PS4 - ECO 310 - Fall 2007 Microeconomic Theory - A...

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ECO 310 - Fall 2007 Microeconomic Theory - A Mathematical Approach Problem Set 4 - Answer Key Question 1: (a) By Shepherd’s Lema ∂E/∂p x = X C so X C = p 1 / 2 y U 2 p 1 / 2 x and Y C = p 1 / 2 x U 2 p 1 / 2 y . By Roy’s Identity X M = X = - ( ∂V/∂p x ) / ( ∂V/∂I ). Inverting the expenditure function gives the indirect utility U = Ip - 1 / 2 x p - 1 / 2 y and so X = I/ (2 p x ), Y = I/ (2 p y ). (b) X = 50, Y = 50, U = 100. (c) X = 41 . 32, Y = 50, U = 100 / (1 * 1 . 21 1 / 2 ) = 90 . 91 (d) Tax revenue R = (1 . 21 - 1) × 41 . 32 = 8 . 677. (e) Here we want to find the additional income necessary to keep the consumer at the old utility of 100. I/ (1 . 21 1 / 2 * 1) = 100 implies I = 110, so the consumer needs 10 units of extra income to provide (Hicks) compensation. The revenue from the tax is less, because it causes a dead-weight loss (DWL). The loss comes from consumers buying less X. The tax not only makes them poorer, but also distorts their consumption patterns causing an additional loss. (f) Income sufficient to buy old quantities = 1 . 21 × 50 + 1 × 50 = 110 . 5, so Slutsky compensation = 10.5 which is greater than Hicks compensation. Given income 110.5, the consumer would actually buy X = 110 . 5 / (2 × 1 . 21) = 45 . 45 < 50 and Y = 110 . 5 / (2 × 1) = 55 . 25 > 50 and get utility 110 . 5 / 1 . 1 = 100 . 45 > 100. (g) Figure 1 shows three different demand curves. Curve AB is the X C for U = 100, Curve DC is the X C for U = 90 . 1, and Curve AC is the X = X M for I = 100. The change in Consumer Surplus (CS) is the area ACHG. You can approximate this with a triangle but you should be able to calculate it exactly as well:
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This note was uploaded on 01/15/2011 for the course ECO 310 at Princeton.

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PS4 - ECO 310 - Fall 2007 Microeconomic Theory - A...

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