Suppose that a decision maker with initial wealth W can pay

Suppose that a decision maker with initial wealth W can pay P for a lottery ticket with a single jackpot prize J. The probability of winning the jackpot is p. Show that a risk averse decision maker prefers buying the ticket to simply keeping their initial wealth W only if the expected value of the lottery winnings (for the decision maker) is positive.

Solution

given W is wealth

P is the price you pay for lottery

J is the winning

p is the probability of winning.

If you win you will get J but for that you need to purchase the lottery for P. that means you will get only J-P.

but if you loose that means whatever you have paid for the lottery is gone that is your P amount wil be gone.

your money will be reduced by P amount . and probability of loosing is 1-p

so expected value of lottery winning is

p*(J-P) + (1-p) * (-P)

here we have used -P in the second expression to represent loss of P amount in case of losing the lottery.

so simplify the equation

p*J-p*P -P + p*P = p*J-P

now that risk averse person will buy only if expected value of lottery is positive that means

(p*J)-P >= 0

so condition for purchasing th lottery by risk averse person

the value of p , J and P should be such that they satisfy the equation (p*J)-P >=0