Question

Two hundred paper mills compete in the paper market. The total cost of production (in dollars) for each mill is given by the formula TC = 500Qmill + (Qmill)2 where Qmill indicates the mills annual production in thousands of tons. The marginal cost of production is MC = 500 + 2Qmill. The external cost of a mill’s production (in dollars) is given by the formula EC = 40Qmill + (Qmill)2 and the marginal external cost of production is MEC = 40 + 2Qmill. Finally, annual market demand (in thousands of tons) is given by the formula Qd = 150,000 – 100P where P is the price of paper per ton. Using algebra, find the competitive equilibrium price and quantity, as well as the efficient quantity. Calculate the magnitude of the deadweight loss resulting from the externality. Illustrate your solution with graphs.

Answer 1

Answer: See explanation

Explanation:

The magnitude of the deadweight loss resulting from the externality is shown below:

MC = 500 + 2Q

MEC = 40 + 2Q

Therefore, the Marginal social cost (MSC) will be:

= MC + MEC

= 500 + 2Q + 40 + 2Q

= 540 + 4Q

Since Demand: Q = 150,000 - 100P, we have to get a function for P which will be:

Q = 150,000 - 100P

100P = 150,000 - Q

P = (150,000 - Q)/100

P = 1,500 - 0.01Q

Total revenue, TR = P x Q

= (1,500 - 0.01Q) × Q

= 1500Q - 0.01Q²

Marginal revenue, MR will be:

= dTR / dQ

= 1,500 - 0.02Q

It should be noted that for when there's no externality, Equilibrium, MC must be equal to MR. Therefore,

1,500 - 0.02Q = 500 + 2Q

2Q + 0.02Q = 1500 - 500

2.02Q = 1,000

Q = 1000/2.02

Q = 495

P = 1,500 - (0.01 x 495)

= 1,500 - 4.95

= 1,495.05

When there's externality, Equilibrium will be:

MR = MSC

1,500 - 0.02Q = 540 + 4Q

4.02Q = 960

Q= 960/4.02

Q = 239

Therefore, P = 1,500 - (0.01 x 239)

= 1,500 - 2.39

= 1,497.61

Then, we will calculate the deadweight loss which will be:

= 1/2 x Difference in price x Difference in quantity

= 1/2 x (1,497.61 - 1,495.05) x (495 - 239)

= 1/2 x 2.56 x 256

= 327.68