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Download Problem Set 6. Competitive Market EconS 526 1
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Problem Set 6. Competitive Market EconS 526 1. Farmers produce corn from labor, a variable input, and land, a fixed input. The labor cost in dollars to produce y bushels of corn is ππ(π¦π¦) = π¦π¦ 2 . There are 100 identical farms which all behave competitively. a. What is the individual farmerβs supply equation of corn? In equilibrium price is equals marginal cost so, ππ = 2π¦π¦. Therefore, we have π¦π¦ = 0.5ππ. b. What is the market supply of corn? ππ = ππππ = 100 β 0.5ππ = 50ππ c. Suppose the demand curve of corn is π·π·(ππ) = 200 β 50ππ. What is equilibrium price and quantity of corn? How much does each individual firm produce? We have supply equals demand 50ππ = 200 β 50ππ. Therefore, p=2 and Y=100. Since p=2, each firm only produces y=1. d. Assume that the market price you calculate in (c) is the long run equilibrium price. What is the equilibrium rental value of land? Each firm will have profit equal to zero. So, if total revenue is py = 2 and labor cost is 1 then rental rate of land is 1 for profit equal to zero. 2. On a tropical island there are 100 boat builders, numbered 1 through 100. Each builder can build up to 12 boats a year and each builder maximizes profit given the market price. Let y denote the number of boats built per year by a particular builder. The cost function for boat builder i is ππ(π¦π¦) = 11 + ππππ such that the first builder has a cost function ππ(π¦π¦) = 11 + 1π¦π¦, the second builder has a cost function ππ(π¦π¦) = 11 + 2π¦π¦, etc. Assume that the quasi-fixed cost $11 is only paid id the firm produces a positive amount of output. If the price of boats is 25, how many builders will choose to produce a positive amount of output? If the price of boats is 25, how many boats will be built per year in total? Firms will enter as long as positive profit is made. The first up to the 24th boat builder will make positive profit. The 25th builder can cover variable cost but not the quasifixed cost so the builder will not build. Therefore, 24 builders will enter the market. Each will produce a maximum of 12 boats. Note that given linear cost, each firm only has an option to produce 0 or a maximum of 12. Therefore there will be 288 boats created (12x24). 3. US consumers have a demand function for umbrellas which has the form π·π·(ππ) = 90 β ππ. Umbrellas are supplied by US firms and UK firms. For simplicity, assume that there is a single representative firm in each country that behaves competitively. The cost function for producing umbrellas is given by ππ(π¦π¦) = 0.5π¦π¦ 2 in each country. a. What is the aggregate supply function for umbrellas? Each individual firm maximizes profit such that p=y so individual supply is y=p. Therefore aggregate output for two firms is Y=p+p = 2p. b. What is the equilibrium price and quantity sold? Supply equals demand so 2p= 90 βp. Therefore, p=30 and Y = 60. c. Now the domestic industry lobbies for protection and Congress agrees to put a $3 tariff on foreign umbrellas. What is the new U.S. price for umbrellas paid by the consumers? The aggregate supply will change Y = p + (p-3). So demand equals supply is now 90-p=2p-3 so that p = 31. d. How many umbrellas are supplied by foreign firms and how many are supplied by domestic firms? US will provide (y=p) 31 while UK provides (y=p-3) 28. Deadline: November4, 2015 beginning of class. See syllabus for penalty due to late submissions.
