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, depend-ing 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:sh411@columbia.edu (S.G. Harrison).
1. Introduction
Much recent work in macroeconomics has focussed on models with multiple, or indeterminate, equilibria.1As 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 neces-sary for indeterminacy.2With 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
1Examples 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.
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, produc-tion 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) Indetermin-acy 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.3I 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 parameteriz-ations. 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;
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 sec-tor-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:
t,nt,kt, andItdenote consumption, labor, capital and investment in period
t, respectively. Also,p
tindicates the relative price of investment and the price of
consumption is normalized to 1. In addition,r
tis the rental rate on capital and
w
tis the wage rate. Lastly, 1/pis the intertemporal elasticity of substitution of
consumption and 1/sis the labor supply elasticity. The"rst-order conditions for this problem are
ct~p!bct`1~p rt`1#(1!d)pt`1
In the consumption sector, the"rm maximizes pro"t subject to the produc-tion funcproduc-tion
f
c,t"(Kac,tN1~c,t a)hckac,tn1~c,t a
wherek
c,tandnc,tdenote the capital and labor the"rm devotes to the
consump-tion sector at timetandais capital's share in production of the consumption good. Also, K
c,t andNc,t denote the economy-wide average capital and labor
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
The"rst-order conditions for this problem are
Similarly, the "rm in the investment sector maximizes pro"t and produces according to
f
I,t"(KaI,tN1~I,ta)hIkaI,tn1~I,ta,
wherek
I,tandnI,tdenote the capital and labor the"rm devotes to the
invest-ment sector at timet. K
I,t and NI,t denote the economy-wide average capital
and labor devoted to the investment sector. Note that the degree of the sector-speci"c externality for the investment sector,h
I, is allowed to di!er from
that of the consumption sector. This"rm's"rst-order conditions are
p
Since capital and labor are used only in the production of the two goods, it must be true that
K
c,t#KI,t"Kt and Nc,t#NI,t"Nt,
whereK
tandNtdenote economy-wide averages of total capital and labor.
2.3. Equilibrium and steady state
An equilibrium is a set of prices,Mp
t,rt,wtN=t/0such that given these prices, the
quantitiesMk
t`1,ct,ntN=t/0solve the household and"rm problems. In addition,
the resource constraints are satis"ed. In equilibrium,
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 forkin terms of the parameters of the model. Given this, the steady-state versions of (3)}(5); and (1) and (6) give two equations inkandn. 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.4Letting
u
trepresents the sunspot shock andx( indicates the log deviation ofxfrom
its steady-state value, I obtain
v
TheJ
i,jare elements ofJ in (8). Speci"cally,
J
Denoting the eigenvalues ofQ
1j1andj2, indeterminacy results when both
j1andj2are inside the unit circle. When the steady state is indeterminate, given ak
0, there is more than onec0that starts o!a path that satis"es the equilibrium
conditions. When more than one equilibrium path exists, equilibria driven by sunspot shocks are possible.
3. Indeterminacy whenh
c"hI"h
In this section, I set b"1/1.01,d"0.025,a"0.3,p"1 ands"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 in-creases (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
c"hI"h, Table 1 reports values ofj1andj2associated with di!erent values
ofh.5Theh"0 case simply corresponds to the standard one-sector model. With
h"0.0773, the equilibrium of the model is still determinate. However,
indeter-minacy results whenh50.0774.6 5See Christiano (1995) for more examples.
Table 1
Indeterminacy whenhc"hI"h
h j1 j2 Indeterminacy?
0 0.93 1.09 No
0.0773 0.46 !1.51 No
0.0774 0.41 !0.82 Yes
0.08 0.81!0.36i 0.81#0.36i 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
cStarting 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 prod-uct of capital will increase with tomorrow's capital stock and agents' expecta-tions will be self-ful"lling.
For the two sector model the condition is
A
cIn 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 whereh
c"hI'0. As Benhabib and
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 whenh
cOhI
In this section, I use the benchmark parameterization and discuss the indeter-minacy properties of the model when h
cOhI. First, I "x hc"0.0774 and
calculatej1andj2under various parameterizations ofhI. Then I"xhI"0.0774
Table 2
Indeterminacy whenhc"0.0774
hI j1 j2 Indeterminacy?
Indeterminacy whenhI"0.0774
h
c j1 j2 Indeterminacy?
0 0.41 !0.82 Yes
0.0773 0.41 !0.82 Yes
0.0774 0.41 !0.82 Yes
0.08 0.41 !0.82 Yes
and repeat the experiment, varyinghc. The main result of this analysis is
Proposition1. 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
c"hI"h. Tables
2 and 3 illustrate this. In Table 2,h
c"0.0774. WithhI"0, the equilibrium of the
model is determinate. The same is true when h
I"0.0773. Once hI50.0774,
indeterminacy results. In Table 3,h
I"0.0774. Regardless of the value ofhc, the equilibrium of the
model is indeterminate. The reader will also notice from the results in Table 3 that as h
c changes, j1 and j2 do not. In other words, the indeterminacy
properties of the model are completely independent of the value of the ex-ternality in the consumption sector. This is easily seen by setting p"1 and solving for the eigenvalues ofQ1. First, note that these depend on the values of the individual elementsQ
1,1andQ2,2and on the product ofQ1,2 andQ2,1. It
turns out that whenp"1, none of these depend onhc. To see this,"rst note that
Q
1,1never depends onhc. Next, settingp"1,Ureduces to
U" i1
(1!k)(a#s)(1#h
c)
,
wherei1is a function of various parameters, but not ofh
and
1,2is independent ofhc. This result is not robust to changes in the value ofp. It
is only true whenp"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 follow-ing. 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 whenpO1
In this section, I present two results regarding the indeterminacy properties of the model under di!erent parameterizations forp, the inverse of the intertem-poral elasticity of substitution of consumption. The"rst result encompasses the fact that Proposition 1 is robust to changes inp. 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
c is not robust to changes in p. Changing hc a!ects the indeterminacy
properties of the model whenpO1.
The"rst result is summarized in the following proposition. Here I seth
c"0.
Proposition2. Aspincreases,the minimumvalue ofh
Inecessary for indeterminacy
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 eachp there is a minimum value ofhI, denotedh.*/, above which indeterminacy results. Withpon the horizontal axis andh
.*/on the vertical axis, the curve is upward sloping. Aspincreases, so does
h
.*/. For example, withp"0.1,h.*/"0.0237; withp"0.5,h.*/"0.0618; with
p"1,h
Fig. 2. Minimum value ofhlneeded for indeterminacy, given ap.
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
pincreases, 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 ofh
con both sides of zero.8
Proposition3. Ashcincreases,thevalue ofhInecessary for indeterminacy changes
diwerently,depending on thevalue ofp. (1)Whenp"1,the indeterminacy proper-ties of the model are completely independent ofh
c.For anyhc,indeterminacy results
withh
I'h.*/,where thevalue ofh.*/does not change withhc. (2)Whenp(1,as
h
cincreases, h.*/ decreases. As returns to scale in consumption increase, lower
returns to scale are needed in investment to get indeterminacy.(3)Whenp'1,as
Fig. 3. Minimum value ofhlneeded for indeterminacy, given ahc.
h
cincreases, h.*/ increases. As returns to scale in consumption increase, higher
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. Withh
con the horizontal axis and h.*/ on the vertical axis, when
p"1, the line is perfectly horizontal. Indeterminacy results whenh
I50.0774,
for any value ofh
c. Whenp"2, the line is upward sloping. Whenp"0.4, the
line is downward sloping.
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 avolatility 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? Whenp(1, consumption is relatively volatile. When returns to scale in consumption increase, the volatility e!ect will domin-ate. 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. Whenp'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. Whenp"1, the two e!ects cancel. The value ofh
cwill not a!ect consumers'intertemporal choices so
the necessary value ofh
Iwill not change.
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 counter-cyclical. 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 cyclical-ity of consumption under di!erent parameterizations ofh
I,hcandp. The second
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 reason-able 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.
Table 4
Time series properties of consumption
hc"h
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 ofp, the values ofp
c/pyand
o(c,y) are reported for three di!erent values ofh
I. In one case, I sethc"hI. In the
other, I seth
c"0.
With regards to the relative standard deviation of consumption, the results in Table 4 illustrate that consumption is more volatile when: (1)hcis high; (2)hIis high; and (3)pis 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
cis high; (2) hIis
high; and (3) pis low.9This can be understood by examination of the house-hold's intratemporal"rst-order condition, which equates the marginal utility of leisure with the value in utility terms of the wage:
c~t pw
t"nst.
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
Fig. 4. Minimum value ofhlneeded for indeterminacy, given ahc.
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
c and that in the investment sector aI, I set ac"0.52 and aI"0.32.
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
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 consump-tion. 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 consump-tion. 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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