Question

In Part (a) of Problems 1 - 3, you can use the results of HW 2 - you do not have to derive the optimal choice from scratch. (The point of these problems is the sketches of the demand functions.) 1) A consumer's utility function is u(x,y) 3 3 (a) Find the consumer's optimal choice for x, y as functions of income prices px, py and income 1. (b) Sketch the demand curve for x as a function of functions of its own price Pr when Py 16, I 256 (c) Sketch the demand curve for x as a function of the other price py when p,-1, 1 81.

Answer 1

u(x, y) = -1/3x^3 - 1/3y^3 Optimal bundle occurs at the point when MRS = P_x/P_y MRS = MU_x/MU_y: MU_x = partial differential U/partial differential x = (-1/3) (-3) x^-4 = x^-4 MU_y = partial differential U/partial differential y = (-1/3) (-3) y^-4 = y^-4 So, MRS = x^-4/y^-4 = y^4/x^4 = (y/x)^4 Now, MRS = P_x/P_y = > (y/x)^4 = (P_x/P_y) y/x = (P_x/p_y)^1/4 y = x y = x[p_x/p_y]^1/4 Budget constraint: P_x middot x + p_y middot y = I P_x middot x + P_y [x (p_x/p_y)^1/4] = I [as y = x (p_x/p_y)^1/4] x [P_x + (p_y^1 - 1/4) P_x^1/4] = I x { P_x + p_y^3/4 P_x^1/4] = I x px^1/4 [p_x^3/4 + p_y^3/4] = I {as P_x^1/4 middot P_x^3/4 = P_x^1/4 + 3/4 = p_x] x = I/P_x^1/4 (P_x^3/4 + p_y^3/4] \r\n P_x = 1 and I = 81 x = I/P_x^1/4 (P_x^3/4 + p_y^3/4) = 81/(1)^1/4 ((1)^3/4 + P_y^3/4) = 81/1 + p_y^3/4