Price and quantity adjustment in a dynamic closed model: The dual stability theorem

LUIGI FILIPPINI Zstituto Uniuersitario, Bergama Unioersitri Cattolica de1 Sacro Cuore Milan

Price and Quantity Adjustment in a Dynamic Closed Model: The Dual Stability Theorem* The paper analyzes dual instability in a dynamic input-output framework. It is divided into three parts: the first gives an informal introduction, the second discusses the similarity with the knife edge, and the last suggests a new interpretation, which guarantees nonnegative solutions and relative stability. It has been assumed that the producers adjust their prices following an adaptive expectation mechanism, and the current level of investment to the expected value defined by a golden rule approach.

In this note attention is confined to the Dual Stability Theorem, conjectured in a dynamic input-output scheme by R. Solow and M. Morishima, but fully proved by D. Jorgenson. The paper is divided into three parts. The first gives the reader an informal introduction and background to the problem, i.e., the dynamic closed input-output model is briefly described with the statements and the implications of the Dual Stability Theorem. The second summarizes the similarity between the Harrod-Domar and the abovementioned instability result. The last deals with the methods suggested to overcome dual stability and, after mentioning the received theories, a new interpretation is proposed which transforms the equilibrium model into a disequilibrium one. This not only guarantees that the model is relatively stable, but also generates nonnegative solutions.

1. Historical Background The first attempt to extend the input-output framework to take account of capital goods was made independently by D. Hawkins (1948) and W. Leontief (1953). Subsequent integration and fur*I am indebted to Professors C. Beretta, L. Johansen, P.C. Nicola, and to an anonymous referee for helpful comments and to Professor D. Jorgenson and to G. Chiesa for valuable conversations on an earlier draft of the paper. I gratefully acknowledge support from C.N.R., Rome (CT 81.03055.10) and CRANEC, Milan. Journal of Macroeconomics, Spring 1983, Vol. 5, No. 2, pp. 185-196 Copyright 0 1983 by Wayne State University Press.

185

Luigi Filippini ther work was carried out by R. Solow (1959), and M. Morishima (1964). The result was a system of price and quantity variables for both the open and the closed economy, under the assumption of fixed technical coefficients, of full utilization and transferability of capital goods from one sector to another. The first version of the closed system shows that the price of each commodity must cover its current cost plus the interest on the value of capital equipment required per unit of output, whereas the quantity produced is derived in terms of the multiplier-accelerator principle, where current inputs include maintenance and replacement costs. In discrete time we have: x(t) = Ax(t) + B[x(t + 1) - x(t)] ;

(1)

p = pA + rpB

(2)

where A = {au} n X n matrix of current input coefficient i at the unit level of activity j; B = {b,} n X n matrix of capital coefficient i at the unit level of activity j; x = {Xi} n column vector of output levels of activity; p = {pi} n row vector of commodity prices; r = rate of interest, assumed constant over time. While the specification of the quantity system was accepted, the price system raised several objections. Without the implicit assumption of perfect foresight and static price expectations by entrepreneurs, the stationary equilibrium of system (2) would almost certainly be unstable if the initial prices and interest rate were not the equilibrium ones. In any case, (2) is not a complete dynamic model of price changes. Solow (1959) suggested a new version of (2), assuming perfect foresight and logical consistency of price movements by the entrepreneur which maximizes the sum of profits and capital gains (or minimizes losses)‘. And he derived the new set of price equations as follows: ‘The price system could be derived from a choice problem about the ownership of a sum of money equal to p(t)B where the available options are to become either an interest earner or an entrepreneur. If the individual chooses the latter option, he will obtain p(t + 1) - p(t + l)A + p(t + l)B, and if he lends, (1 + r)p(t)B. 186

Price, Quantity p(t + 1) = p(t + 1)A + rp(t)B

- [p(t + 1) - p(t)]B

Adjustment .

(3)

The new system formed by (1) and (3), can be reformulated as follows: Bx(t + 1) = [B + C]x(t)

;

(4)

p(t + l)[B + C] = (1 + r)p(t)B where (I - A) = C; or, finally,

(5)

assuming B is nonsingular:”

x(t + 1) = [I + B-‘C]x(t) p(t + 1) = (1 + r)p(t)[I

;

k-9

+ CB-*I-’

.

(7)

Existence, uniqueness and above all the stability properties of the solution depend on its latent roots, assumed distinct, and the eigenvectors of CB-’ and of its inverse BC’. But it is well known that the roots of BC-’ are the CB-“s reciprocals, while the eigenvectors are the same. Existence of such solutions is implied by the nonnegativity of BC’ since A is an input-output matrix, whereas uniqueness is a consequence of assuming its indecomposability. Therefore, as the dominant root of BC-’ is positive from one of Frobenius’ theorems and the corresponding characteristic vector has positive components, the output system will be capable of balanced growth at the rate Al. But output equilibrium proportions are relatively stable3 iff 11 + Ai] < 11 + Al ) for any latent root Ai, i = 2, Since in equilibrium both options must be equally advantageous, we obtain system (3) [see Solow (1959), pp. 33-341. ‘Since B is usually assumed to be nonsingular, several authors have suggested how to circumvent it [see Champbell (1979)]. In any case Morishima’s (1964, pp. 73-76) partition solution may be followed. ‘The relative stability concept introduced by Solow and Samuelson (1953) may be summarized as follows. Given a dynamic system with arbitrary positive initial values, under what conditions will a balanced growth (or constant relative price) time path tend eventually to be established? In a formal way, given a system of linear difference equations: dt

+ 1) = W/,(t)

(for i = 1, 2, 3,

.) ,

where H is an indecomposable square matrix, suppose that z(t) is a reference path (in this case either the balanced growth or the stationary state price solution) and y(t) is a solution starting from an arbitrary initial vector y(0) z 0. Then, the reference path is relatively stable if 187

Luigi Filippini 3, . . . n, whereas for a given rate of interest lower than XI [Morishima (1964), p. 107ff.1,’ the solution of (7) converges to the stationary equilibrium vector (2) iff (1 + r)/]l + hi ] < (1 + t-)//l + A, I. Thus if the price system is stable, then 11 + hiI > II + Al I and vice versa. The above intuitive discussion leads to Jorgenson’s (1960) statement of the Dual Stability Theorem: For systems of order greater than one, if the output system is globally relatively stable, then the price system is unstable in this sense and vice versa.’ M. Morishima (1974) and D. Jorgenson (1961b) have pointed out that the problem of stability is closely related, but not identical, to that of the nonnegativity of the solution of systems (1) and (3), and have proved that stability is a necessary condition for nonnegativity. This is a reason why the Dual Stability Theorem plays an important role in economic theory. It may happen that no system generates nonnegativity solutions for arbitrary nonnegative initial conditions. The dual stability feature of the price and quantity systems taken together in cases where initial values are historically given is a puzzling result. For that reason several authors have confined the model to planning purposes, and discarded it as a representation of the working of free competition. But the theorem exercises its power also when it is used for planning because, given one set of initial values (for price and quantity), it does not leave any degree of freedom.

where 0 is a positive constant independent of i. Therefore the theorem does not refer to the behavior of the solutions themselves, but to how the ratio of the components would change as time passes [see Nikaido (lS70), pp. 14Sff]. 4An economic reason for r 5 A,, is easily derived from (6) and (2) in the case of equality, and for (6) and (7) in the case of inequality. In fact from (2) and (6) ~(1 - A) = rpB ;

(2’)

and (I - A)% = X,Bf

(6’1,

therefore we have r = A,. In the other case we obtain from (6) and (7) (1 + A,)P(f + 1) = (1 + r)p(t) and therefore r < A, [see Lombardini (1968, pp. 21-22)]. “Until now we have followed the accepted specification of the dynamic model, in particular a forward-lag scheme for quantities and a backward-lag for prices. Sev188

Price, Quantity

Adjustment

Many explanations of the theorem have been suggested, from mechanical interpretations based entirely on the mathematical properties of the specification [Intriligator (1971)] to more sophisticated economic ones. Thus it may be worth going to the origin of the model and finding out if there are any intuitive reasons, independent of the disaggregation, leading to the above challenging result.

2. Similarity to the One-Sector Model While the quantity of the static input-output scheme was discovered as the disaggregated Keynesian multiplier by R. Goodwin, the extension to the dynamic case was based on the accelerator principle, and in fact the first reference is the so-called HarrodDomar model. The price equation has not however played such an important role as the quantity one in the aggregate case, and R. Solow (1953) appears as a rare exception. The purpose here is to discuss stability conditions in the onesector model and to discover any similarities with the multi-sectoral scheme. If we use the same notation as adopted above, the system appears as follows: x(t) = ax(t) + P[x(t + 1) - x(t)] ;

(8)

At + 1) = w(t + 1) - P[p(t + 1) - p(t)1+ f%(t)

(9)

where (Y is the marginal propensity to consume and 6 = (1 - IX) the marginal propensity to save, and p is the stock-flow ratio or the accelerator coefficient. Its solution is given by: era1 combinations are therefore possible and not all of them lead to dual instability. If, for example, all the model is specified as a forward-lag scheme, price equations are as follows: p(t) = P(W

+ rp(t)B

- [p(t + 1) - p(t)@

and, therefore,

p(t + 1) = p(t)[(l + r)I - CB-‘1 Stability

conditions

are thus similar to a quantity

system:

x(t + 1) = [I + B-‘C]x(t), for values of A,, A, such that A, = 1 and A, > 1; A, > 1 and Xi > 0. 189

Luigi Filippini x(t) = (1 + y)l X(0) ; and P(t) = [Cl + d/(1

+

r)l’ P(O)

where y = S/p is the Harrodian warranted rate of growth. As is known, it may differ from the present rate. Therefore the quantity solution is unstable; this is what has been called the Harrod-Domar knife-edge problem in the Hahn-Matthews sense. On the other hand the price solution is stable iff r I y,6 a condition already met in the multi-sectoral model. Hence it follows that price and output equation solutions depend on two different conditions. The result leads to the conclusion that inconsistency does not depend on the disaggregation of the model, but on its specification. 3. Methods to Avoid Instability In addition to the imposition of linear restrictions on the initial parameters of the system a number of reformulations of the model have been suggested to overcome dual stability leading to the construction of an explicit theory of disequilibrium.7 D. Jorgenson (1961) and E. Zaghini (1971) have in fact rephrased the model into a disequilibrium scheme, linking directly the price and output system. Th e f ormer has relaxed the strict equality between costs and prices, and between output and utilization; the latter has put forward an interpretation within the framework of a general model of capital accumulation of the Walras-Hicks type.

4. A New Approach One of the aims of this note is to show that the price and quantity system has a nonnegative solution and, as a corollary, is relatively stable. The basic idea underlying the new method starts with the ‘The steady state solution follows solving equation (9) where capital gains disappear; in this case it is stable iff r = y. The equilibrium price level is constant at its initial value P(0) and therefore performs no function. ‘Many other lines of investigation have been suggested; we will just mention them because they refer either to the output system, i.e., the irreversability of capital accumulation by Leontief (1953) and M c M anus (1959) or to the open model, i.e., the optimal capital accumulation interpretation of the model [DOSS0 (1958)]. For further work see Fukuda (1975) and Aoki (1977). 190

Price, Quantity Adjustment specification of the model: a naive accelerator hypothesis implies a zero adjustment period, while the price formation incorporates a full cost pricing with static expectations. In both cases no attention is paid to the fact that adjustments require time and that information on past decisions can help. As a consequence the original dynamic model is transformed into a short-run disequilibrium scheme where markets are no longer cleared and expectations are not always fulfilled. And it is reduced to the original model in the case of market clearance and when expectations are realized. We intend to deal with the solution of the Dual Stability Theorem under the assumption that the price and quantity adjustments are independent of each other. Though it is not crucial in an equilibrium context when a nonsubstitution theorem holds, it requires some explanation in a disequilibrium framework [Hahn (1963)]. In fact, in the specification of the disequilibrium scheme it is assumed that competition operates in such a way as to ensure a uniform rate of profit that the producers expect to earn and moreover that they follow a golden-rule approach in relation to expected growth of investment. The first hypothesis is not crucial as the price system has one degree of freedom, while the second is easily justified as a rule of thumb. If an equal rate of profit among sectors is the main feature of modelling a free-competition process, justification that the price and quantity specification (adjustment) is independent of each other is no longer necessary. Another reason is derived from the fact that excess demand may have no, or sometimes a perverse, effect on prices. In the following sections we introduce the behavioral equations of a typical producer in any sector, first of all for the price and then for the output system.

5. The Price System The basic underlying hypothesis is that producers are supposed to adjust current prices in the direction of the normal prices (defined by a constant rate of profit on equipment costs) following an adaptive expectation mechanism. The typical producer in every sector is assumed to have certain expectations on the rate of profit and on prices. He expects to earn a normal rate and we take it to remain unchanged over time. Fear of entry will ensure that the producers follow a pricing rule assuring them the normal rate of profit in the long run. 191

Luigi Filippini If we now define v as the vector of current net stock rentsthat is profits plus capital gains (or losses)-system (3) can be rewritten as follows:8 p = pA + vB

(10)

and in long run equilibrium the following identity must hold: p = v/r and (10) b ecome equal to (2). Producers do not set current prices equal to long run normal price instantaneously, partly because they have limited information on technology, partly because they are not certain of their expectations. They have therefore some expectation not on input prices themselves but on rents. The behavioral assumption in this paper is that they try to adjust them to a moving equilibrium vector rp(t) which represents the prevailing rents under the hypothesis of no capital gains. Several hypotheses about the formalization of expectations can be suggested; adaptive expectations are assumed here.g The assumption concerning the mechanism by which expectations are formed suggests that they will only be amended or adapted if they are unfilled. The adaptive mechanism can be formalized as follows: ve(t + 1) - v”(t) = [rp(t) where ve is the expected value and negative coefficients of expectations The hypothesis states that the next period is given by the expected proportion of its current deviations value. Thus (11) becomes:

- ve(t)] i

01)

+I is a diagonal matrix of nonless than one. forecast level of rents in the current level amended by some from the moving equilibrium

ve(t + 1) = ~-p(t)+1 - D(I - i)]-’

(12)

where D is the delay operator. Rents expectations are now incorporated in the current price formation. Prices are fixed as we have assumed in such a way to ensure the normal rate of profit in the long run. System (10) is therefore transformed into: “Therefore, v(t + 1) is equal by definition to rp(t) - [p(t + 1) - p(t)]. ‘The use of an adaptive expectation hypothesis is partly due to theoretical reason, but largely due to properties which are convenient for econometric work. 192

Price, Quantity

Adjustment

p(t + 1) = p(t + l)A + ve(t + l)B. Thus by the substitution p(t + 1) = p(t)[(I

(13)

of (12) into (13) we get:

- A)B-‘(I

- $B(I

- A)-’ + rfiB(I

- A)-‘].

Multiplying both sides by (I - A)B-’ and defining z(t) = p(t)(I A)B-‘, (14) may be rewritten in a modified form as: z(t + 1) = z(t)[I - ii + rB(1 - A)-‘fi].

(14) -

(15)

But (15) is a system of simultaneous, homogeneous, linear first-order difference equations. And it is reduced to the long run price system (2) when expectations are fully realized and the adjustment is therefore instantaneous. Under the above-mentioned assumption of I > i and r 5 X1 the specification of the price system has nonnegative solutions p(t) for all t(t = 1, 2, . . .) equal to z(t)B(I - A)-’ as z(t) 2 0” and it is relatively stable from trivial considerations, too.”

6. The Quantity System Producers hold a theory of normal investment I(t), which underlies their expectation mechanism regarding the level of accumulation. Among several possible hypotheses, we assume the most plausible one: growth on the trend, but as entrepreneurs have limited production information, they will approximate the growth rate to the rate of profit following a golden rule approach: i(t) = A Bx(t - 1) where h is the rate of growth

(16)

assumed equal to r.

“‘For a more formal proof see Jorgenson (196lb), p. 277-93 where he proves the following theorem: Given a system of difference equations such as ~(t + 1) = AY(~) + f(t)

for f(t) 2 0 for all for all t iff A has “Since [I - 4 (1978), pp. 34-351 p. 206ffl.

>

t, y(0) 5 0 and A a constant matrix, it has a nonnegative solution at least one element strictly positive in each column. + rB(1 - A)-‘41 2 0 IS indecomposable and primitive [Woods the balanced growth solution is relatively stable [Morishima (lQ64),

193

Luigi Filippini An economic reason for the golden rule assumption follows recalling the properties of the von Neumann model [DOSS0 (1958), p. 2971. No instantaneous adjustment is hypothesized because the current level of investment is explicitly allowed to deviate from the expected level as investment is the outcome of plans made on a longish view. To be more precise, producers are assumed to adjust the current level of investment to the expected value. This is defined by a mechanism which allows for the revision of expectations on the basis of the difference between the current expected investment value and the normal level. Thus we can write (1) as follows: x(t) = Ax(t) + Ie(t + 1)

(17)

where I’ are the expected investment levels. The producers, therefore, adjust investment to its normal value following an adaptive expectation mechanism: I”( t + 1) - I”(t) = &[A Bx( t - 1) - I”(t)] where Q is a diagonal matrix of positive than one. Thus we have:

expectation

Ie(t + 1) = A[1 - (I - &)D]-‘&Bx(t By substituting

08) coefficients

- 1) .

less

(19)

(19) into (17) we get:

x(t) = [(I - A)-‘(1 - @(I - A) + A(1 - A)-%B]x( t - 1) . Multiplying both sides by (I - A) and defining then (20) becomes: u(t) = [I - 4 + XQB(1 - A)-‘]u(t

(20)

u(t) = (I - A)x(t),

- 1)

(21)

But (21) is a system of homogeneous, linear, first-order difference equations and for the same reason discussed in relation to the price equations, the output system is relatively stable and has nonnegative solutions x(t) for all t (t = 1, 2, . . .) as u(t) 2 0 and x(t) = (I - A)%(t). The specification of the dynamic input-output model has some properties. When expectations are realized price equations are reduced to (2). In addition, the steady-state solution of both systems 194

Price, Quantity

Adjustment

is independent of the expectation mechanism and satisfies (2) and (1). Expectations are not required to adjust sufficiently slowly for stability to be maintained, as it is independent of the coefficient values. 7. Conclusions In this paper the dual stability in a dynamic input-output framework has been analyzed and a new specification has been proposed. It was found useful to recapitulate the well-known original problem (Section l), and to discuss the similarity with the HarrodDomar knife edge (Section 2). In other sections we have developed an alternative specification of the dynamic scheme where nonnegative solutions and dual stability properties hold both for prices and quantities. It has been argued that the dual instability property is essentially due to the assumption of myopic perfect foresight in the price and quantity equations. This assumption has been removed. And we have assumed that for a given rate of profit the producers adjust their prices following an adaptive expectation mechanism, whereas they adjust the current level of investment to the expected value defined by a golden rule approach. In addition, the suggested specification shares most dynamic properties with the original model, and dual stability is independent of the value of expectation coefficients. Receioed: ]anuary, 1982 Final tiersion receitied: October, 1982

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Luigi Filippini Hawkins, D. “Some Conditions of Macroeconomic Stability. ” Econometrica 16 (October 1948): 309-22. Intriligator, M. Mathematical Optimization and Economic Theory. Englewood Cliffs, New Jersey: Prentice-Hall, 1971. Jorgenson, D.W. “A Dual Stability Theorem.” Econometrica 28 (October 1960): 892-99. -. “Stability of a Dynamic Input-Output System: A Reply.” Review of Economic Studies 30 (June 1963): 148-49. -. “Stability of a Dynamic Input-Output System. ” Review of Economic Studies 28 (February 1961a): 105-16. -. “The Structure of a Multi-Sector Dynamic Model.” Znternational Economic Review 2 (September 1961b): 276-93. Leontief, W.W. “Dynamic Analysis.” In Studies in the Structure of the American Economy. New York: Oxford University Press, 1953. Lombardini, S. “Competition, Free Entry and General Equilibrium Models. ” Economia internazionale 21 (February 1968): 2-26. McManus, M. “Notes on Jorgenson’s Model.” Review of Economic Studies 30 (June 1963): 141-7. -. “Self Contradiction in Leontief’s Dynamic Model.” Yorkshire Bulletin of Economic and Social Research 9 (May 1959): l21. Morishima, M. Equilibrium, Stability, and Growth. London: Oxford University Press, 1964. Nikaido, H. Zntroduction to Sets and Mappings in Modern Economics. Amsterdam: North-Holland, 1970. Solow, R. M. “Note on the Price Level and Interest Rate in a Growth Model.” Review of Economic Studies 21 (1953): 74-9. -. “Competitive Valuation in a Dynamic Input-Output System. ” Econometrica 27 (January 1959): 30-53. p. “Solow Prices and the Dual Stability Paradox in the Leontief Dynamic System: Comment.” Econometrica 39 (May 1971): 633-34. and P. Samuelson. “Balanced growth under constant returns to scale.” Econometrica 21 (July 1953): 412-24. Woods, J. E. Mathematical economics-topics in multi-sectoral economics. London: Longman, 1978. Zaghini, E. “A Reply. ” Econometrica 39 (May 1971): 634. ---. “Solow Prices and the Dual Stability Paradox in a Leontief Dynamic System.” Econometrica 39 (May 1971): 625-32.

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