Question

Suppose goods ? and ? are perfect substitutes, and the consumer’s utility function is ?(?, ?) = 2? + ?. The consumer’s budget constraint is 10 = ?x? + ?y?. Derive the consumer’s demands for ? and ? in terms of ?? and ??.

Answer 1

We have the following information

U(X,Y) = 2X + Y

The marginal rate of substitution between X and Y is 2/1, 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 = 2

Marginal utility of Y = ∂U/∂Y = 1

Using Lagrangian multiplier

µ = 2X + Y + λ(10 – P_XX + P_YY)

Mathematically, the first-order conditions

∂µ/∂X = 2 – λP_X

2 = λP_X

∂µ/∂Y = 1 – λP_Y

1 = λP_Y

could both hold only if 2/1 = 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).

So, if 2/P_X>1/P_Y, then the consumer will chose

X = 10/P_X

On the other hand if 2/P_X<1/P_Y, then the consumer will chose

Y = 10/P_Y