Question

(a) Write down Jeremy's budget constraint (a formula) and plot all Jeremy's affordable bundles in the graph (his budget set). Find the slope of the budget line (number). Give an economic interpretation for the slope of the budget line (one sentence) (b) Jeremy's utility function is given by U(zi,T2) = V(inx1 +1nT2)2 + 7. Propose a simpler utility function that represents the same preferences (give a formula). Explain why your utility represents the same preferences (one sentence) (c) Plot Jeremy's indifference curve map (graph), find the MRS analytically (give a formula) and find its value at bundle (2,4) (one number). Give an economic interpretation of this number (one sentence) Mark its value in the graph. (If you were not able to solve part c, you can use the utility function V(x1,2)-In 2In r2 for this and the following parts.) (d) Find the optimal bundle (x1, x2) (two numbers). Does Jeremy consume positive amounts of both goods? (e) Find the optimal bundle given p 2, p24 and m 40 assuming U(zi,r2) -2zi +3r2. Does Jeremy consume positive amounts of both goods? Is the optimal bundle at a point of tangency?

Answer 1

P_1 = 2, P_2 = 4, m = 40 Jeremy's budget constraint P_1 x_1 + P_2 x_2 = m 2X_1 + 4X_2 = 40 4X_2 = 40 - 2X_1 X_2 = 10 - 1/2 X_1 dX_2/dX_1 = = 1/2 slope of budget line, U(X_1, X_2) = P(ln X_1 + ln X_2)^2 + 7 U(X_1, X_2) = log (e^(P ln X_1 + ln X_2)^2 + 7) MRS = mu X_1/mu X_2 = 2P(ln X_1 + ln X_2) 1/X_1/2P(ln X_1 + ln X_2) 1/X^2 = X_2/X_1 MRS = P_1/P_2 = > optimality condition X_2/X_1 = 2/4 = > X_2 = 1/2 X_1 2X_1 + 4X_2 = 40 2X_1 + 4 (1/2 X_1) = 40 X_1 = 10 X_1 = 5 U(X_1, X_2) = 2X_1 + 3X_2 = > MRS = 2/3 since MRS > P-1/P_2 = > only X_1 is consumed X_1 = 20 X_2 = 0