Herbie's utility function is U(x, y)  x  8y  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.Herbie has $200 a week to spend.The price of x is $1.The price of y is currently $4 per unit.Herbie has received an invitation to join a club devoted to the consumption of y.If he joins the club, Herbie 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 Herbie would be willing to pay to join this club?

A) $33
B) $21
C) $4.50
D) $16.50
E) None of the above.

Question

Jovanni Palermo · Accepted Answer

Final Answer : D, Explanation :  To find how much Herbie is willing to pay for the club membership, we need to compare his total utility with and without being a club member.Without club membership:Herbie's budget is $200, and the price of x is $1, so he can buy 200 units of x. At this level of consumption, his total utility is:U(x=200, y=0) = 200 - 8(0) = 200With club membership:Herbie can now buy y for $1 per unit. Let's say he decides to buy 50 units of y in addition to the 200 units of x he buys. His total expenditure is 200(1) + 50(1) = $250. However, since he pays $21 to join the club, his net expenditure is $271. At this level of consumption, his total utility is:U(x=200, y=50) = 200 - 8(50) + (1/2)(50)(50) = 1250So the incremental utility from joining the club is:U(club) - U(no club) = 1250 - 200 = 1050Since Herbie is willing to pay up to the amount of incremental utility he receives, the most he would be willing to pay to join the club is $16.50 ($1050/64). Therefore, the correct answer is (D) $16.50.

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