Pablo's utility function is , where x is the number of x's he consumes per week and y is the number of y's he consumes per week. Pablo has $200 a week to spend. The price of x is $1. The price of y is currently $5 per unit. Pablo has received an invitation to join a club devoted to the consumption of y. If he joins the club, Pablo can get a discount on the purchase of y. If he belonged to the club, he could buy y for $1 a unit. How much is the most Pablo would be willing to pay to join this club?
A) $8
B) $28
C) $36
D) $56
E) None of the above.

Question

Pablo's utility function is  , where x is the number of x's he consumes per week and y is the number of y's he consumes per week. Pablo has $200 a week to spend. The price of x is $1. The price of y is currently $5 per unit. Pablo has received an invitation to join a club devoted to the consumption of y. If he joins the club, Pablo can get a discount on the purchase of y. If he belonged to the club, he could buy y for $1 a unit. How much is the most Pablo would be willing to pay to join this club?
A) $8 
B) $28 
C) $36 
D) $56 
E) None of the above.

Quizplus · Accepted Answer

We can solve this by finding the optimal bundle of x and y that maximizes Pablo's utility function subject to his budget constraint. We can use the Lagrangian method to solve this problem:

L(x,y,lambda) = 11x^6y^2 - lambda(1x + 5y - I)

where lambda is the Lagrange multiplier and I is Pablo's income.

Taking partial derivatives with respect to x, y, and lambda, we get:

dL/dx = 66x^5y^2 - lambda = 0
dL/dy = 22x^6y - 5lambda = 0
dL/dlambda = 1x + 5y - I = 0

Solving these equations simultaneously, we get:

x = (lambda/66)^(1/5) * (5/2)^(2/5)
y = (5lambda/22)^(1/2) * (2/11)^(1/2)
lambda = (2/11)^(5/2) * (5/2) * (I/11)^(1/2)

Substituting I = $200 and solving for the optimal bundle of x and y, we get:

x* = 9.02, y* = 7.32

This means that without the club membership, Pablo would consume 9 units of x and 7 units of y, spending $44 in total.

If he joins the club, he can get y at $1 per unit, so he would consume 7 units of y and spend $9 on y. This means he would have $191 left to spend.

To find the most he would be willing to pay for the membership, we can compute the maximum amount he is willing to pay such that he is indifferent between joining the club and not joining. If he pays more than this amount, it would be better for him not to join.

To compute this, we can find the utility of consuming 7 units of y without the club membership:

U(9.02, 7) = 11(9.02)^6(7)^2 = 1,770,258.95

We can then find the maximum amount Pablo is willing to pay as follows:

$191 + P = $44 + $7P

where P is the amount he pays to join the club. Solving for P, we get:

P = $28

This means that if the club membership costs more than $28, it would be better for Pablo not to join, as he would be better off spending that money on buying y without the discount. Therefore, the most Pablo would be willing to pay to join the club is $28, or choice (B).

Pablo's Utility Function Is , Where X Is the Number