The Melitz Model

The Melitz model is a dynamic industry model that incorporates firm productivity heterogeneity into the Krugman (1979) monopolistic competition framework, and focuses on steady state equilibrium only. The original Melitz (2003) model considers a world of symmetric countries, one factor (labor) and one industry, but it can be easily extended to the setting of asymmetric countries.⁴ In each country, the industry is populated by a continuum of firms differentiated by the varieties they produce and their productivity. Firms face uncertainties about their future productivity when making an irreversible costly investment decision to enter the domestic market. Following entry, firms produce with different productivity levels. In addition to the sunk entry costs, firms face fixed production costs, resulting in increasing returns to scale of production. The fixed production costs lead to the exit of inefficient firms whose productivities are lower than a threshold level, as they do not expect to earn positive profits in the future. On the demand side, the agents are assumed to have Dixit-Stiglitz preference over the continuum of varieties. As each firm is a monopolist for the variety it produces, it sets the price of its product at a constant markup over its marginal cost.

There are also fixed costs and variable costs associated with the exporting activities. However, the decision to export occurs after the firms observe their productivity. A firm enters export markets if and only if the net profits generated from its exports in a given country are sufficient to cover the fixed exporting costs. The zero cutoff profit conditions in domestic and exporting markets define the productivity thresholds for firm’s entry into the domestic and exports markets, and in turn determine the equilibrium distribution of nonexporting firms and exporting firms, as well as their average productivities. Typically, the combination of fixed export costs and variable export costs ensures that the exporting productivity threshold is higher than that for production for the domestic market, i.e., only a small fraction of firms with high productivity engage in exports markets. These exporting firms supply both the domestic and export markets.

The remaining of this section describes the detailed specifications of the model. For notational simplicity, the region subscript i is omitted in what follows when it does not lead to confusion.

(1) Demand

There are R countries in the world. In each country, the representative consumer maximizes utility from consumption over a continuum of goods Ω. The utility U, or the aggregate good Q, is described by the CES function,⁵

where q_ω is the quantity of consumption of good ω, and σ is the substitution elasticity across goods. The dual price index of utility P is defined over the prices of each good, p_ω

And the demand for each good is

(2) Production and Trade

There is a continuum of firms in each country, each with a different productivity φ and producing a different variety ω.⁶ Production entails fixed and variable costs, and requires only one factor, labor. Trade is assumed to be costly. A firm must pay a fixed cost in order to export. In addition, there are variable trade costs, which take the form of iceberg transportation costs whereby only a fraction ¹/ _^τij of each unit of good shipped from country i to j ( ^τ_ij =1 for i=j) arrives. Thus, for a firm with productivity φ, the cost of producing q units of good ω and selling them to country j is:

where Wi is the wage rate and serves as the numéraire. ∫_ii is fixed production input and ∫_ii is fixed input to sell a good from country i to country j (i≠j). The fixed costs are assumed to be same for all firms.

Firms are price setters. Given that the demand function is isoelastic, the optimal pricing rule for a firm is to charge a constant markup over the marginal cost:

The profits obtained by firm φ in country i from selling in the domestic market (π_ii) and exporting to country j ( π_ii ) are given by (6).

(3) Entry and Exit of Firms

The distribution of firms across different productivity levels is a result of the entry and exit of firms. Prior to entry, firms are identical. To enter the industry, a firm must incur a sunk entry cost of ∫_e effective labor units. After entry, firms draw their productivity, φ , from a ex-ante distribution g(φ) with support over ( 0,+∞ ). A firm’s productivity remains fixed thereafter. However, a firm will not produce if its expected profits are non-positive. Thus any firm whose productivity is lower than a threshold φ* will choose to exit without even starting production. Similarly, a firm will choose to export to a given country if and only if the net profits generated by the exports are sufficient to cover the fixed exporting costs. The condition defining the threshold is the zero cutoff profit condition:

where φ*_ii is the productivity threshold for production and φ*_ii is the productivity threshold for the least productive firm in country i able to export to country j. To ensure the partitioning of firms by export status, the condition φ*_ii < φ*_ii is assumed to hold for any j≠i. Firms with productivity levels between φ*_ii and the lowest exporting productivity threshold (min φ*_iij ∈ R, j ≠ i ) only produce for their domestic markets. The other firms vary in their exporting partners, depending on their productivity levels and the threshold φ*_ii in specific exporting market.

The surviving firms are assumed to face an exogenous probability of death, δ.⁷ Thus the value of a firm is equal to the stream of future profits discounted by the probability of death if it draws a productivity above the zero-profit productivity cutoff level, or equal to zero if it draws a productivity below the cutoff level.

The number of new entrants in each period is determined by the free entry condition and the general equilibrium. The free entry condition requires the expected value of entering to equal the sunk cost of entering, i.e.,

And in a steady state equilibrium, the mass of firms entering and producing must equal the mass of firms that die. Using M_e to denote the mass of new entrants and M to denote the mass of incumbents, the equilibrium condition is

where G(φ) is the cumulative distribution function of g(φ ) , and 1- G(φ * ) is the ex ante probability of successful entry in the industry.

(4) Firm Average

In equilibrium, the weighted average productivity level of the producing firms in country i as well as that for exporting firms is defined as a function of the cutoff levels, φ*_ij:

where the weights reflect the relative output shares of firms with different productivity levels.⁸

These average productivities are also aggregate as they completely summarize the information in the distribution of productivity levels for all aggregate variables. The aggregate price and demand in country j and total profits earned by firms in country i can also be expressed as functions of these average productivities:

5) Equilibrium

In each country, the representative consumer supplies L units of labor. The equilibrium in the labor market requires that:

where L_p is the labor input for production and L_e is that used in investment by new entrants.

The representative consumer receives labor income and profits, and spends on consumption Q and irreversible investment ∫_e As free entry ensures that total profits are exhausted by the aggregate investment sunk costs of new entrants, i.e., the budget constraint of the consumer is W . L = P . Q . This budget constraint also determines the equilibrium in the goods market in each country.

6) Properties of the equilibrium

Some properties of the equilibrium of the model are worthy mentioning.⁹ First, trade opening leads to the reallocation of market shares and profits among firms. Falling trade costs increase the profits of exporting firms and lower the exporting productivity threshold. As a result, new and less productive firms enter the export markets. Moreover, a reduction of trade costs enables existing exporting firms to increase their sales to foreign markets. In the domestic market, more competition from increased imports results in domestic firms losing a portion of their domestic markets. On the other hand, the expansion of existing high productivity firms into exports and the entry of new firms increase labor demand, driving up the real wage. Reduced profits and rising costs make the less productive firms unable to survive, forcing them to exit. As a result, the most productive firms increase their market shares and profits, while the least productivity firms shrink or exit. Thus, trade opening leads to larger inequalities between firms.

Second, because of the intra-industry resource reallocation, trade liberalization unambiguously increases aggregate productivity in all trading economies. The reallocation of market shares towards exporting firms can boost the aggregate productivity as exporting firms are more productive. The entry of new exporters may also increase average productivity if the new entrants are more productive than the average productivity level. Average productivity in importing countries is also enhanced because of the exit of the least productive non-exporting firms.

Third, trade liberalization always generates a welfare gain in the model. The magnitude of the gain is determined by the interactions between three factors: the decreased number of domestic firms, the increased number of foreign exporters and the increased average productivity of domestic firms. The reduced number of domestic firms supplying to domestic markets causes a negative variety effect for domestic consumers. But this effect is typically overpowered by the increased number of new foreign exporters, so that domestic consumers still enjoy greater product variety. If a large number of domestic firms are replaced by foreign firms, and product variety has a negative impact on welfare, the positive contribution of aggregate productivity gain would more than offset the loss in variety. The net welfare gain from trade liberalization is always positive.

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