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(Y0 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 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
Answer 2.2. Let pl(r) be the market price of the good when the output is Y, c(yi)is the cost of firm i when its output is i. The two firms have the same cost function. The cartel maximizes their total profit max Ti= P(y1+y2)(91+y2)-c(y1)-c(u2) The FOcs are p(Y)+p(Y*)Y=c() We look for a solution for which yi= y*(the symmetric solution). Thus, the FOC P(Y)+P(YY=c We can rewrite(2) as MR(Y)=C where R(Y=P(YY. On the other hand, the Cournot output is determined by MR(Y*)-SP(Yr=c D Figure 5. 1. A market-share Cournot equilibrium In the diagram, point A is the 'competitive solution,, for which each firm takes the market price as given; point B is our solution, for which each firm acts upon a decreas demand and assume equal market share as the others reaction; point C is the Cournot librium From the diagram we can conclude that 2-6
Answer 2.2. Let p(Y ) be the market price of the good when the output is Y, c(yi) is the cost of firm i when its output is yi. The two firms have the same cost function. The cartel maximizes their total profit: max y1, y2 πi ≡ p(y1 + y2)(y1 + y2) − c(y1) − c(y2). The FOCs are p(Y ∗ ) + p0 (Y ∗ )Y ∗ = c0 (y∗ i). (1) We look for a solution for which y∗ 1 = y∗ 2 (the symmetric solution). Thus, the FOC becomes p(Y ∗ ) + p0 (Y ∗ )Y ∗ = c0 Y ∗ 2 . (2) We can rewrite (2) as MR(Y ∗ ) = c0 Y ∗ 2 , where R(Y ) ≡ p(Y )Y. On the other hand, the Cournot output is determined by MR(Y ∗ ) − 1 2 p0 (Y ∗ )Y ∗ = c0 Y ∗ 2 . . ps5-1 p Y B A MR(Y ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 ' Y c D MR(Y ) 2 p'(Y )Y 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 market price as given; point B 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 thecompetitive solutionand the price at the Cournot equilibrium Answer 2.3 (a) For competitive output, firms take price as given in maximizing their own profits P which implies if P>0 0,+∞)ifP=0 That is, the firms supply curve is the horizontal line at P=0. So is the industry The equilibrium industry supply is thus y*= 100 and the equilibrium ce Is (b) Firm 1 maximizes his own profit, given any y maxr;≡P(1+y2)h=(100--v)h, which gives the foc 100-2 Firm 1's reaction function is thus 3n=2(100-y2) (c) By symmetry, the outputs for the two firms should be the same in equilibrium. By the reaction function in(b), we hence have y1 =5(100-1), which gives y1 Therefore, the Cournot equilibrium is 1=y (d)Suppose the two firms collude. They form a monopoly and maximizes their total max≡P(Y)Y=(100-Y)Y, which gives the cartel output: Y*= 50
• 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 πi ≡ P yi, 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 ∗ = 100 and the equilibrium price is P∗ = 0. (b) Firm 1 maximizes his own profit, given any y2 : max πi ≡ P(y1 + y2)y1 = (100 − y1 − y2)y1, which gives the FOC: 100 − 2y1 − y2 = 0. Firm 1’s reaction function is thus yˆ1 = 1 2 (100 − y2). (c) By symmetry, the outputs for the two firms should be the same in equilibrium. By the reaction function in (b), we hence have y1 = 1 2 (100 − y1), which gives y1 = 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 1 (100-12). Firm 2 will take this into consideration when maximizing his own profit max 2=P3n(32)+g2lbp=I (100-y2)y2, which implies y*=50. Then, yi=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 nswer (a) The profit maximization for firm i is max丌 ∑m)m-(k+t)m The FOc is P(Y)+P(Y)yi=k+ti (b) By summarizing (3) from i=l to n, we have nP(Y)+P()Y=mk+∑t This equation determines the industry output Y, which obviously depends on >t rather than the individual tax rates tis (c)Since the total output depends only on > ti and the latter has no change, Y doesn't change for a tax change. Then, by 3), At;= P(Y)Ayi,i.e Pr(Y 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 he 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!
(e) Firm 1 will behave as in (b), and reacts according to his reaction function yˆ1 = 1 2 (100−y2). Firm 2 will take this into consideration when maximizing his own profit: max π2 ≡ P[ˆy1(y2) + y2]y2 = 1 2 (100 − y2)y2, 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 πi = P #[n j=1 yj $ yi − (k + ti)yi. The FOC is P(Y ) + P0 (Y )yi = k + ti. (3) (b) By summarizing (3) from i = 1 to n, we have nP(Y ) + P0 (Y )Y = nk +[n j=1 ti. (4) This equation determines the industry output Y, which obviously depends on Sn j=1 ti, rather than the individual tax rates ti ’s. (c) Since the total output depends only on Sn j=1 ti and the latter has no change, Y doesn’t change for a tax change. Then, by (3), 7ti = P0 (Y )7yi, i.e., 7yi = 7ti P0 (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 identical 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 n 26, resulting th e mondO oly price Pm =2. The monopoly profit is IIm= As long as Im >0, a firm will enter and that is the only firm in the industry. A nswer 2.6. We have (a-y)dy-ncun - nk=5[a-(a-nyn)2]-cnyn nK d-c 2K n+1 n+1 d-c + x2 n+ 11-21-2 n -c)2-nK [1-(1-)2](a-c) K where The implying 1 (a-c)2 Imp 2/3 /3 End
This single firm will be the monopoly and produces at the monopolist output qm = a−c 2b , resulting the monopoly price pm = a+c 2 . The monopoly profit is Πm = (a − c)2 4b − K. As long as Πm ≥ 0, a firm will enter and that is the only firm in the industry. Answer 2.6. We have W(n) = ] nyn 0 (a − y)dy − ncyn − nK = 1 2 a2 − (a − nyn) 2 − cnyn − nK = 1 2 % a2 − a − n a − c n + 12 & − cn a − c n + 1 − nK = n n + 1 a(a − c) − 1 2 n a − c n + 12 − cn a − c n + 1 − nK = n n + 1(a − c) 2 − 1 2 n a − c n + 12 − nK = 1 2 % 2 n n + 1 − n n + 12 & (a − c) 2 − nK = 1 2 1 − (1 − γ) 2 (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 γo = 1 − K (a − c)2 1/3 , implying no = γo 1 − γo = 1 − k K (a−c)2 l1/3 k K (a−c)2 l1/3 = (a − c)2/3 K1/3 − 1. End 2—9