ExchangeEquilibrium

ExchangeEquilibrium - Notes on General Equilibrium in an...

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Notes on General Equilibrium in an Exchange Economy Ted Bergstrom, Econ 210A, UCSB November 30, 2011 From Demand Theory to Equilibrium Theory We have studied Marshallian demand functions for rational consumers, where D i ( p, m i ) is the vector of commodities demanded by consumer i when the price vector is p . In general, the incomes of individuals depend on the prices of goods and services that they have to sell. Therefore in the study of general equilib- rium theory, we need to make incomes depend on the prices of commodities. This is nicely illustrated in the example of a pure exchange economy where there trade but no production. Each consumer initially has some vector of endowments of goods. These goods are traded at competitive prices and in equilibrium the total demand for each good is equal to the supply of that good. A Pure Exchange Economy There are m consumers and n goods. Consumer i has a utility function u i ( x i ) where x i is the bundle of goods consumed by consumer i . In a competitive market, Consumer i has an initial endowment of goods which is given by the vector ω i 0. Where p is the vector of prices for the n goods, consumer i ’s budget constraint is px i i which simply says that the value at prices p of what he consumes cannot exceed the value of his endowment. Consumer i chooses the consumption vector D i ( p ) that solves this max- imization problem. Where x i ( p, m i ) is i ’s Marshallian demand curve, we 1
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have D i ( p ) = x i ( p, pω i ) . Let us denote i ’s demand for good j by D i j ( p ), which is the j th component of the vector D i ( p ). A pure exchange equilibrium occurs at a price ¯ p such that total demand for each good equals total supply. That is, m X i =1 D i p ) = m X i =1 ω i j , which means that m X i =1 D i j p ) = m X i =1 ω i j for all j = 1 , . . . n . This vector equation can be thought of as n simultaneous equations, one for each good. Finding a competitive equilibrium price amounts to solving these n equations in n unknowns. There are two important facts that simplify this task if the number of commodities is small. Homogeneity and a numeraire The first is that the functions D i ( p ) are all homogeneous of degree zero in prices and hence, so is i D i ( p ). To see this, note that if you multiply all prices by the same amount, you do not change the budget constraint (since if px i = i , then it must also be that kpx i = kpω i for al k > 0. Therefore we can set one of our prices equal to 1 and solve for the remaining prices. Since any multiple of this price vector would also be a competitive equilibrium, we lose no generality in setting this price to 1. Walras Law and one Equality for Free The second fact is a little more subtle. It turns out that if demand equals supply for all n - 1 goods other than the numeraire, then demand equals supply for the numeraire good as well. This means that to find equilibrium where there are n goods, we really only need to solve n - 1 equations in n - 1 unknowns. Thus if n = 2, we only need to solve a single equation. If n = 3, we still only need to solve 2 equations in 2 unknowns.
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