Question

Rickie is considering setting up a business in the field of entertainment at children’s parties. He estimates that he would earn a gross revenue of £9,000 or £4,000 with a 50–50 chance. His initial wealth is zero. What is the largest value of the cost which would make him start this business if U(x) = x², for x > 0 and U(x) = –x² for x < 0.

Answer 1

Solution:

Profit = revenue - cost

Then with C as the total cost, with gross revenue = 9000 pounds, profit = (9000 - C) pounds

And with gross revenue = 4000 pounds, profit = (4000 - C) pounds

With initial wealth as 0, here wealth is same as profit.

Rickie will start the business only if the expected utility from business is greater than 0.

Expected utitlity = probability of earning (9000 - C) profit*utility from earning this profit + probability of earning (4000 - C) profit*utility from earning this profit

EU = (1/2)*U(9000 - C) + (1/2)*U(4000 - C) (As, 50-50 chance)

So, we want EU > 0 for Rickie to start the business ... (1)

But which utility function to choose under which case will be soon understood.

Further note that since 9000 > 4000, 9000 - C > 4000 - C, and so if 9000 - C < 0, then 4000 - C is necessarily negative. In same way, if 4000 - C > 0, then profit of 9000 - C will also be positive. But if profit of 9000 - C is positive, 4000 - C isn't necessarily positive.

Since, we have to solve for maximum cost incurrence that will make Rickie start the business, we take the extreme situations: checking for negative wealth/profits

Now, if the wealth (or profit here with initial wealth of 0), x is negative, then with profit as negative from both revenues will give expected utility as negative always, so we reject this case (case: 9000 - C < 0 rejected).

Next case could be when 9000 - C > 0, but 4000 - C < 0, then

With x = 9000 - C, utility function considered is x2, while for x = 4000 - C, utility function considered is -x2

Thus, EU = (1/2)*(9000 - C)2 + (1/2)*(-(4000 - C)2)

EU = (1/2)*(90002 - 2*9000*C + C2) + (1/2)*(-(40002 - 2*4000*C + C2)) (Using property: (a - b)2 = a2 - 2ab + b2)

EU = (81000000 - 18000C + C2)/2 - (16000000 - 8000C + C2)/2

EU = 40500000 - 9000C + 0.5*C2 - 8000000 + 4000C - 0.5*C2

EU = 32500000 - 5000C

As we wish expected utility to be positive, using (1) above, we have:

32500000 - 5000C > 0

32500000 > 5000C

C < 32500000/5000 = 6500

So, required largest value of cost is 6,500 pounds.