Questions 1 and 2 below refer to a consumer who has utility function
U(Xi9X 2) = y/X + y/X2.
(1) (a) For given (p, /), solve the consumer’s utility maximization problem to find the Marshallian demand function. Verify that his indirect utility function is
b. Construct the corresponding expenditure function e(p,u). Hint: use the fact that the expenditure function is the inverse of the indirect utility function.
c. Use Shephard’s lemma to find the Hicksian demand function h(p,u).
(2) Since you will need the Hicksian demand function to answer this question, you may want to verify your answer for (l)(c) by solving the cost minimization problem and constructing the expenditure function and the Hicksian demand function directly.
a. Initially the consumer has income I = 72 and prices are p° = (1,1). Let x° denote the consumer’s optimal consumption bundle at those prices. Assume that the price of good 1 increases so that the new prices are p1 = (2,1). Let x1 denote the consumer’s optimal consumption bundle at the new prices. What are the income and substitution effects for this price change? You may want to refer to the figure on page 44 of the class notes and explicitly find the corresponding points A, B and C in this case.
b. In the same picture, draw the graph of the Marshallian demand and Hicksian demand for good 1 (with x in the horizontal axis and p in the vertical axis). Clearly indicate in your picture, the values of these functions at p? = 1 and p = 2.
c. What is the compensating variation (CV) associated with the price change from p° to p1? What is the corresponding change in consumer surplus (ACS)? You may want to use the graph you drew in (b) to indicate the areas that represent CV and ACS. However, you must also provide a precise computation for CV and ACS.