1. Consider the utility function U(X)=X1.X2. a. Check whether the above function is strictly quasi concave. b. Solve the U-Max problem for the consumer and find X1 and X2 in terms of P1, P2 and M. Make sure you check the second order condition (SOC). c. What does the result in (a) imply for the SOC of utility maximization problem? Prove your claim. d. Derive the indirect utility function. e. Is V (P. M) homogeneous in P and M. Interpret your result? f. Find XI and X2, using Roy's identity and explain your result intuitively. g. Derive the expenditure function, e(P,U) by solving V (P, M) for M. h. Using Sheppard's lemma, find h1 and h2 and explain your result intuitively. i. Verify your results in (h) by solving the expenditure minimization problem. Make sure you check the SOC. j. Verify your result in (g) by finding the expenditure function, using h1 and h2. k. Check whether e (P, U) is concave in P. 1. Discuss briefly, comparing X(P.M) and h(P,U). m. Assume P1-P2-$2 and M-$200. Find the numerical values for the optimum XI, X2 and the maximum utility. n. Assume P1-P2-$2 and the numerical value for the maximum utility in (m). Find the numerical values for h1 and h2, using the results in (h) or (i). Compare your results in (m) and (n) and interpret. o. p. Suppose P1 increases to $4 and P2 stays the same. Find the income and substitution effects of this price change. q. Explain the result in (p), using indifference curves and X(P.M) and h(P,U) demand functions. r. Find the new budget to protect the same utility. s. Find the amount of subvention needed to keep the consumer at the same utility level t. Find the MRS for the (50,50) consumption point using utility function. u. Assume P1-P2-$2. Find the MRS.