Question

Hi Can you please help and show steps on how to approach this question. thank you in advance

1) Coase Theorem Suppose that a rancher is raising cattle (X) next to a farmer. The profits of the rancher are given by (X) 100X - X2 for 0 X 3 100 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 b) Suppose the farmer has the right to dictate how many cattle will be run. How many c) What is the efficient number of cattle to run? (i.e. Solve the social planner's problem) will she choose? cattle will she choose? d) Suppose the government will tax the rancher ST per cow. At what tax rate ST wil the rancher choose to run the efficient number of cattle? e) Suppose the farmer chooses the number of cattle, and the farmer is paid SS per covw 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

a). Suppose the rancher is deciding the cattle to be run. Her objective is to maximise her profit.
The equation of profit is given by:
$\pi(X) = 100X - X^{2}$
We can see this directly, or by differentiating wrt X (where X is the number of cattle)
Profits will be maximised when X = 50.
Thus, if the rancher is in charge, 50 cattle will be run.

b). If the farmer is in charge, the farmer has control over one equation, that is, her utility function. She attempts to maximise her utility. The equation for her utility is given by:
$U(X) = 50(100 - X)$
This equation will be maximised if X = 0.
Thus, if the farmer is in charge, 0 cattle will be run.

c) Using the social planner's problem - a social planner tries to achieve the best possible solution for both parties involved.
The rancher's profits can be written as:
$\pi(X) = X(100 - X)$
The farmer's utility function can be written as:
$U(X) = 50(100 - X)$
The social planner can see that $\pi^{*} = U^{*}$ for a specific value of X.
Thus, to equate profit and utility for both, the social planner will set X = 50 and will let 50 cattle run free.

d). Now, the government taxes the rancher $T per cow. Her profits now become
$\pi = 100X - X^2 - TX$
$\pi = (100-T)X - X^2$
$\pi = X(100-T - X)$

If the rancher lets 49 cows run free, the profits will be
$\pi_{49} = 2499 - 49T$
$\pi_{50} = 2500 - 50T$
To let the efficient number of cattle run free, the profit must be such that $\pi_{50} > \pi_{49}$
2499 - 49T < 2500 - 50T
T < $1.00

For T < $1, the rancher will be profitable letting the efficient number of cows run free and will continue to let them do so.

Hope this helped!