Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue2.Feb2000:

Journal of Economic Dynamics & Control
24 (2000) 273}295

Animal spirits, technology shocks and the
business cycle
Mark Weder*
Department of Economics, Humboldt University Berlin, Spandauer Str. 1, 10178 Berlin, Germany

Abstract
This paper presents a two-sector growth model which allows indeterminacy to occur at
relatively mild degrees of increasing returns. It is shown that economies of scale need only
be present in one sector of the economy, e.g. the investment good producing sector. This
new feature of the model builds on evidence that was recently reported by Basu and
Fernald (1997), (Journal of Political Economy 105, 249}283) and others. The time series
that are generated by the model have properties that are comparable to the real U.S.
postwar data. The sunspot driven model is also able to solve some puzzles of business
cycle research which standard Real Business Cycle models have not been able to
explain. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: E00; E32
Keywords: Sunspots; Technology shocks; Economic #uctuations; Dunlop}Tarshispuzzle

1. Introduction
The last few years have witnessed a revival of business cycle models in which
beliefs of agents (or animal spirits) have played a leading role in explaining
economic #uctuations.1 Most of these models involve strong economy-wide
increasing returns to scale in order for sunspot equilibria to exist. In a recent
paper, however, Basu and Fernald (1997) present evidence that returns to scale

* E-mail: weder@wiwi.hu-berlin.de.
1 See, for example, Farmer (1993), Farmer and Guo (1994), Gali (1994) and Schmitt-Grohe (1997).
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 8 7 - 6

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

are far from evenly distributed across the U.S. economy. In particular, they
report that scale economies are present mainly in the domain of durable goods
production. In the nondurable goods sector of the U.S. economy evidence of
increasing returns to scale cannot be found. Similarily, Harrison (1996) "nds

evidence of modest increasing returns to scale in the U.S. investment good
sector. The consumption goods sector, however, appears to operate under
constant returns.
The innovation of this work is to demonstrate that these empirical "ndings can be used within a two-sectoral optimal growth model with market
imperfections to generate non-uniqueness of rational expectations equilibria.
Moreover, it will be shown that returns to scale in the consumption good
sector are theoretically irrelevant for obtaining indeterminacy. Indeterminacy arises at returns to scale of around 1.07 in the investment sector
alone.2
Although the debate on business cycles was revived as a result of the new
literature on self-fulxlling expectations, it is indisputable that these recent
developments have failed to produce a widely accepted paradigm as of today.
The principal problem in this new literature is the dependency on degrees of
scale economies and market power that are not suggested by (most) recent
empirical studies. Benhabib and Farmer (1996), however, are able to show that
by working with a two-sector optimal growth model the extent of increasing
returns that is needed to obtain indeterminacy can be reduced signi"cantly.3
The main di!erence between their work and the model presented here is that
Benhabib and Farmer do not consider any asymmetry of scale economies of
the sort reported by Basu and Fernald (1997). A further distinction is that they
study an economy with perfect competition and sector speci"c externalities

whereas the model developed here is characterized by Cournot competiton
and internal returns to scale. Unlike the work by Gali (1994), however, in
the present model the variable markup is not a function of the composition
of demand but rather dependent on the degree of economic activity in each
sector. The market structure allows to obtain indeterminacy in the case of
increasing marginal costs where returns to scale originate from overhead costs
only.
The paper proceeds as follows: Section 2 presents the model. The economy's
steady state and the dynamics will be derived in Section 3. In Section 4 the
model is calibrated. This is followed by two exercises; the "rst will establish
parameter constellations at which indeterminacy is possible and the second

2 See Harrison (1996) for a related result.
3 Perli's (1997) two sector home production model is another example which allows to generate
indeterminacy at relatively low returns to scale.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

275

will compute model statistics to assess the model's business cycle properties
(Sections 5 and 6). Section 7 concludes the paper.

2. The model
The economic model developed is a two-sector extension of a baseline Real
Business Cycle model as in, for example, King et al. (1988). Markets for
investment goods and consumption goods are characterized by oligopoly.4
2.1. The household
It will be assumed that the economy consists of a representative household.
The household supplies labor and capital services to the "rms on competitive
markets. A representative agent has expected lifetime utility

C

D

=
E + bt;(C , l )DI ,
(1)
t t t

t/0
where ;( . ) is instantaneous utility, C is a consumption index, l leisure time,
t
t
b the discount factor and I the set of information available at t. The following
t
speci"c functional form for periodic utility is assumed:
B
l1`s with s40,
;(C , ¸ )"log C #
t t
t 1#s t

(2)

where B is a constant. Consumption of the households is de"ned by a CESaggregator over all di!erentiated goods which are normalized to unity:5

AP

1

B

Ct dc
c,t

1@t

, t3(0, 1).
0
The aggregator for the investment good, I , is de"ned as:
t
1
1@h
I"
Ih di
h3(0, 1).
t
i,t
0

The period-by-period budget constraint of the household is given by
C"
t

AP

P

1

0

B

P

p C dc#
c,c,t c,t

1

0

p I di"w ¸ #q K #P
i,i,t i,t
t t
t t
t

(3)

(4)

(5)

4 Gali (1995) also includes oligopoly in the Real Business Cycle framework.
5 Analogously, one could assume that instead of the household a perfectly competitive sector
produces the "nal good from the set of intermediates. The nonperfect substitutability in Eq. (3) is
simply a convenient analytical device for introducing market power in general equilibrium.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

where p
(p ) is the price of the consumption (investment) good c (i). Both
c,c,t i,i,t
prices are taken as given to the households. w is the nominal wage. Furthert
more, the household receives pro"t income from all existing "rms P . It also
t
owns the stock of capital K , which is rented out to the "rms at the rental price
t
q . Factor markets are perfectly competitive. Households are endowed with one
t
unit of time per period which they can either use for leisure or work ¸ :
t
1"¸ #l .
(6)
t
t

The consumer's capital holdings evolve as
K "(1!d)K #I ,
(7)
t`1
t
t
where d is the rate of depreciation. The household maximizes Eq. (1) subject to
Eqs. (3)}(7). As is well known for this class of models, maximization can be
conducted as a two step procedure. The conditional demand on consumption
goods can be derived in the "rst step as
1@(t~1)
p
C
(8)
C " c,c,t
t
c,t
p
c,t
which has a constant price elasticity. Here p ,(:1pt@(t~1) dc)(t~1)@t is the exact

0 c,c,t
c,t
price indices for the consumption goods. The analogous investment good
demand becomes

A B

A B

1@(h~1)
p
I " i,i,t
I.
(9)
i,t
t
p
i,t
Given these conditional demands, it is possible to derive the intertemporal
optimality condition for the household. In symmetric equilibrium, which is the
only case to be considered in this work, the household buys the same amount of
every product and the prices of all goods equal. The price of the consumption
goods in equilibrium is used as the numeraire and, without loss of generality, is
normalized to unity. The budget constraint transforms into
C #p I "w ¸ #q K #P .
(10)
t
t t
t t
t t
t
p can be interpreted as the relative price of investment goods in symmetric
t
equilibrium.
The second step of the household's optimization program consists of computing the optimal path of spending and working. Each household chooses a sequence MC , ¸ , K N= subject to K and to the distribution of technology
0
t t t`1 t/0
innovations (see below). The household's optimal decisions must satisfy
w
B(1!¸ )s! t "0,
t
C
t
p
q #(1!d)p
t`1DI ! t "0 ,
bE t`1
t
C
C
t
t`1

C

D

(11)
(12)

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

277

as well as satisfying the household's budget constraint and the transversality
condition. Eq. (11) describes the household's consumption-leisure trade o! and
Eq. (12) is the standard intertemporal optimality condition.
2.2. The xrms
One signi"cant modi"cation of conventional Real Business Cycle modelling is
considered: consumption and investment goods are produced in two distinct
sectors. Households can move their labor and capital services freely and without
costs between the two sectors. It is assumed that product markets are oligopolistic.
2.3. The consumption goods sector
The part of the economy that produces consumption goods consists of
subsectors or markets of measure one, each producing a di!erentiated product.6
There are N "rms supplying their single good j every period t in their subsector c.
c,t
Each "rm supplies its product on the market under the assumption of Cournot
competition. Costless endogenous entry and exit of "rms will be allowed in
order to drive pure pro"ts to zero.
Firm j has access to the following increasing returns technology:
(13)
> "C "Z (Ka ¸1~a)c!/,
c,j,t
c,j,t
t c,j,t c,j,t
with K capital input, ¸ labor input, and / overhead costs. Z is the state of
c,j,t
c,j,t
t
technology which evolves as7
log Z "o log Z #z , 04o(1 and N(0, p2).
z
t`1
t
t`1
Given the assumption on the form of competition and Eq. (8), the "rm's
program can be written in the speci"c Cournot form
t
~1
C
#C
c,~j,t
c,j,t
p C !w ¸ !q K
(14)
c,t c,j,t
t c,j,t
t c,j,t
C
t
subject to its production function. p
is the price of the "rm's good and
c,j,t
C
is the supply of all other "rms on the market which is taken as given for
c,~j,t
every "rm j. The cost function of a "rm is given by

B

A

max P "
c,j,t

C(w , q , C )"A qa w1~a
t t
t t c,j,t

A

C

B

#/ 1@c
c,j,t
,
Z
t

6 Basically both sectors of the economy are the same in structure. Therefore, only the sector that is
discussed "rst is described in detail.
7 Units have been chosen to make the conditional mean of technology equal to one.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

where A is a constant. Marginal costs are decreasing (increasing) for
c'1 (c(1). Optimality entails
(t!1) p
c,j,t C

A

B

C
A
C #/ 1@c~1
c,j,t
#p "
qaw1~a c,j,t
.
c,j,t cZ t t
#C
Z
c,~j,t
c,j,t
t
t

(15)

The last equation equalizes marginal revenues and marginal costs. At every
period in time the number of active "rms is implicitly determined by a zero
pro"t condition.
In symmetric equilibrium, N C "C "C , N "N and p "p "1
c,t c,j,t
c,t
t c,t
t
c,j,t
c,t
hold, where the last equality follows from the normalization that was already
made in the previous subsection. Inserting the optimal pricing rule into the zero
pro"t condition yields in symmetric equilibrium
C "c
t

B

A

t!1
#1 Z Kac ¸(1~a)cN1~c.
t
t c,t c,t
N
t

(16)

The term (t!1)/N #1 is the inverse of the markup in the consumption goods
t
sector. Note that the markup is decreasing in t which implies that a high
substitutability of the input goods translates into a low degree of market power.
The markup is also decreasing in the number of "rms. That is, the model
predicts a countercyclical pattern of the markup. This behavior is supported by
empirical evidence summarized by Rotemberg and Woodford (1991).8 By combining the optimal markup rule with the conditional demand for labor, it is
possible to derive the (equilibrium) wage rate as

B

A

t!1
Z (Ka ¸1~a)c ¸~1N1~c.
w "(1!a) c 1#
c,t t
t c,t c,t
t
N
t

(17)

The rental rate of capital is given by

A

B

t!1
Z (Ka ¸1~a)cK~1N1~c.
q "ac 1#
c,t c,t
t c,t c,t
t
N
c,t

(18)

2.4. The investment goods sector
There are M "rms supplying their respective investment good j in subsector
i,t
i.9 The market structure and the production technology in the investment goods

8 See also Burda (1985).
9 Again the letter j denotes the individual "rm.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

279

sector are essentially the same as in the consumption goods sector. Each "rm
supplies its product under the assumption of Cournot competition. It operates
with increasing returns technology
(19)
> "I "Z (Ka ¸1~a)g!C,
i,j,t
i,j,t
t i,j,t i,j,t
where K and ¸ are the "rm's capital and labor input and C are overhead
i,j,t
i,j,t
costs. In symmetric equilibrium, the zero pro"t condition is given by

A

B

h!1
(20)
#1 Z Kag ¸(1~a)gM1~g.
t
t i,t i,t
M
t
The optimal inverse factor demands are implicitly determined in symmetric
equilibrium by
I "g
t

B

A

h!1
Z (Ka ¸1~a)gM1~g¸~1
w "(1!a) gp 1#
i,t
t
t i,t i,t
t
t
M
t

(21)

and

A

B

h!1
q "a g p 1#
(22)
Z ( Ka ¸1~a)gM1~gK~1.
t
t
i,t
t
t i,t i,t
M
t
From the zero pro"t condition, it is easy to show that the internal increasing
returns to scales are equal to the markup.

3. The solution method
The following section describes the dynamics of the economy near its steady
state. Since the Second Welfare Theorem does not apply because of the existing
market power, the dynamics cannot be derived by means of the social planner
problem. Therefore, the necessary and su$cient "rst order conditions are
log-linearized.10 The model reduces to the three-dimensional dynamical system

C D CD C D

E[p(
DI ]
p(
0
t`1 t
t
"J KK #R 0 ,
(23)
KK
t`1
t
z
ZK
ZK
t`1
t`1
t
where J is 3]3. The eigenvalues of J must be evaluated at the steady state. The
system contains one predetermined variable, the stock of capital, KK , one
t
endogenous nonpredetermined variable, p( , and one exogenous nonpredetert
mined variable, ZK . Thus, if all eigenvalues of J are inside the unit circle, the
t
10 See also Woodford (1986), Farmer (1993) and Uhlig (1995) for justi"cations of this method.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

rational expectations equilibrium is non-unique. This will be analyzed in the
following section. The calibration method will be applied to check if indeterminacy has realistic relevance.

4. Calibration
Parameter value determination will follow in the Real Business Cycle tradition. Steady-state values of the model will be matched with estimates of average
growth rates and great ratios. A baseline model structure will be de"ned.
Without setting "xed values for all variables, the regions of realistic calibrations
will be shown.
To calibrate the model as close as possible to established Real Business Cycle
theory, parameters are set as proposed in existing studies. Quarterly d is equal to
0.025 and a, the capital share, is set at 0.30.
Basu and Fernald (1995) report estimates for increasing returns from 1.00 to
1.26. However, their preferred point estimate is 1.03. In their work the regression
was restricted by assuming that returns to scale are the same over the economy.
In a more recent work, Basu and Fernald (1997), in turn show that economies of
scale are largely heterogenous across the economy. For durable goods manufacturing, they report signi"cant increasing returns.11 For the production of
nondurables, on the other hand, (insigni"cant) diminishing returns are reported.
A similar picture arises in Harrison (1996). Based on these results, it will be
assumed that the consumption sector in the present model displays close to
constant returns. This is an assumption which will not be changed throughout
this paper.
The markups over marginal cost are given by
N
1
"
MC t!1#N
for the consumption goods sector and by
M
P
"
MC h!1#M
for the investment goods sector. The last two equations each possess one degree
of freedom. For example, if one "xes both the markup and t in the consumption
goods sector, the number of steady state "rms N is uniquely determined. The
same holds for the investment goods sector. Existing empirical literature does

11 Depending on various estimation methods, the point estimate ranges from 1.07 to 1.46.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

281

not o!er information concerning the magnitude of the elasticity of substitution
in Eqs. (3) and (4). In models of monopolistic competition, the elasticity of
substitution and the markup are interdependent since they are exactly inverse to
each other. This is not the case in under Cournot competition. Basu and Fernald
report markup margins from 1.00 to 1.26. Morrison (1990) reports the markup
to be around 1.14. In the remainder, it is simply assumed that the inverse of t (h)
always equals the markup as a normalization.
Kydland and Prescott (1990) report that total consumption expenditures
amount to 80% of output net of government expenditures. If only expenditures
on nondurables and services are considered, the ratio falls to 68%. In the present
model, the ratio of consumption to overall expenditures is generally "xed at
80% which is the same value as in Benhabib and Farmer (1996).12 The steadystate rate of return can be represented by
1
q
r" !d" !1.
b
p
An annual return of four percent conveys a discount rate of b+0.99.13 This
assumption is standard in Real Business Cycle models. C and / do not appear in
the linearized version of the economy.

5. Indeterminacy
5.1. Indeterminacy zones
Indeterminacy is present in the model as long as both roots of the matrix J are
inside the unit circle. A numerical solution is used here. Table 1 considers the
parameters which are not changed in the analysis unless otherwise noted.
Table 1 implies that the markup in the consumption sector is essentially one.
This value is also the measure of increasing returns.14 The assumption can be
justi"ed by comparing it to the empirical work by Basu and Fernald (1997) and
Harrison (1996). The other calibrations were discussed in the previous section.
Table 2 displays regions for indeterminacy for alternative values of scale
economies in the investment goods sector. The labor market follows the Hansen
(1985) construct, that is s"0. Marginal costs are constant in both sectors:
c"g"1.00, thus, for h(1, the markup is variable. The table shows that the
12 This is done by adjusting the preference parameter B. The main results of the paper carry over
to alternative assumptions on the consumption share.
13 Harrison (1996) calibrates the discount factor instead. This minor di!erence shows up in very
small numerical deviations to her work in Tables 2}5.
14 The speci"c value is chosen to avoid division by zero.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

Table 1
Parameters
¸
1/3

C/>
0.80

a
0.30

t
0.9999

d
0.025

b
0.99

Table 2
Roots of model
1/h

EIRS

Root 1

Root 2

Stability

1.05
1.10
1.14
1.144335
1.15
1.20
1.30
1.40
1.50

1.01
1.02
1.02
1.03
1.03
1.04
1.06
1.08
1.10

1.127
1.200
2.387
!0.999
0.766#0.302i
0.972#0.131i
0.990#0.079i
0.995#0.061i
0.996#0.051i

0.911
0.878
0.745
0.630
0.766!0.302i
0.972!0.131i
0.990!0.079i
0.995!0.061i
0.996!0.051i

Saddlepath stable
Saddlepath stable
Saddlepath stable
Indeterminacy
Indeterminacy
Indeterminacy
Indeterminacy
Indetermiancy
Indeterminacy

present model does not require unrealistic scale economies in order to produce
indeterminacy.15 The model is indeterminate at increasing returns to scale in the
investment goods sector of 1.15. Existing one-sector models which were summarized in Schmitt-Grohe (1997) require much higher scale economies. This
result alone indicates an improvement over previous work. Moreover, the
presence of returns to scale is limited to one sector only. The economy-wide
returns to scale (EIRS) amount to 1.03.16 It is generally the case that the
eigenvalues do not change if other assumptions are made on the returns to scale
in the consumption sector. The indeterminacy result depends solely on the scale
economies that are present in the investment goods sector. Finally, if it had been
assumed instead that the markup is constant, for example through monopolistic
competition and by a "xed number of "rms, the constancy of marginal costs
would exclude the possibility of indeterminacy.17

15 The matrix J contains a third root which equals the parameter o which is not reported in the
tables.
16 These are computed by weighting the respective sectoral returns to scale by the respective
output share.
17 See Weder (1996, 1998) for multi-sector models of monopolistic competition.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

283

Table 3
Roots of model
1/h

EIRS

Root 1

Root 2

Stability

1.05
1.07
1.072168
1.10
1.15
1.20

1.01
1.02
1.02
1.02
1.03
1.04

1.200
2.388
!0.998
0.972#0.130i
0.990#0.080i
0.994#0.061i

0.878
0.7453
0.630
0.972!0.130i
0.990!0.080i
0.994!0.061i

Saddlepath stable
Saddlepath stable
Indeterminacy
Indeterminacy
Indeterminacy
Indeterminacy

1/h

EIRS

Root 1

Root 2

Stability

1.15
1.18364
1.20

1.03
1.04
1.04

1.248
!0.991
0.904#0.226i

0.862
0.629
0.904!0.226i

Saddlepath stable
Indeterminacy
Indeterminacy

Table 4
Roots of model

g is set equal 0.90.

Table 3 repeats the analysis for decreasing marginal costs, in particular
g"1/h.18 By assuming decreasing marginal costs the returns to scale that are
needed to produce indeterminacy can be reduced even further. They must be
around 1.07 in the investment sector alone. This value is well within the reported
scale economies in Basu and Fernald (1998). Furthermore, it is marginally lower
than what is reported in Harrison (1996) who considers a model which is closer
in essence to the original Benhabib and Farmer (1996) approach in which
returns to scale are the result of externalities. The economy-wide scale economies are a mere 1.015. It is also of interest that Basu (1995) points out the case
that observed scale economies do not arise from decreasing marginal costs (as in
Farmer and Guo, 1994, or Benhabib and Farmer, 1996), but rather result from
overhead. Table 4 shows that decreasing marginal costs are not generally
needed in the present model: indeterminacy may arise with an upward sloping
marginal costs curve (here c"0.90).
Until now I have demonstrated the results in an indivisible labor environment
only. In Table 5 it is assumed that parameter s varies. The left-hand column of

18 In this case the overhead cost cannot be of positive value and the number of "rms is "xed in the
investment sector. Indeterminacy arises due to a decreasing marginal cost curve. In this case, a close
(but not one-to-one) connection to the model of Benhabib and Farmer (1996) exists.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

(Table 5
Alternative labor supply elasticities
1/h

EIRS

s

ELS

1.072
1.121
1.159
1.202
1.224
1.242

1.015
1.026
1.032
1.040
1.045
1.049

!0
!1
!2
!5
!10
!20

R
2
1
0.40
0.20
0.10

the table indicates the minimum returns to scale that are needed to obtain
indeterminacy. All other remaining parameters are the same as in Table 3. For
lower labor supply elasticities (ELS), the scale economies needed are higher but
are still low when compared to other models. Therefore, the model does not rely
on outright unrealistic labor supply elasticities in order to produce indeterminacy.
Until now only the average returns to scale have been reported. However, the
apparent aggregate returns to scale might be higher due to (strong) procyclicality of sectoral inputs. A useful check on the accuracy of the claim that returns to
scale are indeed modest in the above calculations would be to estimate the
aggregate returns by regressing aggregate output on the bundle of aggregate
inputs.19 The data is taken from a simulated model with 2000 realizations that is
driven by animal spirits shocks alone.20 It is assumed that parameters are set the
same as in Table 2. The returns to scale in the investment sector are 1/h"1.10
which implies average aggregate returns of around 1.02. The regression is as
follows
ln > "b (a ln K #(1!a)ln ¸ )#e .
t
1
t
t
t
This is equivalent to the regressions run by Basu and Fernald (1995). The
procedure yields an estimate bK "1.12. The returns increase by the amount of
1
0.10, however, they can still be considered modest. Other possible calibrations
yield similar results.
In sum the most signi"cant aspect of the present model is that it is able to
yield an indeterminate solution for largely realistic parameter constellations. In
face of recent critique on the animal spirits approach to business cycles, a criticism which centered on the implausible assumptions that were made on the
degree of market imperfections, the present model is able to set out a structure
that allows for the existence of indeterminacy at realistic measures. Moreover,
19 I thank one referee for suggesting this procedure.
20 See the following section for a description of the procedure on how time series are generated.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

285

these increasing returns need only be present in the investment goods sector. In
the present economy increasing returns to scale are due to overhead costs,
a feature which is also supported empirically. The model must still be evaluated
to see how well it is able to replicate stylized business cycle facts, however. This
will be carried out in the next section. Before doing so, an economic reasoning
for the indeterminacy result will be given.
5.2. The economic intuition behind the indeterminacy result
The economic intuition for indeterminacy in the model can be formulated as
follows: suppose agents expect (unrelated to any changes in economic fundamentals) that the future return to capital is going to be high. This will induce
a shift of current resources towards investment goods. However, the expectations must be supported in the new equilibrium, namely at a higher return to
capital. There are several ways to generate an increase in the rental rate at
a higher level of economic activity. All of these cases can be accounted for in the
present in model. First of all, it is assumed that increasing returns are present in
the economy. If returns to scale are the result of a falling marginal cost curve and
not only of overhead costs, an increase in investment goods production can
straightforwardly convey a higher capital return. Second, an increase in investment demand generates also an in#ow of "rms which implies a fall of the
markup (see for example Eq. (18)). If the markup is countercyclical in the
investment goods sector for any given stock of capital, the labor input and
the return to capital can potentially increase at higher activity. That is, even
when marginal costs are constant or increasing, indeterminacy is possible.
Third, labor moves freely across sectors. If the production of investment goods
rises, labor is shifted into the investment goods sector and the return of a given
stock of capital increases.21 If all of these features are combined or work
seperately, the return to capital can increase with economic activity. For all
these situation to arise, the returns to scale in the consumption goods sector are
irrelevant.

6. Business cycle properties
6.1. Population moments
The model must still be judged on how good it can replicate the variability of
the di!erent aggregate macroeconomic time series behavior. In accordance with

21 This last channel is never su$cient in obtaining indeterminacy, however, it contributes to the
result once the economy is in a suboptimal equilibrium.

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M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

the Real Business Cycle approach, the generated model data will be compared
with real data. Table 6 reports population moments for the U.S. economy.
The well-known stylized business cycle fact can be observed: consumption
#uctuates less than output and investment displays a greater volatility than
output. The right part of the table gives cross correlations of the variables. All
variables peak with output. The bottom line displays the autocorrelation of
output growth which measures the persistence of aggregate output. The process
of "rms' entry and exit takes on an important role in the present model. The
procyclical behavior of net business formation is well documented for the U.S.
economy in Audretsch and Acs (1991) or Campbell (1995), and are further
discussed in Devereux et al. (1996).

6.2. Model moments
Sunspot equilibria are de"ned as rational expectations equilibria in which
cyclical behavior arises in response to arbitrary random events that do not have
an e!ect on the fundamental equilibrium conditions of the economy. Once the
sequence of sunspots is generated, the law of motion of the economy, which

Table 6
Selected U.S. business cycle statistics, quartely
Variable
A

B

>
>
>
>
>
>
>
*>

}
C
I
¸
=
IS
CS
}

Relative
volatility
p /p
B A
5.62!
0.69
1.35
0.52
1.14
0.56
0.70
0.99!

Correlation of A(t) and B(t!j) with j"
1

0

!1

0.96
0.82
0.59
0.07
0.72
0.74
!0.72
0.37

1.00
0.85
0.60
0.07
0.76
0.81
!0.89
1.00

0.96
0.84
0.57
0.06
0.74
0.77
!0.73
0.37

The table is taken from King et al. (1988) } deviations from common linear trend, quarterly,
1948:I}1986:IV. Variable de"nitions: >"real gross national output, C"consumption expenditure
on nondurables and services, I"gross "xed investment, ¸"total workers employed (Household
Survey) and ="gross average hourly earnings of production workers. IS"investment expenditures share ("xed invesment) and CS"consumption expenditures share (nondurables and services)
are from Kydland and Prescott (1990), deviations from HP trend, quarterly, 1954:I}1989:IV.
*>"output growth which is taken from Christiano and Todd (1996) whose data set covers the
1947:I}1995:I period.
!indicates that the number is the simple, not relative, standard deviation times 100.

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

287

below includes technology shocks, is given by

C D CD C D

p(
p(
u
t`1
t
t`1
"J KK #R 0
.
KK
t`1
t
z
ZK
ZK
t`1
t`1
t

(24)

Here u
is an i.i.d. expectational error which can be interpreted as animal
t`1
spirits (see Farmer, 1993 or Woodford, 1991).
In the remainder of this section the sample moments of the model will be
reported for various calibrations. First, a baseline calibration is speci"ed which
will not be altered unless noted (Table 7). The value of s implies that the
intraperiod utility is equivalent to that in the Hansen}Rogerson model. The
t calibration follows the notion that increasing returns are only present in
the investment goods sector.
Table 8 reports a version of the model with very low scale economies (1.10)
and which is driven by an i.i.d. animal spirits shock sequence only. The table

Table 7
Parameters

t
0.9999

c
1.00

g
1/k

b
0.99

s
0.00

Table 8
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

Relative
volatility
p /p
B A
1.00
0.53
6.08
5.03
1.41
1.29
0.53
0.41
1.00

Correlation of A(t) and B(t!j) with j"
!1

0

!1

0.96
!0.22
0.86
0.83
!0.83
0.83
!0.22
!0.74
0.21

1.00
!0.37
0.95
0.93
!0.93
0.93
!0.38
!0.69
1.00

0.96
!0.45
0.94
0.92
!0.92
0.92
!0.45
!0.56
0.21

Deviations from linear trend. Variable de"nitions: >"output (> "C #p I ), C"consumption
t
t
t t
expenditures, I"investment (p I ), ¸"hours, P">/¸"productivity, IS"investment share,
t t
CS"consumption share and *> output growth. Statistics are based on 2000 arti"cial realizations
of the model.

288

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

indicates two major counterfactual characteristics of the model. First, consumption and labor productivity are countercyclical. Second investment is much too
volatile. Along other lines, the model performs reasonably well. Most notably it
generates highly autocorrelated time series and realistic relative volatilities of
the remaining variables. The expenditures shares move in correct directions: the
investment share is strongly procyclical and aggregate expenditures on consumption goods are countercyclical.
The countercylical behavior of consumption in two sector models is wellknown from the Benhabib and Farmer (1997) survey. However, this fact is not
limited to multiple equilibria models. Hu!man and Wynne (1997) report the
very same result for a two sector Real Business Cycle model with technology
shocks. Only the introduction of adjustment costs allows the latter authors to
obtain procyclical consumption. Other potential solutions that will be discussed
in the remainder of this work are higher returns to scale, a variable markup and,
combined with greater scale economies, technology shocks. To make Hu!man
and Wynne's (1997) result visible in the present model, Table 9 reports the time
series of the model when it is driven by i.i.d. technology shocks alone (o"0).
The calibration is the same as in the preceeding version. Surprisingly, Tables
8 and 9 o!er a very similar picture of the model economy. Moreover, at least at
low increasing returns to scale, consumption appears to be negatively correlated
with output if technology is i.i.d. Also, as in Hu!man and Wynne (1997),
aggregate labor productivity is countercyclical.
Benhabib and Farmer (1996) point out that a su$ciently higher degree of
returns to scale allows the generation of procyclical consumption in their model.
Therefore, Table 10 considers the case of a model economy observes higher

Table 9
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

Relative
volatility
p /p
B A
1.00
0.54
6.13
5.19
1.29
1.30
0.54
0.38
1.00

Correlation of A(t) and B(t!j) with j"
!1

0

!1

0.96
!0.21
0.87
0.83
!0.83
0.83
!0.21
!0.54
0.28

1.00
!0.37
0.94
0.92
!0.92
0.92
!0.37
!0.69
1.00

0.96
!0.47
0.96
0.94
!0.94
0.94
!0.44
!0.74
0.28

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

289

Table 10
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

Relative
volatility
p /p
B A
1.00
0.62
4.52
3.51
1.10
0.92
0.62
0.70
1.00

Correlation of A(t) and B(t!j) with j"
!1

0

!1

0.98
0.51
0.83
0.74
!0.74
0.74
0.51
!0.23
0.20

1.00
0.42
0.87
0.79
!0.79
0.79
0.46
!0.17
1.00

0.98
0.41
0.89
0.81
!0.81
0.81
0.41
!0.11
0.20

returns to scale of 1.20. This is the same extent of returns to scale as in Benhabib
and Farmer's (1996) simulated model except that scale economies are present in
both sectors in theirs. Animal spirits are the only source of #uctuations. The
dynamics of consumption are substantially altered: the contemporaneous correlation of consumption with output becomes positive and the relative volatility
is very close to what is found in data. This result conforms to the Benhabib and
Farmer (1996) model, although returns to scale are limited to one sector of the
economy here. Labor productivity is procyclical. Also, hours and productivity
are mildly negatively correlated. The prediction of the model is quite close to the
value of !0.34, as reported Canova (1998) for the U.S.economy.22 This is the
socalled Dunlop-Tarshis puzzle which states that real wages (productivity) and
labor input move essentially uncorrelated to each other (see Tarshis, 1939;
Dunlop, 1938). The model is thus able to solve an arduous puzzle of business
cycle research which standard Real Business Cycle models have not been
able to.23
Introducing persistent technology innovations into the last model version
helps to generate an even better match (o"0.90). Both shock sequences are
uncorrelated and of equal size. As can be seen from the last line in Table 11,
output persistence increases and resides at the level that is reported by
Christiano and Todd (1996). Also, the low zerocorrelation of labor and productivity is similar to the predicted one in mentioned empirical studies.

22 McGrattan (1994) who HP "ltered the data reports a correlation of !0.20.
23 It should be noted that these results di!er slightly from those in Benhabib and Farmer (1996). In
particular investment is more volatile and consumption is less procyclical.

290

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

Table 11
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

Relative
volatility
p /p
B A

Correlation of A(t) and B(t!j) with j"
!1

0

!1

1.00
0.63
4.56
3.55
1.11
0.93
0.63
0.70
1.00

0.99
0.50
0.85
0.77
!0.77
0.77
0.50
!0.21
0.34

1.00
0.43
0.86
0.81
!0.81
0.81
0.43
!0.15
1.00

0.99
0.40
0.89
0.83
!0.83
0.83
0.40
!0.09
0.34

Relative
volatility
p /p
B A

Correlation of A(t) and B(t!j) with j"

Table 12
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

1.00
0.63
4.46
3.46
1.08
0.90
0.63
0.70
1.00

!1

0

!1

0.98
0.50
0.85
0.75
!0.75
0.75
0.55
!0.24
0.14

1.00
0.46
0.88
0.80
!0.80
0.80
0.51
!0.19
1.00

0.98
0.40
0.90
0.82
!0.82
0.82
0.47
!0.13
0.14

Table 12 reports the statistics for a version of the model that assumes
a variable markup in the investment good sector, that is, the number of "rms is
nonconstant. In particular, 1/h"1.25 and g"1.20. The model is driven by
animal spirits shocks alone. The table shows that the Benhabib and Farmer
(1996) conjecture that a variable might help to obtain procyclical consumption
does not necessarily hold. Even if the returns to scale are higher as in Table 10,
a comparison with the constant markup case shows that the contemporaneous
correlation of consumption with output is basically unaltered. This can be
understood as follows. If households become more optimistic and begin buying

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

291

investment goods, the (relative) markup declines. This makes investment goods
even more attractive and consumption goods are substituted. Only if returns to
scale are high, the induced wealth e!ect creates a positive correlation.
Table 13 reports the e!ect of the slope of marginal costs onto consumption's
procyclicality. It shows the output-consumption correlation for alternative gs
while holding "xed 1/h"1.20. Procyclical consumption is easier to obtain if
marginal costs are decreasing. One may interprete this "ndings as being similar
to the wealth e!ect that operates in a pure RBC model with technology shocks.
It is this e!ect } a downward shift of the cost function } that generates
procyclical consumption in that model. A falling marginal costs schedule operates similarly.
Finally, an even higher degree of returns to scale is assumed: c"1.50,
combined with constant marginal costs g"1.00 e.g. the case of a variable
markup where increasing returns are solely the result of overhead costs. This
amount of scale economies correponds to the Baxter and King (1991) model.
Animal spirits are the only present driving variable (Table 14). The above extent

Table 13
Model moments
g

Correlation of C(t) and >(t)

1.20
1.15
1.10
1.00
0.90

0.43
0.18
!0.06
!0.36
!0.77

Table 14
Model moments
Variable
A

B

>
>
>
>
>
>
>
¸
*>

>
C
I
IS
CS
¸
P
P
*>

Relative
volatility
p /p
B A
1.00
0.67
3.52
2.70
0.94
0.94
0.67
1.40
1.00

Correlation of A(t) and B(t!j) with j"
!1

0

!1

0.99
0.46
0.83
0.72
!0.72
0.72
0.46
!0.21
0.16

1.00
0.42
0.87
0.76
!0.76
0.76
0.82
!0.27
1.00

0.99
0.36
0.89
0.79
!0.79
0.79
0.36
!0.32
0.16

292

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

Fig. 1.

of returns to scale is admittedly unrealistically high. However the model predictions display close similarities to the ones that have been reported in Table 6 for
the U.S. economy.
Overall, in terms of business cycle statistic, the model is not necessarily worse
than existing (one sector) Real Business Cycle models. Most features of data can
be replicated.
6.3. Impulse response dynamics
This last subsection will illustrate the endogenous propagation mechanism in
the model. I focus on the response of the economy to animal spirits shocks.
Fig. 1 reports the e!ects in the &medium returns to scale' case, e.g. g"1/h"1.2.
The "gure traces the impulse responses associated with a one time animal
spirits innovation. Output, investment and labor rise on impact. Consumption
declines at the time of the shock and then gradually rises thereby overshooting
its steady-state value. The variables exhibit the typical hump-shape response
and they return to their steady-state only very slowly. Hu!man and Wynne
(1997) report the same consumption reaction as a response to a technology
shock in their two sector Real Business Cycle model. Therefore the tendency of
investment and consumption to move initially in opposite directions is not
con"ned to multiple equilibria models or animal spirits shocks. However, the
impact of the shock shows very substantial propagation. The striking feature of
the model is the degree of persistence imparted by a transitory shock: all the

M. Weder / Journal of Economic Dynamics & Control 24 (2000) 273}295

293

persistence is due to endogenous propagation for the white noise animal spirits
shock.24 The model provides an example of an economy with the ability to
signi"cantly propagate temporary shocks.

7. Conclusion
In this paper a stochastic two-sector growth model is developed which allows
indeterminacy to occur at very mild degrees of increasing returns. Furthermore,
it is shown that it is su$cient that these economies of scale are present in only
one sector of the economy. This feature of the model builds on empirical
evidence that was recently reported by Basu and Fernald (1997) and Harrison
(1996). The size of the returns to scale in the consumption good sector is
theoretically irrelevant in generating indeterminacy. The model is also able to
solve some puzzles of business cycle research which standard Real Business
Cycle models have not been able to. Namely, the introduction of animal spirits
at modest returns to scale allows the generation of a low and negative contemporaneous correlation of hours and productivity. Considering more standard
measures of the business cycle, such as the relative volatility of aggregate
variables and comovements, the model performs equally as well as existing Real
Business Cycle models. Finally the paper provides an example of a model which
o!ers a strong endogenous propagation mechanism.

Acknowledgements
I am indebted to Michael Burda for his guidance. I would like to thank Jess
Benhabib, Dalia Marin, Ellen McGrattan, Harald Uhlig, Rolf Tschernig, two
anonymous referees and seminar participants at Heidelberg, EUI-Florence
(EEA Summer School), SUNY at Stony Brook and Toulouse (EEA '97) for
valuable comments. All remaining errors are mine. This research was supported
by the DFG-Sonderforschungsbereich 373 &Quanti"cation and Simulation of
Economic Processes'.

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