Question

Suppose that the budget constraint is given as: PxX + PyY = M and the formulation of a utility function is given as:

χαΥβ U(X, Y)-2 log C + _ 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

Given, budget line: P_xX + P_yY = M
And utility function U(X,Y) = $2logC &plus; \frac{X^{\alpha } &plus; Y^{\beta }}{\delta }$ where C and delta are constants, and 0<alpha,beta<1

The budget line has a slope of $-\frac{P_{x}}{P_{y}}$

The Lagrangean approach transforms a constrained optimization problem into an unconstrained problem of choosing x, y, and the Lagrange multiplier,

to maximize L = u(x; y) + $\lambda$[M - PxX - PyY]

By differentiating L with respect to x, y, and $\lambda$, and setting the derivatives equal to zero, the resulting first order conditions
are:

$\frac{\partial u}{\partial x} - \lambda P_{x} = 0$

$\frac{\partial u}{\partial y} - \lambda P_{y} = 0$

$M - P_{x}x - P_{y}y = 0$
using the given utility function in the formulae above,
$\frac{\alpha X^{\alpha -1}}{\delta }$ $- \lambda P_{x}X$$= 0$

$\frac{\beta Y^{\beta -1}}{\delta }$$- \lambda P_{y}Y = 0$

$M - P_{x}x - P_{y}y = 0$

On solving, we get the required values of X and Y which form tangent to the utility curve and lie on the budget line. The income consumption curve for Px= 1 and Py=1 and M values of 4,5... looks like:

The graph includes the demand functions as well as the ICC curves. The quantities X and Y can be calculated using the method given above, after setting lambda = X = Y.