PS5 - ECO 310&Fall 2007 Microeconomic Theory&A...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 &rm¡s short run supply curve coincides with its marginal cost curve as long as the &rm¡s 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...
View Full Document

{[ snackBarMessage ]}

Page1 / 6

PS5 - ECO 310&Fall 2007 Microeconomic Theory&A...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online