Question

Suppose a consumer has a utility function given by u(x, y) = x + y, so that the two goods are perfect substitutes. Use the Lagrangian method to fully characterize the solution to max(x,y) u(x, y) s.t. x + py ≤ m, x ≥ 0, y ≥ 0, where m > 0 and p < 1. Evaluate and interpret each of the multipliers in this case. What happens to your solution when p > 1? What about when p = 1?

Answer 1

u(x, y) = x + y subject to constraint  x + py ≤ m

A  Lagrangian function is defined as:

L = u(x, y) + (m- x - py) where, is lagrange multiplier

L =  x + y + (m- x - py)

The first order conditions are given by: L/x = 0, L/y=0 and L/ = 0

L/x = 1- = 0 (equation-1)

L/y = 1 - p  = 0 (equation-2)

L/ = m- x - py  = 0   (equation-3)

From equation-1, we get the value, = 1

From equation-. we get = 1/p

Equating the two values of we get 1/p = 1 or p=1. So in equilibrium the price is 1.

• For p = 1, good x =good y

Puttting the value in equation 3

m - x - x =0 gives the value x = m/2 and y = m/2

When price of both the goods are same and the two goods are sunsitute goods, consumer is indifferent to demand x or y. Consumer spends half the income on x and half on y.

• For p<1, and given the goods are substitute goods, the consumer will not demand good x. All the income is spent on consuming y. It is because, x and y provide same level of utility but price p is lesser, so it is cheaper to buy y.
• For p>1, and given the goods are substitute goods, the consumer will not demand good y. All the income is spent on consuming x.