Question

Consider a farmer with utility function U(Y) = ln(Y), where Y is the farmer's income. Supposed the farmer believes there is a 50-50 chance that the next growing season will be abnormally rainy and he must choose which of two crops to plant: (1) wheat, which will produce an income of $28,000 in a normal year but only $10,000 in an abnormally rainy year, and (2) corn, which will produce $19,000 in a normal year but only $15,000 in an abnormally rainy year. (a) Which crop will he choose? (b) Assuming the farmer can plant a portion of each crop, what is the optimal mix of wheat and corn that maximizes the farmer's expected utility (Use a variable c for the fraction of the field planted with corn, set up expected utility function and maximize with respect to c, (c) Supposed wheat crop insurance is available. The insurance costs $4,000 and pays out $8,000 in the event of a rainy growing season. Will the availability of insurance change the farmer's plans.

Answer 1

a) Which crop will he choose

Crop Normal Year Rainly

Wheat $28,000 $10,000

Corn   $19,000 $15,000

Expected Utility = 1/2 in Normal Year + 1/2 in Year

Expected Utility (Wheat) = 1/2 x 28000 + 1/2 x 10000 = 9.725

Expected Utility (Corn) = 1/2 x 19000 + 1/2 x 15000 = 9.734

He gets higher anticipated utility from Corn.

With a blend, the farmer will get 14,000 + 9,500 = 23,500 in a normal year, and 5,000 + 7,500 = 12,500 of every a rainy year. In this way,

Expected Utility (Mix) = (1/2) ln (23,500) + (1/2) ln (12,500)   =   9.74912

He would broaden. Expansion enables him to appreciate a portion of the upside of Wheat in a dry year without bearing every one of the expenses of Wheat if the year is stormy.

b)  Max     E(C)    =(1/2) ln (28,000 +(1-)19,000) + (1/2) ln (10,000 +(1-)15,000)

Max E(C)        =(1/2) ln (19,000+ 9,000) + (1/2) ln (15,000 -5,000)

  Taking the derivative,

            .5(9000)/(19,000+ 9000) -.5(5000)/(15,000- 5000) = 0

Solving: 9(15-5)        =         5(19 + 9)

                        3 - =         19/9 +

                        2 =         8/9

=         4/9

44 Wheat and 56% Corn

C) Let’s assume that the farmer plants the optimal mix (it’s okay if you selected another mix, but I’ll use that for reference here) Under the optimal mix

Crop Normal Year _Rainy

Wheat              .44 ($28,000)   .44($10,000)

Corn                .56($19,000)    .56($15,000)

Optimal mix    $22,960            $12,800

E_{opt mix}(U)         =(1/2) (22,960) + (1/2) (12,800) = 9.74935

Now, with the insurance, and planting wheat

expected utility (Wheat w/ ins) = (1/2) (24,000) + (1/2) (14,000)     = 9.81

9.81>9.75 Yes, the insurance price is low enough to make specializing in wheat optimal.