ISSN (Print) 0473-453X

Discussion Paper No. 1154

ISSN (Online) 2435-0982

UNIONIZATION,
INDUSTRY CONCENTRATION,AND
ECONOMIC GROWTH
Colin Davis
Ken-ichi Hashimoto
Ken Tabata

December 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Unionization, Industry Concentration,
and Economic GrowthColin Davis* Ken-ichi Hashimoto† Ken Tabata‡
December 2021

Abstract
This paper examines how unionization affects economic growth through its impact
on industry concentration in a two-country model of international trade and endogenous
productivity growth. Knowledge spillovers link firm-level productivity in innovation
with geographic patterns of industry ensuring a faster rate of output when industry is relatively concentrated in the country with the greater labor supply. We show that stronger
bargaining power in the relatively large country increases the rate of output growth whenlabor unions are employment-oriented, but decreases the rate of growth when unions
are wage-oriented. We then calibrate the model using labor market data for the UnitedStates and the United Kingdom and study the effects of falling union bargaining power
on industry location patterns, output growth, and national welfare.
Key Words: Labor Union Bargaining Power; Industry Concentration; Knowledge Diffusion; Endogenous Productivity Growth; Endogenous Market Structure
JEL Classifications: F43; O30; O40* The Institute for the Liberal Arts, Doshisha University; (E-mail) cdavis@mail.doshisha.ac.jp†

Graduate School of Economics, Kobe University; (E-mail) hashimoto@econ.kobe-u.ac.jp
School of Economics, Kwansei Gakuin University; (E-mail) tabataken@kwansei.ac.jp
Acknowledgements: This research was financially supported by the Joint Usage/Research Center at ISER
(Osaka University) and KIER (Kyoto University), and the Japan Society for the Promotion of Science: Grantsin-Aid for Scientific Research 19K01638, 20H05631, 20K01569, 20K01631.
‡

1

1 Introduction
There is a great diversity in levels of unionization and collective bargaining over wages acrosscountries. These differences in labor market institutions influence industry location patterns
through their impact on production costs. Although the direct effect of unionization on economic growth is generally ambiguous, industry location patterns are known to have important
implications for economic growth, with greater industry concentration generating knowledge
spillovers that promote faster rates of innovation. In this paper we consider how unionization
affects economic growth through adjustments in industry location patterns.
We introduce an endogenous growth and endogenous market structure framework (Smulders and van de Klundert, 1995; Peretto, 1996, 2018; Young, 1998) that allows for a studyof how unionization affects the geographic location of industry and economic growth. Our
two-country framework consists of two industries. A final good sector employs labor andintermediate goods in the production of a homogeneous good. Firms in the intermediate
sector employ final goods in the production of differentiated varieties and in process innovation aimed at reducing production costs. International trade costs on intermediate goods
ensure that the country with the larger market (i.e., greater labor supply) hosts greater sharesof the intermediate and final good industries, and has more productive intermediate firms.
Imperfect knowledge diffusion between countries links the strength of knowledge spillovers
with the geographic location of industry, generating a positive relationship between industry
concentration and the rate of economic growth.
A key feature of our framework is the endogenous determination of national labor supplies through negotiation between a labor union and a federation of employers. Adopting
the “managerial” labor union introduced by Pemberton (1988), the balance of internal power
between leadership and membership determines union orientation, with important implications for the contract curve that arises from negotiation between the federation and the labor
union. As a result, strengthening the wage orientation of the union lowers the national labor supply. An increase in union bargaining power, however, reduces the labor supply if the2

union is wage-oriented, and expands the labor supply if the union is employment-oriented(Pemberton, 1988; Chang et al., 2007; Chu et al., 2016).
We use the framework to explore how changes in labor union policy affect industry location patterns and the rate of economic growth. Beginning with union orientation, we findthat strengthening the wage orientation of the labor union in the country with the larger labor
supply decreases the large country’s shares of intermediate and final good production, andlowers the relative productivity of its intermediate firms, as the national labor supply contracts. The reduced concentration of industry weakens knowledge spillovers, slowing the rateof growth. In contrast, strengthening the wage orientation of the small country labor union
improves knowledge spillovers by raising industry concentration in the large country, andthus accelerates economic growth.
As seen above, the effects of changes in union bargaining power depend on union orientation. Focusing on the labor union of the large country, if the union is wage-oriented, anincrease in union bargaining power lowers the concentration of industry in the large country,
weakening knowledge spillovers and slowing economic growth. Alternatively, if the union is
employment-oriented, the increase in bargaining power raises industry concentration, thereby
strengthening knowledge spillovers and hastening economic growth. These mechanisms generate similar, but opposite, results for the effects of an increase in labor union bargaining
power in the small country.
We explore the welfare implications of unionization through a numerical calibration of
our model that uses labor market data to estimate union bargaining power for the US andthe UK between 1980 and 2007 through an adaptation of the method introduced by Chu etal. (2016). The numerical analysis suggests that unions in the US and the UK were wageoriented over the period of analysis, and that the general decline in union bargaining power
observed in these countries led to welfare improvements that were for the most part driven
by expansions in national output levels. We also find that a fall in the rate of output growth
generated small welfare losses for both countries.

3A broad empirical literature investigates the relationship between unionization and economic growth (Doucouliagos et al., 2017). Consider, for instance, a selection of studies
examining data for OECD countries. On the one hand, Carmeci and Mauro (2003) andNguyen-Can and Terraz (2021) report that increased union density and greater union centralization have negative effects on per capita income growth. Vernon and Rogers (2013),
on the other hand, conclude that predominant union structure is important, with craft or general unions having negative, enterprise unions having negligible, and industrial unions having
positive effects on labor productivity growth in manufacturing. Furthermore, the results ofStorm and Naastepad (2009) suggest a positive role for labor unions through the effect oflabor market regulation on labor productivity growth. More generally, however, the metaanalysis of Doucouliagos et al. (2017) concludes that there is no significant relationshipbetween unionization and economic growth.
There is a diverse theoretical literature studying how unionization affects economic growth(Shister, 1954; Palokangos, 1996, 2005; Ramos-Parreno and Sanchez-Losada, 2002; Irmen
and Wigger, 2002; Lingens, 2003, 2007). In particular, Chang et al. (2007) introduce a “managerial union” into a model of endogenous growth, and conclude that stronger union bargaining power reduces the labor supply and slows economic growth when the union is wageoriented, but expands the labor supply and hastens growth when the union is employmentorientated. In a similar vein, Chang and Hung (2016) consider a wage-oriented managerial
union in combination with an elastic household labor supply, and show that greater union
bargaining power promotes economic growth as a reduced number of workers increase theirwork hours in response to rising wages, expanding the effective labor supply. These results
are closely linked, however, with a scale effect, whereby the growth rate increases proportionately with the labor force. Indeed, in a scale-neutral endogenous growth and endogenous
market structure framework, Ji et al. (2016) show that the economy absorbs changes in union
bargaining power through adjustments in market entry, leaving no impact on the long-run
growth rate. Our paper extends the growth literature through a consideration of the indi-

4

rect effects of unionization on scale-neutral economic growth that result from relationshipbetween the relative market size and industry location patterns.
Accordingly, our paper is closely related with the new economic geography literaturethat explores the effects of unionization on industry location (Persyn, 2013; Egger and Etzel,
2014). In a two-country model of international trade, Picard and Toulemonde (2006) showthat a symmetric increase in bargaining power across countries leads to a dispersed industry location pattern. Munch (2003) finds that industry tends to agglomerate in the country
with weaker union bargaining power, although industry potentially agglomerates in the country with stronger union bargaining power over an intermediate range for international trade
costs. Because these studies assume a negatively sloped or vertical contract curve, however,
employment-oriented unions are not considered. Thus, our paper contributes to the literature
with a study of how union orientation influences industry location patterns, with an emphasison the implications of unionization for long-run growth.
The remainder of the paper is organized as follows. The next section introduces our
two-country model of trade and endogenous productivity growth. Section 3 provides a characterization of the model’s long-run equilibrium, and Section 4 studies how changes in thewage orientation and bargaining power of labor unions affect long-run economic growth. In
Section 5, we calibrate the model using data for the US and the UK and study the long-run
welfare implications of changes in labor union bargaining power. Section 6 concludes. Proofs
are provided in the appendices.

2 The Model
This section develops a two-country model of trade and productivity growth. We refer to thetwo countries as home and foreign. A final good sector employs labor and intermediate goodsin the production of a homogeneous good for sale to a competitive market. The intermediate
goods sector consists of monopolistically competitive firms that employ final goods in the
production of differentiated varieties and in process innovation that reduces production costs.
5

The countries differ with respect to population size and labor market policy. In each country,
negotiations between a federation of employers and a labor union determine national employment and the wage rate. In what follows, we focus on the home country as we introduce themodel, but analogous conditions are derived for the foreign country.

2.1 Households
The lifetime utility of the representative dynastic household in the home country is

U=

∞
X
ln Ct
,
t
(1
+
ρ)
t=0

(1)

where Ct denotes the consumption of final goods in period t, and ρ > 0 is the subjective
discount rate. The household maximizes lifetime utility (1) subject to the following asset
accumulation equation:
At+1 = (1 + rt )At + wt Lt + bt L̄ − Lt + Πt − Tt − Ct ,(2)

where At is household assets, rt is the real interest rate, wt is the wage rate, L̄ is the inelastic
labor supply, Lt is the number of employed workers, bt is the unemployment benefit provided to unemployed workers, Πt is dividend income derived from the profits of final good
producers, and Tt is a lump-sum tax levied by the government. The representative household
eliminates consumption and employment uncertainty by ensuring equal consumption levels
for all household members regardless of their employment status (Merz, 1995; Andolfatto,
1996). In addition, we adopt the final good as the model numeraire and set its price equal toone.
Solving this dynamic optimization problem, we derive the following Euler equation:
1 + rt+1
Ct+1
=
.
Ct
1+ρ

6

(3)

Perfect international capital mobility ensures a common evolution for home and foreign∗
household consumption, as the interest rate is equalized across countries: Ct+1 /Ct = Ct+1
/Ct∗

= (1 + rt+1 )/(1 + ρ), where variables associated with foreign are indicated with an asterisk.
The foreign household faces a similar utility maximization problem and therefore has similar
demand conditions.

2.2 Government
A relatively passive role is set for the government: distributing unemployment benefits andlevying a lump-sum tax Tt on the household to balance the fiscal budget. The balanced-budget
condition is
Tt = bt L̄ − Lt .

(4)

To avoid degeneration of unemployment benefits along the balanced-growth path, we assume that the unemployment benefit bt is linked proportionately with final output Yt ; that is,
bt = mYt , where m > 0 is a policy that sets the ratio of the unemployment benefit to final
good output in the home country. Similarly, the balanced-budget condition for the foreigngovernment is Tt∗ = b∗t L̄∗ − L∗t , with b∗t = m∗ Yt∗ , where m∗ > 0. Note that we allow
unemployment benefit policies, m and m∗ , to differ across countries.

2.3 Final Good Production
Final goods are produced for an international market characterized by free trade. We adopt
a Cobb-Douglas formation for the aggregate production function. For example, in the homecountry we have
Yt = Ztα Lβt ,7(5)where Zt is an intermediate composite and Lt is labor employment. The factor intensities
associated with the intermediate composite and labor are α ∈ (0, 1) and β ∈ (0, 1). Following
Palokangas (1996, 2005) and Chang et al. (2007), we assume that the production function
exhibits decreasing returns to scale, with α + β < 1, thereby allowing final good producers
to generate positive profits for the labor union and the employer federation to negotiate over.
The intermediate composite is formulated as a CES aggregator over the varieties available
in period t:

Zt = Z nt0

xǫi,t di +

Z n∗t
0

xǫj,t dj

 1ǫ

,

(6)

where xi,t is the demand for intermediate variety i of the nt mass of varieties produced inhome, and xj,t is the demand for variety j of the n∗t mass of varieties produced in foreign.
The total mass of varieties is Nt ≡ nt + n∗t , and the constant elasticity of substitution between
any given pair of varieties is measured by 1/(1 − ǫ), with ǫ ∈ (0, 1).
The profit function of the representative final good producer in the home country isΠt = Yt − wt Lt −Z ntpi,t xi,t di −

Z n∗t
0

0

p∗j,tτ xj,t dj,(7)

where pi,t is the price of intermediate variety i produced in home, p∗j,t is the price of variety
j imported from foreign, and τ > 1 is an iceberg trade cost, under which τ additional units
must be shipped for every unit sold in an export market.
Final good firms set their inputs of xi,t and xj,t to maximize profit (7), generating thefollowing home country demand conditions for varieties produced in home and foreign:
−1

ǫ

−1

1−ǫ
1−ǫ
PZ,t
αYt,
xi,t = pi,tǫ1−ǫxj,t = (τ p∗j,t ) 1−ǫ PZ,t
αYt ,

8

(8)

where the home-country price index for intermediate varieties is

PZ,t =

 Z nt0− ǫpi,t1−ǫ di + ϕ

Z n∗t
0

ǫ
p∗j,t − 1−ǫ dj

− 1−ǫǫ,(9)
ǫ

and PZ,tZt = αYt . We index the level of trade costs using the freeness of trade ϕ ≡ τ − 1−ǫ ∈(0, 1); that is, dϕ/dτ < 0, with ϕ = 0 describing prohibitively high trade costs and ϕ =1 indicating free trade between countries. Final good producers in foreign solve a similar
optimization problem generating analogous demand conditions for intermediate varieties.

2.4 Labor Unions and Collective Bargaining
Central to the labor market in our framework is the negotiations between the labor union
and the employer federation that determine national employment and the wage rate. As in
Pemberton (1988) and Chang et al. (2007), we consider a managerial labor union, in which
union members desire a high wage rate while union leaders aim for a large membership.
Formally, the union’s objective function in the home country is set as
Ot = (wt − bt )χ Lt ,(10)

where χ > 0 measures the weight the union places on incremental wage income from employment (i.e., the wage rate minus the unemployment benefit).
The bargaining problem involves choosing wt and Lt to maximize the following Nashproduct: Otψ Πt 1−ψ , where ψ ∈ (0, 1) denotes the relative bargaining power of the labor
union. Solving this bargaining problem, we obtain the following optimal conditions for thewage rate and employment:Yt,wt − bt = χ wt − βLt
wt Lt= β + ψ(1 − α − β),
Yt

9

(11)
(12)

where we have used (7) and (10). The contract curve (11) describes the locus of points forwhich the union’s indifference curve and the firm’s isoprofit curve are tangent in (wt , Lt )space. The rent division curve (12) indicates the labor union’s negotiated share of income.
The equilibrium level of employment is obtained through substitution of the rent curve(12) and bt = mYt into the contract curve (11):L=1[β + (1 − χ)ψ(1 − α − β)] .
m(13)Hence, we find that employment is decreasing in union wage orientation (χ). The relationshipbetween union bargaining power and national employment is summarized as follows.
Lemma 1 An increase in union bargaining power (ψ) expands employment (L) for χ > 1,
but contracts employment for χ < 1.
Following Chu et al. (2016), we use χ to describe the relative importance that the union places
on wages over membership. When χ > 1, the contract curve has a negative slope, and werefer to the union as wage-oriented. In this first case, a rise in union bargaining power shifts
the rent curve, causing a contraction in national employment. Alternatively, when χ < 1, thecontract curve has a positive slope, and we refer to the union as employment-oriented. In thissecond case, an increase in bargaining power leads to an expansion in national employment.1
Analogous conditions are obtained for the foreign labor market: wt∗ − b∗t = χ∗ (wt∗ −βYt∗ /L∗t ), wt∗ L∗t /Yt∗ = β + ψ ∗ (1 − α − β), and L∗ = (1/m∗ ) [β + (1 − χ∗ )ψ ∗ (1 − α − β)],where we allow union wage orientation (χ, χ∗ ) and bargaining power (ψ, ψ ∗ ) to differ between home and foreign.

2.5 Intermediate Good Production
Intermediate firms employ final goods in the production of horizontally differentiated varieties for supply to the home and foreign markets under monopolistic competition (Dixit and1

In general, the empirical evidence lends more support to a wage-oriented structure for union preferences(Clark and Oswald, 1989).

10

Stiglitz, 1977). Each firm survives for two periods. In the first, firms invest in market entry
and process innovation, and in the second they produce an intermediate product variety.
The production technologies of intermediate firms located in home and foreign areγ
Xi,t = θi,t
Ix,i,t,

∗
∗ γ ∗
Xj,t
= θj,tIx,j,t,(14)∗∗
where Xi,t and Xj,t
are firm-level outputs, Ix,i,t and Ix,j,t
are firm-level inputs of final goods,∗
θi,t and θj,t
are firm-specific productivity coefficients, and γ ∈ (0, 1) is the output elasticity

of productivity. Each intermediate firm produces to meet the combined demands from final
good firms in home and foreign. For example, a home-based intermediate firm sets supply
equal to Xi,t = xi,t + τ x∗i,t .
Under monopolistic competition, the large mass of intermediate varieties sold in eachmarket eliminates strategic interaction between firms as they choose their optimal productionlevels. Thus, each firm maximizes operating profit on sales, πi,t ≡ pi,t Xi,t − Ix,i,t, by setting
price equal to a constant markup over unit cost; that is,pi,t =

1
γ ,
ǫθi,t

p∗j,t =

1
,
∗ γ
ǫθj,t

(15)

for home and foreign firms. Accordingly, optimal operating profit on sales can be obtained as
πi,t = (1 − ǫ)pi,t Xi,t = [(1 − ǫ)/ǫ]Ix,i,t . Substituting the demand conditions (8), the pricing
rules (15), and Xi,t = xi,t + τ x∗i,t into πi,t = (1 − ǫ)pi,t Xi,t yields the following expressions
for the optimal operating profits− ǫπi,t = α(1 − ǫ)pi,t1−ǫ

 ǫ

ǫ
1−ǫ
∗ 1−ǫ
∗
PZ,t Yt + ϕPZ,t Yt ,


ǫ
ǫ
ǫ
∗
∗
∗ − 1−ǫ1−ǫ
∗ 1−ǫ
πj,t = α(1 − ǫ)p j,t
ϕPZ,t Yt + PZ,t Yt ,of home and foreign intermediate firms.

11

(16)

2.6 Process Innovation
Intermediate firms employ final goods both in preparation for market entry and in process
innovation. Specifically, to produce intermediate variety i with productivity θi,t in home inperiod t, a firm must invest a fixed f units of final goods in market entry and a variable IR,i,t−1
units of final goods in process innovation in period t − 1.
The productivities associated with the production technologies of home and foreign intermediate firms are generated by investment in process innovation as follows:

∗
∗
∗
θj,t
= ξKt−1
IR,j,t−1,

θi,t = ξKt−1 IR,i,t−1 ,(17)∗where Kt−1 and Kt−1
are the productivities of home and foreign firms in R&D, and ξ > 0.

Following the in-house process innovation literature (Smulders and van de Klundert, 1995;
Peretto, 1996, 2018), we assume that technical knowledge accumulates within the production technology of each firm. Specifically, national stocks of knowledge are captured by theaverage productivity of the intermediate production technologies employed in each country:
R n∗ ∗Rndj.
θt ≡ (1/nt ) 0 t θi,t di and θt∗ ≡ (1/n∗t ) 0 t θj,t

We then model the productivities of home and foreign firms in process innovation as theweighted average of the productivities of observable technical knowledge in period t − 1,generating an intertemporal knowledge spillover from production to innovation. Knowledge
spillovers in home and foreign are therefore∗
Kt−1 = st−1 θt−1 + δs∗t−1 θt−1
,

∗
∗
Kt−1
= s∗t−1 θt−1
+ δst−1 θt−1 ,(18)where st−1 ≡ nt−1 /Nt−1 and s∗t−1 = 1 − st−1 ≡ n∗t−1 /Nt−1 . The degree of international
knowledge diffusion is regulated by δ ∈ (0, 1), with knowledge spillovers that are completely
national in scope for δ = 0, and perfect international knowledge diffusion for δ = 1.
In period t − 1, firms borrow from households to finance the costs of market entry andprocess innovation. As in Young (1998), the net present values of home and foreign firms in12period t − 1 are

Vi,t−1 =πi,t− (IR,i,t−1 + f ) ,1 + rt∗Vj,t−1
=

∗

πj,t
∗
− IR,j,t−1+f ,1 + rt(19)

where we assume that fixed entry costs are symmetric across countries (f = f ∗ ).
∗Firms choose their optimal levels of investment in innovation, IR,i,t−1 and IR,j,t−1
to maxi-

mize firm value (19) subject to the technology constraint (17). From the first-order conditions∗∗(i.e., ∂Vt−1 /∂IR,j,t−1 = ∂Vt−1/∂IR,j,t−1= 0), we obtain

IR,i,t−1 =ηπi,t,1 + rt∗IR,j,t−1
=

∗
ηπj,t
,
1 + rt(20)

with η ≡ γǫ/(1 − ǫ). These expressions imply that firms located in the same country havesymmetric productivity levels, and therefore set the same prices and employment levels for
production and innovation. Henceforth, we omit the indices i and j, with θi,t = θt and∗θj,t= θt∗ . Moreover, substituting (20) into (19), we rewrite the net present values of homeand foreign firms in period t − 1 as

Vt−1 =(1 − η)πt− f,
1 + rt∗Vt−1=(1 − η)πt∗− f.
1 + rt(21)

We assume that η ≡ γǫ/(1 − ǫ) < 1 in order to satisfy the second-order condition for the
maximization of firm value.

2.7 Market Entry
Firm value drives market entry and exit in the intermediate sector in each country. Specifically, with no costs incurred in the introduction of new product designs, firms enter the intermediate sector when firm value is positive (Vt−1 > 0) and exit when firm value is negative(Vt−1 < 0). Referencing (9), (16), and (21), firm value responds correctly to market entry and∗exit (∂Vt−1 /∂nt < 0 and ∂Vt−1/∂n∗t < 0), and thus the level of market entry in each country13∗adjusts immediately, forcing firm value to zero (Vt−1 = Vt−1
= 0) at all moments in time, inthe spirit of Novshek and Sonnenchen (1987).
A key implication of free market entry and exit, in combination with the integrated financial market, is common scales of production and process innovation across firms in both∗countries.2 Substituting (20) and (21) into Vt−1 = Vt−1
= 0 yields equilibrium firm-level∗
investment in process innovation as IR ≡ IR,t−1 = IR,t−1= ηf /(1 − η). As such, using (18)with (17), the productivity growth rates of intermediate firms in home and foreign are
θt

i
h
Kt−1
−1
ξIR ,
ξIR = st−1 + δ(1 − st−1 )θ̃t−1
θt−1
θt−1
h
i
K∗
θ∗
gθ∗ (θ̃t−1 ) ≡ ∗t = ∗t−1 ξIR = 1 − st−1 + δst−1 θ̃t−1 ξIR ,
θt−1
θt−1
gθ (θ̃t−1 ) ≡

=

(22)

where θ̃t ≡ θt /θt∗ denotes the international productivity differential. These expressions
show that firm-level productivity growth depends solely on knowledge spillovers (Kt /θt andKt∗ /θt∗ ), and is therefore closely linked with national shares of intermediate production (st )and the international productivity differential (θ̃t ).
2.8 Market Equilibrium
We now characterize the market clearing conditions for the intermediate and final good sectors. First, with perfect capital mobility between countries, the total asset holdings of households equals total lending to home and foreign firms: At+1 + A∗t+1 = (IR + f )nt+1 + (IR∗ +∗f )n∗t+1 . Combining this expression with (19) and the free entry conditions Vt−1 = Vt−1=0∗implies that (1 + rt+1 )(At+1 + A∗t+1 ) = πt+1 nt+1 + πt+1
n∗t+1 .

Next, the market clearing condition for intermediate goods is given by

αYtw = nt pt Xt + n∗t p∗t Xt∗ ,
2

(23)

∗
Substituting πt = [(1 − ǫ)/ǫ]Ix,t into (21) and setting Vt−1 = Vt−1
= 0 delivers the equilibrium scale of∗production as Ix,t = Ix,t = [ǫ/(1 − ǫ)][(1 + r)f /(1 − η)].

14

where Ytw ≡ Yt + Yt∗ . The lefthand side is the sum of home and foreign final good firms’
expenditure on intermediate goods. The righthand side is the total revenue of home andforeign intermediate firms.
Lastly, the market clearing condition for final goods is given by

Ytw = Ctw + Nt Ix,t + Nt+1 (IR + f ),(24)

where Ctw ≡ Ct + Ct∗ . Thus, the world supply of final goods is set to meet household
consumption, total employment in intermediate production, and total employment in process
innovation and market entry.

3 Long-run Equilibrium
In this section we derive two implicit conditions for the determination of national shares ofintermediate firms (st ) and the international productivity differential (θ̃t = θt /θt∗ ) in order tocharacterize the steady-state equilibrium of the model. We then derive the long-run rate ofproductivity growth along with national welfare levels. Focusing on steady states that exhibit
a constant international productivity differential and a common rate of productivity growthacross countries (gθ = gθ∗ ), we consider balance growth paths that generate a constant rate of
growth in household consumption (Ct+1 /Ct ).
3.1 Steady-state Industry Location Patterns
The first condition for the determination of st and θ̃t is obtained by recalling that free mar∗ket entry and exit reduces the value of intermediate firms to zero (Vt−1 = Vt−1
= 0).

Then, the integrated financial market implies that all firms have the same level of operating profit; that is, πt = πt∗ at all moments in time. In Appendix A, we combine πt = πt∗ with∗
Yt /Yt∗ = (Zt /Zt∗ )α (Lt /L∗t )β , Yt /Yt∗ = (PZ,t Zt )/(PZ,tZt∗ ), (9), (15), and (16) to solve for the15

equilibrium share of intermediate firms based in home at time t as(θ̃tη − ϕ)yt − ϕ(1 − ϕθ̃tη ),sR,t = η(θ̃t − ϕ)(1 − ϕθ̃tη )(1 + yt )

yt ≡

Yt
=
Yt∗



L
L∗

βǫ
 ǫ−α

θ̃tη − ϕ
1 − ϕθ̃tη! α(1−ǫ)
ǫ−α

,

(25)

where ǫ > α is required to ensure that productivity growth generates a positive rate of growthin final good output (see Section 3.2). This expression is illustrated by the sR curve in Figure
1, and has a strictly positive slope: ∂sR,t /∂ θ̃t > 0.3 An increase in the international productivity differential (θ̃t ) generates an expansion in the home share of intermediate firms throughtwo channels. The first is a direct effect, whereby an improvement in the relative productivityof home-based firms expands their market share (∂sR,t /∂ θ̃t > 0). The second channel is anindirect effect, wherein international trade costs ensure that an increase in θ̃t raises the homecountry’s share of final good production (∂yt /∂ θ̃t > 0) as the relative cost of sourcing home
produced intermediate goods falls. The expansion of the home market relative to foreign
leads to a greater share of intermediate good producers for home (∂sR,t /∂yt > 0). Because
the home share of intermediate firms is bounded (st ∈ (0, 1)), however, there are limits on therange of values for the international productivity differential, outside of which an equilibriumwith active intermediate sectors in both countries is not feasible; that is, θ̃t ∈ (θ̃l , θ̃u ).
Next, we derive a differential equation to describe the dynamics of the international productivity differential. Referencing (22), we havest + δ(1 − st )θ̃t−1gθ (θ̃t )Kt /θtθ̃t+1= ∗= ∗ ∗ =.
Kt /θtθ̃tgθ (θ̃t )1 − st + δst θ̃t(26)

Thus, we naturally find that a constant productivity differential (θ̃t+1 /θ̃t = 1) is synonymouswith equal productivity growth rates for home and foreign intermediate firms (gθ = gθ∗ ).
3

The derivative of (25) with respect to the international productivity differential is
∂sR,t
∂ θ̃t

=



yt
(1 − ϕθ̃tη )2+1(ϕ − θ̃tη )2(1 − ϕ2 )θ̃tη
∂yt
ηϕθ̃tη−1+ η> 0,
1 + yt(θ̃t − ϕ)(1 − ϕθ̃tη )(1 + yt )2 ∂ θ̃twith ∂yt /∂ θ̃t > 0.

16

Figure 1: Determination of the International Productivity Differential (θ̃)
s

1/δ

1

θ̃u

sK
sR
1/2

0

θ̃l

δ

θ̃

1

With all firms employing the same level of final goods in process innovation (IR = IR∗ ),the constant productivity differential and equalized productivity growth rates require thatfirms have access to the same level of knowledge spillovers (Kt /θt = Kt∗ /θt∗ ), allowing us
to derive the following steady-state condition for the home country’s share of intermediatefirms using (18):sK (θ̃) =1 − δ θ̃−1,2 − δ θ̃ − δ θ̃−1(27)

where we now drop the time script to indicate that steady-state values are constant. As shownby the sK curve in Figure 1, when the international productivity differential (θ̃) increases,
a greater share of firms is required for home to offset the improved access of foreign-based
firms to technical knowledge (∂sK /∂ θ̃ > 0). Note that θ̃ ∈ (δ, 1/δ) is required for sK ∈(0, 1). In addition, the slope of the sK curve is closely linked to the degree of knowledge
diffusion. With perfect international knowledge diffusion (δ = 1), the sK curve becomes avertical line at θ̃ = 1, and the international productivity differential vanishes.
In Appendix B, we study the local dynamics around the steady state characterized by theintersection of the sK and sR curves in Figure 1, and obtain the following proposition.
Proposition 1 The international productivity differential (θ̃) converges to a long-run equi-17librium with s ∈ (0, 1) when ∂sK /∂ θ̃ > ∂sR /∂ θ̃.
Proof: See Appendix B.
As a state variable, the international productivity differential is rising (falling) for valuesof θ̃ to the left (right) of the sK curve. Therefore, a stable steady state requires ∂sK /∂ θ̃ >∂sR /∂ θ̃, as depicted in Figure 1. Examining (25), we find that limϕ→0 θ̃l = 0 and limϕ→0 θ̃u =∞. Accordingly, we assume that the freeness of trade (ϕ) is small enough to satisfy θ̃l < δ <1/δ < θ̃u , ensuring the existence of at least one steady state with the slope ranking outlined in
Proposition 1. Hereafter, we focus on equilibria that satisfy these conditions for the existenceof a stable long-run equilibrium.
The long-run international productivity differential (θ̃) is implicitly linked with the relative employment level of the home country (L/L∗ ). Referencing (25), an increase in L/L∗ expands the relative market size of the home country for intermediate producers (∂y/∂(L/L∗ ) >0), causing an upward shift in the sR curve in Figure 1 (∂sR /∂y > 0). As a result, the homecountry’s shares of intermediate firms (s) and final good production (y) both increase. The
rise in s coincides with an increase in the international productivity differential that returns
knowledge spillovers back to equality across countries (K/θ = K ∗ /θ∗ ), as the economy converges to a new long-run equilibrium. These results are summarized in the following lemma.
Lemma 2 An increase in the relative employment of home (L/L∗ ) expands the home country’s shares of intermediate firms (s) and final good production (y), while raising the international productivity differential (θ̃).
From the results of Lemma 2 and the mechanics of Figure 1, referencing (25) and (27), it
becomes clear that equality of national employment levels generates a symmetric equilibriumwith y = 1, s = 1/2 and θ̃ = 1. Accordingly, when home employment rises above foreign
employment (L/L∗ > 1), the home country has greater shares of intermediate firms (s > 1/2)
and final good production (y > 1), and relatively productive intermediate firms (θ̃ > 1).
When the foreign country has a larger relative employment level (L/L∗ < 1), however, it
18

has greater shares of intermediate firms (s < 1/2) and final good production (y < 1), andforeign-based intermediate firms are relatively productive (θ̃ < 1).
As we have seen, knowledge spillovers equalize across countries in the long run. Substituting (27) back into (18), and reorganizing the result, yields the steady-state level of knowledge spillovers as follows:K∗
1 − δ2
K
= ∗ =
.
θ
θ
2 − δ θ̃ − δ θ̃−1(28)

Therefore, we find that K/θ is convex in the international productivity differential (θ̃) witha minimum at θ̃ = 1; that is, ∂(K/θ)/∂ θ̃ = −[δ(1 − θ̃2 )/(1 − δ −2 )θ̃2 ](K/θ)2 . For θ̃ < 1
an increase θ̃ lowers knowledge spillovers, and for θ̃ > 1 the increase in θ̃ raises knowledge
spillovers. Consequently, we find that firm-level productivity in R&D is directly linked withnational shares of intermediate production, with an increase in the geographic concentrationof industry in one country raising the level of knowledge spillovers.
Lastly, before leaving this section, we derive the equilibrium operating profit associatedwith the intermediate firms based in home and foreign asπt =α(1 − ǫ)Ytw
,
Nt

(29)

where we have used πt = (1 − ǫ)pt Xt and πt = πt∗ in the market clearing condition forintermediate goods (23).

3.2 Economic Growth
We now turn to the long-run rate of output growth. First, combining (3), (21), and (29) with∗
Vt−1 = Vt−1
= 0, we rewrite the dynamics of household consumption aswα(1 − ǫ)(1 − η) Yt+1
Ct+1
=
.
Ct
(1 + ρ)f
Nt+1

19

(30)

Figure 2: Long-run Output Growth
g
g(θ̃)

0

1

θ̃

Because we are interested in balanced growth paths that feature a constant rate of growth inconsumption for home and foreign households, this expression implies that final good output
and the mass of intermediate firms grow at the same rate. In addition, given that y ≡ Yt /Yt∗ isconstant in a steady state with a constant international productivity differential (θ̃t+1 /θ̃t = 1),w∗
we find that Yt+1
/Ytw = Yt+1 /Yt = Yt+1
/Yt∗ = Nt+1 /Nt .

Then, substituting (5), (9), (15), (17), and (28) with αYt = PZ,tZt into Yt+1 /Yt delivers
the long-run rate of output growth as follows:
 αǫγ

w
αǫγ
Yt+1
(1 − δ 2 )ξIR ǫ−α,g ≡ w = gθ ǫ−α =Yt
2 − δ θ̃ − δ θ̃−1(31)

where IR = ηf /(1 − η). A casual examination of this expression makes it clear that knowledge spillovers are the key channel through which intermediate and final good production
patterns affect the steady-state rate of output growth. As such, referencing (28), we find thatthe output growth rate is concave in the international productivity differential (θ̃) with a minimum at θ̃ = 1, as depicted in Figure 2. In addition, (31) indicates that the long-run growthrate is scale-invariant as proportionate changes in the labor forces of home and foreign leave
the international productivity differential and the rate of output growth unchanged.
The mechanics of Figure 2, combined with Lemma 2, yield the following result.
Lemma 3 An increase in the relative employment of home (L/L∗ ) depresses steady-state
20

output growth (g) for θ̃ < 1 and accelerates output growth for θ̃ > 1.
Changes in the relative employment of home (L/L∗ ) only affect output growth (g) indirectly through adjustments in the international productivity differential (θ̃). Returning toLemma 2, because ∂ θ̃/∂(L/L∗ ) > 0, an increase in the relative employment of home generates a rightward movement along the g(θ̃) curve in Figure 2. Specifically, the increase in
L/L∗ expands the home country’s shares of intermediate firms and final good production
while raising the relative productivity of home-based firms. As a result, knowledge spillovers(K/θ) fall causing slower output growth when θ̃ < 1 and knowledge spillovers rise leading
to faster output growth when θ̃ > 1.
The level of market entry in the intermediate sector (Nt ) depends on the international
productivity differential (θ̃) and the rate of output growth (g). To emphasize this point, wefirst use (5), (9), (15), and (25) to derive final good output in home and foreign asβα(αǫθtγ ) 1−α (1 − ϕ2 )ν Ntν L 1−α,
Yt =[(1 − ϕθ̃η )(1 + 1/y)]ν

βα(αǫθtγ ) 1−α (1 − ϕ2 )ν Ntν L∗ 1−α
∗
Yt =
,
η
ν

(32)

[(θ̃ − ϕ)(1 + y)]where ν ≡ α(1 − ǫ)/((1 − α)ǫ) < 1. An examination of these expressions shows that∗
Nt+1 /Nt = Yt+1 /Yt = Yt+1
/Yt∗ = g in the long-run equilibrium. A corollary of this result

is that Ct+1 /Ct = g, and referencing (3) the steady-state interest rate is therefore constant:
1 + r = (1 + ρ)g.
Substituting these final output levels into π = α(1 − ǫ)Ytw /Nt = (1 + ρ)f g/(1 − η), wethen obtain the total mass of intermediate firms along the balance growth path as follows:

αǫγ
ǫ−α

Nt = θtΓ(1 + ρ)gf

(
1
 1−ν

β

β

L∗ 1−αL 1−α
+[(1 − ϕθ̃η )(1 + 1/y)]ν [(θ̃η − ϕ)(1 + y)]ν

1
) 1−ν

,

(33)

with Γ ≡ α1/(1−α) (1−η)(1−ǫ)ǫα/(1−α) (1−ϕ2 )ν . Hence, we find that for a given level of home
productivity (θ), our model features the standard tradeoff the arises between market entry andeconomic growth in endogenous growth and endogenous market structure frameworks. In

21

addition, the mass of intermediate firms adjusts to absorb proportionate changes in the labor
forces of home and foreign, ensuring a scale-invariant rate of output growth.
Although productivity growth drives market entry and the expansion of aggregate output
along the balance growth path, adjustments in the international productivity differential (θ̃)
generally have an ambiguous effect on market entry, as an increase in θ̃ expands the final
good output of home, but contracts the final good output of foreign, while generating an
ambiguous effect on the rate of output growth (g), as seen in Lemma 3.

3.3 Social Welfare
Lastly, we solve for the steady-state welfare levels of households in each country. Using(1) and (3), we express the lifetime welfare of home and foreign households along on thebalanced growth path as
1+ρ
U0 =
ρ



1
ln C0 + ln g ,
ρ

1+ρ
U0∗ =
ρ



1
∗
ln C0 + ln g .
ρ(34)Household welfare derives from the current level of consumption (C0 ) and the growth of
consumption (g). In Appendix C, we derive steady-state consumption as follows:

C0 = (1 − α)Y0 +ρf N0 g,2(1 − η)C0∗ = (1 − α)Y0∗ +ρf N0 g,2(1 − η)(35)

where we have set the initial value of home-firm productivity to unity (θ0 = 1), and we haveassumed that initial asset wealth is the same for home and foreign (A0 = A∗0 ). With Y0 , Y0∗ ,and N0 determined by (32) and (33), from the above expressions, we find that labor union
policy influences welfare through its effects on domestic output levels (Y0 and Y0∗ ), market
entry (N0 ), and the rate of output growth (g).

22

4 Labor Unions
This section investigates how changes in the wage orientation and bargaining power of thehome-country labor union affect output growth.

4.1 Wage Orientation of Labor Unions
We begin with an examination of the effects of a rise in the wage orientation (χ) of the homecountry labor union.4 The results are summarized in the following proposition.
Proposition 2 An increase in the wage orientation of the home-country labor union (χ) depresses output growth (g) for L/L∗ > 1, and accelerates output growth for L/L∗ < 1.
Referring back to (13), an increase in χ leads to lower employment in home (dL/dχ < 0),with the labor union’s push for higher wages depressing labor demand. Consequently, the relative employment of home (L/L∗ ) falls, reducing the home country’s shares of intermediatefirms (s) and final good production (y), and lowering the international productivity differential (θ̃), as outlined in Lemma 2. Referencing Lemma 3, the fall in L/L∗ causes an adjustment
in knowledge spillovers (K/θ) that accelerates output growth for θ̃ < 1 and depresses output
growth for θ̃ > 1.

4.2 Bargaining Power of Labor Unions
Next, we consider how an adjustment in the bargaining power (ψ) of the home-country labor
union affects output growth (g). From (13), we find that the effect of a change in bargaining power on national employment depends critically on the slope of the contract curve,
as described by the wage orientation (χ) of the labor union, with dL/dψ < 0 for χ > 1
and dL/dψ > 0 for χ < 1. In the former case, the wage-oriented labor union demands
higher wages as its bargaining power increases, thereby depressing labor demand. In the4

In our framework, raising unemployment benefits m has the same effect as increasing wage orientation χ.

23

latter case, the employment-oriented labor union negotiates greater employment by allowing
lower wages and expanding labor demand.
The effects of changes in bargaining power on labor employment have important implications for the rate of output growth, as summarized in the following proposition.
Proposition 3 If the home-country labor union is wage-oriented (χ > 1), an increase inbargaining power (ψ) slows output growth (g) for L/L∗ > 1, and accelerates output growthfor L/L∗ < 1. If the labor union is employment-oriented (χ < 1), however, an increase in ψ
accelerates output growth for L/L∗ > 1, and slows output growth for L/L∗ < 1.
As we have seen, when the home-country labor union is wage orientated (χ > 1), an
increase in bargaining power (ψ) reduces the home labor supply, leading to a fall in relative
employment (L/L∗ ) that lowers knowledge spillovers (K/θ) and depresses output growthfor L/L∗ > 1, but raises knowledge spillovers and hastens output growth for L/L∗ < 1,following Lemma 3. In contrast, when the labor union is employment orientated (χ < 1),a rise in bargaining power expands the home labor supply, generating an increase in relative
employment (L/L∗ ) that raises knowledge spillovers and accelerates economic growth for
L/L∗ > 1, but lowers knowledge spillovers and slows output growth for L/L∗ < 1.

5 Numerical Analysis
The complex nature of our framework renders a theoretical study of welfare intractable. In
this section, we calibrate the model using data for the income share of labor and employment
in the US and the UK over the period from 1980 to 2007 with the aim of investigating howchanges in labor union bargaining power affect national welfare by inducing shifts in industry
location patterns. We set the US as the home country and the UK as the foreign country.
Table 1 provides a summary of the model parameters. First, the discount rate is fixed toρ = 0.024 following Jones et al. (1993). Then, in the final goods sector, we follow Chu etal. (2016) in assuming values of α = 0.2 and β = 0.5 for the respective factor intensities
24

Table 1: Parameter Values
Parameter

Description

Value

ρ
α
β
ǫ
δ
τ
f
γ
ξ
ψ
ψ∗
χ
χ∗
m
m∗
L̄
L̄∗

Discount rate
Intermediate intensity in final production
Labor intensity in final production
Elasticity of substitution 1/(1 − ǫ)
Degree of knowledge diffusion
Trade cost
Fixed operating cost
R&D elasticity of labor
R&D efficiency
US labor union bargaining power
UK labor union bargaining power
US union wage orientation
UK union wage orientation
US unemployment benefits
UK unemployment benefits
US labor force
UK labor force

0.024
0.2
0.5
0.8
0.5
1.7
1
0.2
0.39650.6274→0.5055
0.6443→0.6091
1.44192.09680.11180.32294.03
1of intermediate goods and labor.5 We set the elasticity of substitution across intermediate
varieties to 1/(1 − ǫ) = 5, yielding a price-cost markup of 1/ǫ = 1.25, which is within therange of estimates presented by Britton et al. (2000) and Gali et al. (2007).
Next, adapting the methodology of Chu et al. (2016), we use (12) with the labor income
share data shown in Figure 3(a) to generate time series data for union bargaining power in the
US (ψ) and the UK (ψ ∗ ). The calibrated data are plotted in Figure 3(c). The solid line showsthat ψ falls from 0.6274 to 0.5055 in the US, and the dashed line indicates that ψ ∗ falls from
0.6443 to 0.6091 in the UK, over the period of analysis. The greater fall in union bargaining
power in the US stems from the larger decline in the labor income share.
We then regress national employment rates, shown in Figure 3(b), on union bargaining
power to calibrate parameter values for union wage orientation and unemployment benefits
in each country.6 The resulting parameter values are χ = 1.4419 and m = 0.1118 for the5

From (12), the lower and upper and bounds of labor income share are β and 1 − α.
Specifically, we estimate the following regression function for each country: Lt /L̄ = B0 + B1 ψt . Referencing (13), the estimates of B0 and B1 yield unemployment benefits as m = β/(B0 L̄) and union wage
orientation as χ = 1 − (mB1 L̄)/(1 − α − β).
625Figure 3: The Effects of Changes in Bargaining Power (ψ, ψ ∗ )
0.7

0.96

0.65

0.94

0.6

0.92

0.55

0.69
0.68
0.67
0.66
1980

1990

2000

2010

0.9
1980

1990

2000

2010

4.2

0.5
1980

1990

2000

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

1.805
0.88

4.15
0.878

1.8

0.876

1.795

4.1
4.05
0.874
4
1980

1990

2000

2010

1980

1990

2000

2010

1.79
1980

1.11

1.02

0.01161.1

1.01990.01151.01981.09
0.01141.01971.08
1.0196
1980

1990

2000

0.0113
2010
1980

1990

2000

2010

1980

-12

30
0.415
0.41

29.5

-12.5

29

-13

0.405
0.4
0.395
1980

1990

2000

2010

28.5
1980

1990

2000

2010

-13.5
1980

In Panels (a) through (c), the solid and dashed lines show values for the US and the UK. Panels (a) and(b) plot HP-filtered trends for labor income shares of GDP and national employment rates. The calibrated
values for union bargaining power are shown in Panel (c). In Panels (d) through (l), the solid lines plot
values for the indicated variable. Data source: OECD Annual Indicators on Unit Labor Costs, US Bureauof Labor Statistics, and UK Office for National Statistics.

26

US, and χ∗ = 1.4419 and m∗ = 0.3229 for the UK, suggesting that labor unions are wageoriented in both countries. Referencing (13), the calibrated parameters yield the time series
data for relative employment plotted in Figure 3(d), where we have normalized the UK labor
force to L̄∗ = 1 and set the US labor force to L̄ = 4.03, matching the relative labor force of
the US in 1980.7
In the intermediate sector, the degree of knowledge diffusion is fixed at δ = 0.5, a valuethat is within the range of estimates reported by Bloom et al. (2013) and that yields a certain
technological proximity between the US and the UK. We adopt τ = 1.7 for the trade cost,based on the estimates of Anderson and van Wincoop (2004) and Novy (2013), generating
a value of ϕ = 0.1197 for the freeness of trade. With respect to process innovation, we
normalize the fixed cost of market entry to f = 1, and then calibrate the intermediate output
elasticity of productivity (γ = 0.2) and the efficiency parameter for process innovation (ξ =0.3965) to generate a target value for output growth of g = 1.02 in 1980, while satisfying thesecond order condition for optimal investment in R&D (η < 1).
The effects of changes in union bargaining power on the key variables of our framework
are plotted in Figures 3(d) to 3(l). Notably, with weaker union bargaining power (ψ < ψ ∗ ),the US has a greater labor supply (L/L∗ > 1), and thus hosts greater shares of final good
production (y > 1) and intermediate good production (s > 1/2), with relatively more productive intermediate firms (θ̃ > 1) across the period of analysis. In addition, we observe thatthe relative labor supply expands and contracts in tandem with the union bargaining power
differential between the UK and the US: ψ ∗ − ψ. Following from Proposition 3, as unions
in both countries are wage-oriented, adjustments in L/L∗ translate directly into changes in
the rate of output growth (g) through the positive correlation between the relative labor supply and the international productivity differential (θ̃). Therefore, a rise in ψ ∗ − ψ increases
output growth, while a fall in ψ ∗ − ψ decreases output growth. Overall, our calibrated model
indicates that changes in union bargaining power generated a 0.0012 percentage point (pp)7

Data source: OECD Labor Market Statistics.

27

Table 2: Simulated Effects on Growth and Welfare
Variables

US

UK

Change in Output Growth: (g1980 − g2007 )-0.0012pp-0.0012pp

Change in Welfare: (U2007 − U1980 )/U1980
Domestic Output Channel (Y0 ):3.654%
3.722%

8.257%
8.396%

Investment Income Channel (N0 g):
Output Growth Channel (g):0.003%-0.072%0.018%-0.157%decrease in output growth between 1980 and 2007, as summarized in Table 2. Lastly, as presented in Figures 3(h) to 3(j), the decline in union bargaining power, and the corresponding
increase in national labor supplies, generates expansions in US output (Y0 ) and UK output(Y0∗ ), inducing a rise in the level of market entry (N0 ).
Turning now to national welfare, Figures 3(k) and 3(l) indicate that changes in union
bargaining power lead to welfare improvements in both the US and the UK. In order toobtain a clearer understanding of the mechanisms driving these welfare improvements, we
decompose welfare adjustments into three channels. In the home country, for example, taking
the total derivative of steady-state utility (34) yields

dU0 =(1 + ρ)f1+ρ
(1 + ρ)(1 − α)dY0 +d(N0 g) + 2 dg.
ρC02(1 − η)C0
ρg

The first term on the righthand side captures the effect of a change in domestic final output(Y0 ), the second term describes the effect of a change in investment income (N0 g), and thethird term describes the effect of a change in the rate of output growth (g).
We quantify the annual effect of each of the channels described above and then aggregate
across the period of analysis to obtain the welfare decomposition shown in Table 2. The
results show that welfare improved by 3.654% in the US and 8.257% in the UK over theperiod of analysis. In both countries, the general decline in union bargaining power caused an
expansion in domestic output that subsequently generated the largest improvement in welfare.
We also observe, however, that the small decrease in rate of output growth brought about a28

significant reduction in the welfare levels of both countries.

6 Conclusion
This paper has examined how unionization affects economic growth through its influence on
industry location patterns. We introduce a two-country model of endogenous growth andendogenous market structure. Intermediate firms invest in process innovation to lower theirproduction costs, and supply differentiated varieties to final good producers. International
trade costs on intermediate goods ensure that the country with the larger market (i.e., greaterlabor supply) hosts a greater share of industry. In addition, imperfect knowledge diffusion
between countries links the strength of knowledge spillovers with the geographic concentration of industry generating a positive relationship between industry concentration and the rateof economic growth.
We use the framework to study the effects of changes in labor union orientation andunion bargaining power on the output growth rate. An increase in the wage orientation of
the labor union in the large country reduces its labor supply, lowering the level of industry
concentration and raising the rate of economic growth. The effect of an increase in union
bargaining power, however, depends on the orientation of the union. When the union is
wage-oriented the labor supply contracts, reducing industry concentration and slowing output
growth. And, when the union is employment-oriented, the labor supply expands, increasing
industry concentration and accelerating economic growth.
In order to explore the welfare implications of unionization, we calibrate our model using
labor market data for the US and the UK between 1980 and 2007. Our numerical analysis
suggests that the general decline in union bargaining power observed in these countries leads
to welfare improvements that are for the most part driven by expansions in national output
levels. We also find that a fall in the rate of output growth leads to small welfare reductions
for both countries.

29

Appendix A
In this appendix, we derive the equilibrium industry location pattern (sR,t ) described by (25).
First, from (21), with free market entry reducing firm value to zero (Vt = Vt∗ = 0), from (21)
we find that profit is the same for all firms: πt = πt∗ . We rewrite this condition using (16)with (15), yt ≡ Yt /Yt∗ , θ˜t ≡ θt /θt∗ and η ≡ γǫ/(1 − ǫ) to obtainηη(1 − ϕθ˜t ) = yt (θ˜t − ϕ)

PZ,t
∗
PZ,t

ǫ
! 1−ǫ

,

(A.1)

where, referencing (9), the price indices for intermediate varieties in home and foreign are− 1−ǫǫ

PZ,t = pi,t Nt

− 1−ǫ
ǫ

1−ǫ
−η
(st + ϕs∗t θ˜t )− ǫ ,∗PZ,t= pi,t Nt

1−ǫ−η(ϕst + s∗t θ˜t )− ǫ .

(A.2)

Substituting (A.2) with s∗t = 1 − st into (A.1) delivers (25). Next, we derive yt ≡ Yt /Yt∗ in∗(25) as follows. Combining (5), αYt = PZ,t Zt and αYt∗ = PZ,tZt∗ , we obtain

Yt
=
Yt∗



Zt
Zt∗

α 

L
L∗

β

Zt PZ,tYt= ∗ ∗ .
∗Yt
Zt PZ,t,Then, removing Zt /Zt∗ from the above two equations yields the following expression:Yt
yt ≡ ∗ =
Yt



L
L∗

β
 1−α

PZ,t
∗
PZ,t

α
!− 1−α,which can be rewritten using (A.1) to obtain yt as shown in (25).

Appendix B
We reduce the model to a dynamic system in three variables: the international productivity
differential (θ̃t ), the ratio of the productivity of the home intermediate production technologyαǫγ/(ǫ−α)in final good output to the mass of intermediate firms (λt ≡ θt30

/Nt ), and the ratio ofhousehold consumption to the mass of intermediate firms (µt ≡ Ctw /Nt ). We then use thereduced system to derive a sufficient condition for the stability of long-run equilibrium.
A key feature of the system is that the dynamic path of θ̃ is determined independently of
λt and µt . Evaluating a linear expansion of (26) around the steady-state at θ̃t+1 /θ̃t = 1 yields(1 − δ θ̃t )(1 − δ θ̃t−1 ) (2 − δ θ̃ − δ θ̃t−1 )2 θ̃t ∂sR,t∂ θ̃t+1=,+
1 − δ2
1 − δ2
∂ θ̃t
∂ θ̃t

(B.1)

where we have used 1 − s + δsθ̃ = (1 − δ 2 )/(2 − δ θ̃ − θ̃−1 ), and the slope of the sR curvein Figure 1 determines the effect of changes in θ̃t on the home share of intermediate firms(st ), as (25) is satisfied at all moments in time. Setting θ̃t as a state variable, stability requires∂ θ̃t+1 /∂ θ̃t < 1. Thus, using (B.1), we rewrite the stability condition as∂ θ̃t+1(2 − δ θ̃ − δ θ̃t−1 )2 θ̃t−1 =−1 − δ2
∂ θ̃t



∂sR
∂sK
−
∂ θ̃
∂ θ̃



,

(B.2)

which provides the slope ranking ∂sK /dθ̃ > ∂sR /dθ̃ that is required for stability, as illustrated in Figure 1 and outlined in Proposition 1.
Next, we derive two differential equations to describe the evolutions of λt and µt , andshow that these variables jump immediately and permanently to their steady-state values.
First, using (32), we obtain the ratio of final good output to the mass of intermediate firms as
Ytw= λ1−ν
Ω,
t
Nt

(B.3)

where Ω ≡ (αǫ)α/(1−α) (1 − ϕ2 )ν {Lβ/(1−α) /[(1 − ϕθ̃η )(1 + 1/y)]ν + L∗ β/(1−α) /[(θ̃η − ϕ)(1 +y)]ν } is constant for a given steady-state value of θ̃. Linking πt = (1−ǫ)pt Xt = [(1−ǫ)/ǫ]IX,t
with (23) gives Nt IX,t = αǫYtw , which in turn is substituted with (B.3) into (24) to arrive at

Nt+1
1−η 
(1 − αǫ)λ1−νΩ − µt .
=
t
Nt
f

31

(B.4)

Then, referencing (31), we express the dynamics of λt as follows:

λt+1 =gfλt ,(1 − η)[(1 − αǫ)λ1−νΩ − µt ]t(B.5)

where the rate of output growth (g) assumes a steady-state value determined by θ̃.
Turning to the dynamics of µt , we use (30), (B.1), (B.2), and (B.3) to obtain the following
differential equation:µt+1 =α(1 − ǫ)Ω(gf )1−ν λ1−νt
µt .
(1 + ρ)(1 − η)1−ν [(1 − αǫ)λt1−ν − µt ]2−ν(B.6)

Together (B.5) and (B.6) describe the dynamics of λt and µt . We evaluate a linear expansion of the system around a steady state for which λt+1 /λt = µt+1 /µt = 1, and thusw
Ct+1
/Ctw = Nt+1 /Nt = g. From (B.4) and (B.5), the steady-state values of λ and µ are(1 + ρ)f g
λ=α(1 − ǫ)(1 − η)Ω

1
 1−ν

,

µ=

[1 − α + (1 − αǫ)ρ]f g.
α(1 − ǫ)(1 − η)Taking a linear expansion of (B.3) and (B.4) around λ and µ, we obtain the following
Jacobian matrixih(2−ν)[1−α+(1−αǫ)ρ]α(1+ǫρ) (1−ν)(1+ρ)µ− 1 + (1 − ν)(1 − αǫ) − 1+ρα(1−ǫ)α(1−ǫ)λ 
1 +
J =
,
ǫ(1−ν)(1+ρ)(1−η)λ1 + (1−ǫ)fg
where the trace and the determinant associated with J are
ǫ(1 − ν)(1 + ρ) (2 − ν)[1 − α + (1 − αǫ)ρ]+,(1 − ǫ)α(1 − ǫ)
(2 − ν)[1 − α + (1 − αǫ)ρ]ǫ(1 − ν)(1 + ρ)
|J| = 1 +1++α(1 − ǫ)(1 − ǫ)α(1 + ǫρ) (1 − ν)(1 − η)(1 + ρ)µ.
1 + (1 − ν)(1 − αǫ) −1+ρα(1 − ǫ)f g

tr(J) = 2 +32

As λ and µ are jump variables, we require two eigenvalues, ω1 and ω2 , with absolute values that are greater than one for saddle point stability. The values of ω1 and ω2 are determined
as a solution to the characteristic equation ψ(ω) ≡ ω 2 −tr(J)ω +|J| = 0. Substituting tr(J),|J|, and ω = 1 into the characteristic equation, we obtain

ψ(1) =

ǫ(1 − ν)(2 − ν)[1 − α + (1 − αǫ)ρ](1 + ρ)+α(1 − ǫ)2
α(1 + ǫρ) (1 − η)(1 − ν)(1 + ρ)µ1 + (1 − ν)(1 − αǫ) −> 0.
1+ρ
α(1 − ǫ)f g

As ψ(1) = (1 − ω1 )(1 − ω2 ) > 0 and tr(J) = ω1 + ω2 > 2, we find that ω1 > 1 and ω2 > 1,
and conclude that λ and µ jump immediately and permanently to their long-run values. Thus,
the slope ranking ∂sK /∂ θ̃ > ∂sR /∂ θ̃ is a sufficient condition for saddlepath stability.

Appendix C
Substituting (4), (12) and Πt /Yt = (1 − ψ)(1 − α − β) into (2), and rearranging, we obtainthe following intertemporal budget constraint:
∞
X
Yt
Ct
= (1 − α)+ (1 + r)A0 .
t
t
(1
+
r)
(1
+
r)
t=0
t=0

∞
X

(C.1)

Along the balanced growth path Ct+1 /Ct = g. Thus, substituting Ct = C0 g t , Yt = Y0 g t , and(1 + r) = (1 + ρ)g into (C.1), we obtain

C0 = (1 − α)Y0 +ρ
(1 + r)A0 .
1+ρ

(C.2)

Because initial assets are the same for home and foreign (A0 = A∗0 ) and the total asset
holdings of households equals total lending to home and foreign firms, A0 + A∗0 = (IR +f )n0 +(IR∗ +f )n∗0 , noting that IR = IR∗ = ηf /(1−η), we obtain A0 = A∗0 = f N0 /(2(1−η)).
Substituting this expression into (C.2) yields (35).

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