AAE 706
The Economics of Risk
EXAM 2
Consider a farmer choosing between two technologies A and B under risk. Farm profit is affected by weather uncertainty, as represented by the random variable e that can take 4 possible values {ej, j = 1, 2, 3, 4}. The 4 states (e1, e2, e3, e4) correspond respectively to poor, fair, good, and excellent weather conditions. The profit generated by each technology under each state of nature is given in the following table:
______
Profitstate e1state e2state e3state e4
($)(poor)(fair)(good)(excellent)
______
Tech. A246 7
Tech. B34510
______
The farmer believes that the relative likelihood of each state is as follows: Pr(e1) = .2; Pr(e2) = .3; Pr(e3) = .4; and Pr(e4) = .1.
1/ You know that the farmer is risk averse and that he/she behaves in a way consistent with the expected utility hypothesis, but you do not know precisely his/her risk preferences. Evaluate the choice between the two technologies A and B using stochastic dominance.
- Is it sufficient to know that the farmer is risk averse to rank the two technology choices? Why or why not?
- Would knowing the following additional information help rank the choices A and B:
. the farmer's utility function is exponential?
. or the farmer's utility function is logarithmic?
. or the farmer's utility function is quadratic?
Why or why not?
2/ The farmer obtains (costlessly) new information about the weather before he/she chooses technology. This new information consists in observing a signal u that has the following likelihood probabilities: Pr(ue1) = .3; Pr(ue2) = .2; Pr(ue3) = .1; and Pr(ue4) = .4.
- How does the new information affect the farmer's beliefs about the relative likelihood of the alternative states of nature?
- How does the new information affect your stochastic dominance results in 1/? Knowing that the farmer is risk averse, what decision would you recommend?
3/ You now learn that the farmer has a logarithmic utility function u = ln(w). Find the maximum risk premium the farmer would be willing to pay to insure against weather uncertainty in 2/ (i.e. after he/she observes the signal u).