Presentation on theme: "Melitz Firm Heterogeneity Helpman Hopenhayn Chaney B J R S."— Presentation transcript:
1 MelitzFirm HeterogeneityHelpmanHopenhaynChaneyB J R S
2 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.
3 Share of manufacturing firms that export CountryYearExporting firms, in per centU.S.A200218.0Norway200339.2France198617.4Japan200020.0Chile199920.9Colombia199018.2Very few plants/firms export:– 18% of U.S. firms in 2002 (the proportion of exporting plants is higher as exporting plantsare 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 countries17.4% of French firms in 1986
4 Share of exports of manufactures CountryYearTop 1% of firmsTop 10% of firmsU.S.A20028196Belgium20034884France44Germany5990Norway5391UK4280
5 Even at the 3-digit sectors there is significant heterogeneity:
7 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 productivityFirms 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
8 DemandRepresentative 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−1Where 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=𝑅Which yields a demand function for each variety 𝜔𝑞 𝜔 = 𝑅 𝑃 𝑝 𝜔 𝑃 −𝜎Where P is the C.E.S. price index𝑃= 𝜔∈Ω 𝑞 𝜔 𝜎− −𝜎
9 ProductionThere 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 𝜑All firms have the same fixed cost, but have different productivity levels indexed by 𝜑.Each firm’s constant marginal cost is given by𝑀𝐶(𝜑)= 𝑤 𝜑Where 𝑤 is the common wage rate normalized to one.
10 𝑟 𝜑 𝑙 𝜑 = 𝜎 𝜎−1 [1− 𝑓 𝑙 𝜑 ] which is increasing in 𝜑 Firm behaviorEach firm faces a residual demand curve with constant elasticity 𝜎All firms select the same profit maximizing markup:𝑝 𝜑 = 𝜎 𝜎−1 1 𝜑Firm revenue and profit are determined by 𝜑 and aggregating variables:𝑟(𝜑)= 𝜎−1 𝜎 𝜎−1 𝑅 𝑃 𝜎−1 𝜑 𝜎−1And𝜋(𝜑)= 𝑟 𝜑 𝜎 −𝑓So 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
11 Aggregation An equilibrium will be characterized by: A mass M of firms/goodsA distribution 𝜇(𝜑) of productivity levels over a subset of (0, ∞)𝑃= 𝜔∈Ω 𝑝 𝜔 1−𝜎 𝑀𝜇 𝜑 𝑑𝜑 −𝜎Given 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−1𝜑 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
12 Firm Entry and Exit Decisions To discover their productivity, firms must pay a fixed cost 𝑓 𝑒Upon 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.
13 Firm Entry and ExitIn 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 >0At that productivity, profits will be 0.𝜋 𝜑 ∗ =0Aggregate productivity 𝜑 is a function of the cutoff:𝜑( 𝜑 ∗ )= 𝜑 ∗ ∞ 𝜑 𝜎−1 𝑔 𝜑 1−𝐺 𝜑 ∗ 𝑑𝜑 1 𝜎−1
14 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 𝜋The net value of entry is the probability of entry times the profits minus the fixed cost.𝑝 𝑖𝑛 𝑣 − 𝑓 𝑒Free entry condition is that this value is 0, or that𝜋 = 𝛿 𝑓 𝑒 1−𝐺( 𝜑 ∗ )
16 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 levelP and R are uniquely identified by average profits and productivityIncreases in profit dispersion across firms shifts up the Zero Cutoff Profit curve.
26 Changes in the Exposure to trade Reasons for change:Increase in the number of trading partnersDecrease in variable trade costs 𝜏Decrease in the fixed export costs 𝑓 𝑥Each of these would causeMarket share reallocations from less productive firms toward more productive firmsExit of the least productive firms from the industryA gain in welfare.
27 Chaney 2005 Heterogeneity and the Gravity Model Develop a model for the extensive margin of tradeDerive a gravity model to examine relationship between trade flows and trade barriers.
28 New assumptions in Chaney vs Melitz Many asymmetric countries with asymmetric trade barriers.No free entry: fixed number of firms per countryPareto distributed productivity shocksA numeraire sector to pin down wages.
30 Productivity distribution Pareto distribution for productivities 𝜑∈ 1, +∞𝑃 𝜑>Φ = Φ −𝜃 𝜃>𝜎−1Note that here productivity is increasing in 𝝋Note that Pareto distributions are invariant to truncation𝑃 𝜑>Φ|𝜑> 𝜑 ∗ = Φ 𝜑 ∗ −𝜃 = 𝑃 𝜑>Φ 𝜑 ∗−𝜃This 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.
31 Ownership structure Without free entry, ownership structure matters. Aggregate world profits (Π) are distributed across workersAggregate income in country I is1+ Π 𝐿 𝐿 𝑖
32 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−𝜎Only 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 𝑖𝑗Entry threshold into j depends on prices in j.Note that throughout I will appeal to constants 𝜆 𝑖 which are defined in Chaney (2008)
33 Aggregate PricesPrices 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
34 Aggregate pricesPrices in j depends on the thresholds of entry into j.Threshold of entry into j depends on prices in j.Therefore𝑃 𝑗 = 𝜆 𝐿 𝑗 𝐿 1 𝜃 𝐿 𝑗 −1 𝜎−1 𝜌 𝑗With 𝜌 𝑗 −𝜃 ≡ 𝑘=1 𝑠 𝑘 τ 𝑘𝑗 −𝜃 𝑓 𝑘𝑗 − 𝜃 𝜎−1 −1𝜌 𝑗 is an aggregate measure of j’s remoteness from the rest of the world, due to both fixed and variable costs.
36 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 −1Whereas 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.
37 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 margins𝑋 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 𝑖𝑗Intensive MarginExtensive Margin
38 Intensive versus extensive margins 𝜁≡ – 𝑑𝑙𝑛 𝑋 𝑖𝑗 𝑑𝑙𝑛 𝜏 𝑖𝑗 = 𝜎−1 + 𝜃− 𝜎− = 𝜃𝜉≡ – 𝑑𝑙𝑛 𝑋 𝑖𝑗 𝑑𝑙𝑛 𝑓 𝑖𝑗 = 𝜃 𝜎−1 − = 𝜃 𝜎−1𝜕𝜁 𝜕𝜎 =0 and 𝜕𝜉 𝜕𝜎 <0Intensive 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.