Midterm 2007 Solutions

Midterm 2007 Solutions - ECO 310, Fall 2007 Midterm...

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ECO 310, Fall 2007 Midterm Examination Solutions October 25 Question 1 [50 points] If necessary, you may use log 2 0 . 69, log 3 1 . 10, log 4 1 . 39, log 5 1 . 61, log 6 1 . 79, and log(1 + z ) z when z is close to 0. Consider the following quasi-linear utility function: U ( x,y ) = log( x + 1) + y. (a) [2 points] Does U * ( x,y ) = U ( x,y + 1) represent the same preference relation as U ( x,y )? Yes. Since U * ( x,y ) + 1 = U ( x,y + 1) is a monotonic tranformation of U * ( x,y ) (b) [2 points] Does U ** ( x,y ) = U ( x + 1 ,y ) represent the same preference relation as U ( x,y )? No, adding to x changes the relationship between x and y, and so is a change in preferences. One way to see this is to calculate the MRS between x and y: ( ∂U/∂x ) / ( ∂U/∂y ). In the previous question this was the same for the utilities. Here for a given x and y it will be different. This implies, among other things, that for the same prices and income, two people with these utility functions will demand different amounts. A consumer chooses x and y to maximize U ( x,y ) subject to the budget and nonnegativity constraints: px + y I, x 0 , y 0 . Note that the price of good y is normalized to 1. In other words, the price of good x and her income are nominated in good y . (c) [12 points] Compute her Marshallian (uncompensated) demand func- tions. Note that boundary solutions may arise. We have to solve the maximization proble: Max log( x +1)+ y s.t. px + y I , x 0 , y 0. As is usual, we assume that the budget constraints bind. Let us find the solutions to the Lagrangian problem (without the inequality constraints) first. (4 points for setting up the problem.) L = log( x + 1) + y + λ ( I - px - y ) 1
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First order conditions: 1 x + 1 - λp = 0 1 - λ = 0 From these, we get p ( x + 1) = 1 or x = 1 p - 1. Substituting in the budget constraint, y = I - 1 + p . We find that x = 1 p - 1 0 if and only if p 1. Therefore for p > 1, x = 0 is binding. We also find that y = I - 1 + p 0 if and only if 1 - I p . Therfore, for
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Midterm 2007 Solutions - ECO 310, Fall 2007 Midterm...

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