Question

Consider the Tragedy of the Commons game from the chapter with two shepherds, A and B, where sA and sB denote the number of sheep each grazes on the common pasture. Assume that the benefit per sheep (in terms of mutton and wool) equals 500-sA-sB implying that the total benefit from a flock of sA sheep is sA(500-sA-sB) and that the marginal benefit of an additional sheep (which is defined by the first partial derivative with respect to sA of the total benefit function) is 500-2sA-sB. Assume that the marginal cost of grazing sheep is 100.

1. Compute the flock sizes and shepherds’ total benefit in the Nash equilibrium.

2. Draw the best-response unction diagram corresponding to your solution.

3. Suppose A’s benefit per sheep drops to 450-sA-sB, which means that his marginal benefit becomes 450-2sA-sB. B’s benefit per sheep does not change. Compute the new Nash equilibrium flock sizes. Show the change from the original to the new Nash equilibrium in your best-response function diagram.

Answer 1

Given:

The benefit per sheep is (500 – sA – sB).

The marginal benefit of an additional sheep is 500 – 2sA – sB and the marginal cost of gazing sheep is 100.

a).

Compute the total benefit of sA as follows.

sA(500 – sA – sB) = 500(sA) – (sA)² – (sA)( sB).

Compute the derivative of 500(sA) – (sA)² – (sA)( sB) as follows.

Thus, the total benefit of sA is sA = 200 – (sB)/2.

Compute the total benefit of sB as follows.

sB(500 – sA – sB) = 500(sB) – (sA)( sA) – ( sB) ².

Compute the derivative of 500(sB) – (sA)( sA) – ( sB) ².as follows.

Thus, the total benefit of sB is sB = 200 – (sA)/2.

b)

Obtain the best response function diagram as shown in the below Figure.

c)

The change in the Nash equilibrium is shown in the below Figure..