Problem set 2 Micro Theory S. Wang Question 2. 1. You have just been asked to run a company that has two factories produc ing the same good and sells its output in a perfectly competitive market. The production function for each factory is Initially, the capital stocks in the two factories are respectively Ki= 25 and K2=100 The wage rate for labor is w, and the rental rate for capital is r. In the short run,the capital stock for each factory is fixed, and only labor can be varied. In long run, both capital and labor can be varied (a) Find the short-run total cost function for each factory (b) Find the company's short-run supply curve of output, and derived demand curve for labor (c) Find the long-run total cost function for each factory and the long-run supply curve of the company (d) If all companies in the industry are identical to your company, what is the long-run industry equilibrium price? (e) Let r=1. Suppose the cost of labor services increases from S1.00 to $2.00 per unit What is the new long- run industry equilibrium price? Can you determine whether the quantity of capital used in the long run will increase or decrease as a result of the increase in the wage rate from S1.00 to S2.00? Question 2.2. Suppose that two identical firms are operating at the cartel solution and that each firm believes that if it adjusts its output the other firm will adjust its output so as to keep its market share equal to What kind of industry structure does this imply
Problem Set 2 Micro Theory, S. Wang Question 2.1. You have just been asked to run a company that has two factories producing the same good and sells its output in a perfectly competitive market. The production function for each factory is: yi = sKiLi, i = 1, 2. Initially, the capital stocks in the two factories are respectively K1 = 25 and K2 = 100. The wage rate for labor is w, and the rental rate for capital is r. In the short run, the capital stock for each factory is fixed, and only labor can be varied. In long run, both capital and labor can be varied. (a) Find the short-run total cost function for each factory. (b) Find the company’s short-run supply curve of output, and derived demand curve for labor. (c) Find the long-run total cost function for each factory and the long-run supply curve of the company. (d) If all companies in the industry are identical to your company, what is the long-run industry equilibrium price? (e) Let r = 1. Suppose the cost of labor services increases from $1.00 to $2.00 per unit. What is the new long-run industry equilibrium price? Can you determine whether the quantity of capital used in the long run will increase or decrease as a result of the increase in the wage rate from $1.00 to $2.00 ? Question 2.2. Suppose that two identical firms are operating at the cartel solution and that each firm believes that if it adjusts its output the other firm will adjust its output so as to keep its market share equal to 1 2 . What kind of industry structure does this imply? 2—1
Question 2.3. Consider an industry with two firms, each having marginal costs equal to zero. The industry demand is P(Y)=100-Y, where Y=y1+y2 is total output (a) What is the competitive equilibrium output? (b) If each firm behaves as a Cournot competitor, what is firm 1s optimal output given firm 2s output? (c)Calculate the Cournot equilibrium output for each firm (d)Calculate the cartel output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm Question 2.4. Consider a Cournot industry in which the firms'outputs are denoted yn, aggregate output is denoted by Y=2ia yi, the industry demand curve is denoted by P(Y), and the cost function of each firm is given by ci(yi)= cyi.For simplicity, assume P(Y<0. Suppose that each firm is required to pay a specific tax ti on output (a) Write down the first-order conditions for firm i (b) Show that the industry output and price only depend on the sum of tax rates (c) Consider a change in each firms tax rate that doesn't change the tax burden on the industry. Letting At i denote the change in firm i's tax rate, we require that Ci Ati=0. Assuming that no firm leaves the industry, calculate the change in firm i's equilibrium output Ay;. Hint: use the equations from the derivations of (a) nd( b)I Question 2.5.(Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let where a>0 and c>0 are two constants Stage 1: All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K>0 Stage 2: All firms that have entered play a Bertrand game
Question 2.3. Consider an industry with two firms, each having marginal costs equal to zero. The industry demand is P(Y ) = 100 − Y, where Y = y1 + y2 is total output. (a) What is the competitive equilibrium output? (b) If each firm behaves as a Cournot competitor, what is firm 1’s optimal output given firm 2’s output? (c) Calculate the Cournot equilibrium output for each firm. (d) Calculate the cartel output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm. Question 2.4. Consider a Cournot industry in which the firms’ outputs are denoted by y1,...,yn, aggregate output is denoted by Y = Sn i=1 yi, the industry demand curve is denoted by P(Y ), and the cost function of each firm is given by ci(yi) = cyi. For simplicity, assume P00(Y ) < 0. Suppose that each firm is required to pay a specific tax of ti on output. (a) Write down the first-order conditions for firm i. (b) Show that the industry output and price only depend on the sum of tax rates Sn i=1 ti. (c) Consider a change in each firm’s tax rate that doesn’t change the tax burden on the industry. Letting ∆ti denote the change in firm i’s tax rate, we require that Sn i=1 ∆ti = 0. Assuming that no firm leaves the industry, calculate the change in firm i’s equilibrium output ∆yi. [Hint: use the equations from the derivations of (a) and (b)]. Question 2.5. (Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let ci(y) = cy, pd (y) = a − y, where a > 0 and c ≥ 0 are two constants. Stage 1: All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K > 0. Stage 2: All firms that have entered play a Bertrand game. 2—2
Question 2.6. Verify the social number of firms to be no=awdla-1 in the section n entry cost
Question 2.6. Verify the social number of firms to be no = (a−c)2/3 K1/3 − 1 in the section on entry cost. 2—3
Answer set 2 Answer 2.1 (a) For each factory with capital stock K C LwL+roy Therefore the short-run cost functions are C1(y) (y) y2+100 100 (b) The firm cares about the total profit from its two factories. The objective of firm is therefore to maximize the total profit (1+y2)-c1(y1)-c2(v) The FOCs give us the well-known equality P=MCI=MC2 We have MCi(y)=25y and MC2(v)=50y. Then p=MCi(v1)and p= MC2(32) imply that p= 25y1 and p= 5092. Thus, y=200 and 32=2. Therefore, the short-run supply function is oOu y=的+y P The labor demands for the factories are L1=+2 P 1/50p 25(2u L2=K2 v2 100(a Therefore the labor demand 12(=2) (c)The cost for each factory is cili Lwl+rKI The lagrange function is ≡uL+r+(v-VKL
Answer Set 2 Answer 2.1. (a) For each factory with capital stock K, c(y,K) ≡ min L {wL + rK | y = √ KL} = w K y2 + rK. Therefore, the short-run cost functions are c1(y) = w 25y2 + 25r, c2(y) = w 100y2 + 100r. (b) The firm cares about the total profit from its two factories. The objective of firm is therefore to maximize the total profit: π = max y1, y2 p · (y1 + y2) − c1(y1) − c2(y2). The FOCs give us the well-known equality: p = MC1 = MC2. We have MC1(y) = 2w 25 y and MC2(y) = w 50 y. Then p = MC1(y1) and p = MC2(y2) imply that p = 2w 25 y1 and p = w 50 y2. Thus, y1 = 25p 2w and y2 = 50p w . Therefore, the short-run supply function is: y = y1 + y2 = 25 2w + 50p w p = 62.5 p w. The labor demands for the factories are: L1 = 1 K1 y2 1 = 1 25 25p 2w 2 = 25 4 p w 2 , L2 = 1 K2 y2 2 = 1 100 50p w 2 = 25 p w 2 . Therefore, the labor demand is L = L1 + L2 = 125 4 p w 2 . (c) The cost for each factory is ci(yi) ≡ min L,K {wL + rK | yi = √ KL}. The Lagrange function is L ≡ wL + rK + λ yi − √ KL , 2—4
implying K The total cost is then c(y)=cIy)+C2(g2)=2vwr(y +92)=2yywr From the profit function T= py-c(y)=(p-2vwr)y, we immediately find the long-run supply function if p> U'=10,oo] if p=2Vu 0ifp<2√ That is, the long-run industry supply curve is horizontal (d) In a competitive market, with a horizontal industry supply curve, the long-run equi librium price must be p=2ywr, whatever the industry demand curve (e) The original long-run equilibrium price is p=2, and the new price is p=2v2.The total capital investment is With an increase in w and p, output y is reduced, implying K will be reduced
implying Li = uw r yi, Ki = u r w yi. The total cost is then c(y) = c1(y1) + c2(y2)=2√wr(y1 + y2)=2y √wr. From the profit function π = py − c(y)=(p − 2 √wr)y, we immediately find the long-run supply function: ys = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ ∞ if p > 2 √wr [0, ∞] if p = 2√wr 0 if p < 2 √wr. That is, the long-run industry supply curve is horizontal. (d) In a competitive market, with a horizontal industry supply curve, the long-run equilibrium price must be p = 2√wr, whatever the industry demand curve is. (e) The original long-run equilibrium price is p = 2, and the new price is p = 2√2. The total capital investment is K = K1 + K2 = u r w (y1 + y2) = u r w y. With an increase in w and p, output y is reduced, implying K will be reduced. p p y s y D . . 2—5