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Unformatted text preview: market for Y is in equilibrium! What we have done here is to show that if we know the market for X is in equilibrium then we can derive that the market for Y is also in equilibrium. What do I mean by excess demands? We can define excess demands for both goods by saying that "market demand market supply = excess demand"1. ZX = X X ZY = Y Y and Walras Law can be written as PX ZX + PY ZY = 0 In this context, we can state Walras Law as: The value of all market excess demands must sum to zero. (11) For those of you who are wondering, Walras Law works for any number of markets (not just two). The statement of Walras Law for the general case of multiple markets is: 1 Of course, if the expression is negative we have excess supply. 110 In an economy with n markets, if (n 1) markets are in equilibrium, then the nth market is also in equilibrium. This implies that we only need to solve for (n 1) equations. Why? Suppose that P1 Z1 + P2 Z2 + P3 Z3 + P4 Z4 + ... + Pn Zn = 0 and further, suppose the first (n 1) markets are in equilibrium. This means that... Z1 = Z2 = Z3 = Z4 = ... = Zn-1 = 0 0 + 0 + 0 + 0 + ... + Pn Zn = 0 so either Pn = 0 or Zn = 0. So the market for good n either results in n being a free good or the market for good n is in equilibrium. PURE EXCHANGE EXAMPLE (2 COBB-DOUGLAS CONSUMERS...again) As promised, we will be going through a couple more examples of how to solve a pure exchange economy using a variety of functional forms in combination. Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a Cobb-Douglas utility function with = 0.4, = 0.6, and = 1while Consumer B has a Cobb-Douglas utility function with = 0.65, = 0.35, and = 1. There are 12 units of each good allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = 7 XB = 5 X = 7 + 5 = 12 GOOD Y YA = 3 YB = 9 Y = 3 + 9 = 12 We solve this particular pure exchange economy as follows: Let's start with Consumer A. This consumer has a utility functio...
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- Spring '10