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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 pricesystem 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
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:
dp_{i}/dt = ｦ_{i}(z_{i}(p))
thus, price adjustment in market i is a function of z_{i}, 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 crosseffects 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:
dz_{i}/dp_{i} < 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 crosseffects 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 dz_{i}/dp_{i} < 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 p_{i}, we still have it that dz_{i}/dp_{i} < 0 under any of the following three cases:
(i) dz_{i}/dp_{i} < 0 after all markets are allowed to adjust fully (as in the "imperfect" case)
(ii) dz_{i}/dp_{i} < 0 when no markets are allowed to adjust (as in the partial market case)
(iii) dz_{i}/dp_{i} < 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 dz_{i}/dp_{i}, the Hicksian approach has often been termed "slope stability".
Let us have n+1 goods, x_{0}, x_{1}, x_{2} .... x_{n} with n+1 prices, p_{0}, p_{1}, p_{2}, ..., p_{n}. Henceforth, unless otherwise noted, we will restrict our attention to a normalized system, thus we can remove our n+1th commodity, x_{0} by Walras's Law and our n+1th price, p_{0}, as numeraire and deal exclusively with an ndimensional system, x_{1}, x_{2}, .., x_{n} and p_{1}, p_{2}, ..., p_{n}. Let z_{i} = x_{i}  e_{i} be excess demand for good i.
Let us specify that the excess demand function for the ith good, z_{i} = z_{i} (1, p_{1}, p_{2}, .., p_{n}), 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:
dz_{1} = a_{11}dp_{1} + a_{12}dp_{2} + ... + a_{1n}dp_{n}
dz_{2} = a_{21}dp_{1} + a_{22}dp_{2} + ... + a_{2n}dp_{n}
....... ...........................................
dz_{n} = a_{n1}dp_{1} + a_{n2}dp_{2} + ... + a_{nn}dp_{n}
where a_{ij} is the partial derivative of excess demand in the ith market with respect to the jth price, i.e. a_{ij} = ｶ z_{i}/ｶ p_{j}. If a_{ij} > 0, then goods i and j are "gross substitutes"; if a_{ij} < 0, they are gross complements. We can assume that a_{ii} < 0 (own effect) is negative. Letting A be an (n ｴ 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 p_{i} led to z_{i} becoming negative (excess supply)  after all the cross effects in other markets worked themselves out. In other words, he wanted to guarantee that dz_{i}/dp_{i} < 0 in the end. To impose that all other markets have adjusted except for the ith market, then we must assume that dz_{j} = 0 for all j = 1, 2, .., n where j ｹ i so that dz is a column vector with dz_{i} is in the ith place and zeroes everywhere else. Consequently, by Cramer's Rule:
dp_{i} = A_{i}/A
where A is the determinant of A and A_{i} 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 A_{i}, that A_{i} = dz_{i}C_{ii}, where C_{ii} is the cofactor of element a_{ii} (i.e. the determinant of the minor obtained by deleting the ith row and column from the original matrix A_{i}  and notice that as i+i is even, then C_{ii} = M_{ii} where M_{ii} is the minor). As a result dp_{i} = dz_{i}C_{ii}/A, or:
dz_{i}/dp_{i} = A/C_{ii}
Thus, in order for dz_{i}/dp_{i} < 0 (i.e. imperfect stability), then it must be that the cofactor C_{ii} 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.
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 dz_{1}/dp_{1} < 0, would be replicated for any dz_{i}/dp_{i} < 0).
Step 1: assume all other prices are rigid (partial case), i.e. dp_{j} = 0 for all j = 2, .., n. Then this entire system dz = Adp reduces to dz_{1} = a_{11}dp_{1 }or simply,
dz_{1}/dp_{1} = a_{11}
Thus, the first condition for stability, is that the own effect (a_{11}) must be negative. As this is true for market 1, it must be true for all markets, thus a_{ii} < 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. dz_{2} = 0. Then this entire system reduces to:
dz_{1} = a_{11}dp_{1} + a_{12}dp_{2}
0 = a_{21}dp_{1} + a_{22}dp_{2}
Reexpressing in matrix form so that we have dz = A*dp for this reduced system where A* is merely the smaller 2ｴ 2 matrix of coefficients a_{ij}, i, j = 1, 2. Then by Cramer's Rule, we know:
dp_{1} = A_{1}/A
where A* is the determinant of the small 2 ｴ 2 coefficient matrix and A_{1} is the determinant of the matrix A* with solution vector dz replacing the first column. Thus:
dz_{1} 
a_{12} 

0 
a_{22} 

dp_{1 }= 
_________ 

a_{11} 
a_{12} 

a_{21} 
a_{22} 
As we can see immediately, A* = a_{11}a_{22}  a_{12}a_{21} and A_{1} = dz_{1}a_{22 }so:
dp_{1} = (dz_{1}a_{22})/(a_{11}a_{22}  a_{12}a_{21})
thus:
dz_{1}/dp_{1} = (a_{11}a_{22}  a_{12}a_{21})/a_{22}
If there is to be stability, then dz_{1}/dp_{1} < 0. We know from condition (i) that a_{22} < 0, thus for stability we need that (a_{11}a_{22}  a_{12}a_{21}) > 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 a_{ij}.
Step 3: Let us allow the first three prices to change while all others remain rigid. Again, assuming the other markets adjust fully, then dz_{2} = dz_{3} = 0. Thus we end up with the following system:
dz_{1} = a_{11}dp_{1} + a_{12}dp_{2} + a_{13}dp_{3}
0 = a_{21}dp_{1} + a_{22}dp_{2} + a_{23}dp_{3}
0 = a_{31}dp_{1} + a_{32}dp_{2} + a_{33}dp_{3}
We can rewrite this in matrix form again as dz = A*dp with these vectors and matrices now being threedimensional. Again, by Cramer's Rule:
dp_{1} = A_{1}/A*
where A* is the determinant of our threemarket matrix and A_{1} is the transformed matrix with the solution vector dz in the first column. This reduces to:
dp_{1} = dz_{1}(a_{22}a_{33}  a_{23}a_{32})/A*
where we don't write out A* because it is a rather horrible term. Nonetheless, rearranging we can see that:
dz_{1}/dp_{1} = A*/(a_{22}a_{33}  a_{23}a_{32})
Notice that the denominator of this expression is merely the determinant of a second order matrix:
a_{22} 
a_{23} 
a_{32} 
a_{33} 
But, by condition (ii) for stability, the determinants all such second order matrices must be positive. Thus, by (ii), (a_{22}a_{33}  a_{23}a_{32}) > 0. Therefore, the only way dz_{1}/dp_{1} < 0 under the threegood 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) a_{11} < 0
(2)
a_{11} 
a_{12} 

a_{21} 
a_{22} 
> 0 
(3)
a_{11} 
a_{12} 
a_{13} 

a_{21} 
a_{22} 
a_{23} 
< 0 
a_{31} 
a_{32} 
a_{33} 
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 multimarket 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 n1, 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 multimarket stability as they were not really "dynamic" in a proper sense but merely "slope conditions" (i.e. we just want to ascertain that z_{i} remains negative after a rise in price p_{i}). 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:
dp_{i}/dt = ｦ_{i}[z_{i}(p(t))]
where prices in the ith market (p_{i}) adjust in response to excess demands in the ith market (z_{i}(p)) depending on the particular form of adjustment in that market (ｦ_{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, p_{i}*, we can "linearize" this expression locally so that we obtain something like:
dp_{i}(t)/dt = k_{i} ・/font> _{j=1}^{n} a_{ij} (p_{j}(t)  p_{j}*)
where k_{i} is the "speed of adjustment" in the ith market, a_{ij} are the partial derivatives of excess demand in market i with respect to price change in market j evaluated at equilibrium (a_{ij} = ｶ z_{i}/dp_{j}  same as before) and (p_{j}(t)  p_{j}*) 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 ndimensional vector of price adjustments (with typical element dp_{i}/dt); K is a diagonal matrix of speeds of adjustments (k_{i} along the diagonal, zeroes everywhere else); A is the old matrix of first partial derivatives (a_{ij} = dz_{i}/dp_{j}) 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) = e^{KAt}(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) ｮ p* as t ｮ ･ , 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^{1}L^{ }D where L is a diagonal matrix with the n eigenvalues l _{i} arrayed along the diagonal and D = [v_{1} v_{2} .... v_{n}] is a modal matrix with eigenvectors v_{i} as columns. We can consequently rewrite this system as:
p(t) = De^{Lt}D^{1}(p(0)  p*) + p*
Obviously, p(t) ｮ p* if the homogeneous part De^{Lt}D^{1}(p(0) ｮ 0 as t ｮ ･ . 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 twomarket system, this is actually rather easy. The characteristic equation of this system would be:
k_{1}a_{11}l 
k_{1}a_{12} 

KA  l I = 
= 0 

k_{2}a_{21} 
k_{2}a_{22}  l 
or:
l ^{2}  l (trKA) + KA = 0
where
trKA = k_{1}a_{11} + k_{2}a_{22}
KA = k_{1}a_{11}k_{2}a_{22}  k_{2}a_{12}k_{2}a_{21}
Now, if k_{1} = k_{2} = 1, then the Hicksian conditions on the alternating minors of A implies a_{11}, a_{22} < 0 and a_{11}a_{22}  a_{12}a_{21} > 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 a_{22} = 1 > 0.
Does the assumption of adjustment coefficients k_{1} = k_{2} = 1 affect anything in this case? Actually, there is no qualitative difference in the 2 ｴ 2 case. Notice that as long as k_{1}, k_{2} > 0, then the trace, trKA = k_{1}a_{11} + k_{2}a_{22} will remain negative as long as a_{11} < 0 and a_{22} < 0, regardless of the values of k_{1} and k_{2}. Similarly, we can factor k_{1}k_{2} in the determinant KA = k_{1}k_{2}(a_{11}a_{22}  a_{12}a_{21}) thus, as long as k_{1}, k_{2} > 0, their values will not affect the sign of the determinant. As a result, we can thus say in the 2 ｴ 2 case, that the Hicksian matrix A is a "Dstable" matrix, which is defined as follows:
DStability: matrix A is "Dstable" 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 Dstability of the Hicksian matrix A to situations beyond the 2 ｴ 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 Dstability). Many examples are available to show this, but the essential point is that in an ndimensional system with n ｳ 3, the Hicksian conditions do not imply the RouthHurwitz (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 (a_{ij} = a_{ji} for all i, j), the Hicksian conditions are sufficient for Dstability.
(2) If A is quasinegative definite (i.e. A + A｢ is negative definite), then the Hicksian conditions are fulfilled and we have Dstability.
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 ArrowMcManus theorem on Dstability, namely:
Theorem: (Arrow and McManus, 1958) matrix A is Dstable 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 ArrowMcManus 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 Dstable. From (2), we see this immediately by the definition of quasinegative definite.
A more interesting result was obtained by Metzler (1945) and generalized by Enthoven and Arrow (1956) which claimed the following:
Proof: (Metzler , 1945; Enthoven and Arrow, 1956) If A is Dstable, then KA is stable for any positive diagonal matrix K. Recall that KA = ﾕ _{i=1}^{n} 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, sgnKA = sgn(1)^{n}. Thus, for any jth order matrix KA_{j}, it must be that sgnKA_{j} = sgn(1)^{j}. Since all the elements of K are positive, k_{i} > 0, then the sign of the determinant of any jth order leading minor KA_{j} is equal to the jth leading minor of A, i.e. sgnKA_{j} = sgnA_{j} = (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.ｧ
Thus, we see that a Hicksian matrix A is necessary for Dstability, but it is certainly not sufficient.
Finally, an interesting theorem provided many years later by Daniel McFadden (1968), employing the FisherFuller 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).
Obviously, the Hicksian conditions on A do not imply stability or Dstability 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 a_{ij} > 0 for all i ｹ j. By Walras's Law, this also implies that a_{ii} < 0 for all i. Thus, crosseffects are strictly positive and own effects are strictly negative. This implies that the matrix A has strictly negative diagonal elements and strictly positive offdiagonal 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 (a_{ij} > 0 for all iｹ j), then the Hicks conditions are necessary and sufficient for a Dstable 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 Dstable matrix A.
Theorem: (Metzlerian ﾞ 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 Dstable matrix.
Proof: First proved by Takashi Negishi (1958). By homogeneity of degree zero of excess demand functions, Euler's Theorem states that:
・/font> _{j=1}^{n} p_{j}(ｶ z_{i}/dp_{j}) = 0 for all i = 1, .., n.
where partial derivatives are all evaluated at equilibrium prices, p_{j}, j = 1, .., n. By gross substitution, ｶ z_{i}/dp_{n} > 0 for all i = 1, .., n1, thus, separating the nth term, we see:
・/font> _{j=1}^{n1 }p_{j}(ｶ z_{i}/dp_{j}) =  p_{n}(ｶ z_{i}/dp_{n}) < 0 for i = 1, .., n1
Thus, isolating the ith market:
・/font> _{jｹ i=1}^{n1 }p_{j}(ｶ z_{i}/dp_{j}) + p_{i}(ｶ z_{i}/dp_{i}) < 0 for i = 1, .., n1
or:
 p_{i}(ｶ z_{i}/dp_{i}) > ・/font> _{jｹ i=1}^{n1 }p_{j}(ｶ z_{i}/dp_{j}) for i = 1, .., n1
Now as, by gross substitution, (ｶ z_{i}/dp_{i}) < 0 for all j and (ｶ z_{i}/dp_{j}) > 0 for all iｹ j, this is equivalent to stating that:
p_{i}(ｶ z_{i}/dp_{i}) > ・/font> _{jｹ i=1}^{n1 }p_{j}(ｶ z_{i}/dp_{j}) for i = 1, .., n1
or, in terms of a normalized matrix A:
p_{i}a_{ii} > ・/font> _{jｹ i=1}^{n1 }p_{j}a_{ij} for i = 1, .., n1
and thus, as we assumed that equilibrium p_{i }> 0 for all i = 1, .., n1 and a_{ii} < 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 Dstable matrix. As the Hicksian conditions are necessary for Dstability, then A is Hicksian.ｧ
The equivalent form of this theorem employing Walras's Law rather than homogeneity is also simply proven:
Theorem: (Metzlerian ﾞ 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 ﾞ 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=1}^{n}p_{i}z_{i} = 0, thus differentiating with respect to p_{j}:
・/font> _{i=1}^{n }p_{i}(ｶ z_{i}/dp_{j}) + z_{j} = 0 for j = 1, .., n
or, evaluating the partial derivatives at equilibrium prices (thus z_{j} = 0), this reduces itself to:
・/font> _{i=1}^{n}p_{i}(ｶ z_{i}/dp_{j}) = 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.ｧ
It might be noted that if A fulfills Walras's Law and/or homogeneity, then there is an h ﾎ R_{+}^{n} such that:
Ah < 0
which is merely one of the lines in our earlier proofs (think of h = [p_{1}, p_{2}, .., p_{n}]｢ . In sum, if matrix A fulfills the properties of (i) gross substitution (a_{ij} > 0 for iｹ j) and (ii) Walras Law and/or homogeneity (there is an h ｳ 0 such that Ah < 0), then (i) A is Hicksian; (ii) A is Dstable.
(C) McKenzie's Conditions and Complementarity
In our version of the HahnNegishiArrowHurwicz 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 Dstable". 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 ｴ n matrix A with real elements is dominant diagonal (dd) if there are n real numbers d_{j }> 0, j = 1, 2, .., n such that
d_{j}a_{jj} > ・/font> _{iｹ j} d_{i}a_{ij}
for j = 1, 2, .., n.
McKenzie went on to prove the following powerful theorem:
Theorem: (Sufficiency) If an n ｴ n matrix A is dominant diagonal and the diagonal is composed of negative elements (a_{ii} < 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 Dstable.
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 Dstable, 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. a_{ij} ｳ 0 for all i ｹ j, with the only restriction that the numeraire a_{0j} > 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:
ｶ x_{i}/ｶ p_{j} = ｶ h_{i}/ｶ p_{j}  (ｶ x_{i}/ｶ m)(x_{j}  e_{j})
where x_{i} is the Marshallian demand for the ith good (i.e. x_{i} = x_{i}(p, m)), h_{i} is the Hicksian demand (i.e. h_{i} = h_{i}(p, u_{0})), m is income and e_{j} is the endowment of the jth good. Using the operators D_{p} for the vector of derivatives with respect to price and D_{y} for the vector of derivatives with respect to income, then we can generalize this to obtain the general Slutky matrix:
D_{p}x = D_{p}h  D_{y}x[x  e]
where D_{p}x is the matrix of price derivatives of Marshallian demands, D_{p}h the matrix of price derivatives of Hicksian demands, D_{y}x is a vector of income derivatives of Marshallian demand and thus D_{y}x[x  e] is the matrix of income effects. As far back as Johnson (1913) and Slutsky (1915), it has been wellknown that the idea of utility maximization of a quasiconcave utility function implies that the matrix of pure substitution terms, D_{p}h is negative semidefinite 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> _{h}D_{p}x^{h} = ・/font> _{h}D_{p}h^{h}  ・/font> _{h}D_{y}x^{h}[x^{h}  e^{h}]
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> _{h}D_{p}h^{h} is negative definite, then, as Hicks (1939: p.317) demonstrates, ・/font> _{h}D_{p}x^{h} will also be negative definite if agents have symmetric income effects, i.e. if ｶ x_{i}^{h}/ｶ m^{h} = ｶ x_{i}^{k}/ｶ m^{k} for all households h, k = 1, .., H and goods i = 1, .., n, so that ・/font> _{h}D_{y}x^{h}[x^{h}  e^{h}] = 0. Now, as it can be shown that ｶ x_{i}/ｶ p_{j} = ｶ z_{i}/ｶ p_{j}, where z_{i} are excess demand functions, then a negative definite ・/font> _{h}D_{p}h^{h} implies a negative definite ・/font> _{h}D_{p}z^{h}. But recall that ・/font> _{h}D_{p}z^{h} evaluated at equilibrium is merely our old matrix A. Thus, if ・/font> _{h}D_{p}z^{h} 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> _{h}D_{p}h^{h} and ・/font> _{h}D_{p}z^{h} 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 a_{ij} > 0 if i ｹ j and both i, j ﾎ J or both i, j ﾎ K, thus goods in each groups are substitute. In contrast, assume that a_{ij} < 0 if i ｹ j and i, j belong to different groups. Let us then define P as:
I_{J} 
0 

P = 

0 
I_{K} 
where each identity matrix I_{J} and I_{K} 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 nonsingular, 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ﾎ J} p_{j}(ｶ z_{i}/ｶ p_{j}) + ・/font> _{jﾎ K} p_{k}(ｶ z_{i}/dp_{j}) = 0
Now, suppose i ﾎ J, then ｶ z_{i}/dp_{j} > 0 for all j ﾎ J and ｶ z_{i}/dp_{j} < 0 for all j ﾎ K. Thus:
・/font> _{jﾎ J} p_{j}(ｶ z_{i}/ｶ p_{j}) =  ・/font> _{jﾎ K} p_{k}(ｶ z_{i}/dp_{j}) > 0.
Now, defining A_{J} as the submatrix of A composed of elements a_{ij} where both i, j ﾎ J and p_{J} as the equivalent subvector of p, then this condition implies that:
A_{J}p_{J} > 0
Now, by assumption, A_{J} fulfills gross substitution (as all pairs i, j ﾎ 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 ｳ 0 such that Ah < 0, then necessarily there is no p_{J} > 0 such that A_{J}p_{J} > 0 is true. As Hicksian conditions were necessary for stability, then A_{J} will not be Hicksian. A reasonable correction to this system was provided in Morishima (1970).
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