Question

Donald loves gambling. His favorite gambling activity is going to the horse tracks and betting on his favorite horse, Betty. Donald's weekly income is $100, which he takes to the tracks. Donald can only purchase $50 lottery tickets on Betty finishing 1st. Betty's probability of winning each race is .5 and in case of a victory Donald gets $100 per every ticket bought while he gets nothing when Betty doesn’t end up in the first place. 1. Suppose that Donald can only bet multiples of $50. Calculate the expected value of buying (i) no tickets, (ii) 1 ticket, and (iii) 2 tickets. Don’t forget to take into account that he has an initial wealth of $100. Is the reward of any of the options more worthy than the other? 2. Donald's utility function is u(x) = x 2 . Find his expected utility from buying (i) no tickets, (ii) 1 ticket, and (iii) 2 tickets. How many tickets will he buy? 3. Suppose Donald’s family wants to stop him from going to the horse tracks. They want to design a conditional cash transfer program in which Donald receives a weekly allowance with the condition of not going to the horse tracks. How much should this allowance be to convince Donald not to buy 2 tickets each week?

Answer 1

1) i) When Donald does not buy any ticket, he retains his weekly income
we have expected value of buying = 0 ( as he is simply not buying anything)
ii) when he purchases 1 ticket,
Expected value of buying = 100 x 0.5 + 0 x 0.5 = $50
iii) when he purchases 2 tickets,
Expectd value of buying = 2(100 x 0.5 ) + 2( 0 x 0.5) = $100
Note that his expected wealth would include his initial income with which he buys ticket and in all 3 cases, Donald's expected wealth ( value of buying ticket + income ) remains at $100.
So his expected wealth remains the same at $100. SO his reward ( expected wealth) from all the 3 options are same. None of the options are actually better for him.

2) Given utility function : u(x) = 2x
here we consider x to be the expected income from buying tickets
i) when he buys no ticket :
expected income = 100 x 0 = 0 ( he retains his weeky income without spending it on tickets)
expected utility = 0
ii) when he buys one ticket:
expected income = 100 x 0.5 + 0 x 0.5 = 50
Expected utility = 100  
iii) when he buys two tickets:
Expected income = 100 x 0.5 + 100 x 0.5 + 0 x 0.5 + 0. 5 = 100
Expected utility = 200
He will buy 2 tickets to maximize his utility to the level of 200

3) The cash transfer should be such that, Donald gets the same value as the expected value he would get from buying 2 tickets , which is as previously determined , 100 ( 100 x 0.5 + 100 x 0.5 + 0 x 0.5 + 0 x 0.5 )
So Donald's family should give him a weekly allowance of $100 to make him indifferent and between going to the horse track and thus prevent him from going there. Here, we are implicitly assuming that donald is risk neutral.