Competition among Large and Heterogeneous Small Firms
aa r X i v : . [ ec on . GN ] M a y Competition among Large and Heterogeneous Small Firms ∗ Lijun Pan † Nanjing University, Osaka University Yongjin Wang ‡ Nankai UniversityMay 25, 2020
Abstract
We extend the model of Parenti (2018) on large and small firms by introducingcost heterogeneity among small firms. We propose a novel necessary and sufficientcondition for the existence of such a mixed market structure. Furthermore, in con-trast to Parenti (2018), we show that in the presence of cost heterogeneity amongsmall firms, trade liberalization may raise or reduce the mass of small firms in op-eration.
Keywords : large firms, small firms, firm heterogeneity, trade liberalization
JEL classification : D4, L10, F11 ∗ Pan gratefully acknowledges the financial support from the Grant-in-aid for Research Activity, JapanSociety for the Promotion of Science (19K13681). † Main address: School of Economics, Nanjing University, Nanjing, China. Affiliated address: Instituteof Social and Economic Research, Osaka University, Japan. Email: [email protected] ‡ School of Economics, Nankai University, Tianjin, China. Email: [email protected].
Introduction
Firm heterogeneity is prevalent in international trade. On the one hand, it is evident thatfirms are heterogeneous in terms of firm size, with a few dominant large exporters and ahost of negligible small firms. On the other hand, small firms also display heterogeneityin the form of productivity differences. This note attempts to address these two typesof firm heterogeneity in a coherent model and examine the impact of trade liberalization.We draw on the model in Parenti (2018) where large and small firms coexist, butgoes beyond that model by introducing cost heterogeneity among small firms. In ourmodel, large firms are treated as established incumbent oligopolists, whereas small firmsare treated as monopolistic competitors who endogenously make entry decisions andface initial uncertainty about their future productivity prior to entry. The heterogeneityamong small firms is in the spirit of Melitz-Ottaviano type’s heterogeneity. First, we propose a novel necessary and sufficient condition for the existence of sucha mixed market structure in a closed economy, accommodating the cost heterogeneityamong small firms. Large and small firms coexist if (i) the marginal cost of large firms isstrictly lower than that of the least efficient small firm in operation, and (ii) consumers’preferences on the differentiated goods are sufficiently high, the number of large firms isnot too high, and the cost of large firms is not too low. The former condition guaranteesthat large firms earn positive profits, and the latter condition guarantees a positive massof small firms in operation.Second, in contrast to Parenti (2018), we show that bilateral trade liberalizationmay raise the mass of small firms selling in each country. Assuming that small firmsshare the same cost and do not export, Parenti (2018) finds that trade liberalizationleads to the expansion of large firms and decreases the mass of small firms selling in eachcountry. However, we introduce cost heterogeneity among small firms and allow the mostefficient ones to export. Therefore, a reduction in trade cost as a result of bilateral tradeliberalization not only reduces the export cost of large firms, but also lowers the tradecost of small firms. The former effect, which is consistent with Parenti (2018), generatesa negative impact on the mass of small firms, whereas the latter effect generates apositive impact on the mass of small firms. The impact of trade liberalization on themass of small firms selling in each country depends on these two opposing effects. Whenthe number and cost of large firms are sufficiently high, the former effect dominates,and trade liberalization decreases the mass of small firms. Otherwise, the latter effectdominates, and trade liberalization increases the mass of small firms. We also find thattrade liberalization may raise the mass of small producers in each country.This paper contributes to the recent studies on the mixed market structure withlarge and small firms. Assuming that small firms have identical cost, Shimomura andThisse (2012) and Pan and Hanazono (2018) identify the condition for the coexistenceof large and small firms and examine the impact of a large firm’s entry in a closedeconomy. Complementary to these works, this paper derives a new coexistence conditionincorporating cost heterogeneity among small firms and investigates the impact of trade See Parenti (2018). See Melitz (2003) and Melitz and Ottaviano (2008). Representative works include Melitz (2003) and Melitz and Ottaviano (2008).
Consider an economy with L consumers, each supplying one unit of labor inelastically.Each consumer consumes two types of goods, a homogeneous good and differentiatedgoods. The homogeneous good is produced under constant returns to scale at unitcost and perfect competition, which implies a unit wage. The differentiated goods aresupplied by a discrete number N of large firms n = 1 , ..., N , and a continuum of smallfirms indexed by i ∈ [0 , M ]. Occupying a substantial market share, each large firmbehaves strategically as an oligopolist. Being negligible in the market, each small firmbehaves non-strategically as a monopolistic competitor. All consumers share the same utility function given by U = α ( Z M x i di + N X n =1 X n ) − β Z M ( x i ) di + N X n =1 ( X n ) ] − γ Z M x i di + N X n =1 X n ) + x , (1)where x i represents the individual consumption of the good produced by small firm i ,and X n represents the individual consumption of the good produced by large firm n . Thenumeraire good is represented by x . The demand parameters α > γ > β > i and large firm n are respectively p i = α − βx i − γ X , ∀ i ∈ [0 , M ] (2) P n = α − βX n − γ X , n = 1 , ..., N (3)2henever x i > X n >
0, and X = R M x i di + P Nn =1 X n is the individual total consumptionof the differentiated good . Inverse demands (2) and (3) can be inverted to the demands for small firm i andlarge firm n : q i = Lx i = αLβ + γ ( M + N ) − Lβ p i + Lβ γβ + γ ( M + N ) P , ∀ i ∈ [0 , M ] (4) Q n = LX n = αLβ + γ ( M + N ) − Lβ P n + Lβ γβ + γ ( M + N ) P , n = 1 , ..., N (5)where P = R M p i di + P Nn =1 P n is the aggregate price. To ensure a positive consumption,i.e., q i ≥ Q n ≥
0, the prices of large and small firms should satisfy p i , P n ≤ β + γ ( M + N ) ( αβ + γ P ) = p max , (6)where p max represents the price at which demand for the good is driven to zero. We consider a 2-stage game. In the first stage, small firms enter the market. In thesecond stage, large and small firms who have entered the market compete in price.
Small Firms
Entry in the differentiated product sector is costly as each firm incursproduct development and production start-up costs. Subsequent production exhibitsconstant returns to scale at a marginal cost c . Following Melitz and Ottaviano (2008),Research and development yield uncertain outcomes for c, and firms learn about this costlevel only after making the irreversible investment f E required for entry. We assume thatthe marginal cost of each small firm is a draw from a common and known distribution G ( c ) with support [0 , c M ] . Since the entry cost is sunk, firms that can cover their marginalcost survive and produce. All other firms exit the industry.Substituting the demand function (4), the gross profit of small firm i with marginalcost c i can be expressed as π i = ( p i − c i )( αLβ + γ ( M + N ) − Lβ p i + Lβ γβ + γ ( M + N ) P ) , ∀ i ∈ [0 , M ] . (7) Large Firms
Large firms are established incumbents and produce at marginal cost C . Substituting the demand function (5), the profit of large firm n can be expressed asΠ n = ( P n − C )( αLβ + γ ( M + N ) − Lβ P n + Lβ γβ + γ ( M + N ) P ) , n = 1 , ..., N. (8)3 .3 The Equilibrium Analysis Small firm i maximizes its profit with respect to its price p i , treating theaggregate price P as given since its market share is negligible. The profit maximizingprice p i and quantity q i then satisfy q i ( c i ) = Lβ [ p i ( c i ) − c i ] , (9)If the profit maximizing price p i is above the price bound p max from (6), then smallfirm i exits. Let c D denote the cost of the small firm who is indifferent about remainingin the industry. This firm earns zero profit as its cost is equal to its marginal cost, thatis, p ( c D ) = c D = p max , and its demand level q ( c D ) is driven to 0. We assume that c M is high enough to be above c D , so that some firms with cost draws between these twolevels exit. All firms with cost below c D earn positive profits (gross of the entry cost)and remain in the industry. Substituting (6), and knowing that p max = c D , the firstorder condition of a small firm with c < c D yields its optimal price p ( c ) and quantity q ( c ): p ( c ) = 12 ( c D + c ) , (10) q ( c ) = L β ( c D − c ) , (11)and the profit can be expressed by π ( c ) = L β ( c D − c ) . (12) Large Firms
Different from a small firm, who treats the aggregate price P paramet-rically, a large firm behaves strategically and internalizes its impact on P . Large firm n maximizes its profit given by (8), yielding the optimal price P n = P = c D + (1 − Θ) C − Θ , n = 1 , ..., N, (13)where Θ = γ/ [ β + γ ( M + N )] ∈ (0 ,
1) represents the internalization by the large firm . Due to the internalization, a large firm’s price depends on the threshold cost c D of smallfirms and the mass M of small firms, which are endogenously determined by the freeentry condition in the first stage . Moreover, incurring the same marginal cost C , largefirms set the same price P .Similar to a small firm, a large firm stops operation if and only if the profit maximizingprice P is above the price bound p max . Since p max = c D , the necessary and sufficientcondition for the existence of large firms is C < c D . We assume that large firms incur zero fixed cost. If the fixed cost is positive, then the condition forthe existence of large firms should be stricter. .3.2 Stage 1 Prior to entry, the expected profit of a small firm is R c D π ( c ) dG ( c ) − f E . Substituting(12), the free entry condition can be rewritten as L β Z c D ( c D − c ) dG ( c ) = f E , (14)which determines the cost cut-off c D .To obtain tractable results, we assume the Pareto parametrization for the cost drawsof small firms, i.e., G ( c ) = ( cc M ) k , c ∈ [0 , c M ] . (15)Given this parametrization, the cut-off cost level c D determined by (14) is then c ∗ D = ( βφL ) k +2 , (16)where φ = 2( k + 1)( k + 2)( c M ) k f E is the technology index.Substituting the optimal prices of small and large firms given by (10) and (13), andusing that p max = c D , (6) can be expressed as( α − c ∗ D ) βγ = M c ∗ D k + 1) + N ( c ∗ D − C ) 1 − Θ( M )2 − Θ( M ) , (17)In equation (17), the LHS is constant and positive, whereas the RHS increases with M .Let M ∗ be the equilibrium mass of small firms. To ensure that M ∗ >
0, the RHS shouldbe smaller than the LHS when M = 0. Therefore, equation (17) uniquely determines apositive mass M ∗ of small firms in operation if and only if( α − c ∗ D ) βγ > N ( c ∗ D − C ) β + γ ( N − β + γ (2 N − . As shown earlier, large firms earn a positive profit if and only if
C < c D . Substitutingthe equilibrium cost cut-off of small firms in (16), we establish the condition for thecoexistence of large and small firms in Proposition 1. Proposition 1
There exists a unique mixed market equilibrium where large and smallfirms coexist if and only if (i)
C < ( βφ/L ) k +2 and (ii) [ α − ( βφ/L ) k +2 ] β > γN [( βφ/L ) k +2 − C ][ β + γ ( N − / [2 β + γ (2 N − . The first condition, which guarantees that large firms earn positive profits, requiresthat the cost of large firms be smaller than the cost cutoff of small firms, which issatisfied if the technology index φ is sufficiently high and the market size L is not toolarge. The second condition, which guarantees a positive mass of small firms, is satisfiedif the consumer’s preference for the differentiated good α is sufficiently high, the cost C of large firms is not too low in comparison with the cost cutoff ( βφ/L ) k +2 of small firms,and the number N of large firms is not too large.5n models with the coexistence of large and small firms, see, e.g., Shimomura andThisse (2012), Parenti (2018), and Pan and Hanazono (2018), it is commonly assumedthat small firms share the same technology. When large firms are single-product andfacing zero fixed cost, their assumption implies large firms operate if and only if C
Similar to the closed economy, we assume that all the large firms sharethe same marginal cost C in production, and the delivered cost of a unit to the foreigncountry is τ C . Let P hD and Q hD represent the domestic levels of profit maximizing priceand quantity sold for a large firm producing in country h . The large firm also exportsoutput Q hX at a delivered price P hX . Analogously to (13), the profit maximizing domesticand export prices are P hD = c hD + (1 − Θ h ) C − Θ h , (22) P hX = τ c hX + (1 − Θ f ) C − Θ f , where Θ h = γ/ [ β + γ ( M h + 2 N )] and Θ f = γ/ [ β + γ ( M f + 2 N )]. A large firm in country h stops selling in the domestic market if and only if P hD is above the price bound p h max .Since p h max = c hD , the necessary and sufficient condition for the domestic operation oflarge firms is C < c hD . Furthermore, it stops exporting if and only if P hX is above the pricebound p f max (where f = h ). Since p f max = τ c hX , the necessary and sufficient conditionfor the existence of exporting large firms is C < c hX . These two conditions indicate that7arge firms share the same cost cutoffs with small firms in both domestic and foreignmarkets.Therefore, large firms in both countries export if and only if
C < min { c HX , c FX } . Entry of small firms is unrestricted in both countries. Small firms choose a productionlocation prior to entry and pay the sunk entry cost. We assume that both countriesshare the same technology for small firms – referenced by the entry cost f E and costdistribution G ( c ). Therefore, the free entry condition is expressed as L β Z c hD π hD ( c ) dG ( c ) + Z c hX π hX ( c ) dG ( c ) = f E . With Pareto parametrization for the cost draws G ( c ) in both countries, the free entrycondition can be rewritten as ( c hD ) k +2 + τ ( c hX ) k +2 = βφL . Since c fX = c hD /τ , the free entry condition (23) can be expressed as( c hD ) k +2 + ρ ( c fD ) k +2 = βφL , (23)where ρ = ( τ ) − k ∈ (0 ,
1) is an inverse measure of trade costs (the “freeness” of trade).This system can be solved for the cutoffs in both countries: c h ∗ D = c f ∗ D = c ∗ D = [ βφL (1 + ρ ) ] / ( k +2) , (24)by which the average cost of small firms selling in each country is ¯ c ∗ D = kc ∗ D / ( k + 1) . By (18) and (22), the aggregate price in country h is given by P h = M h (2 k + 1)2( k + 1) c ∗ D + N c ∗ D + (1 − Θ h ) C − Θ h + N c ∗ D + (1 − Θ f ) τ C − Θ f , Substituting P h , (18) can be expressed as( α − c ∗ D ) βγ = M h c ∗ D k + 1 + N ( c ∗ D − C ) 1 − Θ h − Θ h + N ( c ∗ D − τ C ) 1 − Θ h − Θ h . (25)which uniquely determines the mass of small firms in country h because the RHS strictlyincreases with M h . (25) also implies that M f = M h = M and hence Θ f = Θ h = Θ . Therefore, (25) could be reduced to( α − c ∗ D ) βγ = M c ∗ D k + 1 + 1 − Θ( M )2 − Θ( M ) N [( c ∗ D − C ) + ( c ∗ D − τ C )] , (26)which uniquely determines the equilibrium mass M ∗ of small firms selling in each country.To ensure that M ∗ >
0, we assume that ( α − c ∗ D ) β/γ > N [( c ∗ D − C ) + ( c ∗ D − τ C )][ β + γ ( N − / [2 β + γ (2 N − . .2 Trade Liberalization Based on the above equilibrium analysis, now we examine how bilateral trade liberaliza-tion impacts the mass of small firms.First, (24) indicates that a reduction in trade cost as a result of bilateral tradeliberalization decreases the cost cut-off of small firms, that is, dc ∗ D /dτ > dMdτ = − [ αβγ + (1 + τ ) N C ] kk +2 ρρ +1 1 τ − − Θ2 − Θ N C c ∗ D k +1) + ( Θ2 − Θ ) [( c ∗ D − C ) + ( c ∗ D − τ C )] N , which is negative if and only if2 − Θ( M ∗ )1 − Θ( M ∗ ) αβγ > ( 2 ρ + k + 2 kρ τ − N C, (27)where M ∗ is the equilibrium mass of small firms determined by (26).Proposition 2 establishes the impact of bilateral trade liberalization on the mass ofsmall firms selling in each country. Proposition 2
A decrease in trade costs as a result of bilateral trade liberalizationincreases the mass of small firms selling in each country if ( αβ/γ )[2 − Θ( M ∗ )] / [1 − Θ( M ∗ )] > [(2 ρ + k + 2) τ / ( kρ ) − N C , and decreases the mass of small firms selling ineach country otherwise.
A reduction in trade cost generates two opposing effects on the mass of small firmsselling in each country. First, trade liberalization reduces the export cost and henceenables more small firms to export, which potentially increases the mass of small firms.Second, a decrease in trade cost also generates cost savings on the exported goods of largefirms, which induces large firms to expand export. This generates competitive pressureon small firms and may reduce the mass of surviving small firms. As marginal cost C rises, the cost savings become stronger and lead to more aggressive exporting behaviorby large firms. Furthermore, the aggregation of such aggressive exports increases withthe number N of large firms. Therefore, condition (27) holds as long as the number N or marginal cost C of large firms is not too high.Last, we show that bilateral trade liberalization may also raise the mass of smallproducers in each country. By symmetry, in each country, there are M E G ( c D ) domesticsmall producers and M E G ( c X ) foreign small exporters. Thus, M E G ( c D )+ M E G ( c D /τ ) = M , by which the mass of small entrants in each country can be expressed as M E = M ( c M /c D ) k / (1 + ρ ). The mass of small producers in each country is then expressed by M D = M E G ( c D ) = M ρ . dM D dτ = 11 + ρ [ dMdτ + kρM (1 + ρ ) τ ] . A reduction in trade cost impacts the mass M D of domestic small producers through(a) the change in the mass M of small firms selling in each country and (b) the realloca-tion between domestic producers and foreign exporters within the group of small firms.The change in the mass of small firms selling in each country, according to Proposition2, is positive as long as thenumber and marginal cost of large firms are not too high.Furthermore, trade liberalization invites the most productive small firms to increaseexport, which makes competition tougher and may force some of the least productivesmall producers to exit. This generates a negative impact on the mass of domestic smallproducers. Therefore, if the rise in the total mass of small firms selling in each countryoutweighs the negative impact from the export expansion by the most productive smallfirms, then trade liberalization would also raise the mass of small producers in each coun-try. Otherwise, trade liberalization reduces the mass of small producers. Proposition 3establishes this result with the precise condition. Proposition 3
A decrease in trade costs as a result of bilateral trade liberalization in-creases the mass of small producers if αβ/γ + (1 + τ ) N Ck + 2 > M ∗ c ∗ D k + 1) + M ∗ N [2 c ∗ D − (1+ τ ) C ][ Θ( M ∗ )2 − Θ( M ∗ ) ] + N C (1 + ρ ) τρk − Θ( M ∗ )2 − Θ( M ∗ ) , and decreases the mass of small producers otherwise. Here c ∗ D and M ∗ are the equilibrium values determined in (24) and (26). Owing to theexpansion of small exporters as a result of trade liberalization, the range of parameterswhere trade liberalization increases the mass of small producers narrows down. We conclude by highlighting the distinction between our results and those in Parenti(2018). Take Proposition 2 for instance. Assuming that small firms share the same costand do not export, Parenti (2018) finds that trade liberalization decreases the mass ofsmall firms. In Proposition 2, we go a step further by introducing cost heterogeneityamong small firms and allowing a portion of small firms to export, identifying twoopposing effects of trade liberalization on the mass of small firms. The second effect,which reduces the mass of small firms, is consistent with Parenti (2018), whereas thefirst effect is novel here. The insights of Proposition 2 extend to Proposition 3.
References [1] Melitz, M. J., 2003. The impact of trade on intra-industry reallocations and aggregateindustry productivity. Econometrica, 71(6), 1695-1725. We identify both the case when dM ∗ D /dτ > dM ∗ D /dτ < α = 0 . β = γ = 1 , N = 1, C = 0 . L = 100, f E = 1, and τ = 1 . dM ∗ D /dτ < α = 2 . β = γ = 1 , N = 2, C = 0 . L = 100, f E = 1, and τ = 1 . dM ∗ D /dτ >0.