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Intermediate Microeconomics, 2010 Problem Set 12: Homework Solutions 1. Consider the following quasilinear economy: U ( m;q ) = m + 54 p q | {z } v ( q ) c ( q ) = 1 2 q 2 and Y = 100 (recall, negative consumption of m is allowed). (a) Find the competitive allocation: x ± = [ q ± ;q ± ± ;m ± ] . Here’s the easy way: v 0 ( q ) = c 0 ( q ) ) 54 2 q ² 1 = 2 = q so q S = q D = q ± = 9 and p ± = c 0 ( q ± ) = 54 2 ± 9 ² 1 = 2 = 9 : Then we have m ± = Y ² p ± ± q ± = 100 ² 9 2 = 19 and ± ± = 9 ± 9 ² 1 2 9 2 = 81 2 = 40 : 5 : Here’s the long way. We can begin with the consumer’s problem. We want to choose q and m to maximize U = m + 54 ± q 1 = 2 such that Y = m + p ± q: We can solve for m in the budget constraint, and plug that into the utility function: U = m + 54 ± q 1 = 2 = ( Y ² p ± q ) + 54 ± q 1 = 2 : Now that we have the problem in terms of q only, we can take the derivative: dU dq = ² p + 54 2 q ² 1 = 2 : Setting the derivative to 0 and solving for q , we get q ± = ± 54 2 p ² 2 = 729 p 2 : 1
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We can get m ± = Y ± p ² q ± = Y ± p ² 729 p 2 = 100 ± 729 p : Now that we found the demands, we can deal with the ±rm’s problem. The ±rms maximizes pro±ts ± ( q ) = p ² q ± 1 2 q 2 : Taking the derivative and setting it to zero, we get ± 0 ( q ) = p ± q ± = 0 : This means that q ± = p: This is the supply equation. Let’s check the second derivative:
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