Econometrica, Vol. 68, No. 6 ŽNovember, 2000., 1541᎐1548

INDETERMINACY UNDER CONSTANT RETURNS TO SCALE IN MULTISECTOR ECONOMIES BY JESS BENHABIB, QINGLAI MENG,

1.

AND

KAZUO NISHIMURA1

INTRODUCTION

RECENTLY THERE HAS BEEN a renewed interest in indeterminacy, or alternatively put, in the existence of a continuum of equilibria in dynamic economies that exhibit some market imperfections.2 One of the primary concerns of this literature has been the empirical plausibility of indeterminacy, which arises in markets with external effects or with monopolistic competition, often coupled with some degree of increasing returns. While the early results on indeterminacy relied on relatively large increasing returns and high markups, more recently Benhabib and Farmer Ž1996. showed that indeterminacy can also occur in two-sector models with small sector-specific external effects and very mild increasing returns. Nevertheless, a number of empirical researchers, refining the earlier findings of Hall Ž1990. on disaggregated US data, have concluded that returns to scale seem to be roughly constant, if not decreasing.3 While one can argue whether the degree of increasing returns required for indeterminacy in Benhabib and Farmer Ž1996. falls within the standard errors of the recent empirical estimates, one may also inquire whether increasing returns are at all needed for indeterminacy to arise in a plausible manner. In this paper we argue that in multisector models indeterminacy can arise as a type of coordination problem, even without increasing returns, if there is a small wedge between private and social returns. In simple one-sector models increasing returns, sustained in a market context by external effects or monopolistic competition, can create a coordination problem. In such a setting, if all agents were to simultaneously increase their investment in an asset above the level associated with the initial equilibrium, the rate of return on that asset would tend to increase, justifying the higher level of investment. In a multisector model however, the rates of return and marginal products depend not only on the stocks of assets, but also on the composition of output across sectors. The rate of return on an asset can increase with the stock of the asset even in the absence of increasing returns. For example, consider a two-sector model with a pure consumption and a pure capital good. Increasing the relative price and hence the output of the capital good by moving along the production possibility frontier will increase the marginal product of the capital good if it is relatively more capital intensive. When combined with market distortions and external effects, the consequent rise in the stock of capital may not be enough to offset the initial increase of its marginal product. Both the stock and the marginal product of the capital good would rise simultaneously, mimicking the effect 1 We wish to thank a co-editor, Danyang Xie, and anonymous referees for very useful comments. Support from the C. V. Starr Center for Applied Economics at New York University is gratefully acknowledged. 2 See, for example, Benhabib, and Farmer Ž1994., Benhabib and Perli Ž1994., Benhabib, Perli, and Xie Ž1994., Boldrin and Rustichini Ž1994., Bond, Wang, and Yip Ž1996., Schmitt-Grohe ´ Ž1997., or Xie Ž1994.. 3 See Basu and Fernald Ž1997..

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of increasing returns in the one-sector model. It is therefore possible to have constant aggregate returns in all sectors at the social level, and still to obtain indeterminacy with minor or even negligible external effects in some of the sectors. We illustrate this in the next section in the context of a two-sector endogenous growth model and discuss some extensions. 2.

AN ENDOGENOUS GROWTH MODEL WITH NONLINEAR UTILITY

We consider an economy without fixed factors that exhibits unbounded growth. A representative agent optimizes an additively separable utility function with discount rate Ž r y g . ) 0 and g is the depreciation rate. We have Ž 2.1.

max

⬁

H0 U Ž c . e

yŽ ryg .t

dt

subject to Ž 2.2.

2

yj s e j Ł Ž x i j .

␤i j

Ž j s 1, 2 . ,

is1

Ž 2.3. Ž 2.4. Ž 2.5.

dx 1 dt dx 2 dt

s y 1 y gx 1 y c, s y 2 y gx 2 ,

2

Ý xi j s xi

Ž i s 1, 2 . ,

js1

where we assume that ␤i j ) 0.4 We specify the utility function as UŽ c . s Ž1 y ␴ .y1 c 1y ␴ , ␴ G 0. Note that as in one sector growth models, we do not have a pure consumption good. The first good is both a factor of production and a consumption good. Production is subject to an external effect e j , treated as a constant by the agent, Ž 2.6.

2

e j s Ł x ibji j

Ž j s 1, 2 . .

is1

Therefore the true production functions are Ž 2.7.

2

yj s Ł Ž x i j .

␤ i jqb i j

Ž j s 1, 2 . ,

is1

where under social constant returns Ý2ss1Ž ␤ s j q bs j . s 1. The Hamiltonian associated with the problem given by Ž2.1. ᎐ Ž2.5. is

ž

2

H s U Ž c . q p1 e1 Ł Ž x i1 .

ž

2

is1

q p 2 e2 Ł Ž x i2 . 4

is1

␤ i2

␤ i1

y gx 1 y c

/

y gx 2 q

2

/

Ý wi

is1

ž

2

xi y

Ý xi j

js1

/

.

We assume ␤i j ) 0, which assures that all inputs are used in the production of all goods, for computational and analytical simplicity. It is not difficult to relax this assumption but the notation becomes cumbersome.

INDETERMINACY UNDER CONSTANT RETURNS

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Here pj and wi are Lagrange multipliers, representing utility prices of the capital goods and their rentals, respectively. The static first order conditions for this problem are given by Ž 2.8.

U⬘ Ž c . s cy␴ s p1 ,

Ž 2.9.

ws s pj ␤ s j Ł Ž x i j .

ž

2

␤ i jqb i j

is1

/

Ž x s j . y1

Ž s, j s 1, 2 . .

Note that to derive equation Ž2.9., we use the fact that in equilibrium the inputs x i j generating external effects e j are identical to the inputs chosen by the firm. Under constant returns the unit cost functions are independent of output levels and are invertible. Therefore factor rentals w s Ž w 1, w 2 . are uniquely determined by output prices ps Ž p1, p 2 .. Let xs Ž x 1, x 2 .. The laws of motion for problem Ž2.1. are given by Ž2.3., Ž2.4., where yi s yi Ž x, p ., and by Ž 2.10.

ž / dpi dt

s rpi y wi Ž p .

Ž i s 1, 2 . .

Constant social returns coupled with small external effects imply that some sectors must have a small degree of decreasing returns at the private level. This is in contrast to models of indeterminacy with social increasing, but private constant returns to scale. An implication of decreasing private returns is positive profits. Unless the number of firms is fixed, we must assume that there is a fixed entry cost to determine the number of firms along the equilibrium trajectories. As is clear from Proposition 1 below, the external effects and the degree of decreasing returns required for indeterminacy may be arbitrarily small, and generate only a small amount of profits. If the discounted value of profits along equilibrium trajectories that converge to the balanced growth path is small, a small fixed cost of entry will be sufficient to deter new entrants. 5 Let the growth rate of c and x i along the balanced growth path be ␮. It follows from equation Ž2.8. that prices must then decline at the rate ␴␮. We define discounted variables as Ž 2.11.

␹ i s ey␮ t x i , ␲ i s e ␴␮ t pi ,

␺ i s ey ␮ t yi ,

␻ i s e ␴␮ t wi

for i s 1, 2. Note that ey␮ t c s ey ␮ t Ž p1 .y1r ␴ s Ž␲ 1 .y1r ␴ . Since there are no fixed factors, outputs y are homogenous of degree one in the stocks x, and homogenous of degree zero in prices p, and the factor prices w are homogenous of degree one in prices. Then the equations Ž2.3., Ž2.4., and Ž2.10. can be written as Ž 2.12. Ž 2.13. Ž 2.14. 5

d ␹1 dt d␹2 dt d␲ i dt

s ␺ 1Ž ␹ , ␲ . y Ž g q ␮ . ␹ 1 y Ž ␲ 1 .

y

1

␴

,

s ␺2Ž ␹ , ␲ . y Ž g q ␮. ␹2 , s Ž r q ␴␮ . ␲ i y ␻ i Ž ␲ .

Ž i s 1, 2 . .

The existence of a balanced growth path is easily proved, with small modifications to allow for external effects, along the lines of the proof in Bond, Wang, and Yip Ž1996.. To assure positive prices and quantities, a lower bound to the discount rate is required, as shown in footnote 6 below.

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. The balanced growth path corresponds to the stationary point Ž ␹ 1U , ␹ 2U , ␲ 1U , ␲ U 2 of the above system. The Jacobian of this system is given by

⭸␺

y Ž gq␮. I

⭸␹

Js

⭸␺ ⭸␲

yZ

Ž r q ␴␮ . I y

0

⭸␻ ⭸␲

where Z is a matrix with zeros except for the element of the first row and the first ␴ y1 column, which is Ž1r␴ .␲y1r . 1 Using Ž2.9. and the social constant returns restriction Ý2ss1Ž ␤ s j q bs j . s 1, we find that output prices satisfy Ž 2.15.

2

pj s Ł

ss1

ž /

␤ s jqb s j

ws

Ž j s 1, 2 . ,

␤s j

or Ž 2.16.

ž / ␻s

2

␲js Ł

␤ s jqb s j

Ž j s 1, 2 . ,

␤s j

ss 1

and unit input coefficients are Ž 2.17.

ai j s

Ž 2.18.

s

Ž 2.19.

s

␤i j pj wi

␤i j

ž / Łž / 2

ws

wi

ss1

␤s j

␤i j

2

␻s

ss1

␤s j

␻i

Ł

␤ s jqb s j

␤ s jqb s j

Ž i , j s 1, 2 . .

Let A be the input coefficient matrix with elements a i j . Full employment of factors implies A ␺ s ␹ . Differentiating, we have Ž 2.20.

Ad␺ q S␺ s d ␹

where the elements of the matrix S are wÝ2ss1Ž ⭸ a i jr⭸␻ s . d ␻ s x. In order to obtain the dual relationship of price and output in the context of externalities, we express the price function in terms of input coefficients and Cobb-Douglas exponents. Let Ž 2.21.

a ˆi j s ai j Ž ␤i j q bi j . r␤i j

Ž i , j s 1, 2 . ,

and define Aˆs w a ˆi j x. Prices satisfy Ž2.16.. Differentiating Ž2.16. and using Ž2.19. we obtain Ž 2.22.

d␲ s w Aˆx ⬘d ␻ .

Using equations Ž2.20. and Ž2.22. the Jacobian matrix J becomes Ž 2.23.

Js

w A x y1 y Ž g q ␮ . I 0

⭸␺ ⭸␻

yZ

ˆx Ž r q ␴␮ . I y w A⬘

y1

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provided A and Aˆ are nonsingular Žsee Assumption 1 below.. On a balanced growth path the full employment of factors, A ␺ s ␹ , implies Ž 2.24.

w I y AŽ g q ␮ .x ␹ * s z

where z s Ž a11␲ 1U , a21␲ 1U .. The price equations, ␲ s w Aˆx⬘␻ , imply that on a balanced growth path, Ž 2.25.

ww Aˆx ⬘ Ž r q ␴␮ . y I x ␲ * s 0.

Note that the above relation implies that the matrix ww Aˆx⬘Ž r q ␴␮ . y I x must be singular and it corresponds to the lower right submatrix of the Jacobian J. As expected this always yields a root for J that is identically zero. The vector ␲ will be determined up to a multiplicative constant, while equation Ž2.24. will determine ␹ . The vector ␹ pins down not the level of stocks x, but their discounted values, as is clear from Ž2.11.. The same is true for ␲ which does not pin down the prices p, but their upcounted values. Thus on the balanced growth path quantities x, y, c grow at the rate ␮ , while prices p decline at the rate ␴␮. To determine the growth rate ␮ , we note from equation Ž2.25. that the quantity ˆ Ž r q ␴␮ . corresponds to the inverse of the Frobenius root of the nonnegative matrix A⬘. This is the only root that is associated with the positive eigenvector ␲ *.6 The signs of the roots of J are the same as those of the roots of ww A xy1 y Ž g q ␮ . I x ˆ xy1 x. The system will be locally determinate if J has one negative and wŽ r q ␴␮ . I y w A⬘ root and two roots with positive real parts. Then we can choose the initial prices so that initial conditions lie on the two dimensional manifold spanned by the one dimensional center manifold corresponding to the balanced growth path, and the one dimensional stable manifold corresponding to the negative root.7 If J has two roots with negative real parts however, the system will be indeterminate. In this case, depending on initial stocks, the initial prices can be chosen on the three dimensional space spanned by the one dimensional center manifold corresponding to the balanced growth path, and the two dimensional stable manifold corresponding to the two negative roots. For example, if w A xy1 has one root with negative real part and A⬘ has at least two real positive roots, the system will be indeterminate. In the multisector version of the model, the system will be ˆ has at locally indeterminate if w A xy1 has Ž n y 1. roots with negative real parts and A⬘ least two real positive roots Žsee Proposition 2 below..

ˆ is indecomposable, ␲ * is the unique nonnegative eigenvector of A⬘ ˆ and the Since the matrix A⬘ ˆ s Ž r q ␴␮ .y1 is the largest root in absolute value. Since with associated positive Frobenius root ␭ positive externalities the elements of Aˆ are at least as large as those of A, the Frobenius root of Aˆ ˆy1 ) g q ␮, the will be at least as large as that of A. From these observations it also follows that if ␭ inverse of w I y AŽ g q ␮ .x will be a positive matrix and assure, from equation Ž2.24. that ␹ ) 0. The ˆy1 y g . - r y g. If some externalities are negative, restriction on the discount rate then is Ž1 y ␴ .Ž ␭ the elements of Aˆ may be smaller than those of A. Then, if the Frobenius root of A is less than ␮y1 , we can choose g less than r so that w I y Ž g q ␮ . A xy1 is a positive matrix. 7 A standard alternative method to working with a system that has an identically zero root is to reduce the dimension of the system using ratios of stocks, x irx 1 , as in Mulligan and Sala-i-Martin Ž1993. or Benhabib and Perli Ž1994.. 6

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Let B s w ␤ si x and Bˆs w ␤ si q bsi x. We make the following assumption: ASSUMPTION 1: The matrices B and Bˆ are strictly positi¨ e and nonsingular. Let ⍀ denote the 2 = 2 diagonal matrix with diagonal elements ␻ i , i s 1, 2 and zero off-diagonal elements. Similarly let ⌸ denote the 2 = 2 diagonal matrix with diagonal elements ␲ i , i s 1, 2 and zero off-diagonal elements. Note from Ž2.9. that a i j s pj ␤i jrwi , and therefore we have a ˆi j s pj Ž ␤i j q bi j .rwi. It follows that A s ⍀y1 B ⌸ and Aˆs ⍀y1 Bˆ⌸ , and from Assumption 1 that A and Aˆ will be nonsingular. LEMMA 1: Along the balanced growth path the sign pattern of roots of B is the same as that of Ay1 s ⌸y1 By1⍀ , and the sign pattern of roots of Bˆ is the same as that of ˆ xy1⌸y1. Ay1 s ⍀ w B⬘ Ž . U PROOF: Along the balanced growth path ␻ U i s r q ␴␮ ␲ i . The lemma follows from y1 y1 y1 2 y1 noting that < A < s < ⌸ B ⍀ < s Ž r q ␴␮ . < B < and that every principal minor of w ⌸y1 By1⍀ x of order i will be given by the corresponding principal minor of By1 multiplied by Ž r q ␴␮ . i. If the characteristic equation of By1 is f Ž ␭. s Žy␭. 2 q b1Žy␭. q b 0 s 0, the coefficients bnyi will be the sum of principal minors of order i. Therefore, the characteristic polynomial of w ⌸y1 By1⍀ x will have coefficients Ž r q ␴␮ . i b 2yi . If the characteristic equation of w ⌸y1 By1⍀ x is given by g Ž ␯ . s 0, then Ž r q ␴␮ . y2 g Ž ␯ . s Ž r q ␴␮ . y2 Ž y␯ . 2 q Ž r q ␴␮ . y1 b1Ž y␯ . q b 0 s f Ž ¨ r Ž r q ␴␮ .. . Therefore if ␭ is a root of By1 , then ␭rŽ r q ␴␮ . is a root of w ⌸y1 By1⍀ x and the sign pattern of the roots of B and By1 is the same as that of w ⌸y1 By1⍀ x. The proof that the ˆ is the same as that of w ⍀ w B⬘ ˆ xy1⌸y1 x is identical. inertia of B⬘ Q.E.D. Proposition 1 below gives conditions for indeterminacy that are independent of the utility function. From Lemma 1 the factor intensity difference a22ra12 y a21ra11 is directly related to ␤ 22r␤ 12 y ␤ 21r␤ 11. Therefore we may say that the consumable capital good Žfirst good. is intensive in the pure capital good Žsecond good. from the pri¨ ate perspecti¨ e if ␤ 22 ␤ 11 y ␤ 21 ␤ 12 - 0, but that it is intensive in itself from the social perspecti¨ e if Ž ␤ 22 q b 22 .Ž ␤ 11 q b11 . y Ž ␤ 21 q b 21 .Ž ␤ 12 q b12 . ) 0. The proposition follows ˆ from noting the signs of the determinant and trace of the matrices B and B. PROPOSITION 1: In the two-sector endogenous growth model, if the consumable capital good is intensi¨ e in the pure capital good from the pri¨ ate perspecti¨ e, but it is intensi¨ e in itself from the social perspecti¨ e, then the balanced growth path is indeterminate. PROOF: If the consumable capital good is intensive in the pure capital good from the private perspective, B has negative determinant. This implies that By1 has negative determinant and one negative root. In this case Ay1 y Ž g q ␮ . I has at least one negative root. If the consumable capital good is intensive in itself from the social perspective, Bˆ has ˆ and hence A⬘ ˆ have two positive positive trace and positive determinant. In this case B, ˆ is the Frobenius root Ž r q ␴␮ .y1, which has to be roots. One of the positive roots of A⬘ ˆ xy1 real, and the other one is smaller in modulus. Therefore the positive real root of w A⬘

INDETERMINACY UNDER CONSTANT RETURNS

1547

ˆ will dominate Ž r q ␴␮ .. On the other other than the inverse of the Frobenius root of A⬘ ˆ xy1, w A⬘ ˆ xy1 y Ž r q hand as the inverse of the Frobenius root, Ž r q ␴␮ ., is the root of w A⬘ ␴␮ . I has one zero root and one negative root. Therefore J has one zero root and at least two negative roots. Q.E.D. The analysis of the model above can easily be recast in an n-sector framework. In a ˆ composed of the Cobb-Douglas exponents, will multisector model, the matrices B and B, be of dimension higher than two, with i, j s 1, . . . , n. Indeterminacy will now follow if 2 n-dimensional matrix J has less than n roots with positive real parts. Then the proposition above generalizes as follows: PROPOSITION 2: In the multisector endogenous growth model, if the matrix B has Ž n y 1. roots with negati¨ e real parts and the matrix Bˆ has at least two roots with positi¨ e real parts, the system will be indeterminate. PROOF: From Lemma 1, at the steady state the root structure of the n-dimensional ˆ The system will input matrices A and Aˆ corresponds to the root structure of B and B. be indeterminate if the now 2 n-dimensional matrix J has less than n positive roots. This ˆ x has at least two will happen if w A xy1 has Ž n y 1. roots with negative real parts and w A⬘ ˆ is the Frobenius root real positive roots. Since one of the positive real roots of A⬘ ˆ xy1 Ž r q ␴␮ .y1, all the other roots are smaller in modulus. The real positive roots of w A⬘ Ž . other than the Frobenius root will dominate r q ␴␮ . Therefore J will have at least n q 1 roots with negative real parts. Q.E.D. The result of Proposition 1 holds exactly in a two-sector nonendogenous growth model with fixed labor supply if the first good is a pure consumption good, the second good is a pure capital good, and if utility is linear Žsee Benhabib and Nishimura Ž1998... The reason that we need to resort to linear utility in a two-sector model with fixed labor and a pure consumption good is simple. The existence of multiple equilibrium paths implies that for a given level of the capital stock, there is a continuum of ratios of initial investment to consumption that are consistent with equilibrium. However some curvature in the utility function may destroy the possibility of multiple equilibria if the cost of foregoing current consumption is large relative to future benefits that come from higher initial investment. We can regain some flexibility if the first good is both a consumption good and a capital good, as in the one sector model. In that case increasing the investment level in the second Žpure capital. good does not solely come at the expense of consumption. However if we stick with such a setup, we would have two goods and three factors, one of which is labor. This would significantly complicate the analysis. Switching to an endogenous growth model without a fixed factor avoids this difficulty by making the number of goods and factors equal. Dept. of Economics, New York Uni¨ ersity, 269 Mercer St., New York, NY 10003, U.S.A.; [email protected], Dept. of Economics, The Chinese Uni¨ ersity of Hong Kong, Shatin, N.T. Hong Kong; [email protected] and Institute of Economic Research, Kyoto Uni¨ ersity, Yoshida-Honmachi, Sakyo-ku, Kyoto 606, Japan; [email protected] Manuscript recei¨ ed October, 1997; final re¨ ision recei¨ ed October, 1999.

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