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) = 5N_y + B_y. Xavier is endowed with 4 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

First, we need to find the prices at which nuts and berries trade in this competitive equilibrium. Let P be the price of nuts and Q be the price of berries. Then the budget constraints of Xavier and Yvette are: Xavier: 13P + 4Q = income Yvette: 8P + 6Q = income We know that in equilibrium, the market clears, meaning the total demand for berries and nuts equals the total supply. Thus, Total demand for nuts = 13 + 8 = 21 Total demand for berries = 4 + 6 = 10 Total supply for nuts = 13 Total supply for berries = 6 Therefore, the equilibrium prices can be found by solving the following system of equations: 21P + 10Q = 13 * 21 8P + 6Q = 13 * 6 Solving for P and Q, we get P = 2 and Q = 6.5. Now we can find the optimal choices of Xavier and Yvette by maximizing their utility subject to their budget constraint and the prices of nuts and berries. Xavier's problem is: max N_xB_x s.t. 13P + 4Q = income Using Lagrange multipliers, we can find Xavier's demand functions: N_x = 13/2 B_x = 2 This means that Xavier consumes 13/2 units of nuts and 2 units of berries. Yvette's problem is: max 5N_y + B_y s.t. 8P + 6Q = income Using Lagrange multipliers, we can find Yvette's demand functions: N_y = 0 B_y = 13/2 This means that Yvette consumes 0 units of nuts and 13/2 units of berries. Therefore, the total demand for berries is 2 + 13/2 = 17/2 = 8.5 units, and the total supply is 6 units, so Xavier consumes 8.5 - 6 = 2.5 units of berries. Finally, we check that each person's consumption maximizes their utility, given the prices and their endowment. We can see that Xavier's utility is: U(13/2 * 2, 2 * 2.5) = 13.75 And Yvette's utility is: U(0, 13/2 * 6.5) = 84.125 These are both higher than their initial endowments (Xavier's initial utility is 0, and Yvette's is 32), so their demands are indeed optimal.

Xavier and Yvette Are the Only Two Persons on a Desert