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Unformatted text preview: s. XA + XB = 3.8128 + 8.1872 XA + XB = 12 XA + XB = X YA + YB = 6.7765 + 5.2235 YA + YB = 12 YA + YB = Y So, in both the market for X and the market for Y, demands are equal to supplies and we have market clearing (equilibrium). PURE EXCHANGE EXAMPLE (1 QUASI-LINEAR & 1 LINEAR CONSUMER) Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. UA = 2X + 4Y1/2 A = (XA , YA) = (6,4) 117 UB = X + Y B = (XB , YB) = (4,6) Let's clear up some terminology before we continue... Above, we have the consumers' initial endowments given by: A = (XA , YA) and B = (XB , YB) and their utility functions. What we are trying to find is the equilibrium price ratio and to do this we need the consumers' consumption vectors represented as: CA = (XA , YA) and CB = (XB , YB) to obtain total market demands represented as: X = (XA + XB) and Y = (YA , YB) Definition: An allocation is a specification of a consumption vector for each consumer and CB = (XB , YB) CA = (XA , YA) Definition: A feasible allocation is an allocation which is physically possible given the resources of the economy. An allocation is feasible if XA + XB XA + XB YA + YB YA + YB Now, back to our example... We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = ___2___ 2Y-1/2 118 = YA 1/2 At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA 1/2 = PX PY YA = PX2 PY2 Subbing (1) into (2) we get: PX XA + PY YA = MA PX XA + PX2 = MA PY XA = MA - PX PX PY (3) (1) (2) and we know that MA = 6 PX + 4 PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = 6 PX + 4 PY - PX PX PY XA = 6 + _4PY__ - PX PX PY (4) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = X + Y B = (XB , YB) = (4,6) We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = __1__ 1 119 At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = _1_ = PX 1 PY (1B) So now that we have the equilibrium price ratio. Since only the ratio matters we can use the relationship PX = PY in consumer A's demands above (we can derive consumer B's demands from consumer A's...more in a second) XA = 6 + _4PY__ - PX PX PY =...
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- Spring '10