Question

Richard is deciding whether to buy a state lottery ticket. Each ticket costs \(1, and the probability of winning payoffs is given as follows:

PROBABILITY	RETURN
.5	\)0.00
.25	\(1.00
.2	\)2.00
.05	$7.50

a. What is the expected value of Richard's payoff if he buys a lottery ticket? What is the variance?

b. Richard's nickname is "No-Risk Rick" because he is an extremely risk-averse individual. Would he buy the ticket?

c. Richard has been given 1000 lottery tickets. Discuss how you would determine the smallest amount for which he would be willing to sell all 1000 tickets.

d. In the long run, given the price of the lottery tickets and the probability/return table, what do you think the state would do about the lottery?

Short answer

1. The expected value of Richard's payoff will be $1.025. The variance will be 2.812.

2. Since Richard is a risk-averse person, he would not buy the ticket.

3. The smallest amount at which Richard is willing to sell the 1000 lottery tickets can be found out based on the expected utility of the lottery winning. The amount at which Richard is willing to sell the lottery will be the risk premium deducted from the expected payoff, that is, $1025 minus the expected payoff.

4. The state must increase the price of lottery tickets, reduce payoff, and increase the probability of winning nothing from the lottery.

Step-by-step solution

Expected value of Richard's payoff and variance

The expected value of the lottery is equal to the sum of the total weight of returns by their probability.

The expected value of the lottery can b found out using the probability and the return. Given:

The probability of getting $0.00 as the return is 0.50.

The probability of getting $1.00 as the return is 0.25.

The probability of getting $2.00 as the return is 0.20 and

The probability of getting $7.50 as the return is 0.05.

The expected value of the lottery will be:

$$\begin{gathered} EV=\left(0.5\right)\left(0\right)+\left(.25\right)\left(1\right)+\left(.2\right)\left(2\right)+\left(.05\right)\left(7.5\right) \\ =0+.25+.4+.375 \\ =1.025 \end{gathered}$$

The expected value will be $1.025.

The variance can be found using the expected value obtained.

The variance will be:

$$\begin{gathered} Variance=\left(0.5\right){\left(0-1.025\right)}^2+\left(.25\right){\left(1-1.025\right)}^2+\left(.2\right){\left(2-1.025\right)}^2+\left(.05\right){\left(7.5-1.025\right)}^2 \\ =0.5253+0.000156+0.190+2.096 \\ =2.811 \end{gathered}$$

The variance will be 2.811.

Richard won't buy the ticket

Since Richard is a highly risk-averse person, he won't buy the ticket because the ticket's expected value is $1.025 is higher than the ticket price ($1.00). The expected value of the ticket is not enough to compensate for the risk which he is incurring. The difference between the expected value and the price of the ticket is only $0.025. So this is not a safe amount for a risk aversive person.

A highly risk aversive person won't buy the ticket when the expected value is higher than the price.

Amount at which Richard sells the ticket

With 1000 tickets, Richard's expected payoff is $1025. He does not pay for the tickets, so he cannot lose money. There is a broad range of amounts at which Richard can sell 1000 lottery tickets ranging from 0 to $7500 (7.5(1000). So he can choose any amount among this range which he can avert the risk.

Even though there are several possible amounts at which Richard can sell the ticket, he will sell the ticket at a price at which the risk premium is deducted from the expected payoff. So he will choose an amount at which the risk premium is deducted from $1025.

 Government policies in the long-run

The government policies in response to the present ticket price and the winning probability will be focused on getting a high return. The state is currently incurring a high loss in the current scenario. So the government will try to make it more profitable in the long-run.

To make it profitable,they will increase the price of the lottery tickets and will also try to increase the winning probability to attract more to the lottery. They will reduce the payoff and try to lower the positive probabilities of the payoffs.If the probability of winning is high, more people will purchase the lottery and increase the government's revenue. So the loss can be minimized if the revenue is increasing.

These are some of the possible actions which the government should take in the long-run.