A Global CGE Model with Heterogeneous Firms

I now turn to a specific global CGE model with heterogeneous firms. The CGE model consists of 12 regions, 14 sectors and 5 production factors. Within the 14 sectors, the agriculture and energy sectors produce homogeneous products. In each of these two sectors, there is a representative firm that operates under constant returns to scale technology. The other 12 manufacturing and services sectors produce differentiated products. In these sectors, the production and trade structures of the CGE model closely follow the Melitz model described in Section 2, but with two modifications. First, as in Chaney (2006), I abstract from the dynamic parts of the Melitz model by assuming no entry and exit of firms, no sunk entry costs and no uncertainty about productivity before entry. Thus the CGE model characterizes a static equilibrium rather than a steady-state

equilibrium. This abstraction is mainly due to the computational difficulties associated with multiple corner equilibria that the Melitz model may exhibit when it is extended to a multisector and asymmetric country setting and the intermediate inputs are explicitly considered. The centripetal forces arising from the self-reinforcing forward and backward linkages cause the multiple corner equilibria, and lead the model to behave more like a New Economic Geography model.¹⁰ Second, the CGE model assumes no fixed production costs, but fixed domestic trading costs in sectors with heterogeneous firms. The presence of fixed trading costs leads to increasing returns to scale technology in these sectors. This assumption makes the model more easily compatible with the case when the number of exporting firms is larger than that of firms serving solely their domestic market. Although empirical evidence at the aggregate level strongly supports selection into export markets, there may be in some particular sectors with data pointing to the other direction.¹¹

(1) Demand

In each region of the model, the representative consumer receives income from the supply of production factors to the firms, dividends from the firms and lump-sum transfers from the government. The consumer allocates his disposable income among the consumer goods and savings using the extended linear expenditure system, which is derived from maximizing the Stone-Geary utility function.¹² The consumption/saving decision is completely static. Savings enter the utility function as a “good” and its price is set as equal to the average price of consumer goods.

Investment demand and government consumption are specified as a Leontief function. I assume that in each sector s a composite good Qs is used for household consumption, investment, government consumption and intermediate input. The composite good is a CES aggregation of domestic goods and imports.

where Z^s_ij is the quantity of good s produced in region i sold in the market of region j. The dual price index of composite good s, P^s_j is defined over the aggregate prices of each supplier P_r^s_ij

And the demand function generated from (21) is:

In sectors with homogeneous goods, I follow the standard Armington assumption of national production differentiation, thus σ^s represents the substitution elasticity of good s among different regions in these sectors. The Armington share parameters α^s_ij in these sectors reflect the preference of consumers biasing for home or other regions’ products. In sectors with differentiated goods, σ^s represents the substitution elasticity among varieties of each firm and Z^s_ij is the CES aggregate of the individual varieties that are produced in country i and sold in region j. In these sectors, the Armington share parameters α^s_ij always equal one, meaning that the pattern of bilateral trade flows in these sectors are totally determined by the relative prices of aggregated differentiated goods from each region, P_r^s_ij.

(2) Production and trade

Factor markets: There are five primary factors: capital, skilled labor, unskilled labor, agricultural land and natural resources for the mining sector. It is assumed that factor endowments are fully employed. Land and natural resources are sector-specific but capital and labor are fully mobile across sectors. All primary factors are immobile across countries.

Production technology: Production is modeled using a nesting of CES functions. At the top level, the output Xs is produced as a combination of aggregate intermediate demand and value added. At the second level, aggregate intermediate demand is split into each commodity according to Leontief technology. Value added is produced by a capital-land bundle and aggregate labor. Finally, at the bottom level, aggregate labor is decomposed into unskilled and skill labor, and the capital-land bundle is decomposed into capital and land (for the agriculture sector) or natural resources (for the mining sector). At each level of production, there is a unit cost function that is dual to the CES aggregator function and demand functions for corresponding inputs. The top-level unit cost function defines the marginal cost of sectoral output, Cs.

Firm heterogeneity: In each region and sector, the total mass of potential firms, N^s_i , is fixed. Firms are assumed to get productivity draws φ from a Pareto distribution with low bound φmin and shape parameter γ>σ-1.¹³ Without a loss of generality, the units of quantity can be chosen so that the low bound parameter φmin equals unity. Then the density function g(φ) and the cumulative distribution function G(φ) are:

γ is an inverse measure of firm heterogeneity. The higher γ, the more homogeneous the firms are. Firms do not need to pay a sunk cost to participate in the productivity draw. With the Pareto distribution, the average productivities for non-exporting firms in county i and firms in country i exporting to country j, φ^s_ij , can be expressed as:

where φ^s_ii* is the productivity thresholds for firms in region i entering market j.

Fixed trading costs: In addition to variable costs, firms in the sectors with heterogeneous firms face region-specific fixed costs for their domestic sales and exports. The fixed inputs of these firms are a fixed combination of capital (∫^s_Kij), labor (∫^s_Lij) and intermediate inputs (∫^ts_Xij). Therefore, the expenditure of fixed trading costs, F_ij, is defined as:

where W_i, R_i, and P^t_i are the wage rate, rental rate of capital and price of good t, respectively.

Pricing and cut-off productivity: The model assumes “large group” monopolistic competition under an arbitrarily large number of firms, such that the elasticity of demand for each firm’s output is the substitution elasticity among varieties, σs. This results in a fixed markup as in (5). Then the variety adjusted aggregate prices of domestic sale and exports can be defined as:

where t^s_ij is the tariff rate, and τ^s_ij is ice-berg type variable trade costs and N^s_ij (1 - G(φ^s_ij*)) represents the total mass of firms in sector s and region i that sell in market j .

In sectors producing homogeneous goods, the markup is zero and productivity is fixed and normalized to one. Their producer prices are simply equal to marginal costs.

In sectors with heterogeneous firms, the productivity thresholds for market entry and exporting are:

The total profits of firms in sector s and region i, Π^s_ij, is the residual between the revenue from sales and all production and trading costs.

(4) Equilibrium and Closure

The equilibrium of the good markets requires that the output, X^s_ij, equal the sum of the demand from each markets, i.e.:

Note that for sectors with heterogeneous firms, demand is adjusted by the Dixit-Stiglitz variety effects and the average productivity.

There are three closure rules: the net government balance, investment-savings, and trade balance. I assume that changes in the government budget are automatically compensated by changes in marginal income tax rates. Government expenditures are exogenous in real terms.

Domestic investment is equal to the sum of domestic saving resources, i.e., household saving, government saving, and net foreign saving. As government saving is exogenous, changes in investment are determined by changes in the levels of household saving and foreign saving.

The final closure rule concerns the current account balance. In each region, either the foreign saving or the real exchange rate can be fixed while the other is allowed to adjust, providing alternative closure rules. When foreign saving is set exogenously, the price index of global manufacturing exports is chosen as the numéraire and the equilibrium is achieved by changing the relative price across regions, i.e. the real exchange rate. Alternatively, the GDP price deflator in each region is fixed and foreign saving is endogenous (subject to the constraint of the global balance) to maintain the trade balance. In the simulations conducted in Sections 4 and 5, foreign savings is chosen to be fixed and the manufacturing export price index is the numéraire.

(5) Calibration

The model is calibrated to the GTAP global database (version 6.2). However, some of the information that is central to our model, such as the degree of returns to scale, the shape of productivity distribution, and the magnitude of the fixed and variable trade costs, are not available in the GTAP database. I set these parameters based mainly based on a review of the relevant literature. Table 1 [ PDF 81.7KB | 1 page ] reports some major parameters used in the model. The markup ratios are set to 20%-25% for manufacturing sectors and 30% for services sectors. The choices of markup ratios, together with the optimal pricing rule for monopolistic firms, imply that the substitution elasticity between differentiated varieties is 6.0 for manufacturing sectors and 5.0 for services sectors. Firm productivity is assumed to follow a Pareto distribution. The shape parameters of the Pareto distribution are calibrated to match the profit ratio in total markup, which is estimated to be 64.5% based on French firm data by Arkolakis (2006). Assuming that all regions have access to same technology, the marginal costs C are set equal to unity in all regions for the calibration.

I assume the mass of potential firms in each sector, N^s_i , is proportional to the sectoral output. As fixed production costs, fixed exporting costs and variable trade costs are not available, they are calibrated to the base year’s bilateral trade flows. From the demand function in (18), and using the price function (22), average productivity function (20) and cutoff productivity function (24), we have the following gravity equation determining the bilateral trade flows:

This equation reflects the combined effects of market size (P_jQ_j), stiffness of market competition (reflected in P_j), technology (C_i), number of potential firms (N_i) and trade barriers (t_ij, τ_ij, and F_ij) on bilateral trade patterns. Replacing the variable export costs τ_ij with the share of exporting firms (1 - Gφ^s_ij*) , (27) can be rewritten as:

Based on the empirical findings in Hummels and Klenow (2005), I assume that the extensive margin accounts for 60% of the difference in export values across regions, i.e.

Assuming that 60% of potential firms produce and sell in the domestic market, the shares of exporting firms can be calculated from (29) using the base year trade flows and domestic sales data. The fixed trading costs can also be derived from (28). Given that the shares of firms selling in each market are determined, one can solve their productivity thresholds from (19). I also assume that domestic trade incurs no iceberg costs, i.e. T_ii equals 1. The iceberg trade costs T_ij can thus be obtained from (24).

Zero-trade flows are frequently presented in international trade databases, including the GTAP database. Under an Armington trade structure, the share parameters α^s_ij are zero if there is no trade between their corresponding trading partners. However, for sectors with heterogeneous firms, the zero-trade flow is not allowed in the model because of: (i) the unity of import share parameters α^s_ij for all trade partners, and (ii) the infinity upper bound of firms' productivity distribution. To resolve this dilemma, I modify the benchmark trade data by assigning an arbitrary small value to initial zero-trade cells. This tiny base year trade value leads to very high calibrated fixed and variables trade costs for corresponding trade partners.

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