# 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.

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.

i)

U(X, Y) = In(X) + "

ii) U(X, Y) = min(X, Y) (nt: You can use a diagram or sample values here) 2. Utility Maximization (15 points)

A consumer faces income constraints and has CES preferences of the following form:

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.

3 c. (3 points) Calculate the consumer’s income elasticity of demand.

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 ) are:

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.

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 I=$1000, px=py=$1. 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 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.

6 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.

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