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Melitz Firm Heterogeneity Helpman Hopenhayn Chaney B J R S

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**Firm Heterogeneity: Motivation**

The Helpman-Krugman model features universal exporting by firms in a differentiated product industry: every brand is produced by a single firm in just one country, which exports its output everywhere else in the world; using non-iceberg transport costs or a different demand system can change this outcome. This does not provide a good description of firm-level data. In the data: only a small fraction of firms export; exporters sell most of their output domestically; exporters are bigger than non-exporters; exporters are more productive than nonexporters; exporters pay higher wages than nonexporters.

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**Share of manufacturing firms that export**

Country Year Exporting firms, in per cent U.S.A 2002 18.0 Norway 2003 39.2 France 1986 17.4 Japan 2000 20.0 Chile 1999 20.9 Colombia 1990 18.2 Very few plants/firms export: – 18% of U.S. firms in 2002 (the proportion of exporting plants is higher as exporting plants are more likely to be owned by multi-plant firms) This number has been increasing steadily over the past 10 years: 14.6% of U.S. firms in 1992 – Same pattern holds for both developed and developing countries 17.4% of French firms in 1986

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**Share of exports of manufactures**

Country Year Top 1% of firms Top 10% of firms U.S.A 2002 81 96 Belgium 2003 48 84 France 44 Germany 59 90 Norway 53 91 UK 42 80

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**Even at the 3-digit sectors there is significant heterogeneity:**

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**Melitz Model for Firm Heterogeneity**

Adaption of Hopenhayn (1992) model of industry dynamics to monopolistic competition and general equilibrium ... but abstract from stochastic evolution of productivity at the firm level (Note: product differentiation is crucial in explaining fixed export costs at the firm/product level) Key features: Prior to entry, firms face some initial uncertainty concerning their future productivity Firms must incur a sunk investment to enter the industry and subsequently passively learn about their productivity. (Entry is not otherwise restricted.) Forward looking firms correctly anticipate future market conditions when making entry, exit, and export decisions. No strategic interactions between firms: firms only care about market averages

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Demand Representative consumer has a C.E.S. utility function over a continuum of consumption goods indexed by 𝜔: 𝑈= 𝜔∈Ω 𝑞 𝜔 𝜎−1 𝜎 𝑑𝜔 𝜎 𝜎−1 Where Ω is the set of available products and 𝜎 is the constant elasticity of substitution between goods. Consumers maximize U subject to: 𝜔∈Ω 𝑝 𝜔 𝑞 𝜔 𝑑𝜔=𝑅 Which yields a demand function for each variety 𝜔 𝑞 𝜔 = 𝑅 𝑃 𝑝 𝜔 𝑃 −𝜎 Where P is the C.E.S. price index 𝑃= 𝜔∈Ω 𝑞 𝜔 𝜎− −𝜎

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Production There is a continuum of firms, each choosing to produce a different variety 𝜔. One factor of production, labor, inelastically supplied at its aggregate level L. Labor used is a linear function of the output. 𝑙=𝑓+ 𝑞 𝜑 All firms have the same fixed cost, but have different productivity levels indexed by 𝜑. Each firm’s constant marginal cost is given by 𝑀𝐶(𝜑)= 𝑤 𝜑 Where 𝑤 is the common wage rate normalized to one.

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**𝑟 𝜑 𝑙 𝜑 = 𝜎 𝜎−1 [1− 𝑓 𝑙 𝜑 ] which is increasing in 𝜑**

Firm behavior Each firm faces a residual demand curve with constant elasticity 𝜎 All firms select the same profit maximizing markup: 𝑝 𝜑 = 𝜎 𝜎−1 1 𝜑 Firm revenue and profit are determined by 𝜑 and aggregating variables: 𝑟(𝜑)= 𝜎−1 𝜎 𝜎−1 𝑅 𝑃 𝜎−1 𝜑 𝜎−1 And 𝜋(𝜑)= 𝑟 𝜑 𝜎 −𝑓 So generally plant productivity is measured as revenues per worker: 𝑟 𝜑 𝑙 𝜑 = 𝜎 𝜎−1 [1− 𝑓 𝑙 𝜑 ] which is increasing in 𝜑

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**Aggregation An equilibrium will be characterized by:**

A mass M of firms/goods A distribution 𝜇(𝜑) of productivity levels over a subset of (0, ∞) 𝑃= 𝜔∈Ω 𝑝 𝜔 1−𝜎 𝑀𝜇 𝜑 𝑑𝜑 −𝜎 Given the distribution 𝜇 𝜙 , take the weighted expectation of 𝜑: 𝜑 = 𝐸 𝜑 𝜎− 𝜎−1 = 0 ∞ 𝜑 𝜎−1 𝜇 𝜑 𝑑𝜑 1 𝜎−1 𝜑 is independent of M and summarizes everything we need to know about the distribution of firm productivity necessary for determining aggregate price, quantity, revenue and profits. 𝑃= 𝑀 1 1−𝜎 𝑝 𝜑 , 𝑄= 𝑀 1 1−𝜎 𝑞( 𝜑 ) The welfare level associated with this aggregate outcome is given by the indirect utility at the aggregate price (P) and income level (Y ): 𝑉 𝑃,𝑌 = 𝑌 𝑃 = 𝜎−1 𝜎 𝑌 𝑀 1 1−𝜎 𝜑

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**Firm Entry and Exit Decisions**

To discover their productivity, firms must pay a fixed cost 𝑓 𝑒 Upon entry firms draw 𝜑 from the common distribution 𝑔 𝜑 . A firm which draws a low 𝜑 may decide to immediately exit. Firms face a constant probability 𝛿 of a bad shock that would force them to exit.

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Firm Entry and Exit In a stationary equilibrium, a firm either exits immediately or produces and earns 𝜋 𝜑 until it dies. Given 𝜑, a firm’s value is given by: 𝑣 𝜑 = max 0, ∑ 𝑡=0 ∞ 1−𝛿 𝑡 𝜋 𝜑 = max 0, 1 𝛿 𝜋(𝜑) We can then define the infimum of the set of positive values (surviving) firms: 𝜑 ∗ =inf 𝜑:𝑣 𝜑 >0 At that productivity, profits will be 0. 𝜋 𝜑 ∗ =0 Aggregate productivity 𝜑 is a function of the cutoff: 𝜑( 𝜑 ∗ )= 𝜑 ∗ ∞ 𝜑 𝜎−1 𝑔 𝜑 1−𝐺 𝜑 ∗ 𝑑𝜑 1 𝜎−1

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**Free Entry and the Value of Firms**

𝑝 𝑖𝑛 =1−𝐺 𝜑 ∗ is the ex-ante probability of successful entry. Let 𝑣 and 𝜋 be the average firm value and profit flow across all active firms in the industry. 𝜋 = Π 𝑀 , 𝑣=∑ 1−𝛿 𝑡 𝜋 = 1 𝛿 𝜋 The net value of entry is the probability of entry times the profits minus the fixed cost. 𝑝 𝑖𝑛 𝑣 − 𝑓 𝑒 Free entry condition is that this value is 0, or that 𝜋 = 𝛿 𝑓 𝑒 1−𝐺( 𝜑 ∗ )

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**Equilibrium in a closed economy**

Free entry and zero cutoff profit conditions determine the equilibrium cutoff level and average firm profit: The aggregate productivity level is set by the cutoff level P and R are uniquely identified by average profits and productivity Increases in profit dispersion across firms shifts up the Zero Cutoff Profit curve.

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**Changes in the Exposure to trade**

Reasons for change: Increase in the number of trading partners Decrease in variable trade costs 𝜏 Decrease in the fixed export costs 𝑓 𝑥 Each of these would cause Market share reallocations from less productive firms toward more productive firms Exit of the least productive firms from the industry A gain in welfare.

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**Chaney 2005 Heterogeneity and the Gravity Model**

Develop a model for the extensive margin of trade Derive a gravity model to examine relationship between trade flows and trade barriers.

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**New assumptions in Chaney vs Melitz**

Many asymmetric countries with asymmetric trade barriers. No free entry: fixed number of firms per country Pareto distributed productivity shocks A numeraire sector to pin down wages.

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**Preferences CES/Cobb-Douglas 𝑈≡ 𝑞 0 𝜇 ( Ω 𝑞 𝜔 ) 𝜎 𝜎−1 (1−𝜇)**

𝑈≡ 𝑞 0 𝜇 ( Ω 𝑞 𝜔 ) 𝜎 𝜎−1 (1−𝜇) N asymmetric countries: sizes 𝐿 𝑖 𝑖=1 𝑁 and trade barriers 𝜏 𝑖𝑗 ,𝑓 𝑖𝑗 𝑖𝑗=1 𝑁

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**Productivity distribution**

Pareto distribution for productivities 𝜑∈ 1, +∞ 𝑃 𝜑>Φ = Φ −𝜃 𝜃>𝜎−1 Note that here productivity is increasing in 𝝋 Note that Pareto distributions are invariant to truncation 𝑃 𝜑>Φ|𝜑> 𝜑 ∗ = Φ 𝜑 ∗ −𝜃 = 𝑃 𝜑>Φ 𝜑 ∗−𝜃 This property ensures that when we look at the endogenous selection into the export market, the subset of exporters is Pareto distributed with the same parameter as the distribution of domestic firms. In each country there are 𝐿 𝑛 potential entrants.

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**Ownership structure Without free entry, ownership structure matters.**

Aggregate world profits (Π) are distributed across workers Aggregate income in country I is 1+ Π 𝐿 𝐿 𝑖

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**Zero cutoff profit conditions**

Sales by firm 𝜑 from country 𝑖 in country 𝑗. 𝑟 𝑖𝑗 (𝜑)=𝜇 1+ Π 𝐿 𝐿 𝑗 𝜎 𝜏 𝑖𝑗 𝜎−1 𝜑 𝑃 𝑗 1−𝜎 Only firms with 𝜑≥ 𝜑 𝑖𝑗 ∗ from 𝑖 export to 𝑗, 𝜋 𝑖𝑗 𝜑 𝑖𝑗 ∗ =0 so 𝜑 𝑖𝑗 ∗ = 𝜆 1 𝑓 𝑖𝑗 1 𝜎−1 𝑃 𝑗 𝜎−1 𝐿 𝑗 − 1 𝜎−1 𝜏 𝑖𝑗 Entry threshold into j depends on prices in j. Note that throughout I will appeal to constants 𝜆 𝑖 which are defined in Chaney (2008)

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Aggregate Prices Prices in j depend on which firms are able to enter j, 𝑃 𝑗 1−𝜎 = 𝑘=1 𝑁 𝐿 𝑘 𝜑 𝑘𝑗 ∗ ∞ 𝑝 𝑘𝑗 𝜑 1−𝜎 𝑑𝐺(𝜑) Using the fact the at prices are a markup over marginal cost = 𝑘=1 𝑁 𝐿 𝑘 𝜑 𝑘𝑗 ∗ ∞ 𝜎−1 𝜎 𝜑 𝜏 𝑘𝑗 𝜎−1 𝑑𝐺(𝜑) = 𝑘=1 𝑁 𝐿 𝑘 𝜃 𝜃− 𝜎−1 𝜎−1 𝜎 1 𝜏 𝑘𝑗 𝜎−1 𝜑 𝑘𝑗 ∗𝜎−1−𝜃

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Aggregate prices Prices in j depends on the thresholds of entry into j. Threshold of entry into j depends on prices in j. Therefore 𝑃 𝑗 = 𝜆 𝐿 𝑗 𝐿 1 𝜃 𝐿 𝑗 −1 𝜎−1 𝜌 𝑗 With 𝜌 𝑗 −𝜃 ≡ 𝑘=1 𝑠 𝑘 τ 𝑘𝑗 −𝜃 𝑓 𝑘𝑗 − 𝜃 𝜎−1 −1 𝜌 𝑗 is an aggregate measure of j’s remoteness from the rest of the world, due to both fixed and variable costs.

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**Equilibrium exports and selection**

Firm level exports (r), thresholds of export ( 𝜑 )) and aggregate profits (Π), 𝑟 𝑖𝑗 𝜑 = 𝜆 𝐿 𝑗 𝐿 𝜎−1 𝜃 𝜌 𝑗 𝜏 𝑖𝑗 𝜎−1 𝜑 𝜎−1 𝑖𝑓 𝜑≥ 𝜑 𝑖𝑗 ∗ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝜑 𝑖𝑗 ∗ = 𝜆 4 𝐿 𝐿 𝑗 𝜃 𝜏 𝑖𝑗 𝜌 𝑗 𝑓 𝑖𝑗 1 𝜎−1 Π= 𝜆 5 𝐿

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**Gravity Equation comparing Chaney with Krugman**

Aggregate exports from i to j. 𝑋 𝑖𝑗 = 1+ 𝜆 5 𝜇 𝐿 𝑖 𝐿 𝑗 𝐿 𝜏 𝑖𝑗 𝜌 𝑗 −𝜃 𝑓 𝑖𝑗 − 𝜃 𝜎−1 −1 Whereas with Krugman’s model: 𝑋 𝑖𝑗 = 𝜆 𝐾 𝐿 𝑖 𝐿 𝑗 𝐿 𝜏 𝑖𝑗 𝜌 𝑗 −(𝜎−1) Higher elasticity of trade in Chaney model with respect to variable trade barriers: 𝜃>𝜎−1 This is due to the additional margin of adjustment, the extensive margin of trade. Chaney’s elasticity of trade barriers does not depend on elasticity of substitution. Elasticity of fixed costs is negatively related to the elasticity of substitution. Both the elasticity of trade flows with respect to fixed costs and variable costs are higher in sectors where firms productivity is less dispersed.

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**Intensive versus extensive margins**

If we fully differentiate the expression for aggregate exports, we can evaluate the impact of each type of trade barrier on intensive and extensive margins 𝑋 𝑖𝑗 = 𝑤 𝑖 𝐿 𝑖 ∫ 𝜑 𝑖𝑗 ∞ 𝑥 𝑖𝑗 (𝜑) 𝑑𝐺(𝜑) 𝑑 𝑋 𝑖𝑗 = (𝑤 𝑖 𝐿 𝑖 ∫ 𝜑 𝑖𝑗 ∞ 𝜕𝑥 𝑖𝑗 𝜑 𝜕 𝜏 𝑖𝑗 𝑑𝐺(𝜑)) 𝑑𝜏 𝑖𝑗 −(𝑤 𝑖 𝐿 𝑖 𝑥( 𝜑 𝑖𝑗 ) 𝜕 𝜑 𝑖𝑗 𝜕 𝜏 𝑖𝑗 𝐺′( 𝜑 𝑖𝑗 )) 𝑑𝜏 𝑖𝑗 + (𝑤 𝑖 𝐿 𝑖 ∫ 𝜑 𝑖𝑗 ∞ 𝜕𝑥 𝑖𝑗 𝜑 𝜕 𝑓 𝑖𝑗 𝑑𝐺(𝜑)) 𝑑𝑓 𝑖𝑗 −(𝑤 𝑖 𝐿 𝑖 𝑥( 𝜑 𝑖𝑗 ) 𝜕 𝜑 𝑖𝑗 𝜕𝑓 𝐺′( 𝜑 𝑖𝑗 )) 𝑑𝑓 𝑖𝑗 Intensive Margin Extensive Margin

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**Intensive versus extensive margins**

𝜁≡ – 𝑑𝑙𝑛 𝑋 𝑖𝑗 𝑑𝑙𝑛 𝜏 𝑖𝑗 = 𝜎−1 + 𝜃− 𝜎− = 𝜃 𝜉≡ – 𝑑𝑙𝑛 𝑋 𝑖𝑗 𝑑𝑙𝑛 𝑓 𝑖𝑗 = 𝜃 𝜎−1 − = 𝜃 𝜎−1 𝜕𝜁 𝜕𝜎 =0 and 𝜕𝜉 𝜕𝜎 <0 Intensive Margin: Each firm faces a constant elasticity of demand, so when goods are very substitutable (𝜎 is high) the export of each individual exporter is sensitive to trade barriers. Extensive Margin: As goods become substitutable, the market share of the least productive firms shrink. In the subsequent more competitive market, small differences in productivity translate into large differences in market shares. So the marginal firms export very little, and when they enter, they do not contribute much to aggregate trade. These two forces cancel out for variable trade costs.

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