Then =1-91=1(3+:2)(3+2n 可=-1-3)=1(4-+2)(1-2+a With 1 <a2 re have Ox,>0 O: Thus, in equilibrium, we must have ai=.2. In fact, the two firms must sit in the middle By Proposition 2.1, Pi=p?=c Discussion 1. Nonexistence of Nash equilibrium if n>3 2. Existence of a reactive equilibrium for any n>1 3. What will happen if n identical firms are located on a circle? 5.9. Entry Cost A monopolist industry can be a result of entry costs. Consider 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 Cournot game p(y=a-y where a>0 and c>0 are two constants In stage 2, with n firms in the industry, each firm considers the following problem nax a-c L Us Vh
Then, π∗ 1 = (p∗ 1 − c)ˆz = t 2 3 + x1 + x2 3 2 3 + x2 + x1 6 , π∗ 2 = (p∗ 2 − c)(1 − zˆ) = t 4 3 − x1 + x2 3 1 3 − x2 + x1 6 . With x1 < x2, we have ∂π∗ 1 ∂x1 > 0, ∂π∗ 2 ∂x2 < 0. Thus, in equilibrium, we must have x1 = x2. In fact, the two firms must sit in the middle: x∗ 1 = x∗ 2 = 1 2 . By Proposition 2.1, p∗ 1 = p∗ 2 = c. Discussion: 1. Nonexistence of Nash equilibrium if n ≥ 3. 2. Existence of a reactive equilibrium for any n ≥ 1. 3. What will happen if n identical firms are located on a circle? 5.9. Entry Cost A monopolist industry can be a result of entry costs. Consider 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 Cournot game. Let ci(y) = cy, pd (y) = a − y, where a > 0 and c ≥ 0 are two constants. In stage 2, with n firms in the industry, each firm considers the following problem πi = max yi # a − c −[n j=1 yj $ yi. 2 — 11
In equilibrium of the 2nd stage, n+1 n+1 The zero-profit condition implies the equilibrium number of firms in two stages √R See Problem 1.5 for the result of Bertrand competition in the second stage Let yn denote the outcome of each firm in an n-firm industry. The social welfare is n Let no be the social optimal number of firms. In the above example, the social optimal number of firms, determined by W'(n)=0, is 2/3 Therefore Implying n>n The following proposition is a general result about the entry bias Proposition 2.3. Suppose P(y)<0 and e(y20. Let yn be the symmetric equilib- rium output for a firm. Assume (1) nyn is increasing in n (2)yn is decreasing in n (3)p(nyn)>c(yn)for all n Then.n*>n°-1.■ 5.10. Strategic Investment to Deter Potential Entrants Incumbent firms in an industry often make strategic investments to deter potential entrants. These investments include investments in cost reduction, capacity, and new- product development Consider a two-stage duopoly model
In equilibrium of the 2nd stage, y∗ i = a − c n + 1, π∗ i = a − c n + 12 . The zero-profit condition implies the equilibrium number of firms in two stages: n∗ = a − c √ K − 1. See Problem 1.5 for the result of Bertrand competition in the second stage. Let yn denote the outcome of each firm in an n -firm industry. The social welfare is W(n) = ] nyn 0 pd (y)dy − nc(yn) − nK. Let no be the social optimal number of firms. In the above example, the social optimal number of firms, determined by W0 (no)=0, is no = (a − c)2/3 K1/3 − 1. Therefore, n∗ +1=(no + 1)3/2 , implying n∗ > no. The following proposition is a general result about the entry bias. Proposition 2.3. Suppose p0 (y) < 0 and c00(y) ≥ 0. Let yn be the symmetric equilibrium output for a firm. Assume (1) nyn is increasing in n. (2) yn is decreasing in n. (3) p(nyn) ≥ c0 (yn) for all n. Then, n∗ ≥ no − 1. 5.10. Strategic Investment to Deter Potential Entrants Incumbent firms in an industry often make strategic investments to deter potential entrants. These investments include investments in cost reduction, capacity, and newproduct development. Consider a two-stage duopoly model: 2 — 12
Stage 1. Firm 1 has the option to make a strategic investment k>0 Stage 2. Firms 1 and 2 play a Nash game, choosing strategies 1, 32 E R, resulting profts丌 Let the reaction functions be G1=y1(32, k), 92=2(v1) Suppose there is a Nash equilibrium yi(k), y* (k) in stage 2 satisfying stability condition We have dr2[i(k),2(k)O2(k),(k)]y(k) ak upper oni(m,A)<0.m,边2<0 By equilibrium condition yi=y12(3i),kl, we have 孤-盖 Hence, to reduce the incentive of entry, we need 2k>0, which requires k>0 We also have d1lyi(k), y2(k),k ayi(k), y2(k), k) ay(k), ayi(k), y(k),k d dy ak The first term is the strategic effect; the second term is the direct effect. By equilibrium condition y2=j2[i(32, k)], we find that the strategic effect is positive if 22<0 Example 2.9. Suppose firm 1 is the incumbent and firm 2 is a new entrant. Let C1(y)=c1(k)y, C2(g)=C2y p(y)=a-y where a>0 and c2>0 are constants, and cI(k)>0 and d(k )<0. We have f 1=a-c1(k)-y2
Stage 1. Firm 1 has the option to make a strategic investment k > 0. Stage 2. Firms 1 and 2 play a Nash game, choosing strategies y1, y2 ∈ R, resulting profits π1(y1, y2, k) and π2(y1, y2). Let the reaction functions be yˆ1 = ˆy1(y2, k), yˆ2 = ˆy2(y1). Suppose there is a Nash equilibrium [y∗ 1(k), y∗ 2(k)] in stage 2 satisfying stability condition: ∂yˆ1 ∂y2 ∂yˆ2 ∂y1 < 1. We have dπ2[y∗ 1(k), y∗ 2(k)] dk = ∂π2[y∗ 1(k), y∗ 2(k)] ∂y1 ∂y∗ 1(k) ∂k . Suppose ∂π1(y1, y2, k) ∂y2 < 0, ∂π2(y1, y2) ∂y1 < 0. By equilibrium condition y∗ 1 = ˆy1[ˆy2(y∗ 1), k], we have ∂y∗ 1 ∂k = ∂yˆ1 ∂k 1 − ∂yˆ1 ∂y2 ∂yˆ2 ∂y1 . Hence, to reduce the incentive of entry, we need ∂y∗ 1 ∂k > 0, which requires ∂yˆ1 ∂k > 0. We also have dπ1[y∗ 1(k), y∗ 2(k), k] dk = ∂π1[y∗ 1(k), y∗ 2(k), k] ∂y2 ∂y∗ 2(k) ∂k + ∂π1[y∗ 1(k), y∗ 2(k), k] ∂k . The first term is the strategic effect; the second term is the direct effect. By equilibrium condition y∗ 2 = ˆy2[ˆy1(y∗ 2, k)], we find that the strategic effect is positive if ∂yˆ2 ∂y1 < 0. Example 2.9. Suppose firm 1 is the incumbent and firm 2 is a new entrant. Let C1(y) = c1(k)y, C2(y) = c2y pd (y) = a − y, where a > 0 and c2 ≥ 0 are constants, and c1(k) ≥ 0 and c0 1(k) < 0. We have † yˆ1 = 1 2 [a − c1(k) − y2], yˆ2 = 1 2 (a − c2 − y1). 2 — 13
We have ak>0 and 2u <0. The equilibrium profits are r1=a-2c1(k)+c2 m2={a-22+c(k)2 where K is the entry cost. We have 2<0 6. Competitive Input Market Given input(s)a, let y= f(a) be a production function and R()=pf()f() be the revenue function. The marginal revenue product is MRP≡R(x) Let C()=cf(a) be the cost function. The marginal cost product is MCP≡C(x) We ha MRP=MR× MP and mcp=MC×MP Thus, MRP(a)=MCP(a) For a competitive firm, its demand function is t wd=mrp(a). The supply of factors, such as labor and capital, is determined by the decisions of households Equilibrium is where the demand curve intersects with the supply curve Economic rent is the income received by the supplier over and above the amount required to induce him to offer the input. Transfer earnings is the income required to induce the supply of the input. t Total income of input Economic Rent Transfer Earnings Two special cases: land and unskilled labor Example 2.10. Consider the housing market (1)An earthquake destroys a chunk of housing stock (2) The government imposes a rent ceiling that is lower than the market rate (3)Hong Kong Government restricts the supply of land 2-14
We have ∂yˆ1 ∂k > 0 and ∂yˆ2 ∂y1 < 0. The equilibrium profits are π∗ 1 = 1 9 [a − 2c1(k) + c2] 2 − k, π∗ 2 = 1 9 [a − 2c2 + c1(k)]2 − K, where K is the entry cost. We have ∂π∗ 2 ∂k < 0. 6. Competitive Input Market Given input(s) x, let y = f(x) be a production function and R(x) ≡ pd[f(x)]f(x) be the revenue function. The marginal revenue product is MRPx ≡ R0 (x). Let C(x) ≡ c[f(x)] be the cost function. The marginal cost product is MCPx ≡ C0 (x). We have MRP = MR × MP and MCP = MC × MP . Thus, MRP(xd ) = MCP(xd ). (2.10) For a competitive firm, its demand function is † wd = MRP(x). The supply of factors, such as labor and capital, is determined by the decisions of households. Equilibrium is where the demand curve intersects with the supply curve. Economic rent is the income received by the supplier over and above the amount required to induce him to offer the input. Transfer earnings is the income required to induce the supply of the input. † Total income of input = Economic Rent + Transfer Earnings. Two special cases: land and unskilled labor. Example 2.10. Consider the housing market. (1) An earthquake destroys a chunk of housing stock. (2) The government imposes a rent ceiling that is lower than the market rate. (3) Hong Kong Government restricts the supply of land. 2 — 14
7. Monopsony Monopsony: the only firm in an input market Consider a monopsony: w(a) is the supply function of input, R(a)is the revenue The mcp is t MCP(x)=(x)+ru()=(x)1+ where e is the price elasticity of supply e(x)≡ a du Asε→∞, monopsony→ competitor in the input market The condition(2.10) determines the optimal a Example 2.11. Minimum Wage. Suppose that the government imposes a minimum vage W'min on the labor market. L 8. Vertical Relationships A upstream firm produces output a with cost c(a)and a downstream firm inputs .r to produce output for revenue R(r). Consider a case with R(a)=(a-b c(a)=ca Upstream firm Cost c(x)=cx Input Market Monopolistic supplier Competitive demander Revenue R(x=()x
7. Monopsony Monopsony: the only firm in an input market. Consider a monopsony: w(x) is the supply function of input, R(x) is the revenue function. The MCP is † MCP(x) = w(x) + xw0 (x) = w(x) 1 + 1 ε(x) , where ε is the price elasticity of supply: ε(x) ≡ w x ∂xs ∂w . As ε → ∞, monopsony → competitor in the input market. The condition (2.10) determines the optimal x∗. Example 2.11. Minimum Wage. Suppose that the government imposes a minimum wage wmin on the labor market. 8. Vertical Relationships A upstream firm produces output x with cost c(x) and a downstream firm inputs x to produce output for revenue R(x). Consider a case with R(x)=(a − bx)x, c(x) = cx. Cost ( ) c x cx = Upstream firm Input Market Monopolistic supplier Competitive demander x Downstream firm x Revenue R(x) (a bx)x = − 2 — 15