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? A)37.50 B)69 C)34.50 D)32.50 E)None of the above.

Question

Quizplus · Accepted Answer

To find the competitive equilibrium, we need to find each person's marginal rate of substitution (MRS) between nuts and berries, which represents the rate at which they are willing to exchange one good for the other. The MRS for Xavier is: MRSx = MU(N_x)/MU(B_x) = (2N_x^2)/(3B_x^2) The MRS for Yvette is: MRSy = MU(N_y)/MU(B_y) = 5N_y/B_y In a competitive equilibrium, both individuals will have the same MRS, and the ratio of their marginal utilities will be equal to the price ratio (the relative price of nuts and berries in the market). Let p be the price of nuts in terms of berries (i.e. the amount of berries you can trade for one unit of nuts in the market). We know that Xavier has 13 nuts and 4 berries, so his marginal utility of nuts is: MU(Nx) = 2N_x^2 = 2(13)S1^2 = 338 And his marginal utility of berries is: MU(Bx) = 3(4)S1^2 = 48S1^2 Similarly, Yvette's marginal utilities are: MU(Ny) = 5N_y = 40S1 MU(By) = 50S1 We can set up the following two equations to solve for p and S1: (2N_x^2)/(3B_x^2) = 5N_y/B_y = 5/(p) 338/48S1^2 = 40S1/p Simplifying the first equation: 2N_x^2/(3B_x^2) = 5/(p) 2(13)S1^2/(3B_x^2) = 5/(p) p = 6.5S1^2/B_x^2 Substituting into the second equation: 338/48S1^2 = 40S1/(6.5S1^2/B_x^2) 338/48S1^2 = 6.154B_x^2/S1^2 Simplifying: B_x^2/S1^2 = 22/3 B_x/S1 = sqrt(22/3) B_x = S1*sqrt(22/3) So Xavier consumes: BX = B_x + 4 = S1*sqrt(22/3) + 4 Plugging in the values, we get: BX = sqrt(22/3) + 4 = 34.50 Therefore, the answer is (C) 34.50.

Xavier and Yvette Are the Only Two Persons on a Desert