ps3_sol_fall2011 - Department of Economics Columbia...

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Department of Economics W3211 Columbia University Fall 2011 SO LUTIO NS T O Probl e m S e t 3 Int e rm e diat e Mi c ro ec onomi c s Prof . S e yhan E Arkona c 1 . S teve’s utility for socks (q 1 ) and other goods (q 2 ) is given byU(q 1 ,q 2 ) = 10q 1 .1 q 2 .9 The price of the composite good is p 2 =1 and the price of a pair of socks is p 1 =2. Steve’s income is Y=100. Every year, Steve’s mom buys him 20 pairs of socks. Find the equivalent variation of the gift. What is the difference between the cost of the gift and the equivalent valuation cash amount? Answer: First we must find the bundle that Steve receives after the gift. The Lagrangian with an income of 140 suggests that q 1 =7. However, we know that Steve must consume at least 20 socks (from the gift), so his bundle is the corner solution (20,100).The utility of the bundle (20,100) is 851. When MRS is set equal to MRT we find that q 2 = 18q 1 . We then plug this value for q 2 into the utility function and solve for the bundle that gives the same utility as the gift bundle. 851= 10q 1 .1 (18q 1 ) .9 This bundle is (6.3,113.4). The cost of this bundle is $126, so the EV=26. The difference between the cost of the gift and the EV is 40-26=14. The gift cost $40 whereas Steve would be just as happy with $26 in cash. 2 . For each of the following statements, define all of the underlined terms . Then, explain why the statement is true or false . a . If a consumer views two goods as perfect substitutes then their optimal choice will be a corner solution . b . The substitution effect from a price increase states that the consumer will always choose a smaller amount of that good to consume. However, the income effect states that consumption can move in either direction. c . Suppose Alf and Bo haveconvex indifference curves . Alf likes units of "X’ more than units of "Y’ but Bo likes units of "Y’ much more than units of "X’. Then, in the optimum, Alf’s marginal rate of substitution will be different from Bo’s even if they face the same prices. d . All Giffen goods are normal goods , but not all normal goods are Giffen goods. e . Economists assume that preferences are ordinal . This implies that given two utility functions and one is a monotonic transformation of the other, then they represent the same preferences over bundles of goods.
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Answer: a . Perfect substitutes are goods for which at any point you will trade one good in the same proportion for the other. The specific utility function is U(x,y)=ax+by, and the ICs are linear throughout. A corner solution is when the optimal bundle is all of one good and none of the other (i.e. the solution is at the intercept of the BC and one of the axis). True, since the MRS is constant for p.s.’s, the o ptimal bundle will occur at a corner. False, if they overlap. b . The substitution effect is the change in consumption that occurs following a price change based purely on the fact that relative prices have changed while holding utility fix e d . The income effect is the change in consumption following a price change that results from the person feeling richer or poorer, while holding r e lativ e pri
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This note was uploaded on 01/16/2012 for the course ECON W3211 taught by Professor Elmes during the Fall '09 term at Columbia.

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ps3_sol_fall2011 - Department of Economics Columbia...

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