The total and marginal cost functions for a typical soft coal producer are:
TC = 75,000 + 0.1Q² and MC = 0.2Q
where Q is measured in railroad cars per year. The industry consists of 55 identical producers. The market demand curve is:
Q_D = 140,000 - 425P,
where P is the price per carload. The market can be regarded as competitive.
a. Calculate the short run equilibrium price and quantity in the market. Calculate the quantity that each firm would produce. Calculate producer surplus, consumer surplus, and total surplus at the equilibrium values. Calculate the firm's profit (or loss).
b. The Federal government is considering the imposition of a $15 per carload tax on soft coal. Calculate the short-run equilibrium price and quantity that would exist under the tax. What portion of the tax would be paid by producers and what portion by consumers? Calculate the producer and consumer surplus under the tax and analyze the efficiency consequences of the tax. Calculate the firm's profit (or loss) under the tax. Could the tax be justified despite its efficiency implications?

a.To find market supply curve begin by finding firm's supply curve.Firm's supply curve is MC curve (in this case all of MC lies above AVC):
Solve for Q in terms of MC = P:
MC = 0.2Q
Q = 5P
Market short-run supply is the horizontal sum of firm supply. There are 55 firms in the market, so market supply is 55 times the individual firm's supply.
Q_S = 275P
Equate Q_D and Q_S to determine P and Q.
275P = 140,000 - 425P
700P = 140,000
P = $200
Q = 275(200)
Q = 55,000
Individual firm equates P to MC:
200 = 0.2Q
Q = 1,000
π = TR - TC
TR = (200)(1000)
TR = 200,000
TC = 75,000 + 0.1(1000)²
TC = 175,000
π = 25,000
Producer and consumer surplus:
Solve for P in terms of Q.
Q_S = 275P
P = 0.0036Q
Q_D = 140,000 - 425P
P = 329.41 - 0.0024Q Producer surplus = 0.5(55,000)(200) = 5,500,000
Consumer surplus = 0.5(55,000)(329.41 - 200) = 3,558,775
Total of producer and consumer surplus is
3,558,775 + 5,550,000 = 9,058,775
b.P_b = buyer's price
P_s = seller's price (net of tax)
P_b - P_s = 15 = tax
Q_D = 140,000 - 425 P_b is market demand
Q_S = 275 P_s is market supply
Set supply equal to demand:
140,000 - 425 P_b = 275 P_s
P_b = P_s + 15
140,000 - 425 (P_s + 15) = 275 P_s
140,000 - 425 P_s - 6,375 = 275 P_s
P_s = 190.89
P_b = P_s + 15 = 205.89
Consumers pay:
P_b - P_o = 205.89 - 200 = 5.89
Producers pay:
P_o - P_s = 200 - 190.89 = 9.11
Plug P_s into the supply equation to get quantity:
Q = Q_s = 275 P_s = 275(190.89) = 52,495
(If you plug into the demand equation instead your answer will differ slightly due to rounding.)
Individual firm equates P to MC:
205.89 = 0.2Q + 15
Q = 954.5
π = TR - TC
TR = 205.89(954.50)
TR = 196,522
TC = 75,000 + 0.1Q² + 15Q
TC = 180,424.53
π = 16,097.48
Profit fell from 25,000 to 16,097.48.
Producer and Consumer Surplus:
Demand curve remains: P = 329.41 - 0.0024Q
Solve for P in terms of Q_S.
Q_S = -4125 + 275P
275P = Q_S + 4,125
P = 15 + 0.0036 Q Producer surplus = 0.5(52,497)(205.89) = 5,404,303.67
Consumer surplus = 0.5(52,497)(329.41 - 205.89) = 3,242,214.72
Total of Producer and Consumer Surplus:
= 5,404,303.67 + 3,242,214.72 = 8,646,518.39
Total surplus fell from 9,058,775 to 8,646,518.39.
There is a welfare loss as indicated by the loss in total surplus. The tax could be justified by known externalities of soft coal.