.77777 its a complete question help me please....
1 . Consider a pure exchange economy with two consumers (A and B) and two goods (X and )). The utility functions for the two consumers are represented by: U,(X,, YA)= X,+Y, and Us(X;, Y; ) = min(X, YB) where X, and Y, are the amounts of the two goods consumed by person i. Suppose that the initial endowments owned by the two consumers are X = 80 X B = 20 Y = 20 YB = 30 a. Is the initial allocation of goods Pareto efficient? Explain. b. Derive the contract curve for this economy. C. What would the competitive equilibrium look like in this economy? d. Is the equilibrium calculated above Pareto efficient? Explain.e. Describe all initial allocations that lead to this same nal allocation in the competitive equilibrium. f. Describe all initial allocations that lead to this same competitive price ratio as that obtained with the allocation provided at the start of this problem. 2. A riskaverse investor with W0 to invest has two assets available. One is risk free and pays a return p , while the other is risky and pays a random return of 1'. Assume that the investor choosesa , the proportion of W.J to invest in the risky asset, so as to maximize the expected utility of W = (a - r +(1 a!)- p)- W:I . a. Suppose that r could be either 1-,, orrr , each with probability one half and assume that r,' > r, . Write out the rst-order condition for maximization of expected utility. Is the secondorder condition satised? Explain. Are restrictions required concerning the values ofrk , r} and p for a: to be bounded? b. Suppose that for some parameterization, the optimal value of or is zero. Show that the optimal value of :2 becomes positive if rJ5 is increased. c. Give the Arrow-Pratt measure of relative risk aversion. d. Show that if the investor's measure of relative risk aversion is decreasing, then the optimal a increases with W0. Suppose that there are two goods (X and Y). The representative consumer's preferences over these two goods are given by the utility function U = (X - Y , and income is given by I. a. Let the prices of the two goods be represented by PX and Pr. What are the demand functions for these two goods? Derive the expenditure mction. b. Suppose that the government gives the consumer a voucher that can be used to purchase X, but not I". The value of the voucher is V. Show how this changes the consumer's decision problem. 6. Suppose that instead of a voucher that can only be used to purchase X, the government gives the consumer a voucher that can be used to buy either X or Y. Again, the value of the voucher is V. Keeping the value of V the same as in part (b), are there values of Vfor which the consumer would be indifferent between the two programs? Are there values of V for which the consumer would prefer one program to the other? d. Instead of providing a voucher of value V, the government can reduce the price of X. By what percent would the government have to reduce this
