=i. is computed by simulating the equilibrium state-contingent consumption rule. The
starting values are the consumptions of agents 1 and 2 in the most common state. Equation (22) is
then evaluated for t = 1, . . .,T using an arbitrary value for 0. The resulting
series is used
to compute a series {(A^/At )Aj(si+1) }{=, which, when regressed on the function P(s*;0), yields the
"ex post rational" value of 0 , which can be denoted F(0). 0* is the fixed point of the mapping F().
In practice, this fixed point is found by successive approximations. I specify P(-,0) as a third-order
polynomial. (There was no appreciable change when the order was increased beyond three.)
A more complete description of the method of parameterized expectations can be found in
den Haan and Marcet (1993), or Marcet and Marshall (1994). Marcet and Marshall (1994) gives
conditions under which the approximate solution delivered by this algorithm can be made arbitrarily
close to the true equilibrium if both the polynomial order and the sample size T are sufficiently large.
30
E. A Complete Description of the Simple Model Used in Section 5.
In this appendix I describe the simple two-period model of consumption demand used in
section 5. This example is constructed to yield an aggregate demand function of the simple form
given in equation (16) when there are no adjustment costs.
There are two periods, denoted 1 and 2, and there is a single non-storable consumption good.
There are two classes of agents, denoted I and n, and there are N types of agents in class II. Agents
of class I are endowed with the consumption good only in period
1,
but derive utility from
consumption only in period 2. Agents of class II are endowed with the consumption good in period
2, and derive utility from consuming in both periods 1 and 2. In particular, I assume that agents of
class II maximize a utility function of the form
U(Cj',c2‘) = logCc/) - l o g ^ ) ,
i = 1,2,....N
(27>
where c| denotes the consumption of type i in period t, t=l,2. Let et denote the aggregate endowment
of the consumption good in period t, and let W1 denote the endowment of a class II agent of type
i, to be received in period 2. Notice that Z-,W‘ = e^
Assume that W* is known in period 1 with
certainty. The only asset in this economy is a claim to period 2 consumption. Let it denote the price
of this asset, denominated in units of period
1
consumption.
For any 7t > 0, the optimal action of the class I agents is to exchange their entire endowment
of period
1
consumption for claims to period
2
consumption at price
it.
Class II agents seek
maximize objective function (28) subject to the budget constraint
Cj* + itc2‘ < j t W '.
The first order conditions of this optimization problem are
31
(28)
W‘
7CW‘
c,
(29)
~2~
Only class II agents consume in period 1, so the aggregate demand for period 1 consumption is
(30)
Now, let us assume that adjusting period 1 consumption involves a fixed cost of the form
given in equation (7). For simplicity, I assume that only class II agents face the adjustment cost, and
that the cost is assessed on the agent’s claims to period 2 consumption. Budget constraint (28) must
be replaced by:
C j‘ +
k c
2' + i z A ( C i , c 0') <
7tW ‘.
(3 1 )
If the agent decides that it is optimal to adjust consumption, the optimal choice is governed by firstorder conditions similar to (29):
i _ 7c(Wi - a e 2/N)
Ci
, _ W ‘- a e ^
— — —... —.in ................. . I
C?
=
................. .... ..
2
, , 2)
^
'
2
If the agent does not choose to adjust consumption, then consumption is simply given by:
c , 1 = c0‘;
c2‘ = W i - c 0‘/ 7t.
(33)
Agent i will choose either (32) or (33), whichever yields the highest utility. For any given W* and
Cq, there is a range of rc’s, which I denote (re1, 5?), for which (33) yields the higher utility. This is
the region of inaction for agent i.
Finally, the aggregate demand function is the sum of the
individual demands.
32
REFERENCES
A iy a g a ri, S .R ., and M . G e rtle r,
In d iv id u a l R is k :
1990, "A sse t R e tu rn s w ith T ra n sa ctio n C o s ts and U n in su re d
A Stage I I I E x e r c is e ," w o rk in g paper.
Beaulieu, JJ, 1991, "Optimal Durable and Nondurable Consumption with Transactions Costs,"
Working Paper, M.I.T.
Caballero, R.J., 1993 "Durable Goods: An Explanation for Their Slow Adjustment,"
P o l i t i c a l E c o n o m y 101, 351-384.
Jou rn a l o f
Cochrane, J.H., 1989, "The Sensitivity of Tests of the Intertemporal Allocation of Consumption to
Near-Rational Alternatives," A m e r i c a n E c o n o m ic R e v i e w 79, 319-337.
Cochrane, J.H., and L.P. Hansen, 1992, "Asset Pricing Explanations for Macroeconomics,"
M a c r o e c o n o m i c s A n n u a l, 115-164.
NBER
Constantinides, G. M., 1990, Habit formation: A Resolution of the Equity Premium Puzzle, J o u r n a l
o f P o l i t i c a l E c o n o m y 98, 519-543.
Constantinides, G.M., and D. Duffxe, 1992, "Asset Pricing with Heterogeneous Consumers,"
manuscript
den Haan, W. and A. Marcet, 1993, "Accuracy in Simulations", forthcoming,
R e v i e w o f E c o n o m ic
S tu d ie s .
Eichenbaum, M. and L. P. Hansen, 1990, Estimating Models with Intertemporal Substitution Using
Aggregate Time Series Data, J o u r n a l o f B u s in e s s a n d E c o n o m ic S ta t i s t i c s 8, 53-69.
Eichenbaum, M.S., L.P. Hansen, and K.J. Singleton, 1988, "A Time Series Analysis of
Representative Agent Models of Consumption and Leisure Choice Under Uncertainty,"
Q u a r te r ly J o u r n a l o f E c o n o m ic s 103, 51-78.
Epstein, L.G., and S.E. Zin, 1989, "Substitution, Risk Aversion, and the Temporal Behavior of
Consumption and Asset Returns: A Theoretical Framework," E c o n o m e tr ic a 57, 937-970.
Epstein, L. G. and S. E. Zin, 1991, The Independence Axiom and Asset Returns, working paper,
Camegie-Mellon University.
Ferson, W., and G.M. Constantinides, 1991, "Habit Persistence and Durability in Aggregate
Consumption," J o u r n a l o f F i n a n c i a l E c o n o m ic s 29, 199-240.
Gallant, R., L.P. Hansen, and G. Tauchen, 1990, "Using Conditional Moments of Asset Payoffs to
Infer the Volatility of Intertemporal Marginal Rates of Substitution," J o u r n a l o f E c o n o m e t r ic s
45, 141-179.
G a lla n t, R ., and G . T a u c h e n , 1989
"Sem in o n p a ra m e tric E stim a tio n o f C o n d itio n a lly C o n stra in e d
H eterogeneous P ro ce sse s: A s s e t P ric in g A p p lic a tio n s," E c o n o m e t r i c a 5 7 , 1 0 9 1 -1 12 0 .
Grossman, S.J., and G. Laroque, 1990, "Asset Pricing and Optimal Portfolio Choice in the Presence
of Illiquid Durable Consumption Goods," E c o n o m e t r ic a 58, 25-52.
Grossman, S.J., and R.J. Shiller, 1982, "Consumption Correlatedness and Risk Measurement in
Economies with Non_Traded Assets and Heterogeneous Information," J o u r n a l o f F in a n c ia l
E c o n o m ic s ' 10, 195-210.
Grossman, S.J., A. Melino, and R. Shiller, 1987, "Estimating the Continuous-Time
Consumption-Based Asset-Pricing Model," J o u r n a l o f B u s in e s s a n d E c o n o m ic S t a t i s t i c s 5,
315-327.
Hansen, L. P. and R. Jagannathan, 1991, Implications of Security Market Data for Models of
Dynamic Economies, J o u r n a l o f P o l i t i c a l E c o n o m y 99, 225-262.
Hansen, L.P. and K.J. Singleton, 1982, "Generalized Instrumental Variables Estimation of Nonlinear
Rational Expectations Models," E c o n o m e t r ic a 50, 1269-1286.
He, H., and D. Modest, 1992, "Market Frictions and Consumption-Based Asset Pricing," Finance
Working Paper No. 223, Walter A. Haas School of Business, University of California at
Berkeley.
Heaton, J., 1991, "An Empirical Investigation of Asset Pricing with Temporally Dependent
Preference Specifications," Working Paper # 3245-9l-EFA, Sloan School of Management,
Massachusetts Institute of Technology.
Heaton, J., and D. Lucas, 1994, "Evaluating the Effects of Incomplete Markets on Risk Sharing and
Asset Pricing," manuscript.
Heston, S.L., 1992, "Sticky Consumption," Working Paper #30, Yale School of Organization and
Management.
Lucas, D.J., 1994, "Asset Pricing with Undiversifiable Income Risk and Short Sales Constraints:
Deepening the Equity Premium Puzzle," forthcoming in J o u r n a l o f M o n e t a r y E c o n o m ic s .
Lucas, R. E., Jr., 1978, "Asset Prices in an Exchange Economy,"
E c o n o m e t r ic a
46, 1426-1446.
Luttmer, E.G.J., 1993, "Asset Pricing in Economies with Frictions," Working Paper No. 151,
Department of Finance, J.L.Kellogg Graduate School of Management, Northwestern
University.
Marcet, A., and D.A. Marshall, 1994, "Solving Non-linear Rational Expectations Models by
Parameterized Expectations; Convergence to Stationary Solutions," Working Paper no. 73,
Department of Finance, J.L.Kellogg Graduate School of Management, Northwestern
University.
Marcet, A., and K.J. Singleton, 1990, "Equilibrium Asset Prices and Savings of Heterogeneous
Agents in the Presence of Portfolio Constraints", working paper.
Marshall, D.A., and N.G. Parekh, 1994, "Asset Return Volatility with Fixed Costs of Consumption
Adjustment and Cross-Sectional Heterogeneity," Working Paper.
Mehra, R. and E.C. Prescott, 1985, "The Equity Premium:
E c o n o m ic s 15, 145-161.
A Puzzle,"
J o u r n a l o f M o n e ta r y
Merton, R.C., 1971, "Optimal Consumption and Portfolio Rules in a Continuous-Time Model,"
J o u r n a l o f E c o n o m ic T h e o r y , 3, 373-413.
Ross, S., 1976, "The Arbitrage Theory of Capital Asset Pricing ," J o u r n a l
341-60.
Telmer, C., 1994, "Asset Pricing Puzzles and Incomplete Markets,"
1832.
Weil, P., 1990, "Nonexpected Utility in Macroeconomics,"
29-42.
o f E c o n o m ic T h e o r y ,
J o u r n a l o f F in a n c e ,
13,
48,1803-
Q u a r t e r l y J o u r n a l o f E c o n o m ic s
105,
Table 1
Simulations of Simple Three-Period Model
Panel A: e, = .99779; eh = 1.00221
Range of c for
which standard
equilibrium exists
Range of c for which
adjustment cost
equilibrium exits
p
II
o
Equilibrium interest rates when a = 0: R? = .54%; R° = -.34%
R,
All values c
Does not exist
a = .0000006
c> .19
0 < c< .13
.61%
-.41%
12.4%
a = .0000008
c> .25
0<c< .17
.63%
-.43%
15.0%
a = .000001
c > .31
0 <C<: .19
.65%
-.45%
23.5%
a = .000002
Does not exist
0 < c < .29
.73%
-.52%
40.9%
a = .000004
Does not exist
.19 < c < .41
.86%
-.65%
69.6%
a = .000006
Does not exist
Does not exist
r2
Percent increase in
interest rate spread
Panel B: e4= .99; eh = 1.01
Range of c for
which standard
equilibrium exists
Range of c for which
adjustment cost
equilibrium exits
P
II
o
Equilibrium interest rates when a = 0: R? = 2.1%; R° = -1.9%
All values c
Does not exist
a = .00001
cS .15
0 < c £ .11
2.4%
-2.1%
12.4%
a = .00002
c £ .30
0 <c *£.19
2.6%
-2.3%
23.5%
a = .00004
Does not exist
0 < c ^ .29
3.0%
-2.7%
41.0%
a = .00006
Does not exist
0 < c < .36
3.3%
-3.0%
56.3%
a = .00008
Does not exist
.17
.41
3.6%
-3.2%
69.9%
a = .0001
Does not exist
.34 < c £ .45
3.8%
-3.5%
82.4%
a = .0002
Does not exist
Does not exist
R.
r2
Percent increase in
interest rate spread
Table 1 (continued)
Panel C: e4= .95; eh = 1.05
Equilibrium interest rates when a = 0: Rj = 10.6%; R2 = -9.4%
Range of c for which
adjustment cost
equilibrium exits
p
II
o
R,
Range of c for
which standard
equilibrium exists
All values c
Does not exist
a = .0002
c S .12
0 < c < .09
11.7%
-10.3%
10.1%
a = .0006
c > .36
0 < c < .21
13.6%
-11.8%
27.3%
a = .0008
Does not exist
0 < c < .25
14.4%
-12.4%
34.3%
a = .001
Does not exist
0 < c < .29
15.3%
-13.1%
42.2%
a = .002
Does not exist
.18 < c < .41
18.7%
-15.6%
71.5%
a = .004
Does not exist
Does not exist
Percent increase in
interest rate spread
r2
Panel D: ef = .9; eh = 1.1
Equilibrium interest rates when a = 0: R? = 22.3%; R2 = -18.1%
a =0
Range of c for
which standard
equilibrium exists
Range of c for which
adjustment cost
equilibrium exits
All values c
Does not exist
R.
r2
Percent increase in
interest rate spread
25.5%
-20.1%
12.9%
a = .002
c> .30
0 < c < .19
28.4%
-21.9%
24.5%
a = .004
Does not exist
0 < c £ .29
33.1%
-24.7%
43.1%
a = .006
Does not exist
0 < c £ .34
36.2%
-26.5%
55.2%
a = .008
Does not exist
.19 £ c ^ .35
37.7%
-26.9%
59.9%
Does not exist
Does not exist
oH-»
P
II
8
0 < c < .11
P
II
.15
"Standard equilibrium” denotes a consumption allocation of the form (1). "Adjustment cost equilibrium” denotes an allocation of
the form (2) in the text, "a” denotes the fixed cost of adjusting consumption. Rj and R2 denote the net interest rates in periods 1
and 2 respectively when a = 0: R? = e,/(P(e4-a))-l; R2= e/fpej-l. Rj and R2 denote the net interest rates supporting the
adjustment cost equilibrium (2) at the maximal value of c for which this equilibrium exists: Rj = c£/(p(c4-a))-l; R2= cj/(pc£)-l.
"Percent increase in interest rate spread" equals [(Rj-R^ - (R?-R£)]/ (R?-R£), expressed as a percentage. For all these
computations, P = .999.
Table 2
Panel A: y = 200,000
Inflection points: Alog(c,) = ±.00224
Values of a
Relative risk
aversion
%
.0002
.0004
.0006
.0008
.001
2
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.996
0.068
0.993
ok
1.000
0.120
0.991
ok
1.009
0.174
0.997
0.085%
1.019
0.228
0.995
0.284%
1.036
0.293
0.990
0.644%
5
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.989
0.083
0.991
ok
0.996
0.140
0.998
ok
1.005
0.195
0.998
ok
1.016
0.247
0.997
ok
1.031
0.304
0.991
ok
10
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.979
0.107
0.998
ok
0.988
0.170
0.994
ok
0.999
0.230
0.993
ok
1.012
0.279
0.993
ok
1.025
0.325
0.996
ok
Panel B: y = 400,000
Inflection points: Alog(c,) = ±.00158
Relative risk
aversion
Values of a
.0002
.0004
.0006
.0008
.001
2
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.996
0.066
0.985
ok
1.000
0.115
0.989
ok
1.012
0.195
0.996
0.090%
1.035
0.295
0.997
0.353%
1.102
0.507
0.998
1.645%
5
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.989
0.083
0.993
ok
0.995
0.134
0.992
ok
1.006
0.203
0.997
ok
1.031
0.310
0.997
ok
1.095
0.504
0.997
1.047%
10
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.979
0.112
0.999
ok
0.986
0.161
0.994
ok
0.999
0.229
0.997
ok
1.023
0.325
0.997
ok
1.089
0.520
0.997
ok
Table 2 (continued)
Panel C: y = 500,000
Inflection points: AlogCcJ = ±.00141
Relative risk
aversion
Values of a
.0002
.0004
.0006
.0008
.001
2
Mean IMRS
Std. IMRS
Correlation :
Decentralize
0.994
0.037
0.946
ok
0.997
0.080
0.921
ok
0.998
0.092
0.960
ok
1.010
0.185
0.876
0.110%
1.015
0.210
0.905
ok
5
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.988
0.063
0.979
ok
0.990
0.091
0.969
ok
0.993
0.118
0.985
ok
1.002
0.182
0.947
ok
1.012
0.233
0.885
ok
10
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.978
0.100
0.988
ok
0.982
0.134
0.984
ok
0.985
0.155
0.983
ok
0.994
0.207
0.985
ok
1.008
0.269
0.977
ok
Panel D: y = 600,000
Inflection points: AlogfcJ = ±.00126
Relative risk
aversion
Values of a
.0002
.0004
.0006
.0008
.001
2
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.994
0.029
0.926
ok
0.995
0.049
0.939
ok
0.995
0.056
0.828
ok
0.997
0.085
0.892
ok
1.003
0.132
0.565
ok
5
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.987
0.055
0.977
ok
0.989
0.073
0.967
ok
0.990
0.095
0.966
ok
0.993
0.116
0.898
ok
0.999
0.162
0.730
ok
10
Mean IMRS
Std. IMRS
Correlation
Decentralize
0.976
0.083
0.966
ok
0.979
0.112
0.981
ok
0.983
0.140
0.968
ok
0.986
0.159
0.958
ok
0.994
0.207
0.930
ok
a and y are parameters in adjustment-cost function (8). "Correlation" denotes the correlation coefficient in the simulated data
between PAJAl-i and
In the row labelled "Decentralize", "ok” indicates that the algorithm described in Section C of the
Appendix found no budget-feasible allocation that breaks the proposed equilibrium. A numerical entry in this row indicates the
percentage increase in expected utility associated with a 10% reduction in consumption in the most common state. (No other
consumption perturbation resulted in an increased expected utility.)
xlQ-3
Q u a si-F ix e d C o sts o f C o n su m p tio n A d ju stm e n t
Figure 1: The quasi-fixed adjustment cost function given in equation (7) is plotted as a
function of logCcJ - log(cM). The solid line corresponds to y
=
200,000. The dashed line
corresponds to y = 600,000. For both cases, a(e,/N) is set to .001.
3.5
3
x(t+l)
2.5
2
1.5
1
0.5
1.5
0.5
2.5
x(0
Figure 2: Two indifference curves are plotted in (xt, xt+1) space, where
^
denotes
consumption inclusive of adjustment costs, as in equation (3). (The predetermined value of
lagged consumption c,.! is set to 0.75.) Point
to ct = ct+1. Point
d
a
corresponds to ct = ct.j. Point
corresponds to ct., = c, = ct+1.
b
corresponds
3.5
3-
x(t+ l)
2.5 21.5 10.5 0-
\ viA
1.5
0.5
2.5
x(t)
Figure 3: Allocation
a
can be supported as a competitive equilibrium by a risk-free interest
rate given by the slope of line AA. Line AA gives the frontier of allocations which are
budget-feasible at that interest rate.
3.5
3-
x(t+l)
2.5 21.5 10.5 0
0
F
0.5
1
1.5
2
2.5
x(t)
Figure 4: Allocation/cannot be supported as a competitive equilibrium. The line FF is
tangent to the indifference curve through /. If the risk-free interest rate were given by the
slope of line FF, allocations preferable to / would be feasible.
3
G a m m a = 2 0 0 ,0 0 0 ; A lp h a fro m .0 0 0 2 to .001;
C R R A = 2, 5, 10
0.5
0.45
HJ frontier
0.4
std(IMRS)
0.35
0.3 0.25 0.2
crra=10
0.15
crra=2
0.1
crra=5
00<5.97
0.98
0.99
1
1.01
1.02
1.03
1.04
mean(IMRS)
Figure 5: The means and standard deviations of the intertemporal marginal rate of
substitution (IMRS) implied by the model are plotted for various parameterizations. In this
figure,
y
= 200,000 and (3 = .999. The lower-right contour line displays results when the
coefficient of relative risk aversion (CRRA) equals 2; For the middle contour, CRRA = 5,
and for the upper-left contour, CRRA = 10. Each contour plots mean-standard deviation pairs
for the IMRS for various levels of the adjustment-cost parameter a: a = .0002 corresponds
to the lower-left end of each contour line, rising to a = .001 at the upper-right end of each
contour line. The V-shaped solid line in the middle of the figure is the Hansen-Jagannathan
mean-standard deviation frontier, computed by those authors from monthly asset-return data
from 1959:1 to 1986:12. (See Hansen and Jagannathan (1991, figure 5).) For an IMRS to
satisfy the Hansen-Jagannathan criterion, (mean(IMRS), std(IMRS)) must lie above the
Hansen-Jagannathan frontier.
Gamma = 400,000; Alpha from .0002 to .001; CRRA = 2, 5, 10
*
i
i
i
i
i
0.55
0.5
x""
-
x"
HJ frontier
0.45
x-'^
s''
x" xxX'"-
x ''' x'''
^s'' s'"
std(IMRS)
0.4
/
0.35
/
xx
0.3
x'
0.25
/
0.2
crra=10
0.15 0.1
—
/
/
/
✓
//
/
x'
/
x''
/
x/
/
x' x'
/
//x' //
// //
/
✓
//
/
/ crra=2
crra=5
'' /
n ________ I________ J___________ i___________ i___________ i___________ I___________ I___________
° ‘°(5.96
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
mean(IMRS)
Figure 6: The means and standard deviations of the intertemporal marginal rate of
substitution (IMRS) implied by the model are plotted, as in Figure 5.
The parameterization
is identical to Figure 5, except that y = 400,000. The interpretation of the figure is the same
as for Figure 5.
G a m m a = 5 0 0 ,0 0 0 ; A lp h a from .0 0 0 2 to .001;
C R R A = 2, 5, 10
0.5
0.45
HJ frontier
0.4
0.35
ST
PS
03
S
M
0.25
”
0.2
crra=10
v
0.15
crra=2
0.1
crra=5
0.05
0°975
0.98
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
mean(IMRS)
Figure 7: The means and standard deviations of the intertemporal marginal rate of
substitution (IMRS) implied by the model are plotted, as in Figure 5. The parameterization
is identical to Figure 5, except that y = 500,000. The interpretation of the figure is the same
as for Figure 5.
G a m m a = 6 0 0 ,0 0 0 ; A lp h a from .0 0 0 2 to .001;
C R R A = 2, 5, 10
Figure 8: The means and standard deviations of the intertemporal marginal rate of
substitution (IMRS) implied by the model are plotted, as in Figure 5. The parameterization
is identical to Figure 5, except that y = 600,000. The interpretation of the figure is the same
as for Figure 5.
Asset Price
Aggregate Demand Functions for N = 2, 3, Infinity
Figure 9: The aggregate demand function for the model of section 5 is plotted for four
different cases. The vertical axis gives the relative price 7t of period 2 consumption in units
of period 1 consumption; the horizontal axis gives aggregate demand for period 1
N
consumption, ^
c/, as a function of n . The straight diagonal line gives the demand function
i* l
in the absence of consumption-adjustment costs (a = 0). The remaining three lines give the
demand function when a = .05 for N = 2, 3, and the limiting case as N
—> <
*>.
Working Paper Series
A series o f research studies on regional econ om ic issues relating to the Seventh Federal
R eserve D istrict, and on financial and econ om ic topics.
REGIONAL ECONOMIC ISSUES
Estim ating M onthly R egional V alue Added by C om bining R egional Input
W ith National Production Data
WP-92-8
Philip R. Israilevich and Kenneth N. Kuttner
Local Impact o f Foreign Trade Z one
WP-92-9
David D. Weiss
Trends and Prospects for Rural M anufacturing
WP-92-12
William A. Testa
State and Local G overnm ent Spending—The B alance
B etw een Investm ent and C onsum ption
WP-92-14
Richard H. Mattoon
Forecasting with R egional Input-Output T ables
P.R. Israilevich, R. Mahidhara, and G.J.D. Hewings
WP-92-20
A Primer on Global A uto M arkets
Paul D. Ballew and Robert H. Schnorbus
WP-93-1
Industry Approaches to Environm ental P olicy
in the Great Lakes R egion
WP-93-8
David R. Allardice, Richard H. Mattoon and William A. Testa
The M idw est Stock Price Index—Leading Indicator
o f R egional E conom ic A ctivity
WP-93-9
William A. Strauss
Lean M anufacturing and the D ecisio n to V ertically Integrate
S om e Empirical E vidence From the U .S . A utom obile Industry
WP-94-1
Thomas H. Klier
D om estic C onsum ption Patterns and the M idw est E conom y
WP-94-4
Robert Schnorbus and Paul Ballew
1
Working paperseries continued
To Trade or Not to Trade: Who Participates in RECLAIM?
WP-94-11
Restructuring & Worker Displacement in the Midwest
WP-94-18
ThomasH.KlierandRichardMattoon
PaulD.BallewandRobertH.Schnorbus
ISSUES IN FINANCIAL REGULATION
Incentive Conflict in Deposit-Institution Regulation: Evidence from Australia
EdwardJ.KaneandGeorgeG.Kaufman
WP-92-5
Capital Adequacy and the Growth of U.S. Banks
WP-92-11
Bank Contagion: Theory and Evidence
WP-92-13
Trading Activity, Progarm Trading and the Volatility of Stock Returns
WP-92-16
Preferred Sources of Market Discipline: Depositors vs.
Subordinated Debt Holders
WP-92-21
HerbertBaerandJohnMcElravey
GeorgeG.Kaufman
JamesT.Moser
DouglasD.Evanoff
An Investigation of Returns Conditional
on Trading Performance
WP-92-24
The Effect of Capital on Portfolio Risk at Life Insurance Companies
WP-92-29
A Framework for Estimating the Value and
Interest Rate Risk of Retail Bank Deposits
WP-92-30
Capital Shocks and Bank Growth-1973 to 1991
WP-92-31
The Impact of S&L Failures and Regulatory Changes
on the CD Market 1987-1991
WP-92-33
JamesT.MoserandJackyC.So
ElijahBrewer111,ThomasH.Mondschean,andPhilipE.Strahan
DavidE.Hutchison,GeorgeG.Pennacchi
HerbertLBaerandJohnN.McElravey
ElijahBrewerandThomasH.Mondschean
2
Working paperseries continued
Junk Bond Holdings, Premium Tax Offsets, and Risk
Exposure at Life Insurance Companies
Elijah Brewer IIIand Thomas H. Mondschean
WP-93-3
Stock Margins and the Conditional Probability of Price Reversals
Paul Kofman and James T. Moser
W P -93-5
Is There Lif(f)e After DTB?
Competitive Aspects of Cross Listed Futures
Contracts on Synchronous Markets
Paul Kofman, Tony Bouwman and James T. Moser
Opportunity Cost and Prudentiality: A RepresentativeAgent Model of Futures Clearinghouse Behavior
Herbert L. Baer, Virginia G. France and James T. Moser
The Ownership Structure of Japanese Financial Institutions
Hesna Genay
Origins of the Modern Exchange Clearinghouse: A History of Early
Clearing and Settlement Methods at Futures Exchanges
James T. Moser
The Effect of Bank-Held Derivatives on Credit Accessibility
Elijah Brewer III, Bernadette A. Minton and James T. Moser
Small Business Investment Companies:
Financial Characteristics and Investments
Elijah Brewer IIIand Hesna Genay
W P -93-11
W P -93-18
W P -93-19
W P-94-3
W P -94-5
W P -94-10
MACROECONOMIC ISSUES
An Examination of Change in Energy Dependence and Efficiency
in the Six Largest Energy Using Countries-1970-1988
Jack L. Hervey
W P -92-2
Does the Federal Reserve Affect Asset Prices?
Vefa Tarhan
W P -92-3
Investment and Market Imperfections in the U.S. Manufacturing Sector
Paula R. Worthington
W P -92-4
3
Workingpaper series continued
Business Cycle Durations and Postwar Stabilization of the U.S. Economy
WP-92-6
Mark W. Watson
A Procedure for Predicting Recessions with Leading Indicators: Econometric Issues
and Recent Performance
WP-92-7
James H. Stock and Mark W. Watson
Production and Inventory Control at the General Motors Corporation
During the 1920s and 1930s
WP-92-10
Anil K. Kashyap and David W. Wilcox
Liquidity Effects, Monetary Policy and the Business Cycle
WP-92-15
Lawrence J. Christiano and Martin Eichenbaum
Monetary Policy and External Finance: Interpreting the
Behavior of Financial Flows and Interest Rate Spreads
WP-92-17
Kenneth N. Kuttner
Testing Long Run Neutrality
WP-92-18
Robert G. King and Mark W. Watson
A Policymaker’s Guide to Indicators of Economic Activity
Charles Evans, Steven Strongin, and Francesca Eugeni
WP-92-19
Barriers to Trade and Union Wage Dynamics
WP-92-22
Ellen R. Rissman
Wage Growth and Sectoral Shifts: Phillips Curve Redux
WP-92-23
Ellen R. Rissman
Excess Volatility and The Smoothing of Interest Rates:
An Application Using Money Announcements
WP-92-25
Steven Strongin
Market Structure, Technology and the Cyclicality of Output
WP-92-26
Bruce Petersen and Steven Strongin
The Identification of Monetary Policy Disturbances:
Explaining the Liquidity Puzzle
WP-92-27
Steven Strongin
4
Working paperseries continued
Earnings L o sses and D isp laced W orkers
WP-92-28
Louis S. Jacobson, Robert J. LaLondetand Daniel G. Sullivan
Som e Empirical E vid en ce o f the E ffects on M onetary P olicy
Shocks on E xchange R ates
WP-92-32
Martin Eichenbaum and Charles Evans
An U nobserved-C om ponents M odel o f
C onstant-Inflation Potential Output
WP-93-2
Kenneth N. Kuttner
Investm ent, Cash F low , and Sunk C osts
WP-93-4
Paula R. Worthington
L essons from the Japanese M ain Bank System
for Financial System Reform in Poland
WP-93-6
Takeo Hoshi, Anil Kashyap, and Gary Loveman
Credit C onditions and the C yclical B ehavior o f Inventories
WP-93-7
Anil K. Kashyap, Owen A. Lamont and Jeremy C. Stein
Labor Productivity During the Great D epression
WP-93-10
Michael D. Bordo and Charles L. Evans
M onetary P olicy Shocks and Productivity M easures
in the G -7 Countries
WP-93-12
Charles L. Evans and Fernando Santos
C onsum er C onfidence and E conom ic Fluctuations
WP-93-13
John G. Matsusaka and Argia M. Sbordone
V ector A utoregressions and Cointegration
WP-93-14
Mark W. Watson
T esting for Cointegration W hen S om e o f the
Cointegrating V ectors Are K now n
WP-93-15
Michael T. K Hormth and Mark W. Watson
Technical C hange, D iffu sion , and Productivity
WP-93-16
Jeffrey R. Campbell
5
Working paperseries continued
Economic Activity and the Short-Term Credit Markets:
An Analysis of Prices and Quantities
WP-93-17
Benjamin M. Friedman and Kenneth N. Kuttner
Cyclical Productivity in a Model of Labor Hoarding
WP-93-20
Argia M. Sbordone
The Effects of Monetary Policy Shocks: Evidence from the Flow of Funds
WP-94-2
Lawrence J. Christiano, Martin Eichenbaiun and Charles Evans
Algorithms for Solving Dynamic Models with Occasionally Binding Constraints
WP-94-6
Lawrence J. Christiano and Jonas D.M. Fisher
Identification and the Effects of Monetary Policy Shocks
WP-94-7
Lawrence J. Christiano, Martin Eichenbaum and Charles L. Evans
Small Sample Bias in GMM Estimation of Covariance Structures
WP-94-8
Joseph G. Altonji and Lewis M. Segal
Interpreting the Procyclical Productivity of Manufacturing Sectors:
External Effects of Labor Hoarding?
WP-94-9
Argia M. Sbordone
Evidence on Structural Instability in Macroeconomic Time Series Relations
WP-94-13
James H. Stock and Mark W. Watson
The Post-War U.S. Phillips Curve: A Revisionist Econometric History
WP-94-14
Robert G. King and Mark W. Watson
The Post-War U.S. Phillips Curve: A Comment
WP-94-15
Charles L. Evans
Identification of Inflation-Unemployment
WP-94-16
Bennett T. McCallum
The Post-War U.S. Phillips Curve: A Revisionist Econometric History
Response to Evans and McCallum
WP-94-17
Robert G. King and Mark W. Watson
6
Working paperseries continued
Estim ating D eterm inistic Trends in the
Presence o f Serially Correlated Errors
WP-94-19
Eugene Canjels and Mark W. Watson
S olvin g N onlinear Rational E xpectations
M odels by Parameterized Expectations:
C onvergence to Stationary Solutions
WP-94-20
Albert Marcet and David A. Marshall
The E ffect o f C ostly C onsum ption
A djustm ent on A sset Price V olatility
WP-94-21
David A. Marshall and Nayati G. Parekh
The Im plications o f First-Order Risk
A version for A sset Market R isk Prem ium s
WP-94-22
Geert Bekaert, Robert J. Hodrick and David A. Marshall
A sset Return V olatility with E xtrem ely Small C osts
o f Consum ption A djustm ent
WP-94-23
David A. Marshall
7
@FedFRASER