Online Appendix for Trade Policy under Monopolistic Competition with Firm Selection

Transcription

1 Online Appenix for Trae Policy uner Monopolistic Competition with Firm Selection Kyle Bagwell Stanfor University an NBER Seung Hoon Lee Georgia Institute of Technology September 6, 2018 In this Online Appenix, we examine an compare the entry-externality effect in the Melitz-Ottaviano (MO) an CES moels, respectively, as consiere by Bagwell an Lee (2018a,b). 1 Entry-externality Effects uner MO an CES In the MO moel an in the benchmark close-economy setting, aitional entry generates a positive externality if an only if α 2 c m D < 0 (1) where α is a preference parameter an c m D refers to the critical cut-off cost level in market equilibrium, a function of parameters other than α. As in Bagwell an Lee (2018a), we assume that α > c m D.1 To interpret the sign of (1), we establish now that (1) hols if an only if aitional entry raises the aggregate profit of the economy starting at market equilibrium, N E π NE =N N m > 0. (2) E E The necessary an suffi cient relationship between (1) an (2) can be shown by using the following equations from Bagwell an Lee (2018a), π = (c M ) k (c D ) k+2 2γ(k + 1)(k + 2) N E = 2 (1 + k) γ(c M ) k η (3) (α c D ) (c D ) k+1 (4) Authors aresses: kbagwell@stanfor.eu an seunghoon@econ.gatech.eu. 1 When we iscuss here the MO moel, the notation is unerstoo in the context of that moel as examine by Bagwell an Lee (2018a). 1

2 c D N E = ( N E c D ) 1 = ( 2(1 + k)γ(c M ) k (α(1 + k) kc D ) η(c D ) k+2 ) 1 < 0, (5) where c D refers to the critical cut-off cost level that is associate with a given selection for the level of entry, N E. In these expressions, c M is the scale parameter an k is the shape parameter of Pareto istribution G (c) = (c/c M ) k for c [0, c M ], an α, γ, η are preference parameters as escribe in the main paper. We observe from (4) that α > c m D is equivalent to N E m > 0. Using (3), we erive π = (c M ) k (c D ) k+2 N E 2γ(k + 1) (k + 2) (k + 2) c D c D = π N E Using (3), (4), (5), an (6), we evaluate (2) as follows: N E N E π ˆNE = ˆN m E (k + 2) c D π = π + N E NE =NE N m ( E (k + 2) c D = π 1 + N E c D N E ( α + 2 cd = π (α(1 + k) kc D ) c D. (6) N E ) NE =N m E ) NE =N m E, where c D = c m D when N E = NE m an where α(1 + k) kc D > 0 thus follows from α > c m D. This shows the necessary an suffi cient relationship state above. We may now summarize our fining for the MO moel as follows: starting at the market equilibrium, aitional entry generates a positive externality if an only if it raises the aggregate profit, N E N E π NE =NE m > 0. We now follow Bagwell an Lee (2018b) an examine the moel with CES preferences. 2 In the CES moel, firm-level prouctivity ϕ [1, ) is istribute accoring to a Pareto istribution with shape parameter k, so that G (ϕ) = 1 ϕ k. Uner CES preferences, we claim that N E N E π NE =NE m > 0 hols an this inequality offers a suffi cient conition for EXT > 0 at N E = NE m. To establish this claim, we procee in three steps. First, we observe that, if > π, then CS N E EXT = CS π π + N E > π + N E = N E π, (7) N E N E N E N E an so N E N E π woul then offer a lower boun for EXT. Secon, we show that CS N E > π in fact hols for N E > 0 an thus in particular for N E = NE m. Thir, we confirm that N E N E π NE =NE m > 0. The first step is immeiate from (7). To confirm the secon step, we use Bagwell an Lee s (2018b) fining that CS (σ 1)(1 θ) = N E σ(1 θ) 1 (P ) θ 1 θ N E ɛ ϕ,n E 2 When we iscuss here the CES moel, the notation is unerstoo in the context of that moel as examine by Bagwell an Lee (2018b). 2

3 where (P ) θ σ 1 θ = N E σ 1 k π π = (ϕ ) k (σ 1) f D 1 + k σ Υ (N E ) 1 = (ϕ ) k+1 σ + (1 θ)() σ(1 θ) 1, where Υ > 0 is a constant an where ɛ ϕ,n E ϕ N E N E ϕ can be compute from the last of these expressions. We may now confirm that ( ) CS (1 θ) kσ π = N E σ (1 θ) 1 ɛ ϕ,n E 1 π ( ) (1 + k σ) + (σ (1 θ) 1) + θ = π > 0 k (σ (1 θ) 1) + θ (σ 1) which shows that CS N E > π hols when N E > 0 an thus π > 0. Since NE m > 0, it thus follows that (7) hols at N E = NE m. For the thir step, we show that N E N E π NE =NE m > 0 hols regarless of parameters. In particular, for N E > 0, we fin that π k ϕ N E π = π + N E = π N E π N E N E ϕ N E = (1 k ɛ ϕ,n E ) π = 1 where the above inequality hols by k + 1 σ σ 1 + (1 θ) (σ 1) σ (1 θ) 1 = k σ 1 + k+1 σ k + (1 θ)() σ(1 θ) 1 θ σ (1 θ) 1 > π > 0 k σ 1. It thus follows that N E N E π NE =NE m > 0. Summing up, uner CES preferences, we have shown that N E N E π NE =NE m > 0 hols regarless of parameter values, an that this inequality in turn is a sufficient conition for EXT > 0 at N E = NE m. 2 Conitional Expecte Profit Since the sign of N E N E π NE =NE m seems to play an important role in both moels, we explore further why the two moels show ifferent implications. We rewrite aggregate profit as N π c where N refers to the number of operating firms an π c refers to expecte profit of a firm conitional on its survival. 3 3 We note that π c = π/ (1 G (ϕ )) in the CES moel an π c = π/g (c D ) in the MO moel, where the notation in each case is unerstoo in the context of the moel to which it is applie. 3

4 In both moels, we can separate two channels through which aitional entry affects aggregate profit: N π c = N π c + N π c. (8) N E N E N E In the MO moel, we fin that two opposing forces exist at N E = NE m : i) using N (3) an (5), aitional entry raises the number of operating firms, N E π c > 0; an ii) using (4) an (5), the surviving firms have lower profit ue to the higher competition, N πc N E < 0. However, in the CES moel, the secon channel is absent since the conitional expecte profit is constant, π c = (σ 1) f D / (1 + k σ). 4 This feature of constant conitional expecte profit provies some aitional insight into why the CES moel shows EXT > 0 regarless of parameter values. 3 Role of Enogenous Mark-up We now argue that constant expecte profit conitional on survival can be explaine by the role of constant markup. To make this argument, we follow Mrazova et al (2017) an consier the CREMR (Constant Revenue Elasticity of Marginal Revenue) family of eman functions efine as p cr (q) = β q (q ψ) σ. where β > 0, σ > 1 an q > ψσ. As Mrazova et al (2017) observe, this family inclues CES eman as a special case: when ψ = 0, the elasticity of eman is constant an equal to σ. We focus here on ψ 0. The associate profit-maximization problem is π cr (ϕ) = max { max q [( p cr (q) 1 ϕ ) q f D ], 0 where f D > 0 an the firm s prouctivity is ϕ. We assume that ϕ follows a Pareto istribution with shape parameter k (i.e. G (ϕ) = 1 ϕ k ) where 1 + k σ > 0. We enote the markup of a firm with prouctivity ϕ as } µ (ϕ) pcr (q cr (ϕ)) 1 ϕ p cr (q cr (ϕ)) = 1 σ ψϕ σ B where B β σ ( ) σ an q cr (ϕ) = ( σ β ϕ) σ +ψ refers to the equilibrium output level of a firm with prouctivity ϕ. If ψ > 0, then we have that a more effi cient firm charges a higher markup µ (ϕ) = ψ σ B ϕ > 0 (9) 4 The erivation of π c follows from the expression given above for π an from the fact that 1 G(ϕ ) = (ϕ ) k for the Pareto istribution. 4

5 as in the MO setup. By contrast, if ψ = 0, then the markup is constant with respect to prouctivity as in the CES moel. We calculate the conitional expecte profit with CREMR preferences as follows π cr c = ϕ π cr (ϕ) G (ϕ ϕ > ϕ ) = B = f D (σ 1) 1 σ + k + k B 1 σ + k (ϕ ) k ϕ µ (ϕ) ϕ G (ϕ ϕ > ϕ ) f D ϕ µ (ϕ) ϕ σ k 1 ϕ. (10) where the thir equality hols by integration by parts an the ZCP conition π cr (ϕ ) = 0 µ (ϕ ) (ϕ ) = f D B. The first term in (10) coincies with the conitional expecte profit uner CES preferences. The secon term in (10) shows the role of enogenous markups, a missing channel uner CES preferences. If ψ = 0 (i.e. µ (ϕ) = 0), then it is clear from (10) that π cr c is constant with respect to the critical prouctivity cutoff level, ϕ, as in the CES moel. By contrast, if ψ > 0 (i.e. µ (ϕ) > 0), then π cr c ecreases with ϕ. To establish this latter point, we substitute (9) into (10) an fin that π cr c = f D (σ 1) 1 σ + k + ψ k σ (1 σ + k) (k + 1) from which it irectly follows that, for ψ > 0, π cr c ϕ = ψ k σ (1 σ + k) (k + 1) 1 ϕ 1 (ϕ ) 2 < 0. If we were to embe this analysis into a moel of monopolistic competition ϕ an assume that more entry generates fiercer competition, N E > 0, as is stanar, then for ψ > 0 the expecte profit conitional on survival woul ecrease with aitional entry, πcr c N E < 0, as in the MO moel. For ψ = 0, by contrast, the expecte profit conitional on survival woul be constant with respect to π aitional entry, cr c N E = 0, as in the CES setup. From this perspective, one component of the business-stealing effect in (8) is eliminate uner CES preferences alone among all preferences in the CREMR family. This feature relates to our iscussion in the preceing subsection an provies some reinforcing insight into why the CES moel elivers EXT > 0 regarless of parameter values. 4 References Bagwell, K. an S. Lee (2018a), Trae Policy uner Monopolistic Competition with Firm Selection, manuscript. 5

6 Bagwell, K. an S. Lee (2018b), Trae Policy uner Monopolistic Competition with Heterogeneous Firms an Quasi-linear CES Preferences, manuscript. Melitz, M. J. an G. I. P. Ottaviano (2008), Market Size, Trae, an Prouctivity, The Review of Economic Stuies, 75, Mrazova, M., J. P. Neary an M. Parenti (2017), Sales an Markup Dispersion: Theory an Empirics, manuscript. 6