Local Multi-Market Stability

Tricolor above the Paris Bourse

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"What must we do to prove that the theoretical solution is identically the solution worked out by the market? Our task is very simple: we need only to show that the upward and downward movements of prices solve the system of equations of offer and demand by a process of groping."

(Léon Walras, Elements of Pure Economics, 1874: p.170)


"The laws of change of the price system, like the laws of change of individual demand, have to be derived from stability conditions. We first examine what conditions are necessary in order that a given equilibrium system should be stable; then we make an assumption of regularity; that positions in the neighbourhood of the equilibrium position will be stable also; and thence we deduce rules about the way in which the price-system will react to changes in tastes and resources."

(John Hicks, Value and Capital, 1939, p.32)

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Contents

(1) Introduction
(2) The Hicks Conditions (Slope Stability)
(A) Imperfect Stability
(B) Perfect Stability
(3) The Metzler Conditions (Dynamic Stability)
(A) Dynamic Stability and Stable Matrices
(B) Gross Substitution
(C) McKenzie's Conditions and Complementarity

Selected References

Back

(1) Introduction

Stability in a single market via Walrasian tatonnement or Marshallian quantity adjustments has already been considered. The question that now emerges is whether stability is ensured when multiple markets are considered. The question is of interest because there the interaction between price adjustment in different markets may have substantial implications on the dynamic properties of the system. Specifically, recall that in Walras's tatonnement mechanism:

dpi/dt = i(zi(p))

thus, price adjustment in market i is a function of zi, the excess demand in market i which is, in turn, a function of all other prices, p. Thus, a rise in the price of good i will affect not only own demand in market i but also demands in other markets and thus other prices which, in turn, affect excess demand in market i. The possibility of cross-effects destabilizing the tatonnement mechanism is real, even though Walras himself thought they were of no importance as these cross effects would offset each other - as he writes:

"The consequent changes [in other prices]...exerted indirect influences, some in the direction of equality [of supply and demand in a particular market], and some in the opposite direction that up to a certain point they canceled each other out." (Leon Walras, 1874: p.172)

But this assertion remained unproven and the question was largely forgotten until it was considered by John Hicks (1939).

(2) The Hicks Conditions (Slope Stability)

Recall that in a single market case, if, starting from equilibrium, a rise in price leads to a negative excess demand, then the system is "stable" as all the auctioneer has to to do is to follow the tatonnement rule and lower the price back to equilibrium. Thus, in a partial market, the condition for stability is that, evaluated at equilibrium:

dzi/dpi < 0

i.e. a rise in the own price leads to a fall in excess demand (and thus, from equilibrium, negative excess demand). In multiple markets, this is no longer assured because of the cross-effects mentioned earlier.

To account for cross effects, John Hicks (1939) differentiated between "imperfect" and "perfect" stability in multiple markets. Hicks argued that a system had imperfect stability if the following held: starting from equilibrium, displace the price of a particular good i and allow all other prices to adjust fully so that their respective markets are in equilibrium; if, after all these cross effects have worked themselves out, it is still true that dzi/dpi < 0, then we have "imperfect stability" (as all the auctioneer needs to do is to lower the price of good i).

In contrast, John Hicks characterized a system as perfectly stable if, displacing pi, we still have it that dzi/dpi < 0 under any of the following three cases:

(i) dzi/dpi < 0 after all markets are allowed to adjust fully (as in the "imperfect" case)

(ii) dzi/dpi < 0 when no markets are allowed to adjust (as in the partial market case)

(iii) dzi/dpi < 0 after some markets adjust fully and others do not adjust at all (mix of imperfect/partial cases).

Hicks (1939) set out to show under what conditions a general equilibrium system would have imperfect and perfect stability. Because we are focusing on the sign of dzi/dpi, the Hicksian approach has often been termed "slope stability".

(A) Imperfect Stability

Let us have n+1 goods, x0, x1, x2 .... xn with n+1 prices, p0, p1, p2, ..., pn. Henceforth, unless otherwise noted, we will restrict our attention to a normalized system, thus we can remove our n+1th commodity, x0 by Walras's Law and our n+1th price, p0, as numeraire and deal exclusively with an n-dimensional system, x1, x2, .., xn and p1, p2, ..., pn. Let zi = xi - ei be excess demand for good i.

Let us specify that the excess demand function for the ith good, zi = zi (1, p1, p2, .., pn), is a function of all prices. We have n such equations which can be written out compactly in vector form as z = z(p). Taking the total differential of z = z(p) with respect to prices, we obtain:

dz1 = a11dp1 + a12dp2 + ... + a1ndpn

dz2 = a21dp1 + a22dp2 + ... + a2ndpn

....... ...........................................

dzn = an1dp1 + an2dp2 + ... + anndpn

where aij is the partial derivative of excess demand in the ith market with respect to the jth price, i.e. aij = zi/ pj. If aij > 0, then goods i and j are "gross substitutes"; if aij < 0, they are gross complements. We can assume that aii < 0 (own effect) is negative. Letting A be an (n n) matrix of such first partial derivatives, and dz be the vector of excess demand displacement and dp the vector of price displacement, we can rewrite this system compactly as:

dz = Adp

Recall that for imperfect stability, Hicks wanted merely to guarantee that a rise in a single price pi led to zi becoming negative (excess supply) - after all the cross effects in other markets worked themselves out. In other words, he wanted to guarantee that dzi/dpi < 0 in the end. To impose that all other markets have adjusted except for the ith market, then we must assume that dzj = 0 for all j = 1, 2, .., n where j i so that dz is a column vector with dzi is in the ith place and zeroes everywhere else. Consequently, by Cramer's Rule:

dpi = |Ai|/|A|

where |A| is the determinant of A and |Ai| is the determinant of the matrix obtained by placing the solution vector dz in the ith column. It is easily seen that, expanding by the ith vector of Ai, that |Ai| = dzi|Cii|, where |Cii| is the cofactor of element aii (i.e. the determinant of the minor obtained by deleting the ith row and column from the original matrix Ai - and notice that as i+i is even, then |Cii| = |Mii| where |Mii| is the minor). As a result dpi = dzi|Cii|/|A|, or:

dzi/dpi = |A|/|Cii|

Thus, in order for dzi/dpi < 0 (i.e. imperfect stability), then it must be that the cofactor |Cii| be of opposite sign to the determinant of A. As this must hold for any i, then the Hicksian condition for imperfect stability is that all principal minors of order n - 1 must have an opposite sign to |A|.

(B) Perfect Stability

Let us now turn to the conditions for Hicks's "perfect stability", i.e. allowing some markets to adjust fully and others to remain rigid. Hicks pursued this by a series of steps for a representative market, which we shall follow here for the first market (the conditions that apply for dz1/dp1 < 0, would be replicated for any dzi/dpi < 0).

Step 1: assume all other prices are rigid (partial case), i.e. dpj = 0 for all j = 2, .., n. Then this entire system dz = Adp reduces to dz1 = a11dp1 or simply,

dz1/dp1 = a11

Thus, the first condition for stability, is that the own effect (a11) must be negative. As this is true for market 1, it must be true for all markets, thus aii < 0, for all i = 1, 2, .., n.

Step 2: assume all prices are rigid except those of market 2, but impose the additional condition that prices in the market for good 2 adjust in response to the rise in price of good 1 by bringing market 2 into equilibrium, i.e. dz2 = 0. Then this entire system reduces to:

dz1 = a11dp1 + a12dp2

0 = a21dp1 + a22dp2

Re-expressing in matrix form so that we have dz = A*dp for this reduced system where A* is merely the smaller 2 2 matrix of coefficients aij, i, j = 1, 2. Then by Cramer's Rule, we know:

dp1 = |A1|/|A|

where |A*| is the determinant of the small 2 2 coefficient matrix and |A1| is the determinant of the matrix A* with solution vector dz replacing the first column. Thus:

dz1

a12

0

a22

dp1 =

_________

a11

a12

a21

a22

As we can see immediately, |A*| = a11a22 - a12a21 and |A1| = dz1a22 so:

dp1 = (dz1a22)/(a11a22 - a12a21)

thus:

dz1/dp1 = (a11a22 - a12a21)/a22

If there is to be stability, then dz1/dp1 < 0. We know from condition (i) that a22 < 0, thus for stability we need that (a11a22 - a12a21) > 0, or simply |A*| > 0. As this is required for two markets 1 and 2, it will be required for any pair of markets i, j = 1, .., n so that |A*| > 0 for any coefficient matrix A* of order 2 with elements aij.

Step 3: Let us allow the first three prices to change while all others remain rigid. Again, assuming the other markets adjust fully, then dz2 = dz3 = 0. Thus we end up with the following system:

dz1 = a11dp1 + a12dp2 + a13dp3

0 = a21dp1 + a22dp2 + a23dp3

0 = a31dp1 + a32dp2 + a33dp3

We can rewrite this in matrix form again as dz = A*dp with these vectors and matrices now being three-dimensional. Again, by Cramer's Rule:

dp1 = |A1|/|A*|

where A* is the determinant of our three-market matrix and A1 is the transformed matrix with the solution vector dz in the first column. This reduces to:

dp1 = dz1(a22a33 - a23a32)/|A*|

where we don't write out |A*| because it is a rather horrible term. Nonetheless, rearranging we can see that:

dz1/dp1 = |A*|/(a22a33 - a23a32)

Notice that the denominator of this expression is merely the determinant of a second order matrix:

a22

a23

a32

a33

But, by condition (ii) for stability, the determinants all such second order matrices must be positive. Thus, by (ii), (a22a33 - a23a32) > 0. Therefore, the only way dz1/dp1 < 0 under the three-good case is if the numerator is negative, i.e. |A*| < 0.

So far, we have derived the stability conditions for three cases: one market, two markets and three markets. They were:

(1) a11 < 0

(2)

a11

a12

a21

a22

> 0

(3)

a11

a12

a13

a21

a22

a23

< 0

a31

a32

a33

If a pattern emerges, it should: (1), (2) and (3) are the determinants of the first three principal leading minors of the original Jacobian matrix, A. Note that the determinant of each minor alternates in sign successively. Thus, the Hicksian condition for multi-market stability is that the principal leading minors of the matrix of partial derivatives A alternate in sign or, alternatively stated, that the matrix A be negative definite. Notice that the final minor, of dimension n-1, will of course, will be of opposite sign to |A|, thus the conditions for perfect stability imply imperfect stability. A matrix A which is negative definite is often referred to as a Hicksian matrix.

(3) Metzler Conditions (Dynamic Stability)

(A) Dynamic Stability and Stable Matrices

John Hicks's (1939) exercise was perhaps the first attempt since Walras to pay serious attention to tatonnement and the stability of general equilibrium. Nonetheless, Oskar Lange (1944), Paul Samuelson (1941, 1944, 1947) and Lloyd A. Metzler (1945) objected to the Hicksian conditions for multi-market stability as they were not really "dynamic" in a proper sense but merely "slope conditions" (i.e. we just want to ascertain that zi remains negative after a rise in price pi). True "dynamic stability", with all markets adjusting simultaneously, requires that we set up a system of differential equations and prove these are stable. In this case, relative speeds of adjustment of prices in different markets begin to matter, as they do not in Hicks's original system. Thus, in principle, one needs to consider n differential equations:

dpi/dt = i[zi(p(t))]

where prices in the ith market (pi) adjust in response to excess demands in the ith market (zi(p)) depending on the particular form of adjustment in that market (i). Thus price changes in all markets affect excess demand for good i which in term will induce price changes in the ith market.

Taking a Taylor expansion about the equilibrium price, pi*, we can "linearize" this expression locally so that we obtain something like:

dpi(t)/dt = ki ・/font> j=1n aij (pj(t) - pj*)

where ki is the "speed of adjustment" in the ith market, aij are the partial derivatives of excess demand in market i with respect to price change in market j evaluated at equilibrium (aij = zi/dpj - same as before) and (pj(t) - pj*) is the deviation of market price at time t from equilibrium price in the jth market. We have n such equations, thus we can rewrite the entire system in matrix form as:

dp = KA(p(t) - p*)

where dp is the n-dimensional vector of price adjustments (with typical element dpi/dt); K is a diagonal matrix of speeds of adjustments (ki along the diagonal, zeroes everywhere else); A is the old matrix of first partial derivatives (aij = dzi/dpj) evaluated at equilibrium, p(t) the price vector at time t and p* the equilibrium price vector. The solution to this system of differential is merely:

p(t) = eKAt(p(0) - p*) + p*

where p(0) is the initial set of prices (the set of prices which the system has been displaced to initially). As can be easily proven, for the system to be locally stable, so p(t) p* as t , then all the real parts of the eigenvalues of KA must be negative. These can be found by solving the characteristic equation:

|KA - l I| = 0

for l , which will yield n solutions, l 1, l 2, .., l n. If KA is diagonalizable then KA = D-1L D where L is a diagonal matrix with the n eigenvalues l i arrayed along the diagonal and D = [v1 v2 .... vn] is a modal matrix with eigenvectors vi as columns. We can consequently rewrite this system as:

p(t) = DeLtD-1(p(0) - p*) + p*

Obviously, p(t) p* if the homogeneous part DeLtD-1(p(0) 0 as t . Thus, all the eigenvalues of KA must be negative.

What conditions on K and A guarantee true dynamic stability, i.e. that the real parts of all eigenvalues l 1, l 2, .., l n are negative? In a two-market system, this is actually rather easy. The characteristic equation of this system would be:

k1a11-l

k1a12

|KA - l I| =

= 0

k2a21

k2a22 - l

or:

l 2 - l (trKA) + |KA| = 0

where

trKA = k1a11 + k2a22

|KA| = k1a11k2a22 - k2a12k2a21

Now, if k1 = k2 = 1, then the Hicksian conditions on the alternating minors of A implies a11, a22 < 0 and a11a22 - a12a21 > 0, thus we see immediately that trKA < 0 and |KA| > 0, which is indeed sufficient for true dynamic stability. Of course, the reverse is not true: the Hicks conditions are not necessary for stability. For instance, consider the following matrix A:

-2

4

-1

1

This fulfills dynamic stability as trA = -1 and |A| = 2, but it violates Hicks's conditions as a22 = 1 > 0.

Does the assumption of adjustment coefficients k1 = k2 = 1 affect anything in this case? Actually, there is no qualitative difference in the 2 2 case. Notice that as long as k1, k2 > 0, then the trace, trKA = k1a11 + k2a22 will remain negative as long as a11 < 0 and a22 < 0, regardless of the values of k1 and k2. Similarly, we can factor k1k2 in the determinant |KA| = k1k2(a11a22 - a12a21) thus, as long as k1, k2 > 0, their values will not affect the sign of the determinant. As a result, we can thus say in the 2 2 case, that the Hicksian matrix A is a "D-stable" matrix, which is defined as follows:

D-Stability: matrix A is "D-stable" if KA is stable for any positive diagonal matrix K (i.e. stable for any positive speeds of adjustment).

We would like to extend the D-stability of the Hicksian matrix A to situations beyond the 2 2 case. However, one of the main observations of Samuelson (1941, 1947) and Metzler (1945) is that with multiple markets, the Hicksian conditions collapse. Or rather, when there are three or more markets, the Hicksian conditions are no longer sufficient for stability (much less D-stability). Many examples are available to show this, but the essential point is that in an n-dimensional system with n 3, the Hicksian conditions do not imply the Routh-Hurwitz (necessary and sufficient) conditions for the negativity of eigenvalues.

Nonetheless, under a couple of special cases from Lange (1944) and Samuelson (1941, 1947), we obtain the following:

(1) If A is symmetric (aij = aji for all i, j), the Hicksian conditions are sufficient for D-stability.

(2) If A is quasi-negative definite (i.e. A + A is negative definite), then the Hicksian conditions are fulfilled and we have D-stability.

Case (1) has an interesting economic interpretation as the "pure exchange" case with no trade at equilibrium. In this case, income effects wash out in the aggregate so that matrix A is equivalent to the sum of pure substitution matrices over households which are, in turn, necessarily symmetric. Thus, we can also think of situation (1) as when there is only one "representative" consumer or when all consumers are identical. The second condition does not lend itself so easily to interpretation.

We can see (1) and (2) immediately by applying the Arrow-McManus theorem on D-stability, namely:

Theorem: (Arrow and McManus, 1958) matrix A is D-stable if there exists a positive diagonal matrix C such that A C + CA is negative definite.

Proof: See our mathematical section.

As a result, if we set C = I, then obviously conditions (1) and (2) fulfill the Arrow-McManus condition. From (1), as A is symmetric and, by the Hicksian conditions, negative definite, then A + A is negative definite, and thus A I + IA is negative definite - thus A is D-stable. From (2), we see this immediately by the definition of quasi-negative definite.

A more interesting result was obtained by Metzler (1945) and generalized by Enthoven and Arrow (1956) which claimed the following:

Theorem: (Necessity) If A is D-stable, then A is Hicksian.

Proof: (Metzler , 1945; Enthoven and Arrow, 1956) If A is D-stable, then KA is stable for any positive diagonal matrix K. Recall that |KA| = i=1n l i, i.e. the determinant of KA is the product of the eigenvalues of KA. If KA is stable,then all eigenvalues are negative and, thus, sgn|KA| = sgn(-1)n. Thus, for any jth order matrix KAj, it must be that sgn|KAj| = sgn(-1)j. Since all the elements of K are positive, ki > 0, then the sign of the determinant of any jth order leading minor KAj is equal to the jth leading minor of A, i.e. sgn|KAj| = sgn|Aj| = (-1)j. Thus, the first principal leading minor of A is negative, the second positive, the third negative, and so on. Thus, for all eigenvalues to be negative in the system KA, then A must be Hicksian.

Thus, we see that a Hicksian matrix A is necessary for D-stability, but it is certainly not sufficient.

Finally, an interesting theorem provided many years later by Daniel McFadden (1968), employing the Fisher-Fuller theorem was the following:

Theorem: (McFadden) If A fulfills the Hicksian conditions, then KA is stable for some appropriate positive diagonal matrix K.

Proof: Omitted. See McFadden (1968).

(B) Gross Substitution

Obviously, the Hicksian conditions on A do not imply stability or D-stability and the special cases considered earlier are just too special. More is required and one condition was proposed by Lloyd A. Metzler (1945): namely, if one adds the assumption that all goods are "gross substitutes", then the Hicksian conditions on A are indeed sufficient for stability.

The introduction of gross substitution into stability was initially proposed by Mosak (1944), but it was Metzler (1945) who examined it in the context of true dynamic stability. In short, gross substitution implies that aij > 0 for all i j. By Walras's Law, this also implies that aii < 0 for all i. Thus, cross-effects are strictly positive and own effects are strictly negative. This implies that the matrix A has strictly negative diagonal elements and strictly positive off-diagonal elements. A matrix with such a property is referred to as "Metzlerian". Metzler's claim can be stated in the form of the following theorem:

Theorem: (Metzler) If all goods are gross substitutes (aij > 0 for all i j), then the Hicks conditions are necessary and sufficient for a D-stable matrix A.

Proof: First proved by Metzler (1945). The necessary part proved above already. Sufficiency to be proved later.

Metzler's claim was that gross substitution is required in addition to the Hicksian conditions, actually turned out to be superfluous as the former actually implied the latter. Frank H. Hahn (1958), Takashi Negishi (1958) and K.J.Arrow and L. Hurwicz (1958) went further and demonstrated that the Hicksian alternating minors assumption can actually be dropped. What they argued was that as long as goods are gross substitutes, (i.e. A is Metzlerian) there will be alternating minors anyway - more precisely, with the assumption that our demand functions are homogenous of degree zero and/or Walras's Law, gross substitution implies alternating minors of A. Let us pursue these avenues and, in turn, prove the Metzler theorem of sufficiency of gross substitution for a D-stable matrix A.

Theorem: (Metzlerian Hicksian via Homogeneity) If z(p) is homogeneous of degree zero in prices and p* > 0, then gross substitution implies that A is negative definite and A is a D-stable matrix.

Proof: First proved by Takashi Negishi (1958). By homogeneity of degree zero of excess demand functions, Euler's Theorem states that:

・/font> j=1n pj( zi/dpj) = 0 for all i = 1, .., n.

where partial derivatives are all evaluated at equilibrium prices, pj, j = 1, .., n. By gross substitution, zi/dpn > 0 for all i = 1, .., n-1, thus, separating the nth term, we see:

・/font> j=1n-1 pj( zi/dpj) = - pn( zi/dpn) < 0 for i = 1, .., n-1

Thus, isolating the ith market:

・/font> j i=1n-1 pj( zi/dpj) + pi( zi/dpi) < 0 for i = 1, .., n-1

or:

- pi( zi/dpi) > ・/font> j i=1n-1 pj( zi/dpj) for i = 1, .., n-1

Now as, by gross substitution, ( zi/dpi) < 0 for all j and ( zi/dpj) > 0 for all i j, this is equivalent to stating that:

pi|( zi/dpi)| > ・/font> j i=1n-1 pj|( zi/dpj)| for i = 1, .., n-1

or, in terms of a normalized matrix A:

pi|aii| > ・/font> j i=1n-1 pj|aij| for i = 1, .., n-1

and thus, as we assumed that equilibrium pi > 0 for all i = 1, .., n-1 and aii < 0 for all i, then this is equivalent to stating that the matrix A has a negative dominant diagonal and, thus, that A is a D-stable matrix. As the Hicksian conditions are necessary for D-stability, then A is Hicksian.

The equivalent form of this theorem employing Walras's Law rather than homogeneity is also simply proven:

Theorem: (Metzlerian Hicksian via Walras's Law) If z(p) obeys Walras's Law and p* > 0, then gross substitution implies that A is negative definite (i.e. Metzler matrix Hicksian matrix) and A is a stable matrix.

Proof: This was proved independently by Frank H. Hahn (1958) and Kenenth J. Arrow and Leonid Hurwicz (1958). By Walras's Law, ・/font> i=1npizi = 0, thus differentiating with respect to pj:

・/font> i=1n pi( zi/dpj) + zj = 0 for j = 1, .., n

or, evaluating the partial derivatives at equilibrium prices (thus zj = 0), this reduces itself to:

・/font> i=1npi( zi/dpj) = 0 for j = 1, .., n

Thus, we proceed as we did before for the homogeneity case, albeit interchanging the subscript i for j and thus obtaining column diagonal dominance rather than row diagonal dominance for matrix A.

It might be noted that if A fulfills Walras's Law and/or homogeneity, then there is an h R+n such that:

Ah < 0

which is merely one of the lines in our earlier proofs (think of h = [p1, p2, .., pn] . In sum, if matrix A fulfills the properties of (i) gross substitution (aij > 0 for i j) and (ii) Walras Law and/or homogeneity (there is an h 0 such that Ah < 0), then (i) A is Hicksian; (ii) A is D-stable.

(C) McKenzie's Conditions and Complementarity

In our version of the Hahn-Negishi-Arrow-Hurwicz proofs above, we employed the concept of "diagonal dominance" for the assertion of stability. We proved that if we assumed A was Metzlerian and homogeneity/Walras's Law held, then A would be a negative dominant diagonal matrix and "thus D-stable". This last part was not shown and, indeed, it was not employed in the original proofs. In fact, the concept of diagonal dominance was virtually unknown in economics until it was introduced by Lionel McKenzie (1960). As he defined it:

Diagonal Dominance: A n n matrix A with real elements is dominant diagonal (dd) if there are n real numbers dj > 0, j = 1, 2, .., n such that

dj|ajj| > ・/font> i j di|aij|

for j = 1, 2, .., n.

McKenzie went on to prove the following powerful theorem:

Theorem: (Sufficiency) If an n n matrix A is dominant diagonal and the diagonal is composed of negative elements (aii < 0 for all i = 1, .., n), then the real parts of all its eigenvalues are negative, i.e. A is stable.

Proof: McKenzie, 1960. See mathematical section.

and the following corollary:

Corollary: If A has negative diagonal dominance, then it is D-stable.

Proof: McKenzie, 1960. See mathematical section.

McKenzie's condition of diagonal dominance effectively is the weakest as yet available. As we showed earlier, a Metzlerian matrix which fulfills homogeneity/Walras's Law is diagonal dominant and thus D-stable, but a diagonal dominant matrix is not necessarily Metzlerian (cf. Arrow and Hahn, 1971: p.234). For instance, it can be easily shown that if the matrix A exhibits "weak gross substitution", i.e. aij 0 for all i j, with the only restriction that the numeraire a0j > 0, a considerably weaker condition than the Metzlerian matrix, then A still is dominant diagonal (cf. Takayama, 1974: p.401).

One of the nice features of McKenzie's diagonal dominance is that it does not seem to rule out some degree of complementarity among goods or strange income effects a priori - even though apparently most examples that yield diagonal dominance also exhibit gross substitution. To understand this, let us recall that the Slutsky equation for any two goods is:

xi/ pj = hi/ pj - ( xi/ m)(xj - ej)

where xi is the Marshallian demand for the ith good (i.e. xi = xi(p, m)), hi is the Hicksian demand (i.e. hi = hi(p, u0)), m is income and ej is the endowment of the jth good. Using the operators Dp for the vector of derivatives with respect to price and Dy for the vector of derivatives with respect to income, then we can generalize this to obtain the general Slutky matrix:

Dpx = Dph - Dyx[x - e]

where Dpx is the matrix of price derivatives of Marshallian demands, Dph the matrix of price derivatives of Hicksian demands, Dyx is a vector of income derivatives of Marshallian demand and thus Dyx[x - e] is the matrix of income effects. As far back as Johnson (1913) and Slutsky (1915), it has been well-known that the idea of utility maximization of a quasi-concave utility function implies that the matrix of pure substitution terms, Dph is negative semi-definite and symmetric.

However, these conditions are for individual demand functions. The gross substitution assumption introduced by Metzler is presumed to hold in the aggregate, for market excess demand functions and not necessarily on individual demands. This is troublesome for, as the Sonnenschein (1972, 1973)-Mantel (1974)-Debreu (1974) Theorem shows, there is nothing that guarantees that any Slutsky properties that hold at the individual level will hold at the aggregate level. Intuitively, then, what we seek to examine is the aggregate Slutsky equation:

・/font> hDpxh = ・/font> hDphh - ・/font> hDyxh[xh - eh]

where the individual terms are superscripted by h (for the hth household), and thus we are summing up over households. Now if we assume that ・/font> hDphh is negative definite, then, as Hicks (1939: p.317) demonstrates, ・/font> hDpxh will also be negative definite if agents have symmetric income effects, i.e. if xih/ mh = xik/ mk for all households h, k = 1, .., H and goods i = 1, .., n, so that ・/font> hDyxh[xh - eh] = 0. Now, as it can be shown that xi/ pj = zi/ pj, where zi are excess demand functions, then a negative definite ・/font> hDphh implies a negative definite ・/font> hDpzh. But recall that ・/font> hDpzh evaluated at equilibrium is merely our old matrix A. Thus, if ・/font> hDpzh is negative definite is equivalent to claiming that A alternates in sign. As was shown above, the assumption of "gross substitution" (combined with homogeneity or Walras's Law) implies that ・/font> hDphh and ・/font> hDpzh is indeed negative definite. In a sense, then, gross substitution not only rules out complementarity it also rules out the possibility of strong income effects.

One early attempt at reintroducing complementarity was the notable effort of Michio Morishima (1952). Let us divide all goods into two groups such that any two members of the same group are substitutes, but two members of different groups are complements. If we have n goods, let us then divide the indices of n into disjoint groups, J and K. Assume aij > 0 if i j and both i, j J or both i, j K, thus goods in each groups are substitute. In contrast, assume that aij < 0 if i j and i, j belong to different groups. Let us then define P as:

IJ

0

P =

0

-IK

where each identity matrix IJ and -IK is of order equal to the number of elements in J and K respectively. Notice immediately that P = P-1. Now, by an appropriate permutation of A, we can define M = PAP. Now, if M is a Metzler matrix and Walras Law/homogeneity holds, then all the eigenvalues of M are negative and thus M is stable. However, as long as P is non-singular, then M = PAP has the same eigenvalues as A, thus A will also have all eigenvalues negative. Thus, A can exhibit some degree of complementarity and still be stable. Such an A is referred to as a "Morishima matrix".

However, as Arrow and Hurwicz (1958) indicate, a Morishima matrix runs into trouble because of the numeraire good. Or, rather, the numeraire must be neither in J nor in K. To see why, suppose not. Then, by homogeneity:

・/font> j J pj( zi/ pj) + ・/font> j K pk( zi/dpj) = 0

Now, suppose i J, then zi/dpj > 0 for all j J and zi/dpj < 0 for all j K. Thus:

・/font> j J pj( zi/ pj) = - ・/font> j K pk( zi/dpj) > 0.

Now, defining AJ as the submatrix of A composed of elements aij where both i, j J and pJ as the equivalent subvector of p, then this condition implies that:

AJpJ > 0

Now, by assumption, AJ fulfills gross substitution (as all pairs i, j J are substitutes), but it is obvious that homogeneity and/or Walras's Law is violated. Or rather, if they do hold so that there is an h 0 such that Ah < 0, then necessarily there is no pJ > 0 such that AJpJ > 0 is true. As Hicksian conditions were necessary for stability, then AJ will not be Hicksian. A reasonable correction to this system was provided in Morishima (1970).

Selected References

K.J. Arrow and L. Hurwicz (1958) "On the Stability of Competitive Equilbrium, I", Econometrica, Vol. 26 (4), p.522-552.

K.J. Arrow and M. McManus (1958) "A Note on Dynamic Stability", Econometrica, Vol. 26, p.297-305.

A.C. Enthoven and K.J. Arrow (1956) "A Theorem on Expectations and the Stability of Equilibrium", Econometrica, Vol. 24, p.288-93.

F.H. Hahn (1958) "Gross Substitutes and the Dynamic Stability of General Equilibrium", Econometrica, Vol. 26 (1), p.169-70.

F.H. Hahn (1982) "Stability", in K.J. Arrow and M.D. Intriligator, editors, Handbook of Mathematical Economics, Vol. II. Amsterdam: North-Holland.

J.R. Hicks (1939) Value and Capital: An inquiry into some fundamental principles of economic theory. 1946 Edition, Oxford: Clarendon Press.

O. Lange (1944) Price Flexibility and Employment. 1945 reprint, Bloomington, Indiana: Principia Press.

D. McFadden (1968) "On Hicksian Stability", in J.N. Wolfe, editor, Value, Capital and Growth: Essays in honor of Sir John Hicks. Chicago: Aldine.

L.W. McKenzie (1960) "Matrices with Dominant Diagonal in Economic Theory", in Arrow, Karlin and Suppes, editors, Mathematical Methods in the Social Sciences. Stanford: Stanford University Press

L. A. Metzler (1945) "Stability of Multiple Markets: The Hicks conditions", Econometrica, Vol.13 (4), p.277-92.

M. Morishima (1952) "On the Law of Change in Price-Systems in an Economy which Contains Complementary Commodities", Osaka Economic Papers, Vol. 1, p.101-13.

M. Morishima (1970) "A Generalization of the Gross Substitute System", Review of Economic Studies, Vol. 37, p.177-86.

J. Mosak (1944) General Equilibrium Theory in International Trade. Bloomington, Indiana: Principia Press.

T. Negishi (1958) "A Note on the Stability of an Economy where All Goods are Gross Substitutes", Econometrica, Vol. 26, p.445-7.

T. Negishi (1962) "Stability of a Competitive Economy: A survey article", Econometrica, Vol. 30, p.635-69.

P.A. Samuelson (1944) "The Relation Between Hicksian Stability and True Dynamic Stability", Econometrica, Vol.12, p.256-7.

P.A. Samuelson (1947) Foundations of Economic Analysis. 1983 edition. Cambridge, Mass: Harvard University Press.

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