Question

In: Economics

Suppose you have the following preferences u(x,y) = xa - yb. Calculate the optimal demand functions....

Suppose you have the following preferences u(x,y) = x^a - y^b. Calculate the optimal demand functions. Is good y a normal or inferior good? Please show work.

Expert Solution

We have the following information

U(X,Y) = aX – bY

The marginal rate of substitution between X and Y is –(a/b), which is a constant, independent of the quantities consumed of the goods. The indifference curves between the two goods are straight lines.

Marginal utility of X = ∂U/∂X = a

Marginal utility of Y = ∂U/∂Y = – b

Using lagrangian multiplier

µ = aX – bY + λ(M – P_XX – P_YY)

P_X = Price of X

P_Y = Price of Y

M = Money Income

Mathematically, the first-order conditions

∂µ/∂X = a – λP_X

a = λP_X

∂µ/∂Y = – b – λP_Y

– b = λP_Y

could both hold only if –(a/b) = P_X/P_Y, which would happen by coincidence. Usually, the consumer will choose to be at a corner solution, spending all her money on the good for which a_i/P_i is highest (a_i is coefficient of the ith good in utility function and Pi is the price of ith good).

Demand for Good X

X = M/P_X

Demand for Good Y

Y = – (M/P_Y)

The Good Y is an inferior good. The negative sign in the demand equation for Good Y shows that as income increases the demand for Good Y declines.

Rahul Sunny answered 1 year ago

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