Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol25.Issue5.Apr2001:
Journal of Economic Dynamics & Control
25 (2001) 747}764
Indeterminacy in a model with sector-speci"c
externalities
Sharon G. Harrison*
Department of Economics; Barnard College, Columbia University; 3009 Broadway;
New York, NY, 10027, USA
Received 1 September 1998; received in revised form 1 March 1999; accepted 7 April 1999
Abstract
I examine a model with two sectors of production: consumption and investment. In the
model, indeterminacy of equilibria results due to the presence of small sector-speci"c
externalities in production. In fact, I "nd that indeterminacy results with a certain,
minimum value of the externality in the investment sector, even with no externality in the
consumption sector. I "nd that the indeterminacy properties of the model vary, depending on the form of the utility function. For example, with utility that is logarithmic in
consumption, these properties are completely independent of the value of the externality
in the consumption sector. ( 2001 Elsevier Science B.V. All rights reserved.
JEL classixcation: E00; E32
Keywords: Business cycles; Expectations; Indeterminacy; Production externalities
* Tel: 212-854-3333; fax: 212-854-8947. I gratefully acknowledge input from two anonymous
referees, Jess Benhabib, Lawrence Christiano, David Domeij, Martin Eichenbaum, Roger Farmer,
Duncan Foley, Jang-Ting Guo, Michael Horvath, Kiminori Matsuyama, Kay Robbins, Thomas
Sargent, Alberto Trejos, and several ex-colleagues from Northwestern University. All remaining
errors are, of course, my own.
E-mail address: [email protected] (S.G. Harrison).
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 4 0 - 8
748
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
1. Introduction
Much recent work in macroeconomics has focussed on models with multiple,
or indeterminate, equilibria.1 As this literature has developed, it has contributed
much to our understanding of dynamic macroeconomies. The possible existence
of multiple equilibria has important implications for those of us who study
business cycles, growth and monetary economies. In models of the business
cycle, for example, indeterminacy means that agents' expectations about the
future can be self-ful"lling, and therefore can serve as impulses to aggregate
#uctuations.
In this paper, I examine a model of the business cycle with two sectors of
production: consumption and investment, in which indeterminacy results due
to the presence of sector-speci"c externalities in production. I carry out an
extensive study of the indeterminacy properties of the model and "nd that
indeterminacy results with a certain, minimum value of the externality in the
investment sector, even if there is no externality in the consumption sector. In
addition, I "nd that when there is an externality in the consumption sector, the
size of the externality in the investment sector needed for indeterminacy depends
on this parameter and on the curvature of the utility function.
In models with one sector of production and increasing returns to scale,
indeterminacy results when returns to scale are su$ciently high. For example,
Benhabib and Farmer (1994); Farmer and Guo (1994) and Christiano and
Harrison (1999) specify general equilibrium models in which, with a certain size
externality, and constant internal returns to scale, indeterminacy results. This is
the case because when agents believe, for example, that the return on capital will
rise tomorrow, they act on this by raising tomorrow's capital stock. Given
su$ciently high increasing returns, so that the marginal product, which is equal
to the return on capital, is increasing in capital, the expectation will be ful"lled.
However, in these models, with realistic values of the other parameters, overall
returns to scale must be about 1.5 in order for indeterminacy to result.
More recent work by Benhabib and Farmer (1996) has demonstrated that in
a model with two sectors of production, much lower external e!ects are necessary for indeterminacy.2 With overall returns of about 1.1 in each sector, agents'
beliefs can be self-ful"lling. This follows because in this model, when agents
believe that the return on capital will rise tomorrow they reallocate factors of
1 Examples include Benhabib and Perli (1994); Boldrin and Rustichini (1994); Matsuyama (1991)
and Perli (1998) among others. In#uential early work includes Azariadis (1981); Cass and Shell
(1983); Cooper and John (1988); Diamond (1982) and Murphy et al. (1989), among others.
2 In fact, Benhabib and Nishimura (1998) examine a multi-sector model in which indeterminacy
results with constant returns to scale. In the model, internal returns are slightly decreasing (e.g. 0.93)
and there is a small externality.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
749
production between sectors. This leads to movements in the relative price of
investment that a!ect the return on capital in such a way that the marginal
product of capital need not be increasing in capital for the expectation to be
ful"lled.
Empirical evidence points towards the existence of very small, if any, production externalities at the aggregate level (See, for example Basu and Fernald, 1995
and Burnside, 1996). This evidence leads to rejection of the possibility of
indeterminacy in the one sector model. However, Harrison (1997) provides some
evidence that the externality for investment may be at least as large as 1.1. That
is, indeterminacy in the two sector model is more likely to be empirically
relevant than in the one sector model.
In this paper I present a discrete time version of the Benhabib and Farmer
two sector model. I carry out a comprehensive study of the indeterminacy
properties of the model. To do this, I relax their restriction that the two sectors
experience the same size externality. In addition, instead of restricting the utility
function to be logarithmic in consumption, I use the more general constant
relative risk aversion form. In this way, I can analyze the indeterminacy
properties of the model when these important parameters are allowed to vary.
The main results obtained from allowing these #exibilities are: (1) Indeterminacy results with a certain, minimum value of the externality in the investment
sector, even if there is no externality in the consumption sector; (2) As the
constant of relative risk aversion increases, so does the level of returns to scale
needed for indeterminacy; and (3) As the level of returns to scale in consumption
increases, the level of returns to scale in investment needed for indeterminacy
changes di!erently, depending on the value of the constant of relative risk
aversion.3 I provide numerical examples to support these results and discuss the
intuition for them. I also examine the cyclicality of consumption under di!erent
parameterizations. Lastly, I "nd that these results are robust to allowing the
capital share to di!er across sectors.
The rest of this paper proceeds as follows. Section 2 presents the model.
Sections 3}5 analyze its indeterminacy properties under di!erent parameterizations. Section 6 examines some time series properties of consumption in the
model and discusses a version with di!ering capital shares. Section 7 concludes.
2. The model
The model is a nonstochastic, discrete time version of the model of Benhabib
and Farmer (1996). The household derives utility from consumption and leisure;
3 Weder (1999) "nds the analogous result to (1) in a continuous time version of this model with
logarithmic utility and oligopolistic "rms.
750
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
and there are two sectors of production, consumption and investment. The "rm
in each sector produces with a constant returns to scale technology and a sector-speci"c externality, which is taken as given. While Benhabib and Farmer
restrict the size of the externality to be the same across sectors and utility to be
logarithmic in consumption, I relax these restrictions.
2.1. The household problem
The representative agent acts to maximize his present discounted value
of utility:
A
B
=
c1~p!1 n1`s
! t
max + bt t
1#s
1!p
M
N=
kt`1,nt,ct t/0 t/0
subject to:
c #p I 4r k #w n
t
t t
t t
t t
and
k !(1!d)k "I , k given.
(1)
t`1
t
t
0
Here c , n , k , and I denote consumption, labor, capital and investment in period
t t t
t
t, respectively. Also, p indicates the relative price of investment and the price of
t
consumption is normalized to 1. In addition, r is the rental rate on capital and
t
w is the wage rate. Lastly, 1/p is the intertemporal elasticity of substitution of
t
consumption and 1/s is the labor supply elasticity.
The "rst-order conditions for this problem are
r #(1!d)p
t`1"0
c~p!bc~p t`1
t`1
t
p
t
(2)
c~pw !ns"0.
t
t t
(3)
and
2.2. The xrms' problems
In the consumption sector, the "rm maximizes pro"t subject to the production function
f "(Ka N1~a )hcka n1~a
c,t c,t
c,t c,t
c,t
where k and n denote the capital and labor the "rm devotes to the consumpc,t
c,t
tion sector at time t and a is capital's share in production of the consumption
good. Also, K and N denote the economy-wide average capital and labor
c,t
c,t
devoted to the consumption sector, which are taken as given by the "rm.
The degree of sector-speci"c externality in the consumption sector is h .
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
751
The "rst-order conditions for this problem are
(1!a ) f
af
c,t"w .
c,t"r ,
(4)
t
t
n
k
c,t
c,t
Similarly, the "rm in the investment sector maximizes pro"t and produces
according to
f "(Ka N1~a)hIka n1~a,
I,t I,t
I,t I,t
I,t
where k and n denote the capital and labor the "rm devotes to the investI,t
I,t
ment sector at time t. K and N denote the economy-wide average capital
I,t
I,t
and labor devoted to the investment sector. Note that the degree of the
sector-speci"c externality for the investment sector, h , is allowed to di!er from
I
that of the consumption sector. This "rm's "rst-order conditions are
af
(1!a) f
I,t"w .
p I,t"r , p
tk
t
t
t
n
I,t
I,t
Since capital and labor are used only in the production of the two goods, it
must be true that
K #K "K and N #N "N ,
c,t
I,t
t
c,t
I,t
t
where K and N denote economy-wide averages of total capital and labor.
t
t
2.3. Equilibrium and steady state
An equilibrium is a set of prices, Mp , r , w N= such that given these prices, the
t t t t/0
quantities Mk , c , n N= solve the household and "rm problems. In addition,
t`1 t t t/0
the resource constraints are satis"ed.
In equilibrium,
f "(k kan1~a)1`hc
c,t
t t t
(5)
f "((1!k )kan1~a)1`hI,
I,t
t t t
where
(6)
and
k
n
k " c,t" c,t.
t
k
n
t
t
Substituting into the "rms' "rst-order conditions, we get an expression for the
relative price of investment:
khc
t
(kan1~a )hc~hI.
p"
t (1!k )hI t t
t
(7)
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S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Denoting steady-state values with no subscripts, the steady-state versions of
(1), (4) and (6) can be used to solve for r. This equation and the steady-state
version of (2) can be used to solve for k in terms of the parameters of the model.
Given this, the steady-state versions of (3)}(5); and (1) and (6) give two equations
in k and n. The rest of the steady-state values follow from these.
2.4. Dynamic behavior
The dynamics of this economy are summarized by the three equations
describing the household problem: (1)}(3); and the "ve equations describing the
"rms' problems: (4)}(7). Of these (1) and (2) are intertemporal. I log-linearize this
system of equations around steady state.4 Letting
AB
p(
t
IK
t
r(
kK
u " t , v " t , e "(mK ),
t
t
t
t
w(
c(
t
t
y(
t
n(
t
where m represents the sunspot shock and x( indicates the log deviation of x from
t
its steady-state value, I obtain
AB
(8)
v "Ju
t
t
and
u "Q u #Q e ,
t`1
1 t
2t
where
A
B
dJ #1!d
dJ
2,1
2,2
Q " !J !/(dJ #1!d) p!J !dJ / ,
1
1,1
2,1
1,2
2,2
U
U
/"!b(1!d)J #[ b (1!d)!1]J
1,1
3,1
(9)
and
U"!b(1!d)J #[b(1!d)!1]J #p.
1,2
3,2
4 Some of the notation used here is borrowed from the unpublished appendix to Benhabib and
Farmer (1996).
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
753
The J are elements of J in (8). Speci"cally,
i,j
a(1#h )(1#s)
(a!1)s
J " !ahI(1#s)
I
1,1
, J "
, J "
,
2,1
3,1
(1!k)(a#s)
(a#s)
(1!k)(a#s)
J "
1,2
(1#h )[(1#s)k#(1!a)(1!k)]!(1#s)#(p!1)(1!a)h (1#h )
I
I
c #1,
(1!k)(a#s)(1#h )
c
(1#h )[a(1!k)!ks!1)#(1!p)(1!a)(1#h )]
I
c
J "
2,2
(1!k)(a#s)(1#h )
c
and
h (1#s)!p(1!a)(1#h )
c.
J " c
3,2
(1#h )(a#s)
c
Denoting the eigenvalues of Q j and j , indeterminacy results when both
1 1
2
j and j are inside the unit circle. When the steady state is indeterminate, given
1
2
a k , there is more than one c that starts o! a path that satis"es the equilibrium
0
0
conditions. When more than one equilibrium path exists, equilibria driven by
sunspot shocks are possible.
3. Indeterminacy when h "h "h
c
I
In this section, I set b"1/1.01, d"0.025, a"0.3, p"1 and s"0. This is
one of the parameterizations used in Benhabib and Farmer (1996) and will serve
as the benchmark parameterization of this model. Note that setting s"0 is
equivalent to setting the labor supply elasticity equal to in"nity. Benhabib and
Farmer discuss how varying the elasticity of labor supply a!ects the ease with
which indeterminacy results. In particular, as the labor supply elasticity increases (s decreases), labor is drawn more easily out of leisure and lower
increasing returns are needed for indeterminacy.
Using Benhabib and Farmer's speci"cation as a starting point, i.e. restricting
h "h "h, Table 1 reports values of j and j associated with di!erent values
c
I
1
2
of h.5 The h"0 case simply corresponds to the standard one-sector model. With
h"0.0773, the equilibrium of the model is still determinate. However, indeterminacy results when h50.0774.6
5 See Christiano (1995) for more examples.
6 More precisely, at approximately h"0.077369, the economy undergoes a #ip bifurcation
whereby one eigenvalue is exactly equal to !1 and the other lies within the unit circle. In this case,
the model may converge to a stable two-cycle for any initial condition that lies o! the steady state.
754
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Table 1
Indeterminacy when h "h "h
c
I
h
j
0
0.0773
0.0774
0.08
0.93
0.46
0.41
0.81!0.36i
1
j
2
1.09
!1.51
!0.82
0.81#0.36i
Indeterminacy?
No
No
Yes
Yes
In order to understand the intuition for the idea that indeterminacy results in
this model with much lower returns to scale than in the one sector model,
I compare the intertemporal "rst-order conditions for the one and two sector
models. Examination of the two sector model also requires an understanding of
the production possibilities frontier (PPF) in this case.
In the one sector model the intertemporal "rst-order condition is
A B
p
t`1 "b[r #(1!d)].
t`1
c
t
Starting from a steady-state equilibrium, in which case this equation is satis"ed,
when the return on capital is expected to increase tomorrow, consumption is
sacri"ced for investment, and is less than its steady-state value. Tomorrow's
capital stock and consumption increase. The left-hand side of the equation
increases, relative to steady state. In order to stay in equilibrium, so must the
right-hand side. With high enough returns to scale, tomorrow's marginal product of capital will increase with tomorrow's capital stock and agents' expectations will be self-ful"lling.
For the two sector model the condition is
c
A B
p
[r #(1!d)p ]
t`1 .
t`1 "b t`1
p
c
t
t
In order to understand why it is that indeterminacy can result with lower returns
to scale here, it is important to examine the PPF for the social planner in this
economy. Fig. 1 plots the PPF for the case where h "h '0. As Benhabib and
c
I
Farmer discuss, this PPF is convex. Due to the increasing returns to scale, as
resources are shifted into a good, the marginal product of each factor used in
production of that good increases. (Even if returns to scale are not large enough
to make marginal products increasing in individual inputs, they are increasing in
both inputs.) At the optimal solution, the slope of the PPF is (the negative of )
the relative price of investment, p. Because of the increasing returns to scale,
a shift of resources towards the production of a good lowers the price of that
good. Recall that k represents the share of each factor used in production of the
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
755
Fig. 1. PPF of economy with equal externalities.
consumption good. As k increases, and more of the consumption good is
produced, the curve gets steeper, increasing the relative price of investment.7
Now, when tomorrow's consumption increases and the left-hand side of the
equation increases relative to steady state, tomorrow's relative price of capital
increases as resources are shifted towards consumption. Returns to scale do not
have to be as high as they were in the one sector case. Indeterminacy can result
even if the marginal product of capital is decreasing in capital.
4. Indeterminacy when h Oh
c
I
In this section, I use the benchmark parameterization and discuss the indeterminacy properties of the model when h Oh . First, I "x h "0.0774 and
c
I
c
calculate j and j under various parameterizations of h . Then I "x h "0.0774
1
2
I
I
7 When h Oh it can be shown that the PPF is convex as long as there is increasing returns to
c
I
scale in at least one sector.
756
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Table 2
Indeterminacy when h "0.0774
c
h
I
0
0.0773
0.0774
0.08
j
1
0.93
0.46
0.41
0.81!0.36i
j
2
Indeterminacy?
1.09
!1.51
!0.82
0.81!0.36i
No
No
Yes
Yes
j
Indeterminacy?
Table 3
Indeterminacy when h "0.0774
I
h
c
0
0.0773
0.0774
0.08
j
1
0.41
0.41
0.41
0.41
2
!0.82
!0.82
!0.82
!0.82
Yes
Yes
Yes
Yes
and repeat the experiment, varying h . The main result of this analysis is
c
Proposition 1. Indeterminacy results when there is a certain size externality in the
investment sector, even if there is no externality in the consumption sector.
The size of the externality needed is the same as when h "h "h. Tables
c
I
2 and 3 illustrate this. In Table 2, h "0.0774. With h "0, the equilibrium of the
c
I
model is determinate. The same is true when h "0.0773. Once h 50.0774,
I
I
indeterminacy results.
In Table 3, h "0.0774. Regardless of the value of h , the equilibrium of the
I
c
model is indeterminate. The reader will also notice from the results in Table 3
that as h changes, j and j do not. In other words, the indeterminacy
c
1
2
properties of the model are completely independent of the value of the externality in the consumption sector. This is easily seen by setting p"1 and
solving for the eigenvalues of Q . First, note that these depend on the values of
1
the individual elements Q and Q and on the product of Q and Q . It
1,1
2,2
1,2
2,1
turns out that when p"1, none of these depend on h . To see this, "rst note that
c
Q never depends on h . Next, setting p"1, U reduces to
1,1
c
i
1
,
U"
(1!k)(a#s)(1#h )
c
where i is a function of various parameters, but not of h . As a result, the rest of
1
c
the elements of Q reduce to:
1
(1!k)(a#s)(1#h )i
i
c 2, Q "
3
Q "
2,1
1,2
i
(1!k)(a#s)(1#h )
1
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
757
and
i /(1!k)(a#s)(1#h ) i
c " 4,
Q " 1
2,2
i
U
1
where i , i and i are also not dependent on h , so that the product of Q and
2 3
4
c
2,1
Q is independent of h . This result is not robust to changes in the value of p. It
1,2
c
is only true when p"1. This is discussed in more detail in the next section.
The intuition for how indeterminacy can result with an externality in
the investment sector, even with none in the consumption sector, is the following. When agents expect the return on capital to increase tomorrow, they
need incentive to give up consumption today for investment. As long as
they will be rewarded with productive investment, in the form of increasing
returns in that sector, it will be worthwhile for them to do so. Why is the
same not true for increasing returns to scale in consumption? Higher returns
to scale in consumption will keep them from moving into investment today
and tomorrow's price of capital will not increase enough to keep the economy
in equilibrium.
5. Indeterminacy when pO1
In this section, I present two results regarding the indeterminacy properties of
the model under di!erent parameterizations for p, the inverse of the intertemporal elasticity of substitution of consumption. The "rst result encompasses the
fact that Proposition 1 is robust to changes in p. That is, with any CRRA utility
function, indeterminacy results when there is a certain size externality in the
investment sector, even if there is none in the consumption sector. The result is
that the size of the externality needed in the investment sector changes in
a systematic way with the value of p. The second result is that independence
from h is not robust to changes in p. Changing h a!ects the indeterminacy
c
c
properties of the model when pO1.
The "rst result is summarized in the following proposition. Here I set h "0.
c
Proposition 2. As p increases, the minimum value of h necessary for indeterminacy
I
increases. As agents become more risk averse, higher returns to scale in investment
are needed in order for indeterminacy to result.
Fig. 2 illustrates this property. For each p there is a minimum value of h ,
I
denoted h , above which indeterminacy results. With p on the horizontal axis
.*/
and h on the vertical axis, the curve is upward sloping. As p increases, so does
.*/
h . For example, with p"0.1, h "0.0237; with p"0.5, h "0.0618; with
.*/
.*/
.*/
p"1, h "0.0774; and with p"1.5, h "0.0845.
.*/
.*/
758
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Fig. 2. Minimum value of h needed for indeterminacy, given a p.
l
The intuition for this comes from the idea that increasing returns to scale
leads to two things. The "rst is the bene"t of raised productivity. The second is
the cost in utility terms of the #uctuations resulting from indeterminacy. As
p increases, agents become more risk averse and dislike more the #uctuations
that result from moving resources between the two sectors. Therefore, higher
bene"ts from higher increasing returns to scale are needed to compensate them
for living with these #uctuations.
The next proposition elaborates on the second result. Here I allow for values
of h on both sides of zero.8
c
Proposition 3. As h increases, the value of h necessary for indeterminacy changes
c
I
diwerently, depending on the value of p. (1) When p"1, the indeterminacy properties of the model are completely independent of h . For any h , indeterminacy results
c
c
with h 'h , where the value of h does not change with h . (2) When p(1, as
I
.*/
.*/
c
h increases, h
decreases. As returns to scale in consumption increase, lower
c
.*/
returns to scale are needed in investment to get indeterminacy. (3) When p'1, as
8 Given the results in Harrison (1997), it is not unreasonable to pose negative values for the
externality in the consumption sector.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
759
Fig. 3. Minimum value of h needed for indeterminacy, given a h .
l
c
h increases, h
increases. As returns to scale in consumption increase, higher
c
.*/
returns to scale are needed in investment to get indeterminacy.
Result (1) has already been proven. Fig. 3 illustrates each of these results,
using three values of p. Above each line, indeterminacy results for the given
value of p. With h on the horizontal axis and h
on the vertical axis, when
c
.*/
p"1, the line is perfectly horizontal. Indeterminacy results when h 50.0774,
I
for any value of h . When p"2, the line is upward sloping. When p"0.4, the
c
line is downward sloping.
To understand the intuition for the results it is useful to think about the
meaning of the intertemporal elasticity of substitution, 1/p, which tells us how
consumption responds intertemporally to changes in the interest rate. When the
interest rate increases by 1%, the ratio of tomorrow's to today's consumption
increases by 1/p%. When p(1, the substitution e!ect is stronger than the
income e!ect and a change in the interest rate causes relatively big movements in
consumption. When p'1, the income e!ect is stronger than the substitution
e!ect and a change in the interest rate causes relatively small changes in
consumption. In other words, a higher p is associated with smoother consumption. When p"1, the income and substitution e!ects cancel and consumption
changes by the same percentage as the interest rate.
760
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
The presence of increasing returns to scale in consumption has two opposing
e!ects on the intertemporal choice of consumption. The "rst is a smoothing
ewect. Increasing returns to scale in consumption discourages consumers from
drawing resources out of consumption today, because those resources are very
productive in the consumption sector. In other words, it encourages smooth
consumption. The second is a volatility ewect. Consumers will be encouraged to
take resources out of consumption today because they know that tomorrow
they can get a lot of the consumption good. In other words, it will encourage
volatile consumption.
Now, how the consumers might react to an increase in the level of returns to
scale in the consumption sector? When p(1, consumption is relatively volatile.
When returns to scale in consumption increase, the volatility e!ect will dominate. Consumers will become more willing to sacri"ce consumption today so
lower returns to scale are needed in investment in order for consumption to
change. When p'1, consumption is relatively smooth and the smoothing e!ect
will dominate. Consumers will not want to draw resources out of consumption
so higher returns to scale in investment will be necessary. When p"1, the two
e!ects cancel. The value of h will not a!ect consumers' intertemporal choices so
c
the necessary value of h will not change.
I
6. Discussion
In this section, I address two important issues. The "rst is the well known fact
that some of the time series properties of this model are not consistent with the
data on US business cycles. For example, for reasonable values of the externality
parameters, the model generates a time series for consumption that is countercyclical. This has been documented by various authors, including Benhabib and
Farmer (1996); Schmitt-Grohe (1998) and Weder (1999). I extend the analysis
done by these authors by examining the relative standard deviation and cyclicality of consumption under di!erent parameterizations of h , h and p. The second
I c
issue is that empirical evidence points to di!erent capital shares across sectors.
I discuss a version of the model that incorporates this and "nd that the results in
Propositions 1}3 are not a!ected.
6.1. Time series properties of consumption
It is well known that, though this model results in indeterminacy for reasonable values of the externality parameters, some of the time series properties
implied by the model are counterfactual. In particular, the model produces
a time series for consumption that is countercyclical.
In this subsection, I examine how incorporating the #exibilities allowed in this
paper a!ects the relative standard deviation and cyclicality of consumption.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
761
Table 4
Time series properties of consumption
h "h
c
I
h "0
c
p /p
c y
o(c, y)
p /p
c y
o(c, y)
p"0.5
h "0.1
I
h "0.2
I
h "0.3
I
0.38
0.69
0.95
!0.14
0.58
0.86
0.36
0.58
0.60
!0.24
0.39
0.61
p"1
h "0.1
I
h "0.2
I
h "0.3
I
0.20
0.53
0.64
!0.49
0.31
0.61
0.20
0.42
0.47
!0.51
0.18
0.46
p"2
h "0.1
I
h "0.2
I
h "0.3
I
0.10
0.27
0.40
!0.71
!0.06
0.38
0.09
0.20
0.28
!0.70
!0.10
0.23
Note: Statistics are based on logged data from simulations of length 1000. Fluctuations are driven
only by sunspots.
Table 4 presents the results. For each of three values of p, the values of p /p and
c y
o(c, y) are reported for three di!erent values of h . In one case, I set h "h . In the
I
c
I
other, I set h "0.
c
With regards to the relative standard deviation of consumption, the results in
Table 4 illustrate that consumption is more volatile when: (1) h is high; (2) h is
c
I
high; and (3) p is low. These results re#ect the familiar facts that consumers are
willing to live with more #uctuations when returns to scale are higher and when
risk aversion is lower.
With regards to the correlation of consumption with output, the results
illustrate that consumption is less countercyclical when: (1) h is high; (2) h is
c
I
high; and (3) p is low.9 This can be understood by examination of the household's intratemporal "rst-order condition, which equates the marginal utility of
leisure with the value in utility terms of the wage:
c~pw "ns.
t
t t
When #uctuations in the economy are driven by sunspots, an increase in output
and employment, which leads to a decrease in the wage (the marginal product of
labor), must be accompanied by a fall in consumption. Hence, consumption is
countercyclical. However, as the level of returns to scale in either sector
9 With logarithmic utility, consumption becomes procyclical at approximately h "0.2. This is
I
discussed in Benhabib and Farmer (1996) and Weder (1999).
762
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Fig. 4. Minimum value of h needed for indeterminacy, given a h .
l
c
increases, the marginal product of labor falls less (and eventually rises), and
hence consumption becomes less countercyclical (and eventually procyclical). As
the constant of relative risk aversion falls, consumption also becomes less
countercyclical.
6.2. Diwering capital shares
There is evidence in the empirical literature that the assumption of equal
capital shares does not hold. In this section I relax this restriction and "nd that
Propositions 1}3 are robust. Denoting the capital share in the consumption
sector a and that in the investment sector a , I set a "0.52 and a "0.32.
c
I
c
I
Keeping internal returns to scale constant, this implies labor shares of 0.48 and
0.68, respectively. These numbers are taken from Baxter (1996) and correspond
to her estimates of the input shares in the nondurable and durable goods sectors,
respectively. While I do not present the model here, Fig. 4, which is analogous to
Fig. 3, shows that each proposition carries through. That is, indeterminacy
results even with no externality in the consumption sector; as p increases,
a higher h is needed for indeterminacy; and the lines are sloped as in Fig. 3.
I
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
763
As Benhabib and Nishimura (1998) demonstrate, allowing for di!ering input
shares has some important implications; but doing so does not a!ect the results
in this paper.
7. Summary and conclusion
I have examined a model in which indeterminacy results due to the presence
of very small sector-speci"c external e!ects. With returns to scale of about 1.1 in
each sector, indeterminacy results. Further, indeterminacy results with this level
of returns to scale in investment, even with constant returns to scale in consumption. Given empirical evidence on the sizes of external e!ects, indeterminacy is
more likely empirically plausible in this model than in one sector models.
In addition, I have examined how the indeterminacy properties of the model
change with the shape of the utility function. In particular, indeterminacy is
easier to obtain when consumers are less risk averse with respect to consumption. In addition, in the special case of utility that is logarithmic in consumption,
the level of returns to scale in consumption has no e!ect at all on whether or not
indeterminacy results. Lastly, the countercyclicality of consumption is reduced
as returns to scale increase or risk aversion decreases.
These results provide insight into the mechanisms by which indeterminacy
results in this model. However, they also suggest that further research is
warranted. For example, allowing for non-separable utility or constant elasticity
of substitution in production could add to our understanding of models with
multiple, indeterminate equilibria.
References
Azariadis, C., 1981. Self-ful"lling prophecies. Journal of Economic Theory 25, 380}396.
Basu, S., Fernald, J.G., 1995. Are apparent productive spillovers a "gment of speci"cation error?
Journal of Monetary Economics 36, 165}188.
Baxter, M., 1996. Are consumer durables important for business cycles? Review of Economics and
Statistics 78, 147}155.
Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. Journal of Economic
Theory 63, 19}41.
Benhabib, J., Farmer, R.E.A., 1996. Indeterminacy and sector-speci"c externalities. Journal of
Monetary Economics 37, 397}419.
Benhabib, J., Nishimura, K., 1998. Indeterminacy and sunspots with constant returns. Journal of
Economic Theory 81, 58}96.
Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth.
Journal of Economic Theory 63, 113}142.
Boldrin, M., Rustichini, A., 1994. Growth and indeterminacy in dynamic models with externalities.
Econometrica 62, 323}342.
Burnside, C., 1996. Production function regressions, returns to scale and externalities. Journal of
Monetary Economics 37, 177}201.
764
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Cass, D., Shell, K., 1983. Do sunspots matter? Journal of Political Economy 91, 193}227.
Christiano, L.J., 1995. A discrete time version of the two sector model of Benhabib and Farmer,
Northwestern University, unpublished.
Christiano, L.J., Harrison, S.G., 1999. Chaos, sunspots and automatic stabilizers. Journal of
Monetary Economics, forthcoming.
Cooper, R., John, A., 1988. Coordinating coordination failures in Keynesian models. Quarterly
Journal of Economics 103, 441}463.
Diamond, P., 1982. Aggregate demand management in search equilibrium. Journal of Political
Economy 90, 881}894.
Farmer, R.E.A., Guo, J.T., 1994. Real business cycles and the animal spirits hypothesis. Journal of
Economic Theory 63, 42}72.
Harrison, S.G., 1997. Evidence on the empirical plausibility of externalities and indeterminacy in
a two sector model, Barnard College Working Paper d98-05.
Matsuyama, K., 1991. Increasing returns, industrialization and indeterminacy of equilibrium.
Quarterly Journal of Economics 106, 617}650.
Murphy, K.M., Shleifer, A., Vishny, R.W., 1989. Industrialization and the big push. Journal of
Political Economy 97, 1003}1026.
Perli, R., 1998. Indeterminacy, home production, and the business cycle: a calibrated analysis.
Journal of Monetary Economics 41, 105}125.
Schmitt-Grohe, S., 1998. Endogenous business cycles and the dynamics of output, hours and
consumption. Rutgers University, unpublished.
Weder, M., 1999. Animal spirits, technology shocks and the business cycle. Journal of Economic
Dynamics and Control, forthcoming.
25 (2001) 747}764
Indeterminacy in a model with sector-speci"c
externalities
Sharon G. Harrison*
Department of Economics; Barnard College, Columbia University; 3009 Broadway;
New York, NY, 10027, USA
Received 1 September 1998; received in revised form 1 March 1999; accepted 7 April 1999
Abstract
I examine a model with two sectors of production: consumption and investment. In the
model, indeterminacy of equilibria results due to the presence of small sector-speci"c
externalities in production. In fact, I "nd that indeterminacy results with a certain,
minimum value of the externality in the investment sector, even with no externality in the
consumption sector. I "nd that the indeterminacy properties of the model vary, depending on the form of the utility function. For example, with utility that is logarithmic in
consumption, these properties are completely independent of the value of the externality
in the consumption sector. ( 2001 Elsevier Science B.V. All rights reserved.
JEL classixcation: E00; E32
Keywords: Business cycles; Expectations; Indeterminacy; Production externalities
* Tel: 212-854-3333; fax: 212-854-8947. I gratefully acknowledge input from two anonymous
referees, Jess Benhabib, Lawrence Christiano, David Domeij, Martin Eichenbaum, Roger Farmer,
Duncan Foley, Jang-Ting Guo, Michael Horvath, Kiminori Matsuyama, Kay Robbins, Thomas
Sargent, Alberto Trejos, and several ex-colleagues from Northwestern University. All remaining
errors are, of course, my own.
E-mail address: [email protected] (S.G. Harrison).
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 4 0 - 8
748
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
1. Introduction
Much recent work in macroeconomics has focussed on models with multiple,
or indeterminate, equilibria.1 As this literature has developed, it has contributed
much to our understanding of dynamic macroeconomies. The possible existence
of multiple equilibria has important implications for those of us who study
business cycles, growth and monetary economies. In models of the business
cycle, for example, indeterminacy means that agents' expectations about the
future can be self-ful"lling, and therefore can serve as impulses to aggregate
#uctuations.
In this paper, I examine a model of the business cycle with two sectors of
production: consumption and investment, in which indeterminacy results due
to the presence of sector-speci"c externalities in production. I carry out an
extensive study of the indeterminacy properties of the model and "nd that
indeterminacy results with a certain, minimum value of the externality in the
investment sector, even if there is no externality in the consumption sector. In
addition, I "nd that when there is an externality in the consumption sector, the
size of the externality in the investment sector needed for indeterminacy depends
on this parameter and on the curvature of the utility function.
In models with one sector of production and increasing returns to scale,
indeterminacy results when returns to scale are su$ciently high. For example,
Benhabib and Farmer (1994); Farmer and Guo (1994) and Christiano and
Harrison (1999) specify general equilibrium models in which, with a certain size
externality, and constant internal returns to scale, indeterminacy results. This is
the case because when agents believe, for example, that the return on capital will
rise tomorrow, they act on this by raising tomorrow's capital stock. Given
su$ciently high increasing returns, so that the marginal product, which is equal
to the return on capital, is increasing in capital, the expectation will be ful"lled.
However, in these models, with realistic values of the other parameters, overall
returns to scale must be about 1.5 in order for indeterminacy to result.
More recent work by Benhabib and Farmer (1996) has demonstrated that in
a model with two sectors of production, much lower external e!ects are necessary for indeterminacy.2 With overall returns of about 1.1 in each sector, agents'
beliefs can be self-ful"lling. This follows because in this model, when agents
believe that the return on capital will rise tomorrow they reallocate factors of
1 Examples include Benhabib and Perli (1994); Boldrin and Rustichini (1994); Matsuyama (1991)
and Perli (1998) among others. In#uential early work includes Azariadis (1981); Cass and Shell
(1983); Cooper and John (1988); Diamond (1982) and Murphy et al. (1989), among others.
2 In fact, Benhabib and Nishimura (1998) examine a multi-sector model in which indeterminacy
results with constant returns to scale. In the model, internal returns are slightly decreasing (e.g. 0.93)
and there is a small externality.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
749
production between sectors. This leads to movements in the relative price of
investment that a!ect the return on capital in such a way that the marginal
product of capital need not be increasing in capital for the expectation to be
ful"lled.
Empirical evidence points towards the existence of very small, if any, production externalities at the aggregate level (See, for example Basu and Fernald, 1995
and Burnside, 1996). This evidence leads to rejection of the possibility of
indeterminacy in the one sector model. However, Harrison (1997) provides some
evidence that the externality for investment may be at least as large as 1.1. That
is, indeterminacy in the two sector model is more likely to be empirically
relevant than in the one sector model.
In this paper I present a discrete time version of the Benhabib and Farmer
two sector model. I carry out a comprehensive study of the indeterminacy
properties of the model. To do this, I relax their restriction that the two sectors
experience the same size externality. In addition, instead of restricting the utility
function to be logarithmic in consumption, I use the more general constant
relative risk aversion form. In this way, I can analyze the indeterminacy
properties of the model when these important parameters are allowed to vary.
The main results obtained from allowing these #exibilities are: (1) Indeterminacy results with a certain, minimum value of the externality in the investment
sector, even if there is no externality in the consumption sector; (2) As the
constant of relative risk aversion increases, so does the level of returns to scale
needed for indeterminacy; and (3) As the level of returns to scale in consumption
increases, the level of returns to scale in investment needed for indeterminacy
changes di!erently, depending on the value of the constant of relative risk
aversion.3 I provide numerical examples to support these results and discuss the
intuition for them. I also examine the cyclicality of consumption under di!erent
parameterizations. Lastly, I "nd that these results are robust to allowing the
capital share to di!er across sectors.
The rest of this paper proceeds as follows. Section 2 presents the model.
Sections 3}5 analyze its indeterminacy properties under di!erent parameterizations. Section 6 examines some time series properties of consumption in the
model and discusses a version with di!ering capital shares. Section 7 concludes.
2. The model
The model is a nonstochastic, discrete time version of the model of Benhabib
and Farmer (1996). The household derives utility from consumption and leisure;
3 Weder (1999) "nds the analogous result to (1) in a continuous time version of this model with
logarithmic utility and oligopolistic "rms.
750
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
and there are two sectors of production, consumption and investment. The "rm
in each sector produces with a constant returns to scale technology and a sector-speci"c externality, which is taken as given. While Benhabib and Farmer
restrict the size of the externality to be the same across sectors and utility to be
logarithmic in consumption, I relax these restrictions.
2.1. The household problem
The representative agent acts to maximize his present discounted value
of utility:
A
B
=
c1~p!1 n1`s
! t
max + bt t
1#s
1!p
M
N=
kt`1,nt,ct t/0 t/0
subject to:
c #p I 4r k #w n
t
t t
t t
t t
and
k !(1!d)k "I , k given.
(1)
t`1
t
t
0
Here c , n , k , and I denote consumption, labor, capital and investment in period
t t t
t
t, respectively. Also, p indicates the relative price of investment and the price of
t
consumption is normalized to 1. In addition, r is the rental rate on capital and
t
w is the wage rate. Lastly, 1/p is the intertemporal elasticity of substitution of
t
consumption and 1/s is the labor supply elasticity.
The "rst-order conditions for this problem are
r #(1!d)p
t`1"0
c~p!bc~p t`1
t`1
t
p
t
(2)
c~pw !ns"0.
t
t t
(3)
and
2.2. The xrms' problems
In the consumption sector, the "rm maximizes pro"t subject to the production function
f "(Ka N1~a )hcka n1~a
c,t c,t
c,t c,t
c,t
where k and n denote the capital and labor the "rm devotes to the consumpc,t
c,t
tion sector at time t and a is capital's share in production of the consumption
good. Also, K and N denote the economy-wide average capital and labor
c,t
c,t
devoted to the consumption sector, which are taken as given by the "rm.
The degree of sector-speci"c externality in the consumption sector is h .
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
751
The "rst-order conditions for this problem are
(1!a ) f
af
c,t"w .
c,t"r ,
(4)
t
t
n
k
c,t
c,t
Similarly, the "rm in the investment sector maximizes pro"t and produces
according to
f "(Ka N1~a)hIka n1~a,
I,t I,t
I,t I,t
I,t
where k and n denote the capital and labor the "rm devotes to the investI,t
I,t
ment sector at time t. K and N denote the economy-wide average capital
I,t
I,t
and labor devoted to the investment sector. Note that the degree of the
sector-speci"c externality for the investment sector, h , is allowed to di!er from
I
that of the consumption sector. This "rm's "rst-order conditions are
af
(1!a) f
I,t"w .
p I,t"r , p
tk
t
t
t
n
I,t
I,t
Since capital and labor are used only in the production of the two goods, it
must be true that
K #K "K and N #N "N ,
c,t
I,t
t
c,t
I,t
t
where K and N denote economy-wide averages of total capital and labor.
t
t
2.3. Equilibrium and steady state
An equilibrium is a set of prices, Mp , r , w N= such that given these prices, the
t t t t/0
quantities Mk , c , n N= solve the household and "rm problems. In addition,
t`1 t t t/0
the resource constraints are satis"ed.
In equilibrium,
f "(k kan1~a)1`hc
c,t
t t t
(5)
f "((1!k )kan1~a)1`hI,
I,t
t t t
where
(6)
and
k
n
k " c,t" c,t.
t
k
n
t
t
Substituting into the "rms' "rst-order conditions, we get an expression for the
relative price of investment:
khc
t
(kan1~a )hc~hI.
p"
t (1!k )hI t t
t
(7)
752
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Denoting steady-state values with no subscripts, the steady-state versions of
(1), (4) and (6) can be used to solve for r. This equation and the steady-state
version of (2) can be used to solve for k in terms of the parameters of the model.
Given this, the steady-state versions of (3)}(5); and (1) and (6) give two equations
in k and n. The rest of the steady-state values follow from these.
2.4. Dynamic behavior
The dynamics of this economy are summarized by the three equations
describing the household problem: (1)}(3); and the "ve equations describing the
"rms' problems: (4)}(7). Of these (1) and (2) are intertemporal. I log-linearize this
system of equations around steady state.4 Letting
AB
p(
t
IK
t
r(
kK
u " t , v " t , e "(mK ),
t
t
t
t
w(
c(
t
t
y(
t
n(
t
where m represents the sunspot shock and x( indicates the log deviation of x from
t
its steady-state value, I obtain
AB
(8)
v "Ju
t
t
and
u "Q u #Q e ,
t`1
1 t
2t
where
A
B
dJ #1!d
dJ
2,1
2,2
Q " !J !/(dJ #1!d) p!J !dJ / ,
1
1,1
2,1
1,2
2,2
U
U
/"!b(1!d)J #[ b (1!d)!1]J
1,1
3,1
(9)
and
U"!b(1!d)J #[b(1!d)!1]J #p.
1,2
3,2
4 Some of the notation used here is borrowed from the unpublished appendix to Benhabib and
Farmer (1996).
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
753
The J are elements of J in (8). Speci"cally,
i,j
a(1#h )(1#s)
(a!1)s
J " !ahI(1#s)
I
1,1
, J "
, J "
,
2,1
3,1
(1!k)(a#s)
(a#s)
(1!k)(a#s)
J "
1,2
(1#h )[(1#s)k#(1!a)(1!k)]!(1#s)#(p!1)(1!a)h (1#h )
I
I
c #1,
(1!k)(a#s)(1#h )
c
(1#h )[a(1!k)!ks!1)#(1!p)(1!a)(1#h )]
I
c
J "
2,2
(1!k)(a#s)(1#h )
c
and
h (1#s)!p(1!a)(1#h )
c.
J " c
3,2
(1#h )(a#s)
c
Denoting the eigenvalues of Q j and j , indeterminacy results when both
1 1
2
j and j are inside the unit circle. When the steady state is indeterminate, given
1
2
a k , there is more than one c that starts o! a path that satis"es the equilibrium
0
0
conditions. When more than one equilibrium path exists, equilibria driven by
sunspot shocks are possible.
3. Indeterminacy when h "h "h
c
I
In this section, I set b"1/1.01, d"0.025, a"0.3, p"1 and s"0. This is
one of the parameterizations used in Benhabib and Farmer (1996) and will serve
as the benchmark parameterization of this model. Note that setting s"0 is
equivalent to setting the labor supply elasticity equal to in"nity. Benhabib and
Farmer discuss how varying the elasticity of labor supply a!ects the ease with
which indeterminacy results. In particular, as the labor supply elasticity increases (s decreases), labor is drawn more easily out of leisure and lower
increasing returns are needed for indeterminacy.
Using Benhabib and Farmer's speci"cation as a starting point, i.e. restricting
h "h "h, Table 1 reports values of j and j associated with di!erent values
c
I
1
2
of h.5 The h"0 case simply corresponds to the standard one-sector model. With
h"0.0773, the equilibrium of the model is still determinate. However, indeterminacy results when h50.0774.6
5 See Christiano (1995) for more examples.
6 More precisely, at approximately h"0.077369, the economy undergoes a #ip bifurcation
whereby one eigenvalue is exactly equal to !1 and the other lies within the unit circle. In this case,
the model may converge to a stable two-cycle for any initial condition that lies o! the steady state.
754
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Table 1
Indeterminacy when h "h "h
c
I
h
j
0
0.0773
0.0774
0.08
0.93
0.46
0.41
0.81!0.36i
1
j
2
1.09
!1.51
!0.82
0.81#0.36i
Indeterminacy?
No
No
Yes
Yes
In order to understand the intuition for the idea that indeterminacy results in
this model with much lower returns to scale than in the one sector model,
I compare the intertemporal "rst-order conditions for the one and two sector
models. Examination of the two sector model also requires an understanding of
the production possibilities frontier (PPF) in this case.
In the one sector model the intertemporal "rst-order condition is
A B
p
t`1 "b[r #(1!d)].
t`1
c
t
Starting from a steady-state equilibrium, in which case this equation is satis"ed,
when the return on capital is expected to increase tomorrow, consumption is
sacri"ced for investment, and is less than its steady-state value. Tomorrow's
capital stock and consumption increase. The left-hand side of the equation
increases, relative to steady state. In order to stay in equilibrium, so must the
right-hand side. With high enough returns to scale, tomorrow's marginal product of capital will increase with tomorrow's capital stock and agents' expectations will be self-ful"lling.
For the two sector model the condition is
c
A B
p
[r #(1!d)p ]
t`1 .
t`1 "b t`1
p
c
t
t
In order to understand why it is that indeterminacy can result with lower returns
to scale here, it is important to examine the PPF for the social planner in this
economy. Fig. 1 plots the PPF for the case where h "h '0. As Benhabib and
c
I
Farmer discuss, this PPF is convex. Due to the increasing returns to scale, as
resources are shifted into a good, the marginal product of each factor used in
production of that good increases. (Even if returns to scale are not large enough
to make marginal products increasing in individual inputs, they are increasing in
both inputs.) At the optimal solution, the slope of the PPF is (the negative of )
the relative price of investment, p. Because of the increasing returns to scale,
a shift of resources towards the production of a good lowers the price of that
good. Recall that k represents the share of each factor used in production of the
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
755
Fig. 1. PPF of economy with equal externalities.
consumption good. As k increases, and more of the consumption good is
produced, the curve gets steeper, increasing the relative price of investment.7
Now, when tomorrow's consumption increases and the left-hand side of the
equation increases relative to steady state, tomorrow's relative price of capital
increases as resources are shifted towards consumption. Returns to scale do not
have to be as high as they were in the one sector case. Indeterminacy can result
even if the marginal product of capital is decreasing in capital.
4. Indeterminacy when h Oh
c
I
In this section, I use the benchmark parameterization and discuss the indeterminacy properties of the model when h Oh . First, I "x h "0.0774 and
c
I
c
calculate j and j under various parameterizations of h . Then I "x h "0.0774
1
2
I
I
7 When h Oh it can be shown that the PPF is convex as long as there is increasing returns to
c
I
scale in at least one sector.
756
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Table 2
Indeterminacy when h "0.0774
c
h
I
0
0.0773
0.0774
0.08
j
1
0.93
0.46
0.41
0.81!0.36i
j
2
Indeterminacy?
1.09
!1.51
!0.82
0.81!0.36i
No
No
Yes
Yes
j
Indeterminacy?
Table 3
Indeterminacy when h "0.0774
I
h
c
0
0.0773
0.0774
0.08
j
1
0.41
0.41
0.41
0.41
2
!0.82
!0.82
!0.82
!0.82
Yes
Yes
Yes
Yes
and repeat the experiment, varying h . The main result of this analysis is
c
Proposition 1. Indeterminacy results when there is a certain size externality in the
investment sector, even if there is no externality in the consumption sector.
The size of the externality needed is the same as when h "h "h. Tables
c
I
2 and 3 illustrate this. In Table 2, h "0.0774. With h "0, the equilibrium of the
c
I
model is determinate. The same is true when h "0.0773. Once h 50.0774,
I
I
indeterminacy results.
In Table 3, h "0.0774. Regardless of the value of h , the equilibrium of the
I
c
model is indeterminate. The reader will also notice from the results in Table 3
that as h changes, j and j do not. In other words, the indeterminacy
c
1
2
properties of the model are completely independent of the value of the externality in the consumption sector. This is easily seen by setting p"1 and
solving for the eigenvalues of Q . First, note that these depend on the values of
1
the individual elements Q and Q and on the product of Q and Q . It
1,1
2,2
1,2
2,1
turns out that when p"1, none of these depend on h . To see this, "rst note that
c
Q never depends on h . Next, setting p"1, U reduces to
1,1
c
i
1
,
U"
(1!k)(a#s)(1#h )
c
where i is a function of various parameters, but not of h . As a result, the rest of
1
c
the elements of Q reduce to:
1
(1!k)(a#s)(1#h )i
i
c 2, Q "
3
Q "
2,1
1,2
i
(1!k)(a#s)(1#h )
1
c
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
757
and
i /(1!k)(a#s)(1#h ) i
c " 4,
Q " 1
2,2
i
U
1
where i , i and i are also not dependent on h , so that the product of Q and
2 3
4
c
2,1
Q is independent of h . This result is not robust to changes in the value of p. It
1,2
c
is only true when p"1. This is discussed in more detail in the next section.
The intuition for how indeterminacy can result with an externality in
the investment sector, even with none in the consumption sector, is the following. When agents expect the return on capital to increase tomorrow, they
need incentive to give up consumption today for investment. As long as
they will be rewarded with productive investment, in the form of increasing
returns in that sector, it will be worthwhile for them to do so. Why is the
same not true for increasing returns to scale in consumption? Higher returns
to scale in consumption will keep them from moving into investment today
and tomorrow's price of capital will not increase enough to keep the economy
in equilibrium.
5. Indeterminacy when pO1
In this section, I present two results regarding the indeterminacy properties of
the model under di!erent parameterizations for p, the inverse of the intertemporal elasticity of substitution of consumption. The "rst result encompasses the
fact that Proposition 1 is robust to changes in p. That is, with any CRRA utility
function, indeterminacy results when there is a certain size externality in the
investment sector, even if there is none in the consumption sector. The result is
that the size of the externality needed in the investment sector changes in
a systematic way with the value of p. The second result is that independence
from h is not robust to changes in p. Changing h a!ects the indeterminacy
c
c
properties of the model when pO1.
The "rst result is summarized in the following proposition. Here I set h "0.
c
Proposition 2. As p increases, the minimum value of h necessary for indeterminacy
I
increases. As agents become more risk averse, higher returns to scale in investment
are needed in order for indeterminacy to result.
Fig. 2 illustrates this property. For each p there is a minimum value of h ,
I
denoted h , above which indeterminacy results. With p on the horizontal axis
.*/
and h on the vertical axis, the curve is upward sloping. As p increases, so does
.*/
h . For example, with p"0.1, h "0.0237; with p"0.5, h "0.0618; with
.*/
.*/
.*/
p"1, h "0.0774; and with p"1.5, h "0.0845.
.*/
.*/
758
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Fig. 2. Minimum value of h needed for indeterminacy, given a p.
l
The intuition for this comes from the idea that increasing returns to scale
leads to two things. The "rst is the bene"t of raised productivity. The second is
the cost in utility terms of the #uctuations resulting from indeterminacy. As
p increases, agents become more risk averse and dislike more the #uctuations
that result from moving resources between the two sectors. Therefore, higher
bene"ts from higher increasing returns to scale are needed to compensate them
for living with these #uctuations.
The next proposition elaborates on the second result. Here I allow for values
of h on both sides of zero.8
c
Proposition 3. As h increases, the value of h necessary for indeterminacy changes
c
I
diwerently, depending on the value of p. (1) When p"1, the indeterminacy properties of the model are completely independent of h . For any h , indeterminacy results
c
c
with h 'h , where the value of h does not change with h . (2) When p(1, as
I
.*/
.*/
c
h increases, h
decreases. As returns to scale in consumption increase, lower
c
.*/
returns to scale are needed in investment to get indeterminacy. (3) When p'1, as
8 Given the results in Harrison (1997), it is not unreasonable to pose negative values for the
externality in the consumption sector.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
759
Fig. 3. Minimum value of h needed for indeterminacy, given a h .
l
c
h increases, h
increases. As returns to scale in consumption increase, higher
c
.*/
returns to scale are needed in investment to get indeterminacy.
Result (1) has already been proven. Fig. 3 illustrates each of these results,
using three values of p. Above each line, indeterminacy results for the given
value of p. With h on the horizontal axis and h
on the vertical axis, when
c
.*/
p"1, the line is perfectly horizontal. Indeterminacy results when h 50.0774,
I
for any value of h . When p"2, the line is upward sloping. When p"0.4, the
c
line is downward sloping.
To understand the intuition for the results it is useful to think about the
meaning of the intertemporal elasticity of substitution, 1/p, which tells us how
consumption responds intertemporally to changes in the interest rate. When the
interest rate increases by 1%, the ratio of tomorrow's to today's consumption
increases by 1/p%. When p(1, the substitution e!ect is stronger than the
income e!ect and a change in the interest rate causes relatively big movements in
consumption. When p'1, the income e!ect is stronger than the substitution
e!ect and a change in the interest rate causes relatively small changes in
consumption. In other words, a higher p is associated with smoother consumption. When p"1, the income and substitution e!ects cancel and consumption
changes by the same percentage as the interest rate.
760
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
The presence of increasing returns to scale in consumption has two opposing
e!ects on the intertemporal choice of consumption. The "rst is a smoothing
ewect. Increasing returns to scale in consumption discourages consumers from
drawing resources out of consumption today, because those resources are very
productive in the consumption sector. In other words, it encourages smooth
consumption. The second is a volatility ewect. Consumers will be encouraged to
take resources out of consumption today because they know that tomorrow
they can get a lot of the consumption good. In other words, it will encourage
volatile consumption.
Now, how the consumers might react to an increase in the level of returns to
scale in the consumption sector? When p(1, consumption is relatively volatile.
When returns to scale in consumption increase, the volatility e!ect will dominate. Consumers will become more willing to sacri"ce consumption today so
lower returns to scale are needed in investment in order for consumption to
change. When p'1, consumption is relatively smooth and the smoothing e!ect
will dominate. Consumers will not want to draw resources out of consumption
so higher returns to scale in investment will be necessary. When p"1, the two
e!ects cancel. The value of h will not a!ect consumers' intertemporal choices so
c
the necessary value of h will not change.
I
6. Discussion
In this section, I address two important issues. The "rst is the well known fact
that some of the time series properties of this model are not consistent with the
data on US business cycles. For example, for reasonable values of the externality
parameters, the model generates a time series for consumption that is countercyclical. This has been documented by various authors, including Benhabib and
Farmer (1996); Schmitt-Grohe (1998) and Weder (1999). I extend the analysis
done by these authors by examining the relative standard deviation and cyclicality of consumption under di!erent parameterizations of h , h and p. The second
I c
issue is that empirical evidence points to di!erent capital shares across sectors.
I discuss a version of the model that incorporates this and "nd that the results in
Propositions 1}3 are not a!ected.
6.1. Time series properties of consumption
It is well known that, though this model results in indeterminacy for reasonable values of the externality parameters, some of the time series properties
implied by the model are counterfactual. In particular, the model produces
a time series for consumption that is countercyclical.
In this subsection, I examine how incorporating the #exibilities allowed in this
paper a!ects the relative standard deviation and cyclicality of consumption.
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
761
Table 4
Time series properties of consumption
h "h
c
I
h "0
c
p /p
c y
o(c, y)
p /p
c y
o(c, y)
p"0.5
h "0.1
I
h "0.2
I
h "0.3
I
0.38
0.69
0.95
!0.14
0.58
0.86
0.36
0.58
0.60
!0.24
0.39
0.61
p"1
h "0.1
I
h "0.2
I
h "0.3
I
0.20
0.53
0.64
!0.49
0.31
0.61
0.20
0.42
0.47
!0.51
0.18
0.46
p"2
h "0.1
I
h "0.2
I
h "0.3
I
0.10
0.27
0.40
!0.71
!0.06
0.38
0.09
0.20
0.28
!0.70
!0.10
0.23
Note: Statistics are based on logged data from simulations of length 1000. Fluctuations are driven
only by sunspots.
Table 4 presents the results. For each of three values of p, the values of p /p and
c y
o(c, y) are reported for three di!erent values of h . In one case, I set h "h . In the
I
c
I
other, I set h "0.
c
With regards to the relative standard deviation of consumption, the results in
Table 4 illustrate that consumption is more volatile when: (1) h is high; (2) h is
c
I
high; and (3) p is low. These results re#ect the familiar facts that consumers are
willing to live with more #uctuations when returns to scale are higher and when
risk aversion is lower.
With regards to the correlation of consumption with output, the results
illustrate that consumption is less countercyclical when: (1) h is high; (2) h is
c
I
high; and (3) p is low.9 This can be understood by examination of the household's intratemporal "rst-order condition, which equates the marginal utility of
leisure with the value in utility terms of the wage:
c~pw "ns.
t
t t
When #uctuations in the economy are driven by sunspots, an increase in output
and employment, which leads to a decrease in the wage (the marginal product of
labor), must be accompanied by a fall in consumption. Hence, consumption is
countercyclical. However, as the level of returns to scale in either sector
9 With logarithmic utility, consumption becomes procyclical at approximately h "0.2. This is
I
discussed in Benhabib and Farmer (1996) and Weder (1999).
762
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
Fig. 4. Minimum value of h needed for indeterminacy, given a h .
l
c
increases, the marginal product of labor falls less (and eventually rises), and
hence consumption becomes less countercyclical (and eventually procyclical). As
the constant of relative risk aversion falls, consumption also becomes less
countercyclical.
6.2. Diwering capital shares
There is evidence in the empirical literature that the assumption of equal
capital shares does not hold. In this section I relax this restriction and "nd that
Propositions 1}3 are robust. Denoting the capital share in the consumption
sector a and that in the investment sector a , I set a "0.52 and a "0.32.
c
I
c
I
Keeping internal returns to scale constant, this implies labor shares of 0.48 and
0.68, respectively. These numbers are taken from Baxter (1996) and correspond
to her estimates of the input shares in the nondurable and durable goods sectors,
respectively. While I do not present the model here, Fig. 4, which is analogous to
Fig. 3, shows that each proposition carries through. That is, indeterminacy
results even with no externality in the consumption sector; as p increases,
a higher h is needed for indeterminacy; and the lines are sloped as in Fig. 3.
I
S.G. Harrison / Journal of Economic Dynamics & Control 25 (2001) 747}764
763
As Benhabib and Nishimura (1998) demonstrate, allowing for di!ering input
shares has some important implications; but doing so does not a!ect the results
in this paper.
7. Summary and conclusion
I have examined a model in which indeterminacy results due to the presence
of very small sector-speci"c external e!ects. With returns to scale of about 1.1 in
each sector, indeterminacy results. Further, indeterminacy results with this level
of returns to scale in investment, even with constant returns to scale in consumption. Given empirical evidence on the sizes of external e!ects, indeterminacy is
more likely empirically plausible in this model than in one sector models.
In addition, I have examined how the indeterminacy properties of the model
change with the shape of the utility function. In particular, indeterminacy is
easier to obtain when consumers are less risk averse with respect to consumption. In addition, in the special case of utility that is logarithmic in consumption,
the level of returns to scale in consumption has no e!ect at all on whether or not
indeterminacy results. Lastly, the countercyclicality of consumption is reduced
as returns to scale increase or risk aversion decreases.
These results provide insight into the mechanisms by which indeterminacy
results in this model. However, they also suggest that further research is
warranted. For example, allowing for non-separable utility or constant elasticity
of substitution in production could add to our understanding of models with
multiple, indeterminate equilibria.
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