Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y

Question

Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y represent the quantities of two goods, X and Y.

(a) Calculate the consumer’s marginal utilities for good X and for good Y.

(b) A consumer has diminishing marginal utility for a good if her marginal utility for that good decreases as she consumes more of it, holding constant her consumption of other goods. Does this consumer have diminishingmarginal utility for good X? Does she have diminishing marginal utility for good Y?

(c) Calculate the consumer’s marginal rate of substitution of X for Y.

Now assume that the consumer has $31 to spend on goods X and Y, which have fixed prices of pX=3 and pY=4.

(d) Express the consumer’s budget constraint as an equation. The only variables in the equation should be x and y.

(e) Calculate the first-order condition for the consumer’s optimization problem of dividing her money between goods X and Y in the way that maximizes her utility. (The only variables in the equation should be x and y.)

(f) Calculate the consumption bundle (x,y) that solves the first-order condition for the consumer’s problem.

(g) Calculate the two consumption bundles (x,y) that are on the boundary of the consumer’s optimization problem.

(h) For the three possible solutions that you found in parts (f) and (g), calculate the utility that the consumer would receive fromeach bundle. Whatshould she do to maximize her utility? Justify your answer carefully.

(i) Repeat parts (f), (g), and (h), assuming that, ceteris paribus, the price of X rises to pX=$75.

(j) What should the consumer do to maximize her utility if, ceteris paribus, the price of X rises to pX=100? Justify your answer carefully.

(k) Compare your answers to parts (h), (i), and (j). Discuss the connection between these answers and the Law of Demand.

For the last two parts, assume that the values of prices and the consumer’s wealth are unknown. Use W (for wealth), pX, and pY to represent these three unknown values, all of which must be positive numbers. (l) Repeat parts (d) and (e) with the new assumptions. This should give you a two-equation model of consumer behavior. In this model, which variables are considered exogenous and which are endogenous?

(m) Is it possible for the consumer’s optimal consumption bundle to have x=0? Is it possible for the consumer’s optimal consumption bundle to have y=0? Is this a realistic situation? Justify your answer with possible examples of goods X and Y.

Explanation / Answer

Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y

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