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Unformatted text preview: equilibrium allocation... XB = X - XA = 5 11 / 7 = 24 / 7 YB = Y - YA = 5 22 / 7 = 13 / 7 PURE EXCHANGE EXAMPLE (1 COBB-DOUGLAS & 1 QUASI-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) = (1,19) UB = X0.5Y0.5 B = (XB , YB) = (19,1) We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = ___2___ 2Y-1/2 = 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 123 (3) (1) (2) and we know that MA = PX + 19 PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + 19 PY - PX PX PY XA = 1 + _19PY__ - 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 = X0.5Y0.5 B = (XB , YB) = (19,1) We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.5 X-0.5Y0.5 0.5 X0.5Y-0.5 = YB XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX XB PY Rearranging (1B) we get: PX XB + PY YB = MB 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 + PX XB = MB 2 PX XB = MB 124 XB = MB 2PX (4B) and we know that MB = 19 PX + PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = 19 PX + PY 2PX XB = 19/2 + _PY__ 2 PX (5B) PY YB + PY YB = MB 2 PY YB = MB YB = MB 2 PY (6B) and we know that MB = 19 PX + PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 19 PX + PY 2 PY YB = 1/2 + _19PX__ 2 PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = 1 + _19PY_ - PX + 19/2 + _PY__ PX PY 2 PX = 21/2 + _39PY__ - PX 2PX PY and we know that the fixed supply of X in the economy is the total endowment of X... X = X 125 so, 21/2 + _39PY__ - PX = 20 2PX PY _39PY__ - PX = 19/2 2PX PY 39PY2 19PXPY - 2PX2 = 0 2PX2 + 19PXPY - 39PY2 = 0 So, we need to crack out our quadratic formula... -b + (b2 4ac)1/2 2a -19 + (361 4(2)(-39))1/2 4 -19 + (673)1/2 4 -19 + 25.94224354 4 1.735560886 = PX and so, YA = PX2 PY2 YA =...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
- Spring '10