A consumer's preferences over goods x and y can be represented by the utility function U(x,...

 

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A consumer's preferences over goods x and y can be represented by the utility function U(x, y) = xy. The consumer takes as given the prices of the goods (denoted px and py, respectively) and its income (denoted M). Assume that all prices and quantities are infinitely divisible. (a) (b) (c) (d) (e) Show that the consumer's preferences are homothetic and derive the equation of a representative indifference curve reflecting those preferences. Formally express the consumer's (constrained) utility-maximization problem (UMP), set up the associated Lagrangian and first-order conditions (FOCs), and solve for the demand functions x and y*. Find the equilibrium value of the Lagrange multiplier (*) and provide an economic interpretation of this value. Suppose the consumer's utility function changes to U(x, y) = x 1/3y1/3. Explain (assuming all else equal) whether this change will affect the consumer's equilibrium consumption plan relative to that found in part (b) (be as formal as possible). Now suppose the consumer's utility function changes to u(x, y) = x + y. [Note: calculus methods cannot be applied to this case.] Again, explain (assuming all else equal) whether this change will affect the consumer's equilibrium consumption plan relative to that found in part (b). Determine how much of each good the consumer will purchase if px = 2, Py = 1, and M = 10. Sketch your answer on a well-labeled indifference curve/budget constraint graph. A consumer's preferences over goods x and y can be represented by the utility function U(x, y) = xy. The consumer takes as given the prices of the goods (denoted px and py, respectively) and its income (denoted M). Assume that all prices and quantities are infinitely divisible. (a) (b) (c) (d) (e) Show that the consumer's preferences are homothetic and derive the equation of a representative indifference curve reflecting those preferences. Formally express the consumer's (constrained) utility-maximization problem (UMP), set up the associated Lagrangian and first-order conditions (FOCs), and solve for the demand functions x and y*. Find the equilibrium value of the Lagrange multiplier (*) and provide an economic interpretation of this value. Suppose the consumer's utility function changes to U(x, y) = x 1/3y1/3. Explain (assuming all else equal) whether this change will affect the consumer's equilibrium consumption plan relative to that found in part (b) (be as formal as possible). Now suppose the consumer's utility function changes to u(x, y) = x + y. [Note: calculus methods cannot be applied to this case.] Again, explain (assuming all else equal) whether this change will affect the consumer's equilibrium consumption plan relative to that found in part (b). Determine how much of each good the consumer will purchase if px = 2, Py = 1, and M = 10. Sketch your answer on a well-labeled indifference curve/budget constraint graph.