Question

Suppose that the budget constraint is given as: PX + PyY-M and the formulation of a utility function is given as: U(X, Y)-C2/2 + δΧαΥβ with 0 < α, β < 1 and constants C, δ > 0. Answer for following questions and show all your calculation/ proof. a. Derive the formula of income-consumption curve and draw its graph. b. Derive the demand function of good X and Y respectively as functions of income and price of good X and Y c. Calculate the amount of income spent for good X and Y respectively.

Answer 1

U(x, y) = c^2/2 + delta x^alpha y^beta B C x P_x + y P_y = M b) Now using Lagrangian to find demand func^n L = u(x, y) + lambda [M - xP_x - yP_y] C & delta are constant L = c^2/2 + delta x^alpha y^beta + lambda[M - xP_x - y P_y] FOC: partial differential L/partial differential x = delta alpha (x)^alpha - 1 y^beta - p_x lambda = 0, partial differential L/partial differential y rightdoublearrow delta beta(y)^beta - 1 (x)^alpha - p_y lambda = 0 thus delta alpha (x)^alpha - 1 y^beta/p_x = delta beta (y)^beta - 1 (x)^alpha/p_y rightdoublearrow alpha/beta y/x = p_x/p_y Thus alpha y P_y = beta_x p_x, beta x P_x/alpha y P_y from B C x P_x + beta_x p_x/alpha = M rightdoublearrow X* = alpha M/(alpha + beta)P_x Y* = beta M/(alpha + beta)P_y c) Now Income spent on Good X = P_x X* = alpha M/(alpha + beta) similarly Income spent on Goody = P_y Y' = beta M/(alpha + beta)