Mike Teevee likes to watch television and to eat candy.In fact his utility function is U(x, y)= x²y, where x is the number of hours he spends watching television and y is the number of dollars per week he spends on candy.Mike's mother doesn't like him to watch so much television.She limits his television watching to 36 hours a week and in addition she pays him $1 an hour for every hour that he reduces his television watching below 36 hours a week.If this is Mike's only source of income to buy candy, how many hours of television does he watch per week?

Question

Quizplus · Accepted Answer

We need to maximize Mike's utility function subject to the mother's constraint that he cannot watch more than 36 hours of television per week. Let's set up the Lagrangian:

L(x,y,λ) = U(x,y) + λ(36 - x)

Taking the first order conditions:

∂L/∂x = U1(x,y) - λ = 0
∂L/∂y = U2(x,y) = 0
∂L/∂λ = 36 - x = 0

From the second condition, we get that U2(x,y) = 0, which implies that Mike consumes all of his income on candy. From the third condition, we get that x = 36, which means that Mike doesn't get any additional income from his mother.

Substituting x = 36 into the first condition, we get:

U1(36,y) = λ

In other words, the marginal utility of watching television is equal to the marginal utility of receiving an additional dollar from his mother. Let's rearrange the utility function to get the marginal utilities:

U1(x,y) = 10
U2(x,y) = 1

Substituting these into the first condition, we get:

10 = λ

Therefore, Mike's optimal consumption bundle is (x,y) = (24,12). He watches 24 hours of television per week and spends $12 on candy.

Mike Teevee Likes to Watch Television and to Eat Candy