Question

1. Suppose that David has the following utility function over consumption level c:

U(c) = √c (that’s the square root of c)

Also, suppose David has wealth $1000, but faces the risk of a financial loss $500 with probability 0.2 (20% chance). He does not save anything (he consumes all the wealth he can).

1. What is David’s consumption in the event of no loss? What is his utility in this case?

2. What is David’s consumption in the event of financial loss? What is his utility in this case?

3. What is David’s expected consumption? What is his utility at the expected level of consumption?

4. What is David’s expected utility of consumption? Is this the same as David’s utility from expected consumption from part (c)? Why or why not (if not, explain which is larger)?

Answer 1

Utility function is U = √C

a) When there is no loss, wealth is WN = 1000 and so utility is U = 1000^0.5 = 31.62

b) When there is a loss, wealth is WL = 1000 – 500 = 500 and utility U = 500^0.5 = 22.36

c) Expected consumption = expected wealth EW = 1000*0.8 + 500*0.2 = 900. Utility of this expected

consumption U (EW) = 900^0.5 = 30

d) Expected utility EU = 0.8*(1000^0.5) + 0.2*(500^0.5) = 29.77. We see that expected utility of wealth is smaller than the utility of expected wealth. This shows that David is risk averse.