Question

A customer's income is $20, their per-unit price of Good x equals $5, and at her utility-maximizing bundle, the consumer's MRS equals 4. Then, the per-unit price of good y equals what?

Answer 1

In order to maximize utility a consumer consumes that quantity of X and Y such that indifference Curve is tangent to Budget line. Both of them will be tangent If the consumer is spending all his income and Slope of Budget line = Slope of Indifference curve. As Budget line and Indifference curve are both downward sloping and hence Slope of both of them will be negative. Hence he will choose that amount of X and Y such that Absolute value of slope of both of them are equal.

Budget line is given by :

p_xx + p_yy = Income where p_x = price of x , p_y = price of y

Differentiating with respect to x we get :

p_x + p_y(dy/dx) = 0(As income is given and hence constant)

=> dy/dx = -p_x/p_y (Note we have assumed good x on horizontal(or x) axis and good y on vertical(or y) axis)

Hence Absolute Value of Slope of Budget line = p_x/p_y

As, he will choose that amount of X and Y such that Absolute value of slope of both of them are equal

=> MRS = p_x/p_y

It is given that MRS = 4 , p_x = price of x = 5 and p_y = price of y that we have to calculate

=> 4 = 5/p_y

=> p_y = 5/4 = 1.25

Thus, price of y = 1.25.

Hence, the per-unit price of good y equals $1.25.