Heterogeneous Firms. Notes for Graduate International Trade Lectures. J. Peter Neary. University of Oxford. February 3, 2015

Transcription

1 Heterogeneous Firms Notes for Graduate International Trade Lectures J. Peter Neary University of Oxford February 3, 2015 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

2 Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

3 Introduction Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

4 Introduction Empirical Background The Data Revolution: Micro-Data on Firms. 1990s onwards: Evidence totally orthogonal to traditional theory: Exporting firms are: Rare!: Very few firms export;... and those that do sell most of their output domestically; Larger; More productive:... ex ante ( selection into exporting ), not ex post ( learning by exporting ) (Clerides et al. QJE 1995; Bernard-Jensen JIE 1999). Older Pay higher wages Effects of trade liberalisation: Forces least productive firms to exit (Bernard-Jensen JIE 1999). Encourages market share reallocation towards more productive firms;... and so raises aggregate productivity (Pavcnik REStud 1999, Bernard-Jensen-Schott JIE 2006). J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

5 Introduction Dynamic Industry Equilibrium Monopolistic competition with CES preferences; so what s new? Melitz (2003), Helpman-Melitz-Yeaple (2004) [HMY] Population of ex ante identical firms; but heterogeneous ex post. Firms face two sources of uncertainty: 1 Uncertain productivity ϕ / cost c; quality another interpretation. Drawn from a known distribution with pdf g ( ) 1 c with positive support over (0, ) and associated cdf G ( ) ( 1 c. [g 1 ) ( ) c = G 1 c ] To learn its c, a firm must pay a sunk cost of entry f e. 2 Uncertain lifetime if it chooses to enter. Exogenous probability δ of death i.e., a bad shock that will cause it to exit. HMY and Chaney (AER 2008) ignore the second source, assuming that a successful entrant produces for one period only. This simplifies the model a lot without affecting its main predictions, and many authors have followed them; but it is insightful to solve the free entry case in full. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

6 Selection Effects I: First-Order Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

7 Selection Effects I: First-Order Operating Profits Consider a monopoly firm choosing whether or not to enter a market π (c): Maximum operating profits it can earn c: Exogenous cost parameter (inversely related to productivity) Often, though not always, equal to marginal production cost In some applications, an inverse indicator of quality Other determinants optimally chosen or exogenous; examples later Total profits: Π E = π(c) f E ; fixed cost f E independent of c J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

8 Selection Effects I: First-Order First-Order Selection Result Two assumptions about the maximum profit function: 1 π(c) is a continuous and strictly decreasing function in c 2 π(0) > f E and lim c + π(c) < f E Proposition There exists a c (0, + ) such that π(c ) = f E. For any c c, π(c) f E, and for any c 2 < c 1 c, π(c 2 ) > π(c 1 ) f E. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

9 Selection Effects I: First-Order Application: Which Firms Export? E = (c) f E E f E Exit 1 (c 0 ) Enter 1 c More productive firms select into exporting Very robust result: Not sensitive to CES Counter-examples can be explained in other ways: e.g., Lu (2011) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

10 Selection Effects I: First-Order Simple Result with Many Economic Applications This proposition can be applied to: A monopoly firm in partial equilibrium (time-series or cross-section) A monopolistically competitive industry (cross-section only) Very robust; it holds with: Arbitrary assumptions about demand Arbitrary distribution of variable costs Asymmetric countries Applications: 1 Selection into Production or Exports: Melitz (2003) Details 2 Selection into Marketing: Arkolakis (2010) Details 3 Selection into Worker Screening: Helpman-Itskhoki-Redding (2010) Skip Detailed Applications J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

11 Selection Effects I: First-Order Examples of First-Order Selection Effects 1 Selection into Exports: π (c) Max [ π (x; c)], x π (x; c) = {p (x) τc} x Envelope theorem π c = π c = τx < 0 ˆx = 2 ρĉ, ε 1 ˆπ = (ε 1)ĉ ε p, ρ xp : elasticity and convexity of demand xp p 2 Selection into Marketing: π (c) Max [ π (x, n; c) f(n)], π (x, n; c) = {p (x) c} nx x,n Arkolakis: CES preferences; f (n) = 1 (1 n)1 β 1 β, β (0, ), β 1 Envelope theorem π c = π c = nx < 0 ˆx = 2 ρĉ, ε 1 ε 1 ˆπ = (ε 1)ĉ, ˆn = ĉ nfnn J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65 fn

12 Competition Effects Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

13 Competition Effects Recap: Competition Effects and Subconvexity 4.0 Sub-Convex SC Super- Convex p(x) is subconvex at x 0 IFF: log p(x) is convex in log x p(x) is less convex than a CES demand function with the same elasticity: ρ > ε+1 ε ε is decreasing in sales: [ ε x = ε x ρ ε+1 ε ] Globalization [ x ] leads to competition effects [ p c ] Because the mark-up is decreasing in elasticity: p c = ε ε 1 = ε 1 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

14 Competition Effects Free-Entry Equilibrium with Heterogeneous Firms Maximum operating profits: π(c, λ, β) + c: Firm marginal cost λ: Market aggregate, endogenous to industry, exogenous to firms Interpretation: competition Appears naturally with additive separability (ZKPT, Bertoletti-Epifani) Also nests quasi-linear quadratic preferences (Melitz-Ottaviano) β: Exogenous shock to industry Free-entry condition: Zero-profit condition: π(λ, β) c 0 : c c π(c 0, λ, β) = f π(c, λ, β)g(c) dc = f + f e J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

15 Competition Effects Globalization, Competition and Profits β = k Determination of firm profits: ˆπ = λπ λ π ˆλ + kπ k ˆk π Determination of the level of competition: Combine: π(λ, k) = f + f e ˆλ = π ( ˆπ = kπk π }{{} (1) λπ λ π k π k π λ π λ π } {{ } (2) 1 Direct effect: Tends to raise each firm s profits k π k λ π λ π 2 Indirect effect: Raises all firms profits Increases competition Tends to reduce each firm s profits )ˆk ˆk J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

16 Competition Effects Globalization and the Extensive Margin With additive separability: ( ˆπ = 1 ε ε)ˆk ( ĉ0 = 1 ε ε 0 ε )ˆk Details ε 0 ε(c 0 ): Elasticity faced by the marginal firm ε c c π(c) π ε(c)g(c) dc : Elasticity faced by the average firm 1 CES case: ε 0 = ε = σ No change in threshold firm 2 Subconvex case: Marginal firm has lowest x and so highest ε Competition effect dominates: least efficient firms exit 3 Superconvex case: ε 0 < ε Threshold firm more profitable J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

17 Competition Effects Globalization and the Intensive Margin Firm output: y(c, λ, k) Prices: ŷ = [ ε+1 ερ ε(2 ρ) }{{} ˆp = ε ε + ε ε ε ε+1 ερ ε 2 (2 ρ) }{{} ε 1 ε(2 ρ)]ˆk Same as in homogeneous-firms case ( ), adjusted for ε ε Subconvexity: Outputs more, prices less in larger firms ˆk Details Recall J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

18 Competition Effects Example: Globalization with Linear Demands p i a Y k e y i 1 p( i) a k 1 e ey 2 y i 1 MR( i) a k 1 e ey c i y i y i Linear demands: Substitutability: 0 < e < 1 Market size effect: k Competition effect: Y J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

19 Matching the Size Distribution of Firms Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms CREMR Demands 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

20 Matching the Size Distribution of Firms The Size Distribution of Firms Like many data sets, obviously Pareto ) k x F (x) = 1 ( x x k: Shape or inverse dispersion x: Scale parameter Or is it obviously log normal? x F (x) = Φ ( ) ln x µ σ Or is Pareto OK for right-hand tail? The larger firms that matter... J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

21 Matching the Size Distribution of Firms Model Components Productivity Distribution Demands Technology Sales Distribution J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

22 Matching the Size Distribution of Firms Model Components Productivity Distribution Demands Technology Dixit-Stiglitz (f, c) Sales Distribution J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

23 Matching the Size Distribution of Firms Model Components Productivity Distribution Demands Sales Distribution J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

24 Matching the Size Distribution of Firms Model Components: The Benchmark Case Pareto Productivities CES Demands Pareto Sales J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

25 Matching the Size Distribution of Firms Fitting the Data Pareto Productivities + CES Demands Pareto Sales Helpman-Melitz-Yeaple (2004), Chaney (2006) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

26 Matching the Size Distribution of Firms Fitting the Model Pareto Productivities + CES Demands Pareto Sales Pareto Productivities CES Demands + Pareto Sales J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

27 Matching the Size Distribution of Firms Backing Out Demand Pareto Productivities + CES Demands Pareto Sales Pareto Productivities CES Demands + Pareto Sales Pareto Productivities? Pareto Sales J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

28 Matching the Size Distribution of Firms Which Demands are Consistent with Pareto? Pareto Productivities Pareto Sales CREMR Demands Proposition: Any two imply the third J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

29 Matching the Size Distribution of Firms CREMR Demands CREMR Demands Constant Revenue Elasticity of Marginal Revenue demand function: p(x) = β (x γ)1 θ x CES a special case: γ = 0 p(x) = βx θ, θ = 1/σ CREMR : ˆr = θ 1 θ ˆr Details J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

30 Matching the Size Distribution of Firms CREMR Demands Proposition 1: Intuition Pareto distribution of firm productivities: ( φ ) k G(φ) = 1 g(φ) = k φ (k+1) φ 0 ĝ = (k + 1) ˆφ i.e., Pareto constant elasticity of density φ k 0 MC=MR: ˆφ = ĉ = ˆr CREMR demands: ˆr = θ 1 θ ˆr Implied distribution of sales: θ ˆ g = (k + 1) 1 θ ˆr i.e., constant elasticity of density Pareto J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

31 Matching the Size Distribution of Firms CREMR Demands CREMR Very Different from Other Demands 4.0 SM = 0.5 = 0.3 SC = SM Translog 3.0 = SC Linear = CARA Stone-Geary p(x) = β (x γ)1 θ x ρ(ε) = 2 θ (ε 1) 2 1 θ ε J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

32 Matching the Size Distribution of Firms CREMR Demands CREMR Inconsistent with a Choke Price p(x) = β (x γ)1 θ x lim p(x) = x 0 If both productivity and output follow a Pareto distribution, then demand cannot have a choke price. Most non-ces models of heterogeneous firms assume a choke price: Melitz-Ottaviano (2008) Feenstra-Weinstein (2010) Simonovska (2010) Arkolakis-Costinot-Davidson-Rodriguez-Clare (2012) Feenstra (2014) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

33 Matching the Size Distribution of Firms CREMR Demands CES versus Linear Demands Predicted Size Distributions of Firms from CES and Linear Demands J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

34 Selection Effects II: Second-Order 7J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65 Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order Exports versus FDI Sub- and Supermodularity Exports versus FDI and Supermodularity Supermodularity and Demand 6 Conclusion

35 Selection Effects II: Second-Order Exports versus FDI The Proximity-Concentration Trade-Off Ex post profits: π (t, c) t: Iceberg trade cost c: Marginal production cost, as before Firm has two options: Proximity: FDI avoids the trade cost t Π F = π (0, c) f F Concentration: Exports incur a lower fixed cost: f E < f F Π E = π (t, c) f E J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

36 Selection Effects II: Second-Order Exports versus FDI Exports versus FDI E = (t,c) f E F E F = (0,c) f F f E 1 c f F J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

37 Selection Effects II: Second-Order Exports versus FDI Which Firms Export and Which Engage in FDI? F E 1 ce 1 c F 1 c f E Exit Exports FDI f F Helpman-Melitz-Yeaple (2004) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

38 Selection Effects II: Second-Order Exports versus FDI First- versus Second-Order Selection Effects First-Order Selection Effects Second-Order Selection Effects F E 1 ce 1 c S 1 c F 1 c f E Exit Exports FDI f F Second-order selection effects less robust Only guaranteed if Π F is steeper than Π E This requires that π(c, t) is supermodular J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

39 Selection Effects II: Second-Order Sub- and Supermodularity Supermodularity Definition If the function π (t, c) is differentiable, then it is supermodular in t and c if and only if π tc 0 Intuitively: A higher tariff reduces in absolute value the profit disadvantage of a higher-cost firm. The Matthew Effect : To those who have, more shall be given : A lower tariff is of more benefit to a firm with more sales A lower-cost (more productive) firm usually has higher sales... though not always... Analogous to Hicksian complementarity in consumer theory or strategic complementarity in game theory Supermodularity can be extended to the case where π is not differentiable in t and c: see Mrázová and Neary (2011) Skip examples J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

40 Selection Effects II: Second-Order Sub- and Supermodularity Example 1: Marginal Cost Independent of Output Constant marginal cost c Specific (not iceberg) tariff t Inverse demand function: p (x) So, firm s operating profits equal: π (t, c) Max [{p(x) c t} x] x J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

41 Selection Effects II: Second-Order Sub- and Supermodularity Example of Supermodularity p c 1 c 2 x Less productive firm has higher marginal cost FigEx1a.pdf And therefore... J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

42 Selection Effects II: Second-Order Sub- and Supermodularity Example of Supermodularity p c 1 c 2 MR x 1 x 2 x Less productive firm has higher marginal cost FigEx1b.pdf And therefore lower output, so it benefits less from a tariff reduction. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

43 Selection Effects II: Second-Order Sub- and Supermodularity Example 2: Marginal Cost Varies with Output Suppose instead that marginal cost varies with output: π (t, c) Max [{p(x) t} x C (c, x)] x C (c, x): Total variable costs; C c > 0, C x > 0 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

44 Selection Effects II: Second-Order Sub- and Supermodularity Example of Submodularity p AC 1 AC 2 x Less productive firm is relatively more productive at higher output FigEx2a.pdf So much so that... J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

45 Selection Effects II: Second-Order Sub- and Supermodularity Example of Submodularity p MC 2 AC 1 MC1 AC 2 x Less productive firm is relatively more productive at higher output FigEx2b.pdf So much so that it has lower marginal cost... J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

46 Selection Effects II: Second-Order Sub- and Supermodularity Example of Submodularity p MC 2 AC 1 MC1 AC 2 MR x 2 x 1 x Less productive firm is relatively more productive at higher output FigEx2c.pdf So much so that it has lower marginal cost and higher output J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

47 Selection Effects II: Second-Order Exports versus FDI and Supermodularity Second-Order Selection Effects and Supermodularity In general: Second-order selection effects guaranteed if Π F is steeper than Π E ΠF dc < ΠE dc π c (0, c) < π c (t, c) π ct > 0, i.e., π(t, c) is supermodular In the special case of multiplicative costs : π(t, c) max [p(x) tc] x π tc > 0 Elasticity of Marginal Revenue (EMR) is less then one Both of these are equivalent to: ε + ρ > 3 Proofs J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

48 Selection Effects II: Second-Order Supermodularity and Demand Supermodularity Intuition (a): CES Skip MR highly inelastic: Same elasticity as the inverse demand function: 1/σ Here: EMR = 1/4 This implies a very large response of sales to costs: More productive firms sell a lot more than less productive ones, and so gain more from a reduction in trade costs, enjoying a strong Matthew Effect. Hence the profit function is always supermodular p MR(x) 0 tc p( x) x J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65 1/ x

49 Selection Effects II: Second-Order Supermodularity and Demand Supermodularity Intuition (b): Linear Both demand and MR are linear Both elasticities (as a function of sales) rise as sales rise. p 0 tc Profits must be submodular for very efficient firms: c 0 τc 0 no gain to FDI EMR equals one, and so profits switch from super- to sub-modularity, at half the maximum output. MR(x) p( x) x x So, profits are submodular for larger lower-cost exporters. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

50 Selection Effects II: Second-Order Supermodularity and Demand Supermodularity Intuition (c): Inverse Translog Used in both: Theory: Diewert (1976) Empirics: Agricultural economics Necessary and sufficient for π to be modular in (t, c) at all levels of output MR(x) = β x : EMR = 1 So total variable costs are the same for all firms, irrespective of tariffs Hence, no selection will be observed. p MR(x) 0 tc 1 p( x) x log x x J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

51 Selection Effects II: Second-Order Supermodularity and Demand Summary: Supermodularity and Demand p 0 tc p 0 tc p 0 tc 1/ p( x) x p( x) x ( 1 x) x log p x MR(x) MR(x) MR(x) x x x (a) CES (b) Linear (c) Inverse Translog Selection into FDI by large firms only guaranteed if EMR < 1 CES: EMR < 1: 10% fall in c > 10% rise in sales So more efficient firms have higher profits when they engage in FDI Linear demands: EMR > 1 for larger firms: Reverse selection effects Inverse translog demands: EMR = 1 No selection effects General Condition J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

52 Selection Effects II: Second-Order Supermodularity and Demand Second-Order Selection Effects: Supermodularity Lower-c firms choose FDI IF: π(t, c) supermodular in {t, c} Elasticity of MR < 1 ε + ρ > 3 Clearly: SC SM Recall Demand Functions SM SPT SC 4.0 Super- 3.0 Modular 2.0 Sub- Modular J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

53 Selection Effects II: Second-Order Supermodularity and Demand Can We Have Subconvexity and Supermodularity? 4.0 SM 3.0 SC LES Linear CARA SPT Demand functions represented in {ε, ρ} space by their Demand Manifold Most common demand functions are: Subconvex Competition effects Submodular for high output Reverse selection effects J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

54 Selection Effects II: Second-Order Supermodularity and Demand Can We Have Subconvexity and Supermodularity? 4.0 SM SC AIDS SPT Demand functions represented in {ε, ρ} space by their Demand Manifold Most common demand functions are: Subconvex Competition effects Submodular for high output Reverse selection effects Exception: AIDS/Translog J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

55 Selection Effects II: Second-Order Supermodularity and Demand Postscript: Other Examples of Multiplicative Costs Formally identical: Horizontal FDI: t = τ π(t, c) max [p(x) tc], t 1 x Vertical FDI: t = w [Antràs-Helpman (2004)] w N > w S Choice of Technique: t is penalty for being low-tech [Bustos (2011)] In all cases: Conventional selection effects guaranteed only if π tc > 0 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

56 Conclusion Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

57 Conclusion Conclusion We have learnt so much from recent work on heterogeneous firms but there is lots more to learn! Alternatives to CES? Alternatives to Pareto? And monopolistic competition can never have the last word on superstar firms J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

58 Conclusion Thanks and Acknowledgements* Thank you for listening. Comments welcome! * The research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ ), ERC grant agreement no The contents reflect only the authors views and not the views of the ERC or the European Commission, and the European Union is not liable for any use that may be made of the information contained therein. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

59 Supplementary Material Plan of Lectures 1 Introduction 2 Selection Effects I: First-Order 3 Competition Effects 4 Matching the Size Distribution of Firms 5 Selection Effects II: Second-Order 6 Conclusion 7 Supplementary Material J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

60 Supplementary Material Additive Separability: Detailed Derivations Calculate the elasticities of profits, using the envelope theorem: [ ] π(c, λ, k) max p(y, λ, k) c y where: p(y, λ, k) = λ 1 u (y/kl) y π c = y, π λ = λ 2 u y = λ 1 py, π k = y2 u λk 2 L = y2 k p y Back to Text So: cπ c π = cy π = c λπ p c = (ε 1) λ π = py π = p p c = ε kπ k π = y2 p y (p c)y = ypy yp y = 1 Similarly for the elasticities of aggregate profits: λ π λ π = c k π k π c π(c) π = c c λπ λ (c) π(c) π(c) π g(c) dc = c kπ k (c) π(c) c g(c)dc = c c π(c) π ε(c)g(c) dc = ε π(c) π g(c)dc = 1 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

61 Supplementary Material Globalization and the Intensive Margin: Details Firm output: Back to Text Prices: ŷ = ( kyk y + λy λ y y(c, λ, k) = π c (c, λ, k) k λ ŷ = ˆx = ŷ ˆk = ) ( dλ dk ˆk = kyk y + λy λ y [ ε+1 ερ ε(2 ρ) + ε ε ε ) ( ) 1 ε ˆk = 1 ε ρ ε ˆk ε 1 ε(2 ρ)] ˆk ˆp = ε+1 ερ ε(ε 1) ˆx ( ) 1 ε ρ ε 1 ˆk = ε 1 1 ˆk 2 ρ ε ˆp = ε+1 ερ ε(ε 1) ε 1 2 ρ 1 ˆk ε = ε+1 ερ ε ε(2 ρ) ˆk = ε ε ε+1 ερ ˆk ε 2 (2 ρ) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

62 Supplementary Material CREMR Demands Back to text Proof that CREMR has Constant Revenue Elasticity of Marginal Revenue: r(x) = β (x γ) 1 θ ˆr = (1 θ) x ˆx } x γ r (x) = β(1 θ) (x γ) θ ˆr = θ x ˆx ˆr = θ 1 θ ˆr x γ J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

63 Supplementary Material Supermodularity with Multiplicative Costs: Proofs π(t, c) max x [p(x) tc] x π t = π t = cx Back π tc = x c dx dc = x tc 2p +xp x = ε+ρ 3 2 ρ Elasticity of marginal revenue (EMR) ε MR,x : r(x) xp(x) = x + ε 1 2 ρ x r = xp + p = xp (1 ε) > 0 r = 2p + xp = p (2 ρ) < 0 ε MR,x xr r = 2 ρ ε 1 ε MR,x 1 = ε + ρ 3 ε 1 J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

64 Related Literature Supplementary Material 1 Selection effects: Bertoletti-Epifani (2014), Mrázová-Neary (2011), Bache-Laugesen (2013) 2 Competition effects: ZKPT (2013), Bertoletti-Epifani (2014) Alternatives to CES: Quadratic preferences: Melitz-Ottaviano (2008) Stone-Geary LES: Simonovska (2010) Translog: Feenstra-Weinstein (2010) Negative exponential/cara: Behrens-Murata (2007) Bulow-Pfleiderer: Atkin-Donaldson (2012) QMOR: Feenstra (2014) 3 Matching the size distribution of firm sales: Pareto: Helpman-Melitz-Yeaple (2004), Chaney (2006) Mixture of thin- and fat-tailed Pareto: Edmonds et al. (2012) Log-normal: Head-Mayer-Thoenig (2014), Bee-Schiavo (2014) Piecewise log-normal-pareto: Luttmer (2007), Eaton et al. (2011) J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65

65 Supplementary Material Background Mrázová, M., and J. P. Neary (2011): Selection Effects with Heterogeneous Firms, Discussion Paper No. 588, Department of Economics, University of Oxford. (2013): Not So Demanding: Preference Structure, Firm Behavior, and Welfare, Discussion Paper No. 691, Department of Economics, University of Oxford. Mrázová, M., J. P. Neary, and M. Parenti (2014): Demand, Technology, and the Size Distribution of Firms, in preparation. J.P. Neary (University of Oxford) Heterogeneous Firms February 3, / 65