Question

Clancy has $1,200. He plans to bet on a boxing match between Sullivan and Flanagan. For $4, he can buy a coupon that pays $10 if Sullivan wins and nothing otherwise. For $6 he can buy a coupon that will pay $10 if Flanagan wins and nothing otherwise. Clancy doesn’t agree with these odds. He thinks that the two fighters each have a probability of 1/2 of winning. If he is an expected utility maximizer who tries to maximize the expected value of lnW, where lnW is the natural log of his wealth, it would be rational for him to buy a. 50 Sullivan coupons and no Flanagan coupons. b. 100 Sullivan coupons and no Flanagan coupons. c. 50 Flanagan coupons and no Sullivan coupons. d. 100 Flanagan coupons and no Sullivan coupons. e. 100 of each kind of coupon

Answer 1

The correct option is A- 50 Sullivan coupons and no Flanagan coupon.

Clancy is being given odds of 10 to 4 that Sullivan wins. He thinks that the two fighters each have a probability of π = 0.5 of winning. If Clancy doesn’t bet, he is certain to have $1,200 to spend on consumption goods. His behavior satisfies the expected utility hypothesis and his von Neumann-Morgenstern utility function is π1(ln c1) + π2(ln c2).

Let cW be the consumption contingent on Sullivan winning and cNW be the consumption contingent on Sullivan losing. Betting on Sullivan at odds of 10 to 4 means that if Clancy buys x-coupons on Sullivan winning for $4x, then if Sullivan wins, Clancy makes a net gain of $(10 – 4)x, but if Sullivan loses, Clancy has a net loss of $4x.

Since Clancy had $1,200 before betting, if Clancy bets $4x on Sullivan and Sullivan wins, Clancy would have cW = 1,200 + (10-4)x to spend on consumption. If Clancy bet $4x on Sullivan and he lost, Clancy would lose $4x, and he would have cNW = 1,200−4x. By increasing the number of coupons, x, that Clancy buys, he can make cW larger and cNW smaller. (Clancy could also bet against Sullivan at 10 to 6 odds. If Clancy bets $6x against Sullivan and Sullivan loses, Clancy makes a net gain of $(10 – 6)x and if Sullivan wins, Clancy loses $6x.

If you work through the rest of this discussion for the case where Clancy bets against Sullivan, you will see that the same equations apply, with x being a negative number.) We can use the above two equations to solve for a budget equation. From the second equation, we have x = 300− cNW /4. Substitute this expression for x into the first equation and rearrange terms to find

cW = 1,200 + 6(300− cNW /4)

=3,000 − 3 cNW /2, or

2cW + 3cNW = 6,000, which is the budget constraint.

Then choose Clancy’s contingent consumption bundle (cW, cNW) to maximize U(cW, cNW) = .5ln(cW) + .5ln(cNW) subject to the budget constraint, 2cW + 3cNW = 6,000.

Using Langrange Multipliers, the Lagrangian is L = .5 lncW + .5 lncNW – λ(2cW + 3cNW - 6,000).

The first order conditions are:

(1)∂L /∂cW = 1/(2cW) – 2λ = 0,
(2)∂L /∂cNW = 1/(2cNW) – 3λ = 0,
(3)∂L /∂λ = 2cW + 3cNW - 6,000= 0.

Solve equations (1) and (2) for λ and equate the right-hand sides to get 1/(4cW) = 1/(6cNW), so cW = 3cNW/2. Substitute this into equation (3) to get cNW = 1,000 and using cW = 3cNW/2, cW = 1,500. Above, we found that x = 300− cNW /4, so Clancy should buy x = 300− 1000 /4 = 50 coupons for Sullivan and no coupons for Flanagan.