Fall 2012 Final Exam
ECON 6090, Fall 2012 - Final Exam 135 Minutes. FOUR SIDES TO EXAM SHEET If you believe a question is unclear, please state how you interpret the question and we will take this into account during grading. In proofs, please use formal mathematical language wherever possible. You must show all work for partial credit to be awarded. 1. (25 Points) For some fixed ε > 0, suppose the consumer’s choice rule is for some C(B) = (x1 , x2 , x3 ) : u(x1 , x2 , x3 ) + ε ≥ max
(x1 ,x2 ,x3 )∈B
u(x1 , x2 , x3 )
for some funtion u from R3 into R. In other words, the + agent chooses all items from u(x1 , x2 , x3 ) that come within ε of providing the maximum utility possible given the options in B. (a) (5 points) Suppose we define the preference relation as (x1 , x2 , x3 ) (x1 , x2 , x3 ) if u(x1 , x2 , x3 )+ε ≥ u(x1 , x2 , x3 ). Is this preference relation rational for an arbitrary choice of a continuous function u? If so, prove it. If not, provide a counter-example. (b) (5 points) Does this choice rule satisfy WARP for an arbitrary choice of a continuous function u? If so, prove it. If not, provide a counter-example. Suppose u(x1 , x2 , x3 ) = xα ∗ min{x2 , x3 } for some α ∈ 1 (0, 1). (c) (8 points) Solve the utility maximization problem. (d) (7 Points) Use duality results to compute the Hicksian demand given your result in part (c)
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2. (20 Points) Consider a firm with the following single-output production function that uses capital (k) and labor (l) as inputs f (k, l) = (k γ + Hlγ )1/(2γ) , γ ∈ 0, 1 2
Assume a unit of capital costs r > 0, a unit of labor costs w > 0, and output is sold at a price p > 0. (a) (2 Points) Write the Lagrangian for the profit maximization problem. (b) (2 Points) Find the first order conditions for the profit maximization problem. (c) (8 points) Provide the solution for the profit maximization problem. (d) (5 points) Use Topkis’s theorem to describe k and l as functions of H holding (w, r, p) fixed. (e) (3 Points) Use Topkis’s theorem to
(x1 ,x2 ,x3 )∈B
u(x1 , x2 , x3 )
for some funtion u from R3 into R. In other words, the + agent chooses all items from u(x1 , x2 , x3 ) that come within ε of providing the maximum utility possible given the options in B. (a) (5 points) Suppose we define the preference relation as (x1 , x2 , x3 ) (x1 , x2 , x3 ) if u(x1 , x2 , x3 )+ε ≥ u(x1 , x2 , x3 ). Is this preference relation rational for an arbitrary choice of a continuous function u? If so, prove it. If not, provide a counter-example. (b) (5 points) Does this choice rule satisfy WARP for an arbitrary choice of a continuous function u? If so, prove it. If not, provide a counter-example. Suppose u(x1 , x2 , x3 ) = xα ∗ min{x2 , x3 } for some α ∈ 1 (0, 1). (c) (8 points) Solve the utility maximization problem. (d) (7 Points) Use duality results to compute the Hicksian demand given your result in part (c)
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2. (20 Points) Consider a firm with the following single-output production function that uses capital (k) and labor (l) as inputs f (k, l) = (k γ + Hlγ )1/(2γ) , γ ∈ 0, 1 2
Assume a unit of capital costs r > 0, a unit of labor costs w > 0, and output is sold at a price p > 0. (a) (2 Points) Write the Lagrangian for the profit maximization problem. (b) (2 Points) Find the first order conditions for the profit maximization problem. (c) (8 points) Provide the solution for the profit maximization problem. (d) (5 points) Use Topkis’s theorem to describe k and l as functions of H holding (w, r, p) fixed. (e) (3 Points) Use Topkis’s theorem to