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Unformatted text preview: ECO 310 &Fall 2007 Microeconomic Theory &A Mathematical Approach Problem Set 5 &Answer Key Question 1: (a) Each consumer maximizes U ( x;y ) = y + 10 x & 5 x 2 ; subject to the budget constraint px + y = M : Substituting out y , the objective is F ( x ) M & px + 10 x & 5 x 2 : Now F ( x ) = 10 & p & 10 x; F 00 ( x ) = & 10 < : So the FONC yields the global max, and If p 10 , then F (0) so x = 0 is optimum If p < 10 , then F (0) > and the optimum is x = 1 & p= 10 . The market demand function just the sum over 160 consumers: If p 10 , then X = 0 If p < 10 , then X = 160 & 16 p . (b) Other than the LRTC which is as stated, the other costs as functions of x are meaningful only for x > . The expressions are (5 points): LRTC ( x ) = & if x = 0 4 + x 2 if x > LRAC ( x ) = 4 =x + x SRTAC ( x ) = & if x = 0 1 + x 2 if x > SRAAC ( x ) = 1 =x + x LRMC ( x ) = & 4 if x = 0 2 x if x > SRMC ( x ) = & 1 if x = 0 2 x if x > 1 LRAC 2 + 1 , LRAC 00 3 > . So LRAC ( q ) is minimized at x = 2 and the minimum is 4. SRAAC 2 + 1 , SRAAC 00 3 > . So SRAAC ( q ) is minimized at x = 1 and the minimum is 2. The &rms short run supply curve coincides with its marginal cost curve as long as the &rms pro&t is not lower than & 3 . So, for the &rm to produce any &sh, we need p 2 = 4 & 4 & 3 , which means p 2 . Hence, x = & if p 2 p= 2 if p 2 (c) In the long run, with free entry and exit of &shing &rms, each &rm is price taker and their pro&t is driven to zero. The pro&t & = 1 q ( p & (4 =q + q )) . If p 4 , each &rm can decrease his quantity of output and make positive pro&t. Hence, in the long run, p = 4 , that is at the level of minimum long run average cost. So, if p < 4 , supply equals zero; if p > 4 , the supply equals np= 2 , where n is the number of &rms; if p = 4 , the supply equals 2 n . (d) From the supply curve, p = 4 . Then from the demand curve, X = 16(10 & 4) = 96 . From (c), we know that each &rm produces x = 2 , so there are 48 &rms. Price equals the LRAC at this point, so each makes zero pro&t. The aggregate consumer surplus is the area to the left of the market demand curve, so it equals 1 2 (10 & 4) 96 = 288 . (e) In the short run with 48 &rms, the industry supply curve is X = & if p 2 24 p if p 2 Assuming for the moment that the 48 &rms remain active, the price received by the &rms is given as a function of the market quantity by p = X= 24 : The price paid by the consumers is given as a function of X by p = 10 & X = 16 : With the tax, the equilibrium...
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 '08
 StephenE.Morris

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