The Review of Economic Studies, Ltd.

The Stability of the "Morishima System"Author(s): Charles KennedySource: The Review of Economic Studies, Vol. 37, No. 2 (Apr., 1970), pp. 173-175Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296410 .

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The Stability of the

"; Morishima System"

I. INTRODUCTION1

Metzler [4] proved in 1945 that, for the case where all commodities were gross substi- tutes, the multiple exchange economy would have dynamic (local) stability, if and only if it had Hicksian stability. It was later independently proved by Hahn [2], Negishi [7], and Arrow and Hurwicz [1] that the case of gross substitutes was Hicksian and therefore dyna- mically stable. In 1952, Morishima [5] extended Metzler's theorem to a case in which there was some gross complementarity, and in which there were certain rules governing the signs of the effects of a change in the price of one commodity on the excess demand for other com- modities; and proved that this case also would be dynamically stable if and only if it was Hicksian. In Morishima's own treatment, the sign rules applied not to the entire economy including the numeraire commodity, but to the subeconomy excluding the numeraire. The purpose of this note is to consider the stability of the " Morishima System", as I shall call it to distinguish it from the Morishima case proper, when the sign rules are applied to the entire economy including the numeraire. It will be found that, so long as there is any gross complementarity in the system, the Morishima system cannot be Hicksian and is therefore dynamically unstable.

II. NOTATION AND ASSUMPTIONS

Since the proof will take the form of an adaptation of Negishi's proof for the case of gross substitutes, I shall use his notation as far as possible:

Pi, i = 1, ..., m, is the price of the ith commodity. P0,? i = 1, ..., m, is the equilibrium price of the ith commodity taken to be positive.

Xj(Pjl ..., P,) i = 1 ... m, is the excess demand function of the ith commodity, assumed to be homogeneous of zero degree.

Aij = aXi/OPi, i, j = 1, ..., m, evaluated at P?,j = 1,j..., m.

The sign rules defining the Morishima case are that the sign of A ij is the same as the sign of Aji and that the sign of Aij is the same as the sign of the product of Aik and Aki (i, i

and k distinct), and I shall take these rules as applying to all commodities in the system. If it is defined in terms of the sign rules alone, the system includes as a special case the case where all commodities are gross substitutes. Since this case is already known to be Hicksian and stable, I prefer to exclude it by adopting an alternative formulation, also suggested by Morishima [6], which is equivalent to the definition in terms of the sign rules so long as there is some gross complementarity present in the system:

Definition. The Morishima System is that in which the set of commodities can be divided into two non-empty non-overlapping subsets, such that any two commodities from the same subset are gross substitutes and any two commodities from different subsets are gross complements.

1 I am indebted to Professor Michio Morishima, to Professor Frank Hahn and to my colleague Mr Norman Ireland for assistance in the preparation of this note. Any errors or inaccuracies remain my own responsibility.

M 173

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174 REVIEW OF ECONOMIC STUDIES

I shall use the subscripts i and j to refer to any of the m commodities in the complete set of commodities and the subscripts r and s to refer to any of the (n -1) commodities in one of the subsets (n < m). Thus [A j] is the matrix of A-coefficients for the complete set of commodities; [Ar,] is the matrix of A-coefficients for one of the subsets. It will be useful, after the manner of Hicks, to treat the other subset of commodities as a composite com- modity, which will be designated by the subscript n and can also be used as numeraire.

It is necessary to specify rather carefully what is meant by saying that a matrix is "Hicksian ". I shall say that the matrix [Aij] (i.e. the whole system) is Hicksian if the principal minors of the determinant of [Aij], but not the necessarily singular determinant itself, take the sign (- 1)4, where q is the order of the minor. I shall say that the matrix [Ars] is Hicksian if the principal minors of the determinant of [Ars] and the determinant itself take the sign (- 1).

III. THE STABILITY OF THE MORISHIMA SYSTEM Lemma. If [Aij] is Hicksian, [Ars] is Hicksian.

Proof. The determinant of [Ars] is itself a principal minor of [Aij]. Theorem 1. The Morishima System is not Hicksian.

Proof. The proof follows at the outset the lines of Negishi's proof of his theorem for the case of gross substitutes. Because the excess demand functions are homogeneous of zero degree, we have

s = n-1

Z ArsPs =ArnPno Arn (r = 1, ..., n-1). ...(1) s = 1

Multiplying both sides of (1) by (-1) and replacing (-Ars) by Brs and Arn by Br, we have s = n-1

E BrsPso = Br (r = 1, * . , n -1). ..(2) s = 1

Since the rth commodity is by assumption a gross complement of all the constituent com- modities of the composite commodity (n), Br < 0.

The remainder of the proof uses reductio ad absurdum. Suppose [Aij] is Hicksian. From the Lemma, [Ars] is also Hicksian. If [Ars] is Hicksian, all the diagonal elements of [Ars] will be negative. From the definition of the Morishima System, all the off-diagonal elements Of [Ars] will be positive. From the well-known theorem of Hawkins and Simon [3] for matrices of this type, if [Ars] is Hicksian, all the elements of the inverse matrix of [Brs] will be positive. But this is impossible, since P?>0 and Br<0. Hence [Ars] cannot be Hicksian and [Aij] cannot be Hicksian, and the theorem is proved.

Theorem 2. The Morishima System is dynamically unstable.

Proof. In view of Theorem 1, this theorem will be proved if it can be shown that Morishima's theorem of the equivalence of dynamic stability conditions and Hicksian stability conditions for a subeconomy excluding the numeraire can be extended to an entire economy in which the sign rules hold. To establish this it is necessary to prove that

(a) If the entire economy is Hicksian, every possible subeconomy formed by excluding the ith commodity as nume'raire (i = 1, ..., m) will also be Hicksian. This follows from the argu- ment of the Lemma. (Note that this case occurs only if there is no gross complementarity in the system).

(b) If the entire economy is not Hicksian, every possible subeconomy formed by excluding the ith commodity as nume'raire (i = 1, ..., m) is not Hicksian. If the sign rules apply to the entire economy and there is some gross complementarity in the system, the ith commodity

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THE STABILITY OF THE "MORISHIMA SYSTEM" 175

will be in one of the two subsets into which the set of commodities can be divided. From the argument of Theorem 1, the subset that does not include the ith commodity cannot be Hicksian, and therefore (from the argument of the Lemma) the subeconomy formed by excluding the ith commodity cannot be Hicksian.

Hence, the Morishima theorem can be extended to the entire economy, and the theorem. is proved.'

IV. ECONOMIC IMPLICATIONS

The Morishima System is suggestive because it provides an example of an economy which-to borrow a term from geology-contains afault. The most important lesson to be drawn from the result is that, if there is such afault in the economy, no degree of substitution, between pairs of commodities on one side of the fault or on the other helps in any way to preserve the stability of the economy.

University of Kent at Canterbury CHARLES KENNEDY

First version received December 1968; final version received March 1969

REFERENCES

[1] Arrow, K. J. and Hurwicz, L. " On the Stability of the Competitive Equilibrium ">

Econometrica (1958), 522-552.

[2] Hahn, F. H. " Gross Substitutes and the Dynamic Stability of General Equilibrium" Econometrica (1958), 169-170.

[3] Hawkins, D. and Simon, H. A. " Some Conditions of Macro-economic Stability ",

Econometrica (1949), 245-248.

[4] Metzler, L. A. " Stability of Multiple Markets: the Hicks Conditions ", Econometrica (1945), 277-292.

[5] Morishima, M. " On the Laws of Change of the Price-System in an Economy which Contains Complementary Commodities ", Osaka Economic Papers (1952), 101-113.

[6] Morishima, M. " Notes on the Theory of Stability of Multiple Exchange ", Review of Economic Studies (1957), 203-208.

[7] Negishi, T. " A Note on the Stability of an Economy where All Goods are Gross Substitutes ", Econometrica (1958), 445-447.

1 The instability of the Morishima System was strongly suggested by Arrow and Hurwicz in [1]; but in that paper the authors were less concerned with proving the instability of the system than with establishing its inconsistency, for all values of the price vector, with their own micro-economic assumptions.

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