1. Consider the utility maximization problem max ?^? + ? s.t. ?? + ? = ? (i.e., the price of good ? is 1), where the co...
1. Consider the utility maximization problem max ?^? + ? s.t. ?? + ? = ? (i.e., the price of good ? is 1), where the constants ?, ?, and ? are positive, and the constant ? ∈ (0,1).
A. Find the (Marshallian) demand functions, ?∗(?,?) and ?∗(?,?).
B. Find the partial derivatives of the demand functions w.r.t. ? and ?, and check their signs.
C. Set ? = 1/2. What are the demand functions in this case? What is ?∗? Denote the maximal utility as a function of ? and ? as ?∗(?,?), the value function, also called the indirect utility function. Verify that ??∗ ?? ⁄ = −?∗(?,?).
Homework Answers
A. In order to find Marshallian Demand Functions, we maximize the utility function subject to the given budget constraint.
B. Only the partial derivative of y w.r.t p is negative. Others are positive. Change in x as a change in m is zero.
C. Indirect utility function shows the maximal utility.
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Consider the following utility function over goods 1 and 2,
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Marshallian demand functions and the indirect utility function. (b)
[15 points] Using the indirect utility function that you obtained
in part (a), derive the expenditure function from it and then
derive the Hicksian demand function for good 1. (c) [10 points]
Using the functions you have derived in the above, show that i. the
indirect utility function is homogeneous of degree zero in...
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