Discussion Papers In Economics
And Business

The endogenous decisions of unionization and international trade:
A general oligopolistic equilibrium model analysis

Hosaki SANO

Discussion Paper 18-14

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe endogenous decisions of unionization and international trade:
A general oligopolistic equilibrium model analysis

Hosaki SANO

Discussion Paper 18-14

May 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe endogenous decisions of unionization and international trade:
A general oligopolistic equilibrium model analysis
Hosaki SANO†

Abstract
This paper presents a multi-sector general oligopolistic equilibrium trade model in which unionized
and non-unionized sectors interact. In our model, the proportion of unionized sectors to all sectors is
endogenously determined. We show that the proportion of unionized sectors depends on exogenous
parameters such as productivity, population, the number of firms, union costs, and globalization. The
increase in population raises the proportion of unionized sectors and lowers the competitive wage,
whereas the increase in the number of firms and in the union cost lowers the proportion of unionizedsectors and raises the competitive wage. We also show that trade openness between symmetriccountries raises the competitive wage and lowers the proportion of unionized sectors, whereas theeffect on welfare is ambiguous.
Keywords: labor union, international trade, general oligopolistic equilibrium
JEL classification: F15, F16, L13† Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,

Japan. E-mail: qge007sh@student.econ.osaka-u.ac.jp.

1. Introduction
Although labor unions typically aim to protect workers, the presence of labor unions is unequalamong countries and industries. For example, rates of labor union participation are low in the UnitedStates, Japan, and Korea, while they are high in Iceland, Sweden, and Denmark (OECD 2014).
Similarly, at the industry level, rates of labor union participation are high in the utilities, financial, andinsurance industries, while they are low in the agriculture, real estate, and service sectors in Japan(Ministry of Health, Labour and Welfare 2014). Furthermore, participation rates are high in the utilities
and transportation industry in the United States and in the United Kingdom (U.S. Bureau of Labor
Statistics 2014; Department for Business, Innovation & Skills 2013). In addition to these inter-sector
and cross-country differences in unionization rates, we also observe decreasing trends. In Japan, thenumber of labor union members has changed drastically over time, the rate of participation in unions
decreased to 18% in 2013 from 35% in 1963. This decreasing rate of labor union participation can also
be observed globally (OECD 2014).
This study examines the factors that bring about the above-mentioned differences in the rates oflabor union participation. From a labor protection policy perspective, it is important to emphasize thata labor union is hard to organize. In this regard, we develop a general oligopolistic equilibrium (GOLE)model (Neary 2016) in which there are continuum industries. This model is tractable to analyze firm’s
behavior in continuum industries. Firms are large in their own industry but small in the entire economy.
Each firm behaves strategically in their own industry but treat factor prices and international income
as a parameter. Therefore, it is tractable to analyze that unionized and non-unionized industry interact.
In each industry, firms are unionized or non-unionized. In our model, the rate of unionized firms
to total firms is endogenously determined. This is a novel attempt in the international trade literature.
In addition, we introduce heterogeneous productivity across industries 1 as with Egger and Etzel(2012). Labor unions bargain with firms to raise the wage of workers. We assume the endogenous
decisions of unionization that a union is active (inactive) when the union wage excluded from union
fees exceeds (does not exceed) the competitive wage. We then study which of a number of factors—
union costs, population, number of firms, firm productivity, and globalization—influence theunionization. In particular, we show that globalization: shift to open economy leads deunionization.
Our findings show that several factors encourage the activities of labor unions. The first factor isthe large rents for firms, which are generated by high productivity. If a firm does not have sufficient
rents, a high wage claim brings about negative profits. Consequently, sufficient rents for firms are

We adopt the differences in the productivity across industry to derive the interior solution of the proportion ofunionized firms. If the all firms are identical about productivity, all firms are become unionized or non-unionized. We
can analyze even if we introduce a differences in the cost of union formation or the number of firms instead of thedifferences in the productivity. However, because of availability of data, we introduce a difference of the productivity
in this paper.
1-1-necessary for union activity. Thus, labor unions tend to be organized in high productivity firms.
Figure 1 illustrates the positive correlation between labor union participation rates and laborproductivity in Japan. Industries with high rates of labor union participation (e.g., utilities, financial,
insurance) have relatively high labor productivity, while those with low rates such as the agriculture
and service sectors have relatively low labor productivity (Cabinet Office, Government of Japan 2014).
Indeed, many studies have pointed out the positive effect of labor unions on productivity (see Addisonand Hirsch 1989; Booth 1995). However, this study also theoretically presents the reverse causal
relationship, namely that high productivity attracts labor union activity.
Second, low union costs also accelerate union activity. Union costs involve not only an activity
fund but also efforts for bargaining over wages. Third, we show that when the number of firms is large,
the unionized rate becomes low. A large number of firms leads to more intense competition, which
lowers firms’ profits and thus union activities. Figure 2 shows the correlation between unionparticipation rates and added value (interpreted here as the degree of competition in the industry). In
other words, when competition is not intensive, this indicator becomes high. For example, those in theutilities, financial, and insurance industries are high. Hence, lower competition is expected to attract
the activity of unions in these industries.
In addition, we show that globalization brings about more intense competition among firms andsubsequently decreases their rents, which in turn induces deunionization. Our model points out thatthe recent trend toward globalization has lowered labor union participation rates. This issue ofinternational trade and labor union activity has been widely studied by using international oligopoly
models (Mezzetti and Dinopoulos 1991; Naylor 1998, 1999; Lommerud et al. 2003). However, theseworks have constructed partial equilibrium models, while the competitive wages have been
exogenously given. We extend these works to examine the relationship between international tradeand labor unions by using a general equilibrium model.

-2-Union density by industry (%)

70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
0

1000

2000

3000

4000

5000

6000

7000

Labor productivity by industry (yen/hour)
Figure 1 Union density and labor productivity by industry in Japan, 2012

Union density by industry (%)

70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0

0

100

200

300

400

500

600

700

800

Add value per enterprise place (million yen)
Figure 2 Union density and added value per enterprise place by industry in Japan, 2012
In this body of the literature, several studies have used the GOLE model (Bastos and Kreickemeier
2009; Egger and Etzel 2012; 2014; Kreickemeier and Meland 2013). However, because these studies
analyze unionized labor markets in general equilibrium models, where the proportion of unionizedfirms to total firms is an exogenous parameter, they fail to address the driving force behind the decrease-3-

in rates of unionization. To address this shortcoming, by treating the rates of unionized sectors as an
endogenous parameter, we analyze the trend toward deunionization. Using a general equilibrium
model further allows us to analyze how the labor endowment influences labor unions.
This paper show that several results on the competitive wage are close to Bastos and Kreickemeier(2009). However, we introduce the union cost and heterogeneity of the productivity between sectors
to endogenize the proportion of unionized sector. Therefore, we can analyze simultaneously the effects
on not only the competitive wage but also the proportion of unionized sector.
The remainder of the paper is organized as follows. In Section 2, we set up the GOLE with union
model. In Section 3, we analyze the equilibrium. In Section 4, we extend the setting to an open
economy. In Section 5, we extend this model at some points. Section 6 concludes.

2. The model
2.1 Preferences and consumer demand
We assume that the representative utility function is additively separable over a continuum ofdifferent goods with each subutility function quadratic,1
1
𝑈[{𝑥(𝑧)}] = ∫ [𝑎𝑥(𝑧) − 𝑏𝑥 2 (𝑧)] 𝑑𝑧
2
0

(1)

where 𝑥(𝑧) denotes the consumption of good 𝑧 . The budget constraint of the representative
consumer is given by1(2)∫ 𝑝(𝑧)𝑥(𝑧)𝑑𝑧 ≤ 𝐼
0where 𝑝(𝑧) denotes the price of good 𝑧 and 𝐼 is aggregate income.
Maximizing Eq. (1) subject to the budget constraint for each good provides the inverse demand
function for good 𝑧:
1𝑝(𝑧) = (𝑎 − 𝑏𝑥(𝑧)),𝜆𝑥(𝑧) =

𝜆[{𝑝(𝑧)}, 𝐼] =𝑎 − 𝜆𝑝(𝑧),𝑏(3)𝑎𝜇𝑝 − 𝑏𝐼
𝜎𝑝2

where 𝜆 is the marginal utility of income and 𝜇𝑝 and 𝜎𝑝 are the first and second moments of prices,
respectively. 𝜇𝑝 and 𝜎𝑝 are given by11

𝜇𝑝 ≡ ∫ 𝑝(𝑧)𝑑𝑧 ,𝜎𝑝2 ≡ ∫ 𝑝2 (𝑧)𝑑𝑧.

00

Furthermore, by substituting 𝑥(𝑧) into Eq. (1), we can derive the indirect utility function as
follows:
-4-

̃=
𝑈

𝑎2 − 𝜆2 𝜎𝑝2
.
2𝑏

(4)

Hence, consumer welfare is decreasing in the second moment of prices.
2.2 Technology and production
We choose the consumer’s marginal utility of income as the numéraire and set 𝜆 equal to one, as
is standard in the GOLE literatures (Neary 2016; Bastos and Kreickemeier 2009; Egger and Etzel 2012,2014). Hereafter, wages, prices, union utility, and profits are weighted by the marginal utility of income.
Each industry produces a differentiated good and has 𝑛 symmetric firms apart from union activity.
Hence, firms are relatively large in their industry but are infinitesimal in the economy as a whole.
Firms use labor to produce outputs and compete in quantity in their industry. Output is linear in thelabor input 𝑦 = 𝑙/𝛼(𝑧), where 𝑙 and 𝛼(𝑧) denotes the labor demand and the labor input coefficientin industry 𝑧. In the same industry, firms’ levels of productivity are identical. We assume that the
productivity 𝛼(𝑧) is an increasing function of 𝑧 for industrial heterogeneity. Firms compete under
Cournot competition in each industry. Thus, the profit function of firm 𝑗 is given by𝑛𝜋𝑗 (𝑧) = [𝑎 − 𝑏 ∑ 𝑦𝑖 (𝑧) − 𝛼(𝑧)𝑐𝑗 (𝑧)] 𝑦𝑗 (𝑧),

(5)

𝑖=1

where 𝛼(𝑧)𝑐𝑗 (𝑧) is the marginal cost of labor. We assume that a labor union operates at the firmlevel.2 If labor union 𝑗 is active, their firm pays the union wage 𝑐𝑗 = 𝑤𝑗 to workers, otherwise, theirfirm pays the competitive wage 𝑐𝑗 = 𝑤 𝑐 . There are firms with active union and inactive union in asame industry, therefore we assume that 𝑛𝑈 (𝑧) and 𝑛𝑁𝑈 (𝑧) are the number of active unionizedfirms and inactive unionized firms respectively. Hence, 𝑛 = 𝑛𝑈 (𝑧) + 𝑛𝑁𝑈 (𝑧) holds. By maximizing
profit, the output of firm 𝑗 with active union and firm 𝑘 with inactive union are given by𝑦𝑗𝑈 (𝑧) =𝑎 − 𝑛𝛼(𝑧)𝑤𝑗 (𝑧) + (𝑛𝑈 (𝑧) − 1)𝛼(𝑧)𝑤𝑖 (𝑧) + 𝑛𝑁𝑈 (𝑧)𝛼(𝑧)𝑤 𝑐,𝑏(𝑛 + 1)𝑎 + 𝛼(𝑧)𝑤𝑗 (𝑧) + (𝑛𝑈 (𝑧) − 1)𝛼(𝑧)𝑤𝑖 (𝑧) − (𝑛𝑈 (𝑧) + 1)𝛼(𝑧)𝑤 𝑐
𝑦𝑘𝑁𝑈 (𝑧) =.
𝑏(𝑛 + 1)(6)

2.3 Labor unions
Workers are identical in all respects; however, their wages depend on the characteristics of theirunion. Workers receive the union wage set by their labor union when active in their firm and thecompetitive wage when their labor union is inactive. We assume also that unions are active when they
can set a higher the net union wage than the competitive wage and inactive otherwise. When the union2

For the industry-level union setting, see Section 5.

-5-

is active in a firm, all the workers in that firm belong to the union.
We introduce a Stone–Geary function to represent the union’s preferences. We assume that an
active union requires fixed cost 𝑓 that is measured in utility terms. Hence, union utility can be writtenas𝑉𝑗 (𝑧) = (𝑤𝑗 (𝑧) − 𝑤 𝑐 )𝑙𝑗 (𝑧) − 𝑓.

(7)

where 𝑙𝑗 is labor demand from firm 𝑗 . Each active union unilaterally sets the union wage thatmaximizes union utility for the firm whereas inactive unions do not affect their firms. The fixed costis interpreted as a maintenance cost for the union or a bargaining cost for unilaterally setting wages.
Workers share the fixed cost equally as the union fees. Hence, unionized workers obtain the net union
wage 𝑤(𝑧) − 𝑓/𝑙(𝑧). Summarizing the above, the labor union chooses to be active if its utility is
positive (i.e., the net wage is higher than the competitive wage).

3. Solving the equilibrium in the closed economy
3.1 Game structure
The proportion of unionized sectors and the competitive wage are determined by the outcome of athree-stage game in the equilibrium. In the first stage, unions decide whether to be active or inactive.
As noted in the previous section, a union chooses to be active when the net union wage (the union
wage minus union fees) is higher than the competitive wage. In the second stage, each active union
sets unilaterally its union wage 𝑤 in unionized sectors taking the competitive wage 𝑤 𝑐 as given. In
non-unionized sectors, all workers receive the same competitive wage. Unionized and non-unionizedworkers are identical except their wages. We assume no unemployment. In the third stage, each firm
decides its output taking wages and competitors’ outputs as given. We solve the subgame perfect Nashequilibrium by backward induction.
Firstly, we check the proportion of the unionized firm in an industry. We put the proportion as
𝜙 𝑈 (𝑧) = 𝑛𝑈 (𝑧)/𝑛. By substituting outputs into Eq. (7) and maximizing the union utility function, theunion wage of firm 𝑗 is given by
𝑤𝑗 (𝑧) =𝑎 + 𝑛𝛼(𝑧)𝑤 𝑐 (2 − 𝜙 𝑈 (𝑧))𝛼(𝑧) (1 + 𝑛(2 − 𝜙𝑈 (𝑧))).

Differentiating the union utility by the proportion is given by𝜕𝑉(𝑧)=𝜕𝜙𝑈 (𝑧)22𝑛2 (𝑎 − 𝛼(𝑧)𝑤𝑖 (𝑧))3 > 0.

𝑏(𝑛 + 1) (1 + 𝑛(2 − 𝜙𝑈 (𝑧)))The union utility is an increasing function of 𝜙 𝑈 . We assume that labor unions have perfect-6-

foresight3. The labor unions do not cooperate directly, but understand to achieve the rise of all labor
union’s utility in the same industry becoming active because of 𝜕𝑉(𝑧)/𝜕𝜙 𝑈 (𝑧) > 0. Hence, the alllabor unions are active in a sector 𝑧 as long as 𝑉(𝑧)│𝜙𝑈 (𝑧)=1 > 0 . The sector satisfy
𝑉(𝑧̅)│𝜙𝑈(𝑧̅)=1 = 0 becomes the threshold between unionized sector and non-unionized sector.
Accordingly, we get the union wage of firm 𝑗 given by
𝑤𝑗 (𝑧) =𝑎 + 𝑛𝛼(𝑧)𝑤 𝑐≡ 𝑤(𝑧).
𝛼(𝑧)(𝑛 + 1)

(8)

Unions decide whether to be active by comparing the competitive wage with the net union wage
𝑤 𝑓 . In brief, a union chooses to be active when 𝑤 𝑓 > 𝑤 𝑐 holds. If the union cost 𝑓 = 0, the net
union wage is always larger than the competitive wage and unions always choose to be active. In brief,
the presence of the union cost is a directly negative factor for the incentive to be unionized. Since
production demand does not depend on income, a change in wages does not influence a firm’s outputs
and the outcome of union wage setting. The following equation is an arbitrage condition of union
activity:𝑤 𝑓 ≡ 𝑤(𝑧) −

𝑓
= 𝑤𝑐.
𝑙(𝑧)

We substitute Eqs. (6) and (8) into the above equation. Hence, the condition of labor union activity
is given by𝑏𝑎 − √ (𝑛 + 1)3𝑓
𝑛
𝑤𝑐 <
.
𝛼(𝑧)
Unions have an incentive to be active in sector 𝑧 when this condition is satisfied. Hence, thethreshold 𝑧̃ that divides sectors into unionized and non-unionized is determined endogenously by thiscondition. Therefore, the arbitrage condition of labor union activity is given by𝑏𝑎 − √ (𝑛 + 1)3 𝑓𝑛𝑤𝑐 =.
𝛼(𝑧̃ )(9)Firms in the same industry are identical and produce the same amount of outputs, 𝑦𝑗 (𝑧) = 𝑦𝑖 (𝑧).
Hence, the wage is the same, 𝑤𝑗 (𝑧) = 𝑤𝑖 (𝑧). Then, we substitute Eq. (8) into Eq. (6) to derive output
per unionized and non-unionized firm. We assume 𝑎 − 𝛼(𝑧)𝑤 𝑐 > 0 because output cannot be anegative value. Output is given by𝑦𝑈 =𝑛(𝑎 − 𝛼(𝑧)𝑤 𝑐 ),𝑏(𝑛 + 1)2

𝑦 𝑁𝑈 =

𝑎 − 𝛼(𝑧)𝑤 𝑐.
𝑏(𝑛 + 1)(10)𝑛

The presence of a union lowers the employment level of a firm because 𝑦 𝑈 = 𝑦 𝑁𝑈 𝑛+1 holds. By
For simplification, we set this assumption. Of course, we can analyze it without this assumption. For detail see
Appendix. Here, the important thing is that a ratio 𝜙 𝑈 (𝑧) takes 1 or 0.
3-7-differentiating Eq. (9) with 𝑧, the arbitrage condition of labor union activity is a decreasing functionof 𝑧, 𝑑𝑤 𝑐 ⁄𝑑𝑧 < 0. This means that unions have a large incentive to be active in high productivity
industries, while they have an incentive to be inactive in low productivity industries when thecompetitive wage is sufficiently large. Hence, sectors with 𝑧 ∈ [0, 𝑧̃ ) are unionized and those with𝑧 ∈ [𝑧̃ , 1] are non-unionized (see Figure 3).
Proposition 1
High productivity sectors with 𝑧 ∈ [0, 𝑧̃ ) are unionized and low productivity sectors with 𝑧 ∈[𝑧̃ , 1] are non-unionized.

Figure 3 Arbitrage condition of an active labor union
3.2 Labor market
In a country, there are 𝐿 workers and no unemployment. Therefore, the labor market-clearing
condition is given by
1

𝑧̃

1

𝐿 ≡ ∫ 𝑛𝑙(𝑧)𝑑𝑧 = ∫ 𝑛𝛼(𝑧)𝑦 𝑈 (𝑧)𝑑𝑧 + ∫ 𝑛𝛼(𝑧)𝑦 𝑁𝑈 (𝑧)𝑑𝑧.
0

𝑧̃

0

We substitute the number of workers of unionized and non-unionized sectors into the labor market-8-

clearing condition. By solving this condition, the competitive wage is given by1𝑏𝑛𝑎𝜇1 + 𝑎 ∫𝑧̃ 𝛼(𝑧)𝑑𝑧 − (𝑛 + 1)2 𝐿
𝑛
𝑤 =
,
1
𝜇2 𝑛 + ∫𝑧̃ 𝛼 2 (𝑧)𝑑𝑧
𝑐

𝑤ℎ𝑒𝑟𝑒

1

1

𝜇1 ≡ ∫ 𝛼(𝑧)𝑑𝑧 ,𝜇2 ≡ ∫ 𝛼 2 (𝑧)𝑑𝑧 .

0

(13)

0

Eq. (13) is a decreasing function of 𝑧̃ , 𝑑𝑤 𝑐 ⁄𝑑𝑧̃ < 0.
3.3 Equilibrium
We next consider the equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) , which must satisfy the labor market-clearing
condition and arbitrage condition of labor union activity. Hence, from the arbitrage condition of labor
union activity Eq. (9) and the labor market-clearing condition Eq. (13), we can derive the equilibrium.
Hereafter, we focus on the case of the interior equilibrium.4
In the interior equilibrium case (Figure 4), the equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) is the point of the intersectionof Eqs. (9) and (13). This point is stable because unions have an incentive to be inactive in sector 𝑧 ∈(𝑧̃ ∗ , 1]. If unions are active in sector 𝑧̃ ∗ + 𝛥 (𝛥 is sufficiently small and positive), their net union
wage is lower than the competitive wage. Since the presence of a labor union lowers the wage forworkers, unions become inactive. Therefore, labor unions are inactive in sector 𝑧̃ ∗ + 𝛥. Similarly,
unions have an incentive to be active in sector 𝑧̃ ∗ − 𝛥 because the net union wage is larger than thecompetitive wage. As a result, the intersection point 𝐸 at which 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds is aninterior equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) and stable.

4

For the other cases of the corner equilibrium, see Appendix B.

-9-Figure 4 The case of the interior equilibrium
3.4 Comparative statics
We can analyze graphically how exogenous parameters such as 𝑓 , 𝐿 , and 𝑛 affect theequilibrium values of 𝑧̃ ∗ and 𝑤 𝑐∗ for the case of the interior equilibrium (Figure 4).5
First, we focus on the effect of the fixed union cost 𝑓. A change of 𝑓 affects only the arbitrage
condition of labor union activity Eq. (9). When the union cost increases, Eq. (9) shifts to lower because
Eq. (9) is decreasing function of 𝑓. Hence, higher union cost lowers the threshold value 𝑧̃ ∗ and raises
the competitive wage 𝑤 𝑐 ∗ in Figure 5. At that time, the incentive to unionization weakens by raising
the fixed cost. The decrease in the threshold (i.e., the increase in the proportion of non-unionizedsectors) raises labor demand, because non-unionized sectors absorb larger employment than unionizedsectors. Therefore, the competitive wage increases when the fixed cost increases.
Next, we analyze the effect of the population size 𝐿. A change of 𝐿 affects only the labor marketclearing condition Eq. (13) which is a decreasing function of 𝐿 . Hence, when the population 𝐿
increases, Eq. (13) shifts to upper, the threshold value 𝑧̃ ∗ increases, and the competitive wage 𝑤 𝑐 ∗decreases. The increase in the population raises labor supply and, consequentially, lowers thecompetitive wage. The decrease in the competitive wage widens the wage premium 𝑤 − 𝑤 𝑐 from5

We can also analyze the impact of 𝑓, 𝐿, and 𝑛 by using the implicit function theorem. See Appendix D.

- 10 -

Eq.(8). Hence, The increase in the population raises the incentive to unionization and the thresholdvalue 𝑧̃ ∗.
Finally, we consider the effect of the number of firms. A change of 𝑛 affects Eqs. (9) and (13). Eq.
(9) and is a decreasing function of 𝑛 and Eq. (13) is an increasing function of that. Hence, when thenumber of firms increases, Eq. (9) shifts to lower, Eq. (13) shifts to upper, the threshold 𝑧̃ ∗ decreases,
and the competitive wage 𝑤 𝑐 ∗ increases. The increase in firms causes intense competition in eachindustry, which lowers the rent of each firm. The labor union’s wage claim to firms then becomesmoderate. It is clear that the increase in 𝑛 lowers the union wage from Eq. (8). Therefore, the increasein 𝑛 weakens the incentive to unionization and lowers the proportion of unionized sectors. For thelabor market, the increase in firms raises labor demand. Therefore, the increase in 𝑛 raises thecompetitive wage.
Proposition 2
The increase in population raises the proportion of unionized sectors and lowers the competitive
wage, whereas the increase in the number of firms and in the union cost lowers the proportion ofunionized sectors and raises the competitive wage.

Figure 5 A change in union cost 𝒇- 11 -

This result is close to Bastos and Kreickemeier (2009). Their paper show that deunionization
increases the competitive wage and the union wage and reduces the union wage premium.
Deunionization denotes the decrease in the threshold between unionized and non-unionized sector 𝑧̅
which is an exogenous parameter in their paper. In this paper, the decrease in 𝑧̃ ∗ by a change of theparameter: 𝑓 , 𝐿 , and 𝑛 occurs with the increase in 𝑤 𝑐 ∗ . These processes are different, but therelations of the competitive wage and the proportion of unionized sector are similar. We also show thatthe increase in 𝑤 𝑐 ∗ raises the union wage from Eq. (8) and lowers the union wage premium: 𝑤(𝑧) −𝑤 𝑐 ∗. Hence, the decrease in 𝑧̃ ∗ by a change of the parameter is accompanied by the increase in 𝑤 𝑐 ∗and increases the union wage and reduces the union wage premium. It is constant with their paper.
The comparative statics of the number of firms 𝑛 is also similar with Bastos and Kreickemeier(2009); the increase in 𝑛 raises the comparative wage and the union wage. We show that the increasein 𝑛 raises the competitive wage from proposition 2 but has ambiguous impact on the union wage inthis paper. The increase in 𝑛 has an effect of intensive competition that makes union’s wage clammoderate because we introduce firm-level union setting (They adopt industry-level union setting).
Hence, the effect on the union wage is not clear6.
From the proposition, larger population country has large proportion of unionized sector. This result
may be against the example of large countries such as the United States. However, I guess that othernegative factors on unionization is higher because it is expected that firm’s competition is intense andunion cost is high in the United States. We think that unionization is affected by the degree of the freeentry of the firms and the protection low of the unions as well as population size in reality.

3.5 Welfare
Since welfare depends on the second moment of prices (Eq. (4)), we substitute each price into thesecond moment of prices 𝜎𝑝2 . Hence, 𝜎𝑝2 is represented by
𝜎𝑝2 =1{𝑎2 𝑛(3𝑛 + 2)𝑧̃ + 𝑎2 (𝑛 + 1)2 + 2𝑎(2𝑛 + 1)𝑛2 𝑤 𝑐 𝜇1 + 𝑛4 𝑤 𝑐2 𝜇2(𝑛 + 1)4
1

1+ 2𝑎𝑛𝑤 𝑐 (−𝑛2 + 𝑛 + 1) ∫ 𝛼(𝑧)𝑑𝑧 + 𝑛2 𝑤 𝑐2 (2𝑛 + 1) ∫ 𝛼 2(𝑧)𝑑𝑧}.
𝑧̃

In𝑧̅ ∈ [0, 1], partial differentiation 𝜎𝑝2𝑧̃𝑐in 𝑤 and 𝑧̅ are strictly positive:

𝜕𝜎𝑝2
> 0,
𝜕𝑤 𝑐

𝜕𝜎𝑝2
> 0.
𝜕𝑧̃

(14)

Therefore, the impact of a change in the fixed cost is given by
We introduce industry-level setting in section 5. The effect of the increase in 𝑛 on the union wage isconsistent with Bastos and Kreickemeier (2009).
6- 12 -

̃̃ 𝑑𝜎𝑝2𝑑𝑈
𝑑𝑈
1 𝜕𝜎𝑝2 𝑑𝑤 𝑐 ∗ 𝜕𝜎𝑝2 𝑑𝑧̃ ∗
=
=
−
(
+ ∗
).
𝑑𝑓 𝑑𝜎𝑝2 𝑑𝑓
2𝑏 𝜕𝑤 𝑐 ∗ 𝑑𝑓
𝜕𝑧̃ 𝑑𝑓Raising the fixed cost has two opposite effects on welfare because 𝑑𝑤 𝑐 ∗ /𝑑𝑓 > 0 and 𝑑𝑧̃ ∗ /𝑑𝑓 <0 from Proposition 2. It is shown the effect to welfare by checking which effect is larger.
Unfortunately, we cannot find explicit values of 𝑧̃ ∗ and 𝑤 𝑐 ∗. Therefore, we analyze welfare by using
numerical examples and present the results in Section 4.1.

4. Open economy
4.1 Equilibrium
In this section, we consider international trade between two symmetric countries. We assume thatinternational trade incurs no trade costs and that goods markets are fully integrated, while labormarkets are separated and workers are immobile between the two countries. The foreign and homefirm’s output and wages are the same because the two countries are symmetric in all respects.
The reaction function of firms 𝑗 in country 1, taking the wages, competitors’ output, and foreign
firms’ output as given, is represented by
𝑡
𝑦1𝑗
(𝑧) =

𝑡
𝑡
(𝑧) − 4𝑛𝛼(𝑧)𝑤1𝑗(𝑧) + 2𝑛𝛼(𝑧)𝑤2𝑡 (𝑧)2𝑎 + 2(𝑛 − 1)𝛼(𝑧)𝑤1𝑖𝑏(2𝑛 + 1)

where 𝑦𝑘 and 𝑤𝑘 are the firms’ output and union wage in country 𝑘 ∈ {1,2}, respectively.
The union wage is given by
𝑤𝑗𝑡 (𝑧) =𝑎 + 2𝑛𝛼(𝑧)𝑤 𝑐𝑡≡ 𝑤 𝑡 (𝑧).
𝛼(𝑧)(2𝑛 + 1)

(8t)

The output of the unionized and non-unionized sectors is given by𝑦 𝑈𝑡 (𝑧) =4𝑛(𝑎 − 𝛼(𝑧)𝑤 𝑐𝑡 ),𝑏(2𝑛 + 1)2

𝑦 𝑁𝑈𝑡 (𝑧) =2(𝑎 − 𝛼(𝑧)𝑤 𝑐𝑡 ).
𝑏(2𝑛 + 1)

The arbitrage condition of labor union activity is given by

𝑤

𝑐𝑡

𝑏
(2𝑛 + 1)3𝑓4𝑛.
𝛼(𝑧̃ )

𝑎−√
=

(9t)

We then solve the competitive wage from the labor market-clearing condition:

𝑤

𝑐𝑡

=

1
𝑏
2𝑛𝑎𝜇1 + 𝑎 ∫𝑧̃ 𝛼(𝑧)𝑑𝑧 − 2𝑛 (2𝑛 + 1)2 𝐿
12𝜇2 𝑛 + ∫𝑧̃ 𝛼 2 (𝑧)𝑑𝑧

.

(13t)

Comparing these conditions of autarky and trade openness shows that Eq. (9t) is located over Eq.
(9) when 𝑛 > 2 and Eq. (13t) is located below Eq. (13). Consequently, the shift from autarky to trade- 13 -

openness decreases the proportion of unionized sector 𝑧̃ ∗ and increases the competitive wage 𝑤 𝑐∗
in Figure 6. This result is similar to the effect of the increase in the number of firms.

Figure 6 Effect of a shift to an open economy
Proposition 3
Trade openness raises the competitive wage and lowers the proportion of unionized sectors.
The shift to trade openness has an effect like increasing in the number of firms in the meaning thatcompetition becomes intense. We check this effect focusing the difference of the union wages. We
compare the union wage premium of autarky with that of trade openness. From Eqs. (8) and (8t), it isshown that the union wage premium of autarky is larger than that of trade openness: 𝑤 − 𝑤 𝑐 > 𝑤 𝑡 −𝑤 𝑐𝑡 if the competitive wages are same value: 𝑤 𝑐 = 𝑤 𝑐𝑡 . Trade openness makes the wage claim
moderate and lowers the incentive to be unionized because firm’s competition become intensive.
This effect of shift to trade openness is close to Bastos and Kreickemeier (2009): a shift fromautarky to free trade leads to an increase in the competitive wage and the union wage, however the
proportion of unionized sectors is exogenous parameter and is not change. In this paper, the shift totrade openness raises the competitive wage but has an ambiguous effect on the union wage and welfare.
According to Kreickemeier and Meland (2013), the decrease in trade costs leads to a higher
competitive wage, to higher union wage, and to higher number of unionized workers, whereas we- 14 -

show that trade openness lowers the proportion of unionized sectors in this paper. Because of adoption
the endogenous decisions of unionization, we able to show that trade openness has negative influence
for unionization. Strictly speaking, the effect of trade openness on the number of unionized workers
is ambiguous. However, the unionized workers for a whole may decrease because trade openness
lowers absolutely unionized sectors.
From the above, the proportion of unionized sectors becomes small as a country opening the market.
This result relates a tendency of deunionization to expanse of international free trade in these last few
decades.

4.2 Numerical examples
Eventually, we cannot derive the equilibrium values 𝑧̃ ∗ and 𝑤 𝑐∗ . Then, we analyze how trade
openness and a change in parameters affect welfare by using numerical examples. In table 1, ourexamples show the effects of changes in parameters 𝑓, 𝐿, and 𝑛, where 𝑎(= 60) and 𝑏(= 1) are
fixed and the input labor coefficient 𝛼(𝑧) equals 𝑧 + 1. From Eq. (4), welfare is strictly decreasingin the second moment of prices, 𝜎𝑝2 . As a benchmark with Case 1, we present the increase in 𝑓 inCase 2, the increase in 𝐿 in Case 3, and the decrease in 𝑛 in Case 4.
Case 1Case 2

Case 3

Case 4

𝑓

15

20

15

15

𝐿

15

15

20

15

𝑛

3

3

3

2

Autarky

Case 1Case 2

Case 3

Case 4

Threshold 𝑧̃ ∗

0.48

0.37

0.73

0.84

Competitive wage 𝑤 𝑐∗

28.45

28.81

24.39

24.93

𝜎𝑝2

2440.67 2438.85 2133.35 2458.11

Welfare579.67580.58733.33570.95Trade openness

Case 1Case 2

Case 3

Case 4

0.29

0.17

0.41

0.53

30.58

30.77

27.77

29.25

Threshold 𝑧̃ ∗Competitive wage 𝑤

𝑐∗

𝜎𝑝2

2419.69 2417.94 2111.33

2430.46

Welfare590.15584.77- 15 -

591.03744.33

Table 1 numerical examples
As shown in Table 1, trade openness raises welfare in all cases (Cases 1–4) and welfare rises when
the proportion of unionized sector 𝑧̃ ∗ falls apart from 𝐿. For example, since raising the fixed costraises the competitive wage and lowers the proportion of unionized sectors, raising the fixed cost has
two opposite effects on 𝜎𝑝2 from Eq. (14). We interpret here the effect of a decrease in 𝑧̃ ∗ is larger
than the effect of an increase in 𝑤 𝑐∗. A change of 𝑧̃ ∗ is easy to influence the variation of the price(strictly speaking, the second moment of prices) because the labor cost and the price rise by becoming
unionized. On the other hand, a change of 𝑤 𝑐∗ influences all sectors. So, the change of 𝑧̃ ∗ may belarger on the welfare. In brief, the welfare may rise by the decrease in 𝑧̃ ∗: the increases in 𝑓 and 𝑛,
and trade openness.
The result about a change of 𝐿 is reversed. We interpret the effect of 𝐿 on 𝑤 𝑐∗ is larger than on𝑧̃ ∗ because the increase in 𝐿 drastically lowers the competitive wage. Hence, the increase in 𝑤 𝑐∗lowers the average price and the second moment of prices. In brief, the welfare may rise by the increasein 𝐿

5. Extensions
5.1 The union wage orientation
We introduce the union wage orientation. Its union utility function is given by𝜃(7w)

𝑉𝑗 (𝑧) = (𝑤𝑗 (𝑧) − 𝑤 𝑐 ) 𝑙𝑗 (𝑧) − 𝑓
where 𝜃 expressions the degree of the wage preference of the unions.
Hence, the union wage is given by
𝑤(𝑧) =𝑎𝜃 + 𝑛𝛼(𝑧)𝑤 𝑐.
𝛼(𝑧)(𝑛 + 𝜃)

(8w)

The union wage is an increasing function of 𝜃. The arbitrage condition of union activity and thelabor market-clearing condition are, respectively,1+𝜃𝑛 𝛼(𝑧̃

)2(𝑎 − 𝑤 𝑐 𝛼(𝑧̃ ))𝜃
()(𝑛 + 𝜃)𝛼(𝑧̃ )𝑏 (1 + 𝑛)𝜃

𝑐

𝑤 =

1
𝑏
𝑎𝑛𝜇1 + 𝑎𝜃 ∫𝑧̃ 𝛼(𝑧)𝑑𝑧 − 𝑛 (1 + 𝑛)(𝑛 + 𝜃)𝐿
1(9w)− 𝑓 = 0,𝜇2 𝑛 + 𝜃 ∫𝑧̃ 𝛼 2 (𝑧) 𝑑𝑧

.

(13w)

We use a numerical example because (9w) and (13w) cannot be solved analytically. Table 2
shows the effects of changes in parameters 𝜃, where 𝑎(= 60), 𝑏(= 1), 𝑓(= 15) and 𝑛(= 3) are- 16 -

fixed and the input labor coefficient 𝛼(𝑧) equals 𝑧 + 1.
Case 1 Case 2 Case 3
𝜃

0.75

1

1.25

Autarky

Case 1 Case 2 Case 3

Threshold 𝑧̃ ∗

0.273

0.48

0.540

Competitive wage 𝑤 𝑐∗

29.29

28.45

27.93

Table 2 numerical examples
The increasing in 𝜃 may raise the proportion of unionized sectors and lower the competitive
wage. Labor unions have an incentive to be active when 𝜃 is high because their utility is higher.
Unionized firms reduce the employment by high wage, hence the non-unionized sectors can employ
a lot with lower wages.

5.2 Industry-level union setting
Here, we analyze the case of the industry-level union setting. The industry-level union sets thewage while taking account of industrial rents. Its utility function is given by
𝑉 𝐼 (𝑧) = (𝑤 𝐼 (𝑧) − 𝑤 𝑐𝐼 )𝑛𝑙(𝑧) − 𝑓 𝐼(7i)

where 𝑓 𝐼 is the union cost for industry-level active unions. Workers in the same industry share theunion cost as union dues. Hence, the union wage is given by𝑤 𝐼 (𝑧) =𝑎 + 𝛼(𝑧)𝑤 𝑐𝐼
.
2𝛼(𝑧)

(8i)

It is clear that the union wage is the increasing function of the competitive wage 𝑤 𝑐𝐼 and the union
wage premium is higher than that of the firm-level union if the competitive wages are the same value:𝑤 𝐼 (𝑧) − 𝑤 𝑐𝐼 > 𝑤(𝑧) − 𝑤 𝑐 .
The arbitrage condition of union activity and the labor market-clearing condition are, respectively,𝑏𝑎 − 2√ (𝑛 + 1)𝑓𝐼𝑛
𝑤 𝑐𝐼 =
,
𝛼(𝑧)

𝑤

𝑐𝐼

=

1
2𝑏
𝑎𝜇1 + 𝑎 ∫𝑧̃ 𝛼(𝑧)𝑑𝑧 − 𝑛 (𝑛 + 1)𝐿1

𝜇2 + ∫𝑧̃ 𝛼 2 (𝑧)𝑑𝑧

(9i)

.

(13i)

The result is the same as the propositions of firm-level union setting and independent of the union
formation. The effect of the increase in the number of firm on the competitive wage is consistent with- 17 -

Bastos and Kreickemeier (2009). The increase in 𝑛 raises the union wage because the increase in 𝑛raises 𝑤 𝑐𝐼 and the union wage from Eq. (8i). Since the equilibrium value is not explicit, thecomparisons of the proportion of the unionized sectors and the competitive wage between the firm andthe industry level setting is ambiguous.

5.3 Asymmetric countries
We analyze the case of asymmetric countries. We check how the differences of the population
between countries affect the proportions of unionization and the competitive wages.
Our examples show the effects of changes in parameters 𝐿A and 𝐿B , where 𝑎(= 60), 𝑏(= 1),
𝑓(= 15) and 𝑛(= 3) are fixed and the input labor coefficient 𝛼(𝑧) equals 𝑧 + 1. As a symmetricbenchmark with Case 1, we present the increase in the population of the country A, in Case 2, thedecrease in 𝐿𝐴 in Case 3, and the immigration from country B to A in Case 4 in Table 3.
Case 1 Case 2 Case 3 Case 4
𝐿A

15

16

14

16

𝐿A

15

15

15

14

Threshold 𝑧̃𝐴 ∗

0.285

0.304

0.267

0.298

Competitive wage 𝑤𝐴𝑐∗

30.58

30.26

30.89

30.49

Threshold 𝑧̃𝐵 ∗

0.285

0.290

0.280

0.272

30.58

30.34

30.81

30.65

Competitive wage

𝑤𝐵𝑐∗

Table 3 numerical examples
From these results, the increase in the population in a country may raise the proportion ofunionized sectors and lower the competitive wage in both country. A change of the population of theown country greatly influences.

5.4 Introduction of the minimum wage and unemployment
In contrast to the previous sections, here we analyze the case with unemployment. In this chapter,
we adopt the minimum wage level which is exogenously determined by the government. This setting- 18 -

is close to Egger and Etzel (2012): adopt the differences in the productivity across sectors andexogenous wage which is compared by labor unions.
If the competitive wage is lower than the minimum wage, non-unionized firms have to pay theminimum wage to their workers. Hereafter, we focus the case that the minimum wage is larger thanthe competitive wage. Unions thus decide whether to be active by comparing the minimum wage with
their own net union wage. We can treat unemployment as an endogenous variable by setting theminimum wage as an exogenous variable.
We change the game structure as follows. In the first stage, the government exogenously chooses
the minimum wage level. In the second stage, unions decide whether to be active. In the third stage,
unions set the union wage. In the fourth stage, firms choose the employment level.
Hence, the labor market-clearing condition that determine the endogenous unemployment level isgiven by1𝑧̃

1(1 − 𝑢)𝐿 ≡ ∫ 𝑛𝑙(𝑧)𝑑𝑧 = ∫ 𝑛𝛼(𝑧)𝑦 𝑈 (𝑧)𝑑𝑧 + ∫ 𝑛𝛼(𝑧)𝑦 𝑁𝑈 (𝑧)𝑑𝑧.
0

𝑧̃

0

where 𝑢 is unemployment rate. When the minimum wage equals the equilibrium competitive wage,
unemployment is zero: 𝑢 = 0. Hence, the unemployment rate rises as increasing the minimum wage,
however, when the minimum wage is within a certain range, there is a case that its increase lowers theunemployment rate. From the labor market-clearing condition, the differentiation of 𝑢 by theminimum wage 𝑤
̅ is given by1(𝑎 − 𝐵)2 1
𝑑𝑢= 𝑛𝜇2 ∫ 𝛼 2 (𝑧)𝑑𝑧 − 𝐵𝑑𝛼(𝑧̃ )
𝑑𝑤
̅
𝑤
̅3
𝑧̃
𝑑𝑧̃

𝑏
𝑤ℎ𝑒𝑟𝑒 𝐵 ≡ √ (𝑛 + 1)3 𝑓𝑛Hence, the following relationship is derived:𝑑𝑢≷ 0 𝑖𝑓
𝑑𝑤
̅

𝑤
̅3 ≷

𝐵(𝑎 − 𝐵)2.
1𝑑𝛼(𝑧̃ )
(𝑛𝜇2 + ∫𝑧̃ 𝛼 2(𝑧)𝑑𝑧) 𝑑𝑧̃

The increase in the minimum wage has two effects on unemployment: (i) it increases production
costs and (ii) it reduces the proportion of unionized sector.7 The first effect lowers labor demand,
whereas the second effect increases labor demand because the output of the non-unionized sector islarger than that of the unionized sector. Hence, if the minimum wage is relatively low, its increase
reduces the unemployment rate. In the setting from Egger and Etzel (2012) that all sectors are
exogenously unionized, an increase in the minimum wage raises unemployment. However, theendogenous decision of unionization complicates the effect of the minimum wage on unemployment
in this paper.

7

For more details, see Section 3.

- 19 -

6. Conclusion
This study presents a multi-sector general oligopolistic equilibrium trade model for analyzing how
several factors such as trade openness, firm productivity, number of firms, union costs, andglobalization affect the proportion of unionized sectors, which is treated as an endogenous parameter.
We show that unions in high productivity sectors have a large incentive to be active. For this reason,
firms in high productivity sectors have large rents and can accept high wage demands from unions. If
firms have insufficient rents, however, unionized workers cannot claim a high wage from the firm.
Therefore, in low productivity sectors, unions have an incentive to be inactive. We also show that trade
openness decreases the proportion of unionized sectors, as it leads to intense competition and thus
reduces firms’ rents. Therefore, as globalization advances, the incentive for unions to be active lowers.

Acknowledgments
We would like to thank Kazuhiro Yamamoto, Koichi Futagami, Masaru Sasaki, Yasuhiro Sato, and
Dao-Zhi Zeng as well as the participants of the Applied Regional Science Conference in Okinawa andof the Urban Economics Workshop at The University of Tokyo for their helpful comments andsuggestions.

- 20 -

Appendices
Appendix A: the determinant of 𝜙 𝑈 (𝑧)We can derive the 𝜙 𝑈 (𝑧) without the assumption of labor union’s perfect foresight. Firstly, thefollowing union’s activity results are same. The sector 𝑧 is unionized if 𝑉(𝑧) > 0 for any 𝜙 𝑈 (𝑧).
Similarly, the sector 𝑧 is non-unionized if 𝑉(𝑧) > 0 for any 𝜙 𝑈 (𝑧).
The 𝜙 𝑈 (𝑧) is dependent of the initial value of 𝜙 𝑈 when some sector 𝑧 satisfy 𝑉(𝑧)│𝜙𝑈(𝑧)=1 >𝑈 ∈ (0, 1) satisfy 𝑉(𝑧)│̅̅̅̅
0 and 𝑉(𝑧)│𝜙𝑈 (𝑧)=0 < 0 . In this case, 𝜙̅̅̅̅̅
𝑈 = 0 is unstable
𝜙𝑈 (𝑧)=𝜙
because of 𝜕𝑉(𝑧)/𝜕𝜙 𝑈 (𝑧) > 0. Hence, 𝜙 𝑈 takes 0 or 1. The sector 𝑧 is unionized if the initial̃𝑈 larger than ̅̅̅̅̃𝑈 < ̅̅̅̅
value 𝜙𝜙𝑈 and the sector 𝑧 is non-unionized if 𝜙
𝜙𝑈 . The threshold 𝑧̃ is̃𝑈 however the essential analysis does not change. We can interpret thedecreasing function of 𝜙̃𝑈 = 1.
assumption of labor union’s perfect foresight as 𝜙Appendix B: Analysis of the equilibrium pattern
We can divide the equilibrium into four cases8 by the positional relationship of Eqs. (9) and (13).
First, we divide the cases based on whether Eqs. (9) and (13) intersect. Second, we divide the cases
based on the vertical position relationship of Eqs. (9) and (13).
We define the intersection case as when Eqs. (9) and (13) have an intersection point in 𝑧̃ ∈ (0, 1),
which allows us to derive the interior equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ). We define the non-intersection case as
when Eqs. (9) and (13) do not have an intersection point in 𝑧̃ ∈ (0, 1). In this case, the equilibrium iseither of the corner equilibria, (0, 𝑤 𝑐∗ ) or (1, 𝑤 𝑐∗ ).
There are two intersection cases as follows. The first case is that the slope of Eq. (13) is larger than
the slope of Eq. (9) on the intersection point of Eqs. (9) and (13). Figure 4 represents this case. The
second case is that the slope of Eq. (13) is smaller than the slope of Eq. (9) on the intersection point.
Figure A1 represents this case.
In the case of Figure 4, the equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) is the intersection point 𝐸. This point is stablebecause unions have an incentive to be inactive in sector 𝑧 ∈ (𝑧̃ ∗ , 1]. If unions are active in sector𝑧̃ ∗ + 𝛥 (𝛥 is sufficiently small and positive), their net union wage is lower than the competitive wage.
Therefore, the labor union is inactive in sector 𝑧̃ ∗ + 𝛥. Similarly, unions have an incentive to be active
in sector 𝑧̃ ∗ − 𝛥 because their net union wage is larger than the competitive wage. As a result, point
𝐸 at which 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds is an interior equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) and this equilibrium is

There is another case with two intersections of Eqs. (9) and (13); however, we omit it here because its range is verysmall. For details, see Appendix C.
8- 21 -

stable.
In Figure A1, the intersection point 𝐸′ is unstable because at point 𝐸′ 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ > 0
holds. The corner points 𝐸 and 𝐸′′ are stable because at those points 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds.
Hence, 𝐸 and 𝐸′′ are the stable corner equilibria, (0, 𝑤 𝑐∗ ) and (1, 𝑤 𝑐∗ ).

Figure A1 The case that all sectors are unionized or non-unionized
The two non-intersection cases follow. The first case is that Eq. (9) is always higher than Eq. (13)in 𝑧̃ ∈ [0, 1], which represents the case shown in Figure A2. The second case is that Eq. (9) is always
lower than Eq. (13) in 𝑧̃ ∈ [0, 1], which represents the case shown in Figure A3.
In Figure A2, point 𝐸 is stable because at this point 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds. Hence, theequilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) is the corner equilibrium (1, 𝑤 𝑐∗ ). Similarly, in Figure A3, point 𝐸 is stablebecause at this point 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds. Hence, the equilibrium (𝑧̃ ∗ , 𝑤 𝑐∗ ) is the corner
equilibrium (0, 𝑤 𝑐∗ ).

- 22 -

Figure A2 The case that all sectors are unionized

Figure A3 The case that all sectors are non-unionized- 23 -

For simplicity, we specify that the input labor coefficient 𝛼(𝑧) equals 𝑧 + 1 and derive 𝑤 𝑐 ofthe intersections of Eqs. (9) and (13) with 𝑧̃ = 0 and 𝑧̃ = 1. We compare these values as follows:𝑤 𝑐(𝐸𝑞.

9)│𝑧̃=0 > 𝑤 𝑐(𝐸𝑞.

13)

7
5
│𝑧̃=0 ⇒ 𝐿 > (𝑛 + 1)√𝐻√𝑓 − 𝑎𝐻,
3
6

(15)

𝑤 𝑐(𝐸𝑞.

9)│𝑧̃=0 < 𝑤 𝑐(𝐸𝑞.

13)

7
5
│𝑧̃=0 ⇒ 𝐿 < (𝑛 + 1)√𝐻√𝑓 − 𝑎𝐻,
3
6

(16)

𝑤 𝑐(𝐸𝑞.

9)│𝑧̃=1 > 𝑤 𝑐(𝐸𝑞.

13)

7
𝑎𝑛
│𝑧̃=1 ⇒ 𝐿 > 𝑛√𝐻√𝑓 +𝐻,
63(𝑛 + 1)(17)𝑤 𝑐(𝐸𝑞.

9)│𝑧̃=1 < 𝑤 𝑐(𝐸𝑞.

13)

7
𝑎𝑛
│𝑧̃=1 ⇒ 𝐿 < 𝑛√𝐻√𝑓 +𝐻,
63(𝑛 + 1)(18)

𝑤ℎ𝑒𝑟𝑒 𝐻 ≡where 𝑤 𝑐(𝐸𝑞.

9)and 𝑤 𝑐(𝐸𝑞.

13)𝑛𝑏(𝑛+1),

are the competitive wages of Eqs. (9) and (13) and 𝐻 ≡𝑛/𝑏(𝑛 + 1), respectively. We summarize the above discussion as follows:
Eqs. (15) and (18) hold⇒ Region A; the case of the interior equilibrium (Figure 4),
Eqs. (16) and (18) hold⇒ Region B; the case that all sectors are unionized or non-unionized(Figure A1),
Eqs. (15) and (17) hold⇒ Region C; the case that all sectors are unionized (Figure A2),
Eqs. (16) and (18) hold⇒ Region D; the case that all sectors are non-unionized (Figure A3).
We depict the graph of 𝐿 and √𝑓 in Figure A4.

- 24 -

Figure A4 The division cases of the equilibrium
Appendix C: The case of multiple intersections
There are two intersection points of Eqs. (9) and (13) in this case (Figure A5). Points 𝐸 and 𝐸′′
are stable since at these points 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ < 0 holds. However, point 𝐸′ is unstable since atthis point 𝑑(𝑤 𝑓 − 𝑤 𝑐 )/𝑑𝑧̃ > 0 holds. As a result, the equilibria are 𝐸 and 𝐸′′. The equilibrium of
𝐸 is the same as the case of the interior equilibrium, while 𝐸′′ is the same as the case of the corner
equilibrium.

- 25 -

Figure A5 The case of multiple intersections
Appendix D: Comparative statics
We can also analyze the impact of the parameters in the case of the interior equilibrium (Figure4), using the implicit function theorem. From Eqs. (9) and (13), we define function 𝐹 as follows:1

𝐹(𝑧̃ , 𝑓) ≡𝑏𝑛
1 2
𝑛𝜇2 +∫𝑧̃ 𝛼 (𝑧)𝑑𝑧

𝑛𝑎𝜇1 +𝑎 ∫𝑧̃ 𝛼(𝑧)𝑑𝑧− (𝑛+1)2 𝐿−𝑏𝑛𝑎−√ (𝑛+1)3 𝑓
𝛼(𝑧̃)

= 0.

By using implicit function theorem, we get
𝐹𝑓𝑑𝑧̃ (𝑓)
=−
𝑑𝑓
𝐹𝑧̃

=−

=−

𝑏(𝑛 + 1)3√𝑛𝑓
2𝛼(𝑧̃ )𝑏𝑎 − √𝑛 (𝑛 + 1)3 𝑓 𝑑𝛼(𝑧̃ )𝑎(𝑧̃ )(𝛼(𝑧̃ )𝑤 𝑐 − 𝑎) +1𝑑𝑧̃
𝛼 2 (𝑧̃ )𝑛𝜇2 + ∫𝑧̃ 𝛼 2 (𝑧)𝑑𝑧
𝐹𝑓𝑐 (𝐸𝑞.13)

𝑑𝑤
𝑑𝑧̃

𝑑𝑤 𝑐
−
𝑑𝑧̃

(𝐸𝑞.9),- 26 -

where𝑑𝑤 𝑐 (𝐸𝑞.13)
𝑑𝑧̃

and

𝑑𝑤 𝑐 (𝐸𝑞.9)

are the slopes of Eqs. (9) and (13) in 𝑧̃ , respectively.

𝑑𝑧̃

Since the case of the interior equilibrium

𝑑𝑤 𝑐 (𝐸𝑞.13)
𝑑𝑧̃

>𝑑𝑤 𝑐 (𝐸𝑞.9)𝑑𝑧̃

is satisfied in equilibrium 𝐸(Figure 4), the following equations hold:
𝑑𝑤 𝑐
𝑑𝑧̃
⟹

𝑑𝑧̃ ∗ (𝑓)
𝑑𝑓

∗

< 0,

𝑑𝑤 𝑐 (𝑓)𝑑𝑓> 0,(𝐸𝑞.13)>𝑑𝑧̃ ∗ (𝐿)
𝑑𝐿

𝑑𝑤 𝑐
𝑑𝑧̃

(𝐸𝑞.9)∗> 0,- 27 -

𝑑𝑤 𝑐 (𝐿)𝑑𝐿< 0,𝑑𝑧̃ ∗ (𝑛)
𝑑𝑛

∗

< 0,

𝑑𝑤 𝑐 (𝑛)𝑑𝑛> 0.

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