Question

Sam’s utility function is U(x, y) = 2x + y, where x is the number of x’s he consumes per week and y is the number of y’s he consumes per week. Sam has $200 a week to spend. The price of x is $4. Sam currently doesn’t consume any y. Sam has received an invitation to join a club devoted to the consumption of y. If he joins the club, Sam 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 Sam would be willing to pay to join this club?

a. Nothing b. $100 a week c. $50 a week d. $40 a week e. None of the above.

The answer is b. $100 a week. I need your solution with exactly explain.

Answer 1

Currently only x is consumed. Total amount of x consumed = income/Price of x = 200/4 = 50
So, x = 50 and y = 0
U(50, 0) = 2(50) + 0 = 100

After he joins the club, price of y = $1 and his total income = $200 (given)
The most Sam would be willing to pay to join this club is that which gives him an utility of atleast 100.
Now, U (0, y) = 100 = 2(0) + y
So, y = 100
Thus, he should be able to buy atleast 100 units of y.
Total amount spent on 100 units of y = Price of y*(y) = 1*100 = $100
So, income left = 200 - 100 = $100
Thus, he would be willing to pay at most $100 a week to join this club because any amount greater than $100 would decrease his utility.