09HWS3 - Economics 104A Solution for Problem Set#3 Winter...

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Economics 104A Solution for Problem Set #3 Winter 2009 1. Jerry consumes good 1 and good 2. His utility function is u ( x 1 , x 2 ) = x 1 - 1 x 2 . a) Suppose initially ¯ p 1 = 4, ¯ p 2 = 4, and ¯ I = 100. Find the optimal bundle. Now suppose a quantity tax of $1 is imposed on good 1. Find the substitution, income, and total effects of the tax. Answer: The demand functions are x 1 ( p 1 , p 2 , I ) = I p 1 - s p 2 p 1 for good 1 and x 2 ( p 1 , p 2 , I ) = s p 1 p 2 for good 2 when p 1 p 2 I 2 . Using these demand functions, we get x * = ( x * 1 , x * 2 ) = (24 , 1) as the optimal bundle before tax. Next, the price of good 1 after tax is p 0 1 = 2 which implies Δ p 1 = p 0 1 - ¯ p 1 = 1. Hence, to calculate the (Slutsky) substitution and income effects, we need to increase Jerry’s income by Δ I = Δ p 1 x * 1 = 24, in order to make bundle x * affordable for him after tax. Using the demand functions again, y * = ( y * 1 , y * 2 ) = (24 . 8 - 0 . 8 , 1 . 25) is Jerry’s optimal bundle at p 0 1 , ¯ p 2 , I 0 = ¯ I + Δ I and z * = (20 - 0 . 8 , 1 . 25) if his optimal bundle at p 0 1 , ¯ p 2 , ¯ I . Thus, SE = y * 1 - x * 1 = 0 . 8 - 0 . 8 = - 0 . 09 and IE = z * 1 -
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