Lecture10 - ECON 301 LECTURE#10 PARETO OPTIMAL THEORY PURE...

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1 ECON 301 – LECTURE #10 PARETO OPTIMAL THEORY PURE 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 Cobb-Douglas 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: GOOD X GOOD Y Consumer A $ X A = ¼ $ Y A = ¾ Consumer B $ X B = ¾ $ Y B = ¼ Total $ X = ¼ + ¾ = 1 $ Y = ¾ + ¼ = 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: U A = U A (X A ,Y A ) = X 1/2 Y 1/2 M A = P X ! X A + P Y ! Y A = ¼ P X + ¾ P Y We can figure out consumer A’s marginal rate of substitution as: MRS A = MU X A MU Y A = 0.5 X -1/2 Y 1/2 0.5 X 1/2 Y -1/2 = Y A X A At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRS A = Y A = P X X A P Y (1) P X X A + P Y Y A = M A (2)
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2 Rearranging (1) we get: X A P X = Y A P Y (3) Meaning we can get demands for X A and Y A by subbing (3) into (2) as follows: P X X A + P X X A = M A 2 P X X A = M A X A = M A 2 P X (4) and we know that M A = ¼ P X + ¾ P Y is the endowment income of consumer A, so we sub this in for the M A in (4) to get: X A = ¼ P X + ¾ P Y 2 P X P Y Y A + P Y Y A = M A 2 P Y Y A = M A Y A = M A 2 P Y (6) and we know that M A = ¼ P X + ¾ P Y is the endowment income of consumer A, so we sub this in for the M A in (6) to get: Y A = ¼ P X + ¾ P Y 2 P Y 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: U B = U B (X B ,Y B ) = X 1/4 Y 3/4 X A = 1/8 + _3P Y__ 8 P X (5) Y A = 3/8 + _P X__ 8 P Y (7)
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3 M B = P X ! X B + P Y ! Y B = ¾ P X + ¼ P Y We can figure out consumer B’s marginal rate of substitution as: MRS B = MU X B MU Y B = 0.25 X -3/4 Y 3/4 0.75 X 1/4 Y -1/4 = Y B 3X B At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRS B = Y B = P X 3X B P Y (1B) P X X B + P Y Y B = M B (2B) Rearranging (1B) we get: 3 X B P X = Y B P Y (3B) Meaning we can get demands for X B and Y B by subbing (3B) into (2B) as follows: P X X B + 3 P X X B = M B 4 P X X B = M B X B = M B 4 P X (4B) and we know that M B = ¾ P X + ¼ P Y is the endowment income of consumer B, so we sub this in for the M B in (4B) to get: X B = ¾ P X + ¼ P Y 4P X 1/3 P Y Y B + P Y Y B = M B 4/3 P Y Y B = M B X B = 3/16 + _P Y__ 16 P X (5B)
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4 Y B = 3M B 4 P Y (6B) and we know that M B = ¾ P X + ¼ P Y is the endowment income of consumer B, so we sub this in for the M B in (6B) to get: Y B = 3 (¾ P X + ¼ P Y ) 4 P Y Now that we have the individual demands for each consumer for both goods, we can do
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Lecture10 - ECON 301 LECTURE#10 PARETO OPTIMAL THEORY PURE...

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