Questions:
d. (6 points) Imagine that I=$1000, px=py=$1. What is the maximum income tax the government could impose if they didn't want to reduce consumer utility by more than half? 4. Uncertainty (15 points) a. (5 points) Show graphically that if an individual has diminishing marginal utility of wealth, she will prefer certain income to a fair gamble. Be sure to mark all important points on the graph clearly. 6 b. (10 points) If the individual has utility over wealth (W) = -e-AW , show that the premium that this person would pay to avoid a fair gamble of h is independent of initial wealth.The utility function for an individual is given by U(X, Y) = X-75y-25. Prices for the two goods are Py and P, respectively, and income is I. The uncompensated demand functions for the two goods X and Y are: X(Px, Py,!) = Ap and Y (Px, Pyll) = APy a. (6 points) Derive the compensated demand function for good X, X"(Px,PY,D). b. (3 points) For a small increase in the price of X , what is the total change in the quantity demanded of X ? Express your answer in terms of price(s) and income. c. (6 points) Using the Slutsky decomposition, calculate the substitution and income effects for the price change in part (d). Again, express your answer in terms of price(s) and income.A consumer faces income constraints and has CES preferences of the following form: U(x, y ) = x+ys a. (8 points) Find the consumer's demand for x as a function of prices and income. b. (4 points) Are these preferences homothetic? Explain why or why not. w c. (3 points) Calculate the consumer's income elasticity of demand.a. (6 points) Bob enjoys cookies (x) according to the utility function U(x)=20x-tx, where t is a parameter that reflects how hungry he is. Cookies are costless in Bob's world and so there is no income constraint. Using the envelope theorem, calculate how Bob's maximum utility from eating cookies varies with t. b. (8 points) Are the following utility functions quasi-concave? Show why. i) U(X, Y) = In(X) + In(Y) ii) U(X, Y) = min(X, Y) (Hint: You can use a diagram or sample values here)4. Uncertainty (15 points) a. (5 points) Show graphically that if an individual has diminishing marginal utility of wealth and initial wealth Wo, she will prefer actuarially fair insurance to a potential loss of h dollars when the probability of loss is 50%. Be sure to mark all important points on the graph clearly. b. (10 points) If the individual has utility over wealth U(W) = -e-AW , show that the premium that this person would pay to avoid a fair gamble of h is independent of initial wealth Wo. Note: by "fair gamble" I mean a bet with 50% chance of winning h and 50% chance of losing h.c. (6 points) Using the Slutsky decomposition, calculate the substitution and income effects for the price change in part (b). Again, express your answer in terms of price(s) and income. d. (6 points) Imagine that 1=$1000, px-py=$1. What is the maximum income tax the government could impose if they didn't want to reduce consumer utility by more than half?3. Slutsky Decomposition (21 points) The utility function for an individual is given by U(X, Y) = X.75y-25. Prices for the two goods are Py and P, respectively, and income is I. The uncompensated demand functions for the two goods X and Y' are: X(Px, Py, 1) = ap, and Y (Px. Py.!) = AP a. (6 points) Derive the compensated demand function for good X, X"(Px,Py, U). b. (3 points) For a small increase in the price of X , what is the total change in the quantity demanded of X" ? Express your answer in terms of price(s) and income.2. Utility Maximization (15 points) A consumer faces income constraints and has CES preferences of the following form: U(x, y) = x5 + y. a. (8 points) Find the consumer's demand for x as a function of prices and income. b. (4 points) Are these preferences homothetic? Explain why or why not.1. Preferences and Utility (14 points) a. (6 points) Bob enjoys cookies (x) according to the utility function U(x)=20x-tx , where t is a parameter that reflects how hungry he is. Cookies are costless in Bob's world and so there is no income constraint. Using the envelope theorem, calculate how Bob's maximum utility from eating cookies varies with t. b. (8 points) Are the following utility functions quasi-concave? Show why. D) U(X, Y) = In(X) + Y ii) U(X, Y) = min(X, Y) (Hint: You can use a diagram or sample values here)
