- Email: [email protected]
Aggregate investment in a business cycle model with adjustment costs
Journal of Economic Dynamics and Control 21(1997) 1181-1198
ELSEVIER
Aggregate investment in a business cycle model with adjustment costs Javier Vallb Servicio de Estudios. Banco de Espaiia, Alcaki 50, 28014 Madrid, Spain
Received 17 August 1993
Abstract This paper explains the observed cyclical variability of aggregate investment and the counter movements with respect to its relative price in the US data using a stochastic growth model with adjustment costs. The exogenous disturbances of the model represent preference shocks, productivity innovations and changes on the relative price between consumption and investment goods. Simulations of the model imply that the volatility of those relative price innovations need to be three to four times larger than the standard Solow residual innovations in order to match the model with the data. Keywords:
Investment; Business cycle; Adjustment costs; Simulation E22; E32; E37
JEL classification:
1. Introduction Most of the investment literature is concerned with the problem faced by individual firms (see, for example, Abel, 1990), and few models address the issue of aggregate investment. A prediction of the investment literature concerned with the problem faced by individual firms is that the supply curve of new capital goods must be upward sloping. At the aggregate level, however, we observe that nonresidential investment is negatively correlated with the price of investment goods. This negative correlation can only be reconciled with an individual firm’s upward sloping supply curve for investment goods if supply side fluctuations are large relative to the fluctuations in demand. In this paper, we analyze a general
I am very grateful to Christopher Sims for his advice. Conversations with J. Diaz, H. lchimura and A. Novales are also acknowledged. 0165-1889/97/$17.00 0 1997 Elsevier Science. B.V. Ail rights reserved PII SO165-1889(97)00026-2
1182
J. Yak% /Journal
of Economic Dynamics
and Control 21 (1997) 1181-I198
equilibrium model that simulates the observed variability of aggregate investment as well as some statistical properties on other real variables when exogenous perturbations affect the optimal paths of the variables that solve the assumed structural model. In a general equilibrium model, the price of new capital results from the market clearing condition for the capital goods market. This price determines the rate of investment in each period. When an adjustment cost technology is assumed, the rate of investment is an increasing function of that price. Crosssection studies with US data show a negative correlation between investment and its price (Kydland and Prescott (1982)). We have observed, in US time-series of fixed nonresidential investment and the price deflator of investment goods, that their negative correlation is also quite large, -0.30. A model economy should ideally display these two properties: the optimal supply rule for new capital goods as an increasing function of the corresponding prices and the negative correlation observed in US time-series data. The time series generated by the model reproduce the above two properties of the observed data. They also display different long- and short-run elasticities of investment supply capturing the different rates of adjustment of new capital. The model in this paper includes, besides the usual productivity shock, two additional disturbances: one in preferences and the other in the adjustment cost technology. As is standard in the real business cycle literature, the stochastic fluctuations in productivity are meant to represent unpredictable technological changes or Solow residual innovations. On the other hand, the adjustment cost shock can be viewed as representing changes in productivity embodied in new capital goods or relative price variability coming from changes in the taxation of capital and consumption goods. The shock in preferences represents exogenous factors affecting agents’ willingness to distribute their time between market and nonmarket activities. The model is therefore a version of the neoclassical growth model with a stochastic adjustment cost technology. The technology on the production side uses only one type of capital good while on the output side it is costly to transform one unit of consumption good into one unit of investment good. Once investment has been committed to production the transformation is irreversible and this implies a fixed capital-labor ratio. Finally, capital goods depreciate at a constant rate. The sources of uncertainty in the simulated economy are equal to the number of endogenous variables and we study their relative importance to achieve the empirical properties of the model. In a nonlinear environment like the one studied here, the firm’s and household’s policy functions do not have closed-form solutions. Simulations of the model with selected parameters are run using Sims’ backsolving method (1990). The way the model is solved permits identification of the dynamic effect of changes in the primitive sources of uncertainty on the endogenous variables. At the same time, these simulations allow us to address questions about the
J. VallCs /Journal
of Economic Dynamics and Control 21 (1997) 1181-1198
1183
dynamic effects between the endogenous variables that solve the social planning problem and the prices that come from solving a decentralized equilibrium. The remainder of this paper is organized as follows. Section 2 describes the stochastic growth model with adjustments costs. Section 3 presents the simulation procedure and the results. The discussion on the cyclical and dynamic relation between variables, the importance of adjustment shocks and the effects of prices on investment are also included. Finally, the findings are summarized in Section 4.
2. A stochastic growth model with adjustment costs There is an infinitely alive representative maximizes its expected discounted utility:
Eo
,Eo /3’([c:‘-“J
consumer
in the economy
that
(1 - L,)“~]“-y’/(l - y)),
The utility function displays unit intratemporal elasticity of substitution between consumption and leisure. This elasticity is consistent with the fact that per capita series of labor have shown no significant trend in the US. The leisure share parameter 4 is affected by a random shock or,. Recently, Bencivenga (1992) has shown the importance of preference shocks, to account for the observed variability of consumption and labor when intertemporal effects are absent. The parameter y is the coefficient of risk aversion and b is the discount factor. The technology shows constant returns to scale with two inputs, labor (I,) and capital (K). Investment (I) is the part of the homogeneous output (Y) that is installed at some cost. Once investment is fixed as capital it implies a constant capital-labor ratio. There are two exogenous shocks that affect the technology: dZ, is a Solow residual shock to the quantity produced, whereas 13~~ represents a shock to the adjustment cost. The technology is then written as G(C,, I,, e,,) = (AC: +
m,,z:p
O
A>O,
I
e,,zc- 1Lj’ -=), B>O,
?rrl.
(2)
The production function is of the Cobb-Douglas form with 0:as the capital’s share of output. This corresponds with the observation of a constant capital and
1184
J. Vall&
/ Journal of Economic Dynamics and Control 21 (1997)
118I-1198
labor share of output in the United States since 1955. Both the production and the utility functions have been widely used in the business cycle literature to study fluctuations on aggregate variables (see, for example, Prescott, 1986; and King et al. 1988). The functional form G guarantees that the cost of installing gross investment is convex with a minimum value of zero when investment is zero. The adjustment cost parameter q measures the elasticity of substitution between consumption and investment goods. A value of q equal to one together with the condition that the parameters A and B are equal implies that there is no cost to installing one unit of the investment good. As the parameter q gets bigger the cost of installing one unit of investment good in terms of forgone consumption increases. Capital has a constant depreciation rate 6: K,=I,+(l-6)K,_,,
0<6<1
(3)
then investment at time t becomes productive one period later, at time t + 1. The technology is homogeneous of degree one. Therefore, the distribution of capital between firms is irrelevant (see, for example, Lucas and Prescott, 1971). The random vector O1= (e,,, &, O,,) is assumed to be stationary and identically distributed over time. The vector of random shock follows a lognormal first-order autoregressive distribution, i.e.: log e,, 1 = P log 8, + E,+1, &** N(0, C).
(4)
Usually, a highly persistent shock is necessary to match the optimal paths of a neoclassical growth model with the business cycle properties of actual data. This model allows for serial correlation in each of the shocks. Each element of P is denoted by pij. If contemporaneous correlation between the shocks is not allowed, both the matrix P and C are diagonal. The social planner’s problem is Max EO f /3’([C~‘-“~‘(l - L,)@l]“-y’/(l - y)) r=o s.t.
(AC; + Be,, I:)“” < OatKY- I L: K 1+1 loge,,,
c,,L,
=I,
+(I -6)fG =PiOge,+~+l,&,-N(~,C)
-=
Q't,
Q’t, Qt.
K*+1 and I, are the decision variables in each period t.
J. Vall& 1 Journal of Economic Dynamics and Control 21 (1997) 1181-1198
1185
The state variables of this problem are last period’s stock of capital K,- 1 and the vector of shocks 8,. There exists a solution to the social planner’s problem (5) given by a vector of stationary stochastic processes: K,, I = g(K,, e,), C, = c(L R), L = QK,, 6). Once the shadow price of capital is substituted out there are three first-order conditions in the maximization problem (5): the marginal rate of substitution between consumption and labor equal to its cost; the discounted value of the return on an additional unit of investment next period must be equal to its current cost in terms of consumption goods; and the last equation is the transversality condition on the capital stock. These three equations, together with the budget constraint, form a set of four equations with four unknowns. Given our assumed functional forms, an interior solution is obtained if the transversality condition and the above conditions are satisfied and the serial correlation matrix P has all its eigenvalues inside the unit circle (see, for example, Prescott and Mehra, 1980). When P is diagonal, this means that pij must be between 0 and 1. The nonstochastic steady-state solution for problem (5) gives the well-known positive relation between the interest rate of the economy and the marginal product of capital taking into account the adjustment cost for the new capital, i.e. B [( 1 - 6) + Fk/Gk] = 1. By the assumptions of F and G, total differentiation of this expression gives that increases in the discount factor increase the stationary level of capital and therefore of output and consumption. This result is also obtained when there are no adjustment costs to the growth model (Danthine and Donaldson, 1981). A decrease in the adjustment cost parameter, q, affects the steady state of the economy like a decrease in /?: the cost of capital increases and output and the stock of capital decreases. Nevertheless, capital decreases more than proportionately so that the economy is now more productive and therefore less capital is needed per unit of output. Since we are interested in the dynamic behavior of the price of investment and also of the rental price of capital, we make use of the property that the social optimum is also the solution for a sequence of market equilibrium allocations. The representative consumer sells its labor and rents its capital stock at the competitive prices. The representative firm maximizes profits for each period. Given the sequence of prices for labor and new capital the firm produces consumption and investment goods. Consequently, there are time invariant functions for wages w, = o(K,, O,), rental prices of capital R, = R(K,, 0,) and prices of investment P,, = P,(K,, et), for which the policy functions K(K,, e,), c(K,, 0,) and I(K,, 0,) solve the optimization problems of the household and of the firm operating in competitive markets. The assumed investment adjustment cost function G implies that in equilibrium, the shadow price of new capital goods (G,/Gc) is equal to the current price
1186
J. Vallhs /Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
of investment goods relative to produced consumption goods (P,).i This price will vary not only with exogenous changes in the adjustment cost shock 13~,,but also with the intertemporal effects represented by endogenous changes in the (ZJC,) ratio. Besides the changes in the structural parameters, changes in the uncertainty of the adjustment cost technology also has effects on the capital accumulation. Because the structural shocks are log-normally distributed, changes in the variances of the shocks’ innovations affect not only the variance of the shocks but also their mean. Hence, an increase in the variance of 6sr causes, as a consequence of an increment of uncertainty in the relative prices, an increase not only in the variance of the optimal paths (I&) and (KJY,) but also in their average value. Those level effects, though, are small compared, for example, with the changes generated by the adjustment cost parameter q. In equilibrium, investment is an increasing function of the relative price of new capital, as in Lucas (1967) and Uzawa (1969), because the function G is convex. In Lucas’ model, for example, the firm maximizes the discounted profit function for given sequences of output and factor prices. Nevertheless, in our model the investment good prices have to satisfy for each period the usual asset pricing equation derived from the consumer optimality conditions: the expected value of the intertemporal marginal rate of substitution times the gross yield of capital must be equal to one.
3. Simulation experiments The nonexistence of closed form solutions for the optimal allocation of policies makes it necessary to search for a numerical solution. Once the allocation of real variables is obtained, the sequences of prices are derived from the optimality conditions for the competitive economy. Here we make use of the ‘backsolving’ method (Sims, 1990) to find the solution of the problem in Section 2 (see Appendix). A property of the ‘backsolving’ method is that we use all the nonlinear restrictions without need to linearize them. The stochastic simulation algorithm draws values from the distribution of the expectation error of the only dynamic equation in the first-order conditions and from two of the innovations. Then, it uses the derived stability condition to find the third innovation, and finally using the first-order conditions, the decisions variables are solved. The sequences of prices are derived from the optimality conditions for the competitive economy. This simulation method is an approximation solution because it uses a restriction on the joint distribution of the exogenous shocks and the expectation error ’ Therefore, the ratio of these two variables, also called Tobin’s
t because it already includes the installation cost.
q,
is always equal to one at each
J. VallPs / Journal of Economic Dynamics and Control 21 (1997) 1181-l 198
1187
that comes from a linearized version of the model. Opposite to that, ‘Forward’ simulation methods draw values for all the structural disturbances and satisfy the stability conditions imposing restrictions on the decision rules (see Taylor and Uhlig, 1990 for a description of those simulation methods). Nevertheless, when we do not have a priori notions about the shape of the decision rules we can make restrictions about the distribution of the unobservable shocks 8,. Other empirical work that has used this ‘backward’ solution method (see, for example, Novales, 1992), have imposed a distribution on some function of the shocks like the Lagrangian multiplier or the marginal productivity of consumption. This way of solving avoids the use of linear approximations but has the disadvantage of assuming distributional properties of variables that have no structural interpretation. The model also has the property whereby, unlike other simulated stochastic growth models, the number of unobservables variables (6) is equal to the number of endogenous variables.* 3.1. Evaluation
of the adjustment
cost technology
As a means of evaluating the adjustment cost technology we compare some properties of the vector autoregressive representation (VAR) of actual and simulated data. In particular, we investigate for a range of parameter values of the adjustment cost shock variance (C,,) and of the elasticity of substitution (q), whether the impulse response and the variance decomposition of the estimated VAR for simulated data approximate those properties of the VAR for actual data.3 To obtain the simulated data from the model we fix the discount factor at 0.99 so that the model without adjustment costs displays an approximate annual 4% real interest rate at the stationary level. The leisure share parameter C#Jis 2/3, a standard value that emerges from the literature. The results presented here have assumed a capital share parameter u equal to l/3. The parameters of the adjustment cost function, A and B, are set equal to one so that the relative price of investment goods at the steady state is one when the parameter 4 is one. The capital depreciation parameter 6 is 0.025. We assume that the shocks are very persistent in order to generate the observed autocorrelation in the variables. Their serial correlation parameter is 0.99 and there is no cross correlation (i.e.
2As Ingram et al. (1992) have noted, that property ensures the nonsingularity of the variance-covariance matrix of the simulated data. Furthermore, the above characteristic would allow us to estimate eventually the parameters of the model. 3We will not attempt to develop a formal weighting function that would allow such a comparison (see, for example, Lee and Ingram, 1991).
1188
J. Vallh /Journal
of Economic Dynamics
and Control 21 (1997) 1181-1198
P and C are diagonal). Initially, we fixed the standard deviation of the productivity disturbance, CZ2, to be 0.8%. That is a value similar to the one selected by real business cycle models. The standard deviation of the preference disturbance, Z, 1, has been restricted to have lower variation, 0.2%. For the adjustment cost shock, Za3, we try a range of values from 0.1 to 6%. The estimated VAR corresponds to a system with the same number of variables as disturbances in the model: The system includes, in addition to a constant, the three endogenous variables, consumption, C,, labor, L,, and investment, I,. Because we expect consumption and labor to respond with some lag to exogenous changes in investment, we ordered the VAR with investment in the last position. The simulated data is obtained solving the model for 120 periods. The conditional response of system variables, for up to I2 periods ahead, are presented using a unit shock to the orthogonalized innovations. The responses of the variables are normalized dividing by the standard deviation of the corresponding innovation. The endogenous variables of the model are functions of current and past exogenous shocks. Although the optimal paths of the observed variables are stationary, the reason for the nonstationarity of the sample is the high serial correlation of all the shocks transmitted through the model mechanisms. Therefore, the comparisons are made with the first differences of the simulated and the real data. The actual data have 120 quarterly observations and correspond to the period 1961:3 to 1991:3 for the US economy.4 Investment includes all fixed investment on equipment and structures. Consumption is the result of adding the consumption series of nondurable goods and service goods. The labor series is the ratio of hours worked and time endowment (we assume 112 hours per week and 4.25 weeks a month). The sample average of this labor ratio is around 0.2 and measures hours worked by the employed and the unemployed. Fig. 1 shows the response of investment to the three innovations to the VAR with actual and simulated data. Those simulations are run with the parameter q equal to 1.8 and assuming risk neutral consumers, i.e. the parameter y equal to zero. We figure two simulated responses, one with a standard deviation of the adjustment cost disturbance equal to 1% and the other equal to 3% (this is almost four times the size of the technological disturbance used in the RBC models). The comparison of the responses imply that the model needs a large variation in the adjustment cost shock to obtain in the first periods, after an exogenous impulse in consumption, a small response of investment (first chart) and a high autocorrelation of investment (third chart), similar to the ones in actual data. However with a large size of the adjustment costs disturbance we cannot get the
‘They are from Citibank Economic Database and are measured in per capita terms.
J. Yaks
1.4
07
/Journal
of Economic Dynamics
1189
and Control 21 (1997) 1181-1198
Rk3pCi-S es of ktvesttnent to 0 Ccnscrrptlon lnxwatlm
,
-1\ I
\
I
Slnwloted Responses of Westment to 0 Ltlltx- IrrWatlal
I.t
I
0.0 -
Slm~lotedResponsesof investment to its own tmavotfm
I .*
0.7
0.0
-I .t I
2
3
4
6
6
,
8
9
10
II
12
Fig. 1. Notes: - The VAR system includes a constant and five lags of changes in consumption (MI), changes in labor (AI) and changes in investment (dlnu). Actual means that the VAR is estimated with US quarterly data (61:3-91:3). The other two responses correspond to simulated data with ,?& equal to 1% and 3%, respectively. - All responses are normalized by the standard error of the corresponding innovation.
1190
J. Vallks /Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
positive short-run response of investment to an innovation in labor (second chart). The fact that the exogenous disturbances follow an AR(l) makes the responses of the variables of the VAR system go to zero very quickly after the first period, independent of the size in the adjustment cost shock. That is why the model is not able to reproduce the persistent response of investment found in the data. The variance decomposition for the forecast errors in the VAR system shows that when CS3 is 3% the error of each variable in the long run (after 12 quarters) is explained mostly by its own variation. That is what comes out with actual data. We also find that for the grid of values of the adjustment cost shock, only when that value is 3% does investment have higher power to explain its own error. Lower values of the adjustment cost parameter (q = 1.2) imply negative responses of labor and investment to exogenous shocks in consumption, both in the short run and in the long run. It would also imply, a lower weight of investment to explain its own forecast error. Consequently with the above results, we choose as a benchmark model one in which the adjustment cost disturbance is 3% and the elasticity of substitution between consumption and investment is 1.8. 3.2. Business cycles properties We study the business cycles properties of the adjustment cost economy detrending the simulated and actual data with the Hodrick-Prescott filter (see Prescott, 1986). The actual value for Prt is the deflator for nonresidential investment. Table 1 evaluates the importance of the adjustment cost shock. The model, given the chosen structural parameters and the distribution properties of the exogenous shocks, needs a large variation in the distribution of the adjustment cost technology to obtain the observed cyclical variation in actual output. When Ca3 is as low as the preference shock, 0.2%, output volatility is only 1.2%, whereas when that variance is 3% or 4% we obtain an output variance close to the one in actual data.’ With adjustment costs and Z33 at 3%, the large amount of uncertainty in the model leads to statistics with large standard deviations. The consumption series has a standard deviation that is 0.8 times the standard deviation of output while
‘Note that our measure of output does not correspond with the one Accounts. In the simulations, output Y, is considered to be the outcome Even if output is measured as (C, + Pr, &) it would still differ from the and would show a close picture to output measured by the production
from the National Income of the production function. National Income Accounts function.
J. Vall& / Journal
of Economic Dynamics and Control 21 (I 997) f f Bf - if 98
Table I Business cycle statistics of the adjustment cost economy, Hodrick-Prescott
1191
filter
Simulated economy
US Economy 61:3, 91:3 (in per capita terms)
La3 = 0.01
2x3 = 0.03
z,3 = 0.04
6,.
0.016
0.012 (0.007)
0.014 (0.009)
0.017 (0.012)
QJ,.
0.50
0.83 (0.09)
0.86 (0.056)
0.89 (0.69)
0,
0.86
0.58 (0.34)
0.89 (0.71)
1.08 (0.74)
u&r
3.00
1.86 (1.00)
3.22 (2.93)
3.42 (3.25)
c
0.84
0.98 (0.04)
0.93 (0.24)
0.93 (0.23)
L
0.86
0.63 (0.43)
0.72 (0.38)
0.78 (0.38)
I
0.56
0.85 (0.28)
0.65 (0.99)
0.65 (1.20)
- 0.43
0.52 (0.70)
- 0.34 (1.49)
- 0.58 (0.87)
Correlation with Y
Corr (I, P)
Notes: The structural parameters used for the simulated data are: A = B = 1.0, n = 1.8,6 = 0.025, r#~= 2/3, /I = 0.99 and y = 0.0. The statistics correspond to the means for 20 simulations, each with 120 observations. The standard deviations of the statistics are in brackets.
in the real data it is half the output value. The fact that the preference shock affects both consumption and labor makes them move very closely, whereas in actual data they follow different patterns. 6 The investment series has a large volatility with respect to output, as in actual data, with large values of ZS3. The model is able to reproduce the negative correlation between investment goods and its own price found in the data. This regularity and the above cyclical properties appear also with risk aversion preferences (e.g. y = 1.0). Table 2 evaluates the cyclical properties of the adjustment cost technology. An economy with a low adjustment cost parameter (q) is seen as more productive since it has on average a lower stock of capital per unit of output. In Table 2
6 However the stochastic adjustment cost technology implies a lower variation in labor with respect to output than in a growth model with only a productivity shock. In fact, the empirical labor elasticity of output is below one (0.84) and close to the observed labor share parameter.
1192
J. Valks
/Journal
of Economic
Dynamics
and Control
21 (1997)
II8I-1198
Table 2 Business cycle properties of the adjustment cost technology, Hodrick-Prescott
filter
Economy with adjustment costs (Zs, = 0.03)
Economy without adjustment costs (n = 1.0, 11s3= 0.0)
‘1 = 1.8
z,, = 0.002
c,, = 0.0
?)= 1.2
0s
0.014 (0.009)
0.018 (0.010)
0.016 (0.009)
0.016 (0.008)
GJY
0.86 (0.56)
0.90 (1.72)
0.30 (0.15)
0.50 (0.40)
cl/es
0.89 (0.71)
1.08 (0.72)
0.73 (0.25)
0.92 (0.50)
U&J,
3.22 (2.93)
5.79 (7.80)
3.37 (0.48)
5.45 (3.06)
C
0.93 (0.24)
0.34 (1.06)
0.69 (0.16)
0.68 (0.16)
L
0.72 (0.38)
0.82 (0.25)
0.87 (0.17)
I
0.65 (0.99)
0.67 (0.97)
0.97 (0.02)
0.96 (0.04) 0.97 (0.02)
- 0.34 (1.49)
- 0.75 (0.42)
Correlation
with
Y
Corr (I, P,)
See footnotes to Table 1.
we see three possible costs in terms of the cyclical properties of the model. First, a reduction in q implies an increase in the volatility of all economic series. The same result appears in Basu (1987) in an adjustment cost model of asset pricing. Second, a technology with a low cost of substitution of consumption for investment, provides low comovement of C, and Y, and high relative volatility of I,, compared with actual data. Finally, to get the observed negative comovement of investment with its price the model needs not only large fluctuations of that price but also a high elasticity of substitution. The presence of an adjustment cost shock in this model offsets the relevance of the productivity shock for investment fluctuations.’ Other papers have also found the importance of other real shocks and the implied overstate of the
‘Since the three exogenous disturbances are the driving force of the model, we expect movements in investment to be fuly explained by those fluctuations.
J. Vall& / Journal
ofEconomicDynamics
and Control 2/ (1997) I IBI- I198
1193
Solow residual variability to account for example for the variability of US output (Burnside et al., 1993). However in this model, the large variability of the estimated adjustment cost shock gives a supply side explanation for the negative correlation between prices and quantities of investment goods. A positive exogenous shock to the investment goods market moves resources of the economy to that sector. The amount of investment goods increases with respect to the consumption goods production and the relative value of an extra unit of new capital falls. Therefore, the appearance of adjustment costs increases the amount of capital per unit of output in the economy. The shift in the supply of investment goods by the firms, caused by these exogenous shocks, stimulates an increase in investment and a fall in its price. 3.3. The dynamics of prices Finding good estimates for the rental price of capital has always been considered a difficult problem. In the standard jorgensonian investment equations the marginal product of capital is equal to the rental price. Then the desired stock of capital increases with a lower rental cost. The way that rental price is measured is imposing than the implied return of capital is equal to the one by an alternative asset. Therefore, the rental price is a function of a real interest rate, the depreciation rate and the shadow price of capital (see Abel, 1990). A model of aggregate investment like the one presented here can generate well justified equilibrium prices. The dynamics of the movements of those prices and their effect on capital accumulation is still an open question when both are endogenous and respond to exogenous shocks. In this section we present the comovements of the price of investment and the rental price of capital with aggregate investment through the simulation of the adjustment cost economy. From the dynamic first-order condition (A.l), we obtain that the rental price of capital in steady state, i.e. P,(l/b + 6), must be equal to the marginal productivity of capital, Fk, /Gc,. Outside the steady state we may define the rental price R, as the realized one period gross yield of capital, i.e. Gc,/Gi, [((1 - 6)Gi,+ 1 + Fk,. ,)/Gc,+ r J. The two charts of Fig. 2 show the responses to an impulse in the rental price and investment in a VAR system with those two variables plus the price of investment.’ In a general equilibrium model with adjustment costs the changes in investment no longer respond negatively to an unanticipated increment in the change of the rental price of capital (first chart, Fig. 2). That positive innovation in the
*A decomposition of the residuals in which prices of investment show some sluggishness, not responding initially to disturbances in investment, but with investment series responding immediately to movements in the relative prices seems to be in accordance with the observed dynamics in those two variables.
J. VaN& / Joumol of Economic Dynamics and Control 2/ (1997) 1181-1198 SlmtHOtedResmnses
of Plnv mci Inv to a Rental Price lmovotlon (R)
1 PI”” I””
.
.
.
6
.
I,
1
1
9
3
I
I
r---r II
a
a
.
Slmuloted Responses of R md Plnv to m Investment Imovotlm(lnv)
/\ --_, urn
-0.14
-
[:‘“-Et
0.28
I
3
6
7
3
II
Fig. 2. Notes: - The simulated VAR system includes changes in the rental price (AR), changes in the price of investment (Mine) and changes in investment (dlnv). The parameters are the same ones used in the 2nd column of Table 1. - All the responses are normalized by the standard error of the corresponding innovation
J. Y&k
/Journal
o,fEcunomic
Dynamics and Controro(2/ (1997) 1181-1198
1195
rental price of capital may be caused, for example, by a temporary increase in the capital income tax.g That should have an inmediate effect not only in relative prices but also in the goods allocation. The increase in R, creates more disposable income for the consumers and an upward shift in the demand for investment goods. The firms increase their costs but as new investment goods are purchased the relative price of investment goods declines. Although in steady state R, and PI, move together, outside the stationary solution both prices respond with an opposite sign to an exogenous impulse in investment (second chart, Fig. 2). An interpretation outside the model is that those exogenous innovations may create an upward pressure on the interest rates controlled by the monetary authority that pushes up the rental price. As a consequence, both investment and the rental price of capital will increase. The above two results give a theoretical explanation to the empirical evidence of a small or nonsignificant effect of the rental price in aggregate investment (see for example Gordon and Veitch (1986) for the US). In presence of different simultaneous shocks on the economy such negative effect disappears.
4. Conclusions To reproduce the observed volatility of aggregate investment and the negative correlation with its price in the US data a growth model with large variability in the adjustment costs technology has been constructed. The high cost of substitution of consumption for investment and the large adjustment cost shock variability offsets the importance of productivity shocks, usual in the real business cycle models. However, it gives a supply side explanation for the negative correlation between prices and quantities of investment goods. Such adjustment cost shocks need to have an extremaly large volatility, three to four times larger than the standard Solow residual innovations. The model gives also an explanation for the small effect of the rental price of capital on estimated investment equations found in the literature. An unanticipated innovation of the rental price of capital, when there are several shocks hitting the economy, produces, as an optimal response of the agents, movements of the same sign on investment.
Appendix. Simulation method This appendix describes the ‘backsolving’ method (Sims, 1990) to find the solution for the problem in Section 2. The first-order conditions, the technology ‘It would be a distortionary tax. The government will distribute the revenues in a lump-sum way to the consumers.
1196
J. Vallh
/Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
constraint and the law of motion for the vector of shocks give a set of first-order stochastic difference equations. The only dynamic equation in the first-order conditions corresponds to the derivative with respect to the state variable K,, t: -
(UG/&) + PEt(Uct+~/Gct+l)((l - 4Gi,+, + F/c,)= 0
(A.11
where subindices refer to partial derivatives. Introducing an error term (‘expectation error’) r,rl+1 s.t. E, q,+ r = 0 in (A.l), we obtain a system of six nonlinear, stochastic difference equations in the following way:
(A-2)
H(z,+1,zt) = 5*+1 where z, = (X,, 0,) = (G L,, 5;+
1 =
(0,
0,
?t+
1, &It+
Ll,
1, h+
h,,
br,
1, &3t+
e,J
I).
All the solutions for the problem in Section 2 must satisfy (A.2). An additional equation is required in (A.2) to get a solution path for X,+ 1. There is a restriction on the distribution of [, + 1 that guarantees a stable solution for X,, 1, and uniqueness for given Be and Ze. The steps to solve the model are the following, 1. Linearize (A.2) around the steady state.
raft+
1 =
rl% + 5t+I
(A.3)
where .Zr= z, - z,,, z,, = (C,,, L,,, K,,, 1, 1, 1) is the steady-state vector, ro, rI are 6 x 6 matrices of coefficients. 2. Obtain the generalized eigenvalues and eigenvectors for (r,, r,). There are nonsingular matrices Q and 2 and diagonal matrices Ho, HI such that
rl = QH,Z,
(A.4)
J. Vallh /Journal
ojEconomic
Dynamics and Control 21 (1997) 1181-1198
where Z is the matrix of eigenvectors. The corresponding values 1 are
1197
vector of eigen-
,I(TO, r,) = (J. E Cldet(To, r,) = 0}, = R(H*, HI) = {;I = Hlii/Hoiij*
(A.3
Therefore (A.3) can be expressed as Q&Z5 + I = QHJ&
+ &+I
where
+ i [H,lH,]SH,'Q-'~,+l_, Z&+1 = [H;'H,]'Zz,, S=O
3 The stability of the linearized system is guaranteed when all the roots of that system are lower than the inverse of the discount rate. Since (A.6) has one eigenvalue Ajgreater than l/b for the given functional forms of F,G and U, to eliminate the possibility of solutions of (A.31 which grow at a rate faster than l/p we require the corresponding eigenvalues to be zero, i.e.: Zi.Zl = 0
vt
(A-7)
where Zj. is the row of eigenvectors associated to the eigenvalue Jj. Given the vector of observable variables X,, (A.7) gives an exact relation between E and q. If there were no eigenvalues greater than l/p this would imply that there are many possible solutions for the linearized system, since the mapping between q and E would not be unique. If more than one eigenvalue is greater than l/b the original problem has no solution. As long as we are interested in solutions for the nonlinear system around the steady state (i.e. for a small variance of the exogenous processes) we can use (A.7) as an additional equation to solve the system of first-order conditions (A.2). We can treat one of the elements of innovations &+ 1 as unknown, say {it+ 1, draw values for the remaining <,+ 1 elements and use (A.2), (A.7) and the initial conditions to obtain solution paths for X, + 1.
1198
J. Vallh / Journal of Economic Dynamics
and Control 21 (1997) 1181-1198
References Abel, A., 1990, Consumption and investment in: B. Friedman and F. Hahn, eds., Handbook of monetary economics (North-Holland, Amsterdam). Basu, P., 1987, An adjustment cost model of asset pricing, International Economic Review 28, 609-621. Bencivenga, V., 1992, An econometric study of hours and output variation with preference shocks, International Economic Review 33,449-471. Burnside, C., M. Eichenbaum, and S. Rebelo, 1993, Labour hoarding and the business cycle, Journal of Political Economy 101, 245-273. Danthine, J.P. and J.B. Donaldson, 1981, Stochastic properties of fast versus slow growing economies, Econometrica 49, 1007-1031. Gordon, R. and J. Veitch, 1986, Fixed investment in the American business cycle, 1919-83 in: R. Gordon, ed., The American business cycle, continuity or change (University of Chicago Press, Chicago). Ingram, B.F., N. Kocherlakota, and N. Savin, 1992, Explaining business cycles: a multiple shock approach, Mimeo (Department of Economics, University of Iowa). King, R., C. Plosser, and S. Rebelo, 1988, Production, growth and business cycles 1: The basic neoclassical model, Journal of Monetary Economics 21, 195-232. Kydland, F.E. and EC. Prescott, 1982, Time-to-build and aggregate fluctuations, Econometrica 50, 1345-1370. Lee, B. and B. Ingram, 1991, Simulation estimation of time series models, Journal of Econometrics 47, 197-205. Lucas, R., 1967, Adjustment costs and the theory of supply, Journal of Political Economy 75, 321-334. Lucas, R. and E.C. Prescott, 1971, Investment under uncertainty, Econometrica 39, 659-681. Novales, A., 1992, Solving nonlinear rational expectations models: a stochastic model of interest rates, Econometrica 58, 93-l 11. Prescott, E.C., 1986, Theory ahead of business cycle measurement, Quarterly Review, Fall, 9-21, Federal Reserve Bank of Minneapolis. Prescott, E.C. and R. Mehra, 1980, Recursive competitive equilibrium: the case of homogeneous households, Econometrica 48, 1365-1379. Taylor, J.B. and H. Uhlig, 1990, Solving nonlinear stochastic growth models: a comparison of alternative solution methods, Journal of Business and Economics Statistics 8, l-50. Sims, CA., 1990, Solving nonlinear stochastic optimization and equilibrium models backwards, Discussion Paper 15 (Institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis and University of Minnesota). Uzawa, H., 1969, Time preference and the Penrose effect in a two-class mode1 of economic growth, Journal of Political Economy 77, 628-652.
ELSEVIER
Aggregate investment in a business cycle model with adjustment costs Javier Vallb Servicio de Estudios. Banco de Espaiia, Alcaki 50, 28014 Madrid, Spain
Received 17 August 1993
Abstract This paper explains the observed cyclical variability of aggregate investment and the counter movements with respect to its relative price in the US data using a stochastic growth model with adjustment costs. The exogenous disturbances of the model represent preference shocks, productivity innovations and changes on the relative price between consumption and investment goods. Simulations of the model imply that the volatility of those relative price innovations need to be three to four times larger than the standard Solow residual innovations in order to match the model with the data. Keywords:
Investment; Business cycle; Adjustment costs; Simulation E22; E32; E37
JEL classification:
1. Introduction Most of the investment literature is concerned with the problem faced by individual firms (see, for example, Abel, 1990), and few models address the issue of aggregate investment. A prediction of the investment literature concerned with the problem faced by individual firms is that the supply curve of new capital goods must be upward sloping. At the aggregate level, however, we observe that nonresidential investment is negatively correlated with the price of investment goods. This negative correlation can only be reconciled with an individual firm’s upward sloping supply curve for investment goods if supply side fluctuations are large relative to the fluctuations in demand. In this paper, we analyze a general
I am very grateful to Christopher Sims for his advice. Conversations with J. Diaz, H. lchimura and A. Novales are also acknowledged. 0165-1889/97/$17.00 0 1997 Elsevier Science. B.V. Ail rights reserved PII SO165-1889(97)00026-2
1182
J. Yak% /Journal
of Economic Dynamics
and Control 21 (1997) 1181-I198
equilibrium model that simulates the observed variability of aggregate investment as well as some statistical properties on other real variables when exogenous perturbations affect the optimal paths of the variables that solve the assumed structural model. In a general equilibrium model, the price of new capital results from the market clearing condition for the capital goods market. This price determines the rate of investment in each period. When an adjustment cost technology is assumed, the rate of investment is an increasing function of that price. Crosssection studies with US data show a negative correlation between investment and its price (Kydland and Prescott (1982)). We have observed, in US time-series of fixed nonresidential investment and the price deflator of investment goods, that their negative correlation is also quite large, -0.30. A model economy should ideally display these two properties: the optimal supply rule for new capital goods as an increasing function of the corresponding prices and the negative correlation observed in US time-series data. The time series generated by the model reproduce the above two properties of the observed data. They also display different long- and short-run elasticities of investment supply capturing the different rates of adjustment of new capital. The model in this paper includes, besides the usual productivity shock, two additional disturbances: one in preferences and the other in the adjustment cost technology. As is standard in the real business cycle literature, the stochastic fluctuations in productivity are meant to represent unpredictable technological changes or Solow residual innovations. On the other hand, the adjustment cost shock can be viewed as representing changes in productivity embodied in new capital goods or relative price variability coming from changes in the taxation of capital and consumption goods. The shock in preferences represents exogenous factors affecting agents’ willingness to distribute their time between market and nonmarket activities. The model is therefore a version of the neoclassical growth model with a stochastic adjustment cost technology. The technology on the production side uses only one type of capital good while on the output side it is costly to transform one unit of consumption good into one unit of investment good. Once investment has been committed to production the transformation is irreversible and this implies a fixed capital-labor ratio. Finally, capital goods depreciate at a constant rate. The sources of uncertainty in the simulated economy are equal to the number of endogenous variables and we study their relative importance to achieve the empirical properties of the model. In a nonlinear environment like the one studied here, the firm’s and household’s policy functions do not have closed-form solutions. Simulations of the model with selected parameters are run using Sims’ backsolving method (1990). The way the model is solved permits identification of the dynamic effect of changes in the primitive sources of uncertainty on the endogenous variables. At the same time, these simulations allow us to address questions about the
J. VallCs /Journal
of Economic Dynamics and Control 21 (1997) 1181-1198
1183
dynamic effects between the endogenous variables that solve the social planning problem and the prices that come from solving a decentralized equilibrium. The remainder of this paper is organized as follows. Section 2 describes the stochastic growth model with adjustments costs. Section 3 presents the simulation procedure and the results. The discussion on the cyclical and dynamic relation between variables, the importance of adjustment shocks and the effects of prices on investment are also included. Finally, the findings are summarized in Section 4.
2. A stochastic growth model with adjustment costs There is an infinitely alive representative maximizes its expected discounted utility:
Eo
,Eo /3’([c:‘-“J
consumer
in the economy
that
(1 - L,)“~]“-y’/(l - y)),
The utility function displays unit intratemporal elasticity of substitution between consumption and leisure. This elasticity is consistent with the fact that per capita series of labor have shown no significant trend in the US. The leisure share parameter 4 is affected by a random shock or,. Recently, Bencivenga (1992) has shown the importance of preference shocks, to account for the observed variability of consumption and labor when intertemporal effects are absent. The parameter y is the coefficient of risk aversion and b is the discount factor. The technology shows constant returns to scale with two inputs, labor (I,) and capital (K). Investment (I) is the part of the homogeneous output (Y) that is installed at some cost. Once investment is fixed as capital it implies a constant capital-labor ratio. There are two exogenous shocks that affect the technology: dZ, is a Solow residual shock to the quantity produced, whereas 13~~ represents a shock to the adjustment cost. The technology is then written as G(C,, I,, e,,) = (AC: +
m,,z:p
O
A>O,
I
e,,zc- 1Lj’ -=), B>O,
?rrl.
(2)
The production function is of the Cobb-Douglas form with 0:as the capital’s share of output. This corresponds with the observation of a constant capital and
1184
J. Vall&
/ Journal of Economic Dynamics and Control 21 (1997)
118I-1198
labor share of output in the United States since 1955. Both the production and the utility functions have been widely used in the business cycle literature to study fluctuations on aggregate variables (see, for example, Prescott, 1986; and King et al. 1988). The functional form G guarantees that the cost of installing gross investment is convex with a minimum value of zero when investment is zero. The adjustment cost parameter q measures the elasticity of substitution between consumption and investment goods. A value of q equal to one together with the condition that the parameters A and B are equal implies that there is no cost to installing one unit of the investment good. As the parameter q gets bigger the cost of installing one unit of investment good in terms of forgone consumption increases. Capital has a constant depreciation rate 6: K,=I,+(l-6)K,_,,
0<6<1
(3)
then investment at time t becomes productive one period later, at time t + 1. The technology is homogeneous of degree one. Therefore, the distribution of capital between firms is irrelevant (see, for example, Lucas and Prescott, 1971). The random vector O1= (e,,, &, O,,) is assumed to be stationary and identically distributed over time. The vector of random shock follows a lognormal first-order autoregressive distribution, i.e.: log e,, 1 = P log 8, + E,+1, &** N(0, C).
(4)
Usually, a highly persistent shock is necessary to match the optimal paths of a neoclassical growth model with the business cycle properties of actual data. This model allows for serial correlation in each of the shocks. Each element of P is denoted by pij. If contemporaneous correlation between the shocks is not allowed, both the matrix P and C are diagonal. The social planner’s problem is Max EO f /3’([C~‘-“~‘(l - L,)@l]“-y’/(l - y)) r=o s.t.
(AC; + Be,, I:)“” < OatKY- I L: K 1+1 loge,,,
c,,L,
=I,
+(I -6)fG =PiOge,+~+l,&,-N(~,C)
-=
Q't,
Q’t, Qt.
K*+1 and I, are the decision variables in each period t.
J. Vall& 1 Journal of Economic Dynamics and Control 21 (1997) 1181-1198
1185
The state variables of this problem are last period’s stock of capital K,- 1 and the vector of shocks 8,. There exists a solution to the social planner’s problem (5) given by a vector of stationary stochastic processes: K,, I = g(K,, e,), C, = c(L R), L = QK,, 6). Once the shadow price of capital is substituted out there are three first-order conditions in the maximization problem (5): the marginal rate of substitution between consumption and labor equal to its cost; the discounted value of the return on an additional unit of investment next period must be equal to its current cost in terms of consumption goods; and the last equation is the transversality condition on the capital stock. These three equations, together with the budget constraint, form a set of four equations with four unknowns. Given our assumed functional forms, an interior solution is obtained if the transversality condition and the above conditions are satisfied and the serial correlation matrix P has all its eigenvalues inside the unit circle (see, for example, Prescott and Mehra, 1980). When P is diagonal, this means that pij must be between 0 and 1. The nonstochastic steady-state solution for problem (5) gives the well-known positive relation between the interest rate of the economy and the marginal product of capital taking into account the adjustment cost for the new capital, i.e. B [( 1 - 6) + Fk/Gk] = 1. By the assumptions of F and G, total differentiation of this expression gives that increases in the discount factor increase the stationary level of capital and therefore of output and consumption. This result is also obtained when there are no adjustment costs to the growth model (Danthine and Donaldson, 1981). A decrease in the adjustment cost parameter, q, affects the steady state of the economy like a decrease in /?: the cost of capital increases and output and the stock of capital decreases. Nevertheless, capital decreases more than proportionately so that the economy is now more productive and therefore less capital is needed per unit of output. Since we are interested in the dynamic behavior of the price of investment and also of the rental price of capital, we make use of the property that the social optimum is also the solution for a sequence of market equilibrium allocations. The representative consumer sells its labor and rents its capital stock at the competitive prices. The representative firm maximizes profits for each period. Given the sequence of prices for labor and new capital the firm produces consumption and investment goods. Consequently, there are time invariant functions for wages w, = o(K,, O,), rental prices of capital R, = R(K,, 0,) and prices of investment P,, = P,(K,, et), for which the policy functions K(K,, e,), c(K,, 0,) and I(K,, 0,) solve the optimization problems of the household and of the firm operating in competitive markets. The assumed investment adjustment cost function G implies that in equilibrium, the shadow price of new capital goods (G,/Gc) is equal to the current price
1186
J. Vallhs /Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
of investment goods relative to produced consumption goods (P,).i This price will vary not only with exogenous changes in the adjustment cost shock 13~,,but also with the intertemporal effects represented by endogenous changes in the (ZJC,) ratio. Besides the changes in the structural parameters, changes in the uncertainty of the adjustment cost technology also has effects on the capital accumulation. Because the structural shocks are log-normally distributed, changes in the variances of the shocks’ innovations affect not only the variance of the shocks but also their mean. Hence, an increase in the variance of 6sr causes, as a consequence of an increment of uncertainty in the relative prices, an increase not only in the variance of the optimal paths (I&) and (KJY,) but also in their average value. Those level effects, though, are small compared, for example, with the changes generated by the adjustment cost parameter q. In equilibrium, investment is an increasing function of the relative price of new capital, as in Lucas (1967) and Uzawa (1969), because the function G is convex. In Lucas’ model, for example, the firm maximizes the discounted profit function for given sequences of output and factor prices. Nevertheless, in our model the investment good prices have to satisfy for each period the usual asset pricing equation derived from the consumer optimality conditions: the expected value of the intertemporal marginal rate of substitution times the gross yield of capital must be equal to one.
3. Simulation experiments The nonexistence of closed form solutions for the optimal allocation of policies makes it necessary to search for a numerical solution. Once the allocation of real variables is obtained, the sequences of prices are derived from the optimality conditions for the competitive economy. Here we make use of the ‘backsolving’ method (Sims, 1990) to find the solution of the problem in Section 2 (see Appendix). A property of the ‘backsolving’ method is that we use all the nonlinear restrictions without need to linearize them. The stochastic simulation algorithm draws values from the distribution of the expectation error of the only dynamic equation in the first-order conditions and from two of the innovations. Then, it uses the derived stability condition to find the third innovation, and finally using the first-order conditions, the decisions variables are solved. The sequences of prices are derived from the optimality conditions for the competitive economy. This simulation method is an approximation solution because it uses a restriction on the joint distribution of the exogenous shocks and the expectation error ’ Therefore, the ratio of these two variables, also called Tobin’s
t because it already includes the installation cost.
q,
is always equal to one at each
J. VallPs / Journal of Economic Dynamics and Control 21 (1997) 1181-l 198
1187
that comes from a linearized version of the model. Opposite to that, ‘Forward’ simulation methods draw values for all the structural disturbances and satisfy the stability conditions imposing restrictions on the decision rules (see Taylor and Uhlig, 1990 for a description of those simulation methods). Nevertheless, when we do not have a priori notions about the shape of the decision rules we can make restrictions about the distribution of the unobservable shocks 8,. Other empirical work that has used this ‘backward’ solution method (see, for example, Novales, 1992), have imposed a distribution on some function of the shocks like the Lagrangian multiplier or the marginal productivity of consumption. This way of solving avoids the use of linear approximations but has the disadvantage of assuming distributional properties of variables that have no structural interpretation. The model also has the property whereby, unlike other simulated stochastic growth models, the number of unobservables variables (6) is equal to the number of endogenous variables.* 3.1. Evaluation
of the adjustment
cost technology
As a means of evaluating the adjustment cost technology we compare some properties of the vector autoregressive representation (VAR) of actual and simulated data. In particular, we investigate for a range of parameter values of the adjustment cost shock variance (C,,) and of the elasticity of substitution (q), whether the impulse response and the variance decomposition of the estimated VAR for simulated data approximate those properties of the VAR for actual data.3 To obtain the simulated data from the model we fix the discount factor at 0.99 so that the model without adjustment costs displays an approximate annual 4% real interest rate at the stationary level. The leisure share parameter C#Jis 2/3, a standard value that emerges from the literature. The results presented here have assumed a capital share parameter u equal to l/3. The parameters of the adjustment cost function, A and B, are set equal to one so that the relative price of investment goods at the steady state is one when the parameter 4 is one. The capital depreciation parameter 6 is 0.025. We assume that the shocks are very persistent in order to generate the observed autocorrelation in the variables. Their serial correlation parameter is 0.99 and there is no cross correlation (i.e.
2As Ingram et al. (1992) have noted, that property ensures the nonsingularity of the variance-covariance matrix of the simulated data. Furthermore, the above characteristic would allow us to estimate eventually the parameters of the model. 3We will not attempt to develop a formal weighting function that would allow such a comparison (see, for example, Lee and Ingram, 1991).
1188
J. Vallh /Journal
of Economic Dynamics
and Control 21 (1997) 1181-1198
P and C are diagonal). Initially, we fixed the standard deviation of the productivity disturbance, CZ2, to be 0.8%. That is a value similar to the one selected by real business cycle models. The standard deviation of the preference disturbance, Z, 1, has been restricted to have lower variation, 0.2%. For the adjustment cost shock, Za3, we try a range of values from 0.1 to 6%. The estimated VAR corresponds to a system with the same number of variables as disturbances in the model: The system includes, in addition to a constant, the three endogenous variables, consumption, C,, labor, L,, and investment, I,. Because we expect consumption and labor to respond with some lag to exogenous changes in investment, we ordered the VAR with investment in the last position. The simulated data is obtained solving the model for 120 periods. The conditional response of system variables, for up to I2 periods ahead, are presented using a unit shock to the orthogonalized innovations. The responses of the variables are normalized dividing by the standard deviation of the corresponding innovation. The endogenous variables of the model are functions of current and past exogenous shocks. Although the optimal paths of the observed variables are stationary, the reason for the nonstationarity of the sample is the high serial correlation of all the shocks transmitted through the model mechanisms. Therefore, the comparisons are made with the first differences of the simulated and the real data. The actual data have 120 quarterly observations and correspond to the period 1961:3 to 1991:3 for the US economy.4 Investment includes all fixed investment on equipment and structures. Consumption is the result of adding the consumption series of nondurable goods and service goods. The labor series is the ratio of hours worked and time endowment (we assume 112 hours per week and 4.25 weeks a month). The sample average of this labor ratio is around 0.2 and measures hours worked by the employed and the unemployed. Fig. 1 shows the response of investment to the three innovations to the VAR with actual and simulated data. Those simulations are run with the parameter q equal to 1.8 and assuming risk neutral consumers, i.e. the parameter y equal to zero. We figure two simulated responses, one with a standard deviation of the adjustment cost disturbance equal to 1% and the other equal to 3% (this is almost four times the size of the technological disturbance used in the RBC models). The comparison of the responses imply that the model needs a large variation in the adjustment cost shock to obtain in the first periods, after an exogenous impulse in consumption, a small response of investment (first chart) and a high autocorrelation of investment (third chart), similar to the ones in actual data. However with a large size of the adjustment costs disturbance we cannot get the
‘They are from Citibank Economic Database and are measured in per capita terms.
J. Yaks
1.4
07
/Journal
of Economic Dynamics
1189
and Control 21 (1997) 1181-1198
Rk3pCi-S es of ktvesttnent to 0 Ccnscrrptlon lnxwatlm
,
-1\ I
\
I
Slnwloted Responses of Westment to 0 Ltlltx- IrrWatlal
I.t
I
0.0 -
Slm~lotedResponsesof investment to its own tmavotfm
I .*
0.7
0.0
-I .t I
2
3
4
6
6
,
8
9
10
II
12
Fig. 1. Notes: - The VAR system includes a constant and five lags of changes in consumption (MI), changes in labor (AI) and changes in investment (dlnu). Actual means that the VAR is estimated with US quarterly data (61:3-91:3). The other two responses correspond to simulated data with ,?& equal to 1% and 3%, respectively. - All responses are normalized by the standard error of the corresponding innovation.
1190
J. Vallks /Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
positive short-run response of investment to an innovation in labor (second chart). The fact that the exogenous disturbances follow an AR(l) makes the responses of the variables of the VAR system go to zero very quickly after the first period, independent of the size in the adjustment cost shock. That is why the model is not able to reproduce the persistent response of investment found in the data. The variance decomposition for the forecast errors in the VAR system shows that when CS3 is 3% the error of each variable in the long run (after 12 quarters) is explained mostly by its own variation. That is what comes out with actual data. We also find that for the grid of values of the adjustment cost shock, only when that value is 3% does investment have higher power to explain its own error. Lower values of the adjustment cost parameter (q = 1.2) imply negative responses of labor and investment to exogenous shocks in consumption, both in the short run and in the long run. It would also imply, a lower weight of investment to explain its own forecast error. Consequently with the above results, we choose as a benchmark model one in which the adjustment cost disturbance is 3% and the elasticity of substitution between consumption and investment is 1.8. 3.2. Business cycles properties We study the business cycles properties of the adjustment cost economy detrending the simulated and actual data with the Hodrick-Prescott filter (see Prescott, 1986). The actual value for Prt is the deflator for nonresidential investment. Table 1 evaluates the importance of the adjustment cost shock. The model, given the chosen structural parameters and the distribution properties of the exogenous shocks, needs a large variation in the distribution of the adjustment cost technology to obtain the observed cyclical variation in actual output. When Ca3 is as low as the preference shock, 0.2%, output volatility is only 1.2%, whereas when that variance is 3% or 4% we obtain an output variance close to the one in actual data.’ With adjustment costs and Z33 at 3%, the large amount of uncertainty in the model leads to statistics with large standard deviations. The consumption series has a standard deviation that is 0.8 times the standard deviation of output while
‘Note that our measure of output does not correspond with the one Accounts. In the simulations, output Y, is considered to be the outcome Even if output is measured as (C, + Pr, &) it would still differ from the and would show a close picture to output measured by the production
from the National Income of the production function. National Income Accounts function.
J. Vall& / Journal
of Economic Dynamics and Control 21 (I 997) f f Bf - if 98
Table I Business cycle statistics of the adjustment cost economy, Hodrick-Prescott
1191
filter
Simulated economy
US Economy 61:3, 91:3 (in per capita terms)
La3 = 0.01
2x3 = 0.03
z,3 = 0.04
6,.
0.016
0.012 (0.007)
0.014 (0.009)
0.017 (0.012)
QJ,.
0.50
0.83 (0.09)
0.86 (0.056)
0.89 (0.69)
0,
0.86
0.58 (0.34)
0.89 (0.71)
1.08 (0.74)
u&r
3.00
1.86 (1.00)
3.22 (2.93)
3.42 (3.25)
c
0.84
0.98 (0.04)
0.93 (0.24)
0.93 (0.23)
L
0.86
0.63 (0.43)
0.72 (0.38)
0.78 (0.38)
I
0.56
0.85 (0.28)
0.65 (0.99)
0.65 (1.20)
- 0.43
0.52 (0.70)
- 0.34 (1.49)
- 0.58 (0.87)
Correlation with Y
Corr (I, P)
Notes: The structural parameters used for the simulated data are: A = B = 1.0, n = 1.8,6 = 0.025, r#~= 2/3, /I = 0.99 and y = 0.0. The statistics correspond to the means for 20 simulations, each with 120 observations. The standard deviations of the statistics are in brackets.
in the real data it is half the output value. The fact that the preference shock affects both consumption and labor makes them move very closely, whereas in actual data they follow different patterns. 6 The investment series has a large volatility with respect to output, as in actual data, with large values of ZS3. The model is able to reproduce the negative correlation between investment goods and its own price found in the data. This regularity and the above cyclical properties appear also with risk aversion preferences (e.g. y = 1.0). Table 2 evaluates the cyclical properties of the adjustment cost technology. An economy with a low adjustment cost parameter (q) is seen as more productive since it has on average a lower stock of capital per unit of output. In Table 2
6 However the stochastic adjustment cost technology implies a lower variation in labor with respect to output than in a growth model with only a productivity shock. In fact, the empirical labor elasticity of output is below one (0.84) and close to the observed labor share parameter.
1192
J. Valks
/Journal
of Economic
Dynamics
and Control
21 (1997)
II8I-1198
Table 2 Business cycle properties of the adjustment cost technology, Hodrick-Prescott
filter
Economy with adjustment costs (Zs, = 0.03)
Economy without adjustment costs (n = 1.0, 11s3= 0.0)
‘1 = 1.8
z,, = 0.002
c,, = 0.0
?)= 1.2
0s
0.014 (0.009)
0.018 (0.010)
0.016 (0.009)
0.016 (0.008)
GJY
0.86 (0.56)
0.90 (1.72)
0.30 (0.15)
0.50 (0.40)
cl/es
0.89 (0.71)
1.08 (0.72)
0.73 (0.25)
0.92 (0.50)
U&J,
3.22 (2.93)
5.79 (7.80)
3.37 (0.48)
5.45 (3.06)
C
0.93 (0.24)
0.34 (1.06)
0.69 (0.16)
0.68 (0.16)
L
0.72 (0.38)
0.82 (0.25)
0.87 (0.17)
I
0.65 (0.99)
0.67 (0.97)
0.97 (0.02)
0.96 (0.04) 0.97 (0.02)
- 0.34 (1.49)
- 0.75 (0.42)
Correlation
with
Y
Corr (I, P,)
See footnotes to Table 1.
we see three possible costs in terms of the cyclical properties of the model. First, a reduction in q implies an increase in the volatility of all economic series. The same result appears in Basu (1987) in an adjustment cost model of asset pricing. Second, a technology with a low cost of substitution of consumption for investment, provides low comovement of C, and Y, and high relative volatility of I,, compared with actual data. Finally, to get the observed negative comovement of investment with its price the model needs not only large fluctuations of that price but also a high elasticity of substitution. The presence of an adjustment cost shock in this model offsets the relevance of the productivity shock for investment fluctuations.’ Other papers have also found the importance of other real shocks and the implied overstate of the
‘Since the three exogenous disturbances are the driving force of the model, we expect movements in investment to be fuly explained by those fluctuations.
J. Vall& / Journal
ofEconomicDynamics
and Control 2/ (1997) I IBI- I198
1193
Solow residual variability to account for example for the variability of US output (Burnside et al., 1993). However in this model, the large variability of the estimated adjustment cost shock gives a supply side explanation for the negative correlation between prices and quantities of investment goods. A positive exogenous shock to the investment goods market moves resources of the economy to that sector. The amount of investment goods increases with respect to the consumption goods production and the relative value of an extra unit of new capital falls. Therefore, the appearance of adjustment costs increases the amount of capital per unit of output in the economy. The shift in the supply of investment goods by the firms, caused by these exogenous shocks, stimulates an increase in investment and a fall in its price. 3.3. The dynamics of prices Finding good estimates for the rental price of capital has always been considered a difficult problem. In the standard jorgensonian investment equations the marginal product of capital is equal to the rental price. Then the desired stock of capital increases with a lower rental cost. The way that rental price is measured is imposing than the implied return of capital is equal to the one by an alternative asset. Therefore, the rental price is a function of a real interest rate, the depreciation rate and the shadow price of capital (see Abel, 1990). A model of aggregate investment like the one presented here can generate well justified equilibrium prices. The dynamics of the movements of those prices and their effect on capital accumulation is still an open question when both are endogenous and respond to exogenous shocks. In this section we present the comovements of the price of investment and the rental price of capital with aggregate investment through the simulation of the adjustment cost economy. From the dynamic first-order condition (A.l), we obtain that the rental price of capital in steady state, i.e. P,(l/b + 6), must be equal to the marginal productivity of capital, Fk, /Gc,. Outside the steady state we may define the rental price R, as the realized one period gross yield of capital, i.e. Gc,/Gi, [((1 - 6)Gi,+ 1 + Fk,. ,)/Gc,+ r J. The two charts of Fig. 2 show the responses to an impulse in the rental price and investment in a VAR system with those two variables plus the price of investment.’ In a general equilibrium model with adjustment costs the changes in investment no longer respond negatively to an unanticipated increment in the change of the rental price of capital (first chart, Fig. 2). That positive innovation in the
*A decomposition of the residuals in which prices of investment show some sluggishness, not responding initially to disturbances in investment, but with investment series responding immediately to movements in the relative prices seems to be in accordance with the observed dynamics in those two variables.
J. VaN& / Joumol of Economic Dynamics and Control 2/ (1997) 1181-1198 SlmtHOtedResmnses
of Plnv mci Inv to a Rental Price lmovotlon (R)
1 PI”” I””
.
.
.
6
.
I,
1
1
9
3
I
I
r---r II
a
a
.
Slmuloted Responses of R md Plnv to m Investment Imovotlm(lnv)
/\ --_, urn
-0.14
-
[:‘“-Et
0.28
I
3
6
7
3
II
Fig. 2. Notes: - The simulated VAR system includes changes in the rental price (AR), changes in the price of investment (Mine) and changes in investment (dlnv). The parameters are the same ones used in the 2nd column of Table 1. - All the responses are normalized by the standard error of the corresponding innovation
J. Y&k
/Journal
o,fEcunomic
Dynamics and Controro(2/ (1997) 1181-1198
1195
rental price of capital may be caused, for example, by a temporary increase in the capital income tax.g That should have an inmediate effect not only in relative prices but also in the goods allocation. The increase in R, creates more disposable income for the consumers and an upward shift in the demand for investment goods. The firms increase their costs but as new investment goods are purchased the relative price of investment goods declines. Although in steady state R, and PI, move together, outside the stationary solution both prices respond with an opposite sign to an exogenous impulse in investment (second chart, Fig. 2). An interpretation outside the model is that those exogenous innovations may create an upward pressure on the interest rates controlled by the monetary authority that pushes up the rental price. As a consequence, both investment and the rental price of capital will increase. The above two results give a theoretical explanation to the empirical evidence of a small or nonsignificant effect of the rental price in aggregate investment (see for example Gordon and Veitch (1986) for the US). In presence of different simultaneous shocks on the economy such negative effect disappears.
4. Conclusions To reproduce the observed volatility of aggregate investment and the negative correlation with its price in the US data a growth model with large variability in the adjustment costs technology has been constructed. The high cost of substitution of consumption for investment and the large adjustment cost shock variability offsets the importance of productivity shocks, usual in the real business cycle models. However, it gives a supply side explanation for the negative correlation between prices and quantities of investment goods. Such adjustment cost shocks need to have an extremaly large volatility, three to four times larger than the standard Solow residual innovations. The model gives also an explanation for the small effect of the rental price of capital on estimated investment equations found in the literature. An unanticipated innovation of the rental price of capital, when there are several shocks hitting the economy, produces, as an optimal response of the agents, movements of the same sign on investment.
Appendix. Simulation method This appendix describes the ‘backsolving’ method (Sims, 1990) to find the solution for the problem in Section 2. The first-order conditions, the technology ‘It would be a distortionary tax. The government will distribute the revenues in a lump-sum way to the consumers.
1196
J. Vallh
/Journal
of Economic Dynamics and Control 21 (1997)
1181-1198
constraint and the law of motion for the vector of shocks give a set of first-order stochastic difference equations. The only dynamic equation in the first-order conditions corresponds to the derivative with respect to the state variable K,, t: -
(UG/&) + PEt(Uct+~/Gct+l)((l - 4Gi,+, + F/c,)= 0
(A.11
where subindices refer to partial derivatives. Introducing an error term (‘expectation error’) r,rl+1 s.t. E, q,+ r = 0 in (A.l), we obtain a system of six nonlinear, stochastic difference equations in the following way:
(A-2)
H(z,+1,zt) = 5*+1 where z, = (X,, 0,) = (G L,, 5;+
1 =
(0,
0,
?t+
1, &It+
Ll,
1, h+
h,,
br,
1, &3t+
e,J
I).
All the solutions for the problem in Section 2 must satisfy (A.2). An additional equation is required in (A.2) to get a solution path for X,+ 1. There is a restriction on the distribution of [, + 1 that guarantees a stable solution for X,, 1, and uniqueness for given Be and Ze. The steps to solve the model are the following, 1. Linearize (A.2) around the steady state.
raft+
1 =
rl% + 5t+I
(A.3)
where .Zr= z, - z,,, z,, = (C,,, L,,, K,,, 1, 1, 1) is the steady-state vector, ro, rI are 6 x 6 matrices of coefficients. 2. Obtain the generalized eigenvalues and eigenvectors for (r,, r,). There are nonsingular matrices Q and 2 and diagonal matrices Ho, HI such that
rl = QH,Z,
(A.4)
J. Vallh /Journal
ojEconomic
Dynamics and Control 21 (1997) 1181-1198
where Z is the matrix of eigenvectors. The corresponding values 1 are
1197
vector of eigen-
,I(TO, r,) = (J. E Cldet(To, r,) = 0}, = R(H*, HI) = {;I = Hlii/Hoiij*
(A.3
Therefore (A.3) can be expressed as Q&Z5 + I = QHJ&
+ &+I
where
+ i [H,lH,]SH,'Q-'~,+l_, Z&+1 = [H;'H,]'Zz,, S=O
3 The stability of the linearized system is guaranteed when all the roots of that system are lower than the inverse of the discount rate. Since (A.6) has one eigenvalue Ajgreater than l/b for the given functional forms of F,G and U, to eliminate the possibility of solutions of (A.31 which grow at a rate faster than l/p we require the corresponding eigenvalues to be zero, i.e.: Zi.Zl = 0
vt
(A-7)
where Zj. is the row of eigenvectors associated to the eigenvalue Jj. Given the vector of observable variables X,, (A.7) gives an exact relation between E and q. If there were no eigenvalues greater than l/p this would imply that there are many possible solutions for the linearized system, since the mapping between q and E would not be unique. If more than one eigenvalue is greater than l/b the original problem has no solution. As long as we are interested in solutions for the nonlinear system around the steady state (i.e. for a small variance of the exogenous processes) we can use (A.7) as an additional equation to solve the system of first-order conditions (A.2). We can treat one of the elements of innovations &+ 1 as unknown, say {it+ 1, draw values for the remaining <,+ 1 elements and use (A.2), (A.7) and the initial conditions to obtain solution paths for X, + 1.
1198
J. Vallh / Journal of Economic Dynamics
and Control 21 (1997) 1181-1198
References Abel, A., 1990, Consumption and investment in: B. Friedman and F. Hahn, eds., Handbook of monetary economics (North-Holland, Amsterdam). Basu, P., 1987, An adjustment cost model of asset pricing, International Economic Review 28, 609-621. Bencivenga, V., 1992, An econometric study of hours and output variation with preference shocks, International Economic Review 33,449-471. Burnside, C., M. Eichenbaum, and S. Rebelo, 1993, Labour hoarding and the business cycle, Journal of Political Economy 101, 245-273. Danthine, J.P. and J.B. Donaldson, 1981, Stochastic properties of fast versus slow growing economies, Econometrica 49, 1007-1031. Gordon, R. and J. Veitch, 1986, Fixed investment in the American business cycle, 1919-83 in: R. Gordon, ed., The American business cycle, continuity or change (University of Chicago Press, Chicago). Ingram, B.F., N. Kocherlakota, and N. Savin, 1992, Explaining business cycles: a multiple shock approach, Mimeo (Department of Economics, University of Iowa). King, R., C. Plosser, and S. Rebelo, 1988, Production, growth and business cycles 1: The basic neoclassical model, Journal of Monetary Economics 21, 195-232. Kydland, F.E. and EC. Prescott, 1982, Time-to-build and aggregate fluctuations, Econometrica 50, 1345-1370. Lee, B. and B. Ingram, 1991, Simulation estimation of time series models, Journal of Econometrics 47, 197-205. Lucas, R., 1967, Adjustment costs and the theory of supply, Journal of Political Economy 75, 321-334. Lucas, R. and E.C. Prescott, 1971, Investment under uncertainty, Econometrica 39, 659-681. Novales, A., 1992, Solving nonlinear rational expectations models: a stochastic model of interest rates, Econometrica 58, 93-l 11. Prescott, E.C., 1986, Theory ahead of business cycle measurement, Quarterly Review, Fall, 9-21, Federal Reserve Bank of Minneapolis. Prescott, E.C. and R. Mehra, 1980, Recursive competitive equilibrium: the case of homogeneous households, Econometrica 48, 1365-1379. Taylor, J.B. and H. Uhlig, 1990, Solving nonlinear stochastic growth models: a comparison of alternative solution methods, Journal of Business and Economics Statistics 8, l-50. Sims, CA., 1990, Solving nonlinear stochastic optimization and equilibrium models backwards, Discussion Paper 15 (Institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis and University of Minnesota). Uzawa, H., 1969, Time preference and the Penrose effect in a two-class mode1 of economic growth, Journal of Political Economy 77, 628-652.