Pablo's utility function is U(x, y)= x + 10y - y²/2, 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

Quizplus · Accepted Answer

To solve this problem, we need to find the maximum amount that Pablo is willing to pay to join the club. We can solve this problem by using the concept of marginal utility.

First, we need to find the amount of x and y that Pablo will consume without joining the club. Since the price of x is $1 and Pablo has $200 to spend, he can buy 200 x's per week. Using the budget constraint, we can solve for the amount of y that Pablo will buy without the discount:

$1x + $5y = $200
200 - x = 5y
y = 40 - 0.2x

Next, we need to find Pablo's marginal utility for y. Taking the partial derivative of his utility function with respect to y, we get:

MUy = 10 - 2y - 0.1P1

where P1 is the membership fee for the club. If Pablo joins the club, he can buy y for $1 per unit, so P1 = $1. If Pablo does not join the club, he has to pay $5 per unit for y, so P1 = $0.

If P1 = $0, then MUy = 10 - 2(40 - 0.2x) - 0.1(0) = -6 + 0.4x
If P1 = $1, then MUy = 10 - 2(40 - 0.2x) - 0.1(1) = -5.9 + 0.4x

To maximize his utility, Pablo will consume y until MUy = Py/Px, where Py/Px is the ratio of the price of y to the price of x. Since Py = $5 and Px = $1, Py/Px = 5. Setting MUy equal to 5, we get:

-6 + 0.4x = 5
0.4x = 11
x = 27.5

So without the club membership, Pablo will buy 27.5 units of x and 40 - 0.2(27.5) = 34 units of y. His total utility is:

U($27.5, $34) = 10($27.5) + 10(10)($34) - $5($34)^2/2 = $1,997.50

If Pablo joins the club, he will buy y until MUy = $1/$1 = 1. Setting MUy equal to 1, we get:

-6 + 0.4x = 1
0.4x = 7
x = 17.5

So with the club membership, Pablo will buy 17.5 units of x and 40 - 0.2(17.5) = 36.5 units of y. His total utility is:

U($17.5, $36.5) = 10($17.5) + 10(10)($36.5) - $1($36.5)^2/2 - $28 = $2,005.75

Therefore, the most Pablo would be willing to pay to join the club is the difference between his total utility with the club and without the club:

$2,005.75 - $1,997.50 = $8.25

This is closest to answer choice B, $8.

Pablo's Utility Function Is U(x, Y)= X + 10y -