Trade Unions in an Open Economy – the Caseof Firm-specific Bargaining Power

Marco de Pinto∗

IAAEU Trier and Trier University

Jochen Michaelis†

University of Kassel

July 29, 2016– Preliminary and incomplete –

Please do not cite

Abstract

he bargaining power of trade unions is not identical across firms.The empirical evidence suggests that large and productive firms facestronger trade unions than small and less productive firms. Using aMelitz-type model, we reconsider the impact of unionization on themacroeconomic performance of an economy. For a closed economy,we show that, (i) due to a strong deterrance of market entry, morepowerful trade unions lower the equilibrium cutoff productivity, (ii)more powerful unions may be good for aggregate employment; (iii) butreduce aggregate output. In the open-economy setting, our focus is ontrade liberalization. In a world with uniform (firm-specific) bargainingpower, lower trade costs are neutral for (increase) the unemploymentrate. Aggregate output always goes up.

Keywords: Trade Unions, Bargaining Power, Firm Heterogeneity,International Trade, Unemployment

JEL Classification: F 1, F 16, J 5

∗Institute for Labour Law and Industrial Relations in the European Union (IAAEU),Trier University, Behringstr. 21, D-54296 Trier, Phone: +49651/2014762,Email: depinto@iaaeu.de†University of Kassel, Nora-Platiel-Str. 4, D-34109 Kassel, Phone: +49561/8043562,

Email: michaelis@wirtschaft.uni-kassel.de

1 Introduction

Our work is motivated by the observation that there is a positive relationshipbetween union density and firm size. In Germany, for instance, the fractionof the workforce organized in a trade union increases from 7.6 percent forfirms with less than 10 employees to 33.7 percent for firms with more than2000 employees (Biebeler and Lesch, 2007). Using union membership ratesas a proxy for unions’ bargaining power, we conclude from the empiricalliterature that unions’ bargaining power is increasing in firm size. Since firmsize is strongly correlated with firm productivity, we further conclude thathigh productivity firms will face strong trade unions. In a Melitz-world,where the most productive firms are exporters, the export status of a firmshould thus go hand in hand with a strong trade union. This is confirmed byAgell (2002), who, uses data for 20 OECD countries and shows that tradeopenness is positively and significantly correlated with union density.

In the present study, we design a general equilibrium model to reconsiderthe impact of trade unions on the macroeconomic performance of an openeconomy. How does the wage bargain affect variables such as wages, employ-ment, output, exports, and number of active (exporting) firms? The pointof departure from the literature is the endogeneity of the bargaining powercoefficient; the unions’ bargaining power is firm-specific. More precisely, thebargaining power coefficient is assumed to be increasing in the level of firmproductivity. Our results indicate that such a firm-specific split of the quasi-rents allows for new insights into the impact of the unionization in an openeconomy.

The theoretical literature offers some explanations for the described em-pirical regularity and these explanations may serve as a microeconomic ratio-nale for the endogeneity of the bargaining strength. Agell (2002) argues thatthe process of globalization increases the fraction of the workforce that facesuncertainty about their future position in the wage distribution. As a conse-quence, risk averse workers demand more social insurance, whereby joininga redistributive trade union becomes incentivized. Breda (2015) emphasizesthe costs paid by workers to organize at the firm level, which include sunkcosts invested in the organization process, costs of getting union recogni-tion, and costs of the work invested by union representatives. The incentiveto bear these costs and thus to secure union bargaining power is increasingin the prospects of rent-extraction. The higher the firm productivity, thehigher are the firms quasi-rents, and the higher is the incentive for workersto increase their bargaining power.

Our modeling approach builds on the framework by Egger and Kreick-emeier (2009) who implement efficiency wages into a Melitz (2003) model.

1

We replace efficiency wages by a wage bargain at the firm-level. The unionbargaining strength is divided into a minimum level, which is identical acrossall trade unions, and a firm-specific part, which is increasing in firm produc-tivity. Our focus is on the question how an increase in the minimum level oftrade union power affects the general equilibrium. To highlight the quanti-tative importance of our findings, we also solve our model numerically usingstandard parameter values from the literature.

Our main results are as follows. (i) An increase in the minimum unionbargaining power reduces the cutoff productivity (Melitz lottery) and in-creases the share of low-productivity firms in the economy. Intuitively, thereare two countervailing forces. On the one hand, more powerful unions pushup the wage level. Firms increase their prices, product demand declines,revenues decline, profits decline, and the cutoff productivity goes up. Onthe other hand, given the reduction of profits, the mass of firms enteringthe market decreases. Competition becomes less intense which in turn raisesdemand, revenues and profits of the incumbents. This effect decreases thecutoff productivity. If the unions bargaining power depends positively on thefirms productivity, the latter effect dominates.1 A corollary of this finding:the export cutoff decreases too so that the share of exporting firms shifts up.

(ii) The increase in the bargaining power raises unemployment (as itis well documented by the literature) but the negative employment effectis weakened by the decline in the cutoff productivity. In the benchmarkcase where all unions have the same bargaining power, more powerful tradeunions lead to a decline in aggregate employment via a decline in the massof active firms. In our setting, the decline in the mass of firms is smaller,so that the decline in aggregate employment (increase in unemployment) issmaller. (iii) Regarding aggregate output (which also measures welfare inour model), we find the opposite result. Aggregate output declines in thebargaining power but this reduction is stronger in the case of firm-specificbargaining power. This is a direct consequence of the decline in the cutoffproductivity, which reduces the average productivity of the incumbents suchthat output decreases even further (even if the negative employment effect isless pronounced).

Our paper is related to the strand of literature that investigates the roleof unionization in an open economy setting. Some studies, for instance,Koskela and Stenbacka (2009), Lommerud et al. (2009) and Zhao (1995),focus on the effect of unionization on outsourcing. Similarly, Eckel and Eg-

1In the benchmark case where all firms have the same bargaining power, both effectscancel each other out such that the cutoff productivity and thus firm-selection is unaffectedby the unions bargaining power (see de Pinto and Michaelis, 2014 for this result).

2

ger (2009) analyze the interaction between collective bargaining and foreigndirect investment. The relationship between collective bargaining and themacroeconomic performance in a trade-liberalized economy (mostly focusingon trade in goods) is also analyzed from various perspectives. For example,Kreickemeier and Meland (2013) investigate how trade liberalization affectswages, employment and welfare with (partly) unionized labor markets. Mon-tagna and Nocco (2013) study the effect of different union bargaining levelson the competition between heterogeneous firms. Most recently, Egger etal. (2015) analyze the influence of unionization on inter-industry trade andwelfare in a situation where differences in the degree of unionization betweencountries imply comparative advantages. All these studies focus on the role ofunions bargaining power. However, at least to the best of our knowledge, nostudy considers the empirical regularity that union bargaining power differsacross firms. The present paper aims to fill this gap.2

2 Model

2.1 Production

We consider an open economy model with two symmetric countries. Eacheconomy consists of two sectors: a final goods sector and an intermediategoods sector. In the final goods sector, markets are perfectly competitive andthe final good Y is given by a CES-aggregator over all available intermediategoods:

Y = M− 1σ−1

t

[∫ M

0

q(ω)ρdω +

∫ Mm

0

qm(ν)ρdν

] 1ρ

. (1)

M (Mm) denotes the mass of varieties produced in the home (foreign) coun-try. The mass of all available varieties is given by Mt = M + Mm. q(ω)represents the used quantity of variety ω, which is produced in the homecountry, while qm(ν) stands for the imported quantity of variety ν, which isproduced in the foreign country. ρ ≡ σ/(σ − 1) measures the love of variety,where σ > 1 equals the elasticity of substitution between any two varieties.We choose Y as the numeraire and normalize the corresponding CES priceindex P at unity.

2As we introduce labor market imperfections into an intra-industry trade model withheterogeneous firms, our paper also contributes to the large and still growing strand ofliterature in which the Melitz (2003) framework is extended by different frictions. For theimplementation of efficiency wages see, for instance, Amiti and Davis (2012), Davis andHarrigan (2011) and Egger and Kreickemeier (2012). For the use of search and matchingfrictions compare Helpman and Itskhoki (2010) as well as Helpman et al. (2010).

3

In the intermediate goods sector, firms have to bear fix costs Fe (measuredin units of the final good) to enter the market. After entry, firms draw aproductivity level from a Pareto distribution with G(φ) = 1 − φ−k and thesupport φ ∈ [1,∞], where k denotes the shape parameter of the distribution.Production for the domestic market is given by q = φh, with h denotingemployment. Production for the export market (indexed by x) is associatedwith iceberg transport costs τ ≥ 1: qx = τ−1φhx. Firms can either produceonly for the domestic market or serve the home and foreign market. Totaloutput and employment are given by, respectively, qt = q + Iqx and ht =h+Ihx, where I is an indicator variable which equals one if firms export andzero otherwise.

Both the production for the domestic and export market require (over-head) fix costs F and Fx (measured in units of the final good). Profits fromdomestic and export sales are given by, respectively (see Melitz, 2003):

π =

(p− w

φ

)q − F, (2)

πx =

(px −

τw

φ

)qx − Fx, (3)

with p (px) denoting the price for the variety that is sold in the domestic(export) market and w representing the wage rate. We assume that all em-ployees of a firm receive the wage w, i.e. we do not allow wage differentiationwithin firms. Total profits read πt = π + Iπx.

The goods market in the intermediate goods sector is monopolistic com-petitive and each firm produces one variety of the intermediate good.

2.2 Workers

Both countries are endowed with a mass of identical workers L. Workersinelastically supply one unit of labor at the intermediate goods sector. Weassume that labor units are internationally immobile.

Workers’ expected income is defined as b = uB+(1−u)we, where u ∈ [0, 1]denotes the unemployment rate, B stands for lump-sum tax financed unem-ployment benefits and we is the workers’ expected wage rate. We assumethat B is a constant share s on the expected wage rate, i.e. B = swe, withs ∈ (0, 1). This yields:

b = (1− (1− s)u)we. (4)

4

2.3 Unions

We assume that labor markets in both countries are unionized, i.e. we donot consider an endogenous union formation. We further suppose that unionsact at the firm-level and consider post-entry closed-shop unions, i.e. workerswho are hired at one particular firm must become a member of the respectiveunion. The utility function of the union is given by:

U = htw + (m− ht)b, (5)

with m(> ht) being the union’s membership.There is a Nash-bargaining over w between the firm-specific union and

the firm, while the firm has the right to manage employment. The Nash-product is defined as N = (U −U)γ(πt− πt)1−γ, where γ ∈ [0, 1] denotes theunion’s bargaining power, U = mb is the union’s and πt = −F − IFx is thefirm’s outside option.

It is commonly assumed in the literature that the union’s bargainingstrength is equal across firms. We pursue a different approach and assumethat the bargaining power is divided into two components: a parameter γ,which is identical across all trade unions, and a firm-specific part, which isincreasing in firm productivity. Formally, we assume:

γ(φ, γ) = γ(1− χφ−1

), (6)

where χ is an indicator variable. If χ = 0, bargaining power depends onlyon γ. If χ = 1, bargaining power increases in productivity. Note that γ canbe interpreted as the bargaining power of the most productive firm because1− φ−1 → 1 if φ→∞.

2.4 Timing

The timing of events is as follows:

1. Firms decide about market entry, i.e. paying the entry costs Fe anddrawing a productivity level. Conditional on entry, firms decide aboutproduction.

2. Unions and firms Nash-bargain over wages.

3. Firms decide about employment (which is equivalent to the choice ofthe profit-maximizing price).

4. The final goods are produced.

5

3 Closed Economy

In this section, we analyze the consequences of collective bargaining withouttrade, i.e. we assume I = 0 and Mt = M .

3.1 Partial Equilibrium

The partial equilibrium is determined by solving the aforementioned gameby backwards induction and taking the macroeconomic variables b and M asgiven. At stage four, the final goods producer maximizes Y (which is equal

to its profits) by choosing q(ω) subject to PY =∫M0q(ω)p(ω)dω. Using the

autarky level of (1) yields:

q(ω) = p(ω)−σY

M. (7)

Firms in the intermediate goods sector maximize profits over p subjectto the final goods producer’s demand (7) at the third stage. Due to theCES assumption, profit-maximizing prices are a constant mark-up over (firm-specific) variable costs:

p(φ,w) =1

ρ

w

φ. (8)

Inserting (8) into (7) leads to the profit-maximizing output q(φ,w) = p(φ,w)−σY/M .This pins down labor demand as h(φ,w) = φ−1p(φ,w)−σY/M . Revenues aredefined as r(φ,w) = q(φ,w)p(φ,w) and the profit function reads:

π(φ,w) = (1− ρ)r(φ,w)− F. (9)

The Nash-wage bargaining takes place at stage two. Maximizing theNash-product over w subject to the firm’s optimal choices at stage threeleads to the bargained wage:

w(φ) = θ(φ, γ)b, (10)

θ(φ, γ) ≡ σ − 1 + γ(φ, γ)

σ − 1> 1, (11)

with θ representing the wage markup.

Lemma 1 For χ = 1, the bargained wage is increasing in the firm’s produc-tivity because high-productive firms face unions with higher bargaining power,which bargain higher wage-markups. For χ = 0, wages are identical for allfirms because unions have the same bargaining power and bargain the samewage markup.

6

At stage one, firms enter the market as long as expected profits are highenough to cover entry costs. Due to free entry, we get:

1

δ

∫ ∞1

(1− ρ) (r(φ,w(φ))− F ) dG(φ) = Fe, (12)

where δ denotes the exogenously given death probability of firms. Once afirm has entered the market and received a productivity, it starts producingif and only if profits are non-negative. The marginal firm with productivityφ has zero-profits:

(1− ρ)r(φ, w(φ)

)− F = 0. (13)

3.2 General Equilibrium

3.2.1 Cutoff Productivity

The equilibrium cutoff productivity is determined by the free-entry condition(12) and the zero-profit cutoff condition (13). As shown in the AppendixA.1.1, we can rewrite both equations as:

Y

M=

σ

1−G(φ)

(δFe + (1−G(φ))F

), (14)

Y

M=

(θ(φ, γ)

φ

)σ−1σF

1−G(φ)

∫ ∞φ

(φ

θ(φ, γ)

)σ−1dG(φ). (15)

Combining (14) and (15), we obtain:

E ≡

(θ(φ∗, γ)

φ∗

)σ−1 ∫ ∞φ∗

(φ

θ(φ, γ)

)σ−1dG(φ)− (φ∗)−k − δFe

F= 0, (16)

which implicitly pins down the equilibrium cutoff productivity φ∗ = φ∗(γ),where ∗ indicates an equilibrium outcome.

3.2.2 Labor and Goods Market

The labor market equilibrium is characterized by the expected wage rate we,the expected income b and the unemployment rate u. By definition, theexpected wage rate is given by:

we =(

1−G(φ))−1 ∫ ∞

φ

w(φ)dG(φ). (17)

7

In the equilibrium,(1− u)Lwe = W = ρY (18)

holds, where we have used the fact that total wage income W is a constantshare ρ of total revenues R = PY = Y due to monopolistic competition.Total employment is given by:

(1− u)L = M(

1−G(φ))−1 ∫ ∞

φ

h(φ)dG(φ)

= ρσY(

1−G(φ))−1 ∫ ∞

φ

φσ−1w(φ)−σdG(φ),

(19)

where the second line follows from inserting the technology and the optimaloutput of firms. Substituting (19) into (18) yields:

we = ρ−(σ−1)(

1−G(φ))[∫ ∞

φ

φσ−1w(φ)−σdG(φ)

]−1. (20)

By combining (17) and (20) as well as inserting (10), we can solve for theequilibrium expected income as:

b∗(φ∗, γ) = ρ[Γ1(φ

∗, γ) · Γ2(φ∗, γ)

] 1σ−1

, (21)

Γ1(φ∗, γ) ≡

(1−G(φ∗)

)−1 ∫ ∞φ∗

φσ−1θ(φ, γ)−σdG(φ), (22)

Γ2(φ∗, γ) ≡

(1−G(φ∗)

)−1 ∫ ∞φ∗

θ(φ, γ)dG(φ). (23)

Inserting (10) and (21) into (17), we obtain the equilibrium expected wage:

we∗(φ∗, γ) = ρΓ1(φ∗, γ)

1σ−1 Γ2(φ

∗, γ)σσ−1 . (24)

With b∗ and we∗ at hand, we can compute the equilibrium unemploymentrate. Substituting (24) and (21) into (4) and rearranging yield:

u∗(φ∗, γ) =1

1− s

(1− 1

Γ2(φ∗, γ)

). (25)

Finally, we use the equilibrium condition (18) to calculate equilibriumoutput as:

Y ∗(φ∗, γ) =(

1− u∗(φ∗, γ))we∗(φ∗, γ)

L

ρ. (26)

Note that we use aggregate output per capita Y ∗/L as a measure for welfareas usually done in these class of models (see Egger and Kreickemeier, 2009).3

3To complete the determination of the general equilibrium, we can use (12) to pin

8

3.3 The Role of the Bargaining Power

3.3.1 Analytical Solution

In this section, we analyze the effect of an increase in the union’s bargainingpower. Specifically, we consider a rise of the parameter γ, i.e. all unions inthe economy become more powerful, and distinguish between an economywhere firm-level trade unions have the same bargaining strength (χ = 0) andan economy where the unions’ bargaining power is firm-specific (χ = 1).

Regarding the effect on the equilibrium cutoff productivity φ∗, we find:

Proposition 1 For χ = 1, φ∗ is decreasing in γ, i.e. dφ∗/dγ < 0. For

χ = 0, φ∗ is not affected by γ.

Proof 1 See Appendix A.1.2

An increase in γ raises wage payments of all operating firms. This has twocountervailing effects. On the one hand, profits of the least productive firms,c.p., decline such that they are forced out of the market. On the other hand,expected profits decrease which lowers the incentive to enter the market. Themass of operating firms decline, which increases the demand for varietiesproduced by the incumbent firms. Profits raise and more low-productivefirms are able to produce without a loss. If the bargaining power is firm-specific (χ = 1), the second effect dominates the first such that the cutoffproductivity decreases. If the bargaining power is equal across firms (χ = 0),both effects cancel out and the cutoff productivity remains constant.

An increase in the unions’ bargaining power has the following impact onthe equilibrium expected wage rate we∗.

Proposition 2 For χ = 1, an increase in γ has an ambiguous effect on we∗.For χ = 0, an increase in γ does not affect we∗.

Proof 2 See Appendix A.1.3

In a world where unions have identical bargaining strength, an increase inthe bargaining power is fully absorbed by a decrease in the unemploymentrate (see de Pinto and Michaelis, 2014). As such, the equilibrium expectedwage rate remains constant. With firm-specific bargaining power, the effectis not clear-cut. The expected wage rate, c.p., increases because of the rise of

down the equilibrium mass of operating firms M∗. The equilibrium mass of entrants isthen given by M∗e = (1−G(φ∗)−1M∗.

9

the bargained wage, but, c.p., declines because of the reduction of the cutoffproductivity.

The effect on the equilibrium unemployment rate u∗ can be summarizedas:

Proposition 3 For χ = 1, the effect of an increase in γ on u∗ is parameter-dependent. For χ = 0, an increase in γ unambiguously increases u∗.

Proof 3 Differentiating (25) with respect to γ leads to:

du∗

dγ=∂u∗

∂φ∗︸︷︷︸>0

dφ∗

dγ+∂u∗

∂γ︸︷︷︸>0

. (27)

As shown in Appendix A.1.4, ∂u∗/∂φ∗ > 0 and ∂u∗/∂γ > 0 hold. Insertingthe respective expressions imply:

du∗

dγ

{<≥

}0⇔ ε

{<≥

}1− k + 1

kφ∗︸ ︷︷ ︸

>1

,

ε ≡ γ

φ∗

dφ∗

dγ< 0,

which proves the first part of the Proposition. If χ = 0, we have dφ∗/dγ = 0(see Lemma 1), which proves the second part of the Proposition.

An increase in γ raises the bargained wage. Operating firms respond tosuch an increase of their marginal costs by a price hike, which, c.p., reducesdemand and employment and thus increases the unemployment rate. If thebargaining power is firm-specific (χ = 1), there is a countervailing effect. Anincrease in γ reduces the cutoff productivity (as argued above), which, c.p.,raises the mass of active firms and decreases the unemployment rate. Thestrength of the latter effect is measured by the bargaining power elasticityof the cutoff productivity ε. If this elasticity would be sufficiently high inabsolute terms, the unemployment rate could eventually decrease in γ.

Finally, an increase in γ affects aggregate output (and hence welfare):

Proposition 4 For χ = 1, an increase in γ has an ambiguous effect on theequilibrium output. For χ = 0, an increase in γ unambiguously decreasesequilibrium output.

10

Proof 4 Differentiating (26) with respect to γ leads to:

dY ∗

dγ=dY ∗

du∗︸︷︷︸<0

du∗

dγ+dY ∗

dwe∗︸ ︷︷ ︸>0

dwe∗

dγ. (28)

If χ = 1, the sign of (28) is ambiguous because the employment effect isparameter-dependent (see Proposition 3) and the effect on the expected wagerate is not clear cut (see Proposition 2). This proves the first part of theProposition. If χ = 0, the employment effect is negative (see Proposition 3)and the expected wage does not depend on γ (see Proposition 2), which provesthe second part of the Proposition.

Intuitively, a higher unemployment rate reduces equilibrium output becausethe production factor labor is used less, while higher expected wage rateincreases Y ∗ because consumers have more wage income. The effect on ahigher bargaining power thus depends on the the effects on u∗ and we∗ andtheir interdependencies.

3.3.2 Numerical Solution

To quantify the impact of an increase in γ on we∗, u∗ and Y ∗, we solve ourmodel numerically. For that purpose, we rely on calibrations by Bernardet al. (2007) and set σ = 3.8, δ = 0.025 and Fe = 2. Additionally, wetake the results of the structural estimations by Balistreri et al. (2011) intoaccount and set k = 4.6 as well as F = 0.25 (which is the average value ofestimated fix costs in the US and Europe). For the replacement ratio, weassume s = 0.6. The country’s labor endowment L is normalized at unitywithout loss of generality.

The simulations results for the labor market are illustrated in Figure 1.We find that the equilibrium expected wage rate monotonously decreases inγ if the bargaining power is firm-specific (χ = 1). The decline in the cutoffproductivity and the resulting reduction of we is strong enough to overcom-pensate the increase in we due to higher bargained wages. In the limitingcase of γ = 1, the expected wage rate is about 11% lower compared to asituation where all firms have identical bargaining strengths (χ = 0). Theunemployment rate monotonously increases in γ. The positive employmenteffect of a decrease in the cutoff productivity is not high enough to com-pensate for the negative employment effect of higher wages. Compared tothe χ = 0 case, however, the unemployment rate is significantly lower. Forγ = 1, for example, this reduction is about 50%.

11

Figure 1: Labor Market

Figure 2 depicts the simulations results for the goods market. As theexpected wage rate decreases and the unemployment rate rises, it is notsurprising that the equilibrium output montonously declines in γ. In a worldwith firm-specific bargaining power, however, this reduction is up to 72%lower, compared to a world where unions have identical bargaining powers.This is because the negative employment effect is substantially mitigated, asshown before.

12

Figure 2: Goods Market

13

4 Open Economy

To analyze to effects of trade unions in an open economy, we consider thefull model with bilateral trade between both countries in the following. Weare in particular interested on the implications of an increase in the unions’bargaining power and on how trade liberalization affects the equilibriumoutcomes.

4.1 Equilibrium

The partial equilibrium is determined by solving the four-stage game analo-gously to the closed economy. Demand for the domestic and export marketare given by, respectively:

q(ω) = p(ω)−σY

Mt

, (29)

qx(ω) = px(ω)−σY

Mt

. (30)

Prices, output and employment for the export market are shares of therespective domestic outcomes (which are identical to the closed economycounterparts when replacing M with Mt): px(φ,w) = τp(φ,w), qx(φ,w) =τ−σq(φ,w) and hx(φ,w) = τ−(σ−1)h(φ,w) (see Melitz (2003) for the same re-sult). Revenues and profits from exporting thus read rx(φ,w) = τ−(σ−1)r(φ,w)and πx(φ,w) = (1 − ρ)τ−(σ−1)r(φ,w) − Fx. The solution of the Nash-bargaining yields the same wage rate as before such that the bargained wageis given by (10).

The zero-profit cutoff condition is identical to (13), while the free-entrycondition changes to:

1

δ

∫ ∞φ

((1− ρ)r(φ,w(φ))− F ) dG(φ)

+1

δ

∫ ∞φx

((1− ρ)τ−(σ−1)r(φ,w(φ))− Fx

)dG(φ) = Fe,

(31)

where φx denotes the cutoff productivity for exporting defined by:

πx(φx, w(φx)) = (1− ρ)τ−(σ−1)r(φx, w(φx))− Fx = 0. (32)

Firms with productivities lower than φx serve only the domestic market, whilefirms with productivities higher than φx additionally export. We can define

14

the ex-ante probability that a firm exports as (see Egger and Kreickemeier,2009):

α(φ, φx) ≡1−G(φx)

1−G(φ), (33)

such that the mass of exporters is given by Mx = αM .The determination of the general equilibrium is conceptually identical to

the procedure described in Section 3.2. As shown in Appendix A.2.1, we cancompute two Eqs. which simultaneously determine the equilibrium cutoffand export cutoff productivity as a function of the bargaining power andtrade costs: φ∗ = φ∗(γ, τ, Fx) and φ∗x = φ∗x(γ, τ, Fx). These Eqs. read:

E1 ≡

(θ(φ, γ)

θ(φx, γ)

φx

φ

)σ−1

− F

τσ−1Fx= 0, (34)

E2 ≡ E(φ, γ) +

(θ(φ, γ)

φ

)σ−1

τ−σ∫ ∞φx

(φ

θ(φ, γ)

)σ−1

dG(φ)− φ−kxFxF

= 0,

(35)

with E given by (16). Note that we are not able to find closed-form solutions

for φ∗ and φ∗x. When analyzing the effects on both productivities analytically,

we focus on the partial effects only, i.e. we look at the effects on φ for anygiven level of the export cutoff productivity, φ(φx), respectively on the effects

on φx for any given level of the cutoff productivity, φx(φ).The equilibrium outcomes in the labor and goods market are given by

(see Appendix A.2.1):

b∗(φ∗, φ∗x, γ) = ρ[(1 + α(φ∗, φ∗x))

−1(Γ1(φ∗, γ) + Γ3(φ

∗, φ∗x, γ))Γ2(φ∗, γ)

] 1σ−1

,

(36)

we∗(φ∗, φ∗x, γ) = ρ(

1 + α(φ∗, φ∗x))− 1

σ−1(

Γ1(φ∗, γ) + Γ3(φ

∗, φ∗x, γ)) 1σ−1

Γ2(φ∗, γ)

σσ−1 ,

(37)

u∗(φ∗, γ) =1

1− s

(1− 1

Γ2(φ∗, γ)

), (38)

Y ∗(φ∗, φ∗x, γ) =(

1− u∗(φ∗, γ))we∗(φ∗, φ∗x, γ)

L

ρ. (39)

where Γ3 is defined as:

Γ3(φ∗, φ∗x, γ) ≡ τ−(σ−1)

(1−G(φ∗)

)−1 ∫ ∞φ∗x

φσ−1θ(φ, γ)−σdG(φ). (40)

15

4.2 The Role of the Bargaining Power

As in the closed economy, we can analyze the effects of an increase in γ. Forthe export cutoff productivity φ and the export cutoff productivity φx, weobtain:

Proposition 5 For χ = 1, an increase in γ increases φx for any given levelof φ and decreases φ for any given level of φx. For χ = 0, an increase in γdoes neither affect φx nor φ.

Proof 5 See Appendix A.2.2

An increase in the union’s bargaining power raises the firms’ marginal costs.Holding φ constant, this implies that fewer firms manage to export withouta loss and φx increases. The intuition for the impact of γ on φ is identical tothe one in the closed economy. Note that the effect on the equilibrium cutoffproductivity (export cutoff productivity) φ∗ (φ∗x) includes the repercussion

effect through changes in φx (φ). As shown in Appendix A.2.2, dφx/dφ > 0holds. This implies that an increase in γ has two countervailing effects on theexport cutoff productivity. On the one hand, φx increases as argued above.On the other hand, competition and firm-selection becomes less intense (φ

decreases) which reduces φx. The net effect is ambiguous. The same argu-mentation is true for the impact on the cutoff productivity. However, thesign of dφ/dφx is parameter-dependent (see Appendix A.2.2) such that it isambiguous whether the repercussion effect through the increase in the exportcutoff productivity would mitigate or amplify the decline in φ.

As in the closed economy case, we are not able to predict the effect onthe equilibrium expected wage rate we∗ (see (37)). With respect to theequilibrium unemployment rate, we obtain the same result as postulated inProposition 3, with the exception that we cannot analytically compute thesign of dφ∗/dγ as argued before. Note that u∗ does not depend on the exportcutoff productivity such that no additional effects from the selection into theexport market work on the equilibrium unemployment rate.

The numerical solution confirms our findings from the closed economysetting, both in qualitative and quantitative terms. In particular, we findthat the equilibrium export cutoff productivity φ∗x increases in γ and that

the equilibrium cutoff productivity φ∗ decreases in γ. Taking the repercussioneffects into account thus does not alter our finding in Proposition 5.

16

4.3 Trade Liberalization

4.3.1 Analytical Solution

How does trade liberalization by either a reduction of variable trade costsτ or fix trade costs Fx affect the equilibrium outcomes? In this section,we answer this question and distinguish again between an economy where allfirm-level unions have identical bargaining power (χ = 0) and economy wherethe bargaining strength depends on the firm’s productivity (χ = 1). Thisprovides deeper insights about the role of trade unions in an open economysetting.

Regarding the cutoff productivity φ and the export cutoff productivityφx, we find:

Proposition 6 For χ = 1, a decrease in τ or Fx decreases φx for any givenlevel of the cutoff productivity and increases φ for any given level of the exportcutoff productivity. For χ = 0, the same results hold.

Proof 6 see Appendix A.2.3

Trade liberalization implies that more firms export, i.e. φx decreases. Thisraises competition and firm-selection becomes more severe, i.e. φ increases.Since both cutoff productivities are interdependent, repercussion effects mustbe taken into account in the equilibrium (as explained for case of an increasingbargaining power). To simplify the following argumentation, we assume thatthese repercussion effects do not change the sign of the initial effects:

Assumption 1 Trade liberalization increases the equilibrium cutoff produc-tivity φ∗ and decreases the equilibrium export cutoff productivity φ∗x.

This assumption corresponds to the usual finding in the literature.The impact of trade liberalization on the expected wage rate is ambiguous

because we do not know (analytically) the sign of dwe∗/dφ∗. With respectto the equilibrium unemployment rate u∗, we find:

Proposition 7 For χ = 1, a decrease in τ or Fx increases u∗. For χ = 0, adecrease in τ or Fx has no effect on u∗.

Proof 7 Differentiating (38) with respect to φ∗ and taking Assumption 1into account leads to:

du∗

dτ=du∗

dφ∗︸︷︷︸>0

dφ∗

dτ︸︷︷︸<0

< 0,du∗

dFx=du∗

dφ∗︸︷︷︸>0

dφ∗

dFx︸︷︷︸>0

> 0, (41)

which proves the first part of the Proposition. For the second part, note thatΓ2(χ=0) = θ.

17

Suppose that unions have identical bargaining power. In this case, theassumption of CES implies that the equilibrium unemployment rate is in-dependent of the cutoff productivity and thus the distribution of firms. Assuch, neither τ nor on Fx have an effect on u∗ (see Eckel and Egger, 2009).In contrast, the distribution of firms play a role for the equilibrium unem-ployment rate if the bargaining power is firm-specific. An increase in firm-selection raises the unemployment rate because, intuitively, unions bargainhigher wages (on average) which reduces labor demand. Since trade liberal-ization increases the equilibrium cutoff productivity, unemployment rises.

Regarding equilibrium output Y ∗ (and hence welfare), the effect of trade

liberalization is ambiguous because it is not clear how the increase in φ∗

affects Y ∗, see the analog result in the closed economy.

4.3.2 Numerical Solution

Since the results of changes in τ and Fx are qualitatively and quantitativelysimilar, we report only the former. Fix trade costs are set to Fx = 0.22, whichis the average value of estimated fix trade costs in the US and Europe byBalistreri et al. (2011). The upper bound of the union’s bargaining power, γ,is assumed to be 0.5. Figure 3 shows that the equilibrium cutoff productiv-ity increases if trade is liberalized, while the equilibrium cutoff productivityincreases. This confirms Assumption 1. For any given level of τ , the cutoffproductivity is lower if the union’s bargaining power is firm-specific, whichwas explained in Section 3.3.2. We call the difference in the χ = 1 and χ = 0curves level effect in the following.

18

Figure 3: Trade Liberalization and Firm-selection

The effects on the labor market are illustrated in Figure 4. The equilib-rium expected wage rate increases if τ is reduced irrespective of the union’sbargaining power structure. The level effect is explained by the reductionof we∗ if the union’s bargaining power is firm-specific. With respect to theequilibrium unemployment, the level effect is relatively strong, i.e. u∗ is upto 50% lower if we account for differences in the unions’ bargaining strengths.In addition, we see that trade liberalization has a negative employment ef-fect in this case, but this effect is relatively weak. In a world where unionshave identical bargaining power, the unemployment increasing effect of tradeliberalization disappears.

19

Figure 4: Trade Liberalization and the Labor Market

Finally, we simulate the effect on equilibrium output respectively welfare(see Figure 5). As a result, trade liberalization raises welfare in any cases.This implies that the negative employment effect does not outweigh the pos-itive effect of higher competition. We further see that welfare is higher ifunions have firm-specific bargaining powers. This is mainly driven by thehigher level of employment which raises output.

20

Figure 5: Trade Liberalization and the Goods Market

5 Conclusion

In the present paper, we have designed a general equilibrium model to analyzethe impact of trade unions on the macroeconomic performance of an openeconomy. The point of departure from the literature is the endogeneity of thebargaining power coefficient, the unions’ bargaining power is firm-specific andit is assumed to increase in the level of firm productivity. We have addressedtwo questions: first, how does the wage bargain affect variables such as wages,employment, output, exports, and the number of active/exporting firms, andsecond, how does trade liberalization affect the open economy equilibrium.

Our analysis delivers new insights into the interaction of the rent-shiftingeffect of a stronger union bargaining power and the impact on the degreeof competition via the entry of new firms. We show that the overall effectof stronger trade unions on aggregate employment may be positive, whereasthe impact on aggregate output will always be negative. Trade liberalizationreduces aggregate employment but increases aggregate output.

Due to the firm-specific bargaining power, wages differ between firms. Ouranalysis, however, is quiet silent on the question how the wage distributionis affected. Further research is needed to tackle this point.

21

A Appendix

A.1 Closed Economy

A.1.1 Zero-profit Cutoff and Free-entry Condition

Using r(φ, w(φ)) = p(φ, w(φ))−(σ−1)Y/M , (8) and (10), we can rewrite (13)as:

Y

M= ρ−(σ−1)

(θ(φ, γ)

φ

)σ−1

bσ−1σF. (A.1)

The CES price index is defined as:

P =(

1−G(φ)) 1σ−1

[∫ ∞φ

p(φ,w(φ))−(σ−1)dG(φ)

]− 1σ−1

. (A.2)

Using P ≡ 1, (8) and (10), we obtain:∫ ∞φ

(φ

θ(φ, γ)

)σ−1dG(φ) = ρ−(σ−1)(1−G(φ))bσ−1.

Solving this Eq. with respect to b and inserting the resulting expression into(A.1) yields:

Y

M=

(θ(φ, γ)

φ

)σ−1σF

1−G(φ)

∫ ∞φ

(φ

θ(φ, γ)

)σ−1dG(φ), (A.3)

which is identical to (15) in the main text.

Using r(φ, w(φ)) = p(φ, w(φ))−(σ−1)Y/M and noting that only firms with

φ ≥ φ are operating, we can rewrite (12) as:

Y

M

∫ ∞φ

p(φ,w(φ))−(σ−1)dG(φ) = σ(δFe + (1−G(φ))F

). (A.4)

Due to P ≡ 1, we can combine (A.2) and (A.4) to obtain:

Y

M=

σ

1−G(φ)

(δFe + (1−G(φ))F

), (A.5)

which is identical to (14) in the main text.

22

A.1.2 Proof of Proposition 1

Recall the (implicit) condition of the equilibrium cutoff productivity φ∗ (16):

E ≡

(θ(φ∗, γ)

φ∗

)σ−1 ∫ ∞φ∗

(φ

θ(φ, γ)

)σ−1dG(φ)− (φ∗)−k − δFe

F= 0.

Totally differentiating (with dk = dF = dFe = dδ = dσ = 0) yields:

dφ∗

dγ= − ∂E/∂γ

∂E/∂φ∗. (A.6)

Differentiating E with respect to φ∗ implies:

∂E

∂φ= −

(1− ∂θ

∂φ

φ

θ

)(θ(φ, γ)

φ

)σ−1

φk(σ − 1)

∫ ∞φ

(φ

θ(φ, γ)

)σ−1φ−k−1dφ,

where we have used the Pareto distribution and suppressed the equilibriumindication ∗ for notational simplification. Using (11) and (6), we get:

∂θ

∂φ

φ

θ=

γ

σ − 1 + γ(1− φ−1)1

φ,

which is smaller than one because φ(σ − 1 + γ) > 2γ holds for usual values

of σ. Therefore, we conclude that ∂E/∂φ < 0.Differentiating E with respect to γ yields:

∂E

∂γ= k

(θ(φ, γ)

φ

)σ−1 [(σ − 1)

A1(φ, γ)

θ(φ, γ)

∂θ

∂γ+∂A1(φ, γ)

∂γ

],

A1(φ, γ) ≡∫ ∞φ

(φ

θ(φ, γ)

)σ−1φ−k−1dφ.

Using (11) and solving the partial derivatives on the RHS, we obtain:

∂E

∂γ= k

(θ(φ, γ)

φ

)σ−1σ − 1

γ

[A2(φ, γ)− A3(φ, γ)

],

A2(φ, γ) ≡∫ ∞φ

(φ

θ(φ, γ)

)σφ−k−2dφ,

A3(φ, γ) ≡∫ ∞φ

(φ

θ(φ, γ)

)σφ−k−2

θ(φ, γ)

θ(φ, γ)dφ.

23

Since θ(φ, γ) > θ(φ, γ), we have A3 > A2 and can thus conclude that

∂E/∂γ < 0. Using (A.6), we find dφ/dγ < 0 as postulated in the maintext.

If χ = 0, (13) can be rewritten as Y/M = (σFk)/(k−σ+ 1). Combiningthis with (A.5) yields:

φ∗(χ=0) =

(F

δFe

σ − 1

k − σ + 1

) 1k

, (A.7)

which is independent of the cutoff productivity as postulated in the maintext.

A.1.3 Proof of Proposition 2

Differentiating (24) with respect to γ yields:

dwe∗

dγ=

1

σγΓ

1σ−1

1 Γσσ−1

2

(Γ1

dγ

γ

Γ1

+ σΓ2

dγ

γ

Γ2

). (A.8)

Observing (22) and (23), we obtain:

dΓ1

dγ=∂Γ1

∂φ

dφ

dγ+∂Γ1

∂γ, (A.9)

dΓ2

dγ=∂Γ2

∂φ

dφ

dγ+∂Γ2

∂γ. (A.10)

The partial derivatives can be calculated as:

∂Γ1

∂φ= φk−1k

(k

∫ ∞φ

(φ

θ(φ, γ)

)σφ−k−2dφ−

(φ

θ(φ, γ)

)σ

φ−k−1

), (A.11)

∂Γ1

∂γ= −φkk

∫ ∞φ

θ(φ, γ)−σ−1∂θ

∂γφσ−k−2dφ < 0, (A.12)

∂Γ2

∂φ=

γ

σ − 1

k

k + 1φ−2 > 0, (A.13)

∂Γ2

∂γ=

1

σ − 1

(1− k

k + 1

1

φ

)> 0, (A.14)

where we have used the Pareto distribution, (11) and (6) to rewrite (23) as:

Γ2(φ∗, γ) =

1

σ − 1

(σ − 1 + γ

(1− k

k + 1

1

φ∗

)). (A.15)

24

Inserting (A.13) and (A.14) into (A.10), we see that the sign of dΓ2/dγis parameter-depended:

dΓ2

dγ=

1

σ − 1

[1− k

k + 1

1

φ

(1− γ

φ

dφ

dγ

)], (A.16)

sign

[dΓ2

dγ

]= sign

[γ

φ

dφ

dγ−(

1− k + 1

kφ

)]. (A.17)

The sign of dΓ1/dγ depends on the sign of ∂Γ1/∂φ. However, we cannotanalytically solve the integral in (A.11) such that we are not able to clarify

under which parameter constellations ∂Γ1/∂φ and thus dΓ1/dγ would have apositive or negative sign. Therefore, the sign of dwe∗/dγ is ambiguous, whichproves the first part of Proposition 2.

For χ = 0, we can rewrite (22) and (23) as: Γ1(χ = 0) = θ−σφ(χ =

0)σ−1k/(k−(σ−1)) and Γ2(χ = 0) = θ. This implies we(χ = 0) = ρφ(k/(k−(σ − 1)))1/σ−1, which does not depend on γ. This proves the second part ofProposition 2.

A.1.4 Proof of Proposition 3

Differentiating (25) with respect to γ yields:

du∗

dγ=

1

1− s1

Γ2Γ2︸ ︷︷ ︸≡D>0

dΓ2

dγ. (A.18)

This shows that ∂u∗/∂φ∗ = D∂Γ2/∂φ∗ > 0 (see (A.13)) and ∂u∗/∂γ =

D∂Γ2/∂γ > 0 (see (A.14)). Moreover, it indicates that:

sign

[du∗

dγ

]= sign

[dΓ2

dγ

],

where the RHS is given by (A.17). This proves Proposition 3.

A.2 Open Economy

A.2.1 Derivation of the General Equilibrium

To compute the export cutoff productivity, we combine (13) and (32) andobtain:

r(φ, w(φ))

r(φx, w(φx))=

F

τσ−1Fx.

25

Inserting subsequently the profit-maximizing revenue, price and bargainedwage leads to (34).

Adopting the same steps as in Appendix A.1.1, we can rewrite the free-entry condition (31) and the zero-profit cutoff condition (13) as:

Y

Mt

=1

1 + α(φ, φx)

σ

1−G(φ)

(δFe + (1−G(φ))F + (1−G(φx))Fx

),

(A.19)

Y

Mt

=1

1 + α(φ, φx)

(θ(φ, γ)

φ

)σ−1σF

1−G(φ)[∫ ∞φ

(φ

θ(φ, γ)

)σ−1dG(φ) + τ−σ

∫ ∞φx

(φ

θ(φ, γ)

)σ−1dG(φ)

],

(A.20)

where we have used the definition of the price index:

P =(

(1−G(φ))α(φ, φx))) 1σ−1

[∫ ∞φ

p(φ,w(φ))−(σ−1)dG(φ) + τ−(σ−1)∫ ∞φx

p(φ,w(φ))−(σ−1)dG(φ)

]− 1σ−1

.

Combining (A.19) and (A.20) yields the (implicit) determination of the cutoffproductivity as given by (35).

To determine the labor market equilibrium, we use (17), (18) and thedefinition of total employment

(1− u)L = M(

1−G(φ))−1(∫ ∞

φ

h(φ)dG(φ) + τ−(σ−1)∫ ∞φx

h(φ)dG(φ)

).

(A.21)

Following the same approach as in Section 3.2.2, we obtain (36), (37) and(38). Solving (18) with respect to Y leads to the equilibrium output as givenby (39). To close the model, we can use (14) to pin down the equilibrium massof available varieties M∗

t . The equilibrium mass of firms M∗ is determined

by rearranging Mt = M +Mx = (1 + α(φ, φx))M , which also pins down theequilibrium mass of exporters M∗

x .

26

A.2.2 Derivation of Proposition 5

Computing partial derivatives from (34) and (35) yields:

∂E1

∂φ=σ − 1

φ

(θ(φ, γ)

θ(φx, γ)

φx

φ

)−(σ−1)1− ∂θ(φ, γ)

∂φ

φ

θ(φ, γ)︸ ︷︷ ︸<1

> 0, (A.22)

∂E1

∂φx= −σ − 1

φx

(θ(φ, γ)

θ(φx, γ)

φx

φ

)−(σ−1)1− ∂θ(φx, γ)

∂φx

φx

θ(φx, γ)︸ ︷︷ ︸<1

< 0, (A.23)

∂E1

∂γ= −σ − 1

γ

(θ(φ, γ)

θ(φx, γ)

φx

φ

)−(σ−1) (φ(φx, γ)−1 − θ(φ, γ)−1

)︸ ︷︷ ︸

<0

> 0, (A.24)

∂E2

∂φ=∂E

∂φ︸︷︷︸<0

+∂

∂φ

(θ(φ, γ)

φ

)σ−1

︸ ︷︷ ︸<0

τ−σ∫ ∞φx

(φ

θ(φ, γ)

)σ−1dG(φ) < 0, (A.25)

∂E2

∂φx= −τ−σ

(θ(φ, γ)

θ(φx, γ)

φx

φ

)σ−1

+ kφ−k−1x , (A.26)

∂E2

∂γ=∂E

∂γ︸︷︷︸<0

+τ−σk

(θ(φ, γ)

φ

)σ−1σ − 1

γ

[A2(φx, γ)− A3(φx, γ)

]︸ ︷︷ ︸

<0

< 0.

(A.27)

Using (A.22), (A.23) and (A.24), we find (by totally differentiating (34)):

dφxdγ

= − ∂E1/∂γ

∂E1/∂φx> 0,

dφx

dφ= − ∂E1/∂φ

∂E1/∂φx> 0.

Totally differentiating (35) and observing (A.25), (A.26) and (A.27), we alsoget:

dφ

dγ= −∂E2/∂γ

∂E2/∂φ< 0,

while the sign of dφ/dφx is ambiguous (which is driven by the ambiguoussign of (A.26)). This proves Proposition 5.

27

A.2.3 Derivation of Proposition 6

Computing partial derivatives from (34) and (35) yields:

∂E1

∂τ= (σ − 1)τ−(σ−1)−1

F

Fx> 0, (A.28)

∂E1

∂Fx= τ−(σ−1)

F

F 2x

> 0, (A.29)

∂E2

∂τ= −στ−σ−1

(θ(φ, γ)

φ

)σ−1 ∫ ∞φx

(φ

θ(φ, γ)

)σ−1dG(φ) < 0, (A.30)

∂E2

∂Fx= −φ−kx F−1 < 0. (A.31)

Using (A.28), (A.29), (A.30) and (A.31) as well as (A.22), (A.23), (A.25)and (A.26), we find (by totally differentiating (34) and (35)):

dφxdτ

= − ∂E1/∂τ

∂E1/∂φx> 0,

dφxdFx

= −∂E1/∂Fx

∂E1/∂φx> 0,

dφ

dτ= −∂E2/∂τ

∂E2/∂φ< 0,

dφ

dFx= −∂E2/∂Fx

∂E2/∂φ< 0,

which proves the first part of Proposition 6. For the second part, note thatif χ = 0, the export cutoff productivity is given by:

φx(χ=0) = τ

(F

Fx

) 1σ−1

φχ=0.

28

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