Strict diagonal dominance in asymptotic stability of general equilibrium
© Hsu and Shih; licensee Springer. 2013
Received: 14 June 2013
Accepted: 8 August 2013
Published: 28 August 2013
Stability of general equilibrium is usually depicted by a dynamic process of price adjustment which makes the flow of prices eventually come to rest at certain prices, so that the supply and demand of every commodity tend to equal each other. Here we construct a dynamical system of a competitive economy and find that the strict diagonal dominance of the Jacobian matrix of the excess demand function at its equilibrium guarantees the asymptotic stability.
MSC:37N40, 91B55, 93D20.
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
Economists look for conditions that make the equilibrium under consideration stable. For example, in , Arrow, Block and Hurwicz establish the stability on the assumption that the underlying excess demand function encoding the dynamics satisfies some gross substitute conditions; in , Uzawa also supposes that a gross substitute condition prevails and that the stability is guaranteed by using Lyapunov’s stability theorem; in , Bear finds the equivalence of the stability of ‘lagged’ systems and ‘first-order aggregation’ of lagged systems. Our condition is that the excess demand function has a strictly diagonally dominant Jacobian matrix at its equilibrium. Conditions of this sort can also be found in [4, 5], where Hadar assumes the excess demand function to have a strictly diagonally dominant Jacobian matrix at not only a single point, but everywhere. By using a contraction technique, Hadar is able to show that the dynamical system has a unique equilibrium which is globally stable. In spite of the stronger conclusion, the assumption made in [4, 5] imposes too strong restriction on the excess demand function as stated by Hadar. Here we construct a dynamical system of a competitive economy and find that the strict diagonal dominance of the Jacobian matrix of the excess demand function at its equilibrium guarantees the asymptotic stability. Our analysis is based on the estimation of the range of eigenvalues and solving recurrences and concludes the local stability of the general equilibrium.
We begin with a mathematical formation of the relations between supply, demand, and prices in an economy. Consider an economy with commodities . Each commodity has its own price . The vector is called a price system. While we say vectors and are demand and supply, respectively, the components and represent, respectively, the demand and supply of commodity i, . A vector is called an excess demand, while we think for some demand and some supply . Suppose that a price system p is given. Each consumer or producer in the economy will make consumption or produce commodities according to the price system p, resulting in a demand and a supply corresponding to p. The excess demand function is defined by . If there is a price system such that , meaning that the demand in each market is equal to the supply, then the state of prices and excess demand is called a general equilibrium of the economy.
In this dynamical system, a state variable represents a state of the economy with the first n components of x standing for the price system and the remaining ones standing for the excess demand.
every component of is not zero;
- (ii)the Jacobian matrix is strictly diagonally dominant, i.e.,
for all .
Assumption (i) means that none of the commodities has a zero equilibrium price. Assumption (ii) means that each ‘own-good’ price effect dominates the sum of the effects of the respective price on all other markets. Assumption (iii) means that a price raise (decrease) of some commodity should reduce (increase) its excess demand. As demonstrated by the following theorem, will be asymptotically stable while the adjustment speeds of the prices lie in a suitable range.
Theorem 1 There is a such that the equilibrium of system (1) is asymptotically stable whenever .
To show this, let us introduce the following simple result.
Lemma 1 Suppose that A is a complex Jordan canonical form. If A is invertible, then there exists a upper triangular matrix B such that .
Then a computation shows that . □
For a square complex matrix A, the notation denotes the collection of all eigenvalues of A.
Hence (8) holds for , completing the induction and proving the claim.
Since M is similar to N, (8) and (7) imply that goes to the zero matrix as k goes to infinity, showing that every eigenvalue of M lies in the open unit disc, so that the equilibrium of system (2) is stable and attracting.
meaning that the equilibrium of system (1) is stable. By the continuity of F at , (11) shows that as . This completes the proof. □
This work was supported by the National Science Council of the Republic of China.
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