Melitz Model

Melitz Model is the canonical heterogeneous-firms general-equilibrium model of international trade, developed by Marc J. Melitz in his landmark 2003 Econometrica paper, "The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity." Whereas Krugman's (1980) New Trade Theory assumed that all firms within a monopolistically competitive industry are symmetric and therefore predicted that every firm in an industry exports to every trading partner, plant-level longitudinal datasets compiled in the 1990s (Bernard & Jensen, 1995, 1999) revealed vast, persistent heterogeneity within narrowly defined industries: only a small minority of firms export, and exporting firms are substantially more productive, larger, and more capital-intensive than non-exporters even prior to entering foreign markets. By synthesizing the Dixit–Stiglitz Model of monopolistic competition with Hugo A. Hopenhayn's (1992) dynamic industry equilibrium model of uncertain firm-level productivity draws, Melitz demonstrated that opening to trade induces an intra-industry selection and market-share reallocation process—forcing the least productive firms to exit while expanding the market shares of the most productive exporters—that generates endogenous aggregate productivity gains even when individual firm-level productivity remains fixed.

Consider an economy endowed with L units of inelastic labor (with the competitive wage normalized to w = 1) where consumers maximize a symmetric Dixit–Stiglitz constant-elasticity-of-substitution (CES) utility function over a continuum of differentiated varieties ω ∈ Ω with elasticity of substitution σ = 1 / (1 − ρ) > 1:

U = Q = [ ∫_ω∈Ω q(ω)^ρ dω ]^{1 / ρ}

As derived in the Dixit–Stiglitz framework, consumer optimization yields isoelastic variety demand q(ω) = Q[p(ω)/P]^−σ and expenditure r(ω) = R[p(ω)/P]^1−σ, where R = PQ = L denotes aggregate expenditure and P = [∫_ω∈Ω p(ω)^1−σ dω]^1/(1−σ) is the ideal aggregate price index.

Unlike the symmetric Krugman (1980) model, firms in the Melitz model differ in their labor productivity φ > 0. Prior to entering the market, a prospective firm is uncertain of its productivity and must pay an irreversible sunk entry cost f_e > 0 (in units of labor) to unveil its idiosyncratic productivity parameter φ, which is drawn from a common known cumulative distribution G(φ) with density g(φ) on (0, ∞) (frequently parameterized in empirical work following Chaney [2008] as a Pareto distribution G(φ) = 1 − (φ_min / φ)^k with shape parameter k > σ − 1).

Once a firm observes its productivity draw φ, it decides whether to exit right away or remain active. Producing q units of output in any period requires a per-period fixed overhead production cost f > 0 and a constant marginal labor requirement 1 / φ:

l(q, φ) = f + q / φ

Facing isoelastic demand with elasticity σ, every active firm sets its price at a constant markup 1 / ρ = σ / (σ − 1) over its marginal cost 1 / φ:

p(φ) = 1 / (ρ φ) = [ σ / (σ − 1) ] (1 / φ)

Substituting p(φ) into the demand and profit expressions reveals that firm revenue r(φ) and variable profit r(φ) / σ are power functions of φ^{σ − 1}:

r(φ) = R (P ρ φ)^{σ − 1} and π(φ) = r(φ) / σ − f

Consequently, for any two active firms with productivities φ₁ and φ₂, their relative revenues and variable profits depend only on their productivity ratio:

r(φ₁) / r(φ₂) = ( φ₁ / φ₂ )^{σ − 1}

More productive firms charge lower prices (p(φ) ∝ φ⁻¹), sell larger quantities (q(φ) ∝ φ^σ), earn larger revenues (r(φ) ∝ φ^σ−1), and generate higher net profits. Active firms face an exogenous Poisson hazard δ ∈ (0, 1) of a bad shock forcing exit in each period.

Upon drawing φ, a firm chooses to produce if and only if its per-period profit is non-negative (π(φ) ≥ 0). Because π(φ) is increasing in φ, there exists a unique survival productivity cutoff φ* defined by π(φ*) = 0:

r(φ*) = σ f

Firms with draws φ < φ* exit upon drawing without producing, whereas firms with φ ≥ φ* remain active, yielding the truncated equilibrium productivity distribution μ(φ) = g(φ) / [1 − G(φ*)] for φ ≥ φ*.

A defining analytical insight of Melitz (2003) is that the entire distribution of heterogeneous firms can be summarized by a single weighted harmonic-mean aggregate productivity φ̃(φ*):

φ̃(φ*) = [ (1 / (1 − G(φ*))) ∫_φ*^∞ φ^{σ − 1} g(φ) dφ ]^{1 / (σ − 1)}

Because r(φ) / r(φ*) = (φ / φ*)^{σ − 1}, average revenue across all active firms equals r̄ = r(φ̃) = [φ̃(φ*) / φ*]^{σ − 1} r(φ*). Combining this with the cutoff condition r(φ*) = σf yields the Zero Cutoff Profit (ZCP) Condition, which expresses average profit π̄ across surviving firms as a decreasing function of the cutoff φ*:

π̄ = π(φ̃) = f [ ( φ̃(φ*) / φ* )^{σ − 1} − 1 ] (ZCP)

Meanwhile, prospective entrants continue paying the sunk entry cost f_e until the expected present discounted value of post-entry profits (weighted by the probability of successful entry p_in = 1 − G(φ*) and discounted by the exit rate δ) equals f_e, giving the Free Entry (FE) Condition:

π̄ = ( δ f_e ) / [ 1 − G(φ*) ] (FE)

Because 1 − G(φ*) decreases with φ*, the Free Entry curve slopes upward in (φ*, π̄) space, while the Zero Cutoff Profit curve slopes downward (under mild regularity conditions on G(·), including the Pareto distribution), guaranteeing a unique autarky equilibrium (φ_a*, π̄_*a).

Now suppose the world consists of n + 1 symmetric countries that open to international trade. Serving the domestic market incurs the standard overhead cost f, whereas exporting to each of the n foreign markets entails two additional trade costs:

1. Per-Unit Iceberg Variable Trade Cost (τ > 1): Delivering 1 unit of a variety to a foreign consumer requires shipping τ > 1 units, raising export marginal cost to τ / φ and export price to p_x(φ) = τ / (ρ φ) = τ p_d(φ).
2. Per-Market Fixed Export Cost (f_x > 0): Establishing foreign distribution networks, regulatory compliance, and marketing in each destination market requires a recurring per-period fixed labor cost f_x > 0.

Because production technologies exhibit constant marginal cost 1 / φ, a firm's domestic pricing and export pricing decisions are separable, yielding domestic profit π_d(φ) and per-market export profit π_x(φ):

π_d(φ) = r_d(φ) / σ − f and π_x(φ) = r_x(φ) / σ − f_x = τ^{1 − σ} r_d(φ) / σ − f_x

These two profit conditions define two distinct productivity cutoffs—a domestic survival cutoff φ* (where π_d(φ*) = 0) and an export entry cutoff φ_x* (where π_x(φ_x*) = 0)—linked by the structural ratio:

φ_x* / φ* = τ ( f_x / f )^{1 / (σ − 1)}

Under the empirically realistic assumption that fixed and variable export barriers are sufficiently large that τ^{σ − 1} f_x > f, we obtain partitioning of firms by export status (φ_x* > φ*): only the most productive subset of surviving firms (φ ≥ φ_x*) find it profitable to pay the fixed cost f_x to enter export markets, while intermediate-productivity firms (φ ∈ [φ*, φ_x*)) serve only the domestic market.

In the open-economy equilibrium, the Zero Cutoff Profit condition shifts upward because surviving firms with φ ≥ φ_x* earn additional export profits n π_x(φ):

π̄ = f k(φ*) + p_x n f_x k(φ_x*) (Open-Economy ZCP)

where k(φ*) = [φ̃(φ*) / φ*]^{σ − 1} − 1 and p_x = [1 − G(φ_x*)] / [1 − G(φ*)]. Because the Free Entry condition π̄ = δf_e / [1 − G(φ*)] is unchanged by trade, the upward shift of the ZCP curve raises the domestic survival cutoff relative to autarky:

φ* > φ_a*

This inequality represents the Melitz selection and reallocation mechanism, operating through two simultaneous channels:

1. Product Market Competition Channel: Foreign exporters enter the domestic market with high-productivity varieties, lowering the aggregate price index P and shifting residual demand inward for every domestic firm.
2. Factor Market (Real Wage) Channel: High-productivity domestic exporters expand production to serve foreign markets, increasing aggregate labor demand and bidding up the real wage w / P.

Squeezed by both tougher product-market competition and higher real labor costs, low-productivity firms in the interval [φ_a*, φ*) can no longer cover their fixed overhead cost f and are forced to exit. Market share and labor are reallocated from these exiting low-productivity firms (and shrinking domestic-only firms) toward high-productivity exporters (φ ≥ φ_x*), raising weighted-average industry productivity φ̃_t and generating unambiguous aggregate welfare gains (W = 1 / P > 1 / P_a).

Building on the Melitz model, Thomas Chaney (2008) solved the multi-country asymmetric model in closed form by assuming that firm productivities follow a Pareto distribution G(φ) = 1 − φ^−k (with k > σ − 1). Under Pareto heterogeneity, bilateral aggregate trade flows X_ij from country i to country j decompose cleanly into two margins:

• Intensive Margin: How much each existing exporter sells when variable trade costs τ_ij fall, with elasticity σ − 1.
• Extensive Margin: How many new firms cross the export cutoff φ_x,ij* as τ_ij falls, with elasticity k − (σ − 1).

Summing the intensive and extensive elasticities reveals that in a Pareto-Melitz world, the aggregate trade elasticity with respect to variable trade costs in the gravity model of trade equals the Pareto tail shape parameter k rather than the substitution elasticity σ − 1 (X_ij ∝ τ_ij^−k f_x,ij^{−[k/(σ−1) − 1]}). Furthermore, Costas Arkolakis, Arnaud Costinot, and Andrés Rodríguez-Clare (2012) proved that across a broad class of canonical quantitative trade models—including Armington, Eaton–Kortum (2002), Krugman (1980), and the Pareto-Melitz model—the total real income gain from trade relative to autarky is summarized by the ACR welfare formula W_j / W_j,autarky = λ_jj^−1/ε, where λ_jj is the domestic expenditure share and ε is the aggregate trade elasticity (ε = k in Pareto-Melitz).

How does the Melitz model generate aggregate productivity gains without changing any individual firm's productivity?

In the baseline Melitz model, each firm's technical productivity draw φ is permanently fixed after entry. Aggregate industry productivity φ̃ rises when trade opens purely through composition and market-share reallocation: the least productive firms (φ ∈ [φ_a*, φ*)) exit the industry completely, intermediate non-exporting firms lose market share, and the most productive exporting firms (φ ≥ φ_x*) expand their output and employment shares.

Why do only a fraction of firms export in the Melitz model, unlike in Krugman (1980)?

In Krugman (1980), there is no fixed per-market export cost (f_x = 0) and all firms have identical productivity. Under CES demand, price is never driven to infinity by finite iceberg transport costs τ, so every firm sells at least a small positive quantity in every foreign market. In the Melitz model, entering a foreign market requires paying a positive fixed cost f_x > 0; only firms with productivity φ ≥ φ_x* generate sufficient variable export profit r_x(φ) / σ to cover f_x.

Can the number of domestically consumed varieties fall when a country opens to trade in the Melitz model?

Yes. Because opening to trade raises the domestic survival cutoff (φ* > φ_a*), the mass of active domestic producers M falls, and if foreign fixed export barriers f_x are relatively high, the influx of foreign exported varieties n M_x may be smaller than the number of exiting domestic varieties. Nevertheless, Melitz (2003) proved that the higher average productivity (and lower prices) of the surviving and imported varieties always outweighs any reduction in variety count, so the aggregate price index P falls and consumer welfare W = 1 / P rises.

1. Melitz, M. J. (2003). The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica, 71(6), 1695–1725. https://doi.org/10.1111/1468-0262.00467
2. Bernard, A. B., & Jensen, J. B. (1999). Exceptional exporter performance: Cause, effect, or both? Journal of International Economics, 47(1), 1–25. https://doi.org/10.1016/S0022-1996(98)00027-000027-0)
3. Hopenhayn, H. A. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica, 60(5), 1127–1150. https://doi.org/10.2307/2951541
4. Chaney, T. (2008). Distorted gravity: The intensive and extensive margins of international trade. The American Economic Review, 98(4), 1707–1721. https://doi.org/10.1257/aer.98.4.1707
5. Arkolakis, C., Costinot, A., & Rodríguez-Clare, A. (2012). New trade models, same old gains? The American Economic Review, 102(1), 94–130. https://doi.org/10.1257/aer.102.1.94