Coursenotes_ECON301

164 1 a contract curve yields allocations xa xb ya yb

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Unformatted text preview: URE EXCHANGE ECONOMY EXAMPLE Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a square root utility function while Consumer B has a CobbDouglas utility function with = 0.25, = 0.75, and = 1. There is one unit of each good allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = XB = X = + = 1 GOOD Y YA = YB = Y = + = 1 We solve this particular pure exchange economy as follows: Let's start with Consumer A. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UA = UA(XA,YA) = X1/2Y1/2 MA = PX XA + PY YA = PX + PY We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = 0.5 X-1/2Y1/2 0.5 X1/2Y-1/2 = YA XA At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA = PX XA PY PX XA + PY YA = MA 166 (1) (2) Rearranging (1) we get: XA PX = YA PY (3) Meaning we can get demands for XA and YA by subbing (3) into (2) as follows: PX XA + PX XA = MA 2 PX XA = MA XA = M A 2 PX (4) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + PY 2 PX XA = 1/8 + _3PY__ 8 PX PY YA + PY YA = MA 2 PY YA = MA YA = M A 2 PY (6) (5) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (6) to get: YA = PX + PY 2 PY YA = 3/8 + _PX__ 8 PY (7) 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 income that constrains their utility as follows: UB = UB(XB,YB) = X1/4Y3/4 167 MB = PX XB + PY YB = PX + PY We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.25 X-3/4Y3/4 0.75 X1/4Y-1/4 = YB 3XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX 3XB PY Rearranging (1B) we get: PX XB + PY YB = MB 3 XB PX = YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + 3 PX XB = MB 4 PX XB = MB XB = MB 4 PX (4B) and we know that MB = PX + PY is the en...
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