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nakungth nakungth
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5 years ago
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:
   QD = 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?
Textbook 

Microeconomics


Edition: 8th
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CanihCanih
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5 years ago
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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.
   QS = 275P

Equate QD and QS 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.
   QS = 275P
   P = 0.0036Q
   QD = 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.
   Pb = buyer's price
   Ps = seller's price (net of tax)
   Pb - Ps = 15 = tax
   QD = 140,000 - 425 Pb is market demand
   QS = 275 Ps is market supply
Set supply equal to demand:          
   140,000 - 425 Pb = 275 Ps
   Pb = Ps + 15
   140,000 - 425 (Ps + 15) = 275 Ps    
   140,000 - 425 Ps - 6,375 = 275 Ps    
   Ps = 190.89    
   Pb = Ps + 15 = 205.89
Consumers pay:    
   Pb - Po = 205.89 - 200 = 5.89
Producers pay:    
   Po - Ps = 200 - 190.89 = 9.11
Plug Ps into the supply equation to get quantity:    
   Q = Qs = 275 Ps = 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 QS.
   QS = -4125 + 275P
   275P = QS + 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.
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nakungth Author
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4 years ago
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3 weeks ago
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