ECON
ECON326-hw-3-solutions.pdf

# ECON326-hw-3-solutions.pdf - Economics 326 HW 3 Solution...

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Economics 326 HW 3 Solution Due: 28 November 2017 13:15pm Name: 1. Consider a utility function u ( x 1 , ..., x n ) whose continuous partial derivatives satisfy n i =1 x i ∂u ∂x i = a for some constant a and for all x 1 > 0 , x 2 > 0 , ..., x n > 0. Show that the function v ( x 1 , ..., x n ) = u ( x 1 , ..., x n ) - aln ( x 1 + x 2 + ... + x n ) is homogeneous of degree 0. (Hint: Use Euler’s theorem- f is homoegenous of degree k if, and only if, the following equation holds for all ( x 1 , ..., x n ) n i =1 x i ∂f ( x 1 , ..., x n ) ∂x i = kf ( x 1 , ..., x n ) ) Answer. n i =1 x i ∂v ( x 1 , ..., x n ) ∂x i = n i =1 x i ∂u ( x 1 , ..., x n ) ∂x i - a ( 1 x 1 + x 2 + ... + x n x 1 + 1 x 1 + x 2 + ... + x n x 2 + ... + 1 x 1 + x 2 + ... + x n x n ) = n i =1 x i ∂u ( x 1 , ..., x n ) ∂x i - a x 1 + x 2 + ... + x n x 1 + x 2 + ... + x n = a - a = 0 Thus, Euler equation holds for v ( x 1 , x 2 , ..., x n ) with homogeneity of degree 0. 2. A firm minimizes its cost of production C ( L, K ) = WL + RK subject to the production function Q = F ( K, L ) = 4 L 1 4 K 1 4 where W is the unit cost of labor and R is the unit cost of capital.

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(a) Solve for the cost-minimizing demand functions for labor and capital.
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• Fall '15
• michaelsampson
• Economics, Utility, Mathematical optimization, Constraint, Lagrangian mechanics, Leonhard Euler, Xn

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