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T o b in 's q a n d A s s e t R e tu r n s :
I m p lic a tio n s f o r B u s i n e s s
C y c le A n a l y s i s
Lawrence J. Christiano and
Jonas D.M. Fisher
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
October 1995 (W P -95-14)
FEDERAL RESERVE BANK
OF CHICAGO
October 1995
T o b i n ’s q a n d A s s e t R e t u r n s :
I m p l i c a t i o n s for B u s i n e s s C y c l e A n a l y s i s
Law rence J. Christiano*
Federal Reserve Bank of Minneapolis,
Northwestern University,
National Bureau of Economic Research,
and Federal Reserve Bank of Chicago
Jonas D . M. Fisher*
University of Western Ontario
ABSTRACT
The marginal cost of plant capacity, measur^1 1 the price of equity, is significantly procyclical. Yet, the
price of a major intermediate input into expanding plant capacity, investment goods, is countercyclical.
The ratio of these prices is Tobin’s q. Following convention, we interpret the fact that Tobin’s q differs
from unity at all, as reflecting that there are diminishing returns to expanding plant capacity by installing
investment goods ( “adjustment costs”). However, the phenomenon that interests us is not just that Tobin’s
q differs from unity, but also that its numerator and denominator have such different cyclical properties. We
interpret the sign switch in their covariation with output as reflecting the interaction of our adjustment cost
specification with the operation of two shocks: one which affects the demand for equity and another which
shifts the technology for producing investment goods. The adjustment costs cause the two prices to respond
differently to these two shocks, and this is why it is possible to choose the shock variances to reproduce the
sign switch.
These model features are incorporated into a modified version of a model analyzed in Boldrin, Christiano
and Fisher (1995). That model incorporates assumptions designed to help account for the observed mean
return on risk free and risky assets. We find that the various modifications not only account for the sign
switch, but they also continue to account for the salient features of mean asset returns.
We turn to the business cycle implications of our model. The model does as well as standard models
with respect to conventional business cycle measures of volatility and comovement with output, and on one
dimension the model significantly dominates standard models. The factors that help it account for prices
and rates of return on assets also help it account for the fact that employment across a broad range of sectors
moves together over the cycle.
*Prepared for the Economics and Finance Conference, held at Washington University in St. Louis on Septem
ber 15-16, 1995. We are grateful to Michele Boldrin, Martin Eichenbaum, and Narayana Kocherlakota for
their advice. Christiano thanks the National Science Foundation, and Fisher thanks the SSHRC for financial
support. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve
Bank of Minneapolis or the Federal Reserve System.
1
In tro d u c tio n
The price of a marginal unit of plant capacity, as measured by the price of equity, is sig
nificantly procyclical. Yet, the price of a major input into expanding plant capacity, new
investment goods, is countercyclical. In this paper we provide a quantitative account for
this
s ig n s w i t c h p h e n o m e n o n .
We do so in a model that does at least as well as previous
models in accounting for key features of (a) asset returns and (b) the business cycle. The
features of asset returns that we have in mind include the observed low average return on
risk free assets and the high average return on equity. The features of business cycles we
have in mind include standard measures of comovement and volatility as well as measures
of persistence.
To capture (a) and (b), we build on the recent model by Boldrin, Christiano and Fisher
(1995) (BCF). In particular, we assume consumption and investment goods are produced
by distinct production technologies and that there are limitations on the mobility of factors
of production across sectors. In addition, we adopt the habit persistence specification of
preferences proposed in Constantinides (1990) and Sundaresan (1989).
We show that the sign switch phenomenon can be accounted for by a combination of
separate shocks to the two production technologies and the “adjustment cost” model of
investment analyzed in Lucas and Prescott (1971) and the references they cite. According to
this model, the more quickly a firm attempts to incorporate new investment goods into an
existing plant, the less effective those goods are on the margin at expanding plant capacity.
After verifying that our model can account for the sign switch phenomenon and that
it continues to accoimt for mean asset returns as in BCF, we examine its business cycle
characteristics. We find that, not only does the model do about as well as standard models
on the business cycle statistics usually emphasized, but the model actually makes a step
forward on one particularly important business cycle fact. Perhaps the
d efi ni ng
characteristic
of business cycles is comovement: activity across a broad range of sectors moves up and
down together over the business cycle (see Lucas (1981, p.217) and Sargent (1979, p.215).)
Standard real business cycle models are consistent with this fact in that they imply that
the outputs of the consumption sector and the investment goods sector are both procyclical.
1
However, we report evidence suggesting that employment across these sectors is procyclical
as well.1 Standard real business cycle models are inconsistent with this evidence. They
have the property that consumption is smoothed over the cycle: in a boom, consumption
rises relatively little, as the improvement in technology is partially offset by a reallocation of
factors of production out of consumption and into investment goods sectors. The opposite
occurs in a recession. This is why standard models have the dubious implication that hours
worked in the production of consumption goods is countercyclical. Although this is a feature
of most real business cycle models, it is not a feature of ours.
The following section provides a brief, nontechnical overview of the analysis. After that,
we document the empirical properties of equity prices and prices and quantities of investment
goods. Then, we formally describe our model and present the quantitative analysis. Finally,
we present concluding remarks.
2
O v e rv ie w
o f th e A n a ly s is
In what follows we first discuss the cyclical properties of investment prices, and then we go
on to explain how our model accounts for these properties. We then discuss the sign switch
phenomenon. Finally, we discuss the business cycle implications of the model.
I n v e s t m e n t G o o ds P r i c e s
The time series behavior of the price of an important component of investment goods,
producers’ durable equipment, has recently been documented and analyzed by Greenwood,
Hercowitz and Krusell (1992). They show that the price deflator of this good, as measured
by Gordon (1990), is counter cyclical. In addition, they document that, starting in particular
in the 1980s, this price index exhibits a downward trend. These trend and cyclical character
istics are a feature of household durable goods too. Together, these two components account
for about 65 percent of the value of overall investment activity. The remaining components
of investment—investment in structures and residential investment—also exhibit a down
ward trend in their price starting in the 1980s, but they do not display the same significant
1See also Benhabib, Rogerson and Wright (1991, ftn.14), Murphy, Schleifer and Vishny (1989).
2
countercyclicality. As a result, the price index of overall investment activity is only slightly
countercyclical.
We follow Greenwood, Hercowitz and Krusell (1992) in interpreting these features of the
price data as reflecting that investment dynamics are driven by both demand and supply
shocks. Demand shocks are modeled as arising from a technology shock that is common
across investment and consumption goods sectors. A boom triggered by this kind of shock
generates a relative shift in demand towards investment goods for consumption-smoothing
reasons and so produces a rise in their price. Supply shocks are modeled as arising from a
disturbance that is specific to the technology for producing investment goods. Innovations
in this shock generate a negative covariance between the price of investment goods and
output. We parameterize the variances of our two shocks so that the model reproduces
the observed weak countercyclicality of investment goods prices and also reproduces the
estimated variance of the aggregate Solow residual reported in Christiano (1988).
We accommodate the trend in the relative price of investment goods by the assumption
that the investment-specific technology shock is a random walk with a positive drift. The
implication that disturbances from this source are permanent is consistent with the notion
that they represent shocks in the rate of arrival of innovations. At the same time, we posit
that the economy-wide shock is transitory and has no trend. This is the only shock affecting
the sector producing consumption goods, and its transitory nature captures the notion that
the only disturbances to that technology are shocks to the weather, or natural disasters,
or perhaps even labor disputes .2 The drift in the investment-specific technology shock is
set to reproduce the observed rate of growth in consumption. The single autoregressive
parameter in the stationary economy-wide shock is selected to reproduce the slight negative
autocorrelation in the growth rate of the measured Solow residual reported in Christiano
(1988).
2In effect, th e m odel tak es th e positio n th a t we know how to bake b read or serve a h e a rty m eal a b o u t
as well as we did cen tu ries ago. P erm a n en t shifts in th e technology for producing con su m p tio n goods are
viewed as b ein g em bodied in cap ital.
3
The Sign Switch
To see how adjustment costs help account for the sign switch, it is useful to first consider
the benchmark case in which there are no adjustment costs. Then, the price of equity—we
identify this with the marginal cost of new plant capacity—and the marginal cost of new
investment goods are identical, i.e., Tobin’s
q
is identically equal to unity. Thus, in the
absence of adjustment costs, the weak countercyclicality of investment goods prices would
be shared by equity prices and there would be no sign switch.
Under adjustment costs, the price of equity becomes procyclical. This is because adjust
ment costs have the effect of (i) reducing the response of equity prices to investment-specific
technology shocks (these shocks make equity prices countercyclical), and (ii) enhancing their
response to aggregate shocks (a force for procyclicality). The reason for (i) is that with
adjustment costs, an investment-specific technology shock triggers two opposing effects on
equity prices. On the one hand, the fall in the price of investment goods exerts a down
ward pressure on the price of equity. On the other hand, the higher level of investment is
associated with a drop in the margin *1 *ffoctiveness with which investment goods enhance
new plant capacity, and this exerts upward pressure on the price of equity. The reason for
(ii) is that with adjustment costs, an aggregate technology shock triggers two reinforcing
effects on equity prices. This type of shock generates rises in both the price and quantity
of investment goods. The rise in the price generates upward pressure on equity prices, and
under adjustment costs the rise in investment does too.
Our adjustment cost formulation is controlled by a single curvature parameter. We set
this parameter so that the model reproduces the observed positive correlation between equity
prices and output.
O th er M odel Im plications
We have enough free parameters so that our model can exactly capture the sign switch
phenomenon. To test the model, we examine other implications. First, our adjustment cost
formulation generates an elasticity of investment to Tobin’s q , an object for which there exist
empirical estimates. We compare our model’s implications with these estimates.
4
Second, the progress that our model makes on the comovement puzzle - the fact that
sectoral employment moves together over the business cycle - reflects two assumptions, (i)
Following BCF, we assume factors of production must be allocated prior to the realization
of the current period shocks. This assumption is intended to capture the various real-world
frictions that make it difficult for workers to quickly leave a job in one sector and start work
in another. In our model, it is simply not possible to immediately shift factors of production
across sectors in the period of a shock,
(ii) There is also little incentive to shift labor
resources out of the consumption goods sector in the periods after a positive technology
shock. Employment in the production of consumption goods rises in the periods after a
favorable shock in the investment goods sector because the associated wealth effect makes
consumption goods more valuable. Employment also rises in the periods after an aggregate
shock. The expansion in the supply of consumption goods in the period of the shock has the
effect of raising the value of consumption goods in subsequent periods because of the effects
of habit persistence in the utility function. The transient nature of the aggregate shock then
ensures that employment must be high to satisfy that desire. Employment in the production
of consumption goods is procyclical because it is procyclical relative to each of the shocks in
the model.3
Third, we document that our model implies low risk aversion on the part of households.
As in BCF, steady state relative risk aversion with respect to a bet on wealth is restricted to
unity. In addition, our model can account for the observed equity premium by assuming a
very small degree of relative risk aversion with respect to bets on consumption. However, the
ability of the model to account for the equity premium with low relative risk aversion with
respect to consumption reflects some seemingly counterfactual implications for equilibrium
consumption growth. It implies consumption growth is negatively autocorrelated—in the
data it is positively autocorrelated—and it overstates the innovation in consumption.
To understand how these implications of the model help account for its success in ex
plaining the mean equity premium, it is useful to repeat an observation in BCF. They argue
3A n o th e r fe a tu re of o u r m odel environm ent helps account for (ii); nam ely, our assu m p tio n th a t u tility
is lin ear in leisure. W ith in a ce rta in class of utility functions, th is assu m p tio n m axim izes th e likelihood of
positive com ovem ent of secto ral em ploym ent. We discuss th is issue in d etail below.
5
that the key to getting an equity premium in a model like ours lies in generating an equilib
rium process for the capital gains component of the return on equity which has the “right”
pattern. In practice this translates into the requirement that (i) households have a strong
desire to buy assets when consumption is high and to sell assets when consumption is low;
and (ii) the nature of the technology has the effect of frustrating these desires. Habit per
sistence in preferences delivers (i) and the limitations on the mobility of resources deliver
(ii). Another way to enhance (i) is to construct a model environment in which equilibrium
consumption growth is negatively autocorrelated. This feature particularly enhances the
motive to smooth consumption and, thus, to buy assets when there is a positive innovation
in consumption.
3
P ric e D a ta
In this section, we present our analysis of the dynamic properties of postwar U.S. data
on share prices and the price of new investment goods. Our results are that the price
of aggregate investment goods is slightly countercyclical and displays a downward trend,
particularly beginning in the 1980s. The cyclical behavior of the price of equity differs
sharply from that of investment. Equity prices are significantly procyclical.
3.1
N e w I n v e s t m e n t G o o d s Prices
We study the components of U.S. investment reported in the National Income and Product
Accounts (NIPA). We also consider the annual price indexes for consumer durables and for
business equipment reported in Gordon (1990) for the period 1947-1983. Investment price
indices were divided by the implicit price deflator for household consumption of nondurables
and services and then logged prior to analysis. We now consider the trend and business cycle
characteristics of these data .4
Trend
4For a re la te d discussion, see F isher (1994a).
6
Figure 1 graphs the price data, together with their Hodrick-Prescott (HP) trend, for the
period 1947QI-1995QI. Our broadest measure of investment is the NIPA measure of business
fixed investment plus consumer durables. Note from Figure la that the associated implicit
price deflator displays roughly no trend until the 1980s, whereupon it takes a sharp turn
down. In interpreting this, note the difference between the Gordon price series for household
durables and business equipment and the associated implicit price deflators from the NIPA.
Gordon’s are the series that fall sharply throughout the sample in Figures lb and lg. He
argues that the difference reflects the failure of the NIPA data to properly take into account
quality improvement in consumer and producer durables. This suggests that, despite the
apparent lack of trend prior to the 1980s in Figiue la, investment good prices probably
were falling then. The behavior of the price of residential investment (Figure le) and of
structures investment (Figiue If) suggests, though, that the fall in aggregate investment
prices probably was slower before the 1980s than after.
Figure 2 displays the ratio of the various categories of investment to Gross Domestic
Product. In each case, the solid line depicts expenditure shares, that is, the numbers are
formed as a ratio of nominal investment to nominal GDP. The dashed line depicts the ratio
in real terms. Note in Figure 2 a that the ratio in value terms of our broad measure of
investment is roughly stationary, while the ratio in real terms trends up from about 21
percent of GDP in the early 1950s to about 27 percent of GDP now. Thus, the fall in the
price of investment goods in the 1980s has been offset by a simultaneous increase in real
output. This is also a feature of components of investment, for example consumer durables
(Figure 2b) and business equipment in the 1980s (Figure 2g). Investment in structures
appears to be an exception, with quantity not rising by enough to offset the reduced price
in the 1980s (Figure 2f). Figures 2h and 2i indicate that the sum of private consumption of
nondurables and services, and government purchases, expressed as a ratio to total output,
is roughly stationary. However, the share of the components does not appear stationary.
We infer from Figures 1 and 2 that, to a first approximation, the aggregate data display
balanced growth in expenditure share terms, but that the quantity of investment goods grows
7
more rapidly than the quantity of consumption goods.5 This abstracts from other important
features of the data, including the significant upward trend in the price index of important
components of investment prior to the 1980s.
B usiness Cycles
Now consider the business cycle properties of the price data.
Figure 3 displays the
deviations of the (logged) prices from their HP trend (solid line) together with the associated
deviations for log GDP (dashes). Casual inspection confirms the Greenwood, Hercowitz
and Krusell (1992) finding that the price of equipment is strongly countercyclical. Figure
4 displays the associated cross correlation functions and associated plus and minus two
standard deviation error bands. First, note that the contemporaneous correlation is negative,
though not significantly so, for our broad measure of investment (see Figure 4a). Durables are
countercyclical—though significantly so only when correlated with output one quarter in the
past—but the correlation between fixed investment prices and output is not significant. This
reflects the very different cyclical behavior of equipment verstis structures and residential
investment. Equipment is significantly countercyclical, whereas residential investment is
strongly procyclical and structures are acyclical.
The data suggest that there are interesting differences in the business cycle properties of
the components of investment. Further analysis of these differences is beyond the scope of
this paper .6 Our model recognizes only one form of investment, and we calibrate it based on
our point estimate for the correlation between the price of aggregate investment and output,
which is —0.15.
3.2
S t o c k Prices
We consider the cyclical behavior of the S & P 500, Dow Jones and New York Stock Exchange
stock price indexes for various industries, as supplied in Citibase .7 In each case, the price
5T h is is co n sisten t w ith th e resu lts of th e form al sta tistic a l analysis presented in E ichenbaum a n d H ansen
(1990) an d F ish er (1994b).
im p lic a tio n s of th e p rocyclicality in residential investm ent price deflator are explored in F ish er (1995).
7T h ese indexes are b e st th o u g h t of as th e p ro d u ct of price and quantity. W e assum e th a t m ost of th eir
business cycle v a ria tio n reflects v ariatio n s in price.
8
index was deflated by the same implicit price deflator for consumption of nondurables and
services used to deflate the price indexes of new investment. All data were logged and HP
filtered prior to analysis. Table 1 reports the volatilities of the price data, divided by the
volatility of output, which is roughly 1.8 percent. In addition, the correlations between the
price indexes and the cyclical part of U.S. GDP are also reported. Note that, with two
exceptions, these correlations are significantly positive. The two exceptions are the S&P 500
data for the transportation and utilities industries. The dynamic correlations with output
are presented in Figure 5. Note that the largest correlations are between the stock price and
next quarter’s GDP. These correlations are almost all near 0.5.
4
M odel E conom y
This section presents our model economy. There is a single representative household and
two production sectors. One produces the consumption good, and the other the investment
good. There are two technology shocks: a logarithmic random walk shifts the production
function for investment goods, and a stationary first order autoregressive shock shifts both
production functions. Households and firms are competitive.
In what follows we first present the household problem and a discussion of risk aversion.
Then we consider the problem of the firm and equilibrium. We also discuss various features
of the equilibrium of the model.
4.1
Household Problem
Prior to the realization of the date t random variables, the household evaluates consumption
and leisure henceforth according to
OO
(1 )
[log( C t+j - X t + j ) + 77(1 -
H c,t+ j - H ht+j) } ,
3=0
where 77 is a positive scalar,
H Cit, H l t
goods-producing industries, and
Xt
denote employment in the consumption and investment
denotes the habit stock, which is assumed to evolve
according to
(2 )
X t+1 = h X t + b C t .
9
Here,
Ct
Et
is the expectation operator conditioned on all variables dated
and earlier. Also,
t
denotes consumption, and we specify that utility is linear in leisure following Rogerson
(1988) and Hansen (1985).
The household budget constraint is
(3)
c t + s; + sl + Bt
< (1 +
for
x
=
t =
c,i,
0,1,.... Here,
and
in period
t,
Bt
+ (1 +
rlt)S U
Sf
+ (1 +
denotes date
r {_ x) B t- i
w xH c<t + w lt Hi't ,
purchases of shares of equity in industry
t
x,
for
denotes purchases of risk-free debt. The rate of return on equity purchased
(1 + r®t+1), depends upon the period
return on debt, (1 -)- r { ) , depends on the date
rate in industry
+
x
t
state variables are
+ 1 state of nature, while the rate of
state of nature. Also,
t
, which is a function of the date
The household’s date
t
t
denotes the wage
w*
state of nature.
S 1_x, S \ _ v B t~ \ , X t -
In addition, the house
hold knows the values of all prices and rates of retiun for each date and state of nature. The household’s problem at time
H c, t , H i , t , C t , S f , S l , B t .
t
is to select values for its time
t
choice variables,
We capture the notion that there is a degree of precommitment in
the labor supply decision by imposing a particular information constraint on the variables.
In particular, we require that households choose
t
H c<t, H l t
prior to the realization of the date
state of nature, while the remaining choice variables are selected afterward. We refer to
this restriction on the allocation of work effort as the
l i m i t e d l abor m o b i l i t y
assumption. The
household’s objective is to maximize (1) subject to (2)—(3) and the condition that the future
choice variables satisfy the same information constraints. The household’s intratemporal
first order necessary conditions for labor are
(4)
E t - \ W XACtt = rj, x = c , i .
Its intertemporal first order conditions are
(5)
for
(6)
EtPc,t+i ( 1 + r ex t + 1 )
x = c,i,
EtPc,t+i
+ r{),
where
Pct+i =
= 1=
0A-C,t+1
Act
’
10
and Ac>t is the derivative of (1) with respect to
p c,t+i
is the value, in date
t
Ct,
when
E t- \
is replaced by
E t.
The variable
consumption units, of a unit of date t + 1 consumption indexed
by state of nature and scaled by the conditional probability of that state of nature.8
4.2
Risk Aversion
Evaluations of models of asset prices often focus on the implications for risk aversion. One
measure of risk aversion is the amount a person is willing to pay to avoid an unanticipated
gamble. Two types of gamble are of interest: “gambles on wealth” and “gambles on con
sumption.” These are differentiated according to whether agents can use credit markets
to mitigate the effects of the outcome of the gamble. With a gamble on wealth, agents
have full access to credit markets in the period of the gamble. Constantinides (1990) argues
that habit persistence agents have little aversion to gambles like this because they have a
relatively painless way of dealing with the state of the world in which they lose. The fall
in the present value of consumption that must occur with the loss of a bet on wealth can
be accommodated by reducing consumption slowly so that the habit stock has a chance to
fall. By specifying
(3
to be close to unity and formulating habit persistence in terms of the
logarithm, the steady state level of relative risk aversion in wealth is unity in our model (for
further discussion, see BCF.)
A gamble on consumption has the property that agents have no access to credit markets
in the period of the gamble. As a result, the full amount of a loss or gain must be absorbed
by current consumption. Agents then have full access to credit markets in the periods after
the gamble.
We suspect that risk aversion over consumption gambles is harder to measure (or intro
spect on) than risk aversion over wealth gambles. Still, it is useful to define a measure of
risk aversion over consumption gambles precisely so that we can report on this aspect of the
model in the results section. For tractability, we define this concept of risk aversion relative
to a slightly simpler environment, in which hours worked is fixed, the rate of return on sav
8In p a rtic u la r, let gt{s') d en o te th e d a te t conditional probability th a t sta te of n a tu re s' will be realized
in p erio d t + 1. T h e n gt(s')pc,t+i(V) is th e value— d en o m inated in d a te t consum ption u n its— of a u n it of
c o n su m p tio n in d a te t + 1, sa te of n a tu re s'. Here, pCtt+\(s') is th e value of p Cit+ iin s ta te of n a tu re s'.
11
ing is constant, and there is no uncertainty. Thus, suppose a household has the following
preferences:
OO
(7 ) x ^ i o g f a - x , ) ,
t=o
where the habit stock evolves as before. At date 0, the household has a given stock of
wealth, Wo (= ^ - l + B_i), and habit,
X q,
and seeks to optimize (7) subject to the following
intertemporal budget constraint:
The solution to this problem is characterized by
( C , - X , ) = Q 1 ‘,
7 = /3 ( l+ r ) ,
where
(8)
A'o)
(? -* )
Therefore, the value function for this problem is, apart from an additive constant,
(9)
=
For further details, see BCF.
Let
Co, C i , ...,
be the solution to this problem. Now, suppose the household is confronted
with the following gamble: it is given
and must give up
fiCo
[i C q
units of consumption goods with probability 1/2
consumption goods also with probability 1/2. We measure relative
risk aversion in consumption, RRAC, by the fraction,
u,
of
C q the
sacrifice with probability one in order to avoid this gamble. That is,
log (C0(l
- v ) - X 0)
(10) = i {log (C0(l
+ log (C0(l
Here,
W\
+
u
solves
0 v ( W u h X o + b C 0( l - v ) )
- f i ) - X 0)
+ n ) - X 0)
household is willing to
+
+
Pv
pv
(W ,
h X o + b C 0( l - m))
(Wl5 h X o +
bC0{\ +
//))} .
= (1 + r) Wo — Co and is unaffected by the outcome of the gamble. We solve
this problem on a steady state growth path by factoring Co from (10) and setting
and
W q/ C
q
function of
X q/ C
q
to their steady state values. Evidently, the measure of risk aversion we use is a
fj,.
12
4.3 F i r m s
There are consumption goods producing and investment goods producing firms. Each has a
one period planning horizon. Whatever physical capital the firm uses in production in period
t +
1 must be put into place by the end of period
t.
This capital is produced by combining
previously installed capital with new investment goods. To finance the purchase of these
inputs, the firm issues debt and equity in period
t.
There are separate equity markets for the
two types of firms, and the two types of equity command different, competitively determined,
state-contingent rates of return. Since the competitive rate of return on debt is known at
the time it is issued, in equilibrium there can be only one rate of return for that financial
instrument. When period
t +
1 occurs, the firm observes the state of nature and, hence, the
prevailing wage rate. It then enters competitive labor markets to hire that amount of labor
which maximizes cash flow in that state of nature. The firm’s cash flow is the value of its
production, plus its undepreciated stock of capital, net of expenses. The firm’s objective at
date
is to maximize the date t value of cash flow at t + 1, summed across all possible states
t
of nature.
There are several prices relevant to the firm’s capital decisions. There are the prices
of the raw materials used in period
t
to produce end of period
t
capital—i.e., the price of
new investment goods and of previously installed capital. Also, there is the price (actually,
marginal cost) of end of period
t
capital. This is, in general, different from the date
t
+ 1
price of previously installed capital because the latter reflects the realized state of nature in
period
t
+ 1. In sum, these prices are
• PfeX)t ~ price of previously installed capital in sector
• Pi,t
~ price of new investment goods.
• Pk'x,t ~ price of newly produced capital in sector
t
x = i,c.
x
=
i,
c, available for production in
+ 1.
Each of these prices is taken as parametric by the firm.
statement of the firm problem in each sector.
13
We now provide a formal
The technology for producing consumption goods in t + 1 is
(11)
CM
<
where 6 t+ 1 is a covariance stationary shock to technology:
(12)
0t +1
=
pOt
+
£ t + 1,
0 < p < 1. The technology for producing new investment goods in t + 1 is
(13) I c,t+ i
+ Ii,t+ i
< V t+ iK ° t 4-i(exp(0t+i)-ffi,t+i)1_a
where
(14)
Vt + i = e x p ( p
Here,
st
and
pt
+
p t+ i ) V t .
are zero-mean, random variables which are independent of each other and
over time and which have standard deviations
transformation between
I c,t+i
and
I ltt+i
cre
and crM, respectively. The linear rate of
implicit in (13) guarantees that, in equilibrium,
the prices of new investment goods for the consumption and investment goods sectors are
equalized.
The technology for producing end of period
(15)
K x<t+ i
<
Q x ( y, z )
t
capital,
K x t+ u
for industry
x
is
,
where
(16)
for
Q x(y,z)
x = c,i
and
ai
ip <
+
i/ip
a 2z'1’
1. In (15)—(16),
y
denotes previously installed capital and z denotes
new investment goods. When
ip =
1, (15) corresponds to the conventional linear capital
accumulation equation. When
ip <
1, then the marginal product of new investment goods
is decreasing in the flow of investment. The technology described in (15)-(16) is a special
case of the adjustment cost formulation posited in Lucas and Prescott (1971) and in the
references they cite. We choose the constants, 0^ > 0 and a2 > 0, to guarantee Qf =
Q2 —
1
in nonstochastic steady state. This has the effect of making the nonstochastic steady state
properties identical to what they are when
ip =
14
1, regardless of the actual value of
ip. Also,
it has the effect of forcing Tobin’s
q
to be unity on a steady state growth path. Finally, as
discussed further below, it has the effect of normalizing Tobin’s
The financing constraint faced by a firm in industry
(17)
Pkx,ty
+
and its period
(18) 7r?+1 =
<
Pi,tZ
t
x
q
at unity when
ip
= l.9
is
S x>t + B X't ,
+ 1 cash flow constraint is
Y x,t+1
+ (1 -
6 ) Q X (y , z ) P kx,t+1 - w xt+ l H x,t+l
- (1 +
r eX it+l) S f
- (1 +
r{)B t >
0.
Here, Y x^+1 is the firm’s gross output, given by (11) or (13), measiued in date t + 1 consump
tion units.10 Also, we assume that if
period
t +
Qx
is the amount of capital used by the firm dining
1, then (1 —8 ) Q X remains at the end of the period, when it is made available
for sale. The firm’s profit function is the value of 7r*+1 denominated in imits of the date
consumption good, summed across all possible data
variable,
p c,t+ i ,
t +
1 states of nature:
firm also makes use of
P^t+\-
The firm’s objective is to find
,t >-Dx,t
EtPc,t + i
S Xjt, B x>t, z , y , H x^+1
to solve
max 7rf+1,
PiX,t -f-1
subject to the relevant production technology and (17)—(18).
9T h e form ulas for a j an d a.2 are
ai
=
[(1 - 5 ) e x p ( - / 2 / ( l - a ) ) ] 1 - ^ ,
a-2
=
[ l - ((1
-6)exp(-fl/(l
- a ) ) ) 1^]
10T h u s, for th e co n su m p tio n industry,
Yc,t+i = Q° (V,z)a (exT>(9t+ i) H c,t+ i y - a ,
a n d for th e in v estm en t goods indu stry ,
Yi,t+i = Piit+\Vt+ i Q c (y, z ) a (exp(9t+i )H itt+ i ) l ~a ■
The
is given by (6) in equilibrium and is viewed as parametric by the firm. Other
variables viewed as exogenous by the firm are Pfca. t+1,ty^+1,r® f+1, r / for
(19) _ max
E tp c,t+i^t+i-
t
15
x = i,c.
The
x = i
There are a variety of useful ways to write the efficiency conditions associated with this
problem. The first order condition for hours worked is
(20)
= iu£,
m p l Xtt+1
where
m p l Xit+ 1
denotes the marginal product of labor, denoted in period
t +
1 consumption
units. Let the marginal value to the firm of an extra unit of K x>t+1 be denoted by
(21)
Vx,t ~ EtPc,t +1 \pl'Pkx,t+\
where
m p k Xyt+ 1
d" (1
^)-ffcI,t+l] )
denotes the marginal product of capital, denoted in period t + 1 consumption
units. The first order conditions associated with
(22)
VxjQ^t
=
VXit,
z , y , S Xtt, B x>t,
P i , t \ V i , t Q i tt = Pkx,t^, E tp c,t+ i(l
+ r i,t+i) =
where A > 0 is the multiplier on the constraint (17), and
with respect to its
i th
argument
i —
Q*t
are
E tp c,t+ i(l + r {)
= A,
is the partial derivative of
Qx
1,2.
Let the marginal cost of producing
K x t +\
by a firm in industry
x
be denoted by
Pk'x,t-
It
is readily established that
TD
P
P
jrkXlt
'
Household optimization ensures, via (5) and (22), that A = 1 in equilibrium. This, together
k'x,t ~ Q h ~
Q it
with (23) implies
(24)
Vx,t
Pk'x,t >
b c,
i.e., the marginal value of end-of-period
t
capital is equated to its marginal cost.
4.4 E quilibrium
We adopt the normalization that the number of firms of each type and the number of
households is one, and we assume that all agents of each type behave identically. A sequenceof-markets equilibrium is then defined in the usual way. Market clearing implies that, in a
symmetric equilibrium, the demand for previously installed industry
denoted above by
z,
x
capital in period £,
equals the supply, (1 —8 ) K Xit■ Similarly, the demand for period
t
new
investment goods by industry x, denoted by y, is 7I t.
We proceed now to discuss various features of the equilibrium, including the sign switch,
equity premium, Tobin’s
q,
and comovement of employment.
16
4.4.1
T h e Sign Sw itch
We now endeavor to provide insight into how it is that our model can account for the sign
switch observations: the fact that the price of equity—which we identify with
cyclical, while the price of new investment goods,
Pi t
P itt,
Pk'x,t—is
pro
is slightly countercyclical. Consider
first. Investment productivity shocks alone create a negative covariance between
P itt
and
output, and shocks to aggregate productivity by themselves create a positive covariance be
tween
P i it
for
and er^ so that the model generates a slightly countercyclical p f.
ae
and output. Thus, it should be no surprise that we can select relative magnitudes
Now consider
costs (i.e.,
ip
Pk'x,t-
If Qf.t = 1, as in the conventional formulation without adjustment
= 1), then obviously
switch. However, when
ip < 1,
Pk'x,t
=
Pi t ,
and there is no way to account for the sign
then there is a wedge between these two prices. The wedge
has the effect of reducing the impact on
of investment-specific technology shocks and of
Pk'x,t
increasing the impact of aggregate technology shocks. Consider a positive investment shock
first. Not surprisingly, in our computational experiments we find that this generates a fall in
equilibrium
P^t
and an increase in
two offsetting effects on
in
I x>t
Pk'x,t
Pk>x,t-
I Xtt.
The first relation in (23) indicates that this triggers
The fall in P,)t has the effect of driving P ^ t down, but the rise
has the opposite effect, by driving
falls proportionally less than does
Q x2 l
Pl t
down. In view of this, it is not surprising that
after an investment technology shock. Consider
now a positive shock to aggregate technology. This triggers an increased demand for capital
for consumption-smoothing reasons. Not surprisingly, this results in a rise in
rise in I x
By reducing
more than the rise in
the rise in
P i <t.
Ix t
has the effect of driving
By reducing the impact on
P k>
xt
costs in effect reduce the source of countercyclicality in
Pk'x,t
Pi t
and also a
up proportionally
of investment shocks, adjustment
Pk >
x.
This is why the model predicts
that this variable is procyclical.
In our quantitative analysis, we study an aggregate price index, which we obtain by
combining our two equity prices as follows:
1
p _K c,t+ i p
r i/c',£5
-nt'.i — ”is — -n
- U "f” rs
tvf+i
J\ t+1
r-j
where
K t+ 1
=
K c>t+1
+
K iit+ 1-
17
4.4.2
T o b in ’s q
Tobin’s
q,
the ratio of the marginal value to the firm of
K x^+i
divided by the marginal cost
of a new investment good, is
(25)
=
•* i,t
= \
V2, t
which is unity when
(1 - S ) K
al
+ a2
lx,t
a2
if) = 1,
*
x,t
since 0,2 = 1 in that case. The sign switch phenomenon can
be stated in terms of the elements of Tobin’s
q:
the numerator is procyclical, while the
denominator is countercyclical.
4.4.3
T h e E q u ity P re m iu m
To discuss the equity premium, it is convenient to first obtain an expression for the rate
of return on equity. Linear homogeneity guarantees that, in equilibrium, maximized profits
(19) are zero. The cash flow constraint (18) then guarantees
wf+ l
= 0 in each date and state
of nature. Using this and (20)-(24), one gets the following equilibrium condition after some
algebra:
(26)
1 + r | 1+1 = m P k .‘+i + ( 1
) _ (1 + r ' b f .
,t
Here, 7 f =
B * /S f
Euler equation,
denotes the firm’s debt to equity ratio. The household’s intertemporal
E tp c,t+ i(l
+ r (+1) = b implies
E tp Ctt+iE t ( l + r^+ l ) = l - C o v t ( p c, t + i , l + r t + i )
or, using (5) and (26)
,
,
E t ( l + r e t+1)
(27) -----— —f --------1 =
1+ n
~Covt
(
m p k Xtt+1 + ( l - 8 ) P kxtt+1\
p Cyt+1, ----------------- ---------------\
Ek>x,t
)
(1 + 7 t ),
where the object on the left of the equality is the date t premium on equity in industry
is, approximately,
E tr l
x
and
<+1 —r { . BCF argue that a key channel by which a change in model
specification impacts upon the equity premium operates via its impact on the equilibrium
stochastic process for capital gains,
P kx,t+i/Pk'x,t-
via changes in the stochastic processes for
p c,t+i
The alternative channels, which operate
and
m p k x t + i / P k>
x t,
exert very little direct
effect on the conditional covariance. BCF stress that the combination of habit persistence
preferences and limited factor mobility are effective in producing the sort of stochastic process
for
P kxtt+ i / P k ' x,t
that results in a sizeable equity premium.
18
As is well known, -yf is indeterminate in a model like ours. Equilibrium is consistent with
any state-date contingent pattern for 7 f , although the equilibrium quantity allocations in the
model are unique. To make the analysis interesting, we must therefore fix 7 * exogenously.
We do so by setting 7 f = 7 t = 7 - A numerical value is assigned to 7 in the next section.
We define the overall return on equity as r®+1:
\^°)
re
— P ^ K c,t + i e
r t+ 1
—
r>
Kt
r c,t+ 1 +
1 PkljK ht+ i
r>
K ,t -\-1
e
r i,t+ U
where K t+ 1 = Pfc',tAc,t+ 1 + P k>.,tKi,t+1 -11
4 .4 .4
C om ovem ent
To understand our model’s implications for comovement, it is useful to consider the bench
mark case where
b = h =
0 and the utility of leisure is a power function, separable from
consumption. In this case, equilibrium in the labor market associated with the consumption
good sector implies via (11), (20), and the appropriate analog of (4)
(29)
where
Ot n .c,t
v
> 0 and £ > 0. The specification in ( 1 ) corresponds to £ = 0. Because the
employment decision is made prior to the realization of the date
(30)
Hc,t
=
v{
1-
t
shocks
H c<t - H itty .
It is easily verified that this equation must hold even when the limited labor mobility as
sumption is dropped so that the date t labor decision is contingent upon the date t exogenous
shocks. Thus, without habit persistence, getting comovement in labor is impossible, with or
without limited labor mobility: if £ > 0, then
Hl t
and
Hct
must move in opposite directions.
The case £ = 1 is also inconsistent with comovement, since employment in the consumption
sector is predicted to be constant. Still, relative to preferences based on alternative values
of £, £ = 1 appears to be the most favorable to comovement.
In our quantitative results below, which are based on £ = 1 , we find that to get comove
ment, habit persistence and limited labor mobility are both required.
n T h is is ju s t [(1 + r® t+1) 5 c,t + (1 + rf)t+1)5 iit] / ( S c,t + Si,t), after m aking use of th e firm s’ first order
co n d itio n s a n d our re stric tio n s on th e d e b t to equity ratio.
19
5
A s s ig n in g V a lu e s to th e P a r a m e te r s
In this section, we explain how we assigned values to our model’s parameters. As a prelim
inary evaluation of the model, we report on its trend properties and on its implications for
Tobin’s
q.
There are 11 model parameters, three preference parameters, and four each of technology
and the exogenous shocks:
AM ,
(31)
7,
Pi &ei Pi G\i-
We find it convenient to consider 5 parameters,
0 , a , 8 , j , p,
and the remaining 6 ,
h, b, p,
cre, p , 07 , separately. Loosely, the first set controls the steady state properties of the model,
while the second set controls the business cycle and asset pricing properties.
5.1
P a r a m e t e r s Controlling b t e a d y State
We set
(32)
0 =
0.99999, a = 0.36, 8 = 0.021,7 = 2/3, p a / { l
The indicated value of
0
- a) =
0.004.
was selected to maximize the model’s ability to account for the
observed low risk free rate. The value of a was chosen so that the model’s implication for the
share of GDP earned by capital coincides with an empirical estimate of that quantity based
on data for the 1970s and 1980s taken from the NIPA, as reported in Christiano (1988, ftn.3).
However, as emphasized there, this is the midpoint of a relatively large range of values for a,
determined by the details of how one measures capital income in the NIPA. Also, Christiano
and Eichenbaum (1992, p.441) report that the sample average of [1 —( K t+ 1 —I t ) / K t] is
0.021. By setting
8 =
0.021 in the model, this empirical sample average is reproduced along
the model’s nonstochastic steady state growth path .12 The value of 7 was selected to match
12C h ristia n o an d E ichenbaum (1992) also re p o rt an e stim ate of a . However, th e ir e stim a te exploits pro p
erties of th e s tru c tu re of th e ir m odel, w hich are not shared by our model.
20
the corresponding empirical estimate of the debt to equity ratio reported in Benninga and
Protopapadakis (1990). The linear form of preferences for the representative agent was
chosen to enhance the model’s implication for the volatility of labor. Finally
is the steady state growth rate of consumption in the model, and
ft
Jia/{ 1
—a )
was selected so that
it coincides with the corresponding sample average reported in Christiano and Eichenbaum
(1992, p.441). The parameter
r]
just controls scale, and we set it to 1.
The properties of the model along a steady state growth path are reported in Table 2 (see
the column marked “Calibrated”). A corresponding set of estimates for the U.S. economy
is reported for the entire postwar period and for the period starting the 1980s. We report
results for two sample periods because of the evidence described earlier, which suggests the
official estimates may underestimate the trend fall in the price of investment goods before
the 1980s.
Two empirical measures of consumption are reported,
C
and
C,
with the latter including
government purchases. Given our level of abstraction, it probably makes sense to identify
consumption in the model with the sum of household and government consumption.13 With
this measure of consumption, the model evidently understates consumption’s expenditure
share in output, and correspondingly overstates investment, by about 12 percent of output.
To some extent, this mismatch between model and data reflects that our empirical measure
of government consumption includes government investment. From the perspective of the
model, it makes more sense to include this in our measure of investment. The evidence
suggests that this consideration would not entirely close the gap between the model and
data. Government gross fixed capital formation (including military) has taken a declining
share of GDP. It peaked at about 7.5 percent of output at around the time of the Korean
war and has been falling steadily since then. In the decade after 1975, the ratio stabilized at
about 3.5 percent of output.14 So, at best, these considerations can account for only a part of
the discrepancy between the empirical and model expenditure shares. Given the imprecision
13As is well know n, th is in te rp re ta tio n is form ally rationalized by th e assu m p tio n th a t p riv ate a n d public
co n su m p tio n m e perfect su b stitu te s. U nder these circum stances, innovations in governm ent co n su m p tio n
w ould be m irro red by eq u al red u ctio n s in p rivate consum ption. Interestingly, th e gross featu res of postw ar
U.S. d a ta a p p e a r co n sisten t w ith th is view (com pare F igures 2h an d 2i.)
14T h is is based on an analysis of th e governm ent investm ent d a ta stu d ied in C h ristian o (1988).
21
in the estimated value of
a,
there is probably room for reducing it in order to improve the
model’s implications for expenditure shares. This is consistent with the information in the
column marked
“a =
0.28” which reports results for a lower value of
a
that is within the
range of estimates reported in Christiano (1988). In the analysis in the next section, we
report results for this reduced value of
a.
However, we do not comment on them because
they correspond closely to the results based on
a
= 0.36.
Now consider the growth statistics in the Table 2. Consistent with the model, variables
measured in consumption units grow less rapidly than does real investment. However, the
difference in growth rates based on the entire postwar period is not as great as the model
predicts. For the model to capture this, we would need to introduce growth into our aggregate
technology shock too. Note how different, however, the period since the 1980s is. There is a
sharp decline in the growth rate of the investment price deflator and a corresponding sharp
rise in the growth of real investment (see Figures 1 and 2.) With such a relatively short
period there is, of course, a danger of confounding trend and business cycle movements.
Still, the reduction in the price trend spans two business cycles (see Figure 1). Our model’s
assumption that all growth originates in the investment sector is not a bad approximation
to the experience of the past decade.
5.2
P a r a m e t e r s Controlling Business Cycles a n d Asset Pricing
The remaining parameters are
h, b, xjj, p,
er£, <rM. Values for these six parameters were set based
on the following six moments of the data:
(33)
p(Y ,P i)
= -0.15,
p ( Y , P k0
= 0.30,
p(A Solow ) =
-0.10, <r(AS o l o w )
=
0.018,
and
(34)
re - r f =
Here,
p (x , y)
6.63,
rf =
denotes the correlation between the logged, HP filtered variable
ilarly filtered variable, y;
variable,
x;
1.19.
and
a(x)
p(x)
and the sim
denotes the first order autocorrelation of the untransformed
denotes the corresponding standard deviation. Also,
gate GDP, measured in base year prices, and
x
A Solow
22
Y
denotes aggre
denotes the logarithmic first difference
of the Solow residual, computed using a simple aggregate production function.15 The first
two statistics in (33) characterize the sign switch. The two statistics in (34) are taken from
Cecchetti, Lam, and Mark (1993) (CLM).
Conditional on a set of feasible values for
h, b,
values for
were selected so that
ip, p,
the model exactly reproduces the four statistics in (33). The model’s implication for these
statistics was computed by Monte Carlo simulation. In simulated data sets, time series on
the growth rate of the Solow residual, A S o l o w , were computed using the same algorithm
used in the data. Thus, an aggregate production function was used for this calculation, even
though there does not exist an aggregate production function relationship between aggregate
inputs and aggregate outputs in our model.
We can define a mapping from feasible
h,b,
first compute the four parameters,
v
= [(re —r^),r^]' as follows. For given
ip, p , a £, a ^ ,
as described above. W ith the model
h,b
now fully parameterized, its implied value of
to
u
was computed by Monte Carlo simulation.
In particular, we simulated 500 artificial datasets, each of length 120 observations. In each
data set we computed the sample average of the annualized risk free rate and the equity
premium on annualized equity returns. The model’s implied value of
by the mean of these 500 sample averages. Denote this mapping by
the set of feasible
h, b
u
was approximated
v = f(h ,b).
We define
as the set of points in the unit box having the property that
<
Xt
is as close as possible to
vt,
Ct
and AC)t < 0 are never observed in the Monte Carlo simulations used to evaluate /.
We chose feasible values of h,
the sample estimates of
(35)
C (b, h) = \vT
-
Also, the 2 x 2 matrix
so that the model’s implied
v
provided in CLM. Our distance metric is
v
f(b,
b
h)]' V
Vt
f l [ur
-
f(b,
C (b , h ),
where
h )].
is the CLM estimate of the underlying sampling variance in
Op.
Let
(36)
J = C ( b T , h T ),
15 T h e ag g reg ate Solow resid u al is
z t = ----- — ,
w here Y t = Ct + Ic,t + Ii,t, K t = K C)t + K i yt) H t = Hi,t 4- H c,t• In section 6.2 below, we explain our ratio n ale
for in te rp re tin g th ese m easures of Yt and Kt as th e “base” year m easures of o u tp u t and capital.
23
where
bT , h T
values of
minimizes £(6, h) over the feasible values of b, h. In practice, we could not find
which set
b, h
J
= 0.
This procedure for determining the parameters in effect treats the statistics in (33) as
though they were known with perfect certainty. Presumably, a procedure which took into
account the sampling uncertainty in (33) would “sacrifice” a bit on hitting the elements in
(33) that are estimated the least precisely, in exchange for doing better on (34). We have
not explored such statistical estimation procedures.
We obtained the following results:
(37)
b =
0.55,
h =
0.0,
J =
4.23.
The corresponding estimates of
(38)
if)
i p , p , a e ,cr^
= 0.40, p = 0.52, a e = 0.017,
To see how / and £ vary with
b
and
0.028.
=
h,
are16
consider Figure 6. It displays the empirical equity
premium/risk free rate combination in (33) and 5 and 1 percent confidence intervals about
this point based on the estimates of CLM. In addition, there are four lines with stars. Each
line corresponds to a particular value of
stars correspond to
b
= 0.4 to
b
h,
as indicated. Starting from the lower left, the
= 0.6, in increments of 0.025. For
values of b up to 0.6 were feasible. The optimal point,
the figure. As the figure makes clear, increases in
that is the reason why
h
h
b
h
= 0.2 and 0.3, not all
= 0.55, h = 0.00, is also indicated in
sharply increase the risk free rate, and
= 0 at the optimum.
It is interesting to compare these results to those in BCF. That paper uses the same
estimation strategy, but comes up with different estimates for
b
and
h:
0.35 and 0.40 respec
tively. These differences reflect differences in the specification of the models: in BCH there
is only an aggregate technology shock, which is specified to be a random walk with drift,
and leisure enters log-linearly in utility, rather than linearly. In one respect, the two sets of
estimates of
b
and
h
are similar. Ours imply a steady state ratio of the habit stock to con
sumption equal to 0.55, whereas the BCF estimates imply a value of 0.58. Not surprisingly,
we report below that the implications for steady state risk aversion in consumption are also
quite similar. The implications for steady state risk aversion in wealth are identical.
16T h e im plied values of a\ a n d a 2 are 0.98 and 0.12, respectively.
24
Though both sets of estimates imply roughly the same magnitude for the habit stock,
their difference lies in how sensitive the habit stock is to recent consumption. In BCF’s
estimates the habit stock is relatively insensitive, whereas in our estimates the habit stock
is very sensitive to recent consumption. We suspect that this is an important part of the
explanation for the difference in estimation results.
An important finding in BCF is that the magnitude of the equity premium is decreasing
in the autocorrelation of consumption growth. The BCF estimation procedure appears to
have exploited this fact by selecting a positive value of
h
in order to produce negative
autocorrelation in equilibrium consumption growth. With a positive value of
h,
the surge
in consumption in the period of a shock leaves the habit stock relatively unaffected in the
subsequent period. As a result, the value of consumption in that period is not particularly
high and so households cut back their consumption from the high level in the previous period.
This reduction is what produces the negative autocorrelation in consumption growth in the
BCF model.
In our model, there are other sources of negative persistence in consumption growth,
and so the estimation strategy has less need to manipulate
b
and
h
to accomplish this. Our
aggregate technology shock is stationary, and the estimation strategy chooses a low value
for its autocorrelation in order to reproduce the negative autocorrelation in the growth rate
of the Solow residual. Negative autocorrelation in equilibrium consumption growth is a
consequence of this.
To help evaluate our parameter estimates, we computed the implied elasticity of invest
ment with respect to Tobin’s
q . 17
That quantity is 1.66 in our model. For comparison, Abel
(1980) reports estimates of this quantity that range from 0.27 to 0.52. Relative to Abel’s
estimates, we have understated the degree of adjustment costs (i.e., overstated
17T h e elasticity of in v estm en t in in d u stry x w ith respect to T obin’s q is, using (25)
d lo g /x ,t
d log qf
1 -i f)
25
ip).
6
Q u a n tita tiv e R e s u lts
We quantify the mechanisms in our model that enable it to account for the salient features
of asset prices and returns (Table 3). We then go on to examine our model’s implications for
business fluctuations and for risk aversion (Tables 4-6). We compare the model’s business
cycle implications with the corresponding empirical evidence. We show, for example, that
employment across a wide variety of sectors is strongly procyclical. This similarity is partic
ularly striking because the trends in these sectors are very different (Figures 7-8). Though
the model does not replicate the diversity in trends, it does replicate the procyclicality of
employment across sectors.
6.1
Financial M a r k e t s
Table 3 presents various statistics which capture the implications of our model for financial
variables. The column marked “calibrated” reports results for the model calibrated in the
previous section. The columns to the right of that report results based on various perturba
tions of the calibrated model, obtained by Monte Carlo simulation. The first column presents
the corresponding sample estimates.
The Sign Switch
Consider the phenomenon that the model was specifically designed to address, the sign
switch. That it exactly reproduces the statistics we use to characterize that phenomenon is
not surprising—the parameter values were picked in part to accomplish just that. Table 3
is constructed to help assess the role played in accounting for the sign switch by two model
features: the assumption of adjustment costs in the installation of investment goods, and the
multiple shock assumption. (The intuition about how these factors are supposed to work is
reviewed in the overview section above.)
The column marked
“ip
= 0.9” is suggestive of what happens when adjustment costs in
the investment function are shut down. In this case, the wedge between the price of equity
and the price of investment goods is essentially eliminated. As a result, both have roughly
the same correlation with output. That correlation turns out to be nearly zero, because
26
the two shocks in the model have roughly offsetting effects, in terms of their impact on
the cyclicality of these prices. To gauge the role played by the multiple shock assumption,
consider the column marked “erM = 0” so that the investment-specific technology shock is
set to zero. In this case there are only sources of procyclicality in the two prices, and so it is
not surprising that there is a strongly positive correlation between equity prices and output,
and between the price of investment goods and output.
Further insight into our model’s account for the sign switch may be obtained from Figure
9. This figure displays the response of the model variables to a one-standard deviation in
novation in the aggregate technology shock (solid line) and in the investment-specific shock
(dashed line). Consider the response to the aggregate shock. This response produces a
sharp rise in both the investment goods price, P,, and the price of equity,
P k'.
Consistent
with the intuition in the overview and the discussion in the model section, the jump in the
equity price exceeds that in the price of investment goods. Now consider the response to
an investment-specific technology shock. As anticipated by our earlier discussion, Figure 9h
shows that
P k> falls
relatively little, by comparison with P,.
Investment adjustment costs have the effect of muting the response of the price of equity
to investment shocks and amplifying their response to aggregate shocks. This is why the
price of equity is more procyclical than the price of investment goods, and is at the heart of
our model’s account of the sign switch phenomenon.
Although the model accounts well for the cyclical comovement with output of investment
and equity prices, it does not account well for the magnitude of their cyclical volatility. In the
data, the standard deviation of equity prices is a little below 10 percent, while the standard
deviation of investment good prices is a little above 1 percent. In the model, these two prices
have roughly the same standard deviation, equal to the midpoint between the two empirical
standard deviations. Interestingly, the Shiller (1981) “excess volatility” puzzle stands here.
Despite its (counterfactually, as we will see) high volatility in interest rates, the model still
cannot account for the observed high volatility of stock prices.
T h e E q u i t y P r e m i u m a n d R i s k Free R a t e
27
For convenience, our model’s implications for the mean return on assets reported in Figure
6 are reproduced in Table 3. The table also shows that the equity premium is reduced by a
factor of 10 by eliminating habit persistence in preferences (see the column,
ub — h =
0”), or
by dropping the limited labor mobility assumption, that the labor supply decision to each
sector is determined prior to the realization of the current period shock (see the column,
“Full Labor”). This is consistent with BCF’s conclusion that limitations on factor mobility
and habit persistence can produce an equity premium in a business cycle model. By contrast,
the model’s ability to account for the risk free rate is, if anything, hurt by habit persistence
and limited labor mobility.
Other features that are important for the model’s ability to replicate the equity premium
are the persistence of the aggregate technology shock and the standard deviation of its
innovation. By contrast, the investment specific shock has essentially nothing to do with the
equity premium (see the “cr^ = 0” column). This latter is not surprising, in view of (27).
Abstracting from the (relatively small) impact of future consumption on the marginal utility
of present consumption, the equity premhim would be zero if there were only investment
specific shocks. This is because there is no contemporaneous impact on consumption from
these shocks.
Regarding the aggregate shock, consider the impact of reducing the standard deviation in
the innovation in aggregate technology, <re, from its value in the calibrated model to 0.0085
(see the column marked “a e = 0.85%,
b
= 0.55”). This value of a e is of independent interest
because it equates the standard deviation of the innovation in equilibrium consumption,
(1 —a)cr£, with an estimate of the corresponding empirical magnitude.18 The drop in the
value of cr£ causes the equity premium to fall to 1.45 percent, and it also results in a fall in
the risk free rate to 1.93 percent.
The equity premium is decreasing in the persistence of the aggregate shock,
p.
The
intuition for this is explored extensively in BCF and is based on the reasoning associated
18A regression of p e r c a p ita co n su m p tion grow th on one lag of itself w ith estim a tio n p erio d 1947:2-1995:1
p ro d u ces a fitte d resid u al w ith sta n d a rd error 0.0054. W hen lagged per c a p ita G D P g ro w th is also included
in th e regression, th e s ta n d a rd erro r d rops to 0.0051. T h e d a ta used for these regressions were ta k e n from
C itib ase. C o n su m p tio n is m easured as consum ption of nondurables an d services (G C N Q + G C S Q ), a n d th e
p o p u la tio n v ariab le is Q P O P .
28
with the permanent income hypothesis. When
p
is small, the innovation in consumption
due to an aggregate shock is temporary, producing a large increase in the demand for new
capital for consumption-smoothing reasons, and this in turn generates a big rise in
Pk>.
The
resulting high capital gain is the reason equity is a bad hedge against risk, and this is what
underlies the high equity premium.
Table 3 shows that the model does less well on the second moment properties of asset
returns. In particular, it overstates by a factor of two the volatility of the equity premium
and the return on equity. It overstates by a factor of three the volatility of the risk free
rate.19
6.2
Business Cycle Implications
Table 4 presents the business cycle implications of our model. The format of that table
corresponds to that of Table 3. The results in the “Data” column are based on data from
the NIPA, which are computed using base year prices. To put our simulated output and
investment data on a comparable basis, we measure them in “base” year prices too. The
base year in our simulations is the initial observation, when the state of investment-specific
technology,
Vt ,
is set to unity, and the model is assumed to be on a steady state growth path,
so that the relative price of investment goods is one. Thus, base year output in the model
is the simple sum of the physical quantity of consumption and investment goods produced.
Similarly, base year investment is the quantity of investment goods.
S t a n d a r d B u s i n e s s C y c le S t a t i s t i c s
The first 7 rows of Table 4 report the model’s performance in relation to standard business
cycle statistics. Its performance is roughly comparable to that of standard models. It shares
a problem with standard models in that it understates the volatility of hours worked.
19W e are in v estig atin g w ays of accounting for th e low volatility in th e risk free rate. O ne way, inspired by
C am p b ell an d C o ch ran e (1995), is to m ake assum ptions w hich have th e im plication th a t w hen equilibrium
c o n su m p tio n d rops close to h a b it, th e u n certain ty in fu tu re consum ption relative to h a b it rises. T hese two
p h en o m en a e x e rt opp o sin g effects on th e risk free rate. If they cancel, as in C am pbell an d C ochrane (1995),
th e n th e risk free r a te is co n sta n t.
29
The table suggests both shocks play an important role in the model’s business cycle
implications. If the investment-specific shock is too small (V M = 0”), then the relative
volatility of consumption is too high. If the aggregate shock is too small, then the relative
volatility of consumption is too low (see
“cre
= 0.0085”). Not surprisingly, adjustment costs
reduce the volatility of hours worked (see
“ip
= 0.9”). The limited labor mobility assumption
increases the relative volatility of consumption. Again, this is not surprising.
Comovement
We begin by characterizing the salient characteristics of employment comovement over
the cycle. Doing so is complicated to some extent by the fact that some sectors produce
both consumption and investment goods, and so the available employment data to not come
neatly categorized according to whether they correspond to
Hi
or
H c.
To overcome this
complication, we report results for 10 different measures of hours worked. Among these
there are some that clearly correspond primarily to
products”), and others seem clearly related to
Hi
Hc
(for example, “food and kindred
( “construction”).
Figure 7 reports the logged data and the associated HP filter trends. W hat is perhaps
most striking about these data is the
lack
of uniformity: some trend up and some trend
down and one ( “food and kindred products,” Fig. 7g) even does both. A sharply different
picture emerges when one considers the deviations of these data from their HP trend. These
are reported in Figure 8. Also reported in Figure 8 is the deviation of logged GDP from
its HP trend. The picture that emerges in Figure 8 is one of great similarity among the
variables: all comove positively with GDP. They differ in terms of amplitude, but there is
little visual evidence of a phase shift. Table 5 reports the correlations with output, and the
relative volatility to output, and associated standard errors. In all cases the correlations are
large, positive, and statistically significant.
Table 4 reports the model’s implications for comovement (see
p ( Y , H c)
and
For ease of comparison, we report the empirical correlation for construction
p(Y ,H i).)
( Hi )
and for
food and kindred products (H c). Note that the model implies employment in both sectors
is procyclical, with the degree of procyclicality being less strong in the consumption sector.
30
To gain insight into how the model manages to deliver this comovement, consider the
response of variables to the aggregate shock, reported in Figure 9. Note that consumption
rises by nearly 1.1 percent in the period of the shock. In a standard model, the impact of an
aggregate shock on consumption would be much lower, as workers and capital are predicted
to instantaneously switch out of the consumption sector and into the investment sector.20
Thus, the specification of technology in our model prevents a countercyclical response of
hours worked in the consumption sector in the period of the shock. The transient nature of
the technology shock, together with habit persistence, ensures that the employment response
in the consumption sector remains strong and positive in subsequent periods. Habit persis
tence implies that because consumption was high in the period of the shock, the value of
consumption is high in the periods after. Our estimate,
h =
0, plays an important role here.
The transience of the shock implies that to supply consumption in those periods requires
high labor effort. And this is exactly what happens according to Figure 9f.
Now consider the response to an investment shock. The response of employment in the
consumption sector to this shock is positive—presumably reflecting a wealth effect—but very
close to zero. The fact that hours wuik^J in the consumption sector is procyclical relative to
both shocks guarantees that our model is able to account for the comovement phenomenon.
To see what happens in our model when these restrictions on intersectoral factor mobility
are relaxed, consider the “Full Labor” column in Table 4. That column reports the dynamic
implications of a model parameterized just like our calibrated model, with the only excep
tion that employment responds flexibly to the shocks. Note that now employment in the
consumption sector is countercyclical. Evidently, to get comovement, habit persistence is
not enough, and the limited labor mobility assumption is needed too. In section 3 we showed
that with the limited mobility assumption only, and not habit persistence, comovement is
not possible either. That is, to get comovement in our framework,
both
habit persistence
and the limit labor mobility assumption are required.
Several features of our framework have played an important role in delivering our comove
ment result. These include the transitory nature of the aggregate technology shock, our small
20To b e concrete, b akery equ ip m en t an d bakers in th e “food and kindred p ro d u cts” sector are p red icted
to tra n sfo rm in sta n ta n e o u sly into bulldozers and bulldozer drivers in th e “co n stru ctio n secto r.”
31
estimated value of
h,
and, as explained in the previous section, the linear specification of
utility in leisure. These considerations help explain why BCF do not find comovement. Their
aggregate technology shock is a random walk, their estimated value of h is positive, and they
adopt a log-linear specification of utility in leisure.
Persistence
One indicator that our calibrated model introduces persistence is that its implied growth
rate of the Solow residual has autocorrelation —0.1, while equilibrium output growth has
autocorrelation 0.02 (see Table 4). Christiano (1988) shows that a standard business cycle
which reproduces a pattern like this in the Solow residual implies first order autocorrelation
in output growth equal to roughly —0.1. Another indicator of endogenous persistence can
be seen in the parameterization,
ae
= 0.0085,
b
= 0.65. In this case, the growth rate of
the Solow residual is essentially uncorrelated over time. Yet, the autocorrelation in output
growth is 0.11 (see Table 4).
6.3
I m p l i c a t i o n s for R i s k A v e r s i o n
It is useful, for purposes of assessing the plausibility of our model, to document its impli
cations for RRAC, as defined in (10). This is done in Table 6, which reports the values of
(xlOO) associated with various values of
/i
and the various values of
b
u
we consider, along a
steady state growth path. We report risk aversion for our calibrated model and the various
perturbations on it studied above. In addition, for comparison, we report consumption risk
aversion implied by the parameterization considered in Constantinides (1990) and for the
production and exchange models studied in BCF.
Note that for our calibrated model
(b =
0.55), a household would be willing to pay
2.5 percent of one period’s consumption in order to avoid a fair bet on 10 percent of that
consumption.
This is a low level of risk aversion, particularly by comparison with the
levels of risk aversion required in other studies that seek to account for the equity premium.
Moreover, our model’s implication for the risk free rate and equity premium is very close to
the empirical values of these variables (recall Figure 6.) So, it is natural to investigate how
32
it is that our model manages to do this, w ith so little risk aversion. The explanation lies in
two apparently counterfactual im plications of the model.
W h y is R R A C s o L o w in th e C a lib r a te d M o d e l?
F irs t, note from Table 4 that our model implies the autocorrelation of consumption
growth is —0.14, whereas the corresponding em pirical estimate is 0.19. Th is im plication of
the model reflects the importance of the aggregate shock, the transitory nature of which
enhances the model’s ability to account for the equity premium for a given level of R R A C.
The transitory nature of the aggregate shock to technology implies that households have a
strong smoothing motive when there is a positive innovation. To investigate the quantitative
importance of these considerations, we examined a version of the model in which p — 0.99,
so that the aggregate technology shock is almost a random w alk. Now, consumption growth
is also virtu a lly a random walk. Predictably, we found that this change reduces the equity
premium to 3.46 percent (see Table 3, b = 0.55, p = 0.99.) To offset this, we raised b to 0.60.
According to Table 6, w ith this specification of u tility, households are w illing to give up 2.9
percent of a period’s consumption to avoid a 10 percent gamble, up only a little from 2.5
when b = 0.55. Although this is a higher level of risk aversion, it is perhaps not beyond the
realm of em pirical plausibility. Interestingly, this version of our model preserves the basic
features of the calibrated model: its ability to account for the sign switch, comovement, and
the basic features of the business cycle.
The second reason our calibrated model can account for the observed equity premium
w ith so little risk aversion is that it overpredicts the innovations in consumption. In the
model, the standard deviation in the innovation in consumption is ju st (1 — a )c re ~ .011,
for reasons discussed above. B u t, in the data this quantity is about one-half of one percent,
0.005. Not surprisingly, this feature of the model also enhances its ability to account for the
equity premium. To investigate how important it is, we studied a version of the model w ith
(1 —c t ) a e = .0054 (see the indicated column in Tables 3 and 4). Not surprisingly, the equity
premium is reduced substantially w ith this change, down to 1.45 percent. We then increased
b
to 0.65, and this returned the equity premium up to 4.10 percent. Interestingly, we still
can account qualitatively for the sign switch and comovement observations. T h is value of b
33
im plies higher risk aversion in consumption (see Table 4 ), but presumably not a level that
economists w ill find im plausibly high. Note that this change also raises the autocorrelation
of consumption growth to nearly zero. T h is reflects that increasing b enhances the motive
to smooth consumption.
W h at L evel o f R R A C D o es it Take
?
Thus, relatively sm all increases in the model’s im plication for risk aversion in consumption
can move it in the direction of being more consistent w ith the consumption data. B u t we
have not moved the model all the way. W hat sort of risk aversion in consumption would that
im ply? The answer is in B C F . They study a pure exchange economy in which equilibrium
consumption growth is modelled based on U .S. consumption data. They accoimt for the
equity premium w ith b = 0.58 and h = 0.30. According to Table 6, w ith these parameters,
a household is w illing to give up 6.7 percent of consumption to avoid a fa ir, 10 percent
gamble. Some w ill perhaps view this as a high degree of risk aversion. Does this mean that,
necessarily, to accotxnt for the observed equity premium, high R R A C is required? The answer
may be yes. B u t, there are at least three reasons to think that the answer might actually be
no.
A ll of these reasons build, in different ways, on the notion that the inform ation observed
by the economic analyst and that observed by households differ in some way.
F irs t, from the analysis above, it is clear that the details of the consumption process
m atter a lot for determining how much R R A C is required to account for the equity premium.
Y et, there is little confidence in the quality of this data (see W ilcox (1992).) Gibbons (1989)
cites this low quality as a reason for ignoring consumption altogether in evaluating asset
pricing models. The range of uncertainty about the consumption data when these quality
considerations are integrated w ith the usual sampling uncertainty may include parameterizations of consumption which permit accounting for the observed equity premium w ith low
R R A C.21
21One indicator of data uncertainty is the fact that the first order autocorrelation in the growth rate of
BCF’s consumption data is 0.34. This reflects that their measure of consumption and their sample period dif
fer from ours. This high level of consumption autocorrelation underlies their estimate of RRAC. Presumably,
they would have reported a lower RRAC, had they used our data set.
34
Second, suppose all the features of the univariate stochastic process underlying the con
sumption data were known accurately.
Quah (1990) has shown that, even a process in
which the univariate representation is a first order autoregression in growth rates w ith posi
tive A R (1 ) parameter is consistent w ith an unobserved components representation in which
transitory shocks play a very large role. The analysis in this paper has exhibited various
em pirical considerations that make such representations plausible, although the calibrated
model fails to reproduce crucial features of the type of statistical environment contemplated
in Quah (1990). T h is is because the univariate representation of equilibrium consumption
exhibits negative persistence in its growth rate. The parameterization, a e = 0.0085, b = 0.65,
does exhibit features of Quah’s environment: in this case, the univariate representation of
consumption resembles a random walk because its growth rate is nearly uncorrelated over
time; yet the innovation in consumption entirely reflects its transitory component (see Table
4). In the statistical environment like the one studied by Quah, as long as agents observe
the two underlying components driving consumption, their demand for equity may be driven
in an im portant way by the transitory component, possibly leading to a large premium on
equity. We are currently exploring this possibility further.
T h ird , as is well known, various transformations are applied to the data, which are
likely to have the im plication that measured consumption displays more persistence than
the actual consumption choices made by agents. The fact that the data are aggregated over
time is perhaps the prime example of this possibility.22 Thus, agents could be living in an
environment w ith relatively little persistence in consumption, which could be reflected in a
high equity premium (like in our a £ = 0.0085, b = 0.65 model), even though published data
exhibit substantial persistence due to time aggregation. For a quantitative investigation of
this idea in a closely related context, a discussion of the “Deaton paradox” for consumption,
see Christiano (1989).
22 Another possibility is seasonal adjustment, which is thought by some to have the effect of smoothing the
data. However, some preliminary analysis suggests this may not be an important part of the explanation of
the gap between our model and the data. We generated 1,000 observations from the calibrated model. The
first order autocorrelation of consumption growth in this data is —0.0773. We then applied the version of the
Census X -ll seasonal adjustment procedure implemented in RATS’ EZ-X11 program to seasonally adjust
the data (we used the “multiplicative adjustment” option). The first order autocorrelation of the growth
rate of the seasonally adjusted artificial consumption data is —0.0382. We concluded that the smoothing
implicit in seasonal adjustment does very little to increase first order autocorrelation in growth rates.
35
To summarize this discussion of risk aversion, our model can account for the mean equitypremium and risk free rate w ith low risk aversion. On the one hand, we argued that the
model’s apparent counterfactual im plications for consumption played an im portant role in
this result. On the other hand, the model’s im plications are not v e r y far off. Moreover, we
discussed a variety of considerations that could in principle reconcile the model w ith the data.
These considerations make us optim istic that a model can be found that accounts for the key
first moments of asset prices without having counterfactual im plications for consumption.
7
C o n c lu d in g R e m a r k s
A number of researchers have argued that asset pricing data contain useful inform ation for
macroeconomists. For example, the early work of H all (1978) and Hansen and Singleton
(1982,1983) showed how to use asset pricing data to test im plications of equilibrium models
and estimate their parameters.
Data on asset prices are not only useful for evaluating
models, but also for providing guidance about how to further develop them. In view of these
considerations, it is surprising that b^Cuess cycle researchers have made relatively little use
of asset pricing data. In this and a previous paper w ith Michele Boldrin, we took a few steps
in this direction.
In B C F , we explored modifications in a standard business cycle model that could ac
count for the observed high equity premium and low risk free rate. In this paper, we were
interested in understanding an observation that we in itia lly found puzzling: equity prices are
procyclical, while investment prices are (weakly) countercyclical. Although the literature on
Tobin’s q prepares one for the possibility that these two prices are not identical, we were
nevertheless surprised find that their business cycle dynamics are so very different.
In this paper we incorporated the features proposed in B C F to account for key aspects
of the first moment properties of asset returns, together w ith additional features designed
to account for the business cycle properties of asset prices. After establishing that there is a
param eterization of our model that can account for the price and rates of return on assets,
we tinned to see what this parameterization implies for business cycles and for risk aversion.
We find that the model does at least as well as standard business cycle models in ac
36
counting for conventional business cycles facts. On two dimensions, we find that the model
actually represents a step forward relative to the standard business cycle model. F irs t, as
in B C F , we find that the modifications designed to account for mean asset returns help
confer an internal propagation mechanism to the model. Second, in our model environment,
the modifications also allow the model to be consistent w ith the fact that employment is
procyclical across a broad range of sectors.
The basic features that we use to account for the asset pricing phenomena are habit
persistence preferences and lim itations on the ability to quickly move factors of production
both cross-sectionally and intertem porally. These same lim itations, by slowing the economy’s
ability to respond to shocks, have the effect of introducing persistence. A t the same tim e,
lim itations on intersectoral m obility, coupled w ith habit persistence, have the effect of making
employment across sectors move up and down together over the cycle.
The results in this paper and in B C F support the notion that the same frictions needed
to account for the salient features of asset prices and returns are also useful in understanding
the salient features of business cycles.
37
A
T h e P l a n n e r ’s P r o b l e m
The quantities in a competitive equilibrium of our model can be computed by solving the
following planner problem: maximize
OO
£ o £ / ? {h i(G - X t ) + r j ( T - H c>t - H i>t)
t =0
+ A x>t [ X t+1 - h X t - b C t}
+ A c,t [ ^ ( e x p ^ O ^ t ) 1-" -
+ A m \V t K * t (e x p (^ )^ ,i)1_Q ~ h ,t ~ I ii.t,
+ a ,« [O ' ((1 - 6 ) K c,„ [ , , ) - K CJ+,\
+ «i,i [0‘ ((1 - i ) K , , u h ,t) - K k w ] }
subject to
9 t
=
V t
= exp(/2 +
p Q t-x
+
e t ,
£ t
P t ) V t-
~
i
N (
,
0, c r 2 )
P t
~ -A(0, c r 2 )
Here, Ay t, y = x , c , i and f2y f, y = c , i are Lagrange m ultipliers. The planner is assumed to
choose H c<t and H t t prior to the realization of the date t state of nature, while the remaining
choice variables (including the m ultipliers) are selected afterward.
To solve the planner problem we first have to transform it into its stationary form. The
variables in the transformed problem are c t , k Ctt + i , k ^ t+ i , x t+ i , i c^ ,i i,t , AC)f, A,)t, AI)t,u>Cit and
dJi't-
These are defined as follows (not all variables are transformed in the same w ay):
-
Ct
Ct ~ ya/(l-a) ’
_
Ic,t
*Clt “ yV(l-a) >
L-
-
,*+i
Kc’t+l ~ yl/(l-a) ’
■
i
W
_ _ K i ,t+ 1
1 ~ yl/(l-a) ’
_
Ii,t
- yV(l-a) ’
38
X t+i
Xt+1 ~ ya/il-a)
\ =
_T/a/
Ac,i
V t (1_a^AA c>t,
\
_t/1
—
V*/(1_q )aA t^ ,
\
— T/a^
1“a^A
AX|t —
Vt
l\Xjt
In what follows it w ill also be helpful to use the following:
Y itt
Y<*
ya/(l-a) ’
y^1
y*'1
t/ 1 /(1 - q )
Vf
where Y c>t - K “t ( e x p ( 9 t ) H c<ty
“ and Y i>t = VtK ° t ( e x p ( 9 t ) H itt ) 1
The first order conditions for an interior solution to the stationary version of the planner
problem can be rearranged to form the following system of equations:
(39) \ x,t - ( 3 E
Ai , t
1
c t+i
- e x p ( -(J L t+ i)x t+ i
1
-P E
Q2,l
(40)
exp(—/}t+ i)(A Iif+1/H-
Qt = 0
V c ,t+ l
,Q +1 —6xp( —£ l t + l ) x
+ e xp (—/2f+i ) ( l - 6 ) — ^— Q ch t+ 1
t+ l
b^x,t+ i ) a '~
^C,t-f1
=
0
V 2 ,i+ 1
(41)
Ai,t+i
V2,f+1
1
(42) £7
\ c t - e x p ( - f tt)x t
(43) E
r2 / =
+ exp (—M«+i)(l \ ^t,t+l
V2 ,t
- b \ x<t)
- M i - ®)jr- - v n:
(1 - a ) j f ~ ~ 9 n:
0
/
=o
c,t
= o
(44)
0
(45) i c ( “I- %x,t
V i,t
In these expressions
0
denotes the information set that includes realizations of the tech
nology shocks up to and including time t , and f2* denotes the information set identical
to Qf except that the time t realizations of the technology shocks are excluded.
Also,
Q Jt, j = 1,2 denote partial derivatives of Q * w ith respect to its first and second argu
ments, respectively, evaluated at the time t values of the arguments, for x = c , i . Fin ally,
At =
1 - a ) and /2t+ 1 = j j t / a .
39
Making use of the resource constraints implicit in the formulation of the planner problem,
we can collapse the model to the following seven equations
E [ Vj(kCtt,kc,£+l> kCjt+27
^ x , t , ^x,t+i) ^ i , t ,
ki,t+1>
2> xti%t+lj %t+2jH ct^H ct+\, Hi,t, Hi ,*+l>
i @t>@ t + 1 , f^t, (J't+i ')|
= 0,
E \ V j { k c,ti k c , t +li k c , t + 2 , k i , t , k i , t + 1) ^-4,4+2) 2+,
j = 1,2, 3, 6,7;
2-4+2) H Ctt , -He,4+1> -Hf.t) -Hf^+l,
'^i ,4>AIi4+ i,Aijt,Ai]4+x,^4,04+x,//4,^4+i)|f2(] =0,
j = 4, 5.
Here v j ( • ) , j = 1 , 2 , 3 , 6 , 7 are equations (39), (40), (41), (44), and (45) respectively, and
Vj (•),
j = 4 , 5 are equations (42) and (43), respectively.
These equations can be solved
using the methods described in Christiano, Fisher and Valdivia (1995).
We can use the multipliers from a solution to the planner problem to compute the relative
prices studied in the main text. First, the relative price of the new investment good is given
by
p —
-n,4 =
•'V*
Ai,t
^c,4
AC)t V t
1
Second, the prices for K c, t + \ and K i t + i are
Pk'c,t =
a c,4
A c,4
a 4,4
u’ct 1
Pk[,t =
AC)t Vt’ ^ ’c A c,t
^4,4
1
A ctVt’
respectively. Third, the prices for installed capital are
P
_ Q l,4^c,4 _ LOct Q l,t
fcc,t =
Act ~
p
M
_ ^?1,4^*,4 _ ^4,4^1,4
= A ct ~ A~t~ K '
Notice that each of these prices will trend downward if V t has a positive trend. To use these
formulas we require expressions for \ c , t , u c t and u h t . These can be computed from the first
order conditions from the planner problem. The expressions derived in this way are given
by
Ac.t
y c,t A j 4
^c,4 =
exp(jlt)xt
b \ x ,ti
,
•, and
Q h
A,;
'4 ,4
Wi,t" Qit'
40
R eferences
Abel, Andrew, 1980, “Em pirical investment equations: An integrative fram ework,” C a m e g ie R o c h e s t e r C o n f e r e n c e S e r i e s o n P u b lic P o lic y ,
vol. 12, pp. 39-91.
Benhabib, Jess, Richard Rogerson, and Randall W right, 1991, “Homework in macroeco
nomics: Household production and aggregate fluctuations,” J o u r n a l o f P o l i ti c a l E c o n
om y,
vol. 99, pp. 1166-1187.
Benninga, Simon and A ris Protopapadakis, 1990, “Leverage, time preference and the ‘equity
premium puzzle,’ ” J o u r n a l o f M o n e ta r y E c o n o m ic s , vol. 25, pp. 49-58.
Boldrin, Michele, Lawrence J . Christiano, and Jonas D.M . Fisher, 1995, “Asset pricing
lessons for modeling business cycles,” National Bureau of Economic Research working
paper 5262.
Cam pbell, John Y ., and John H. Cochrane, 1995, “B y force of habit: A consumptionbased explanation of aggregate stock market behavior,” National Bureau of Economic
Research working paper 4995.
Cecchetti, Stephen G ., Pok-sang Lam , and Nelson C . M ark, 1993, “The equity premium
and the risk-free rate: Matching the moments,” J o u r n a l o f M o n e ta r y E c o n o m ic s , vol.
31, pp. 21-45.
Christiano, Lawrence J ., 1988, “W hy does inventory investment fluctuate so much?” 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 ,
vol. 21, pp. 247-280.
Christiano, Lawrence, J ., 1989, “Comment on ‘consumption, income, and interest rates:
Reinterpreting the time series evidence,” ’ in O livier Jean Blanchard and Stanley F is
cher, editors, N B E R M a c r o e c o n o m ic s A n n u a l 1 9 8 9 , pp. 216-233.
Christiano, Lawrence J ., and M artin Eichenbaum, 1992, “Current real-business-cycle the
ories and aggregate labor-market fluctuations,” A m e r ic a n E c o n o m ic R e v ie w , vol. 82,
pp. 430-450.
41
Christiano, Lawrence J., Jonas Fisher, and Victor Valdivia, 1995, “A linearization method
for solving dynamic, general equilibrium models,” unpublished manuscript.
Constantinides, George, 1990, “Habit formation: A resolution of the equity premium puz
zle,” 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 , vol. 98, pp. 519-542.
Eichenbaum, Martin, and Lars Peter 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 t a t i s t i c s , vol. 8, pp. 53-69.
Fisher, Jonas, 1994a, “A dynamic, general equilibrium empirical analysis of real and mon
etary aspects of the business cycle,” unpublished Northwestern University Ph.D. dis
sertation.
Fisher, Jonas, 1994b, “Relative prices, complementarities, and co-movement among com
ponents of aggregate expenditures,” University of Western Ontario research report
9405.
Fisher, Jonas, 1995, “W hy does residential investment lead business investment over the
business cycle?” manuscript, University of Western Ontario.
Gibbons, Michael R., 1989, “On the volatility of bond prices,” C a r n e g ie - R o c h e s te r C o n f e r
e n c e S e r i e s o n P u b lic P o l i c y , vol. 31, pp. 139-175.
Gordon, Robert J., 1990, T h e m e a s u r e m e n t o f d u ra b le g o o d s p r ic e s , University of Chicago
Press.
Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell, 1992, “Macroeconomic implications
of investment-specific technological change,” Institute for Empirical Macroeconomics
discussion paper 76, Federal Reserve Bank of Minneapolis.
Hall, Robert E., 1978, “Stochastic implications of the life cycle permanent-income hypoth
esis: Theory and evidence,” J o u r n a l o f P o litic a l E c o n o m y , vol. 86, pp. 971-987.
Hansen, Gary, 1985, “Indivisible labor and the business cycle,” J o u r n a l o f M o n e t a r y E c o
n o m i c s , vol. 16, pp. 309-327.
42
Hansen, Lars Peter, and Kenneth Singleton, 1982, “Generalized instrumental variables
estimation of nonlinear rational expectations models,” E c o n o m e tr ic a , vol.
50, pp.
1269-1286.
Hansen, Lars Peter, and Singleton, Kenneth J., 1983, “Stochastic consumption, risk aver
sion, and the temporal behavior of asset returns,” J o u r n a l o f P o l i ti c a l E c o n o m y vol.
91, pp. 249-265.
Lucas, Robert E., Jr., and Edward Prescott, 1971, “Investment under uncertainty,” E c o n o
m e t r i c a , vol. 39, pp. 659-681.
Lucas, Robert E., Jr., 1981, “Understanding business cycles,” reprinted in S tu d ie s in
B u s i n e s s - C y c l e T h e o r y , M IT Press.
Murphy, Kevin M., Andrei Shleifer, and Robert W. Vishny, 1989, “Building blocks of
market clearing business cycle models,” N B E R M a c r o e c o n o m ic s A n n u a l 1 9 8 9 .
Quah, Danny, 1990, “Permanent and transitory movements in labor income: An explana
tion for ‘excess smoothness’ in consumption,” J o u r n a l o f P o litic a l E c o n o m y , vol. 98,
pp. 449-475.
Rogerson, Richard, 1988, “Indivisible labor, lotteries and equilibrium,” J o u r n a l o f M o n e ta r y
E c o n o m ic s , vol. 21, pp. 3-16.
Sargent, Thomas J., 1979, M a c r o e c o n o m ic th e o r y , Academic Press.
Shiller, Robert, 1981, “Do stock prices move too much to be justified by subsequent changes
in dividends?” A m e r i c a n E c o n o m ic R e v ie w , vol. 71, pp. 421-436.
Sundaresan, Suresh M., 1989, “Intertemporally dependent preferences and the volatility of
consumption and wealth,” R e v ie w o f F in a n c ia l S tu d ie s , vol. 2, pp. 73-89.
Wilcox, David W., 1992, “The construction of U.S. consumption data: Some facts and their
implications for empirical work,” A m e r ic a n E c o n o m ic R e v ie w , vol. 82, pp. 922-941.
43
Table 1: Business Cycle Properties of Stock Prices
Second Moment Statistics
Dow Jones
Industry
S&P 500
------ corr(p, y )
corr(p,y)
. °JL
.
5.22
0.30
Composite
5.14
0.36
(0.55) (0.10)
(0.53) (0.09)
Capital Goods
6.06
0.39
(0.56) (0.09)
0.21
Utilities
4.66
(0.64) (0.11)
6.42
0.30
Finance
(1.03) (0.12)
Industrial
5.33
0.35
(0.55) (0.09)
Transportation 5.82
0.19
(0.79) (0.15)
NYSE
corr(p,y)
5.20
(0.53)
na
0.37
(0.09)
na
6.82
(1.40)
6.73
(0.93)
5.62
(0.75)
7.45
(1.08)
0.30
(0.09)
0.38
(0.10)
0.35
(0.12)
0.36
(0.10)
Notes: (i) Data source— CITIBASE. The sample period is 1947I-1995I for NYSE Composite, Dow Jones, and 1966I-1995I for the
other NYSE variables. The sample period is 19471—19951 for all but two of the S&;P 500 data series. It is 19701—19951 for S&P 500 finance
and transportation. Data on gross domestic product (GDP) cover the period 19471—19951. Stock price data are deflated by implicit price
deflator for consumption of nondurables and services.
(ii) Statistics— All data were logged, and then Hodrick-Prescott filtered prior to analysis. a p denotes the standard deviation of the
(detrended) stock price, a y denotes the standard deviation of output, and c o r r ( p , y ) denotes the correlation between p and y . Numbers
in parentheses denote the standard errors of <j p / ( tv and corr(p,y), computed as in Christiano and Eichenbaum (1992). For estimation of
the relevant zero-frequency spectral density a Bartlett window, truncated at lag four, was used.
44
Table 2: Sample First Moments, Macroeconomic Variables
Data
1947:1-1995:1 1983:1-1995:1
~ C ---- 0.56
0.59
Y*
C
0.76
0.75
Y*
hi
0.24
0.24
Y*
Ph'K
10.6
y*
1.46
AY*
1.85
1.84
1.68
AC
1.58
1.86
AC
3.75
AI
2.35
1.11
APJ
1.81
-2.54
A Pi
-0.57
Model
Calibrated a = 0.28
0.64
0.72
0.36
11.3
1.60
1.60
0.28
9.1
1.60
1.60
4.44
1.60
-2.84
4.44
1.60
-2.84
Notes: (i) Data from “Data” column are taken from Citibase, except the capital-output ratio, which was taken from Christiano (1988)
and covers the period 1956:111-1984:1. Data in “Calibrated Model” column are properties of the nonstochastic steady state. A x ~ first
difference of log of otherwise untransformed variable.
(ii) U.S. data— C ~ consumption of nondurables (GCNQ) and services (GCSQ ); C ~ consumption of nondurables and services,
plus government purchases (GGEQ ); / ~ business fixed investment (G IFQ ) plus household durable purchases (GCDQ ). The preceding
variables are measured in 1987 dollars. Y * ~ current dollar Gross Domestic Product (GDP) divided by implicit price deflator for
consumption of nondurables and services { { G C N + G C S ) / ( G C N Q + G C S Q ) ) ; Pi I ~ current dollar business fixed investment (G IF)
plus current dollar durable purchases (G CD ), divided by implicit price deflator for consumption of nondurables and services; Pi ~
business fixed investment plus household durable goods deflator ((G C D + G IF)/(G C D Q -i-G IFQ )), divided by implicit price deflator for
consumption of nondurables and services. Growth rate results are for per capita variables, obtained by scaling by the propulation
between 16 and 64 years old (PAN17-PAN19). The growth rate results extend only to the period 1995:1, reflecting the availability of the
population data in our version of Citibase. A ll mnemonics correspond to variable names in Citibase.
(iii) Calibrated model data—Y * ~ C + P {I.
45
Table 3: Prices and Rates of Return
<7e = 0.85%
Statistic Data Calibrated Full Labor b = h = 0 b = 0.55 b = 0.65
2.24
3.35
6.66
1.96
7.82
re
8.56
(0.02)
(0.01)
(0.03)
(0.06)
(0.08)
rJ
1
.
6
4
1
.
9
3
2.56
1
.
6
5
2.78
1.19
(0.05)
(0.07)
(0.03)
(0.03)
(0.09)
(0.81)
_ yf
0
.
3
2
1
.
4
5
4.10
0
.
5
8
5.78
6.63
(0.12)
(0.07)
(0.04)
(0.03)
(0.15)
(1.78)
34.5
8.67
13.8
41.2
11.3
CT
19.53
(0.04)
(0.1)
(0.1)
(0.03)
(0.2)
9.02
2.42
16.0
18.2
3.58
5.27
a rS
(4e-2)
(0.03)
(0.01)
(0.01)
(0.74)
(0.1)
17.4
30.1
10.7
8.28
36.4
19.02
(Tj.e —y/
(0.02)
(0.1)
(0.1)
(0.2)
(0.03)
(1.73)
3.73
4.85
3.22
3.11
1.11
5.16
° p,
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
4.07
1
.
3
7
2
.
5
0
1
.
6
9
4.77
9
.
3
1
a P k>
(0.01)
(0.01)
(4e-3)
(0.01)
(0.01)
-0.64
-0.42
-0.78
-0.75
-0.15
P(Y,Pi ) -0.15
(4e-4)
(4e-3)
(3e-3)
(3e-3)
(0.01)
0.04
-0.08
-0.08
-0.06
0.30
0.30
p(X,Pk>)
(4e-4)
(4e-3)
(0.01)
(0.01)
(4e-3)
46
0
8.46
(0.08)
2.77
(0.08)
5.69
(0.15)
40.9
(0.2)
18.2
(0.1)
36.0
(0.1)
4.11
(0.01)
4.69
(0.01)
0.77
(3e-4)
0.82
(2e-4)
CTit =
ip
= 0.9 P = 0
8.95
(0.08)
2.80
(0.09)
6.15
(0.15)
42.7
(0.2)
18.2
(0.1)
38.1
(0.2)
5.25
(0.02)
5.13
(0.02)
-0.09
(0.01)
0.00
(0.01)
11.1
(0.1)
3.76
(0.10)
7.31
(0.17)
49.4
(0.3)
23.9
(0.1)
42.3
(0.2)
5.65
(0.02)
5.39
(0.02)
-0.07
(0.01)
0.33
(4e-3)
p = 0.99
b = 0.55 b = 0.60 a = 0.28
5.47
(0.06)
2.01
(0.06)
3.46
(0.11)
29.7
(0.1)
11.8
(0.04)
27.0
(0.1)
4.31
(0.02)
3.53
(0.01)
-0.38
(0.01)
0.18
(4e-3)
8.18
(0.08)
2.43
(0.07)
5.75
(0.15)
39.8
(0.2)
16.3
(0.1)
35.9
(0.1)
5.10
(0.02)
4.52
(0.01)
-0.28
(0.01)
0.18
(4e-3)
7.98
(0.08)
2.26
(0.08)
5.72
(0.15)
40.9
(0.2)
18.1
(0.1)
36.1
(0.1)
5.18
(0.02)
4.77
(0.01)
-0.08
(0.01)
0.37
(4e-3)
Notes: (i) The “Data” column contains estimates of the mean and standard deviation of the risk-free return and the equity premium,
with standard errors in parentheses over the period 1892-1987 for U.S. data. These estimates are taken from Cecchetti, Lam and Mark
(1993). These authors do not report the analogous values for the return to equity; (ii) p ( x , y ) denote the correlation between variables
x and y; (iii) Except for the correlations, all statistics are in percent terms; (iv) Rates of return are annualized; (v) Results for the
models are based on 500 replications of sample size 120, and Monte Carlo standard errors are reported in parentheses. The latter are the
standard deviation, across replications, of the associated statistics, divided by \/500; (vi) Prices and output in the model and the data
are logged and HP filtered prior to analysis, rates of return are not filtered. Pi is measured by the ratio of 1987 dollar business fixed
investment plus consumption of durables to the implicit price deflator for consumption of nondurables and services. o p k, p ( Y , P k>) are
taken from Table 1, and the fact that a y — 1.79 percent.
47
Table 4: Business Cycle Statistics
a e = 0.0085
Statistic
Data
aY
1.79
o c/o y
0.47
(T i/ay
2.91
(Th / (TY
0.82
p (y ,c )
0.78
p(Y ,I)
0.70
P {Y ,H )
0.81
p
0.52
(y , h c )
p (Y ,H i)
0.79
p(A Y )
0.36
p(A C )
1 0.19
p(AS)
-0.10
a{A S)
1.80
Calibrated
Full Labor
b = h = 0
p = 0.99
b = 0.55
b = 0.65
a fl = 0
1.90
(0.01)
1.19
(4e-3)
0.99
(2e-4)
^ = 0.9
2.65
(0.02)
P = o
b = 0.55
b = 0.60
2.32
(0.02)
2.32
(0.02)
0.61
(4e-3)
1.80
(0.01)
0.14
(4e-4)
0.61
(4e-3)
1.01
(2e-4)
0.26
(2e-4)
0.46
(4e-3)
1.95
(0.01)
0.44
(le-3)
2.09
(0.01)
0.54
(4e-3)
1.92
(0.01)
0.29
(le-3)
1.80
(0.01)
0.14
(4e-4)
0.63
(4e-3)
2.14
(0.01)
0.20
(le-3)
a = 0.28
1.92
(0.01)
2.16
(0.01)
0.56
(4e-3)
1.87
(0.01)
0.20
(le-3)
2.09
(0.01)
0.27
(2e-3)
2.05
(0.01)
0.29
(2e-3)
2.14
(0.01)
0.53
(3e-3)
1.89
(0.01)
0.15
(4e-4)
2.06
(0.01)
0.18
(le-3)
1.90
(0.01)
0.34
(3e-3)
2.06
(0.01)
0.19
(le-3)
0.55
(4e-3)
0.95
(le-3)
0.63
(3e-3)
0.43
(0.01)
0.98
(3e-4)
0.14
(0.01)
0.53
(4e-3)
0.95
(le-3)
0.69
(3e-3)
0.31
(0.01)
0.97
(4e-4)
0.72
(3e-3)
0.32
(0.01)
0.97
(4e-4)
0.68
(3e-3)
1.00
(3e-6)
1.00
(4e-6)
0.48
(3e-3)
0.48
(0.01)
0.96
(le-3)
0.80
(2e-3)
0.49
(4e-3)
0.95
(le-3)
0.38
(4e-3)
0.62
(0.01)
0.94
(le-3)
0.75
(2e-3)
0.62
(0.01)
0.95
(le-3)
0.75
(3e-3)
0.63
(4e-3)
0.91
(le-3)
0.60
(3e-3)
0.23
(0.01)
0.70
(3e-3)
-0.38
(4e-3)
0.93
(2e-3)
NA
(NA)
0.13
(0.01)
0.69
(3e-3)
0.13
(0.01)
0.75
(3e-3)
0.75
(3e-3)
0.43
(3e-3)
0.53
(3e-3)
0.31
(0.01)
0.81
(2e-3)
0.07
(0.01)
0.67
(3e-3)
0.16
(0.01)
0.75
(2e-3)
0.13
(0.01)
0.75
(3e-3)
0.27
(0.01)
0.68
(3e-3)
0.02
(4e-3)
-0.14
(4e-3)
-0.05
(4e-3)
0.16
(4e-3)
-0.003
(4e-3)
-0.23
(4e-3)
0.10
(4e-3)
-0.12
(4e-3)
0.11
(4e-3)
-0.09
(4e-3)
-0.12
(4e-3)
-0.15
(4e-3)
0.26
(4e-3)
-0.14
(4e-3)
-0.13
(4e-3)
-0.25
(4e-3)
0.12
(4e-3)
0.01
(4e-3)
0.12
(4e-3)
0.01
(4e-3)
0.01
(4e-3)
-0.14
(4e-3)
-0.10
(4e-3)
1.80
(0.01)
-0.13
(4e-3)
1.92
(0.01)
-0.10
(4e-3)
1.80
(0.01)
-0.03
(4e-3)
1.45
(0.01)
-0.03
(4e-3)
1.45
(0.01)
-0.23
(4e-3)
1.24
(4e-3)
-0.04
(4e-3)
-0.29
(4e-3)
2.04
(0.01)
0.01
(4e-3)
1.69
(0.01)
0.01
(4e-3)
0.33
(le-3)
48
1.80
(0.01)
1.70
(0.01)
-0.12
(4e-3)
1.64
(0.01)
Notes: (i) Figures in the “Data” column are based on U.S. data, covering the period 1947:1-1995:1, taken from Citibase. Consumption
is measured by consumption of nondurables and services, GCNQ+GCSQ, measured in 1987 dollars, divided by GPO P, a not-seasonallyadjusted measure of population (including armed forces overseas). Investment is business fixed investment, G IFQ , plus consumption
of durable goods, GCDQ, measured in 1987 dollars, and scaled by GPOP. H c is measured by LWH20X, which is employment in the
industry in Table 5 that looks most (to us) like a consumption goods industry. Similarly, H i is measured by LW HCX in Table 5; (ii)
With the exception of the correlations and the relative volatilities, all the statistics are reported in percentage terms; (iii) Results for the
model are based on 500 replications of sample size 120, and Monte Carlo standard errors are reported in parentheses; (iv) p ( x ) means
the first order autocorrelation of the variable x , p ( x , y ) means the correlation between variables x and y, and cr(x). means the standard
deviation of x . A x means the first difference of the log of (otherwise untransformed) x . p ( A S ) denotes the first order autocorrelation of
the growth rate of the model implied Solow residual, and a ( A s ) denotes the standard deviation of the growth rate of the model implied
Solow residual, (v) Variables without A have been logged and HP filtered prior to analysis; (vi) the entry NA (Not Applicable) signifies
that the indicated number is not defined. The discussion in Section 4.4.4 indicates that when b = h = 0, then H c is a constant.
49
Table 5: Business Cycle Properties of Hours Worked
Hours worked, Ht
1. Household
(lhours)
2.Total private
(lwhx)
2.a. Goods producing
(lwhgx)
Construction
(lwhcx)
Manufacturing:
Durable goods
(lwhdx)
Non durable goods
(lwhnx)
Food and kindred products
(lwh20x)
Apparel and textiles
(lwh23x)
2.b. Service producing
(lwhpx)
Retail trade
(lwhrx)
T — —3
r = -2
corr(Ht,Yt-T)
T = •- 1 r = 0 T = 1
0.18
0.09
0.40
0.08
0.46
0.09
0.49
0.08
0.45
0.09
0.65
0.05
0.71
0.05
0.73
0.05
0.64
0.07
0.48
0.09
0.55
0.08
0.25
0.72
0.06
0.75
0.05
0.41
0.13
0.69
0.07
0.61
0.07
0.79
0.06
(Th /(TY
0.82
0.06
1.18
0.05
2.26
0.22
0.11
0.26
0.10
0.11
3.18
0.23
0.25
2.96
0.14
1.43
0.18
0.69
0.08
1.94
0.25
0.66
0.05
0.82
0.05
0.11
0.26
0.11
0.35
0.10
0.10
0.12
0.12
0.40
0.08
0.15
0.56
0.08
0.38
0.11
0.10
0.51
0.65
0.09
0.11
0.81
0.03
0.90
0.03
0.92
0.02
0.79
0.05
0.91
0.02
0.86
0.03
0.52
0.13
0.73
0.06
0.80
0.05
0.88
0.03
r = 2
T= 3
0.51
0.09
0.70
0.81
0.05
0.93
0.05
0.93
0.05
0.83
0.07
0.69
0.07
0.84
0.08
0.81
0.08
0.77
0.69
0.10
0.12
0.92
0.04
0.77
0.05
0.52
0.82
0.08
0.56
0.09
0.46
0.12
0.12
0.60
0.07
0.85
0.06
0.81
0.06
0.35
0.39
0.14
0.08
0.10
0.11
0.82
0.08
0.11
0.11
0.65
0.11
0.64
0.10
0.33
0.11
0.73
0.66
0.47
0.09
0.11
Notes: (i) H ~ measure of hours worked, taken from Citibase, with mnemonic indicated in parentheses in column 1.
Y t ~ GDP in 1987 dollars, from Citibase. Sample period— all data are quarterly, and with the exception of lhours, they cover
the period 1964:1-1995:1. Lhours covers the period 1947:I-1993:IV. Numbers in the second row of each block are standard deviations,
computed using the procedure described in note (ii) to Table 1.
(ii) Sources and definitions— data from 1 ~ household survey, manhours employed per week; data from 2 ~ establishment survey,
indexes of aggregate weekly hours of production or nonsupervisory workers on private nonagricultural payrolls by industry.
50
Table 6. Measures of Risk Aversion in Consumption.
Model
Calibrated
B C F Calibrated
Perturbed, Calibrated
Perturbed, Calibrated
Perturbed, Calibrated
B C F, Exchange Economy
b
h
X
0.55
0.35
0.60
0.65
0.80
0.58
0.0
0.40
0.0
0.0
0.0
0.30
0.55
0.58
0.60
0.65
0.80
0.82
1%
0.02
0.03
0.03
0.04
0.12
0.16
10%
2.5
2.7
2.9
3.6
6.7
7.3
20%
9.8
9.7
10.3
11.9
18.6
12.4
Notes: Entries in the last three columns are the value of 100 x u associated with the indicated row value of b and the column value
of fi. The value of u solves (4.10). The model column indicates what motivated the particular parameterization considered.
51
0 065 -
Fig . 2f: Stru ctu res Investm ent
on
0060 -
oio -
0055 -
ooe -
0 050 0 045 0 040 0 035 0030 0 025 -*
ooe oo/ -
ooe 005 0 04 -
Fig . 2g: Equipm ent Investm ent
Fig . 2i: Governm ent P u rch a se s
0 300 -
I
I
I
" " 1
0»0
-i
I'
l' ,
I I, l ^
I
'
,
\ .
F ig u re 4: C o rre latio n B e tw e e n Q u arte rly P ric e of In ve stm en t at
t
and Output at
In ve stm en t p rice an d output d ata logged and d etren d ed b y H P filter
Fig. 4b: Household durable investment
Fig. 4c: Fixed investment
Fig. 4d: Nonresidential investment
Fig. 4e: Residential investment
Fig. 4f: Structures investment
Correlation
Correlation
Correlation
Fig. 4a: Fixed investment and durable goods
Correlation
Fig. 4g: NIPA equipment investment
t-k
Figure 5: Correlation Between Stock Price M easure at
t and Output at t-k
Correlation
Price and output data logged and detrended by HP filter
S&P500: Finance vs output
Correlation
NYSE: Finance vs output
k
S&P 500: Utilities vs output
Correlation
NYSE: Utilities vs output
k
k
k
S&P 500: Capital good vs output
Correlation
Dow Jones vs output
k
S&P 500: Industrial vs output
Correlation
NYSE: Industrial vs output
k
k
S&P 500: Transportation vs output
Correlation
NYSE: Transportation vs output
k
k
k
Figure 6: Assigning Values to Preference Parameters, b and h
14
1 2
b = 0.475
1 0
8
b = 0.450
6
b = 0.425
4
h = 0.3
b = 0.400
/j
0 . 2
2
• Corresponds to a particular h,
---1
0
2
- 1
3
Risk free rate (A R % )
4
i
1
5
i
6
b
combination
i
8
Fig. 7a: Household hours vsrsu s trend
Fig. 7d: Construction hours versus trend
Fig. 7b: Total Private hours versus trend
no 41) -
Fig. 7e: Durable goods hours versus trend
4*0 -
*n 4>0466 “
4tO4)) 4JO-
Fig. 7J: Retail trade versus trend
Fig. 71: Service-producing hours versus trend
41 47 46 46 44 41 4.
M
1/
IV
0050
Fig. la : Household hours vartut output
Figure 9: Impulse Response Function for Calibrated Model
% Deviation
Fig. 9a: Response of C
% Deviation
Fig. 9b: Response of I
Fig. 9h: Response of P^
% Deviation
Fig. 9c: Response of h
% Deviation
Fig. 9d: Response of \Q
% Deviation
Fig.9e: Response of lj
------ Aggregated Shock
- - - • Investment Shock
0
@FedFRASER