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ps8sol_09 - Cornell University Economics 3130 Problem Set 8...

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Cornell University Economics 3130 Problem Set 8 Solutions 1. Suppose there are two consumers A and B comprising the entire market for good x 1 . There is also a good x 2 . A’s consumption bundle is x A = ( x A 1 , x A 2 ) and B’s is described in similar fashion. Their utility functions are u A 1 = x A 1 + 200 3 x A 2 - ( x A 2 ) 2 and u B 2 = x B 1 + 400 3 x B 2 - ( x B 2 ) 2 . The price of good x 1 is 1 and the price of good x 2 is p . Individuals have incomes I = ( I A , I B ) Assume that both individuals have sufficient income so that an interior solution characterizes their optimal choices. (a) What are the (Marshallian) demands for A and B ? What is the market demand X 2 ( p, I )? x A 1 = I A - 100 p 3 + p 2 2 x A 2 = 100 3 - p 2 x B 1 = I B - 200 p 3 + p 2 2 x B 2 = 200 3 - p 2 X 1 = I A + I B - 100 p + p 2 X 2 = 100 - p (b) What are the indirect utility functions for A and B? V A ( p, I A ) = I A - 100 3 p + 1 4 ( p ) 2 + 10000 9 V B ( p, I B ) = I B - 200 3 p + 1 4 ( p ) 2 + 40000 9 (c) What is A’s price elasticity of demand for good 2? What is B’s? What is the price elasticity of market demand for good 2? ε A = - ∂x A 2 ∂p p x = - - 1 2 6 p 200 - 3 p = 3 p 200 - 3 p ε B = 3 p 400 - 3 p ε = p 100 - p Now, suppose that the supply curve for good 2 is y s 2 = - 20 + 2 p and that the market is competitive. (d) What is the equilibrium price p and quantity ( Q )? 100 - p = - 20 + 2 p p = 40 Q = 60 1
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(e) Suppose now that the government introduces a subsidy of $18 per unit and that new firms do not enter. What is the new equilibrium quantity? What is the price paid by consumers? What is the price received by producers? With a subsidy of 18, p S = p D + 18. Substituting in, one finds p D = 28, p S = 46, and Q = 72.
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