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: y i = null K i L i ,i=1, 2. Initially, the capital stocks in the two factories are respectively K 1 =25 and K 2 =100. Thewagerateforlaborisw, 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 increaseinthewageratefrom$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 = y 1 + y 2 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 firm1behavesasafollowerandfirm 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 y 1 ,...,y n , aggregate output is denoted by Y = null n i=1 y i , the industry demand curve is denoted by P(Y), and the cost function of each firm is given by c i (y i )=cy i . For simplicity, assume P 00 (Y) < 0. Supposethateachfirm is required to pay a specifictax of t i 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 null n i=1 t i . (c) Consider a change in each firm’s tax rate that doesn’t change the tax burden on the industry. Letting ?t i denote the change in firm i’s tax rate, we require that null n i=1 ?t i =0. Assuming that no firm leaves the industry, calculate the change in firm i’s equilibrium output ?y i . [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 c i (y)=cy, p d (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 n o = (a?c) 2/3 K 1/3 ?1 in the section on entry cost. 2—3 Answer Set 2 Answer 2.1. (a) For each factory with capital stock K, c(y,K) ≡ min L {wL+ rK | y = √ KL} = w K y 2 + rK. Therefore, the short-run cost functions are c 1 (y)= w 25 y 2 +25r, c 2 (y)= w 100 y 2 + 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 y 1 ,y 2 p · (y 1 + y 2 )?c 1 (y 1 )?c 2 (y 2 ). TheFOCsgiveusthewell-knownequality: p = MC 1 = MC 2 . We have MC 1 (y)= 2w 25 y and MC 2 (y)= w 50 y. Then p = MC 1 (y 1 ) and p = MC 2 (y 2 ) imply that p = 2w 25 y 1 and p = w 50 y 2 . Thus, y 1 = 25p 2w and y 2 = 50p w . Therefore, the short-run supply function is: y = y 1 + y 2 = null 25 2w + 50p w null p =62.5 p w . Thelabordemandsforthefactoriesare: L 1 = 1 K 1 y 2 1 = 1 25 null 25p 2w null 2 = 25 4 null p w null 2 ,L 2 = 1 K 2 y 2 2 = 1 100 null 50p w null 2 =25 null p w null 2 . Therefore, the labor demand is L = L 1 + L 2 = 125 4 null p w null 2 . (c) The cost for each factory is c i (y i ) ≡ min L,K {wL+ rK | y i = √ KL}. TheLagrangefunctionis L ≡wL+ rK + λ null y i ? √ KL null , 2—4 implying L i = null w r y i ,K i = null r w y i . The total cost is then c(y)=c 1 (y 1 )+c 2 (y 2 )=2 √ wr(y 1 + y 2 )=2y √ wr. From the profitfunctionπ = py ?c(y)=(p? 2 √ wr)y, we immediately find the long-run supply function: y s = ? ? ? ? ? ? ? ? ? ? ? ∞ 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 equi- librium 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 = K 1 + K 2 = null r w (y 1 + y 2 )= null 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 Answer 2.2. Let p(Y) be the market price of the good when the output is Y, c(y i ) is the cost of firm i when its output is y i . The two firms have the same cost function. The cartel maximizes their total profit: max y 1 ,y 2 π i ≡p(y 1 + y 2 )(y 1 + y 2 )?c(y 1 )?c(y 2 ). TheFOCsare p(Y ? )+p 0 (Y ? )Y ? = c 0 (y ? i ). (1) We look for a solution for which y ? 1 = y ? 2 (the symmetric solution). Thus, the FOC becomes p(Y ? )+p 0 (Y ? )Y ? = c 0 null Y ? 2 null . (2) We can rewrite (2) as MR(Y ? )=c 0 null Y ? 2 null , where R(Y) ≡p(Y)Y. On the other hand, the Cournot output is determined by MR(Y ? )? 1 2 p 0 (Y ? )Y ? = c 0 null Y ? 2 null . . ps5-1 p Y B A )(YMR ? ? ? ? ? ? 2 ' Y c D YYpYMR )(')( 2 1 ? . C . Figure 5.1. A market-share Cournot equilibrium In the diagram, point A is the ‘competitive solution’, for which each firm takes the marketpriceasgiven;pointB is our solution, for which each firm acts upon a decreasing demand and assume equal market share as the other’s reaction; point C is the Cournot equilibrium. From the diagram, we can conclude that 2—6 ? The equilibrium output at B is lower than the output at the ‘competitive solution’ and the output at the Cournot equilibrium. ? The equilibrium price at B is higher than the price at the ‘competitive solution’ and the price at the Cournot equilibrium. Answer 2.3. (a) For competitive output, firms take price as given in maximizing their own profits: max π i ≡Py i , which implies y ? i = ? ? ? ? ? +∞ if P>0 [0, +∞ ) if P =0. That is, the firms’ supply curve is the horizontal line at P =0. So is the industry supply curve. The equilibrium industry supply is thus Y ? =100and the equilibrium price is P ? =0. (b) Firm 1 maximizes his own profit, given any y 2 : max π i ≡P(y 1 + y 2 )y 1 =(100?y 1 ?y 2 )y 1 , which gives the FOC: 100?2y 1 ?y 2 =0. Firm 1’s reaction function is thus ?y 1 = 1 2 (100?y 2 ). (c) By symmetry, the outputs for the two firms should be the same in equilibrium. By the reaction function in (b), we hence have y 1 = 1 2 (100?y 1 ), which gives y 1 = 100 3 . Therefore, the Cournot equilibrium is y ? 1 = y ? 2 = 100 3 . (d) Suppose the two firms collude. They form a monopoly and maximizes their total profit: max π ≡P(Y)Y =(100?Y)Y, which gives the cartel output: Y ? =50. 2—7 (e) Firm 1 will behave as in (b), and reacts according to his reaction function ?y 1 = 1 2 (100?y 2 ). Firm 2 will take this into consideration when maximizing his own profit: max π 2 ≡P[?y 1 (y 2 )+y 2 ]y 2 = 1 2 (100?y 2 )y 2 , which implies y ? 2 =50. Then, y ? 1 =25. In summary, the competitive industry output is the highest, the Stackelberg industry output is the second, the Cournot industry output is the third, and cartel output is the lowest. Answer 2.4. (a) The profit maximization for firm i is max π i = P null n null j=1 y j null y i ?(k +t i )y i . The FOC is P(Y)+P 0 (Y)y i = k + t i . (3) (b) By summarizing (3) from i =1 to n, we have nP(Y)+P 0 (Y)Y = nk + n null j=1 t i . (4) This equation determines the industry output Y, which obviously depends on n null j=1 t i , rather than the individual tax rates t i ’s. (c) Since the total output depends only on n null j=1 t i and the latter has no change, Y doesn’t change for a tax change. Then, by (3), nullt i = P 0 (Y)nully i , i.e., nully i = nullt i P 0 (Y) , where Y is determined by (4). Answer 2.5. This is from Example 12.E.2 on page 407 of MWG (1995). Once n identi- cal firms are in the industry, they play a Bertrand game. As we know, if n≥ 0, the result is the competitive outcome, i.e., p ? = c and the profit without including the entry cost K is zero for all the firms. This means that each firm loses K in the long run. Knowing this, once one firm has entered the industry, all other firms will stay out. Therefore, more intense competition in stage 2 results in a less competitive industry! 2—8 This single firm will be the monopoly and produces at the monopolist output q m = a?c 2b , resulting the monopoly price p m = a+c 2 . The monopoly profitis Π m = (a?c) 2 4b ?K. As long as Π m ≥ 0, a firm will enter and that is the only firm in the industry. Answer 2.6. We have W(n)= null ny n 0 (a?y)dy?ncy n ?nK = 1 2 null a 2 ?(a?ny n ) 2 null ?cny n ?nK = 1 2 null a 2 ? null a?n a?c n+1 null 2 null ?cn a?c n+1 ?nK = n n +1 a(a?c)? 1 2 null n a?c n +1 null 2 ?cn a?c n +1 ?nK = n n +1 (a?c) 2 ? 1 2 null n a?c n +1 null 2 ?nK = 1 2 null 2 n n +1 ? null n n+1 null 2 null (a?c) 2 ?nK = 1 2 null 1?(1?γ) 2 null (a?c) 2 ? γ 1?γ K, where γ ≡ n n+1 . Then, 0= ?W ?γ =(1?γ)(a?c) 2 ? K (1?γ) 2 , implying γ o =1? null K (a?c) 2 null 1/3 , implying n o = γ o 1?γ o = 1? null K (a?c) 2 null 1/3 null K (a?c) 2 null 1/3 = (a?c) 2/3 K 1/3 ?1. End 2—9