Question

Coase theorem question. Specifically e & f

Suppose that a rancher is raising cattle (X) next to a farmer. The profits of the rancher are given by π(X) = 100X −X2 for 0 ≤ X ≤ 100 and the utility of the farmer is given by: U(W,X) = W(100 − X) for 0 ≤ X ≤ 100 where W is her level of wealth. Assume initially W = 50.

a) Suppose the rancher has the right to run as many cattle as she likes. How many cattle will she choose?

b) Suppose the farmer has the right to dictate how many cattle will be run. How many cattle will she choose?

c) What is the efficient number of cattle to run? (i.e. Solve the social planner’s problem)

d) Suppose the government will tax the rancher $T per cow. At what tax rate $T∗ will the rancher choose to run the efficient number of cattle?

e) Suppose the farmer chooses the number of cattle, and the farmer is paid $S per cow by the rancher. (The amount paid to the farmer enters her wealth.) How many cows will the farmer choose to run?

f) Bonus: Why do the answers to c) and e) differ?

Answer 1

c).

Consider the given problem here the profit function of “Rancher” and the utility function of “farmer” are given in the question. So, here the social planer will try to maximize the sum of “profit” and “utility” function.

=> S(X) = A(X) + U(X) = 100*X – X^2 + W*(100-X) = 100*X – X^2 + 50*(100-X).

=> S(X) = 100*X – X^2 + 50*100 – 50*X = 50*X – X^2 + 50*100.

=> the FOC for maximization is given by, “dS/dX = 0”.

=> 50 - 2*X = 0, => X = 25. So, the socially optimum level of cattle is given by “X=25”.

e).

Now, the farmer chooses the numbers of cattle and the farmer is paid “$S” per cow, => the new utility function is given by.

=> U(X) = W*(100 – X) + S*X = 50*(100 – X) + S*X = 5000 – 50*X + S*X= 5000 + (S-50)*X

=> U(X) = 5,000 + (S-50)*X, => dU/dX = (S-50) > 0, => farmer will try to get as much cattle as possible, => the optimum number of cattle is “100”. Now, if “dU/dX < 0”, => additional cattle decrease the farmer’s level of utility, => the optimum level of cattle is given by “0”.

So, here the final decision depends on the value of “S”, => whether it is more or less than “50”.

f).

Now, the socially planner will choose the numbers of cattle in such a way that the sum of “profit” and “utility” function together will maximum. Now, if the decision goes to farmer then the farmer will try to choose the numbers of cattle such that only the utility function will be maximum. So, that is the big reasons that change the value of cattle.