14.581 International Trade — Lecture 14: Firm Heterogeneity Theory (I) — Melitz (2003) 14.581 Week 8 Spring 2013 14.581 (Week 8) Melitz (2003) Spring 2013 1 / 42
Firm-Level Heterogeneity and Trade What’s wrong with previous theories? Nineties have seen a boom in the availability of micro-level data Problem: previous theories are at odds with (or cannot account for) many micro-level facts: Within a given industry, there is …rm-level heterogeneity 1 Fixed costs matter in export related decisions 2 Within a given industry, more productive …rms are more likely to export 3 Trade liberalization leads to intra-industry reallocation across …rms 4 These reallocations are correlated with productivity and export status 5 14.581 (Week 8) Melitz (2003) Spring 2013 2 / 42
Firm-Level Heterogeneity and Trade What does Melitz (2003) do about it? Melitz (2003) will develop a model featuring facts 1 and 2 that can explain facts 3, 4, and 5 This is by far the most in‡uential trade paper in the last 10 years Two building blocks: Krugman (1980): CES, IRS technology, monopolistic competition 1 Hopenhayn (1992): equilibrium model of entry and exit 2 From a normative point of view, Melitz (2003) may also provide “new” source of gains from trade if trade induces reallocation of labor from least to most productive …rms (more on that later) 14.581 (Week 8) Melitz (2003) Spring 2013 3 / 42
Today’s Plan A Refresher on Monopolistic Competition 1 Krugman (1979) 1 An Important Special Case: CES Utility 2 Melitz (2003) 2 Krugman (1980) meets Hopenhayn (1992) 1 Selection into Exports and the Impact of Trade 2 14.581 (Week 8) Melitz (2003) Spring 2013 4 / 42
Monopolistic Competition Basic idea Monopoly pricing: Each …rm faces a downward-sloping demand curve No strategic interaction: Each demand curve depends on the prices charged by other …rms but since the number of …rms is large, each …rm ignores its impact on the demand faced by other …rms Free entry: Firms enter the industry until pro…ts are driven to zero for all …rms 14.581 (Week 8) Melitz (2003) Spring 2013 5 / 42
Monopolistic Competition Graphical analysis p AC MC D MR q 14.581 (Week 8) Melitz (2003) Spring 2013 6 / 42
Krugman (1979) Endowments, preferences, and technology Endowments: All agents are endowed with 1 unit of labor Preferences: All agents have the same utility function given by U = R n 0 u ( c i ) di where: u ( 0 ) = 0, u 0 > 0, and u 00 < 0 (love of variety) σ ( c ) � � u 0 cu 00 > 0 is such that σ 0 � 0 (why?) IRS Technology: Labor used in the production of each “variety” is l i = f + q i / ϕ where ϕ � common productivity parameter 14.581 (Week 8) Melitz (2003) Spring 2013 7 / 42
Krugman (1979) Equilibrium conditions Consumer maximization: 1 p i = λ � 1 u 0 ( c i ) Pro…t maximization: 2 � � � w � σ ( c i ) p i = � σ ( c i ) � 1 ϕ Free entry: 3 � � p i � w q i = wf ϕ Good and labor market clearing: 4 q i = Lc i nf + R n q i L = ϕ di 0 14.581 (Week 8) Melitz (2003) Spring 2013 8 / 42
Krugman (1979) Equilibrium conditions rearranged Symmetry ) p i = p , q i = q , and c i = c for all i 2 [ 0 , n ] c and p / w are simultaneously characterized by � � 1 p σ ( c ) w = (PP): σ ( c ) � 1 ϕ w = f p q + 1 ϕ = f Lc + 1 (ZP): ϕ n can then be computed using market clearing conditions 1 n = f / L + c / ϕ 14.581 (Week 8) Melitz (2003) Spring 2013 9 / 42
Krugman (1979) Graphical analysis p/w Z P (p/w) 0 Z P c 0 c 14.581 (Week 8) Melitz (2003) Spring 2013 10 / 42
Krugman (1979) Gains from trade revisited p/w Z P Z’ (p/w) 0 (p/w) 1 Z P Z’ c 1 c 0 c Suppose that two identical countries open up to trade This is equivalent to a doubling of country size (which would have no e¤ect in a neoclassical trade model) Because of IRS, opening up to trade now leads to: 1 1 Increased product variety : c 1 < c 0 ) f / 2 L + c 1 / ϕ > f / L + c 0 / ϕ Pro-competitive/e¢ciency e¤ects : ( p / w ) 1 < ( p / w ) 0 ) q 1 > q 0 14.581 (Week 8) Melitz (2003) Spring 2013 11 / 42
CES Utility Trade economists’ most preferred demand system Constant Elasticity of Substitution (CES) utility corresponds to the case where: U = R n σ � 1 σ di , 0 ( c i ) where σ > 1 is the elasticity of substitution between pair of varieties This is the case considered in Krugman (1980) What is it to like about CES utility? σ � 1 Homotheticity ( u ( c ) � ( c ) is actually the only functional form σ such that U is homothetic) Can be derived from discrete choice model with i.i.d extreme value shocks (See Feenstra Appendix B) Is it empirically reasonable? 14.581 (Week 8) Melitz (2003) Spring 2013 12 / 42
CES Utility Special properties of the equilibrium Because of monopoly pricing, CES ) constant markups: � � 1 p σ w = σ � 1 ϕ Because of zero pro…t, constant markups ) constant output per …rm: w = f p q + 1 ϕ Because of market clearing, constant output per …rm ) constant number of varieties per country: L n = f + q / ϕ So, gains from trade only come from access to Foreign varieties IRS provide an intuitive reason why Foreign varieties are di¤erent But consequences of trade would now be the same if we had maintained CRS with di¤erent countries producing di¤erent goods 14.581 (Week 8) Melitz (2003) Spring 2013 13 / 42
CES Utility Special properties of the equilibrium Decentralized equilibrium is e¢cient Decentralized equilibrium solves: R n 0 p i ( q i ) q i di max q i , n nf + R n q i subject to : ϕ di � L . 0 A central planner would solve: R n σ � 1 σ di max 0 ( q i ) q i , n nf + R n q i subject to: ϕ di � L . 0 1 � 1 Under CES, p i ( q i ) q i ∝ q σ ) Two solutions coincide i This is unique to CES (in general, entry is distorted) This implies that many properties of perfectly competitive models will carry over to this environment 14.581 (Week 8) Melitz (2003) Spring 2013 14 / 42
Melitz (2003) Demand Like in Krugman (1980), representative agent has CES preferences: � Z � σ σ � 1 σ � 1 σ d ω U = ω 2 Ω q ( ω ) where σ > 1 is the elasticity of substitution Consumption and expenditures for each variety are given by � p ( ω ) � � σ q ( ω ) = Q (1) P � p ( ω ) � 1 � σ r ( ω ) = R (2) P where: � Z � Z 1 1 � σ ω 2 Ω p ( ω ) 1 � σ d ω P � , R � ω 2 Ω r ( ω ) , and Q � R / P 14.581 (Week 8) Melitz (2003) Spring 2013 15 / 42
Melitz (2003) Production Like in Krugman (1980), labor is the only factor of production L � total endowment, w = 1 � wage Like in Krugman (1980), there are IRS in production l = f + q / ϕ (3) Like in Krugman (1980), monopolistic competition implies p ( ϕ ) = 1 (4) ρϕ CES preferences with monopoly pricing, ( 2 ) and ( 4 ) , imply r ( ϕ ) = R ( P ρϕ ) σ � 1 (5) These two assumptions, ( 3 ) and ( 4 ) , further imply π ( ϕ ) � r ( ϕ ) � l ( ϕ ) = r ( ϕ ) � f σ 14.581 (Week 8) Melitz (2003) Spring 2013 16 / 42
Melitz (2003) Production Comments: Higher productivity ϕ in the model implies higher measured productivity 1 � � r ( ϕ ) l ( ϕ ) = 1 f 1 � ρ l ( ϕ ) More productive …rms produce more and earn higher revenues 2 � ϕ 1 � σ � ϕ 1 � σ � 1 q ( ϕ 1 ) and r ( ϕ 1 ) q ( ϕ 2 ) = r ( ϕ 2 ) = ϕ 2 ϕ 2 ϕ can also be interpreted in terms of quality. This is isomorphic to a 3 change in units of account, which would a¤ect prices, but nothing else 14.581 (Week 8) Melitz (2003) Spring 2013 17 / 42
Melitz (2003) Aggregation By de…nition, the CES price index is given by � Z � 1 1 � σ ω 2 Ω p ( ω ) 1 � σ d ω P = Since all …rms with productivity ϕ charge the same price p ( ϕ ) , we can rearrange CES price index as � Z + ∞ � 1 1 � σ p ( ϕ ) 1 � σ M µ ( ϕ ) d ϕ P = 0 where: M � mass of (surviving) …rms in equilibrium µ ( ϕ ) � (conditional) pdf of …rm-productivity levels in equilibrium 14.581 (Week 8) Melitz (2003) Spring 2013 18 / 42
Melitz (2003) Aggregation Combining the previous expression with monopoly pricing ( 4 ) , we get 1 1 � σ / ρ e P = M ϕ where � Z + ∞ � 1 σ � 1 ϕ σ � 1 µ ( ϕ ) d ϕ ϕ � e 0 One can do the same for all aggregate variables σ σ � 1 q ( e R = Mr ( e ϕ ) , Π = M π ( e ϕ ) , Q = M ϕ ) Comments: These are the same aggregate variables we would get in a Krugman 1 (1980) model with a mass M of identical …rms with productivity e ϕ But productivity e ϕ now is an endogenous variable which may respond 2 to changes in trade cost, leading to aggregate productivity changes 14.581 (Week 8) Melitz (2003) Spring 2013 19 / 42
Melitz (2003) Entry and exit In order to determine how µ ( ϕ ) and e ϕ get determine in equilibrium, one needs to specify the entry and exit of …rms Timing is similar to Hopenhayn (1992): There is a large pool of identical potential entrants deciding whether to 1 become active or not Firms deciding to become active pay a …xed cost of entry f e > 0 and 2 get a productivity draw ϕ from a cdf G After observing their productivity draws, …rms decide whether to 3 remain active or not Firms deciding to remain active exit with a constant probability δ 4 14.581 (Week 8) Melitz (2003) Spring 2013 20 / 42
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