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10HWS4

# 10HWS4 - Economics 104A Solution for Problem Set#4 Winter...

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Economics 104A Solution for Problem Set #4 Winter 2010 1. Let u ( x 1 , x 2 ) = 2 x 1 + x 2 be a consumer’s utility function. Let the prices of good 1 and good 2 and the consumer’s income be ¯ p 1 = 1, ¯ p 2 = 2, and ¯ m = 100, respectively. a) What is the income effect caused by a quantity tax of t = 1 on good 1? Explain your answer. Answer: Note first that the utility function implies MRS = 1 / x 1 . It thus follows that the optimal quantity of good 1 at prices p 1 and p 2 is determined by 1 / x 1 = p 1 /p 2 . This shows that the optimal quantity of good 1 is independent of the consumer’s income. Thus, the income effect of a quantity tax of t = 1 on good 1 is 0. As mentioned above, the consumer’s utility function implies that the MRS does not depend on good 2. Hence a change in income only changes the optimal quantity of good 2 but not the optimal quantity of good 1. b) Use CV to determine how much in dollar terms the quantity tax hurts the consumer. Answer: From the answer to part a, the optimal bundle at original prices and income has quantity of good 1 equal to ( p 2 /p 1 ) 2 = 4 and quantity of good 2 equal to ( m/p 2 ) - ( p 2 /p 1 ) = 48. The utility level at this bundle is 52. Let m 0 be the new income that the consumer would need to order to achieve utility level 52 after the quantity tax has been imposed. Then, m 0 satisfies 2 p 2 p 1 + t + m 0 p 2 - p 2 p 1 + t = 52 . Solving the above equation, we get m 0 = 102. Thus, CV = m 0 - m = \$2 . c) Use EV to determine at most how much of the income would the consumer be willing to give up to prevent the quantity tax, assuming he does not lie. Answer: After the quantity tax has been imposed, the optimal bundle for the consumer has quantity of good 1 equal to ( p 2 /p 1 + t ) 2 = 1 and quantity 1

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of good 2 equal to ( m/p 2 ) - ( p 2 /p + t ) = 49. Thus the utility level at this
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10HWS4 - Economics 104A Solution for Problem Set#4 Winter...

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