Country Size, Technology and Manufacturing Location

Endowments

There are two regions denoted Home and Foreign. Two goods are produced in a homogenous agriculture (A) sector and differentiated manufacturing (M) sector. Labor is the only primary factor of production. Labor is mobile within each country and between the agriculture and manufacturing sectors. Using s_L to denote labor shares, the size of the labor endowment in Home is s_LL^w, the size of Foreign is (1 − s_L)L^w and λ = (1 − s_L)/s_L is the ratio of endowments. Foreign variables and parameters carry an asterisk and world aggregates are denoted by superscript w.

Geography enters via iceberg transportation costs in the manufacturing sector. Locally consumed goods have zero transport costs. A fraction of a shipped good melts away in transit. For one unit of a good to arrive at its destination τ ∈ [1, ∞) units of the good must be shipped.

Preferences

Consumers have Cobb–Douglas preferences for the A- and M-sectors with a Dixit–Stiglitz (1977) constant elasticity of substitution (CES) sub-utility for M-sector varieties:

(1)

A price index for the entire basket of industrial varieties is defined in the usual way:

(2)

where and . The final destination price paid on imported and domestic goods is p_i and n^w = n + n* measures the aggregate mass of firms.

Varieties produced in the M-sector are indexed by i. Elasticity of substitution is restricted to σ > 1. The indirect utility function for a representative consumer with aggregate income Y is:

(3)

Market demand for variety i is obtained by applying Roy's identity to (3):

(4)

The expression for Foreign demand is isomorphic.

The Dixit–Stiglitz CES sub-utility imparts a “love of variety” to preferences. Utility is increasing in the number of goods available. So (4) implies that total consumption of good i is a function of its own price and total expenditure weighted by the price index of all other M-sector goods.

Production and Market Structure

Manufacturing sector

The M-sector is characterized by monopolistic competition and increasing returns. Firms produce using labor and an intermediate input that is a composite of all M-sector goods. The intermediate input creates cost-linkages between manufacturing firms that are an agglomeration mechanism.

The production technology for Home is represented by the total cost function:

(5)

where is the producer price index for the M-sector composite. Marginal costs are incurred in terms of labor where a_M is the Home unit labor requirement and q is a scaling parameter that is constant across countries. Firms also pay a fixed cost F, that is Cobb–Douglas with expenditure shares of (1 − µ) for labor and µ for an intermediate composite good of all M-sector varieties. The producer price index, , is identical to the consumer price index because the expenditure shares are equal, thus .

Firm demand for each industrial variety is derived by applying Shepard's lemma to (5):

(6)

Because the intermediate good appears only as a component of fixed costs, demand is invariant to firm scale. F is normalized to unity so that each firm requires one unit of the composite good. Total firm profits are given by:

(7)

where operating profits are π_i = p_ix_i − a_Mqw_Mx_i.

Each firm maximizes monopoly profits in its own differentiated variety. The first order conditions for profit maximization imply a markup rule of p_i(1 − 1/σ) = a_Mqw_M. Following the markup rule, operating profits of each firm become π_i = p_ix_i/σ. The numerator is the value of total sales for each variety where . A higher elasticity of substitution (higher σ), lowers profits.

Labor markets

From the cost function, total labor demand in the manufacturing sector for Home and Foreign is

(8)

(9)

Labor is supplied inelastically to the sector with the highest wages. Manufacturing wages adjust to clear the labor market. If neither country is fully specialized and both sectors are in operation, factor price equalization implies wages are unity. Then labor employed in the manufacturing sector is just a function of parameters and other endogenous variables. Labor market clearing can be ignored when solving for the equilibrium. Under full specialization, wages and labor allocations in (8) and (9) are endogenous.

Relative Productivity

Labor productivity, the inverse of a_M, is country specific. One interpretation of this specification is to define the firm as mobile with implicit, identical managerial or firm specific technology across countries. Remaining productivity differences measured by a_M and are outside the firm's control. A second interpretation sidesteps a global definition of the firm. Manufacturing firms are defined by the cost function (5) alone and not internationally mobile. Firms enter in response to profit opportunities and exit in response to losses within each country. Changes in industrial structure are accomplished by allocating labor between agriculture and manufacturing.

A measure of relative unit labor requirements in Home and Foreign is constructed. With A-sector unit labor costs normalized to unity, differences in the variable costs of production (a_M and ) are the sole parameters of interest. Using notation in Baldwin et al. (2003), relative productivity is indexed by χ

(10)

Units of and q = (1 − 1/σ) are chosen to set foreign prices equal to wages. Then Home unit labor requirement in terms of χ are . Productivity differences correspond to deviations of χ from unity.²

Using (10) and the normalization for foreign unit costs in the markup rule of firms gives the following expressions for domestic and import prices:

(11)

The first price subscript denotes the country of origin; the second denotes where the good is consumed. Note that export prices are simply domestic prices scaled by τ. At the individual variety level, Home's local and export prices are decreasing in relative productivity. The ratio of Home export prices to local Foreign prices is . Thus Home's competitiveness depends multiplicatively on transport costs, relative productivity and relative wages. Substituting these price expressions into the price indices reduces the Δ and Δ* terms in equation (2) to the following compact formulas:

(12)

(13)

where s_n ≡ n/n^w and φ = τ^1−σ. The parameter φ is an index of trade costs on the interval [0,1]. When φ = 0 trade costs are prohibitively high (τ → ∞). φ = 1 implies completely free trade (τ = 1). The location of production impacts the price index in the presence of transport costs. Increasing a country's share of firms, s_n or (1 − s_n) respectively, lowers the price index as local consumers avoid paying transport costs on a larger fraction of varieties. High productivity in only one country can lower the price index in both countries. This is true regardless of transport costs for both Home and Foreign when at least some firms are operating in the most productive location.

Short-run Equilibrium and Market Clearing

The geographical allocation of industry is taken as given and there is no firm entry and exit in the short run. Consumers maximize utility, firms maximize profits and markets clear.

The goods market clearing condition for manufacturing requires the total value of sales to equal the total value of demand for each variety:

(14)

Income for a representative Home consumer is the sum of labor earnings and firm profits Y = [w_ML_M + w_AL_A] + s_nn^wΠ. Total M-sector sales are the share of expenditure in consumer and firm demand for each country, µE and µE*. Adding wage income to firm expenditure on intermediate goods and simplifying yields total expenditure:

(15)

(16)

Total manufacturing sales must be equal to the total world manufacturing budget share, µE^w, for markets to clear according to (14). This implies µ[s_nπ + (1 − s_n)π*]n^w = E^wµ/σ. An expression for world expenditure is obtained by summing (15) and (16) vertically and simplifying,

The first two terms are world wage income and the third is total world operating profits. This can be solved for world expenditure E^w = (Wage Income)/(1 − b).

Substituting consumer and firm demand for individual varieties into π = px/σ and summing across sales in both Home and Foreign markets for a representative firm, operating profits are

(17)

The expression for B represents shares of world sales expenditure weighted by the local price index.

Substituting (17) into Home and Foreign expenditure defined in (15) and (16) and dividing through by E^w gives Home and Foreign expenditure shares in terms of the spatial allocation of wage income and firm operating profits:

(18)

(19)

(20)

where is Home's share of the wage bill. The size of each market in expenditure terms is directly related to its shares of manufacturing (s_n) and labor (s_L).

The expression for s_E can be solved explicitly in terms of the division of manufacturing firms as:

(21)

Closed form expressions for total profits in terms of the mass n^w, and allocation of firms s_n, and wages are found by substituting (12), (17) and (21) into total profit equation (7). These Home and Foregin total profit equations and the labor equations in (8) and (9) are solved for long-run equilibria in the next section.

Non-full Specialization and Wages

To fix ideas, the model is simplified by imposing a non-full specialization condition. This assumption is relexed in section 4. If the A-sector good is produced in both locations, factor price equalization prevails and manufacturing wages are . Labor market clearing can be ignored and the profit equations (7) are two equations in two unknowns (s_n, n_w). This requires a non-full specialization (NFS) condition of (1 − µ) > max{s_L, (1 − s_L)}L^w. The A-sector must be a large enough share of total consumption that no single country can be fully specialized in the M-sector. As a result, the Agricultural good is always produced in both countries.

Firm Entry and Exit

In the long run, profits in each country are driven to zero by the entry and exit of firms. Atomistic and myopic firms will enter the M-sector in either location when profits in (7) are positive and exit when profits are negative.

The NFS condition admits three feasible and stable equilibria: (1) full agglomeration at Home when n* = 0, Π = 0 and profits to entry in Foreign are negative, (2) full agglomeration in Foreign when n = 0, Π* = 0 and Home profits are negative and (3) an interior case when Π = Π* = 0 and firms are active in both locations. Assessing the stability conditions of these three equilibria is the subject of the rest of this section.

An Illustrative Example

Total profits are two nonlinear equations in two unknowns (s_n and n^w). Completely solving the model for all stable and unstable equilibria requires numerical simulation. Fortunately, most of the critical points in the model can be found without extensive numerical methods. Before proceeding to derive more analytical results, it will be useful to present a fully solved and informative case. Consider a stylized dichotomy of global integration where a relatively large country opens to trade with a small country that has a productivity advantage in manufacturing. How will the location of manufacturing respond as these two countries move from autarky to free trade?

A large labor force is a larger market for goods. When trade costs are high, profits of the representative firm will be higher by producing for a large local market and exporting abroad. This market access effect is clear from expenditure equations (15), (16), and (21). A production–cost effect is the second agglomeration force. A larger share of firms in one country will lower its producer price index, raise profits and encourage firm entry. Relative productivity differences enter through this channel, but the price lowering impact is on locally consumed and exported goods. This productivity advantage raises profits and supports a larger number of firms in equilibrium. Trade costs will moderate the relative strength of these forces when Home has high manufacturing productivity and Foreign is large.

In Figure 1, Home has a modestly higher productivity. The relative productivity index is χ = 1.1 and Foreign is twice as large, λ = 2. Panel (a) plots the the share of world manufacturing in Home against trade costs. All stable equilibria are shown as solid lines. Panel (b) shows the contemporaneous share of employment in manufacturing in Home and Foreign. There are three phases of integration as trade costs fall: (1) dispersion, (2) full agglomeration in the large Foreign market followed by (3) full agglomeration in the more efficient Home market.

Beginning from autarky, high trade barriers naturally favor the larger market of Foreign. A market crowding effect keeps a small share of manufacturing firms in Home. As trade costs fall, Foreign's initial advantage is self-reinforcing. Higher consumer demand in Foreign results in a higher share of firms. This creates higher local demand for intermediate goods, lowers the Foreign price index and raises profits. More firms in Foreign enter manufacturing leading consequently to a shift in manufacturing employment. This process continues until manufacturing fully agglomerates in Foreign and profits are driven to zero. The level of trade costs where agglomeration in Foreign becomes sustainable is denoted by . No manufactured goods are produced in Home even though it has higher manufacturing productivity.

This would be the end of the story with size asymmetry alone, but this economy continues to respond to falling trade costs. Agglomeration in Home eventually becomes stable (profitable) as trade costs fall. In this case though, the sustain point for Home, , comes after the sustain point for Foreign. As a result, when history has favored Foreign and the rents to agglomeration are high, firms will have no incentive to establish manufacturing operations in Home.

At an even lower level of trade costs, the rents to agglomeration in Foreign vanish. Despite the larger local market in Foreign, transport costs are low enough that it is cost effective to manufacture goods in Home and export to Foreign. This occurs for levels of trade costs above . The agglomeration is catastrophically³ reversed and a Ricardian pattern emerges. Panel (b) reflects this change in labor market shares. Nearly all of Home's labor endowment is engaged in manufacturing for very low trade costs.

While this bifurcation diagram is rich with features, they are not always robust to variation in size and technology. Figure 2 repeats the analysis of Figure 1 when Home has even higher relative productivity of χ = 1.3. Now lowering trade costs from autarky benefit Foreign initially, but not enough that Foreign captures all manufacturing industry. Home achieves full agglomeration quickly as trade costs fall with a commensurate increase in manufacturing employment.

In the following sections a more analytical approach is used to answer a number of questions. For what relative productivity can the initial size effect be overcome as in Figure 2? Can the sustain points defined above be ordered? How do the equilibria respond to variation in relative productivity? Will free trade always favor the country with the highest manufacturing productivity, regardless of size?

The Mass of Firms with Full Agglomeration

Productivity differences in variable costs will effect the equilibrium number of firms for agglomeration outcomes, but labor endowments will not. A more productive labor force, will support a larger number of firms and varieties. Deriving this result is straightforward.

Beginning with the case for Home, profits for a given level of φ are determined by s_n and n^w. When s_n = 1 firms will enter the Home market until total profits are driven to zero. The equilibrium number of firms, denoted by n₁ is found by setting s_n = 1 and imposing the zero profit condition in Home:

(22)

which yields n₁ = (b/1 − b)^1/(1−a)χ^a/(1−a). The equilibrium number of firms is increasing in Home's relative productivity and it is invariant to trade costs. The corresponding condition for Foreign is n₀ = (b/1 − b)^1/(1−a). These two values for aggregate firms are related as follows:

(23)

The country with higher manufacturing productivity will support a larger number of firms.

Free Trade Produces a Ricardian Allocation of Industry

Free trade implies φ = 1. Imposing this condition will equalize the price index in both countries. The location of firms is determined by evaluating the difference in Home and Foreign total profits:

(24)

The denominator is positive for any allocation of firms. When χ > 1, Home has a productivity advantage and the difference in profits will be positive. For the case χ < 1, the converse is true.

Agglomeration and Intermediate Trade Costs

Full agglomeration is a stable, long-run equilibrium in Home if there are no positive profits to entering the Foreign market. The formal objective is to find the values of trade costs that support agglomeration. These are the “sustain points” referred to in earlier. For the case of agglomeration in Home, the condition, Π*(n₁, s_n = 1) < 0 must be satisfied or firms will have incentive to enter the market in Foreign. This is equivalent to requiring that agglomeration rents Ω are positive as follows:

(25)

where the third equality follows from the zero profit condition.

A closed-form expression for agglomeration sustaining relative productivity as a function of trade costs and relative endowments exists. Setting Foreign profits to zero and rearranging terms yields:

(26)

where the ratio of Foreign to Home labor endowments is λ = (1 − s_L)/s_L. The levels of of trade costs and relative productivity that solve this equation are denoted as χ_s and φ_s. While exact solutions must be found numerically, the functional form for χ in terms of exogenous parameters permits further quantitative and graphical analysis. The condition for Foreign is found by substituting 1/λ for λ in (26) and then inverting the entire the right hand side.

Figure 3 plots the locus of sustain points defined in (26) for two cases of size asymmetry. Agglomeration in Home is stable for all ordered pairs (φ, χ) above the locus. Above this line, the profits of an atomistic firm entering to Foreign are negative. Agglomeration in Foreign is stable below the locus.

The value and ranking of the critical thresholds of φ required to sustain agglomeration are determined by relative productivity. These sustain points occur where the horizontal line for χ = 1.1 cuts the two loci. The location, ranking and number of sustain points depends on relative productivity. In panel (a), the parameters are set to match the scenario in the example of the Firm Entry and Exit subsection. The overlap of the stability ranges for φ is easy to observe and the sustain points can be ordered as .

Agglomeration Rents

Many advanced, industrialized countries are relying on high productivity to prevent the erosion of manufacturing employment. In the case where Foreign is relatively large, Figure 3(a) indicates that nearly complete trade liberalization is required for Home to exploit its productivity advantage. Alternatively, the productivity advantage could be high enough that no level of trade costs supports agglomeration in Foreign. Panel (b) depicts such a scenario when the size advantage of Foreign is reduced to λ = 3/2 and relative productivity remains at χ = 1.1. Now there is a single sustain point and Home is the only stable agglomeration outcome.

More formally, the expression for χ^s in (26) has a positive second derivative with respect to φ_s and attains a minimum at the point:

(27)

where the min operator is required for values of . The minimum relative productivity that sustains agglomeration in Home is:

(28)

Below this level of relative productivity, a bifurcation occurs and full agglomeration is never stable in Home. A corresponding maximum level of can be derived for Foreign. The value of is increasing in λ. Even slight size asymmetry requires that χ > 1 must hold to sustain agglomeration when Home is small. Conversely, large countries can sustain agglomeration for somewhat severe levels of disadvantage.

The rents to agglomeration in Home decrease when χ falls. As defined, is the level of χ below which the rents to agglomeration are completely overwhelmed by the gains from superior technology in Foreign. This result is summarized in the following proposition:

Relation to Symmetric NEG Models

Fully symmetric NEG models are included as a special case of this model when χ = λ = 1. Asymmetry resolves some of the fundamental indeterminacy in standard NEG models. When trade costs fall, the pattern of agglomeration is predicted by relative size and technology rather than chance events. The model does display a mild form of hysteresis when trade costs are increased. The qualitative pattern of manufacturing location is reversed, but at different threshold levels of trade costs.

This simple specification shares the same “genome” with a large class of vertical linkage models examined in Ottaviano and Robert-Nicoud (2005). The qualitative behavior examined above should be the same in less tractable models. The results apply directly to regional NEG models such as Forslid and Ottaviano (2003) where cross-country, skilled labor migration, rather than vertical linkages are the agglomeration mechanism.⁴

Relaxing the NFS Condition

The NFS condition can be relaxed for the sake of realism at the expense of simplicity. Wages and the allocation of labor within each country are potentially endogenous without factor price equalization. The NFS condition is a restriction on the share of manufactured goods in consumer and firm expenditure. If µ is larger than Home's share of world labor but smaller than Foreign's share, full manufacturing specialization in Home is possible.

Consistent with Ricardian predictions, full specialization only occurs in the country with a higher productivity in manufacturing. For example, say Home is small and more productive. The entire manufacturing sector is too large to fit entirely in Home, but low trade costs combined with higher productivity can make entry profitable. For low enough trade costs specialization in Home will be inevitable. Home manufacturing wages will necessarily exceed those in agriculture to attract labor. Foreign will still maintain some manufacturing and all agricultural employment at equalized wages of unity.

Qualitative Results

Figure 4 replicates Figures 1 except that µ = 0.5. A higher share of intermediate goods in fixed costs strengthens the production–cost linkages. As before, Foreign's initial size advantage in terms of market and supplier access is amplified as trade costs fall from autarky levels. Manufacturing shares will shift from Home to Foreign. This also effects the share of manufacturing employment in each country.

Similar to before, at some level of lower trade costs full specialization, rather than full agglomeration, in Home becomes possible but not inevitable. As trade costs fall further, full agglomeration in Foreign is no longer sustainable and Foreign loses part of it's manufacturing core to Home. Labor shares follow suit, but now Foreign retains some manufacturing employment. Home devotes its entire labor endowment to manufacturing and imports all of it agricultural good from Foreign. Given these small adjustments to the nature of equilibria, the patterns and intuition of section three are qualitatively preserved.

Analytical Results

Some of the tractability of section 3 can be salvaged. The agglomeration sustaining locus for Home is lost because wages, the labor allocation and number of firms in Home are changing with trade costs. The situation with respect to the Foreign version of equation (26) is different. The condition in (26) assumes wages are unity in both countries. When manufacturing agglomerates in Foreign, these assumptions continue to hold. For the levels of trade costs (φ) and technology (χ) where it is exactly satisfied, a firm can enter the Home market at the prevailing wage and turn a profit in the short run. In the long run, more firms enter, compete for labor and bid up the wage rate until Home is fully specialized in manufacturing.

Proposition 3. Full agglomeration in Home is welfare improving for all if and only if φ > χ^1/(a−1). Otherwise, Labor in in the periphery will experience a welfare loss.

Unlike symmetric models, agglomeration is not always welfare improving for a representative agent in the host country. The well-being of the periphery depends positively on the productivity level abroad and negatively on trade costs. There are ranges of trade costs where an improvement in Home technology would not necessarily benefit Foreign. Consider the example from Figure 1 where χ = 1.1. Agglomeration of all manufacturing shifts from Foreign to Home for φ > 0.77 but it is not welfare improving in Foreign until φ > 0.9. A similar calibrated result is obtained by Eaton and Kortum (2002). Welfare is increasing when a trading partner's technology improves for nearby countries, but less so for countries that are far away.