Question

Suppose a consumer had a utility function given by: U= 2X + 8Y. If the price of Good X (Px) is $7 and the price of Good Y is $5 then what is the utility maximizing quantity of Good X the consumer will purchase with a budget of $12? (Round to the nearest two decimal places if necessary.)

Answer 1

Solution:

Utility function of the consumer: U(X, Y) = 2X + 8Y

Px = $7, Py = $5, Income, M = $12

Thus, budget line for this individual becomes: Px*X + Py*Y = M

7X + 5Y = 12

Notice, that given the preferences of this consumer, he/she treats goods X and Y as substitutes. In case of substitutes, utility maximizing condition for any consumer is as follows:

If MRSxy > Px/Py, consumer consumes only good X, so, Y = 0

If MRSxy < Px/Py, consumer consumes only good Y, so, X = 0

If MRSxy = Px/Py, consumer consumes anywhere on the budget line (including the extremes/intercepts, which are the two above cases)

where, MRSxy is the marginal rate of substitution of good X for good Y, and Px/Py is the price-ratio.

MRSxy = Marginal utility of good X (MUx)/Marginal utility of good Y (MUy)

MUx = $\partial U/\partial X$ = 2

MUy = $\partial U/\partial Y$ = 8

So, MRSxy = MUx/MUy = 2/8 = 0.25

Given the prices, price-ratio becomes Px/Py = 7/5 = 1.4

Clearly, MRSxy = 0.25 < 1.4= Px/Py

Thus, as already stated, this consumer will consume only good Y

So, with X = 0, using the budget line we get, 7(0) + 5Y = 12

Y = 12/5 = 2.4

Consumer consumes (X,Y) = (0, 2.4), that is 0 units of good X and 2.4 units of good Y to maximize the utility, subject to the budget line.