Fickle Consumers, Durable Goods, and Business Cycles

Journal of Economic Theory  ET2413 journal of economic theory 81, 3757 (1998) article no. ET982413

Fickle Consumers, Durable Goods, and Business Cycles Mark Weder Department of Economics, Humboldt University Berlin, 10178 Berlin, Germany Received February 14, 1998

This paper develops a three sector dynamic general equilibrium model with the specific aim of evaluating the role of consumer durables in the generation and propagation of business cycles. The model displays indeterminacy at commonly considered realistic parameter restrictions and basically constant returns to scale. If calibrated at modest increasing returns to scale, the sunspots driven model version produces a close match to moments of aggregate U.S. data. Journal of Economic Literature Classification Numbers: E00, E32.  1998 Academic Press

1. INTRODUCTION There are undisputably various mechanisms that play a role in producing business cycles. The aim of this paper is to study the role of durable goods and sunspots in the generation and propagation of economic fluctuations.1 Specifically the objective is to explore whether and if, then to what extent, exogenous shifts in spirits drive economic growth. Changes in consumer confidence are important leading indicators in (commercial) business cycle forecasting. Kantona [17] points out that these variations can be only partially explained by available economic statistics. The primary channel through which changes in consumer sentiment may affect aggregate spending is by the purchase of consumer durables. The Survey Research Center at the University of Michigan poses (among others) the following questions for their consumer attitude index: ``Now about big things people buy for their houseI mean furniture, house furnishings, a refrigerator, a stove, a TV, and 1 The terms sunspots and animal spirits will denote the same concept in this work. In line with Cass and Shell [9], Farmer [12] and others, it describes equilibrium situations that depend on extraneous uncertainty only because agents believe it to be so.

37 0022-053198 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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MARK WEDER

things like that. Do you think now is a good or bad time to buy such large items?'' ``Do you expect to buy or build a house for your own use during the next twelve months?'' ``Do you expect to buy a car during the next twelve months? Does anyone else in the family expect to buy a car during the next twelve months?'' Purchases of households' durables are often discretionary in that they can be postponed if economic conditions are not favorable and vice versa. If consumer confidence worsens, expenditures on durables are likely to bear the brunt of the dip. While consumption on nondurables is rather smooth, households' expenditures on durable goods account for a disproportionate share of output fluctuations. It is argued in this paper that a theory of the business cycle which places animal spirits at the center of explaining economic fluctuations should not neglect durable goods in the analysis. Changes in consumer spirits may spur spending drops or boosts in durables which push the economy into self-fulfilling alternating periods of contractions and expansions. Moreover, Matsusaka and Sbordone [20] report that the hypothesis that consumer attitude Granger-causes GNP cannot be rejected. They claim that roughly 20 0 of U.S. GNP innovations are the result of upward and downward impulses of consumer bullishness. Table I reports cyclical behavior for several variables of the U.S. economy. All time series are quarterly and are linear detrended. The table summarizes stylized business cycle facts. Consumption of nondurables, being by far the largest component of aggregate output, is a relatively smooth series. The bulk of the volatility in GNP is due to investment expenditures and household investment in consumer durables. All variables are procyclical and are highly autocorrelated. Finally, the contemporaneous correlation of hours and productivity is essentially zero. This paper attempts to construct a theoretical business cycle model that takes these stylized facts into account. It allows for extraneous noise to come through the channel of consumer durables in generating cycles. The model is inherently a dynamic general equilibrium model, however, it features several modifications to the standard version. Baxter [3] is a related general equilibrium approach to explain these facts. However, her model assumes complete markets and perfect competition, thereby excluding sunspots from having a potential role in explaining cycles. The present model is a three sector variant of the Real Business Cycle model. The novel aspect is the feature of durable goods which can be used either for consumption or investment purposes. Production occurs in two

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DURABLE GOODS AND BUSINESS CYCLES TABLE I a Cyclical Behavior of U.S. Output and Production Inputs Quarterly, 1948:I1986:IV Relative volatility

Variable A Output Output Output Output Output Output Output Hours 2

B Output Nondurable consumption Durable consumption Investment Hours 1 Hours 2 Productivity Productivity

Correlation of A(t) and B(t&j ) with j=

_ B _ A

1

0

&1

1.00 0.69 1.97 1.35 0.52 0.69 1.00 1.45

0.96 0.82 0.45 0.59 0.07 0.25 0.78 &0.38

1.00 0.85 0.39 0.60 0.07 0.34 0.76 &0.34

0.96 0.84 0.32 0.57 0.06 0.37 0.70 &0.26

a Table taken from King, Plosser and Rebelo [18]. Moments of durables, hours 2, and productivity are from Canova [8].

stages. In the first stage, differentiated intermediate goods are created by a number of monopolistic competitors. Production takes place with an increasing returns to scale technology. In the second stage, intermediate goods are welded together by perfectly competitive final goods sectors that operate under constant returns. Durable consumption and investment goods are produced with the same intermediate inputs. However, every monopolistic supplier of these goods faces a nonconstant markup, depending on the relative shares in demand. The monopolist cannot price discriminate on the respective demands thus the markup varies with the composition of demand. It will be shown that the equilibrium of the economy is indeterminate at close to constant returns to scale. In the case of indeterminacy there is an infinite number of paths that converge to equilibrium. Random noise, which is unrelated to the model's fundamentals, can cause economic fluctuations. If calibrated at modest increasing returns to scale, the sunspot driven model produces a close match to moments of aggregate U.S. data. The paper is organized as follows: Section 2 presents the theoretical model. The equilibrium dynamics are developed in Section 3. Section 4 concludes the paper.

2. THE MODEL This section introduces the theoretical model. The household sector is considered first.

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2.1. Preferences The economy consists of one representative household-agent. At date 0 the agent's preferences are: E

_



} &

: ; tU(C t , S t , l t ) I0 t=0

; # (0, 1),

(1)

where C t denotes consumption, S t , the stock of durable goods, l t leisure and ; the discount factor. 2 I0 is the set of information that is available to the household at 0. E[ . | I0 ] is the expectations operator conditional on I0 . The household is endowed with one unit of time which she can either use for work L t or for leisure: 1=L t +l t . A specific functional form for instantaneous utility is assumed as U(C t , S t , l t )=

1 1 C* 1&o + (1&L t ) 1&/ 1&o t 1&/

o{1

or U(C t , S t , l t )=log C *+ t

1 (1&L t ) 1&/ 1&/

o=1.

The consumption bundle C * t is given by a CES-aggregator * * 1* C* t #[C t +(`S t ) ]

with *1, /0, o>0, and `>0. 1o is the elasticity of intertemporal substitution of the consumption bundle and 1/ is the elasticity of intertemporal substitution of leisure. There are positive initial endowments of capital K 0 and of the durable stock S 0 . The consumer's capital holdings evolve as K t+1 =(1&$ K ) K t +I t

$ K # [0, 1].

(2)

K t denotes the stock of capital and I t denotes investment. $ K is the rate of depreciation. Similarily the stock of durable consumption goods is augmented according to S t+1 =(1&$ S ) S t +D t

$ S # [0, 1]

(3)

2 Preferences are defined over the flow of services of the durables stock. However, by using an appropriate normalization, the above formulation results.

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DURABLE GOODS AND BUSINESS CYCLES

41

where D t refers to the investment in new durable consumption goods and $ S is the rate at which the durable stock depreciates. Installation requires one period for both stocks. The period-by-period budget constraint of the household is given by w t L t +q t K t +6 t C t + p t D t + p t I t

(4)

where the right-hand side is the agent's spending. Here p t is the price of both final durable goods. 3 The left-hand side of (4) indicates income. The household owns the stock of capital and rents its services to the firms at the rental price q t . Labor services are sold at w t . Both factor prices are taken as given for the household as factor markets are assumed to be perfectly competitive. Furthermore the household receives profit income from all firms, 6 t . Profits are taken as given to the household. At t=0 the agent chooses sequences of nondurables and durables consumption and leisure to maximize (1) subject to (2) through (4). The Bellman equation for the household is given by 4 v(K t , S t , Z t )=

max Ct , Lt , Kt+1 , St+1

u(C t , S t , l t )+;E[v(K t+1 , S t+1 , Z t+1 | It ] (5)

subject to (2), (3), and (4) and the initial conditions of the two stocks. Define *t as the current value Lagrange multiplier associated with the household's budget constraint. The first-order conditions entail &* t =0 [C *t +(`S t ) * ] (1&o)*&1 C *&1 t

(6)

&(1&L t ) &/ +* t w t =0

(7)

* t p t =;E[* t+1 q t+1 +(1&$ K ) * t+1 p t+1 ] | It ]

(8)

* t p t =;E[(1&$ S ) * t+1 p t+1 +[C *t+1 +(`S t+1 ) * ] (1&o)*&1 ` *S *&1 t+1 | It ] (9) in addition to the household's budget constraint and the usual transversality conditions. Equations (6) and (7) describe the households consumption leisure trade-off. Euler equation (8) gives the relation between consuming nondurables and savings in capital. Equation (9), which describes optimal consumption of durables, can be interpreted as follows: Suppose a unit decrease of the household's investment in durable consumption goods at t. This amount is now available for nondurable consumption. Therefore, the 3

The formulation already considers the symmetric case that will be derived later. C t is the numeraire. 4 Since, in addition to the above assumptions, the state of technology Z t follows a simple first order autoregressive process, the unique solution of the problem exists.

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MARK WEDER

utility changes in this period by u C (C t , S t , l t ) p t . However, the next period's utility rises at u S (C t+1 , S t+1 , l t ). Moreover, there is an additional indirect effect on utility. As a consequence of the lower investments, the future utility decreases due to the depreciation of the stock of durables by (1&$ S ) ;u C (C t+1 , S t+1 , l t+1 ) p t+1 . Along the optimal path small reallocations must leave welfare unchanged.

2.2. Technology There are three final goods in the economy. Investment goods and durable consumption goods are produced in the same sector. Final goods are produced through the use of a range of differentiated intermediate inputs. 2.2.1. Nondurable Consumption Goods Sector A perfectly competitive sector produces the final consumption goods according to a constant returns technology given by Ct =

\|

1

0

x vt, j dj

+

1v

v # (0, 1)

(10)

where x t, j is the quantity of intermediate input j used in production of the final consumption good. The number of intermediates is fixed and normalized to one. 1(1&v) is the elasticity of substitution between any two intermediate goods. The associated cost function for a competitive firm is C(C t , p t, c )=C t

_|

t 0

p v(v&1) dj t, c, j

&

(v&1)v

(11)

with the exact price index p t, c . The prices of the intermediate goods are taken as given. A set of suppliers that are monopoly producers of their own intermediate good have access to the following technology ' x t, j =Z t[k :t, j l 1&: t, j ] &,

: # (0, 1),

'>0,

,0,

(12)

where Z t is the economy wide total factor productivity, k t, j capital input, l t, j labor input and , overhead costs of firm j. Each firm takes factor prices as given. Technology follows log Z t+1 =\ log Z t +(1&\) log Z+z t+1

\ # (&1, 1)

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43

DURABLE GOODS AND BUSINESS CYCLES

with Z#1, Z 0 given and z t+1 is N(0, _ 2z ). The cost function associated with this technology is given by C(q t , w t , x t, j , Z t )=Aq :t w 1&: t

\

x t, j +, Zt

+

1'

.

(14)

Intermediate suppliers face a constant elasticity of demand which can be deduced from the conditional demand function x Ct, j =

\

p t, c, j p t, c

+

1(v&1)

Ct .

(15)

Profit maximization requires that each monopolistic firm sets its price p t, c, such that max 6 ct, j = p t, c, j x( p t, c, j )&C(q t , w t , x( p t, c, j ), Z t ).

j

(16)

pt, c, j

This results in 1 p t, c, j = . C(q t , w t , x t, j , Z t )x t, j v

(17)

The last equation is the standard monopoly pricing policy of constant markup equals prices over marginal cost. v is the inverse of the markup. Due to the lack of evidence of significant pure profits in most industrial economies, it is further assumed that pure profits average to zero in the model. This will be the case in both sectors of the economy. In this case the inverse of v is also the extent of increasing returns to scale in this sector. 5 2.2.2. Durable Goods Sector Final durable goods are supplied by perfect competitors. One group of firms produces consumption durables and the other is engaged in the production of investment goods. Both final goods industries operate with a constant returns technology using the same set of intermediate goods. Final durable investment goods are produced according to It=

\|

1

0

x %t, i di

+

1%

% # (0, 1).

(18)

5 See Weder [25] for a short discussion of the connection between markups, zero profits and returns to scale under zero-profit assumption.

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MARK WEDER

Final durable consumption goods technology is given by Dt =

\|

1

0

x _t, i di

+

1_

_ # (0, 1).

(19)

From the last two equations, the following demand functions for each intermediate good x t, i can be deduced x It, i =

p 1(%&1) It t, x, i 1 %(1&%) ( 0 p t, x, { d{) 1{

(20)

xD t, i =

Dt p 1(_&1) t, x, i 1 _(1&_) ( 0 p t, x, { d{) 1_

(21)

and

Intermediate goods are supplied by monopolistic competitive firms. Each supplier produces one differentiated good, x t, i . Intermediate firms are not able to price discriminate between investment and (durable) consumption demand from the final goods sector, as is also the case in Gali [13]. Market clearing at the firm level requires that I x t, i =x D t, i +x t, i

(22)

holds. Intermediate firms technology is given as # x t, i =Z t[k :t, i l 1&: t, i ] &

#>0 0

(23)

which is analogous to the nondurables sector's case. Intermediate firms are price takers on factor markets. Profit maximization together with the nonprice discrimination assumption imply a nonconstant markup: 1 x It, i 1 xD t, i + p t, x, i _&1 x t, i %&1 x t, i #1+ t . = C(q t , w t , x t, i , Z t ) 1 xD 1 x It, i t, i + +1 x t, i _&1 x t, i %&1 x t, i

(24)

It can be easily checked that if both % and _ approach unity, so does + t . The inverse of + t measures the returns to scale in the durable goods sector. 2.3. Symmetric Equilibrium The model is analyzed in symmetric equilibrium which is given by the following equations Market clearing in nondurables requires that I xD t, i +x t, i =x t, i =x t

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DURABLE GOODS AND BUSINESS CYCLES

45

or alternatively that x It, i x It It It = = = #r t x t, i x t D t +I t X t

(25)

holds. Market power in the durable sector is measured by 1 1 (1&r t )+ r t +1 _&1 %&1 . +t = 1 1 (1&r t )+ r _&1 %&1 t

(26)

Profit maximization requires that prices of final durable goods equal unit costs. Since input prices are the same for both durables' final industries, the prices of final durable goods must be the same. Without loss of generality, the price of nondurables is normalized to unity. In symmetric equilibrium: p t, c, j #1 \j

and

p t, x, i # p x, t \i.

(27)

It is assumed that households can move their input factors between sectors freely and without costs so that the wage rate and the return to capital are equalized. Factor market clearing implies # ' L (1&:) =:'vZ t K :'&1 L t,(1&:) q t =:#+ t p t, x Z t K :#&1 t, x t, x t, c c

(28)

and (1&:) #&1 (1&:) '&1 =(1&:) 'vZ t K :' w t =(1&:) #+ t p t, x Z t K :# t, x L t, x t, c L t, c

(29)

to hold. Due to the assumption on factor intensities, factor proportions will be the same in both intermediate sectors: K t, x L t, x = #} t Kt Lt

and

K t, c L t, c = #1&} t Kt Lt

(30)

where we denote the fraction of resources absorbed by the durables sector by } t . The price of durable goods can be derived as p t, x =

} 1&# v' t (K : L 1&: ) '&#. + t # (1&} t ) 1&' t t

(31)

p t, x can be interpreted as the relative price and will be henceforth denoted by p t . The production in the two sectors of the economy is given by (1&:) # & X t =} #t Z t K :# t Lt

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MARK WEDER

and (1&:) ' &,. C t =(1&} t ) ' Z t K :' t Lt

(33)

Finally, aggregate ``nominal'' output is given by w t L t +q t K t +6 t =Y t =C t + p t D t + p t I t .

(34)

A measure of ``real'' output would set the relative price to its constant steady state value. The model is completely specified by this collection of equations. To solve for the dynamics, it is necessary to derive an approximated version. This will be done in the next section.

3. THE EQUILIBRIUM DYNAMICS This section will examine the dynamics of the model. The steady state, that is, the long run behavior of the model, is considered first. 3.1. The Steady State The deep parameters (or primitives) of the model are summarized in Table II. The steady state of the economy is derived as follows. Kydland and Prescott [19] report an average ratio of 0.64 for overall investment expenditures to consumption expenditures on durables for the U.S. economy over the 19541989 time-span. Therefore this ratio is set in the model at r=0.64.

(35)

In line with King, Plosser, and Rebelo [18], it is assumed that the portion of time endowment that is spent working is 20 0: L=0.20.

(36)

The markup equation is determined by primitives 1 1 (1&r)+ r+1 _&1 %&1 . += 1 1 (1&r)+ r _&1 %&1

(37)

The two transition equations for the stocks have their steady state version in the form of $ K K=I

and

$ S S=D.

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47

DURABLE GOODS AND BUSINESS CYCLES TABLE II Deep Parameters r

;

v

_

%

'

#

$S

$K

L

:

/

o

*

The first Euler equation implies ; &1 =qp &1 +1&$ K

(39)

in steady state. Durable goods output is given by (1&:) # (1&:) # &=+#K :# =+#} #K :#L (1&:) #. X=K :# x Lx x Lx

The rental price of capital in steady state entails q=:+#p} #&1K :#&1L (1&:) #.

(40)

From these equations it is possible to get the fraction of resources that is absorbed by the durables sector } as }=

r( ;

&1

:$ K . +$ K &1)

(41)

The steady state capital stock is then determined by # (1&:) # 1(1&:#) ) . K=(r+#$ &1 K } L

(42)

The relative price of durable goods becomes p=

} 1&# v' (K :L 1&: ) '&#. +# (1&}) 1&'

(43)

After having determinated }, K and p from deep parameters, all remaining steady state variables can be computed. Furthermore, since it is assumed that profits average to zero, 6 c =6 x =6=0

(44)

holds as well. Finally, from the household's first order conditions (6) and (9), the constant ` can be computed as `=

_

pC *&1( ; &1 &(1&$ S )) S *&1

&

1*

.

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3.2. Dynamics The model is solved by deriving a log linear approximation of equations. The system reduces to a stochastic matrix difference equation of the form E[* t+1 | It ]

* t

_& _ & S t p^ t K t Z t

=J

E[S t+1 | It ] E[ p^ t+1 | It ] E[K t+1 | It ]

.

(46)

E[Z t+1 | It ]

Since * t+1 and p^ t+1 are the only endogenous nonpredetermined variables, and given that technology is a stationary process, indeterminacy requires that at most one eigenvalue of J &1 is outside the unit circle. In this case, a multiplicity of equilibria arises: the economy's fundamentals are not sufficient to determine one unique solution. The model may provide an example of an economy in which fluctuations are driven by nonfundamental random variables, i.e., sunspots. 3.3. Sunspot Zones This subsection discusses the parameter combinations in which the model exhibits indeterminacy. Table III summarizes the calibrated parameters of the model version that will be fixed. The economy's parameters are set at values that are standard in the Real Business Cycle literature. The calibration implies constant marginal costs in both sectors. The figures for the respective rates of depreciation are taken from Baxter [3]. The same holds for the calibration of *. Baxter's justification for the choice of * is to generate realistic comovements. Our value of * corresponds to the lower end of her specification. The choice of the discount factor entails a rate of interest (measured as q&$ K ) of 5.5 0 per annum. The : value implies a capital share of 30 0. Further, it is assumed that the sector which is producing the nondurables is perfectly competitive. 6 This is consistent with evidence reported by Basu and Fernald [2] and Harrison [15]. Estimates of the elasticity of substitution lie in the range 0.025. The specification of o=1 implies log utility. Hansen's [14] indivisible labor model is used. That is, all employment fluctuations occur at the extensive as opposed to the intensive margin (/=0). Figure 1 shows the combinations for which J &1 has three or four eigenvalues inside the unit circle while noting that \ # (0, 1). If % is sufficiently smaller than _, four eigenvalues of J &1 are smaller than one in absolute 6

To avoid divisions by zero, set v=0.9999.

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DURABLE GOODS AND BUSINESS CYCLES

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TABLE III Deep Parameters r

;

'

#

$S

0.64

0.988

1.00

1.00

0.039

$K

L

:

o

*

0.017

0.20

0.30

0.75

&1

value (shaded area). It appears that changes in the markup are responsible in generating indeterminacy. Moreover, the implied returns to scale must only be very low: a _&% combination of (0.99, 0.945) corresponds to scale economies of 1.022 in the intermediate durables sector. This value is even lower than the one reported in Benhabib and Farmer's [4] two sectoral model. Moreover, the scale economies must only be present in one sector of the economy. The average returns to scale in the economy are even lower. In the above mentioned case, the sector-weighted measure amounts to a mere 1.006. The type of an asymmetric sectoral distribution of returns to scale is in line with Basu and Fernald's [2] evidence for the U.S. industry. Harrison [15] reports related empirical (and theoretical) results. Therefore, the findings in this paper are appealing not only on theoretical grounds. The requirement %<_, which is only present in the above case

FIGURE 1

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MARK WEDER

can be rationalized insofar as that the production of industrial final goods demands a more sophisticated technology. It can be shown that by allowing for a decreasing marginal costs curve, the required extent of scale economies can be reduced even further. In Fig. 2 the above exercise is repeated, however, #=1+ is set such to allow for diminishing marginal costs. A _&% combination of (0.99, 0.96), encompasses returns to scale in the durable sector of 1.019 while preserving indeterminacy. Economy-wide increasing returns entail 1.004 in this case. Indeterminacy can now also arise for cases in which %>_ while allowing somewhat higher returns to scale. Assuming scale economies in the durable sector of about 1.08 (i.e., _=%=0.92), allows to abandon the variable markup case altogether and indeterminacy results from the same mechanism as in Benhabib and Farmer [5] or Weder [25]. The darker area in Fig. 2 indicates the region in which all eigenvalues of J &1 are inside the unit circle. The consumption share on nondurables is 0.69 in the U.S. economy (excluding the government sector, see Kydland and Prescott, [19]). With the above assumptions and modest returns to scale, the model's share entails 0.67. The model's investment and durable consumption shares of output are 21 and 12 0, respectively. Both of these ratios match those in U.S. data: The respective values are 19 0 for fixed investments and 11 0 for consumption durables (see again Kydland and Prescott, [19]). The capital output ratio is 3.01 in the model. This is close to the value of 3.32 that has been reported by Cooley and Prescott [11]. The steady state behavior of the model is very robust to changes in market structure and

FIGURE 2

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DURABLE GOODS AND BUSINESS CYCLES

51

returns to scale. Consequently, Salyer's [23] critique on the fallacious long run behavior of the BenhabibFarmer [6] indeterminacy model is not applicable to the present three sector model. 3.3.1. Interpretation Overall the defining characteristic of the model is that indeterminacy is possible at levels of market imperfections which are basically negligible. Yet, how does the model generate indeterminacy? Three factors contribute interdependently to this case. The work by Benhabib and Nishimura [7] and others show that multisector growth models display indeterminacy at relatively low levels of returns to scale. Harrison [15] and Weder [25] point out that it is only the returns to scale in the investment goods sector that matter for indeterminacy to occur. Gali [13] demonstrates that nonconstant markups can also be responsible for this behavior: The returns to capital can be a decreasing function of output.7 Both of these features are present in the model. The original element which limits the increasing returns requirement of this model, however, is the presence of two consumption goods. The essential difference between the demand of durables and nondurable consumption goods is that the intertemporal price of the durables enters the household's decision making. Its price path will affect the demand. The relevant intertemporal margin is the user costs (or the rental equivalent price). Assume that the household expects the user costs' time path to be upward sloping. She henceforth increases investment expenditures in durable consumption goods S t to reap capital gains. If the markup is declining as a result of this change in the composition of demand (e.g., %<_), the relative price p t declines.8 The rental equivalent price path can become increasing and expectations are self-fulfilling. Indeterminacy requires % to be strictly smaller than _ as long as marginal costs are constant (i.e., #=1). Baxter [4] reports a slightly negative correlation between output and the relative price for the United States. Therefore, booms could be explained by the above argument. Moreover, changes in expenditures on household durables could be a decisive factor in generating recessions as well. 3.4. Model Moments This section presents the dynamic properties of several versions of the model developed in this paper. Table IV summarizes the calibrated 7

The mentioned dependency of the model's indeterminacy on the slope of the marginal cost curve supports this fact. 8 To see this, check Eq. (35) and set #='=1. If the firms's cost function is decreasing in output, for example #>1, the effect is simply enforced. If #='>1, an activity shift towards the durable sector is also associated with a lower relative price. However, if #='<1 or #{', analytical solutions are no longer available.

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52

MARK WEDER TABLE IV Deep Parameters r

;

v

_

%

'

#

0.64

0.988

0.9999

0.975

0.90

1.00

1.00

$S

$K

L

:

/

o

*

0.039

0.017

0.30

0.00

0.75

&1

0.20

parameters of the model version that will be simulated first. (See also Table V.) The assumption on / corresponds to the case of infinite labor supply elasticity as in Hansen [14]. The specification o=0.75 implies that agents are somewhat less risk averse as in the log case. The calibrated numbers for % and _ yield returns to scale in the durables sector of 1.05 and economywide returns of scale of slightly more than 1.01. The model is driven by white noise sunspot shocks. In line with King, Plosser, and Rebelo [18], Benhabib and Farmer [5] and others, unfiltered statistics are considered. The series correspond to ``real'' outputs. If statistics for U.S. data (Table I) are compared to statistics for the sunspot model economy, the numbers suggest that the model cannot account for the observed variability of the aggregates. Consumption on nondurables is much too volatile and households' expenditures on durables exhibit more variance than investment. The crossmovements display only limited resemblence to data. In particular, nondurable consumption exhibits a puzzling dip at the zero correlation. However, Huffman and Wynne [16] show that the failure of explaining crossmovements is not restricted to multisectoral models with multiple equilibria. Multisector TABLE V Cyclical Behavior of Multiple Equilibria Model Relative volatility

Variable A Output Output Output Output Output Output Hours

B Output Nondurable consumption Durable consumption Investment Hours Productivity Productivity

Correlation of A(t) and B(t&j ) with j=

_ B _ A

1

0

&1

1.00 1.79 10.33 7.79 1.28 0.28 0.23

&0.22 0.36 &0.11 &0.42 &0.24 0.30 0.29

1.00 &0.97 0.97 0.94 0.99 &0.97 &0.98

&0.22 0.15 &0.30 &0.09 &0.23 0.23 0.24

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DURABLE GOODS AND BUSINESS CYCLES

versions of the Real Business Cycle model often exhibit the same peculiar characteristics. The demand driven constant returns version of the Baxter and King [4] one sector Real Business Cycle model also predicts incorrect crosscorrelations. Finally, the table shows that the autocorrelations of all variables appear to be far too low. To circumvent these problems, a higher degree of returns to scale is assumed (Table VI). In particular, the parameterization of %=0.80 and _=0.90 implies returns to scale in the durable sector of 1.17. Economywide scale economies are also still very low at 1.04. These figures are not outright unrealistic when considering recent empirical work. The prediction of the model now improves dramatically. All output components are procyclical. The model possesses a strong internal propagation mechanism: output is highly autocorrelated even though the model is driven by white noise shocks. The pronounced shortcoming of the multiple equilibrium version remains consumption of nondurables, which is still too volatile and observes too little correlation with output. To alter this, the presence of slightly higher scale economies are considered (Table VII). _=0.90 and %=0.725 imply returns to scale in the durable sector of 1.20 and economy-wide returns to scale of 1.05. The model is similar to the simulated Benhabib and Farmer [5] two-sector model in its size of the returns to scale. The model performs even better at matching contemporaneous covariances. To consider a more noncommittal version of the model, the above economy is driven by stochastic technology as well (Table VIII). Both shocks are uncorrelated and of equal size. The technology parameter value \ is set at 0.95. The model's predictions are again similar to the reported stylized facts. The correlation of nondurable consumption and output increases in the presence of technology shocks. However, this improvement coincides with a worsened performance along the labor market. TABLE VI Cyclical Behavior of Multiple Equilibria Model Relative volatility

Variable A Output Output Output Output Output Output Hours

B Output Nondurable consumption Durable consumption Investment Hours Productivity Productivity

Correlation of A(t) and B(t&j ) with j=

_ B _ A

1

0

&1

1.00 0.98 3.60 5.17 0.60 0.58 0.98

0.83 0.53 0.86 0.16 0.48 0.93 0.58

1.00 0.26 0.78 0.60 0.85 0.83 0.41

0.83 0.36 0.81 0.32 0.58 0.82 0.41

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54

MARK WEDER TABLE VII Cyclical Behavior of Multiple Equilibria Model Relative volatility

Variable A Output Output Output Output Output Output Hours

B Output Nondurable consumption Durable consumption Investment Hours Productivity Productivity

Correlation of A(t) and B(t&j) with j=

_ B _ A

1

0

&1

1.00 0.94 2.99 4.63 0.60 0.59 0.99

0.81 0.66 0.83 0.16 0.41 0.93 0.59

1.00 0.35 0.88 0.62 0.84 0.84 0.41

0.81 0.48 0.80 0.29 0.53 0.82 0.41

Table IX reports a version of the model in which the elasticity of substitution in the consumption bundle is lower: *=&10. The remaining parameters are set as in Table VI, that is, the returns to scale in durables productions are 1.17. The model is driven by animal spirits shocks only. For this version of the model the relative volatilities are very similar to what is observed in U.S. data. However, the greatest strength of the multisector multiple equilibrium model is found along the labor market. First, hours become relatively volatile. Second, the contemporaneous correlation of hours and productivity is essentially zero. It is therefore relatively uncomplicated to solve the DunlopTarshispuzzle. Common Real Business Cycle models are generally unable to generate similar crossmovements unless technology shocks are complemented by alternative (demand) disturbances as in the McGrattan [21] model with government. TABLE VIII Cyclical Behavior of Multiple Equilibria Model Relative volatility

Variable A Output Output Output Output Output Output Hours

B Output Nondurable consumption Durable consumption Investment Hours Productivity Productivity

Correlation of A(t) and B(t&j) with j=

_ B _ A

1

0

&1

1.00 0.98 2.45 3.75 0.50 0.68 1.36

0.90 0.75 0.77 0.29 0.54 0.93 0.56

1.00 0.60 0.79 0.58 0.79 0.90 0.44

0.90 0.71 0.80 0.34 0.56 0.92 0.53

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55

DURABLE GOODS AND BUSINESS CYCLES TABLE IX Cyclical Behavior of Multiple Equilibria Model Relative volatility

Variable A Output Output Output Output Output Output Hours

B Output Nondurable consumption Durable consumption Investment Hours Productivity Productivity

Correlation of A(t) and B(t&j) with j=

_ B _ A

1

0

&1

1.00 0.81 2.53 4.80 0.83 0.49 0.59

0.68 0.51 0.72 0.17 0.26 0.80 0.43

1.00 0.26 0.85 0.75 0.87 0.55 0.07

0.68 0.35 0.69 0.28 0.36 0.65 0.25

In sum, the multiple equilibria model developed here, does not perform inferiorly in predicting economic fluctuations when compared to the pure Real Business Cycle version as in Baxter [3]. In fact, it appears that the labor market side of the model performs superior to its antecedent. 3.5. Impulse Response Dynamics This last subsection will illustrate the endogenous propagation mechanism in the model. The calibration is the same as in Table VII. Figure 3 traces the

FIG. 3.

Impulse Response Dynamics.

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56

MARK WEDER

impulse response dynamics associated with a one time sunspot innovation. All variables rise on impact. The variables exhibit the typical hump-shaped response. This result is encouraging since Huffman and Wynne [16] as well as Weder [25] report that consumption declines at impact in standard two-sector Real Business Cycle models. The striking feature of the model is the degree of persistence imparted by the transitory shock: All the persistence is due to endogenous propagation for the white noise shock.

4. CONCLUSION The model developed in this paper contributes to the debate on whether and if, then to what extent, nonfundamental disturbances cause economic fluctuations. It has been shown here that consumer durables can be modelled straightforwardly in dynamic general equilibrium models with indeterminacy. It is of particular importance that indeterminacy can be obtained at very close to constant returns to scale. Moreover, if returns to scale are sufficiently strong, the cyclical characteristics of the model match the aggregate fluctuations of U.S. aggregate data. The model is capable of generating the degree of persistence seen in data without having to introduce serial correlations into the driving shock. The propagation of shocks does not operate along the intertemporal substitution of leisure as a response to technological disturbances Therefore the joint behavior of hours and productivity is no longer strongly correlated, a case which is suggested by empirical evidence. The so-called DunlopTarshis puzzle does not pose a complication for the model. In line with the work by Benhabib and Nishimura [7] it is shown that in multisector models, returns to scale can be of very low size and still be able to generate fully specified general equilibrium models with indeterminacy. The critique that has been raised on the first generation of models (for example, by Aiyagari [1] on Benhabib and Farmer [6]) is thus not applicable to the newer versions.

ACKNOWLEDGMENTS The author acknowledges helpful comments from Jess Benhabib, Michael Burda, Stefan Profit, and seminar participants at Humboldt University, UCLA, and the Multiple Equilibria Indeterminacy Conference at New York University. I take full responsibility for all remaining errors.

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REFERENCES 1. R. S. Aiyagari, The econometrics of indeterminacy: An applied study, a comment, Carnegie Rochester Conf. Ser. Public Policy 43 (1996), 273284. 2. S. Basu and J. G. Fernald, ``Returns to Scale in U.S. Production: Estimates and Implications,'' Board of Governors of the Federal Reserve System International Finance, Discussion Papers *546 1996. 3. M. Baxter, Are consumer durables important for business cycles? Rev. Econ. Statist. 78 (1996), 147155. 4. M. Baxter and R. G. King, ``Productive Externalities and Business Cycles,'' Institute of Empirical Macroeconomics, Discussion Paper *53, 1991. 5. J. Benhabib and R. E. A. Farmer, Indeterminacy, and sector-specific externalities, J. Monet. Econ. 37 (1996), 421443. 6. J. Benhabib and R. E. A. Farmer, Indeterminacy and increasing returns, J. Econ. Theory 63 (1994), 1941. 7. J. Benhabib and K. Nishimura, Indeterminacy and sunspots with constant returns, J. Econ. Theory 81 (1998), 5896. 8. F. Canova, ``Detrending and Business Cycle Facts,'' European University Institute, Working Paper, 1992. 9. D. Cass and K. Shell, Do sunspots matter? J. Polit. Econ. 91 (1983), 193227. 10. T. F. Cooley and E. C. Prescott, Economic growth and business cycles, in ``Frontiers in Business Cycle Research'' (T. F. Cooley, Ed.), Princeton Univ. Press, Princeton, NJ, 1995. 11. R. E. A. Farmer, ``The Macroeconomics of Self-Fulfilling Prophecies,'' MIT Press, Cambridge, MA, 1993. 12. J. C. Fuhrer, What role does consumer sentiment play in the U.S. macroeconomy? New Eng. Econ. Rev. (1993), 3244. 13. J. Gali, Monopolistic competition, business cycles, and the composition of aggregate demand, J. Econ. Theory 63 (1994), 7396. 14. G. D. Hansen, Indivisible labor and the business cycle, J. Monet. Econ. 16 (1985), 309328. 15. S. G. Harrison, ``Production Externalities and Indeterminacy in a Two Sector Model: Theory and Evidence,'' working paper, Department of Economics, Northwestern University, 1996. 16. G. W. Huffman and M. A. Wynne, ``The Role of Intratemporal Adjustment Costs in a Multisector Economy,'' working paper, Department of Economics, Southern Methodist University, 1997. 17. G. Kantona, ``Psychological economics,'' Elsevier, AmsterdamNew York, 1975. 18. R. G. King, C. I. Plosser, and S. T. Rebelo, Production, growth, and business cycles: I. The basic neoclassical model, J. Monet. Econ. 21 (1988), 195232. 19. F. E. Kydland and E. C. Prescott, Business cycles: Real facts and a monetary myth, Fed. Res. Bank Minn. Quart. Rev. 10 (1990), 318. 20. J. G. Matsusaka and A. M. Sbordone, Consumer confidence and economic fluctuations, Econ. Inquiry 33 (1995), 296318. 21. E. R. McGrattan, A progress report on business cycle models, Fed. Res. Bank. Minn. Quart. Rev. 18 (1990), 216. 22. R. Perli, ``Indeterminacy, Home Production, and the Business Cycle: A Calibrated Analysis,'' working paper, Department of Economics, University of Pennsylvania, 1995. 23. K. D. Salyer, The macroeconomics of self-fulfilling prophecies: A review essay, J. Monet. Econ. 35 (1995), 215242. 24. J. Shea, Do supply curves slop up? Quart. J. Econ. 108 (1993), 132. 25. M. Weder, ``Animal Spirits, Technology Shocks, and the Business Cycle,'' working paper, Humboldt University Berlin, Sonderforschungsbereich 373, 1997.

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