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4. Uncertainty (15 points) . (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. 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.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,!) = Ap and Y(Px, Pyll) = AP, a. (6 points) Derive the compensated demand function for good X, X"(Px,Py,D). 4 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.2. Utility Maximization (15 points) A consumer faces income constraints and has CES preferences of the following form: U(x, y ) = x +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. w c. (3 points) Calculate the consumer's income elasticity of demand.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. U(X, Y) = In(X) + In(Y) ii) U(X, Y) = min(X, Y) (Hint: You can use a diagram or sample values here)