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# Consider a consumer with Cobb-Douglas utility function U(X, Y) = X1/3Y2/3, income M = $1500, who faces market prices PX = $1 and PY = $4.

1. Consider a consumer with Cobb-Douglas utility function U(X, Y) = X1/3Y2/3, income M = $1500, who faces market prices PX = $1 and PY = $4.

a) Carefully express this consumer's optimization problem using the specifics of this problem in place of the generic utility function, prices, and income in the following:

max U(X, Y) subject to PXX + PYY ≤ M. X, Y

b) Compute the absolute value of the marginal rate of substitution for this consumer and simplify as far as possible, recalling that the formula is |MRS(X, Y) | = MUX(X, Y) / MUY(X, Y). Show your work.

c) Determine whether a typical indifference curve has a diminishing (absolute value of the) marginal rate of substitution as X increases (and Y decreases). Justify your answer. Hint: inspect |MRS(X, Y)| that you calculated above when X increases (↑) and Y decreases (↓).

d) If indifference curves have a diminishing (absolute value of) marginal rate of substitution, find the interior solution by solving the usual two equations that characterize an interior solution:

PXX + PYY = M (BL)

(X, Y) / MUY(X, Y) = PX / PY. (ES)

If indifference curves have a constant or increasing (absolute value of) marginal rate of substitution, find the corner solution by determining which corner of the budget line gives a higher utility score. Show your work.

2. Repeat problem 1 for a consumer with utility function U(X, Y) = X1/3 + Y1/3, but with the same income and facing the same prices as in problem 1.

3. Repeat problem 1 for a consumer with utility function U(X, Y) = [2X + Y]2 , but with the same income and facing the same prices as in problem 1.