Question

5. Suppose you are thinking of starting a business. You assume there's a 10% chance of being massively profitable with $1 million in profit, a 60% chance of being fairly successful with $100,000 in profit, and a 30% chance of the business collapsing, leaving $ 200,000 in losses (negative profit). Not starting the business would leave you with zero profits. (a) Calculate the expected value of the profit in this business venture. Is it worth it to start the business, given this metric? (b) Given the following utility function: u(x) = { 21/2 27 - 12/1/2 r> < 0 Calculate the expected utility of the profit in this business. Is it worth it start the business given this metric?

Answer 1

Solution:

1. Given the probabilities for each respective level of earnings, we can find the expected value of profit in the business venture as follows:

Expected profit = Sum(probability*profit value)

Expected profit = 10%*1m + 60%*(100000) + 30%*(-200000)

Expected profit = 0.1*1000000 + 0.6*100000 - 0.3*200000

Expected profit = 100000 + 60000 - 60000 = 100000

If not starting a business, profits will be 0

Since, the expected profit with business start = $100,000 > $0, it is worthy to start the business.

2. Similar to above case, expected utility of profit = sum(probability*utility value of profit)

Note that for negative value of profit, utility = -|x|^1/2

Then, for loss of 200,000, we would have utility = -|(-200000)|^1/2 = -(200000)^1/2

Expected utility = 0.1*(1000000)^1/2 + 0.6*(100000)^1/2 + 0.3*(-(200000)^1/2)

Expected utility = 0.1*1000 + 0.6*316.227766 - 0.3*447.213595

Expected utility = 100 + 189.73666 - 134.164079 = 155.57 (approximately)

Without business start, expected utility = (0)^1/2 = 0

Since, expected utility with business is greater than expected utility without business, it is worthy to start the business.