ECO2704 Lecture Notes: Melitz Model

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1 ECO2704 Lecture Notes: Melitz Model Xiaodong Zhu University of Toronto October 15, / 22

2 Dynamic Industry Model with heterogeneous firms where opening to trade leads to reallocations of resources within an industry Opening to trade leads to Reallocations of resources across firms Low productivity firms exit High productivity firms expand so there is a change in industry composition High productivity firms enter export markets Improvements in aggregate industry productivity No change in firm productivity Consistent with empirical evidence from trade liberalizations? 2 / 22

3 The theoretical model is consistent with a variety of other stylized facts about industries Heterogeneous firm productivity Ongoing entry and exit Co-movement in (gross) entry and exit due to sunk entry costs Exiting firms are low productivity (selection effect) Explains why some firms export within industries and others do not Contrast with traditional theories of comparative advantage Exporting firms are high productivity (selection effect) No feedback from exporting to productivity 3 / 22

4 Single factor: labor, numeraire, wage normalized to 1 Firms enter market by paying sunk entry cost (f e ) Firms observe their productivity (ϕ) from distribution G(ϕ) Productivity is fixed thereafter Once productivity is observed, firms decide whether to produce or exit Firms produce horizontally-dierentiated varieties, with a fixed production cost (f d ) and a variable cost that depends on their productivity Firms face an exogenous probability of death (δ) per period due to force majeure events 4 / 22

5 Preferences and demand [ˆ ] σ Q = q(k) σ 1 σ 1 σ dk,σ > 1 k Ω Let R be the total expenditures of the representative consumer. Then, ( ) p(k) σ q(k) = Q P ( ) p(k) 1 σ r(k) = p(k)q(k) = R P Here p(k) is the price of variety k and P is the price index faced by the consumer, [ˆ ] 1 P = p(k) 1 σ 1 σ dk k Ω 5 / 22

6 Technology The amount of labour needed for a firm to produce q(k) units of variety k is f d + q(k)/ϕ(k) So the total cost of production is and the marginal cost is f d + q(k)/ϕ(k) 1/ϕ(k) All firms with the same productivity behave in exactly the same way. So we will use productivity ϕ rather than variety k to identify a firm. 6 / 22

7 Profit maximization Each firm produces a differentiated variety (monopolistic competition): π(ϕ) = max q(ϕ) {p(ϕ)q(ϕ) f d + q(ϕ)/ϕ} Constant markup pricing: which implies the following: p(ϕ) = σ σ 1 ϕ 1 = (ρϕ) 1 (1) r(ϕ) = ( ϕ) σ 1 P σ 1 R, q(ϕ) = [ ϕ] σ P σ 1 R (2) π(ϕ) = r(ϕ) σ f d (3) 7 / 22

8 Entry and exit Prior to entry, a firm incurs a sunk cost of entry, f e. Upon entry, it draws a productivity ϕ randomly from a distribution G(.). The firm will stay in the market only if Current value of firm ϕ is v(ϕ) = t=0 π(ϕ) 0 or r(ϕ) σf d { } π(ϕ) (1 δ) t max{π(ϕ),0} = max,0 δ Let ϕ be the cutoff productivity such that π(ϕ ) = 0 and p in (ϕ ) = 1 G(ϕ ) the probability that a firm will stay. Then, productivity distribution of producing firms is given by the following density function: µ(ϕ) = { g(ϕ) p in (ϕ ) ifϕ ϕ 0 otherwise (4) 8 / 22

9 Equilibrium in a closed economy A stationary equilibrium is a vector (ϕ, M, M e, P, R) of cutoff productivity, mass of producing firms, mass of entrants, price level and aggregate expenditures such that price, revenue, quantity and profit for each firm ϕ are given by equation (1) to (3), the distribution of producing firms is given by µ(ϕ) in equation (4), and the following conditions hold: Zero profit condition: π(ϕ ) = 0 Free entry condition: ˆ π(ϕ) p in (ϕ ) µ(ϕ)dϕ = f e δ Stationarity condition: Market clearing condition: p in (ϕ )M e = δm R = L 9 / 22

10 Price index [ˆ P = P = M 1/(1 σ) ρ 1 [ˆ ] 1 p(ϕ) 1 σ 1 σ µ(ϕ)mdϕ ] ϕ σ 1 µ(ϕ)dϕ = M 1/(1 σ) (ρϕ) 1 (5) Here [ˆ ϕ = ] 1 ϕ σ 1 σ 1 µ(ϕ)dϕ is a weighted average of firm productivities. 10 / 22

11 Profit function and entrants expected value From (2), (3), (5) and the market clearing condition: π(ϕ) = ( ) ϕ σ 1 L ϕ σm f d Entrants expected value: ˆ π(ϕ) v = p in (ϕ ) µ(ϕ)dϕ = p in (ϕ ) π(ϕ) δ δ = p in(ϕ ) δ [ ] L σm f d 11 / 22

12 Mass of producing firms Zero profit condition= Free entry condition = Thus, ( ϕ ϕ p in (ϕ ) δ ) σ 1 L σm = f d [ ] L σm f d = f e (6) [ (ϕ ) 1 σ 1] f d = δf e ϕ p in (ϕ ) 12 / 22

13 Cutoff productivity in equilibrium [ (ϕ ) 1 σ 1] f d = δf e ϕ p in (ϕ ) ˆ [ ( ) ϕ σ 1 = H(ϕ ) ϕ 1] g(ϕ)dϕ = δf e /f d (7) ϕ LHS is a decreasing function, so ϕ increases in fixed production cost f d, but decreases in entry cost f e 13 / 22

14 Number of firms in equilibrium From equation (6), M = L σ[f d +δf e /p in (ϕ )] 14 / 22

15 Size distribution in equilibrium Firm ϕ employs z(ϕ) =q(ϕ)/ϕ number of worker for production ( ) p(ϕ) σ ( ) ϕ σ 1 z(ϕ) = Q/ϕ = σρf d P ϕ Thus, the average firm size is z = σρf d ( ϕ ϕ ) σ 1 = σρ[f d +δf e /p in (ϕ )] 15 / 22

16 Properties of Closed Economy Equilibrium Increasing fixed production cost f d will raise average productivity raise average firm size but lower the number of producing firms Increasing sunk entry cost f e will lower average productivity lower average firm size and lower the number of producing firms 16 / 22

17 n + 1 identical countries, so identical wage, price level and income across countries wage is normalized to 1 To produce, a firm will first incur a fixed production cost f d If the firm decides to export, it will also incur a fixed export cost f x Iceberg trade cost τ 17 / 22

18 Trade costs Same markup pricing for exports: Revenue functions: p x (ϕ) = τ ϕ = τp d(ϕ) Profit functions: r d (ϕ) = ( ϕ) σ 1 P σ 1 R π d (ϕ) = r d(ϕ) σ f d r x (ϕ) = τ 1 σ r d (ϕ) π x (ϕ) = r x(ϕ) σ f x π(ϕ) = π d (ϕ)+nπ x (ϕ) 18 / 22

19 Cutoff productivities The cutoff productivity below which a firm will now produce: ϕ d π d (ϕ d ) = 0 Nominal income remains the same as in closed economy: R = L Price level is lower than that in the closed economy = π d (ϕ) < π a (ϕ) = ϕ a < ϕ d Trade increases competition and forces some inefficient firms to exit The cutoff productivity for exporting: max { ϕ } d,ϕ x π x (ϕ x) = 0 If τ σ 1 f x > f d, then ϕ d < ϕ x : not all producing firms export, and exporting firms are more productive 19 / 22

20 Open economy results The opening of trade leads to: Rise in the zero profit cutoff productivity Rise in average firm revenue and profit Low productivity firms between ϕ a and ϕ d exit Intermediate productivity firms betwen ϕ d and ϕ x contract Only firms with productivities greater than ϕ x enter export markets and expand All of the above lead to a change in industry composition that raises aggregate industry productivity 20 / 22

21 Subsequent literature Helpman, Melitz and Rubinstein (2004) Export Versus FDI with Heterogeneous Firms," American Economic Review, 94, Introduces both exports and FDI as alternative means of serving a foreign market Introduces an outside sector to tractably characterize equilibrium with many asymmetric sectors Antras and Helpman (2004) Global Sourcing," Journal of Political Economy, 112(3), Combines the Melitz model with the Antras (2003) model of incomplete contracts and trade Bernard, Redding and Schott (2007) Comparative Advantage and Heterogeneous Firms," Review of Economic Studies, 73(1), Incorporates the Melitz model into the framework of integrated equilibrium of Helpman and Krugman (1985) 21 / 22

22 Subsequent literature Chaney, Thomas (2008) Distorted Gravity: the Intensive and Extensive Margins of International Trade," American Economic Review, September. Provides a simplified static version of the Melitz model without ongoing firm entry and with an outside sector Examines the model s implications for the extensive and intensive margins of international trade Arkolakis, Costas, Klenow, Peter, Demidova, Svetlana and Andres Rodriguez-Clare (2009) The Gains from Trade with Endogenous Variety," American Economic Review, Papers and Proceedings, 98 (4), Solves the Chaney version of the model without an outside sector Derives a sufficient statistic for welfare of the same form as that in Eaton and Kortum (2002) 22 / 22

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