DP2015-29

Effects on the Cross-Country Differencein the Minim um Wage on International
Trade, Growth and Unemployment *Ch ih iro INABA
Katsufumi FUKUDA
July 14, 2015

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

Effects on the cross-country difference in the minimum
wage on international trade, growth and unemployment ∗Chihiro Inaba †Department of Economics, Kobe University
Katsufumi Fukuda ‡Hiroshima University
May 2015
Abstract
We construct a dynamic general-equilibrium North-South growth model withinternational trade with both homogenous and heterogeneous firms, endogenous
northern economic growth, and unemployment. Unemployment is emerged from theimbalance between the endogenous labor supply and the firms’ labor demand underbinding the minimum wage policy. The north produces two goods, high-tech good andlow-tech good, while the South produces only low-tech good by the scarcity of technology.
Both goods are traded between the countries. The production of the high-tech goodneeds R&D activity for variety creation, which is a source of economic growth.
In this setting, we analyze the southern policy change that increases the southernminimum wage, and show that the increase in the southern minimum wage affects the
structure of international trade and the northern growth rate and unemployment.

I would like thank to Noritsugu Nakanishi, Yunfang Hu, and Davis Colin for theirhelpful comments and suggestions. Needless to say, any errors remaining in this paper
are the responsibility of the authors.
† 2-1, Rokkodai-cho, Nada-ku, Kobe,Hyogo, 657-8501, Japan ,Email: chihiro.inaba01@gmail.com‡ Email: 2katsufumi.fukuda@gmail.com∗

1. Introduction
Recently, some developing countries have rapidly grown, and affect the economy of
the advanced countries through international transactions. Especially, China has grownquickly since the 1980s and the volume of international trade has also been increasing.
Per capita real GDP in terms of U.S. dollar was 307.35 in 1980 and is 6077.65 in 2012 1.
According to Autor, et al. (2013), the amount of U.S. manufacturing imports from China
increased from 2.9% in 1991 to 11.7% in 2007. While the expenditure share of the U.S.
on Chinese goods was 0.6% in 1991, the rate raises up to 4.6% in 2007. The competition
between Chinese goods and the U.S. goods is fierce and affects the labor market in the
U.S. Autor et al. (2013) have examined the effects of import competition on the U.S.
labor market, and showed that its competition has a negative effects on employment
from 1990 to 2007. A situation like this occurs even in EU. Bloom et al. (2011) show thatthe import from China to EU has grown dramatically from 1980 to 2007. Moreover, they
showed that it leads low productive firms exit and there exists a positive effect oninnovation and a negative effect on employment in manufacturing. The importcompetition with China also contributes to 15% of increases in innovation in EU
between 2000 and 2007. An increases in imports by 10% decreases employment by 3.5%.
They showed that import from China affects economic growth in developed countries.
However, with globalization in China, the government of China has increased theminimum wage. The labor law regarding minimum wage in China was first settled in
1994 to protect vulnerable workers from the most egregious aspects of intensively
competitive markets. According to Fang and Lin (2013) and Wang and Gunderson(2011), the minimum wages in China have increased recently. Wang and Gunderson

1

The data is from World Economic Outlook Databases in 2013.

(2011) showed that the increase in the minimum wage has the negative effects on
Chinese economy. In other developing countries such as Thailand, Vietnam and so on,
the local governments increase the legal minimum wages in the 2010s. These countries
also contribute to the expansion of the world trade, and largely affect the economy of theadvanced countries, i.e., the U.S. and the EU.
How does the recent tide of increasing the minimum wage in developing countries
affect the economy of the advanced countries through globalization? To examine therelationship of the minimum wages between the developing country and the advanced
country, we construct a two country endogenous growth model. Let us denote the U.S.
and the EU by the North and China (or developing countries) by the South, respectively.
We examine the effects of increases in the minimum wage in the South on innovation in
the North through changes in employment in both countries. In the North, there arethree sectors: low-technology and high-technology goods sectors, and R&D sector. In the
low-tech sector which exists in both countries, there is a continuum of firms withidentical technologies. In this sector, a fixed number of firms produces goods with labor.
Monopolistically competitive profit is redistributed to consumer in a lump sum
distribution.
The high-tech producer is well-differentiated and supplied by monopolisticcompetitors. We consider two cases of the high-tech sector: homogenous firms andheterogeneous firms. In the case of homogenous firms, once a firm succeeds in R&D
activity of creating a high-tech brand (variety), it can sell the variety monopolisticallyand permanently. In the case of firm heterogeneity, we use the framework of 𝑎𝑎̀ la

Melitz (2003). Before entering the market, each firm incurs the sunk cost to find out
productivity. In addition, each firm must pay the sunk cost to enter each market. Basedon the realization of productivity, each firm exclusively chooses to serve both themarkets, the northern market, and exit.
In the both cases, the R&D sector is a perfectly competitive sector. Following Bloom,
et al. (2014), the R&D sector exists only in the North. We assume that there areinter-temporal knowledge spillovers, and then endogenous growth with variety
expansion occurs in this model. Production of any goods and R&D activities uses labor.
The governments in both countries implement the minimum wage policy. The labor is
supplied inelastically. The demand of the workers for each activity also depends on theminimum wage. If the minimum wage is set above the equilibrium wage,
unemployment occurs.
Our paper is related to the following two research branches of literature. First is
the analysis of international trade between the advanced country and the developing
country. Bloom et al. (2014) constructed a North-South model with trade and growth to
examine the import competition from China. They assumed that a constant fraction of
the patent of goods are produced in the South and allow to exporting to the North. They
further assume that increases in its fraction imply trade liberalization and showed that
trade liberalization leads to a higher growth rate in their model. They considered theperfectly competitive labor market. Thus, they did not considered the effects on labormarkets in the North. Autor et al. (2013) empirically showed that import from China
negatively affects employment in U.S.
Secondly, our research is also related to the relationship between international
trade and the minimum wage. The minimum wage is one of the most important factors
which generates labor market rigidities and then, the involuntary unemployment.
Davis (1998), who has first dealt with the relationship between labor market rigiditiesand international trade, showed that bad foreign labor market institutions exert apositive spillover on domestic workers. On the other hand, Felbermayr et al. (2009)
empirically showed that trade partner's real wage rigidity positively affects
unemployment in own country for 20 OECD countries between 1982 and 2003.
Egger, Egger and Marksen (2012) constructed a two country static model of tradewith firm heterogeneity, where there exists unemployment in both countries. Moreover,they examined an increase in the minimum wages in one country affects negatively oninternational trade and employment in both countries. The one-side increase in theminimum wage in the foreign country decreases the number of varieties the country
produces, which decreases the available varieties in the world. The decrease in theavailable varieties reduces the domestic country's demand for goods, and enforces firm
to exit in from the domestic market. As a result, the domestic country is also harmed by
the increase in the foreign country's minimum wage. However, they did not consider theeffects of minimum wage on economic growth through employment. Moreover, whether
their arguments are consistent with the context of North-South model is not clear.
Sener (2006) examined the effects of changes in the minimum wage on
unemployment in the EU and economic growth in the EU and the U.S. However, hismodel considers only unemployment in one country not in both. Grieben (2009)
constructed a North-South R&D based growth model of trade. He examined the effects
of globalization through increases in population in the South on innovation in bothcountries and the northern employment. But, there exists no southern unemployment
in his model.
The remainder of the paper is as follow. Section 2 explains the base of the modeland states the characteristic of the steady state equilibrium. Section 3 analyzes theeffects of the minimum wage on international trade, growth and unemployment. Section
4 concludes with summary and future research.

2. The model
There are two countries in the world, North and South. The households in bothcountries obtain the utility from leisure and consumption of two goods: the high-tech
good and the low-tech good. In each countries, the worker's wage is protected by theminimum wage set by the government, w N , w S . Under the minimum wage, eachworker decides the allocation of working. Firms also decide how much workers they
employ for profit maximization. If the minimum wage is higher than the equilibrium
wage where the labor supply and the labor demand is equalized, the unemployment
occurs.

2.1 The household
We consider the representative household. The household lives infinitely and obtain theutility from consumption of two types of differentiated goods: the high-tech goods Di

and the low-tech goods Y i , i = N, S. He inelastically supplies Li for working and earns
the wage w i , i = N, S . w i is the minimum wage set by the government.

The North The northern consumer uses a part of the wage income for the consumption
and saves the residual of the income. The northern household maximizes the following
lifetime utility:∞

U(t) = � e−ρ(τ−t) [α log DN (τ) + β log Y N (τ)]dτ , α, β ∈ (0,1), α + β = 1t(1)where ρ is the subjective discount rate. DN (t) and Y N (t) are the sub-utility of theconsumption for the high-tech good and the low-tech good at time t. The sub-utility of
the high-tech good has the following quantity index given by:mD (t)

DN (t) = ��

0

σ
σ−1
σ−1
N (κ) σxDdκ�,σ > 1(2)

N
where xD(κ) is the consumer's quantity consumed of a product κ at time t, and mD (t)is the number of available variety of the high-tech good in an economy at time t. The

parameter σ measures the degree of product differentiation. The sub-utility of the
low-tech good also has the following quantity index given bymY

Y N (t) = ��

0

σ
σ−1
σ−1
N (κ) σxYdκ�, 𝜎𝜎 > 1(3)

where xYN (κ) is the consumer’s quantity consumed of a product κ at time t, and mY isthe number of available variety of the low-tech good in an economy.

The northern household maximizes the utility subjected to the following budget
constraint:Ȧ N (t) = rN AN (t) + w N LN (t) − E N (t)where AN (t) is the stock or saving of the household and E N is defined as theexpenditure for the consumption goods, Di and Y i (i = N, S):
E i ≡ PDN DN + PYN Y N

where PDN is the price index of the high-tech good and PYN is the price index of the

low-tech good. Since the consumptions of the high-tech good and the low-tech good have
the Cobb-Douglas form, each expenditure for both goods is
PDN DN = αEN , PYN Y N = βE N

The budget constraint for the high-tech goods is

(4)

mD

�

0

N
N
pN
D (κ)xD (κ) dκ = αE

(5)With the sub-utility in Eq. (2), solving the static optimization problem yields thedemand function of the high-tech good:
N (κ)
xD
=

where

PDN (t) = ��−σNpN
D (κ) αE
{PDN (t)}1−σ

mD

0

1−σ
pN
dκ�
D (κ)

(6)

1
1−σ

Similarly, the budget constraint for the low-tech goods is
mY

�

0

N
N
pN
Y (κ)xY (κ) dκ = βE

(7)With the sub-utility in Eq. (3), solving the static optimization problem yields thedemand function of the low-tech good:where

xYN (κ) =PYN = ��

mY

0

−σ
N
pN
Y (κ) βE
{PYN (t)}1−σ

1−σ
pN
dκ�
Y (κ)

(8)

1
1−σ

To solve the dynamic optimization problem, we define the following Hamiltonian:ℋ ≡ e−ρ [α log DN (t) + β log Y N (t)] + λ[rN (t)AN (t) + w N (t)LN − E N (t)]Substituting Eq. (4) into DN (t) and Y N (t) in the Hamiltonian and solving the
Hamiltonian, we obtain a general Euler equation:
Ė N= rN − ρEN

The South Different from the northern household, the southern consumer cannot savethe income for the next time. In other words, the southern household sorts the wage
income into the consumption of the high-tech good and the low-tech good at every time.

The demands of the consumptions goods are the same ones in the North, Eq. (6) and (8).

2.2 Firms
2.2.1 Low-tech good
Both countries produce the low-tech goods and export them to each other. The low-techgoods is differentiated and sold under monopolistic competition market. Each firm in
the differentiated sector is able to produce a unique variety of low-tech good. Countryi, i = N, S is endowed with a fixed mass miY of these varieties. For simplicity, we assume
that m� Y = mN
Y =

mSY
γ

, where γ ≥ 0 is the exogenous parameter. The productivity of the

low-tech goods' producer is homogenous, and all varieties are produced and exported.
The production of the low-tech good uses one unit of labor. The profit of low-tech good
becomesjj

πiY (κ) = piY (κ)xYi(κ) − w i xYi (κ) + piY (κ)xY − w i xY (κ), i, j = N, S, i ≠ j

All firms face the demand of the low-tech good in Eq. (8) and set the price as
piY (κ) = piY =The operating profit is
πiY =

σw i
, i = N, S
σ−1

(w N )1−σ β(EN + E S ), i = N, Sσm
� Y [(w N )1−σ + γ(w S )1−σ ](9)Since free entry is not allowed, the positive profit, miY πiY, occurs in equilibrium. The

profit is distributed into all the households. The labor demand in the low-tech good's
production equals that the number of variety times the amount of production,

miY xYi, i = N, S . Therefore, the wage income from the low-tech good production isexpressed asN
N S
w N LY = mN
Y πY + mY πY =β(EN + E S )1 + γωσ−1(10)where ω ≡
is

wN
wS

. Similarly, the southern wage income from the low-tech good production

w S LSY =β(EN + E S )
1
+1
γωσ−1

(11)

2.2.2 High-tech goods}
The high-tech industry is only located in the North and supplies the goods for the Northand the south. The high-tech good is also well-differentiated but the variety of thehigh-tech good is created by R&D activity. Differentiated with low-tech goods, themeasure of the high-tech good increases over time. Each firm in the high-tech industry
owns one variety. The high-tech goods are produced only by using labor and sold undermonopolistic competition. In this section, all the firms are homogenous.
Production Each high-tech firm earns the following profit:
S
S
N
N
N N
N S
πD (κ) = pN
D (κ)xD (κ)xD (κ) − w xD (κ) + pD (κ)xD (κ) − w xD (κ)

From profit maximization, the monopolistic price of the variety is set aspN
D (κ) =

σw N
σ−1

(12)

Using Eq. (13), we calculate the operation profit:
πN
D (κ) =σ−σ (σ − 1)σ−1 (w N )1−σ αEN (t){PDN (t)}1−σ(13)

Innovation We assume that the patent term of a variety is permanent. The value of firm
is defined as the sum of the present discount value of profit:∞τV ≡ � πD (τ)e− ∫t r(s)ds dτtDifferentiating the above equation, we obtain no-arbitrage condition. The no-arbitrage
condition requires that the return from risk-free asset market equals to the one ofequity security which consists of capital gain (or loss) and dividend.
V̇ (t) πD+= r(t)
V(t) V(t)Rearranging the no-arbitrage condition, the value of firm is calculated as
V i (t) =πDV̇ (t)
r(t) −V(t)(14)

In the innovation sector, firms engage in R&D activities. The unit labor
requirement associated with creating a variety is bI (t). In other words, it takes bI (t)units of labor at time t to create one variety. Each individual firm treats bI (t) as aparameter, but it can change over time due to knowledge spillovers. We assume perfectknowledge spillover among firms, therefore,bI (t) =1mD (t)(15)

Free entry is allowed in the innovation sector. Under an equilibrium that a newvariety is continuously created, excess profit must equal to the cost of R&D activity.
Therefore, the following free entry condition is established:
V(t) = bI (t)w N(16)

Steady-state equilibrium We examine the steady state equilibrium of the model. The
steady state equilibrium is defined as an equilibrium where all endogenous variables
grow at constant (not necessarily identical) rates over time. Following Grossman andHelpman (1991), the world expenditure is normalized. Therefore, the market interest

rate rN (t) must be constant over time and equal the discount rate in any steady-state
equilibrium, rN (t) = ρ.

We define the (constant) growth rate of creating variety as g ≡ ṁ D /mD . From Eq.

(14) and (16), the steady-state-equilibrium growth rate of variety g is obtained as:
g=

α
−ρ
σw N

(17)

Since the labor supply is exogenous as LN and LS , the unemployment, U N and U S ,are calculated by subtracting the labor demand from the labor supply. In the South, thelabor is used only by the low-tech sector:
LDS =βγωσ−1(1 + γωσ−1 )w S

Therefore, the southern unemployment is
U S = LS −βγωσ−1(1 + γωσ−1 )w s

(18)

In the North, labor force is used by R&D activity, production in the high-tech sector,
and production in the low-tech sector:
LDN = g +=α(σ − 1)β(σ − 1)+Nσwσ(1 + γωσ−1)αβ(σ − 1)+−ρNw
σ(1 + γωσ−1 )w N

Therefore, the Northern unemployment isαβ(σ − 1)U N = LN − � N +− ρ�w
σ(1 + γωσ−1 )w N

(19)2.2.3 Effects of the minimum wage
This section analyzes how a change in the minimum wages affects international trade,growth, and unemployment. As explained in introduction, we have the interest on theeffects of the policy change in the developing country on the advanced countries.

Therefore, we investigate the role of the developing country's minimum wage,w S . Whenw S increases, the northern relative wage rate w N /w S decreases, and then ω decreases.
The change in ω affects the low-tech good in both countries through the price index.
With reducing ω, the low-tech goods produced in the North relatively become cheaper
than that of the South, and the North has an advantage on the low-tech good. This∂LDN

advantage increases the labor demand of the low-tech good in the North ( ∂wS > 0), andraises the northern wage income in low-tech sector, which increases the northern
expenditure. Therefore, the effect of the increase in the southern minimum wage on thenorthern unemployment is negative:γβ(σ − 1)2 ωσ−2
∂U N
=
<0
∂w S σ(w S )2 (1 + γωσ−1 )2In other hand, since the South is suffered from disadvantage in low-tech sector, the∂LDSsouthern labor demand decreases: ∂wS < 0. Similarly, the effect of the increase in thesouthern minimum wage on the southern unemployment is positive:∂U S
βγωσ−1[(γωσ−1 + σ)w S ] > 0=S(1 + γωσ−1 )2 (w S )3∂w

As seen in Eq. (17), the northern growth rate g does not depend on w S . Therefore,

the increase in the southern minimum wage does not affect the northern growth. The
main reason is the assumption of homogenous firms. In the high-tech sector, all firms
produce goods and export them to the South. Although the increase in w S changes thesizes of the market, each firms' decision is not changed. Moreover, since the decrementof the southern expenditure is offset by the increment of the northern expenditure in

the model, R&D activities of firms are not affected by increasing w S . Therefore, thegrowth rate is constant even if the southern minimum wage increases.

2.2.4 Firms heterogeneity and the minimum wage
In this section, we consider the case of heterogeneous firms in the high-tech sector.
Since the household optimization and the profit maximization of the low-tech firms are
the same as the case of homogenous firms, we omit the explanation of these problems.
In the high-tech sector, a firm first has to develop a new variety as the case ofhomogenous firms. After having incurred the fixed cost, the firm receives a patent to
exclusively produce the new variety and learns the labor productivity associated with
its production. The productivity is drawn from a probability density function, so
different firms have different marginal costs of production. As Melitz (2003), theproductivities of firms are heterogeneous among firms.
After having developed a new variety and learned its labor productivity, the firm
decides whether or not to incur the one-time fixed costs of selling the variety in the local
and foreign markets. We think of these market-entry costs as reflecting the costs ofadapting the variety to market-specific standards, regulations, and norms. The firmneeds to draw a sufficiently high productivity to justify the entrance to the domestic
market and an even more favorable productivity to justify entering the foreign market.
Therefore, the decisions of R&D activities vary with the productivity of firms.

Innovation To develop a new variety, a firm needs to create FI units of knowledge in theinnovation sector. Thus, the cost of developing a new variety is w N bI (t)FI at time.
Knowledge development is also involved in adapting a variety to market-specificstandards, regulations, and norms. To sell a new variety in the local market, a firmneeds to create FN units of knowledge at cost w N bI (t)FN and to sell a new variety inthe southern market, a firm needs to create FS units of knowledge at cost w N bI (t)FS .

Once a firm has developed a new variety, it learns the labor productivity φ associatedwith its production. The labor productivity φ is drawn from a probability density
function g(φ) with support (1, ∞) and the corresponding cumulative distributionfunction G(φ). Once drawn, the labor productivity of a firm associated with producing aparticular variety does not change over time. We assume that the labor productivity is

subject to the Pareto distribution, that is,g(φ) = kφ−k−1 , G(φ) = 1 − φ−k , φ ∈ (1, ∞), k > 2(20)

where k is the shape parameter.

Due to the heterogeneity in unit labor requirements, there are three types of firms:

nonproducing firms, domestic firms, and exporting firms. Firms that get sufficiently
unfavorable draws (too low φ ) choose not to produce and exit, firms that getintermediate draws choose to just produce only for the domestic market, and firms that
get the most favorable draws choose to produce for both the domestic and the foreign
markets.

Figure 1: High-tech firm’s decision and productivity
Figure 1 describes the relationship between R&D decisions and firm’s productivity.
For a firm that develops a new variety, let φN denote the cutoff of labor productivity atwhich the firm is indifferent between selling in the local market with incurring the fixed
cost w N bI (t)FN and immediately shutting down production. Similarly, let φS denotethe cutoff of labor productivity at which the firm is indifferent between selling in thelocal market only and incurring the additional fixed cost w N bI (t)FS to export its
high-tech good.

Production Given a draw of φ > φN from the common density function g(φ) firm’s
profits from local sales of variety are given byN
πN (φ) = pN
D (φ)xD (φ) −

wN N
x (φ)
φ D

N (φ)
is the demand for a sold variety φ in the North and pN
where xD
D (φ) is thecorresponding price. From the pricing rule, the price of the high-tech good sold in thenorthern market becomespN
D (φ) =σw N
(σ − 1)φ(21)

Substituting the price into profit, we obtain operating profits of firms selling thedomestic market:
πN
D (φ) =σσ (σ − 1)σ−1(w N )1−σ αEN (t)φ1−σ {PDN (t)}1−σ(22)

Similarly, given a draw φ < φS, additional profits from exports are given byS(φ) −πSD (φ) = pSD (φ)xD

wN S
x (φ)
φ D

S
(φ) is the foreign demand for the exported variety φ and pSD (φ) is the pricewhere xDof the exported variety φ. The corresponding price of the high-tech good sold in thesouthern market ispSD (φ) =σw N(σ − 1)φ(23)

Substituting for the price, the operating profits earned by an exporting firm are
πSD (φ) =σ−σ (σ − 1)σ−1 (w N )1−σ αES (t)1−σφ1−σ �PDS (t)�(24)Market entry From the profits of firms in selling in domestic market and exporting to
the foreign market, we consider firm's decision for market entry. Similar to the case ofhomogenous firms, the no-arbitrage condition of supplying to i country becomesV̇ i πiD (φ)+= rN (t), i = N, S
V i V i (φ)

where V i (φ), i = N, S, is the value of firm selling the high-tech good in the market i.
Solving for V i (φ) yields

V i (φ) =πiD (φ), i = N, S
V̇ i (φ )rN (t) + i
V (φ)

Given that the firms with the cutoff levels, φN andφS , are indifferent among exit,

entering the local market, and exporting, respectively, the costs of entry have to equal
the benefits of entry:
V i �φi � = w N bI (t)FS , i = N, S

(25)

Substituting Eq. (22) and (24) into Eq. (25), we obtain the northern market entry
conditionσ−σ (σ − 1)σ−1 (w N)1−σ (φN )αEN
= w N bI (t)FṄ I (t)bN{PD (t)}1−σ �rN (t) −�bI (t)(26)

Similarly, the southern market entry condition is obtained asσ−σ (σ − 1)σ−1 (w N )1−σ (φS )αES
= w N bI (t)FS
̇
1−σ
(t)
b
�PDS (t)�
�rN (t) − I �bI (t)

With Eq. (26) and (27), the ratio of the cutoff for domestic market and exporting is

(27)

σ−1

φN
� S�
φ

1−σ

FN ES (t) PDN (t)=��
FS E N (t) PDS (t)(28)

Next, we determine the incentive to develop new variety. We assume that there is
free entry into variety innovation. Since any firm can develop a new variety, the ex ante
expected benefit of developing a new variety must equal the cost of variety innovation.
Therefore, the free entry condition of firms is∞∞� [V N (φ) − w N bI (t)FN ]g(φ) dφ + � [V S (φ) − w N bI FS ]g(φ) dφ = w N bI FI
φN

φS

Rewriting the free entry condition, we obtain the expected value of firms:
E[V] ≡whereΔ≡

σ−σ (σ − 1)σ−1 (w N )1−σ αΔ= w N bI (t)F�rN (t) − ḃ I /bI

∞
∞
EN
g(φ)
ES
g(φ)
σ−1
�
φ
dφ
+
�
φσ−1
dφ
1−σ
N
N
1−σ
S
1 − G(φ )1 − G(φN ){PD (t)}φNφS�P (t)�

(29)

(30)

D

and
F� ≡

∞
∞
FI
g(φ)
g(φ)
N
+
�
F
dφ
+
�
FS
dφ
N
N
1 − G(φ )1 − G(φ )1 − G(φN )
φN
φS
k

φN
= FI (φN )k + FN + FS � S �
φ(31)

Δ is the average profit of producing firms, and F� is the ex ante expected fixed cost ofdeveloping a profitable variety measured in units of knowledge created.

The flow of new varieties is determined by the labor, LR , devoted to R&D divided by

the labor units required for successful innovation bI (t)F� . Therefore, the flow of newvariety is obtained as

̇ LR
mD (t) = �
bI F

(32)

Steady state equilibrium As the case of homogenous firms, we define the steady-state
equilibrium as an equilibrium where all endogenous variables grow at constant (notnecessarily identical) rates over time. The world expenditure is normalized. Therefore,
the market interest rate rN (t) must be constant over time and equal the discount rate

in any steady-state equilibrium,rN (t) = ρ. It then follows from studying Eq. (26) and (27)that, as the growth rates of the endogenous variables PDi , i = N, S and bI (t) are all

constant over time, F� must grow at a constant rate. But, as FI is a constant, Eq. (31)implies that F� cannot grow at a constant rate unless the cutoff φN and φS is constant.

Hence F� , φN and φS are all constants in any steady-state equilibrium.
The price index for local and export market is
1−σ

σw N
�
σ−1

PDN (t)1−σ = �

1−σ

σw N
�
σ−1

PDS (t)1−σ = �kmD (t)(φN )σ−1
1+k−σ

(33)
k

kmD (t)(φS )σ−1 φN
� S�
1+k−σ
φ

(34)

Substituting Eq. (33) and (34) into Eq. (28) yields the ratio of cutoff of local and export
markets:
k

φN
FN ES
� S� =φ
FS E N(35)

Furthermore, substituting the price index in Eq. (33) and (34) into the northern cutoff
condition in Eq. (26) and the free entry condition in Eq. (29) yields(1 + k − σ)αEN= w N FN
σk(ρ + g)

(36)
k

αEN
ES
φN
�1 + N � = w N �FI (φN )k + FN + FS � S � �σ(ρ + g)E
φ(37)

From Eq. (37), the cutoff level depends on the expenditures in the northern country andthe southern country, E N and E S . We solve these expenditures from the household'sbudget constraint. In the steady state equilibrium, the expenditure and the saving isconstant, Ė i = Ȧ N = 0, i = N, S. Therefore, the expenditure in both countries is expressedas

E N = ρAN + w N LN ,E S = w S LS

where ANequals the northern investment to firms and Li is labor demand. Since

the southern households do not save, the southern expenditure equals to the total
income earned from the low-tech sector. Therefore, the southern expenditure is
calculated asES =βγωσ−1
1 + γωσ−1(38)

Since the world expenditure is one, the northern expenditure is calculated by
EN = 1 −βγωσ−1
1 + (1 − β)γωσ−1=1 + γωσ−1
1 + γωσ−1(39)

Substituting Eq. (39) into (36), the northern growth rate isg=(1 + k − σ)[1 + (1 − β)γωσ−1]−ρ
σkw N FN (1 + γωσ−1 )(40)Next, let us consider the cutoff of firm productivity. Substituting Eq. (35), (37), (39),and (40) into Eq. (36), we obtain the equation for the domestic cutoff φN:(1 + k − σ)FI (φN)k − (σ − 1)FN
βγωσ−1=(σ − 1)FN
1(1 − β)γωσ−1(41)Figure 2 depicts the determination of φN. The LHS of Eq. (41) does not depend on

φN , and it has the horizontal line in figure 2. On the other hand, the RHS is increasing
function of φN. To confirm the exist of φN, the following condition must be satisfied.
(1 + k − σ)FN
> (φN )k(σ − 1)FIWe obtain the cutoff value of the domestic sale asφN = �1/k

(1 + γωσ−1 )(σ − 1)FN�
[1 + (1 − β)γωσ−1 ](1 + k − σ)FI

(42)

Figure 2: Determination of the northern cutoff φN

Finally, we solve the unemployment. The unemployment is solved by subtracting
the labor demand from the labor supply. The unemployment in the North and the SouthisU N = LN −(1 + k − σ){1 + (1 − β)γωσ−1 } − ρσkw N FN (1 + γωσ−1 )(1 + k − σ)[1 + (1 − β)γωσ−1 ](σ − 1)[α(1 + γωσ−1 ) + β ]−σw N (1 + γωσ−1 )U S = LS −γωσ−1 (σ − 1)
σw S (a + γωσ−1 )

(43)

(44)

Effects of the minimum wages As section 2.2.3, we analyze effects of an increase in thesouthern minimum wage w S . An increase in w S means a decrease in ω. In the south,
the lower ω increases the cost of producing the low-tech good and the price. Since thesouthern workers earn the wage only from the low-tech sector, the southern income andexpenditure clearly decrease. On the other hand, the lower ω has the northern

low-tech sector sell the good cheaper than the south and the profits of the northernlow-tech firms increase.
Since the northern profits in low-tech sector are given back to the northern
consumer, the northern expenditure increases. Figure 2 describe the effects of anincrease in w S on the domestic cutoff φN . A decrease in ω drags down theleft-hand-side of Eq. (41). A new domestic cutoff is smaller than the previous one. The
northern country has a relatively advantage on the low-tech good by lower ω, and thenorthern workers in the low-tech good earn higher wage income. If other variables (i.e.,φN) are constant, the increase in the wage income raises the northern expenditure, E N.

Since the firms in the high-tech sector are relatively attracted to the local market, thefirm's entry in the local high-tech sector and the cutoff productivity decreases. The
lowering φN∗ means not only the increase in the number of firms but the decrease in

the average productivity because the inefficient firms can entry into the market. Since
the effect of increasing he wage income in the low-tech sector is larger than that ofdecreasing the average productivity, we obtain ∂E N / ∂ω < 0.

Figure 3: An increase in w s and the domestic cutoff

The cutoff of local market also affects the cutoff productivity of exporting. With Eq.

(35), (36), and (37), the relationship between φNand φS is
φS =
∂φS

(σ − 1)FS (φN )k(1 + k − σ)FI (φN )k − (σ − 1)FN

(45)

From Eq. (45), we obtain ∂φN < 0. When the southern minimum wage increases, theexpenditure in the South decreases. The exporting firms face the decrease in thesouthern demand of the high-tech goods. Therefore, exporting firms exit and choose to
be the local firms.
As seen in Eq. (40), we can affirm that the increase in w S (the decrease in ω) raise

the growth rate of varieties:∂g∂ω< 0. From Eq. (26), the increase in the northern

expenditure and the decrease in the northern price for the high-tech sector index
promotes R&D activities. Therefore, the growth rate increases. The result is notobserved in the case of the homogenous firms. In the case of the homogenous firms, oncea firm succeeds at developing a new variety, the firm can sell for both domestic andforeign markets. Since an increase in w S changes activities of all the firms, it does notaffect the growth rate in the north. However, the heterogeneities of firms have each firmchoose the number of R&D and the supply destinations. These choices are different withthe productivity of the firms. Since an increase in w S reduces the cutoff for domestic
firms, more potential firms enter into the domestic market and engage in R&D. As aresult, the growth rate increases in the case of firm heterogeneities.
The change in the foreign minimum wage also affects the unemployment rate.
Differentiating Eq. (43) with respect to ωσ−1, we obtain∂U Nβρσw N kβγ=+>0∂ωσ−1 (1 + k − σ)[1 + (1 − β)γωσ−1 ]2 σw N [1 + (1 − β)γωσ−1 ]2Since both the northern expenditure and the growth rate increase with increasing w S ,the northern labor demand in all sectors increase and the unemployment decreases.

The effect on the southern unemployment is solved by Eq. (44) as∂U S σ − 11γ(σ − 1)(w N )σ−1
=
�
+
�>0
(w S )σ∂w S
σ (w S )2

The unemployment in the South clearly increases when w S increases.

An unilateral increase in the minimum wage asymmetrically affects the twocountries. We focus on the southern policy of the minimum wage. When the government
in the South raises the minimum wage, the southern economy is harmed by thedisadvantage on the exporting, which results in decreasing the expenditure andincreasing the unemployment. On the other hand, the North has an advantage on the
sector which competes with the South, which induces higher expenditure, speeding up
the growth rate, and reduction in the unemployment. This result is similar with Davis(1998). He concludes that, through international trade, a country setting the minimum
wage is harmed while the other country with a flexible wage obtains positive effects.
Unlike Davis (1998), we allow that the minimum wage is bound in both countries andtake into account growth effect.
However, the result of our model is not simple. In the low-tech sector, the North� Y units of varieties. Since theproduces m� Y units of varieties and the South does γmhousehold in the North consumes both countries' varieties, they are suffered from theincrease in the price of the low-tech good made in the South. The increase in the price
may reduce the utility of the northern household. In the high-tech sector, the number ofthe exporting firms decreases. Since the North has more advantage on the high-tech
sector than the South, the contraction of exporting is bad news for the North. Finally,
the effects of the foreign minimum wage also increase the northern growth rate. The
increase in the growth rate means the household in both countries obtain more varietiesof the high-tech sector in the future. This effect may increase the welfare of thesouthern household.

3. Conclusion
We constructed a North-South-R¥&D-based growth model of trade in two cases:
homogeneous firms and heterogeneous firms. We examined effects of southernminimum wage on the northern growth rate and unemployment in both countries. In
both cases, higher southern minimum wage leads to the higher relative price ofsouthern low-tech goods, which in turn decreases the southern expenditure. On theother hand, the north takes advance in the low-tech sector, which increases thenorthern expenditure. In the case of homogenous firm, firm decision are not affected by
the change of the southern minimum wage. Therefore, the northern growth rate does
not change.
In the case of firm heterogeneity, firms' decisions are different with the productivity.
Since an increase in w S attracts to the domestic sell, more potential firms enter into

the domestic market and engage in R&D. The increase in the entry of firms encourages

R&D, which increases the northern growth rate. In both cases, the northern
unemployment unambiguously decreases and the southern unemployment increases.
We consider the future work. Although we assume that R&D is conducted only inthe northern country, the recent developing countries also engage in R¥&D activity.
Introducing R¥&D into the southern country, we need analyze the relationship between
the minimum wage and economic growth through international trade. Moreover, recent
empirical papers show that trade liberalization in the South affects the wage inequality
in the north, we need incorporate unskilled worker and skilled worker into our model.
Finally, we need analyze the effects of competition between China and Mexico in theU.S. market on the growth rate and employment in U.S.

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