The global correspondence principle

Economics Letters 34 (1990) l-4 North-Holland

The global correspondence principle Murray C. Kemp University of New South Wales, Kensington, NS W 2033, Australia

Yoshio Kimura and Makoto Tawada Nagoya City University, Nagoya 467, Japan. Received 26 December 1989 Accepted 2 February 1990

The Global Correspondence Principle is generally invalid if there are more than two goods. However, attenuated forms of the Principle, valid for any number of goods, can be found by restricting the Jacobian of the excess-demand functions.

1. Introduction Samuelson (1971) showed that if the world economy is globally stable (and if there are just two commodities) then, qualitatively, the ultimate effect on the terms of trade of an international transfer depends only on the impact of the transfer on world excess demands; in particular, the ultimate effect of the transfer is independent of the size of the transfer and of the stability or instability of the initial equilibrium. Later [Samuelson (1983)], he dubbed this finding the Global Correspondence Principle (GCP). Recently Bhagwati et al. (1987) have noted that the GCP is valid for any parametric disturbance. It is a pity that the GCP was first expounded in a context of comparative statics. For the probability of encountering an economy in unstable equilibrium is quite small and this has suggested to some that the scope of the Principle is correspondingly small. In fact the Principle can be given a much broader setting: If a system is globally stable then, given any Walrasian disequilibrium, newly created or not, the price of each commodity in excess demand (excess supply) will approach a level above (below) that currently prevailing. Since the restated GCP must be valid for all points of time, it implies that prices are monotone functions of time. The real problem with the GCP is quite different. Both in the initial paper by Samuelson and in the subsequent contribution of Bhagwati et al., only two commodities are recognized. As Samuelson (1971) himself noted, it is an open question whether the GCP is valid for more ample economies. It is this question that we seek to answer in the present paper. For mathematical convenience only, we revert to the comparative-statical setting of the problem. Let pi(t) be the price of the ith good in terms of the 0th good at time t, and let E be an unspecified parameter. Then we may write the set of n + 1 excess demand functions

Jql, Pl(&.., 0165-1765/90/$03.50

p,(t);

E) =E,(p,(t)

,..., p,(t);

c), j=O, l,...,n.

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

2

M. C. Kemp et al. / The global correspondence principle

In an initial equilibrium, possibly unstable, E;(P:,...,

p,“; P)=o,

i=l,...,

n.

(1)

[By Walras’ Law, E,( py, . . . , p,“) = 0 also.] This equilibrium is disturbed when E jumps from co to e’=e’+Ae. Let(p: ,..., pi)=(py ,..., pi)+(Apl ,..., Ap,) be the new stable price vector to which the economy moves; and let AE, be the immediate (constant-price) impact of AC on the excess demand for the ith good. Then the GCP states that, for n = 1, Ap, . AE, > 0 if AE, # 0; and the unresolved question is whether Ap,-AE,>O

if

AE,#O,

i=l,...,

n

n>l,

(2a)

or, a weaker version, whether t

Ap,-AEj>O

if, for some i, AE,#O,

j=l

In fact, the simple counter conditions (on conditions are

n > 1.

(2b)

GCP is invalid as soon as more than two goods are recognized; section 2 contains a example. Attention then switches to the more prosaic problem of finding sufficient the excess-demand functions) for modified versions of (2a) and (2b) to hold. Such developed in section 3.

2. The GCP is invalid if n > 1 Suppose that n = 2. In fig. 1 we display the two post-disturbance loci Ei( pl, p2; cl) = 0 (i = 1, 2), drawn on the assumption that 6E,/ap, < 0, aEi/ap2 > 0 (i = 1, 2) and that the locus E, = 0 cuts the

P,A E,=O

/r E,=O

Fig. 1

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h4. C. Kemp et al. / The global correspondence principle

locus E1 = 0 three times, first from below, then from above and finally from below again. Evidently prices are globally stable. Since AC can shift both curves, in any manner, the initial equilibrium prices can lie anywhere in the positive quadrant. In a special case, AC shifts just one locus and the initial prices lie on the remaining locus. However, whatever the nature of the shift, prices follow a cyclical path of adjustment. In particular, the initial prices might be represented by PI, where the first good is in excess supply, and yet Ap, > 0; or they might be represented by P2, where the first good is in excess demand, and yet Ap, -Z 0. Thus if n > 1 the GCP is invalid both in the form (2b) and in the stronger form (2a). Indeed it is invalid even when, before and after the disturbance, there is a unique and stable equilibrium.

3. Attenuated forms of the GCP for n > 1 The GCP is invalid for n > 1. However attenuated versions of it, valid for arbitrary n, can be found by restricting the Jacobian of the excess-demand functions. Suppose that the economy is in an initial equilibrium, that it is disturbed by a shift of excess demand from the numeraire to the first good and that, out of equilibrium, prices are governed by the tgtonnement i= l,...,

pi=f;(E,(p)),

(3)

p,) and fi is an increasing and sign-preserving function.

wherep=(p,,..., Theorem 1.

n,

Suppose that, for all p > 0, aE,/ap,>O,

i,j=I

,...,

n, iZj.

(4)

Then a shift of excess demand from the numeraire to the first good causes a monotone increase of all prices. Proof. As soon as the excess demand for the first good increases, p1 begins to rise, as required by (3). From (3) and (4), the increase of p1 induces an excess demand for goods 2,. . . , n and therefore forces up their prices too. To verify that all prices move monotonely, suppose the contrary, that there is a non-empty subset K of N= {l,..., n } such that the excess demands for goods in K are negative. From the continuity of excess demand it follows that for any j of K there is tj such that Ej( p( tj)) = 0, where p(t) denotes the solution of (3) at time t. Let tm = min.,EKtjand K,={j~K:t~=t”},sothat Ej(p(t”))

>O

if

j4K,,

if

jsK,,.

and Ej(p(t”))=O

(5b)

If j e K,, (3) and (5a) imply that pi is increasing at P. Together with (4) and (5b), this in turn implies that, just after t*, Ej70 and p,>O for jEK,,. 0

hf. C. Kemp et al. / The global correspondence principle

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Corollary I. Suppose that (4) is replaced by the requirement that the Jacobian of the excess-demand functions ( aEi/i3pj) is a Morishima matrix and that the two subsets R and S of N which characterize the Morishima case are everywhere the same. Then Theorem I remains valid, mutatis mutandis. Proof.

It suffices to consider the case in which 1 E R. For notational brevity, let

A= and p = Aq, so that (3) is transformed into d=A-‘f(E(Aq)).

(3’)

Bearing in mind that A -’ = A the Jacobian of (3’) is J = AflaE/lzlp)A, where f^ is a diagonal matrix the jth diagonal element of which is &‘( .) > 0 and aE/ap is the n x n matrix (aEi/apj). Now it is a matter of simple calculation that J satisfies (4) and that pR = qR and ps = - qJ, where, for example, pR is a subvector obtained from p by extracting pi (j E R). Applying Theorem 1, it becomes clear that the prices of goods in R (respectively S) rise (fall) throughout the process of adjustment. The rest of the proof is also clear. 0 Finally we remark that, if ( aEi/apj)

is irreducible, the strong inequalities of (4) may be weakened.

Bhagwati, J.N., R.A. Brecher and T. Hatta, 1987, The global correspondence principle; A generalization, American Economic Review 77, 124-132. Samuelson, P.A., 1971, On the trail of conventional beliefs about the transfer problem, in: J.N. Bhagwati et al., eds., Trade, balance of payments and growth: Papers in international economics in honor of Charles Kindleberger (North-Holland, Amsterdam) 327-351. Samuelson, P.A., 1983, Foundation of economic analysis, enlarged edition (Harvard University Press, Cambridge, MA).