Heterogeneous Firms. Notes for Graduate Trade Course. J. Peter Neary. University of Oxford. January 30, 2013

Transcription

1 Heterogeneous Firms Notes for Graduate Trade Course J. Peter Neary University of Oxford January 30, 2013 J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

2 Plan of Lectures 1 Empirical Background 2 Overview of the Melitz Model 3 Equilibrium in Autarky 4 Effects of Trade 5 Extensions J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

3 Plan of Lectures Empirical Background 1 Empirical Background 2 Overview of the Melitz Model 3 Equilibrium in Autarky 4 Effects of Trade 5 Extensions J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

4 Empirical Background Empirical Background The Data Revolution: Micro-Data on Firms. 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 and 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 January 30, / 29

5 Plan of Lectures Overview of the Melitz Model 1 Empirical Background 2 Overview of the Melitz Model Dynamic Industry Equilibrium Continuum CES Preferences 3 Equilibrium in Autarky 4 Effects of Trade 5 Extensions J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

6 Overview of the Melitz Model Dynamic Industry Equilibrium 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. 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 ( 1c ) = 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 January 30, / 29

7 Overview of the Melitz Model Continuum CES Preferences Continuum CES Preferences CES preferences [ with a continuum of goods: little new: 1/θ, U = ω Ω dω] q (ω)θ 0 < θ = σ 1 σ < 1, σ = 1 1 θ > 1 Optimal consumption: q (ω) = Optimal expenditure: r (ω) p (ω) q (ω) = R = PQ = ω Ω r (ω) [ dω Price index: P = ω Ω p (ω)1 σ dω [ ] p(ω) σ P Q Q = U ] 1 1 σ [ ] p(ω) 1 σ P R J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

8 4 Effects of Trade J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29 Plan of Lectures Equilibrium in Autarky 1 Empirical Background 2 Overview of the Melitz Model 3 Equilibrium in Autarky Production Entry Equilibrium Selection in Autarky: Figure Productivity of Entrants Average Productivity, Prices and Profits ZCP (Zero Cutoff Profit) Condition FE (Free Entry) Condition Industry Equilibrium: Figure Equilibrium in Autarky: Recap General Equilibrium in Autarky: Details

9 Production Equilibrium in Autarky Production Firms have different productivities ϕ (inverse of variable costs c). All infra-marginal firms make positive profits; otherwise little new: Labour the only factor, with w = 1: TC (c) = f + cq (c) Profit maximisation implies: p (c) = σ 1 σ c, q (c) = ( σ σ 1 c) σ P σ 1 R Price-cost margin: p (c) c = σ 1 1 c = 1 σ p (c) Revenue: r (c) = ( σ σ 1 c) 1 σ P σ 1 R = Variable profit: ( σ 1 σ 1 c P ) σ 1 R ( ) σ 1 r (c) cq (c) = [p (c) c] q (c) = 1 σ r (c) = σ 1 1 σ c P Rσ Total profit: ( ) σ 1 π (c) = 1 σ r (c) f = σ 1 1 σ c P Rσ f Higher productivity firms have higher output, revenue and profits. They also charge lower prices. Ratios (rankings) of p, q, and r depend only on productivities: p(c 1 ) p(c 2 ) = c 1 c 2 q(c 1 ) q(c 2 ) = ( c2 c 1 ) σ and r(c 1 ) r(c 2 ) = ( c2 c 1 ) σ 1 J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

10 Entry Equilibrium in Autarky Entry To be or not to be? Depends on firm s expected value given ϕ: { V (ϕ) = max 0, } } (1 δ) t π (ϕ) = max {0, 1 δ π (ϕ). 0 This is steady-state analysis; no learning-by-doing etc. Probability of death acts just like a discount factor. So: Entry occurs IFF V (ϕ) 0 π (ϕ) 0 ϕ ϕ where: ϕ : π (ϕ ) = 0 (1) So far, very like homogeneous-firms model; but there: π = 0 holds for all firms (since they are homogeneous); This pins down q for all firms... and market-clearing pins down mass of firms n. Here: π (ϕ ) = 0 is one equation in ϕ and (through P) n. It determines q (ϕ ) but there are many other q (ϕ)... So: we need more equations... J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

11 Equilibrium in Autarky Equilibrium Selection in Autarky Equilibrium Selection in Autarky: Figure c f (c * ) 1 c 1 1 Exit Enter J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

12 Equilibrium in Autarky Productivity of Entrants Productivity of Entrants Equilibrium distribution of firm productivities: Ex ante, it continues to be g (ϕ). Ex post, distribution of entrants productivities is different: µ (ϕ). Probability of a bad draw is G (ϕ ); so probability of entry is 1 G (ϕ ) Hence distribution of ϕ on [ϕ, ), conditional on successful entry, is: µ (ϕ) = { g(ϕ) 1 G (ϕ ) if ϕ ϕ 0 otherwise (2) Aggregate price: P = where: [ 0 p (ϕ)1 σ Mµ (ϕ) d ϕ] 1 1 σ = M 1 σ 1 [ p ( ϕ) M: mass of firms. ϕ (ϕ ) ϕ ϕ σ 1 1 σ 1 µ (ϕ) d ϕ] Average productivity; (strictly, a weighted symmetric mean ). J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

13 Equilibrium in Autarky Average Productivity, Prices and Profits Average Productivity, Prices and Profits Proof that P = M 1 σ 1 p ( ϕ) [ P = ω Ω p (ω)1 σ dω p(ϕ 1 ) p(ϕ 2 ) = ϕ 2 ] 1 σ 1 = [ 0 p (ϕ)1 σ Mµ (ϕ) d ϕ] 1 1 σ Recall: ϕ 1 so: p (ϕ) = p ( ϕ) ϕ [ ϕ P = M 1 σ 1 ] 1 ϕ1 σ 1 σ 0 p ( ϕ)1 σ µ (ϕ) d ϕ ϕ 1 σ = M 1 σ 1 p ( ϕ) ϕ [ 0 ϕσ 1 µ (ϕ) d ϕ ] 1 σ 1 = M 1 σ 1 p ( ϕ) from defn. of ϕ (Careful with exponents!) QED ϕ is a sufficient statistic for the industry. In particular: Average profits π = π ( ϕ); Proof: π ϕ π (ϕ) µ (ϕ) d ϕ; BUT: π (ϕ) = 1 σ r (ϕ) f = 1 ( ) σ ϕ r (ϕ) µ (ϕ) d ϕ f ; BUT: r(ϕ) σ 1 r( ϕ) = ϕ ϕ = 1 σ r ( ϕ) ( 1 ϕ ) σ 1 ϕ ϕ σ 1 µ (ϕ) d ϕ f ; BUT: Recall defn. of ϕ. = 1 σ r ( ϕ) f QED J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

14 Equilibrium in Autarky ZCP (Zero Cutoff Profit) Condition ZCP (Zero Cutoff Profit) Condition Given π = 1 σ r ( ϕ) f and ϕ (ϕ ), π is a function of ϕ. [ ] r( ϕ) It s even neater than that, recalling that r(ϕ ) = ϕ(ϕ ) σ 1; ϕ [ ϕ(ϕ ) ] σ 1 π = 1 σ r (ϕ ) ϕ f ; BUT: π (ϕ ) = 1 σ r (ϕ ) f = 0; [ { ϕ (ϕ π = } ) σ 1 ϕ 1] f [ZCP] (3) Slope in { π, ϕ } space depends on two competing effects: 1 Higher ϕ raises average productivity, and so profits, of surviving firms: ϕ (ϕ ) > 0. A selection effect, not a firm-level productivity effect 2 Higher ϕ means tougher competition: profits are decreasing in rivals productivity. (2) dominates for many distributions: ZCP downward-sloping. e.g., fat-enough tails: lognormal, exponential, etc. (1) and (2) exactly cancel for Pareto: G (ϕ) = 1 ( b ϕ ) k: ZCP flat. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

15 Equilibrium in Autarky FE (Free Entry) Condition FE (Free Entry) Condition Expected PV of profits must equal sunk cost of entry: 0 v (ϕ) g (ϕ) d ϕ = f e; BUT: v (ϕ) = 1 δ π (ϕ) for ϕ ϕ ; otherwise = 0; 1 δ ϕ π (ϕ) g (ϕ) d ϕ = f e ; BUT: Recall distribution of entrants productivities; 1 δ [1 G (ϕ )] ϕ π (ϕ) µ (ϕ) d ϕ = f e ; BUT: Recall defn. of π; π = δf e 1 G (ϕ ) [FE] (4) So: Average industry profits rise with δ (think Grim Reaper), f e (think larger firms) and ϕ (higher productivity cutoff). (3) and (4): Two equations in two unknowns, π and ϕ ; So they are determined by f e, f, δ, and g (ϕ) only. Melitz shows that FE must be cut by ZCP only once from above; So equilibrium is unique and stable. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

16 Equilibrium in Autarky Industry Equilibrium in Autarky Industry Equilibrium: Figure FE f e ZCP * J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

17 Equilibrium in Autarky Equilibrium in Autarky: Recap Equilibrium in Autarky: Recap The story so far: p (ϕ) = σ 1 σ 1 ϕ P = M 1 σ 1 p ( ϕ) = M 1 σ 1 π (ϕ) = 1 σ r (ϕ) f r (ϕ) = r ( ϕ) = ( ) σ 1 σ 1 σ ϕp R ( σ 1 σ ϕp ) σ 1 R = (M 1 1 σ π = 1 σ r ( ϕ) f ) σ 1 R = R M σ σ 1 1 ϕ This implies that: π = 1 R σ M f σ, f given; π and ϕ determined by (3) and (4). Finally: R determined by aggregate budget constraint: R = wl = L. N.B. This is the first, and only, place where GE appears: see next page. BUT: it is crucial: without induced rise in real wage in GE, the selection effects of trade would not arise in the model. Only remaining unknown is M, which is therefore: M = L ( π+f )σ. Compare Krugman: M = L cy+f from LME; = L f σ Since y = (σ 1) f c cy + f = (σ 1)f f. from CES. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

18 Equilibrium in Autarky General Equilibrium in Autarky: Details General Equilibrium in Autarky: Details Stationary equilibrium: Aggregate variables stay constant. So, mass of new entrants each period M e must be such that mass of successful entrants equals mass of exiters: [1 G (ϕ )] M e = δm. Successful entrants and failing incumbents have the same productivity distribution; So: equilibrium distribution µ (ϕ) is not affected by simultaneous entry and exit. Finally, how come R = L? What happened to profits? The trick is that sunk entry costs also require labour. So: L = L p + L e ; aggregate labour used for production and investment (in entry). But: wl p = R Π; while: wl e = wm e f e = w δm 1 G (ϕ ) f e = M π = Π So, with w = 1, L = R Π + Π = R. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

19 5 Extensions J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29 Plan of Lectures Effects of Trade 1 Empirical Background 2 Overview of the Melitz Model 3 Equilibrium in Autarky 4 Effects of Trade Trade Costs Firms in Home and Export Markets Equilibrium Selection in Trade: Figure The ZCP Locus with Trade Industry Equilibrium in Autarky and Trade: Figure Adjustment to Trade Liberalisation Comparing Trade and Autarky Comparing Trade and Autarky (cont.)

20 Trade Costs Effects of Trade Trade Costs Now: replicate the home economy without trade costs: n: # foreign countries; all identical, so FPE holds, w = 1 in all. Budget constraint is now: R = (n + 1) L. All firms will export, trade does not affect average productivity. (3) and (4) continue to determine the same equilibrium π and ϕ. [Think Krugman 1979: n + 1 > 0 ˆM = n + 1.] So, we need trade costs; 2 kinds in fact: 1 Iceberg variable costs: τ 2 Fixed cost of exporting f x ; incurred after firm learns its ϕ. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

21 Effects of Trade Firms in Home and Export Markets Firms in Home and Export Markets Firms charge constant mark-ups in both domestic and export markets. ( ) σ 1 So: Domestic revenue: r d (ϕ) = σ 1 σ ϕp R as before ( ) σ 1 Revenue in export market j: r X (ϕ) = τ 1 σ σ 1 σ ϕp j Rj Symmetry Total exporter revenue: ( 1 + nτ 1 σ) r d (ϕ) Profits on domestic sales: π d (ϕ) = r d (ϕ) σ f. This = 0 determines threshold ϕ as before. Profits in an export market: π X (ϕ) = r X (ϕ) σ f X = τ1 σ r d (ϕ) σ f X. This = 0 determines a new threshold ϕx. r Threshold ratio: d(ϕx ) r d (ϕ ) = τσ 1 f Xf BUT: r d(ϕx ) r d (ϕ ) ( ) 1 fx σ 1 ϕ ϕ X = τ i.e., sorting as in data (ϕ X > ϕ ) IFF τ f ( = ϕ ) σ 1 X ϕ (5) ( fxf ) 1 σ 1 > 1 τ σ 1 f X > f. We assume this holds from now on. (A little unsatisfactory... ) J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

22 Effects of Trade Equilibrium Selection in Trade Equilibrium Selection in Trade: Figure c x c (c * ) 1 * ( c 1 x ) c 1 1 f f f x Home Sales Only Export Exit Enter J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

23 Effects of Trade The ZCP Locus with Trade The ZCP Locus with Trade Probability of entry = 1 G (ϕ ) as before; Probability that an entrant exports: p X = 1 G(ϕ X ) 1 G (ϕ ) Average expected profits of an entrant: π = π d ( ϕ) + p X nπ X ( ϕ X ) Here: ϕ X is the weighted average productivity of exporters; so: [ { π(ϕ ϕ(ϕ )= } ] [ σ 1 { ) ϕx (ϕ ϕ 1 f +p X n } σ 1 ) 1] ϕ X (ϕ ) f X [ZCP t ] (6) This is clearly greater than in autarky; i.e., π (ϕ ) > π a (ϕ ) for any arbitrary ϕ. i.e., ZCP shifts up relative to autarky. Finally: FE locus is unaffected. So: ϕ > ϕ a and π (ϕ ) > π a (ϕ a) J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

24 Effects of Trade Industry Equilibrium in Autarky and Trade: Figure Industry Equilibrium in Autarky and Trade ( * a ) FE ) a ( * t a ( * a ) ZCP - Trade f e * * a t ZCP - Autarky * J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

25 Effects of Trade Adjustment to Trade Liberalisation Adjustment to Trade Liberalisation Result that ϕ > ϕ a matches the data: Trade causes marginal firms to exit; Selection effect of trade; Why? NOT a competition effect through demand side. Remember: CES: Firm size and therefore π fixed by costs. Ans.: Labour market adjustment is crucial (though in the background). Increase in profitable opportunities for relatively more productive firms more entry Increase in labour demand Increase in real wage w P ; i.e. fall in P Least productive firms in autarky can no longer make profits. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

26 Effects of Trade Comparing Trade and Autarky Comparing Trade and Autarky Mass of active home firms falls: M < M a ; proof: Total revenue of domestic producers: R = L Average revenue: r = 0 r (ϕ) µ (ϕ) d ϕ = σ ( π + f + p X nf X ) BUT: R = M r M = L σ( π+f +p X nf X ) < M a (Recall that M a = L σ( π a +f ) and π > π a. What about total number of varieties? # firms selling in any one market: M t = (1 + np X ) M Likely (except for very high τ) that M t > M a i.e., gains from variety. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

27 Effects of Trade Comparing Trade and Autarky (cont.) Comparing Trade and Autarky (cont.) ( ) ϕ > ϕa r d (ϕ) = ϕ σ 1 ( ) σ ϕ f < ra (ϕ) = ϕ σ 1 σ ϕ f a i.e., all firms earn less on home market. However: r a (ϕ) < r d (ϕ) + nr X (ϕ) for ϕ > ϕx (harder to prove... ) i.e., exporting firms gain. J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

28 Plan of Lectures Extensions 1 Empirical Background 2 Overview of the Melitz Model 3 Equilibrium in Autarky 4 Effects of Trade 5 Extensions J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29

29 Extensions Extensions Quadratic preferences Variable mark-ups, competition effects: Melitz-Ottaviano (REStud 2007); but, partial equilibrium. Combine with HO: Bernard-Redding-Schott (REStud 2008). Quality: Baldwin-Harrigan (AEJ 2011) and many more: Predict that price rises not falls with productivity. Zeroes in the Trade Matrix: Helpman-Melitz-Rubinstein (QJE 2008): Link with gravity model; avoids prediction that every firm serves every export market. Other types of sorting: FDI: HMY (AER 2004) R&D: Bustos (AER 2011) Wages: Egger-Kreickemeier (IER 2009), Helpman-Itskhoki-Redding (Em 2010). In all cases: Trade-off between fixed and variable costs; Only more productive firms select into higher fixed-cost activity. Or, is that true? See next file and Mrázová-Neary (2012)... J.P. Neary (University of Oxford) Heterogeneous Firms January 30, / 29