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Pset5 Answers Econ310 Fall 07 - ECO 310 Fall 2007...

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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 p x + y = M : Substituting out y , the objective is F ( x ) ± M ° p x + 10 x ° 5 x 2 : Now F 0 ( x ) = 10 ° p ° 10 x; F 00 ( x ) = ° 10 < 0 : So the FONC yields the global max, and If p ² 10 , then F 0 (0) ³ 0 so x = 0 is optimum If p < 10 , then F 0 (0) > 0 and the optimum is x = 1 ° p= 10 . The market demand function just the sum over 160 consumers: If p ² 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 > 0 . The expressions are (5 points): LRTC ( x ) = ° 0 if x = 0 4 + x 2 if x > 0 LRAC ( x ) = 4 =x + x SRTAC ( x ) = 1 + x 2 SRAAC ( x ) = 1 =x + x MC ( x ) = 2 x LRAC 0 2 + 1 , LRAC 00 3 > 0 . So LRAC ( q ) is minimized at x = 2 and the minimum is 4. SRAAC 0 2 + 1 , SRAAC 00 3 > 0 . So SRAAC ( q ) is minimized at x = 1 and the minimum is 2. The ±rm²s short run supply curve coincides with its marginal cost curve so long as the price does not fall below the minimum SRAAC . Thus p = 2 x when p ² 2 . Or x = ° 0 if p ³ 2 p= 2 if p ² 2 1

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Observe that if p = 2 , the ±rm is indi/erent between producing 0 and 1 (it merely loses its sunk cost, and only just recovers its avoidable cost if it produces 1). (c) In the long run, with free entry and exit of ±shing ±rms, the industry²s supply curve is a horizontal line at the level of the minimum LRAC , that is, at p = 4 . To be more pedantic, it consists of all points with p = 4 and x = 2 , 4, 6, 8, 10, : : : , but not noting this is not a problem. (d) From the supply curve, p = 4 . Then from the demand curve, X = 16(10 ° 4) = 96 . Each ±rm produces x = 2 at the bottom point of its LRAC , 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 .
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