Question

Jim Hawkins lives in a two-state world where his crop is sure to be good if the weather is wet (state 1) but will fail entirely if the weather is dry (state 2). He can install irrigation canals to get some crops in the dry state, but only at the cost of reducing his income in the wet state. Explicitly: D = (W+150)(100-W)/300 where W is his income in the wet state and D is his income in the dry state. He starts out with W=100 (so that D=0).

1) If his utility function is v(c) = ln(c), explain why if there is any chance at all that it won't rain, he will dig at least some canals.

2) Based on his probability assessment, wet state income of 75 is optimal for him.  What does he believe that the probability of rain is?

3) It turns out that there is another island that he can trade with.  That island has the same weather (same two-state world), and they will trade him 3 wet claims for every dry claim that he gives them - in other words, the price ratio for dry to wet is 3 --- each dry claim is worth 3 wet claims.  Assuming that he can trade with the other island, what is his optimal production?

4) Explain geometrically why he is better off trading with them. Which direction in your state claim picture corresponds to increasing utility?

Answer 1

1) The utility function is v(c) = ln(c)

If c = 1, then v(1) = ln(1) = 0 means that Jim has no sense of happiness when producing commodity c. Therefore, if c = 1, it may be because there is no rain and Jim will dig at least some canals.

2) If W = 75 is the best

D = (75 + 150) (100-75)/300=18.75

W/D ratio is 4

W / W + D = 75 / 93.75 = 4/5 = 0.8

3) On another island, the price ratio is 3, that is, W/D=3. When W is optimal at 75, D = 18.75

W = 18.75 * 3 = 56.25

4) W / W + D = 56.25 / 18.75 + 56.25 = 3/4 = 0.75 in case 2