Sam's utility function for his monthly consumption of goods X and Y is U(X,Y) = 100X - X2 + Y. (Assume this function applies only for bundles that contain no more than 100 units of X; bundles with greater amounts of X are not relevant for this question). For this utility function, MUX = 100 - 2X, and MUY = 1. Sam's monthly income is $3,000, and the price of Y is $1.
a. What is Sam's uncompensated demand function for X, as a function of Px?
b. Suppose the price of X is initially $20, and that it rises to a price greater than $20. What is Sam's compensating variation for this price change (expressed as a function of Px)?
c. What is Sam's compensated demand function for good X, fixing the utility level he achieved at the original price of good X (Px=$20)? How does it compare to your answer in part (a)? Why?
d. Compute the change in consumer surplus for t is change to a price Px greater than $20. How does it compare to your answer in part (b)?