Question

(Tragedy of the Commons) Canterbury is a small pastoral town with a grassy area known as the “commons.” The town’s two farmers may freely graze their cattle on the commons. In the spring each farmer simultaneously and independently buy (identical) cattle for $10 a head. The farmers must send their cattle to the commons during the spring and summer to feed and fatten them up. At the end of summer the cattle are sold at the market price of $1 per hundred pounds. (The farmers are price-takers in the larger cattle market.) The problem is that the commons is a small area that can only feed so many cattle before the grazing becomes poor. To capture this idea, let Q be the total number of cattle sent to the commons. The total weight (in hundreds of pounds) of all the cattle at the end of the season is given by W(Q) = 100Q − 10Q^2 , so that weight of a single cow is W(Q)/Q = 100 − 10Q. Let qi be the number of cattle farmer i buys in the spring and sends to the commons. Assume that a farmer can send a “fractional” cow so that qi can be any number greater than or equal to zero. (You can think of this as the farmer sending the cow to the commons for only a fraction of the season.) Naturally, Q = q1 + q2. Thus, the payoff functions are u1(q1, q2) = (100 − 10q1 − 10q2) q1 − 10q1 and u2(q1, q2) = (100 − 10q1 − 10q2) q2 − 10q2.

(a) Write the game in normal form.

(b) Find farmer 1 and 2’s best response function. It’ll help to notice that the game is symmetric.

(c) Find the Nash equilibrium number of cattle each farmer sends to the commons (i.e., head of cattle purchased): (q*1 , q^2 ).

(d) Show that the equilibrium is inefficient because too many cattle are sent to the commons in equilibrium. That is, show that q*1 + q*2 exceeds the quantity that would maximize the joint payoff. (The joint payoff is (100Q − 10Q^2)− 10Q = 90Q − 10Q^2 .)

(e) What is the intution behind part (d)? Can you think of other contexts, possibly including ones we covered in class, to which the same type of analysis applies? That is, where there is "too much" of an activity in equilibrium

Answer 1

(a)

(b)

Farmer1's Utility is given by:

Best response of Farmer 1 given Farmer 2 chooses q2

Farmer2's Utility is given by:

Best response of Farmer 2 given Farmer 1 chooses q1

We can see that Best responses are symmetric of the form

(c)

Putting Best Response of Farmer 2 in Farmer 1's Best response for the Nash Equilibrium

Putting q1=6 in Farmer 2's BR:

Hence Nash Equilibrium of the game is

(d)

Total Cattles in Nash Equilibrium: Q* = 6

Now lets find the socially optimal Number of cattles which will maximize the joint payoff:

Joint Payoff:

Lets find optimal number of total cattles on the commons:

FONC:

Optimal Number of cattles in the commons should be 4.5

Hence, Nash Equilibrium is inefficient which puts 1.5 more cattles in the commons a total of 6