PROBLEM SET 3 Question 1 Suppose that in a competitive industry with 100 identical firms the short run cost function of each firm is given by: C(q)=16+q 2 a) Derive and graph the AC, AVC, and MC function of each firm. b) What is the supply of each firm (in the short run)? Draw it s graph. c) What is the total supply of the industry? Draw its graph. d) Suppose that the total demand of the industry is Q D =520-2p. What is the demand that each firm faces? e) What is the equilibrium price and the quantity that each firm produces at the equilibrium? f) Calculate the profit/loss of each firm. g) What is the market equilibrium price and quantity? h) Graph the equilibrium for each firm and for the market on the same coordinate plane. Question 2 Suppose that the production function of a firm is given by: q=f(l,k)= LK In the short run the capital is fixed at K and the price of capital is r=4 a) Calculate K and the price of labor w if the short run cost function is: C(q)=4+q 2. b) In the long run both capital and labor are variable. What is the optimal allocation of capital and labor for producing q=20 units of output? c) Draw the graph of the solution to the optimal allocation problem. d) Graph the long run expansion path. e) Derive the long run total cost function of the firm and draw its graph Question 3 In a market with identical firms the short run cost function is given by 1
C(q)=30+4q+q 2 a) Derive and graph the fixed cost (F) variable costs (VC), and marginal cost (MC) functions b) If there are 100 firms in the market derive and draw the market supply function Q. c) If the market demand is given by Q D (p)= 500-20p find the equilibrium price and equilibrium quantity. d) How much will each firm produce. Calculate the revenue, cost and profit of each firm. Question 4 A firm has the following production function: q=f(l,k)= LK In the short run capital is fixed at K=4. The price of capital and price of labor are r=1 and w=4 respectively. a) Find the cost of production for producing 40 units of output in the short run. b) Derive the equation of the short run cost function. c) Derive the equation of short run AC, AVC and MC cost functions. d) Fill in the following table using the cost functions. q AC AVC MC 0-2 4 6 8 e) Using the values from the table above draw the graphs of AC, AVC, MC cost functions. f) What is the optimal allocation of capital and labor (in the long run) for producing 20 units of output? g) What is the cost of production in the long run? Show the optimal solution graphically. h) Derive the equation of long run expansion path and long run cost function and draw their graphs. 2
Question 5 Below is the production information for Hormsbury Corp., which operates in a perfectly competitive market with hundreds of identical firms. K=1 K=2 K=3 K=4 K=5 L=1 8 12 15 16 18 L=2 15 18 22 26 28 L=3 18 24 28 30 32 L=4 20 26 31 34 36 L=5 21 28 32 36 38 Capital and labor are the only inputs; output is 0 if K=0 or L=0. r=3 and w=2. a) Assuming that in the short run capital is fixed at K =3 fill in the blanks in the following table. q TC AC AVC MC L=1 15 0.133 L=2 22 L=3 28 0.536 L=4 31 0.258 L=5 32 19 b) If the market price is p=0.667 how many units does the firm produce? What is the profit of the firm? c) If there is free entry and exist in the long run will the price rise, fall or stay constant? Explain why? In the long run each firm in this market has the following average costs. K=1 K=2 K=3 K=4 K=5 L=1 0.525 0.567 0.633 0.675 0.694 L=2 0.467 0.556 0.591 0.615 0.678 L=3 0.502 0.524 0.536 0.600 0.656 L=4 0.525 0.538 0.558 0.618 0.688 L=5 0.565 0.612 0.689 0.734 0.767 d) What will be the equilibrium price in the long run assuming no entry barriers? e) If the demand function is Q D =4507-15p how many firms will be active in the long run? 3
Question 6 Suppose that a firm uses only capital (K) and labor (L) for production. The production function of the firm is given by: q=f(l,k)=kl Price of capital is r=4 euro and capital is fixed at K=5 in the short run. a) Does the production function exhibit increasing, decreasing or constant returns to scale? Explain why? b) Calculate the price of labor (w) if the cost of producing 100 units of output is 200 euro in the short run. For part c you may take w=16 if you could not solve part b c) What will be the optimal allocation of labor and capital for producing 100 units of output in the long run? Calculate the cost of producing 100 units algebraically. Show the optimal solution graphically Question 7 A firm has which uses only capital (K) and labor (L) for production has a Cobb-Douglas production function of the form q=f(l,k)=l a K b. Price of capital and labor are r and w respectively. Currently the firm optimally produces q=q units of output at a cost of C=C by choosing an optimal combination (L, K ) of its inputs. a) Suppose at a later time period the price of capital increases to r > r. Given the price of labor w as before draw a graph and explain the relation between the new optimal allocation (L, K ) and the new cost of production C=C if the firm wants to maintain its output level at q=q as before. (No points for a guess without the correct graph ) b) Suppose further that instead of an increase in price of capital, the price of labor increases to w > w. Given the price of capital r as before draw a graph and explain the relation between the new optimal allocation (L, K ) and the new production level q=q if the firm wants to maintain it s cost at C=C. (No points for a guess without the correct graph) Question 8 In a competitive market with many firms the long run cost function of each firm is: C(q)=q 2 +3q+f if q> 0, C(q)=0 if q=0. 4
The market demand is given by Q d = 1010-34p (a) Interpret f (f>0). Is there a value of f so that there wont be any supply? For the rest of the question take f=4. b) How many units will each firm produce? c) Calculate the equilibrium price and the equilibrium quantity and draw the graph of the equilibrium. d) Calculate the number of firms which will be active in the long run in the market. Question 9 (Theory Question ) a)define what a decreasing returns to scale (DRS) production function is b) State (need not to prove) if the following production functions exhibit IRS, CRS or DRS. 1)F 1 (L, K) = 5L + 6K 2) F 2 (L, K) = 4min(2L, 3K) 3) F 3 (L, K) = 5L 0.6 K 0.3 c) State the condition for a competitive firm to shut down i) in the SR, ii) in the LR. d) Sketch the graphs of the AC, AVC and MC functions of a competitive firm which makes a loss but still continues to operate in the SR. Also shade the area of the loss of the firm on your graph. e) Explain what economies of scale means. What are the possible reasons for economies of scale? draw a cost function and label the regions for which the firm enjoys economies of scale, no economies of scale and suffers from diseconomies of scale. f) Explain why in general the costs of production are lower in the LR than in the SR. g) Explain what economies of scope means and give an example. Question 10 A firm that functions in a competitive industry with 50 identical firms has the following short run cost function: C(q) = q 2 +3q+9 a) Derive the equation of AC, AVC and MC functions of the firm. At what level of output does AC reach its minimum? b) Derive the equation of the supply function of each firm and the total supply of the industry. 5
Market demand is given by Q D =150-20p where price is in euro and quantity is in 100,000 units c) Calculate the equilibrium price, equilibrium quantity, the quantity that each firm produces, and the profit (loss) of each firm. d) Draw two different graphs; one showing the market equilibrium and one showing the firm equilibrium (graph of firm equilibrium should show the supply of each firm and the residual demand that each firm faces) e) Calculate the consumer surplus (CS), producer surplus (PS) and the total welfare (W). f) Suppose that the government puts a price ceiling of p c = 4 euro. Calculate the CS, PS and the deadweight loss ( W) under the price ceiling policy. Question 11 In a market the demand for a particular product is given by Q D = 150 2p (price in euro and quantity in million units). The domestic supply is given by Q d S = 3p and the world price of the good is p=20 euro. a) Calculate the CS, PS and the W if there is free trade. b) Analyze the welfare effect of a ban on imports? (Calculate the CS, PS, W and dead-weight loss if there is a ban on imports and explain how the CS and PS change) c) Suppose further that instead of a ban on imports the government collects a tariff of T=5 euro from foreign suppliers. Discuss the effects of the tariff. (Calculate the CS, PS, total welfare, and dead-weight loss if there is tariff of T=5 euro and explain how the CS and PS change) Question 12 Suppose that the demand for a good is given by : Q d = 1200 20p (where price is in euro cents and quantity in millions of units per year) The supply of this good is given by: Q S =-200+30p a) Calculate the CS, PS and the total welfare (W) if there is no government intervention in the market. b) Suppose government levies a tax of T=50 cents from suppliers. Analyze in detail the welfare effects of this tax. 6
c) Suppose instead of a tax government subsidizes the suppliers by a subsidy of s=50 cents. Analyze the welfare effects of this subsidy in details. d) Suppose further that instead of a subsidy, government puts a price ceiling of p c =25 cents. Analyze the welfare effects of the price ceiling policy. 7