Question

A factory does the production around the river.
On the other hand, a fisherman does the fishing in the river.
Now, if a factory produce one unit, the benefit of fisherman decrease by 6 due to the pollution.
A factory has the right to use the river, price of goods=P=20,   C（Q）=Q² ※cost=C

（1）what is the production quantity Qm to maximize the benefit of factory?

（2）when the fisherman negotiates with the factory, the production quantity in the factory decreases by social optimum point. at the same time, the fisherman pays 2/3 of benefits to the factory . How much does the benefit of factory increase, compared with Qm.
※ negotiation cost is zero.

Answer 1

1) The profit-maximizing level for the factory is where the marginal benefit of the factory is equal to the marginal cost. Here, Marginal revenue is equal to the price, therefore MR = 20 and Cost = Q², therefore marginal cost to factory, MCfa = dC/dQ = 2Q

Therefore, to get Qm, we equate MR = MC

2Qm = 20 => Qm = 20/2 = 10 units

Therefore, production quantity Qm that maximizes the benefit of the factory is equal to 10 units.

2) Marginal cost to fisherman, MCf = 6, therefore social marginal cost, MSC = MCfa + MCf = 2Q + 6

Now, the socially optimum point of production: MR = SMC

=> 2Qs + 6 = 20

=> 2Qs = 20-6 = 14

=> Qs = 14/2 = 7

Therefore, the socially optimum level of production, Qs is equal to 7 units.

change in output = 10-7 = 3

Benefit to fisherman = 3*6 = $18

Amount paid to the factory by the fisherman = 2/3 (18) = 2*6 = $12

At social optimum

Revenue to the factory = Qs *P = 7*20 = $140

Cost to the factory = Qs² = 7*7 = $49

Profit to the factory = (Revenue - cost) + benefit paid from fisherman = (140-49) + 12 = $103

At private optimum

Revenue to the factory = Qm *P = 10*20 = $200

Cost to the factory = Qm² = 10*10 = 100

Profit to the factory = (Revenue - cost) = 200 -100 = 100

Change in benefit to the factory = $(103-100) = $3

The benefit of factory increase by $3 after negotiation.