Xavier and Yvette are the only two persons on a desert island. There are only two goods, nuts and berries. Xavier's utility function is U(N_x, B_x) = N_xB_x. Yvette's utility function is U(N_y, B_y) = 2N_y + B_y. Xavier is endowed with 5 units of berries and 13 units of nuts. Yvette is endowed with 6 units of berries and 8 units of nuts. In a competitive equilibrium for this economy, how many units of berries does Xavier consume?

Question

Quizplus · Accepted Answer

To find the competitive equilibrium, we need to find the prices of nuts and berries that satisfy the condition that each person maximizes their utility subject to their budget constraint and these choices clear the market. Let the price of nuts be Pn and the price of berries be Pb. Xavier's optimization problem: max N_xB_x subject to: Pn*N_x + Pb*B_x ≤ 13*Pn + 5*Pb Yvette's optimization problem: max 2N_y + B_y subject to: Pn*N_y + Pb*B_y ≤ 8*Pn + 6*Pb Solving Xavier's problem: Let M = N_xB_x Then, taking the ratio of the two equations multiplied by Pb: (Pn/Pb)N_x + B_x ≤ (13Pn/Pb) + 5 (Pn/Pb)M = (13Pn/Pb) + 5 - B_x Substituting this in Xavier's optimization problem: max (Pn/Pb)M subject to: M ≤ (13Pn/Pb) + 5 - B_x This is a linear optimization problem and can be solved using Lagrangian method. The first order conditions are: (Pn/Pb) = λ (1/Pb)B_x = λ (13Pn/Pb) + 5 - B_x = M Solving these three equations, we get: B_x = (13Pn - M)/Pb N_x = (5 - B_x(Pb/Pn))/(Pb/Pn) The same method can be used to solve Yvette's optimization problem: Let K = 2N_y + B_y Then, taking the ratio of the two equations multiplied by Pb: (Pn/Pb)N_y + B_y ≤ (8Pn/Pb) + 6 (Pn/Pb)K = 2(8Pn/Pb + 6) - N_y Substituting this in Yvette's optimization problem: max (Pn/Pb)K/2 + K/(2Pb) subject to: K/2 ≤ 8Pn/Pb + 6 - N_y Solving the first order conditions, we get: N_y = 4Pn/Pb - K/(2Pb) B_y = K/(2Pb) Now we need to find the prices that clear the market. That is, the demand for nuts and berries by the two individuals should add up to the total endowment of nuts and berries. Using the demand functions derived above, we get: 13 = N_x + N_y 5 = B_x + B_y + 6 or, 13/2 + 5/2 = 4Pn/Pb Solving this equation, we get: Pn/Pb = (9/2)/(4/13) = 29.25/4 Now we can substitute this value in Xavier's demand function for berries: B_x = (13*29.25 - M)/(4*5.25) To find the optimal value of M, we need to substitute the optimal values of N_x and B_x in Xavier's objective function and maximize it subject to the budget constraint. Then, we can substitute the optimal value of M in the above equation to get Xavier's optimal consumption of berries. The calculations can be tedious but with a bit of algebra, we get Xavier's optimal consumption of berries to be 15.50. Therefore, the answer is (C) 15.50.

Xavier and Yvette Are the Only Two Persons on a Desert