Then, π ? 1 =(p ? 1 ?c)?z = t  2 3 + x 1 +x 2 3  2 3 + x 2 +x 1 6  , π ? 2 =(p ? 2 ?c)(1? ?z)=t  4 3 ? x 1 +x 2 3  1 3 ? x 2 +x 1 6  . With x 1 <x 2 , we have ?π ? 1 ?x 1 > 0, ?π ? 2 ?x 2 < 0. Thus, in equilibrium, we must have x 1 = x 2 . 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 c i (y)=cy, p d (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 π i =max y i # a?c? n [ j=1 y j $ y i . 2—11 In equilibrium of the 2nd stage, y ? i = a?c n+1 , π ? i =  a?c n +1  2 . 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 y n denote the outcome of each firm in an n-firm industry. The social welfare is W(n)= ] ny n 0 p d (y)dy?nc(y n )?nK. Let n o be the social optimal number of firms. In the above example, the social optimal number of firms, determined by W 0 (n o )=0, is n o = (a?c) 2/3 K 1/3 ?1. Therefore, n ? +1=(n o +1) 3/2 , implying n ? >n o . The following proposition is a general result about the entry bias. Proposition 2.3. Suppose p 0 (y) < 0 and c 00 (y) ≥ 0. Let y n be the symmetric equilib- rium output for a firm. Assume (1) ny n is increasing in n. (2) y n is decreasing in n. (3) p(ny n ) ≥c 0 (y n ) for all n. Then, n ? ≥ n o ?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: 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 y 1 ,y 2 ∈ R, resulting profits π 1 (y 1 ,y 2 ,k) and π 2 (y 1 ,y 2 ). Let the reaction functions be ?y 1 =?y 1 (y 2 ,k), ?y 2 =?y 2 (y 1 ). Suppose there is a Nash equilibrium [y ? 1 (k),y ? 2 (k)] in stage 2 satisfying stability condition:     ??y 1 ?y 2 ??y 2 ?y 1     < 1. We have dπ 2 [y ? 1 (k),y ? 2 (k)] dk = ?π 2 [y ? 1 (k),y ? 2 (k)] ?y 1 ?y ? 1 (k) ?k . Suppose ?π 1 (y 1 ,y 2 ,k) ?y 2 < 0, ?π 2 (y 1 ,y 2 ) ?y 1 < 0. By equilibrium condition y ? 1 =?y 1 [?y 2 (y ? 1 ),k], we have ?y ? 1 ?k = ??y 1 ?k 1? ??y 1 ?y 2 ??y 2 ?y 1 . Hence, to reduce the incentive of entry, we need ?y ? 1 ?k > 0, which requires ??y 1 ?k > 0. We also have dπ 1 [y ? 1 (k),y ? 2 (k),k] dk = ?π 1 [y ? 1 (k),y ? 2 (k),k] ?y 2 ?y ? 2 (k) ?k + ?π 1 [y ? 1 (k),y ? 2 (k),k] ?k . The first term is the strategic e?ect; the second term is the direct e?ect. By equilibrium condition y ? 2 =?y 2 [?y 1 (y ? 2 ,k)], we find that the strategic e?ect is positive if ??y 2 ?y 1 < 0. Example 2.9. Suppose firm 1 is the incumbent and firm 2 is a new entrant. Let C 1 (y)=c 1 (k)y, C 2 (y)=c 2 y p d (y)=a?y, where a>0 and c 2 ≥ 0 are constants, and c 1 (k) ≥ 0 and c 0 1 (k) < 0. We have ? ?y 1 = 1 2 [a?c 1 (k)?y 2 ], ?y 2 = 1 2 (a?c 2 ?y 1 ). 2—13 We have ??y 1 ?k > 0 and ??y 2 ?y 1 < 0. The equilibrium profits are π ? 1 = 1 9 [a?2c 1 (k)+c 2 ] 2 ?k, π ? 2 = 1 9 [a?2c 2 +c 1 (k)] 2 ?K, where K istheentrycost.Wehave ?π ? 2 ?k < 0.  6. Competitive Input Market Given input(s) x, let y = f(x) be a production function and R(x) ≡ p d [f(x)]f(x) be the revenue function. The marginal revenue product is MRP x ≡ R 0 (x). Let C(x) ≡ c[f(x)] be the cost function. The marginal cost product is MCP x ≡ C 0 (x). We have MRP = MR×MP and MCP = MC×MP . Thus, MRP(x d )=MCP(x d ). (2.10) For a competitive firm, its demand function is ? w d = 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 o?er 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(x) is the supply function of input, R(x) is the revenue function. The MCP is ? MCP(x)=w(x)+xw 0 (x)=w(x)  1+ 1 ε(x)  , where ε is the price elasticity of supply: ε(x) ≡ w x ?x s ?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 w min on the labor market.  8. Vertical Relationships Aupstream firmproduces output x with cost c(x) and adownstream firminputs x to produce output for revenue R(x). Consider a case with R(x)=(a?bx)x, c(x)=cx. Cost ( )cx cx= Upstream firm Input Market Monopolistic supplier Competitive demander x Downstream firm x Revenue R(x) (a bx)x=? 2—15