Jess Benhabib New York University Department of Economics New York, NY 10003 Kazuo Nishimura Tokyo Metropolitan University Faculty of Economics 1-1-1 Yakumo, Meguro-ku Tokyo, Japan I. INTRODUCTION Various authors have investigated the dynamic theory of factor demands by considering the "costs of adjustment" faced by the firm. See Eisner and Strotz [5], Treadway [15], Lucas [8], Lucas and Prescott [9], Mortensen [12], Brock and Scheinkman [3] and Scheinkman [14]. In such models the firm maximizes the present value of its profit stream under the constraint that changing the levels of factor inputs involves costs of adjustment. One can then obtain optimal time paths for investments, that is for the accumulation or decumulation of the stocks of factor inputs. Under certain assumptions on the structure of the model, the optimal time paths of the stock of factors asymptotically approach some long-run equilibrium value. In particular Brock and Scheinkman [3] and Scheinkman [14] have studied conditions under which factor lev~ls globally converge to some steady state equilibrium. In this paper we are concerned with models that are "unstable," that is models where the optimal time paths of investment are such that factor levels do not converge to the steady state values. We will construct and analyze an example where the time paths of factor levels and of investments are, or approach, a closed orbit. 1 Let us stress that the orbital paths constructed will be completely endogenous: they will not depend on exogenous fluctuations in factor prices, the output price or technology, nor on money illusion or irrational behavior. The investment decisions of the firm will be optimal decisions.

The possibility

*We are grateful to Professors W. A. Brock and M. J. P. Magill for suggestions and constant encouragement. Also, we would like to thank Cyrus Sorooshian for his help with the innumerable calculations in the paper. 1Recently Magill [10] has studied the causes of spiralling of optimal paths but not the convergence to closed orbits. See section III, footnote 4. R. M. Goodwin et al. (eds.), Nonlinear Models of Fluctuating Growth © Springer-Verlag Berlin Heidelberg 1984

74

of such cyclical investment, derived from the optimal behavior of the firm, could form a basis for a new approach to business cycle theory. In the following section we briefly sketch the model, essentially following Treadway [15]. In section III we construct an example where the steady state values of the factor stocks are totally unstable; that is it is never optimal to approach the steady state values if we do not start with them. In section IV we use bifurcation theory to show the existence of optimal paths for factor levels and investments that are closed orbits. We briefly discuss the stability of the orbits and the "structural stability" of the system. Then we generalize our results to the case of "perfect foresight equilibrium" by allowing prices to respond to the quantity supplied. Finally in section V we offer some heuristic explanations for our results. II. THE MODEL Following Treadway [15] we have a firm operating in competitive output and factor markets, facing a rate of interest r, a vector of factor prices g, and an output price of unity. It maximizes the present value of profits:

f

Max

OO

o

e -rt (F[x(t),y(t) ] - wx(t) - gy(t))dt

(1)

subject to n.x. 1 1

x.1 (0)

1, ... ,n

i

where n i is the rate of depreciation of the i'th factor, x is the input vector, y is the investment vector and w is a vector of "current account" costs which from here on we set equal to zero since it simplifies analysis without affecting any of our results. The output function F(x,y) satisfies the following: (A.1)

Let

Q =

n

F :Q x R

7

{XE.Rnlx > O} and let ~ be the interior of~. 2

•

n

R is of class C on Q x R .

To solve (1) we write the Hamiltonian, H

=

e-rt(F[x(t),y(t)] - gy(t) + q(t)[y(t) - nx(t)])

Then

75

and using the Maximum Principle we obtain the necessary conditions, x.

(2)

1

(3)

o

i

= 1, ... ,n

(4)

We define a steady state as a vector (i,y,q) that satisfy (2), (3) and (4) such that xi'Yi = 0 for i = 1, ... ,no We make the following assumption:

. .

There exists a steady state solution evaluated at

(x,y)

(i,y,q)

to (2), (3) and (4)

is non-singular.

Using equation (4), the above assumption allows us to express Yi's as a function of the vector (q,k) in the vicinity of the steady state (i,y,q) and we obtain from (2) and (3) a system of 2n differential equations in (x,q). Furthermore, using (4), we obtain (5)

a

[.g.] = -

a2 F -1 a2 F [--"2] [ayax] ay

(6)

The local stability of the steady state can be studied from the roots of the Jacobian of equations (2) and (3) and is given by + (r+n)I

(7)

It is easily shown that the roots of J are symmetric around r/2 and that for r sufficiently "close" to zero the real parts of the roots come in pairs of opposite sign. (See Treadway [15], pages 850-851.) In the latter case the steady state is saddle-point stable and for any initial value of x(O) the firm can choose q(O) so that the optimal path returns to the steady state. The optimality of the path (q(t),k(t)) approaching the steady state is assured if the transversality conditions,

76

lim e -rt q.(t) [-x(t). - x(t). ] > 0

t+oo

1.

1.

1.-

i

1, ... ,n

(8)

are satisfied where x(t) is any non-negative feasible path, provided that F(x,y) is concave. (See Pitchford and Turnovsky [13], pages 28, 29. )

In the next section we will give an example of a steady state that is not saddle-point stable. In the context of our investigation locally unstable steady states are not, as Treadway [15] puts it, "irrelevant"; in section IV we will show, using Hopf's Bifurcation Theorem, how the existence of closed orbits can be deduced from the local behavior of the optimal path in the neighborhood of the unstable steady state. III. EXAMPLES OF INSTABILITY We follow the notation of the previous section. tion, subject to adjustment costs, is given by2

The output func-

F(x,y) (9)

Note that the first term is simply strictly concave a Cobb-Douglas production function. Let the discount rate rO and the factor prices be· as follows: gl = 2.2037,

g2 = 1.5222, rO

(10)

0.06075

Since it substantially simplifies calculation we follow Treadway [15} in setting the depreciation of each good equal to zero.

We obtain

the following steady state values: 2For all results that follow we require the output function F(x,y) to have the form given by (9) only at the steady state (Y1'Y2,i 1 ,i2 , Q1,Q2). At steady state values, it is easily shown that the function F(x,y) is concave. To preserve concavity elsewhere, we can smoothly "bend" the output function in any manner we wish. This does not affect the results of the following sections which rely only on the properties of the steady state under consideration.

77

A

A

0,

Y1'Y2

xl

aF - aY2

q2

( 10 )2 9 aF aX 1

0.7,

x2 0.25,

( 10 ) 3 9 aF aX 2

1. 5,

q1

0.15

Let us note at this point that labor could be explicitly introduced into the model. We could write the Cobb-Douglas production function 1/2 1/3 Y as 0.5x 1 x 2 £ where £ represents the amount of labor. If we assume that there are no adjustment costs to labor, then given the wage rate the amount of labor used at each moment would be determined by its marginal value product.

If factor stocks x 1 (t) and x 2 (t) change through time so would the demand for labor. 3 However, for the results of this and the following sections we would only need the steady state value of labor.

By proper choice of units we can set this value equal to unity

and the steady state calculations would not be affected. The values of the parameters have been chosen so that the Jacobian matrix J, given in (7), reduces to

J

=

~

Q + rI

a2F -1

[-]

oQ,J

(11)

ay2

and where Q

a 2F a2F [dxay] [-2] ay A

It can easily be shown that the roots of J are given by the roots of Q + rI and - Q.

Since the roots of Q, evaluated at the steady state,

are -0.06075 + 0.3109i and the discount rate is 0.06075, the roots of the matrix

J

are A1 ,A 2

=

0 + 0.3109i, A3 ,A 4

=

0.06075 ~ 0.3109i.

We constructed the above example with zero real parts for A1 and A2 so that we can apply it to the Hopf Bifurcation Theorem (see the next section).

To obtain a totally unstable steady state we slightly modify

the example and set gl = 1.4814, g2 = 1.1851 and rO = 0.1.

For these

values the steady state vector (~,y,q) does not change and in particular the roots of the matrix Q are the same. the roots of

J

become A1 ,A 2

=

However, since rO is different,

0.03925 + 0.3109i, A3 ,A 4

3Alternatively, we could set the output function as

=

0.06075 ~ 0.3109i.

0.5xi/2x~/3

+ £y

so that the amount of labor used, determined by its marginal value product, would be independent of the values of xl and x 2 .

78

Now all the four roots of J have positive real parts.

This implies

that for any initial (x 1 (0),x 2 (0)) there is no choice of (q1(0),q2(0)) that would steer the system to (x,y,q). The steady state is totally unstable. 4 In the next section we will use our first example to establish the existence of a closed orbit solution to equations (2), (3). IV. ORBITAL PATHS OF FACTOR ACCUMULATION Theorem 1 (the Hopf Bifurcation Theorem)

.

Let x = F(x,~), x = (xl"" ,x n ) be a real system of differential equations with real parameter~. Let F(x,~) be Cr in x and ~ for x in a domain G and I~I < c. For I~I < c let F(x,~) possess a Cr family of stationary solutions x = x(~) lying in G: F(x(~),~)

For

~

=

= 0

0, let the matrix Fx(x(O),O) have one pair of pure imaginary

roots, a(~) ! B(~)i; a(O) = 0, B(O) f 0 and a(O)/d~ f O. Then there exists a family of real periodic solutions x = x(t,S),~ = ~(S) which has properties ~(O) = 0, and x(t,O) = x(O), but x(t,S) f x(~(S)) for sufficiently small S f o. x(t,s) is Cr. ~(s) and T(s), where T(s) = 2ll/IB(0)1 is the period of the orbit, are Cr - 1 . Proof See Hopf [6] and section 2, pages 197-198, of "Editorial Comments" by Hopf's translators, N. L. Howard and N. Koppel in Marsden and McCracken [11]. Hopf states and proves the theorem for F(x,~) analytic. Howard and Koppel revise Hopf's proof and provide the Cr version given above. Theorem 2 Consider the optimal problem (1) with parameters given by (9), (10). Then there exists a continuous function r = res), rO = reO) and a continuous family of optimal paths (x[t,r(s)] ,q[t,r(s)]) that are nonconstant closed orbits in the positive orthant for sufficiently small 4The analysis of Magill [10] shows that spiralling of optimal paths occurs because of the strong skew-symmetric forces at equilibrium. The skew-symmetric forces are represented by the off diagonal-matrices of the larger matrix [J] and are the causes of spiralling, though by themselves are not enough to give the existence of closed orbits.

79 A

E

A

f 0, and which collapse to the stationary point values (x(rO),q(r O)

as E

-+

O.

Proof From equation (2) we observe that for any steady state, y(r)

= O.

Using the implicit function theorem we can also show that the stocks are locally differentiable functions of r, that is, x appendix).

=

x(r) (see the

It is shown in the appeUtiix that the real parts of the pure A

imaginary roots of the Jacobian J, given by equation (11), are not stationary with respect to r.

Thus, we can apply Theorem 1 and obtain

orbits in (q(·,r),x(·,r». Note from Theorem 1 that the amplitude of the orbit that bifurcates from the steady state is zero at the bifurcation value rO grows with the deviation of r from rOo

=

0.0675 and

Since the steady state

(q,x) is

positive, the orbits that bifurcate from it must also be in the positive orthant for r sufficiently close to rOo

This establishes the existence

of positive orbits. Finally, to establish that the orbit is an optimal path we simply note that the orbit (q(' ,r),k(' ,r»

is positive and bounded.

Applying

the transversality conditions given by (8) we establish optimality. Q.E.D. Theorem 2 establishes that for some initial values of x, and appropriately chosen initial values of q, the optimal path is a closed orbit. To study the convergence of optimal paths from initial levels of stock in the neighborhood of the orbit we use the concept of stable manifolds. Definition Let the dz/dt h(z) possess a periodic orbit solution z = yet) of least period p, p > 0, where h(z) is of class C1 on an open set. Let C : z

=

yet), 0 < t

solutions z as t

-+

00

=

~

p.

Let the points z in the neighborhood of C, on

z(t) of dz/dh

=

h(z), and which satisfy dist(C,z(t»

constitute a d-dimensional manifold.

-+

0

Then the periodic orbit

is said to have a d-dimensional locally stable manifold. The orbits that result from Hamiltonian systems are attracting if for any given initial values of the state variables x(O) in the neighborhood of the orbit, there exists initial values for co-state variables q(O) such that the path (x(t),q(t» orbit.

there is an orbit.

asymptotically converges to the

In other words, a 2n-dimensional system has a "stable" orbit if n~dimensional

stable manifold in the neighborhood of the

In our discounted Hamiltonian system this cannot be guaranteed

and after the initial bifurcation we may end up with either an

80

n-dimensional or an (n -l)-dimensional stable manifold (see Benhabib and Nishimura [2], Theorems 3, 4, 5).5 If the orbits exist for r > ~ (alternatively, r < where r is the bifurcation value of the discount rate, then the stable manifold has dimension n (alternatively, n -1). Whether the orbits occur for ~ > r or ~ < r depends on the interaction of the first and higher order terms of the Taylor expansion around the stationary point of the differential equations describing the motion of the system. Th~re also exists the degenerate (non-generic) possibilitYi the effect of all higher order terms vanish at the bifurcation value of the parameter r, in which case the family of orbits form a "center" and exist only for r = r.

r)

A

Let us also note that if we slightly perturb one of the parameters of the system, we can still obtain the existence of orbits by adjusting the bifurcation value of r. For instance consider a small change in n i , the depreciation of good i. Let a(r,n i ) + r ~ b(r,ni)i be roots of the Jacobian J with a + r = 0, b 1 0, for n i = 0, as in our example of the previous section where non-constant closed orbits exist for r + a 1 0. We would expect a(n.1 ,r) to change with n.. However, since air + 1 1 0, 1 as shown in the appendix, the implicit function theorem implies that for n i in the neighborhood of its initial value, there always exists a value of r such that a(ni,r) + r = 0. However, to show the persistence of orbits for a general and unrestricted perturbati~n of all the parameters of the system requires more complex arguments in terms of "structural stability." (For "structural stability" arguments in relation to the Hopf Bifurcation see Arnold [1]. See also Hirsh and Smale [6], Chapter 16.) In general it can be shown that the subcritical orbits (those existing for r > ~) and the supercritical orbits (those existing for r > that arise from the Hopf Bifurcation will persist under small perturbations of the system.

r)

The results of Theorem 2 were derived under the assumption of fixed factor and comrwodity prices. We can now show that such an assumption is not necessary for our results. Suppose we have m identical firms in the industry, producing the same good and facing an aggregate demand curve, p(x,y) = 1 - b(mF(x,y», b ~ 0. The representative firm's profit appearing in (1) will be p(x,y)F(x,y) - wx - gy. Following Lucas and Prescott [9], we define a "perfect foresight equilibrium" price path, p : [O,e») + [0 ,00), that equates demand and supply through time and that 5(For the stability of orbits bifurcating from stationary points as they lose their stability, see Bruslinskaya [4] and also Marsden and McCracken [11], sections 4, 4A.)

81

is perfectly foreseen by the firms in the industry. The case analyzed earlier corresponds to b = 0, that is, p = 1. From the first argument in the previous paragraph we know that we can show the existence of closed orbits if we slightly perturb one of the parameters of our system. Thus for small positive values of b the results of Theorem 2 will hold. Note that in this case the price p(x,y) will also follow an orbital path. A similar argument can be constructed to hold for the factor prices g. (See the discussion in the following section.)

v.

FINAL REMARKS

Oscillations in the demand for factors with fixed output and factor prices may at first seem to be inconsistent with intertemporal profit maximization. We show that along an orbital path, the net marginal productivities of factors, that is, net of adjustment costs, will be fluctuating. It may appear that the firm can do better by shifting its purchases of factors towards periods where they have a high net productivity at the margin. But if the future is discounted, a larger output, and therefore a larger revenue, that only comes tomorrow may not compensate for the loss of the smaller revenue today.6 In fact, persistent oscillations occur for discount rates above a critical bifurcation value that depends on technology. From Theorem 1 we see that the amplitude of the orbital path of factor levels is zero when the discount rate is equal to the bifurcation value and grows with the distance of the discount rate from the bifurcation value. While a positive discount rate may be necessary to explain the orbital path of factor stocks it is by no means sufficient. The source of oscillatory behavior must be found in the asymmetrical effects of investments on the marginal productivities of the input stocks. This can be seen by considering the matrix J of the example in section III, given by (11). The submatrices Q, which determine the roots of J and therefore the nature of the motion around the steady state, contain the a

v ln fact one would have to compare costs and gains generated by the reallocation of investments not only between two points in time, t1 and t 2 , but along the orbital path at every instant between those two time

points as well. A marginal change in one stock at t1 affects the whole path up to t 2 . We must also remember that the orbits are not necessarily symmetric; the path halfway along the orbit from (x 1 ,x 2 ) to (i 1 ,i 2 ) is not necessarily that from

(x 1 ,x2 )

to

(x 1 'x 2 )

in reverse.

82

cross-partial terms of the function F(x,y). tions playa crucial role.

Thus technological condi-

(See footnote 4.)

Finally, let us consider the oscillatory behavior of factor stocks and output when we allow the output price to respond to the quantity produced, as we did at the end of the last section.

If the output is

not perishable and there are no storage costs (a possibility that we did not consider earlier), intertemporal arbitrage would dampen the cyclical variations in the price of output.

A constant price level

could emerge, while output and factor stocks continue along their orbital path as in the case previously considered.

However, the fact that output

is storable with zero costs does not guarantee a constant price level.

A positive discount rate would induce the firms to accept a lower price today rather than wait until tomorrow for the higher price.

Thus oscil-

lations in price as well as output can be preserved in a market with rational, profit-maximizing firms. APPENDIX In this appendix, we will show by actual calculation that the real A

parts of the pure imaginary roots of the matrix J, given by (11) in section III, are not stationary with respect to r. roots of J are (a + r) (a + r)

=

~

That is, if the

bi, -a + bi (see sections II and III) and

f O.

0 at the steady state, we will show that [d(a + r)]/dr

It is obvious that even if [d(a + r)]/dr

=

0 turned out to be the case,

a highly unlikely event, a small change in the parameters of the function F(x,y), given in section III, would remedy the situation.

Never-

theless, we carried out the calculations. While labor does not explicitly enter the calculations below, we may think of the model as not involving any adjustment costs for labor Ct S y and a production function with labor of the form Ax 1 x 2 + R,. The production function, that is, the first term of F(x,y) in section III, 1/2 1/3 1/2 1/3 oY 0.5x 1 x 2 ,can then be replaced by 0.5x 1 x 2 + N In such a case labor does not affect any of the calculations and its marginal value product is independent of the levels of xl and x 2 . There is no conceptual difficulty in extending the calculations below to a production function of the form

xi/2x~/3R,Y,

ate together with xl and x 2 . what more tedious.

where the level of labor would fluctu-

However, the calculations would get some-

Consider steady state values Y1'Y2 for r and

x2

= 0.06075.

=

0, xl

=

(10/9)2,

x2

=

(10/9)3

First we show that steady state values of Y1' Y2' ~1

can be written as functions of r.

Since depreciations are

83

assumed to be zero, steady state values of Y1'Y2 are identically zero. Setting q1

=

0 in equation (3), we use the implicit function theorem

to show that steady state values of ~1'~2 can be written as differentiable functions of r in the vicinity of ~1 = (10/9)2, ~2 = (10/9)3. Solving for d~l/dr, d~2/dr, we obtain: - 5.2610 -11. 2746 We would like to show that the real parts of the pure imaginary

J,

roots of the matrix to r.

given by (11), are not stationary with respect

The calculations are tedious and we will provide a brief sketch

below. It can be shown that the roots of the matrix J given by (7), which A

becomes J with our specific parameter values of section III, has the same roots as the matrix

Z

=

[

+ rFyx)_ ] F-1(FXy - Fyx + rFyy) -+~~ F- 1 (FXX ________________ __________ O

-Ly~y

I

(See Tr;adway [15], pages 850-851.) roots will be (a + r)

~

We know from section II that the

- a + bi.

bi and

We also know thay the alge-

braic product of the roots must be identically equal to the determinant of Z.

If we differentiate this identity with respect to r we obtain

one equation in da/dr and db/dr; the values of xl' x 2 ' Y1' Y2' dx 1 /dr, dx 2 /dr that appear when we differentiate the determinant of Z are evaluated at the steady state values given above. state values for a and b are known; a

=

Note also that steady

0.06075 and b

=

0.3109.

Unfortunately, to obtain a second equation in da/dr and db/dr we cannot use the equality of the sum of the roots and the trace, since the sum of the roots equals 2r and both a and b cancel out.

So instead

we use the fact that a restricted sum of principal minors of order 2 of the matrix Z always equals the restricted sum of the matrix Z always equals the restricted sum of the pairwise products of the roots:

L j>i

I

a ii

a ..

lJ

L

j>i a ..

Jl

A.A. 1

J

a ..

JJ

where aij's elements and Ai's are the roots of Z.

Fortunately, the

84

zeroes in Z simplify the algebra. Differentiating the above identity, we obtain another equation in da/dr, db/br and we then solve to obtain da/dr

- 29.1491

db/dr

- 0.3377

The rate of change with respect to r of the real parts of the pure imaginary roots of Z, (a + r), is d(a + r) dr

=

da/dr + 1

- 28.1491

Thus [d(a + r)]/dr f 0 and the hypothesis of Theorem 1 holds for our example in section II. REFERENCES 1. Arnold, V. 1., "Lectures on Bifurcation and Versal Families,"

Russian Mathematical Surveys, 27 (1972), 54-123.

2. Benhabib, J. and K. Nishimura, "The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multi-Sector Models of Optimal Economic Growth," Journal of Economic Theory, 21 (1979), 421-444. 3. Brock, W. A. and J. A. Scheinkman, "On the Long-Run Behavior of a Competitive Firm," in Equilibrium and Disequilibrium in Economic Theory, ed. G. Schwodiauer, D. Reidel Publishing Company, Dordrecht, Boston, 1978. 4. Bruslinskaya, N. N., "Qualitative Integration of a System of n Differential Equations in a Region Containing a Singular Point and a Limit Cycle," Society Mathematics Doklady, 2 (1961), 9-12. 5. Eisner, R. and R. Stroz, "Determinants of Business Investment," in Impacts of Monetary Policy (Commission on Money and Credit), Prentice-Hall, Englewood Cliffs, N.J., 1963. 6. Hirch, M. W. and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. 7. Hopf, E., "Bifurcation of a Periodic Solution From a Stationary Solution of a System of Differential Equations," in J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. 8. Lucas, R. E., "Optimal Investment Policy and the Flexible Accelerator," International Economic Review, 8 (1967), 78-85. 9. Lucas, R. E. and E. C. Prescott, "Investment Under Uncertainty," Econometrica, 44 (1976), 841-865. 10. Magill, M. J. P., "On Cyclical Motion in Dynamic Economics," Journal of Economics, Dynamics and Contr 1, 1 (1979), 199-218.

85

11. Marsden, J. E. and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical SCiences, No. 10, SpringerVerlag, New York, 1976. 12. Mortensen, D. T., "Generalized Costs of Adjustment and Dynamic Factor Demand Theory," Econometrica, 41 (1973), 657-665. 13. Pitchford, J. D. and S. J. Turnovsky, Applications of Control Theory to Economic Analysis, North-Holland, New York, 1977. 14. Scheinkman, J. A., "Stability of Separable Hamiltonians and Investment Theory," Review of Economic Studies, 45 (1978), 559570. 15. Treadway, A. B., "The Rational Multivariate Flexible Accelerator," Econometrica, 39 (1971), 845-855.