Discussion Paper No. 1052

PRODUCTIVITY GROWTH,
INDUSTRY LOCATION PATTERNSANDLABOR MARKET FRICTIONSColin Davis
Ken-ichi Hashimoto

April 2019

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

Productivity Growth, Industry Location Patterns
and Labor Market Frictions
Colin Davis
Doshisha University∗

Ken-ichi Hashimoto
Kobe University†

April 2019

Abstract
This paper constructs a two-country model of international trade to study how labormarket frictions affect industry location patterns, unemployment rates, and fully endogenous productivity growth. We show that when the larger country offers subsidies to labor
search costs or reduces unemployment benefits, the domestic unemployment rate falls,causing greater industry concentration and faster productivity growth, but higher unemployment for the smaller country. When similar labor market policies are implemented
in the smaller country, however, the resulting fall in domestic unemployment leads tolower industry concentration and slower productivity growth, while lowering unemployment in the larger country.
JEL Classifications: E24; J64; O41; R11
Keywords: Labor market frictions; Industry location; Imperfect knowledge spillovers;
Endogenous productivity growth∗

The Institute for the Liberal Arts, Doshisha University, Karasuma-Higashi-iru, Imadegawa-dori, Kamigyo,
Kyoto, Japan, 602-8580, cdavis@mail.doshisha.ac.jp.
†
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Japan, 657-8501,
hashimoto@econ.kobe-u.ac.jp.
Acknowledgements: The authors are grateful for helpful comments from Taiji Furusawa, Tatsuro Iwaisako,
Takaaki Morimoto, Yoshiyasu Ono, Yuki Saito, Hajime Takatsuka, and participants of the JSIE 78th Annual Meeting and seminars held at ISER. The authors also acknowledge financial support from JSPS throughGrants-in-Aid for Scientific Research (A), (C), and (S): grant numbers 15H05728, 16H02016, 16K03624, and19K01638.

1 Introduction
In an increasingly integrated world economy, firms are less constrained by national borders as
they search for the best geographic locations for production. The shift toward more flexible
location patterns often collides with economic policy on two fronts. Regional policy tends
to emphasize employment and innovation-based growth, hand in hand, as demonstrated by
the Europe 2020 Integrated Guidelines for the Economic and Employment Policies of Member States. At the same time, however, policy makers also tend to support the existence ofinnovation and production clusters across regions (Martin 2008) as having a positive effecton growth. Further research is needed to shed light on how industry location patterns affectunemployment and growth in order to clarify the links between regional policy objectives.
With this objective in mind, we develop an endogenous market structure and endogenous
growth framework (Aghion and Howitt 1998; Laincz and Peretto 2006; Etro 2009) to consider the links between industry location patterns, national unemployment rates, and long-run
productivity growth in a two-country model of international trade. In particular, we combine
the process innovation framework of Smulders and van de Klundert (1995) and Peretto (1996)
with the equilibrium unemployment framework of Mortensen and Pissarides (1999). First, in
an intermediate sector, monopolistically competitive firms produce horizontally differentiatedvarieties for sale to final good firms and invest in process innovation with the aim of reducing
future production costs. Then, in the final good sector, firms employ workers to manage theassembly of intermediate products into final goods for both consumption by households andemployment in the production and innovation activities of intermediate firms.
Production and innovation are footloose between countries (Martin and Rogers 1995), buttrade in intermediate goods is subject to trade costs. Thus, the country with the larger market
size, as measured by the size of the employed labor force, attracts greater shares of both intermediate and final good production through a standard home market effect (Krugman 1980).
In addition, technical knowledge spillovers from production into the innovation activities ofintermediate firms diminish with distance. As a result, all intermediate firms locate their in2

novation activities in the country with the larger market, as the greater share of intermediate
production leads to higher knowledge spillovers that lower the cost of process innovation.
The labor market of each country features a matching process in which final good firms
incur search costs as they look for unemployed workers and then engage in Nash bargaining
to determine the division of profit with their employees. Free entry and exit in the final good
sector reduces firm value to the capitalized cost of hiring a worker, and national labor markets
are subsequently linked through a common interest rate that is determined in an international
financial market. Domestic labor market policies then influence unemployment rates in bothcountries through the response of capitalized hiring costs to adjustments in the interest rate.
We use the framework to consider the effects of economic integration. An improvement ininternational knowledge diffusion directly raises knowledge spillovers into innovation, whilea reduction in trade costs raises knowledge spillovers indirectly by increasing the concentration of intermediate production in the larger country. In both cases, the rate of productivity
growth accelerates. The interest rate also rises, however, increasing the capitalized cost of
hiring workers and adversely influencing the unemployment rates of the two countries. Thus,
improved economic integration leads to a higher rate of economic growth at a cost of lower
employment levels.
The framework is also used to study two types of national labor market policies: subsidies
to the search costs associated with hiring workers and unemployment benefits for workers
without jobs. Search subsidies allay hiring costs, placing downward pressure on the unemployment rate, while unemployment benefits reduce the incentive to look for work placing
upward pressure on the unemployment rate. Beyond these direct effects on employment levels, however, lies the indirect effect of changes in industry concentration on labor market
outcomes through adjustments in the interest rate. In the larger country, either an increase in
search subsidies or a decrease in unemployment benefits increases the number of employedworkers and expands market size, raising the country’s share of intermediate production.
As a result, knowledge spillovers improve and productivity growth accelerates. The interest

3

rate once again rises, however, placing upward pressure on the unemployment rates of bothcountries. Overall, the unemployment rate rises in the smaller country, but falls in the larger
country as the direct effect dominates the indirect effect.
The effects of national labor market policies implemented in the smaller country operate through similar mechanisms with opposite results. An increase in search subsidies, or adecrease in unemployment benefits, expands the market size of the smaller country as employment rises, attracting intermediate production and reducing industry concentration in thelarger country. Hence, lower knowledge spillovers lead to slower productivity growth. Theinterest rate also falls, causing the unemployment rate to decrease in each country.
An analytical study of national welfare effects proves to be intractable within our framework. But we are able to use simple numerical examples to present several welfare results.
On the one hand, these numerical examples suggest that improved economic integration,
through lower trade costs or a higher knowledge diffusion, is always welfare improving forboth countries. On the other hand, increases in search subsidies tend to benefit the welfare of
the implementing country, while hurting the welfare of the other country. Raising unemployment benefits tends to improve the welfare of both countries. In this manner, the numericalexamples indicate the potential for national labor market policy to influence the welfare ofneighboring countries through the effects of shifts in industry location on long-run growth.
This paper contributes to the theoretical literature examining the relationship between
innovation-based endogenous growth and unemployment in open economy settings. Building on the closed economy models of Aghion and Howitt (1994) and Mortensen (2005),
this literature generally adopts the quality ladders framework (Grossman and Helpman 1991)
of creative destruction. Sener (2001) demonstrates that trade liberalization promotes economic growth, but raises unemployment for low-skilled workers as faster innovation leads
to a higher job separation rate. Moreover, Stepanok (2016) shows that a reduction in trade
costs may hurt or benefit employment depending on the size of the R&D sector relative to theeconomy. Similarly, introducing a North-South product cycles framework, Stepanok (2018)4

finds that stronger intellectual property rights lower unemployment in the advanced North,with the effects of trade liberalization on employment depending on the outside options ofworkers in negotiations with employers. While the above literature mainly focuses on theimplications of trade liberalization for growth and unemployment, the endogenous market
structure and endogenous growth framework developed in this paper enables a study of theinterdependence of national labor markets that arises when industry location patterns shift in
response to changes in domestic labor market policy, allowing us to provide new insights into
the relationship between growth and unemployment in open economic settings.
The remainder of the paper is organized as follows. In Section 2 we introduce our theoretical model of industry location, labor market frictions, and endogenous productivity growth.
Section 3 then characterizes the balance growth path that arises in long-run equilibrium. Section 4 applies the model to a study of the effects of improvements in economic integration
and adjustments in labor market policy on unemployment rates, industry location, productivity growth, and national welfare levels. We conclude the paper in Section 5. All proofs arerelegated to the Appendix.

2 The Model
We now introduce our theoretical framework in which two countries, home and foreign, eachproduce intermediate and final goods. Final goods are produced using labor and intermediate
goods both for consumption by households and for employment in the innovation and production activities of intermediate firms. Labor is the sole factor of production. We present
the framework with a focus on the home country, with analogous conditions for the foreigncountry. Foreign variables are indicated by an asterisk.

2.1 Households
In each country, there is a dynastic representative household with a large number of members
L, of whom (1 − u)L are employed and uL are unemployed, with u ∈ (0, 1) denoting the5unemployment rate. As such, the household receives an expected wage income of w(1 − u)Land unemployment benefits of buL, where w is the wage rate and b is the unemployment
benefit. The household eliminates consumption and employment uncertainty by providing
perfect consumption insurance, with equal consumption levels for all household members,
regardless of whether they are employed or not (Merz 1995; Andolfatto 1996).
The lifetime utility of the representative household takes the following constant intertemporal elasticity of substitution formulation:

W =

Z ∞
0

e−ρt

C(t)1−1/σ − 1
dt,
1 − 1/σ(1)

where C is home consumption of the final good, ρ is the subjective discount rate, and σ ∈(0, 1) is the intertemporal elasticity of substitution.1 The household selects an expendituresaving path with the aim of maximizing lifetime utility subject to the flow budget constraint

Ȧ(t) = r(t)A(t) + (1 − u(t))w(t)L + u(t)b(t)L − C(t) − T (t),(2)

where A is asset wealth, r is the interest rate, and a dot over a variable denotes time differentiation. Government policies are funded through a lump-sum tax T levied on the household.
The optimal expenditure-saving path follows the standard Euler condition for the evolution of household expenditure:˙C(t)= σ(r(t) − ρ),
C(t)

(3)

where with a perfectly integrated international financial market, interest rates and expendituredynamics are common across countries; that is, Ċ/C = Ċ ∗ /C ∗ = σ(r − ρ). In order tosimplify the model exposition, henceforth we suppress time notation where possible.
1

See Hall (1988), Guvenen (2006), and Chiappori and Paiella (2011) for empirical estimates of the intertemporal elasticity of substitution that are consistent with σ ∈ (0, 1).

6

2.2 Final good Sector
The final good sector produces a homogeneous good for supply to an international market
characterized by free trade. Selecting the final good as the model numeraire, we set the priceto unity, under the assumption that aggregate demand is sufficiently large to ensure that bothcountries always have active final good sectors.
Each final good firm is managed by a single worker and employs intermediate goods inproduction. The technology of a representative firm in the home country is

y=

Z n

ε

x (i)di +

Z n∗

∗ε

∗

x (i )di

0

0

∗

α/ε

,

(4)

where α ∈ (0, 1), x(i) and x(i∗ ) are the input quantities associated with the masses of intermediate varieties n and n∗ produced in home and foreign, and ε ∈ (0, 1) is the degree of
product differentiation between any given pair of varieties.
The profit of a final good producer located in the home country isΠY = y −Z np(i)x(i)di −Z n∗τ p(i∗ )x(i∗ )di∗ ,

(5)

0

0

with p(i) and p(i∗ ) denoting the prices of intermediate varieties produced in home and foreign. International trade in intermediate goods involves iceberg trade costs whereby τ > 1
units must be shipped for every unit sold in an export market (Samuelson 1954).
Profit maximization generates the following firm-level demands in the home country forthe intermediate varieties produced in home and foreign:1p(i)− 1−ε αy,x(i) = R n
R n∗
−ε
−ε
∗ ) 1−ε dj ∗1−ε dj + ϕ
p(j)
p(j
0
0

i∈n

(6)

i∗ ∈ n∗(7)1(τ p(i∗ ))− 1−ε αyx(i ) = R n
,
R n∗
−ε
−ε
∗ ) 1−ε dj ∗1−ε dj + ϕ
p(j)
p(j
0
0
∗

where the level of trade costs is indexed by ϕ ≡ τ −ε/(1−ε) ∈ (0, 1), which as a decreasing

7

function of trade costs captures the freeness of trade: ϕ = 0 describes prohibitively hightrade costs and ϕ = 1 indicates free trade. Substituting the demand conditions into the profit
function (5) yields the profit for a final good producer located in home as ΠY = (1 − α)y. In
addition, with a single worker employed at each firm, the number of producing firms matches
the level of employment (1−u)L, and aggregate final good output in home is Y ≡ y(1−u)L.

2.3 Intermediate Sector
Firms in the intermediate sector employ final goods in both production (IX ) and in process
innovation (IR ) as they supply unique differentiated varieties for sale to final good producers in a market characterized by monopolistic competition (Dixit and Stiglitz 1977), while
investing in process innovation to improve their production technologies (Smulders and vande Klundert 1995; Peretto 1996). Although intermediate firms do not incur costs with market
entry, at each moment in time they do face fixed costs (IF ) measured in units of final goods.
The technology of an intermediate firm i with production located in home isX(i) = z(i)γ IX (i),(8)

where X is output, z is firm-level productivity, IX is the quantity of final goods employed in
production, and γ ∈ (0, 1) is the output elasticity of productivity. Given the CES production
technology employed in the final good sector, intermediate firms maximize profit by setting
price equal top(i) =

1,εz(i)γ(9)where 1/ε > 1 is the constant markup and 1/z γ is the unit production cost. Matching supply
with the combined demands of home and foreign final good producers, home-based intermediate production is X = x(1 − u)L + τ x∗ (1 − u∗ )L∗ . As a result, the optimal operating profit
on sales (πX = pX − IX = (1 − ε)pX) of an intermediate firm with production located in8home is−ε−εα(1 − ε)ϕp(i) 1−ε Y ∗α(1 − ε)p(i) 1−ε Y+.
πX (i) = R n
R n∗
R n∗
Rn
−ε
−ε
−ε
−ε
∗ ) 1−ε dj ∗∗ ) 1−ε dj ∗1−ε dj + ϕ
1−ε dj +
p(j
p(j
ϕ
p(j)
p(j)
0
0
0
0

(10)

As introduced above, each period intermediate firms employ IR units of labor in process
innovation with the aim of reducing production costs. The evolution of the productivity offirm i is then governed byż(i) = KIR (i).

(11)

Importantly, the productivity of R&D investment depends on knowledge spillovers from production technologies into the innovation process. Adapting the specification of Baldwin andForslid (2000), we model the knowledge spillovers received by an intermediate firm with
innovation located in home as
1
K=
N

Z n

Z n∗
∗ ∗
∗
z(j)dj + λz (j )dj ,
0

(12)

0

where N ≡ n + n∗ represents the total mass of intermediate varieties. Knowledge spillovers
are measured as a weighted average of the productivities of the production technologies observable by the firm, with the degree of international diffusion λ ∈ (0, 1) leading to a greater
weighting for knowledge spillovers from production technologies located in proximity to theinnovation activity of the firm. This specification is supported by an empirical literature that
documents the existence of international knowledge spillovers that diminish with distance(Bottazi and Peri 2003; Thompson 2006; Mancusi 2008).
In order to simplify the analysis, we assume that although the production processes associated with individual production lines are unique, initial productivity levels are symmetricacross the intermediate firms operating in both countries. As such, we henceforth drop thefirm index i with z = z ∗ , and define the level of knowledge spillovers into innovation activity

9located in home as K = zk, wherek = s + λs∗ ,(13)with s ≡ n/N and s∗ ≡ n∗ /N denoting the shares of intermediate firms with production
located in home and foreign.
The total per-period profit of a firm with production located in home is composed ofoperating profit on sales (10), less the cost of final goods employed in process innovation(IR ) and the per-period fixed cost (IF ); that is,ΠX = πX − IR − IF .

(14)

Firm value (Q) is therefore equal to the present discounted value of future profits:

Q=

Z ∞

Rt

′

′

e− 0 r(t )dt ΠX (t)dt.

(15)

0

The firm selects its level of investment in innovation with the aim of maximizing firm
value at each moment in time. This optimization problem is resolved using a current-value
Hamiltonian function: H = ΠX + qKIR , with q capturing the internal value of the firm’s
stock of technical knowledge in home. The optimal level of investment in process innovation
then satisfies the following static and dynamic efficiency conditions:

q=

1
,
kz

r=

εγπX k k̇ ż
− − ,
1−ε
k z

(16)

where the small market shares that arise under monopolistic competition lead firms to ignore
the effects of productivity improvements on the intermediate good price indexes and the levelof knowledge spillovers (12) when determining their optimal investment levels.
Free market entry and exit drives firm value to zero in the intermediate sector. In partic-

10

ular, the time derivative of firm value (15) yields a no-arbitrage condition (rQ = ΠX + Q̇)which equates the rate of return on market entry with the risk free rate of return. Firms enter
the market when firm value is positive (Q > 0), decreasing per-period profit (14) and lowering firm value, but exit the market when firm value is negative (Q < 0) increasing per-period
profits and raising firm value (Novshek and Sonnenchen 1987). The adjustment is immediate(Q̇ = 0), with the number of firms jumping to a level that satisfies Q = 0. Setting per-period
profit equal to zero (ΠX = 0) then yields a free market entry condition

πX = IR + IF ,(17)

which determines the equilibrium scales for production and process innovation.

2.4 Labor Market
The labor market features two types of workers: those looking for jobs and those employedwith final good firms. Similarly, there are two types of final good producers: those with job
vacancies and those employing workers. Defining the vacancy rate v ∈ (0, 1) as the numberof firms with vacancies relative to the overall labor force, the number of vacant final goodfirms searching for workers is vL.
Vacant final good firms incur time dependent search costs h while looking for an employee. As a result of this search effort, at each moment in time vacant firms are matchedwith unemployed workers according to the Poisson arrival ratem≡

M(vL, uL),vL

where the matching function M(vL, uL) is homothetic and increasing in both of its arguments (Pissarides 2000). The matching rate for vacant firms is therefore written as m(θ) ≡M(1, 1/θ), with θ ≡ v/u describing labor market tightness. We assume that m(θ) is continuously differentiable in (0, ∞) with m′ (θ) < 0.
11

Denoting the value of a producing final good firm by J, and a vacant firm by V , theno-arbitrage conditions associated with each firm type are˙
rJ = ΠY − w − δ(J − V ) + J,

(18)

rV = −(1 − △)h + m(θ)(J − V ) + V̇ ,(19)

where δ ∈ (0, 1) is the exogenous probability of an adverse shock that leads to job separation.
The government supports the job matching process through the provision of a partial subsidy△ ∈ (0, 1) to the search costs of vacant firms.
Similarly, denoting the present values of the income streams associated with employment
and unemployment by E and U, the no-arbitrage conditions for each worker type arerE = w − δ(E − U) + Ė,(20)

rU = b + θm(θ)(E − U) + U̇ ,(21)

where θm(θ) captures the matching rate of unemployed workers looking for jobs, and isincreasing in market tightness: ∂[θm(θ)]/∂θ > 0, as M(vL, uL)/(uL) = M(θ, 1) = θm(θ).
Given the operating profit of final good firms and labor market tightness, wages arecontinuously renegotiated between the firm and its employee through Nash bargaining (Pissarides 2000). More specifically, wages are set to maximize the Nash product (E − U)β (J −V )1−β , where β ∈ (0, 1) is the employee’s bargaining power, and J(w) and E(w) are determined through the no-arbitrage conditions (18) and (20). The first order condition yields(1 − β)(E − U) = β(J − V ), which regulates the balance between the net values of employment and operating a final good firm.
Vacant firms incur the net search cost (1 − △)h at each moment in time, with an averagetime frame of 1/m(θ) required to find a worker. But free market entry and exit in the final
good sector drives the value of vacant firms to zero (V = 0). As a result, from the no-arbitrage

12

conditions (18) and (19), we haveJ=

(1 − △)h,m(θ)˙(r + δ)J = ΠY − w + J.

(22)
(23)

Thus, the value of a producing firm is reduced to the expected cost of hiring a worker.
Following the equilibrium unemployment literature (Mortensen and Pissarides 1999; Pissarides and Vallanti 2007; Miyamoto and Takahashi 2011; Hashimoto and Im 2019), we
assume that search costs are determined proportionately with the output of final good producers: h = h̄y, with h̄ ∈ (0, 1). In addition, we set unemployment benefits as a percentage of the wage rate: b = b̄w with b̄ ∈ [0, 1). Then, using ΠY = (1 − α)y with
(1 − β)(E − U) = β(J − V ), in Appendix A, we derive the equilibrium wage rate for thehome country as
β 1 − α + (1 − △)h̄θ y.
w=
1 − (1 − β)b̄(24)Thus, the wage rate depends on labor market tightness and the output of final good firms (y).
Given the matching and separation processes discussed above, the evolution of the unemployment rate follows: u̇ = δ(1 − u) − θm(θ)u, with the first and second terms respectively
describing the flows into and out of unemployment. Setting this differential equation equal
to zero yields the steady-state employment rate as

1−u=

θm(θ)
.
δ + θm(θ)(25)

Long-run employment is therefore directly linked with labor market tightness through theBeveridge curve (Pissarides 2000). Importantly, an increase in θ raises the matching rateθm(θ) for workers looking for employment, causing the employment rate to improve.

13

2.5 Production and Innovation Location Patterns
We model location patterns following the footloose capital literature that builds on Martin
and Rogers (1995). In particular, the freedom of intermediate firms to shift production andinnovation independently between countries intrinsically links national shares of final good
production, intermediate production, and innovation activity.
Looking first at the pattern of final good production, we take the ratio of the production
functions (4) for firms in home and foreign, and substitute in the demands for intermediate
goods (6) and (7) to obtain
y
=
y∗



s + ϕs∗ϕs + s∗α(1−ε)
 ε(1−α).

(26)

Next, the free movement of intermediate production leads to the equalization of operating∗
profit across home and foreign; that is, πX = πX
, where we assume that both countries

produce intermediate goods. Using the pricing rule (9) with operating profit (10) and thefinal good production ratio (26), the home share of intermediate production is found as
Y /Y ∗ − ϕ,s=(1 − ϕ)(1 + Y /Y ∗ )∗Y /Y =

(1 − u)L(1 − u∗ )L∗ ε(1−α)
ε−α

,

(27)

where we will find that ε > α is required for a positive rate of productivity growth understable market entry (see Section 3.2). Thus, the country with the larger employed labor force
attracts the greater share of intermediate production through a standard home market effect(Krugman 1980); a result that is magnified by a fall in trade costs (Baldwin et al. 2003).
Substituting the production shares back into operating profit (10) yields πX = α(1 − ε)(Y +Y ∗ )/N for all firms.
Turning now to the innovation location decision, we show that all intermediate firms locate process innovation in the country with the larger market. With constant national employment levels, intermediate production shares (27) and knowledge spillovers (13) are constant14

in the steady state (k̇ = 0). Accordingly, the no-arbitrage condition for investment in process innovation (16) indicates that with symmetric initial productivity levels (z = z ∗ ), thelocation with a higher level of knowledge spillovers (k) attracts all innovation activity, andall firms experience the same rate of productivity growth (ż/z = ż ∗ /z ∗ ). Thus, innovation
concentrates fully in the country with the larger employed labor force.

2.6 Government
National governments support unemployment benefits (b) and subsidies (△) to search costs(h) through a lump-sum tax levied on household income. The balanced government budget
in the home country is

T = buL + △hvL,(28)

where the first term on the righthand side is the total payment of benefits to unemployedworkers, and the second term is the total subsidies provided to vacant final good firms.

2.7 Market Clearing Conditions
Given the behavior of households, firms, and governments discussed above, the asset market
adjusts perfectly to ensure that at all moments in time we have

A + A∗ = J(1 − u)L + J ∗ (1 − u∗ )L∗ .

(29)

The righthand side represents the total supply of asset value, with J denoting the value of an
operating final good firm and (1 − u)L measuring the number of operating firms. Combining
the government budget constraint (28) and the household budget constraint (2) with (22),(25), and (29) yields a market clearing condition for final goods:

Y + Y ∗ = C + C ∗ + vhL + v ∗ h∗ L∗ + N(IX + IR + IF ),
15

(30)

where we have used the fact that all intermediate firms have the same scales of production
and innovation (see Appendix B for the derivation).

3 Long-run Equilibrium
This section establishes a steady-state equilibrium with a balanced growth path, along which
output (Y ) and consumption (C) grow at the same rate, and the interest rate (r), the unemployment rate (u), and labor market tightness (θ) are constant over time. In order to simplify
the analysis, we assume that relative labor endowments and labor market frictions ensure thathome always has a larger market ((1−u)L > (1−u∗ )L∗ ), with greater shares of intermediate
and final good production (s > 1/2), and that home therefore hosts all process innovation.
First, to derive the balanced rate of output growth, we combine the intermediate gooddemands (6) and (7) with final good production (4) and x(i) = ϕ1/ε x(i∗ ) to obtainα(1−ε)αα(1−ε)αγy = (αε) 1−α (s + ϕs∗ ) ε(1−α) N ε(1−α) z 1−α .

(31)

We then show that all macroeconomic variables (Y, Y ∗ , C, C ∗, N) grow at the same rate:
ẏ
ẏ ∗
αεγ
Ẏ ∗
Ċ
Ċ ∗
Ṅ
Ẏ
= ∗ =
= ∗ = = ∗ =
=
g,
Y
Y
C
C
y
y
N
ε−α

(32)

where g ≡ ż/z denotes the productivity growth rate, and we have used ṡ = 0 (see Appendix
C for the derivation).

3.1 National Labor Market Equilibrium
The long-run equilibrium associated with the labor market of each country is described by
two job creation conditions which implicitly determine labor market tightness for home andforeign. For example, combining ΠY = (1 − α)y, h = h̄y, and J˙/J = ẏ/y = σ(r − ρ) with16

(22), (23), and (24) yields a job creation condition for the home country:(1 − △)h̄ ((1 − σ)r + σρ + δ)
(1 − α)(1 − β)(1 − b̄) − β(1 − △)h̄θ,=m(θ)
1 − (1 − β)b̄

(33)with an analogous expression for job creation in the foreign country. The lefthand side of (33)
captures the expected capitalized cost of hiring a worker, while the righthand side describes
the net profit derived from employing the worker (Pissarides 2000).
For a given level of the interest rate, domestic labor market policy determines labor market
tightness. From (33), it is then immediate that:
Lemma 1 A decrease in search costs (h̄), unemployment benefits (b̄), labor market power(β), or the separation rate (δ), or an increase in the search subsidy (△), causes a rise inlabor market tightness (θ).
Within each country, the job creation condition (33) regulates the influence of domestic labormarket policy on labor market tightness. First, returning to (19), an increase in the searchsubsidy (△), or alternatively a decrease in the search cost (h̄), lowers the cost of finding a
new worker, temporarily increasing the value of a vacant firm (V ). As a result, firms enter the
final good sector, causing labor market tightness to rise until vacant firm value has returned tozero through a fall in the value of an operating firm (J), as shown in (22). Second, a decreasein employee negotiating power through either a fall in unemployment benefits (b̄), or a direct
decrease in labor market power (β), leads to lower wages that reduce the expected cost of
hiring a worker. As such, labor market tightness rises.2 Finally, a rise in the job separation
rate lowers the values of both operating firms and employed workers, causing the number ofvacancies and employed workers to decrease and labor market tightness to fall.
2

See Peretto (2012) and Chu et al. (2016) for theoretical studies of the relationship between union powerand economic growth.

17

3.2 Long-run Productivity Growth
The domestic labor markets of home and foreign are linked through the interest rate, which isdetermined in the international financial market and is influenced by industry concentration
through the level of knowledge spillovers. Following Smulders and van de Klundert (1995),
we describe the international financial market equilibrium using supply and demand conditions for investment funds. First, equating (3) and (32), the household supply of investment
funds is described by a preference conditionr=

αεγ
g + ρ,σ(ε − α)(34)

within which a positive relationship arises between the interest rate and the rate of productivity growth, as households require a higher return on savings to compensate them for delaying
consumption. Second, the demand for investment funds from intermediate firms is captured
with a technology condition that is obtained by reorganizing the second expression of (16)with (11) and (17):r=εγIF k (1 − ε(1 + γ))g
−
.
1−ε
1−ε

(35)

An increase in the interest rate puts downward pressure on the productivity growth rate as
optimal investment in process innovation falls for a given level of knowledge spillovers (k).
The preference and technology conditions are illustrated in Figure 1. Matching the demand (34) and supply (35) for investment funds, we derive the equilibrium rate of productivity growth as follows:g = σ(ε − α)(εγIF k − (1 − ε)ρ)ψ −1 ,(36)

where ψ = αε(1 − ε)γ + σ(ε − α)(1 − ε(1 + γ)). This expression yields the following lemma
outlining the requirements for positive productivity growth under stable market entry.
18

Figure 1: Equilibrium Productivity GrowthrPreference

Technologyg0Lemma 2 Positive productivity growth requires k ∈ ((1 − ε)ρ/(εγIF ), 1).
In Appendix D, we show that a stable level of market entry into the intermediate sector requires a ψ > 0 in (36) to ensure that firm value responds correctly to market entry and exit(dΠX /dN < 0) in the spirit of Novshek and Sonnenchein (1987); a condition that is synonymous with a greater slope for the preference condition than the technology condition.
Therefore, the expression εγIF k − (1 − ε)ρ dictates the range of knowledge spillovers overwhich positive productivity growth arises in long-run equilibrium. With the conditions forstable market entry satisfied, we find that an increase in the level of knowledge spillovers
raises the rate of productivity growth. For k ≤ (1 − ε)ρ/(εγIF ), however, there is no productivity growth, as intermediate firms do not find it optimal to invest in process innovation.
In addition, we find that the rate of economic growth is not biased by a scale effect, since
proportionate increases in the populations of each country are fully absorbed by an expansionof the level of market entry, and therefore do not influence the innovation rate. To see this,using (16), (17), (27), and (31), with πX = α(1 − ε)(Y + Y ∗ )/N, we obtain the entry levelassociated with current productivity (z) as follows:

N(z) =ε(1−α)
α(1−ε)
ε(1−α)αεγε(αεα ψ 1−α ) ε−α (1 + ϕ) ε−α z ε−α ((1 − u)L) ε−α + ((1 − u∗ )L∗ ) ε−α((σ(ε − α) + αεγ)IF − σ(ε − α)ρ/k)19ε(1−α)ε−α, (37)

where the denominator is positive under the conditions outlined in Lemma 2. Thus the framework is consistent with the empirical literature that supports endogenous market structure andendogenous growth frameworks that are not biased by scale effects (Laincz and Peretto 2006;
Ha and Howitt 2007; Madsen et al. 2010).

4 Policy Analysis
We now use the model to evaluate the effects of improvements in economic integration andadjustments in the domestic labor market policies of home and foreign on unemployment
rates, industry location patterns, and long-run productivity growth.

4.1 Economic Integration
Beginning with an analysis of the effects of improvements in economic integration, we obtainthe following results.
Proposition 1 An improvement in economic integration, through either lower trade costs (τ )or a higher degree of knowledge diffusion (λ), accelerates the rate of productivity growth (g),but raises the unemployment rates of both countries (u and u∗ ).
Proof: See Appendix E.
Improved economic integration affects the location pattern of intermediate production,and subsequently unemployment rates and productivity growth, through two channels: aknowledge spillover effect and a job creation effect. Addressing first the knowledge spillover
effect, from (13), an increase in the degree of knowledge diffusion (λ) directly raises thelevel of knowledge spillovers (k). Similarly, from (27), an increase in the freeness of trade(ϕ) strengthens the home market effect, causing the home share of intermediate production(s) and the level of knowledge spillovers (k) to both rise. In either case, the technology
condition (35) shifts to the right in Figure 1, with greater knowledge spillovers allowing fora faster rate of productivity growth (36).
20Turning next to the job creation effect, the resulting increase in the interest rate (r) induces
a fall in labor market tightness and raises the unemployment rates of both home and foreign,as the capitalized cost of hiring workers increases in (33). The job creation effect has an
ambiguous influence on the concentration of industry in the home country (s), however, as
the fall in employment could be greater for either country. Overall, the knowledge spillover
effect dominates, and the home share of intermediate production increases, allowing us toconclude that policies designed to promote economic integration lead to faster economic
growth, but at a cost of higher unemployment rates.

4.2 National Labor Market Policy
In this section, we consider how changes in national labor market policy affect unemployment
rates and long-run productivity growth. Looking first at the labor market policy of the homecountry, we obtain the following proposition.
Proposition 2 In the larger home country, increases in search subsidies (△) and decreases
in unemployment benefits (b̄) both accelerate productivity growth (g), while lowering theunemployment rate of home (u) and raising the unemployment rate of foreign (u∗ ).
Proof: See Appendix E.
Referring to the job creation condition (33), and the results of Lemma 1, the direct effectof either an increase in search subsidies, or a decrease in unemployment benefits, is greaterlabor market tightness and a rise in the number of employed workers in home. The increase inhome employment raises the home share of intermediate production, shifting the technology
condition to the right in Figure 1, and enabling a faster rate of productivity growth throughthe knowledge spillover effect. The interest rate also rises, however, indirectly putting downward pressure on employment in both countries through the job creation effect. The overall
result is greater industry concentration in home and faster productivity growth, while the unemployment rate rises in foreign but falls in home as the direct effect dominates the indirect
job creation effect.
21

We next derive the following results for the labor market policy of the foreign country.
Proposition 3 In the smaller foreign country, increases in search subsidies (△∗ ) and decreases in unemployment benefits (b̄∗ ) both dampen productivity growth (g), while reducing
the unemployment rates of home and foreign (u and u∗ ).
Proof: See Appendix E.
Changes in foreign labor market policy have similar, but opposite, effects to those of thehome country. Through the job creation condition (33), the direct effect of an increase in
search subsidies, or a decrease in unemployment benefits, is an increase in foreign labor market tightness and a rise in foreign employment that reduces the concentration of intermediate
production in the home country. Accordingly, the technology condition shifts leftward in Figure 1 through the knowledge spillover effect, depressing the rate of productivity growth. Theinterest rate also falls, however, indirectly putting upward pressure on the employment rates
of both home and foreign through the job creation effect. Overall, the home share of intermediate production falls and productivity growth slows, with reductions in the unemployment
rates of both countries.

4.3 National Welfare
We briefly discuss how economic integration and labor market policy influence national welfare levels. With consumption and employment uncertainty eliminated through the household
provision of perfect consumption insurance (Merz 1995; Andolfatto 1996), all household
members in a given country have the same level of welfare. For example, lifetime utility (1)is used to derive the steady-state welfare of home at time zero as follows:
1−1/σ

W0 =

1
(ε − α)C0−,(1 − 1/σ)((ε − α)ρ − (1 − 1/σ)αεγg) (1 − 1/σ)ρ

(38)

with current household consumption determined from the budget flow constraint (2) as C0 =w(1 − u)L − △hvL + κΘ, where κ ∈ [0, 1] is the exogenous home share of total asset wealth,22and Θ ≡ (ΠY − w)(1 − u)L + (Π∗Y − w ∗ )(1 − u∗ )L∗ − (1 − △)hvL − (1 − △∗ )h∗ v ∗ L∗
denotes investment income earned on total asset holdings (see Appendix F). We also derive
an analogous set of expressions for the national welfare of foreign.
Improvements in economic integration and adjustments in labor market policy affect welfare through a growth effect (g) and a level effect (C0 ). Looking first at the growth effect,
as we have seen from (36), the rate of productivity growth is positively linked with industry
concentration (s) through knowledge spillovers (k). Accordingly, the growth effect causes
the welfare of both countries to improve with either an increase in the labor market tightnessof home (θ) or a decrease in the labor market tightness of foreign (θ∗ ).
Turning to the level effect, we combine (22), (23), (24), (25), and (33) to rewrite current
consumption as follows:κ(1 − △∗ )h̄∗ Y ∗ κ∗ (1 − △)h̄δ△h̄
Y + (σρ + (1 − σ)r)Y,−
C0 = 1 − α −m(θ)
m(θ∗ )Ym(θ)
where we have normalized current productivity (z = 1). The first term captures the contribution of labor to final good output less the cost of subsidizing search efforts in the domestic
labor market, while the second term describes the net investment income earned on asset
wealth held in the final good firms of each country. From this expression, we find that labormarket tightness influences current consumption through four channels: the cost and benefit of search subsidies, the interest rate, relative final good production, and aggregate final
good production. The ambiguous and opposing nature of these channels, combined with thegrowth effect, renders a general analysis of national welfare intractable.
As an alternative, we present several numerical examples, specifying the matching rate
as m(θ) = 0.15θ−0.6 following evidence presented by Petrongolo and Pissarides (2001). For
the demand parameters, we set ρ = 0.05, σ = 0.5, and κ = 0.5 , where the intertemporalelasticity of substitution is consistent with estimates by Hall (1988), Guvenen (2006), andChiappori and Paiella (2011). Technology parameters are set as α = 0.75, ε = 0.8, γ =

23

0

.2
.4
.6
.8
1
Degree of Knowledge Diffusion

−30
−35
−40
0

.2
.4
.6
Home Search Subsidy

.8

0

.2
.4
.6
.8
Home Unemployment Benefits

.2
.4
.6
.8
Foreign Search Subsidy

−50

−30

−40

−40

−40

−30

−30

0

−20

.8

−20

.2
.4
.6
Freeness of Trade

−20

0

−45

−50

−50

−40

−40

−30

−30

−20

−20

Figure 2: National Welfare Effects

0
.2
.4
.6
.8
Foreign Unemployment Benefits

The solid line measures home welfare and the dashed line measures foreign welfare. Benchmark parameters are α = 0.75, β = 0.5, γ = 0.5, δ = 0.01, ε = 0.8, κ = 0.5, λ = 0.15, ρ = 0.05, σ = 0.5, ϕ = 0.12,b̄ = b̄∗ = 0.001, h̄ = 0.05, h̄∗ = 0.08, △ = △∗ = 0.05, IF = 0.4, L = 1.08, and L∗ = 1.

0.5, and IF = 0.4, with the degree of product differentiation matching with values for theelasticity of substitution 1/(1 − ε) reported by Bernard et al. (2003) and Broda and Weinstein(2006). For the labor market, we set benchmark parameter values to β = 0.5, δ = 0.01, h̄ =0.05, h̄∗ = 0.08, △ = △∗ = 0.05, b̄ = b̄∗ = 0.001, L = 1.08, and L∗ = 1, ensuring that homealways has a greater share of intermediate production. Following the mid-range of estimates
for the degree of knowledge diffusion provided by Bloom et al. (2013), we assume λ = 0.15.
The level of trade costs is set to τ = 1.7 following Anderson and van Wincoop (2004) andNovy (2013), yielding ϕ = 0.12 for the freeness of trade. The benchmark parameter set
generates unemployment rates of u = 0.063 and u∗ = 0.082, an intermediate production
share of s = 0.62, and a productivity growth rate of g = 0.045.
The numerical examples are presented in Figure 2, where the solid line measures home
welfare and the dashed line measures foreign welfare. The ranges of policy parameters measured on the horizontal axes are limited to ensure that intermediate production shares andmatching rates lie in the unit interval. First, improved economic integration raises the wel24

fare of both countries, suggesting that the positive growth effects dominate the ambiguous
level effects associated with decreases in trade costs and increases in knowledge diffusion.
Second, adjustments in labor market policies have symmetric effects for home and foreign,with an increase in search subsidies improving the welfare of the implementing country,
while lowering the welfare of the other country. Increases in unemployment benefits raise
the welfare of both countries up to a point, after which they become detrimental to the implementing country. These numerical results suggest that the level effects of labor market policy
adjustments tend to dominate the growth effects over the assumed parameter set. Importantly,
we find that there is potential for national labor market policies to influence the welfare ofneighboring countries through the effects of shifts in industry location on long-run growth.

5 Conclusion
We construct a two-country model of industry location and trade and investigate the relationships between national labor market frictions, unemployment rates and fully endogenous
productivity growth. With perfect capital mobility, firms are free to shift their production
and innovation activities between countries. Trade costs and imperfect knowledge spillovers
result in the partial concentration of production and the full concentration of innovation in
the country with the larger market size as measured by the employed labor force. Focusing
on national labor market policies, we consider how unemployment benefits and subsidies tofirm-level search costs affect productivity growth through adjustments in national unemployment rates, shares of industry, and knowledge spillovers from production into innovation. We
show that when the larger country offers subsidies to search costs or reduces unemployment
benefits in the domestic labor market, the resulting decrease in the unemployment rate causes
an increase in the concentration of industry and accelerates productivity growth as the levelof knowledge spillovers rises. If similar domestic labor market policies are implemented in
the smaller country, productivity growth is dampened as the unemployment rate decreases
leading to lower industry concentration and a fall in knowledge spillovers.
25

Appendix A: Derivation of the Wage Condition
First, taking the difference of (20) and (21), we have(Ė − U̇ ) =β(r + δ + θm(θ))J+ b − w.
1−βInto this expression, we substitute the time derivative of (1 − β)(E − U) = β(J − V ) with(22), (23), and V̇ = 0. Then, rearranging the result obtains

w = (1 − β)b + β(ΠY + (1 − △)hθ).

Finally, using b = b̄w, h = h̄y, and ΠY = (1 − α)y yields the wage condition (24).
Appendix B: Market Clearing Condition
Aggregating the household budget constraints (2) of home and foreign givesȦ + Ȧ∗ = r(A + A∗ ) + (1 − u)wL + (1 − u∗ )w ∗ L∗ − △hvL − △∗ h∗ v ∗ L∗ − C − C ∗ .

Next, time differentiation of (29) yields
˙ − u)L − J u̇L + J˙∗ (1 − u∗ )L∗ − J ∗ u˙∗ L∗ .
Ȧ + Ȧ∗ = J(1Then, combining the above two expressions with u̇ = δ(1 − u) − θm(θ)u, (22), (23), and(29), we haveΠY (1 − u)L + Π∗Y (1 − u∗)L∗ = C + C ∗ + vhL + v ∗ h∗ L∗ .
Finally, using ΠY = (1 − α)y, Π∗Y = (1 − α)y ∗, and pXN = α(Y + Y ∗ ) = N(IX + IR + IF ),we obtain the market clearing condition for final goods (30).

26Appendix C: Balanced Growth PathFirst, using h = h̄y, v = uθ, Y = y(1 − u)L, and pXN = α(Y + Y ∗ ) = N(IX + IR + IF )in (30), we have

Y



 




uh̄θ
C∗
u∗ h̄∗ θ∗ Y ∗
Y∗
−
=C 1+
.
+
(1 − α) 1 +Y
1 − u 1 − u∗ Y
CIn the steady state, the interest rate (r), the unemployment rate (u), and labor market tightness(θ) are constant, ensuring that the relative production scale of final goods (Y /Y ∗ ) is also
constant. Also, as Ċ/C = Ċ ∗ /C ∗ = σ(r − ρ) from (3), relative consumption (C/C ∗ ) does
not change over time. Thus, we find that output and consumption grow at the same rate
Ċ/C = C˙ ∗ /C ∗ = Ẏ /Y = Ẏ ∗ /Y ∗ = ẏ/y = ẏ ∗/y ∗ . Then, as intermediate firm production(IX ) and innovation (IR ) scales are constant, given the interest rate (r), α(1 + Y ∗ /Y )Y =N(IX + IR + IF ) yields Ṅ/N = Ẏ /Y .
Appendix D: Stable Market Entry
Combining the second expression of (16) with k̇ = 0, (11), (14), and (34), we solve for the
per-period profit of an intermediate firm as

ΠX =

ψπX
σ(ε − α)ρ+− IF ,(1 − ε)(αεγ + σ(ε − α)) (αεγ + σ(ε − α))kwhere πX = α(1 − ε)(Y + Y ∗ )/N. Thus, given (Y + Y ∗ ), as dπX /dN = −πX /N < 0,
ψ = αε(1 − ε)γ + σ(ε − α)(1 − ε(1 + γ)) > 0 is required to ensure dΠX /dN < 0. Then,the numerator of (39) yields the required range for k described in Lemma 1.

Appendix E: Comparative Statics
Combining (33), (34), and (36), we describe the steady-state system implicitly in terms of θand θ∗ using the following allocation curves for home and foreign: ΓK = ΓJ −(ρ+δ)/(1−σ)

27and ΓK = Γ∗J − (ρ + δ)/(1 − σ), whereαεγ (εγIF k(θ, θ∗ ) − (1 − ε)ρ),αε(1 − ε)γ + σ(ε − α)(1 − ε(1 + γ)
(1 − α)(1 − β)(1 − b̄) − β(1 − △)h̄θ m(θ),ΓJ =(1 − σ)(1 − (1 − β)b̄)(1 − △)h̄(1 − α)(1 − β)(1 − b̄∗ ) − β(1 − △∗ )h̄∗ θ∗ m(θ∗ )∗.
ΓJ =(1 − σ)(1 − (1 − β)b̄∗ )(1 − △∗ )h̄∗

ΓK =We take the total derivatives of the allocation curves to obtain:






∂ΓK
J
− ∂Γ
∂θ
∂θ
∂ΓK
∂θ

∂ΓK
∂θ ∗







∂ΓK

 dθ   − ∂ϕ
=

∂Γ∗J
∗
∂ΓK
K
dθ
− ∂θ∗
− ∂Γ
∂θ ∗
∂ϕ

K
− ∂Γ
∂λ

∂ΓJ
∂△

∂ΓJ
∂ b̄

0

K
− ∂Γ
∂λ

0

0

∂Γ∗J
∂△∗



 dϕ  dλ 
0  d△ 

,




∂Γ∗J


d
b̄


∂ b̄∗


 d△∗ 




∗
db̄

where
∂ΓK
α(εγ)2 IF ∂k ∂s ∂u
=
> 0,
∂θ
ψ
∂s ∂u ∂θ∂ΓKα(εγ)2 IF ∂k ∂s ∂u∗
=
< 0,
∂θ∗
ψ
∂s ∂u∗ ∂θ∗(1 − α)(1 − β)(1 − b̄) − β(1 − △)h̄θ ∂m(θ)/∂θ − β(1 − △)h̄m(θ)
∂ΓJ
< 0,
=
∂θ
(1 − σ)(1 − (1 − β)b̄)(1 − △)h̄(1 − α)(1 − β)(1 − b̄∗ ) − β(1 − △∗ )h̄∗ θ∗ ∂m(θ∗ )/∂θ∗ − β(1 − △∗ )h̄∗ m(θ)∗
∂Γ∗J
=
< 0.
∂θ∗
(1 − σ)(1 − (1 − β)b̄∗ )(1 − △∗ )h̄∗
While the slope of the home allocation is strictly positive in (θ, θ∗ ) space, the foreign
allocation curve is generally ambiguous. In order to ensure the existence of long-run equilibrium, we focus on cases for which the foreign allocation curve has a negative slope; thatis, ∂ΓK /∂θ∗ − ∂ΓJ /∂θ∗ > 0. Then the determinant of the Jacobian matrix given above isstrictly positive: |J| = −(∂ΓK /∂θ)(∂Γ∗J /∂θ∗ ) − (∂ΓJ /∂θ)((∂ΓK /∂θ∗ ) − (∂Γ∗J /∂θ∗ )) > 0.
28

Cramer’s rule then yields the following comparative static results for each proposition.
For Proposition 1 we havedθ∂ΓK ∂Γ∗J 1
=
< 0,
dϕ
∂ϕ ∂θ∗ |J|dθ∗∂ΓK ∂ΓJ 1=< 0,
dϕ∂ϕ ∂θ |J|dθ∂ΓK ∂Γ∗J 1
=
< 0,
dλ
∂λ ∂θ∗ |J|dθ∗∂ΓK ∂ΓJ 1=< 0,dλ∂λ ∂θ |J|α(εγ)2 IF ∂k ∂s
∂ΓK
=
> 0,
∂ϕ
ψ
∂s ∂ϕ

∂ΓK
α(εγ)2 IF ∂k
=
> 0.
∂λ
ψ
∂λ

where

Next, the results for Proposition 2 are
dθ
∂ΓJ
=
d△
∂△



∂ΓK
∂Γ∗J
−
∂θ∗
∂θ∗



1
> 0,
|J|

dθ∗
∂ΓJ ∂ΓK 1
=−
< 0,
d△
∂△ ∂θ |J|

∂ΓJ
dθ
=
db̄
∂ b̄



∂Γ∗J
∂ΓK
−
∂θ∗
∂θ∗



1
< 0,
|J|

dθ∗
∂ΓJ ∂ΓK 1
> 0,
=−
db̄
∂ b̄ ∂θ |J|

where
∂ΓJ
(1 − α)(1 − β)(1 − b̄)m(θ)
> 0,
=
∂△
(1 − σ)(1 − (1 − β)b̄)(1 − △)2 h̄
(1 − β)β(1 − α + (1 − △)h̄θ)m(θ)
∂ΓJ
=−
< 0.
∂ b̄
(1 − σ)(1 − (1 − β)b̄)2 (1 − △)h̄
Finally, the results for Proposition 3 aredθ∂Γ∗J ∂ΓK 1
=
−
> 0,
d△∗
∂△∗ ∂θ∗ |J|

dθ∗
∂Γ∗J
=
d△∗
∂△∗



∂Γ∗J ∂ΓK 1
dθ
=− ∗
< 0,
db̄∗
∂ b̄ ∂θ∗ |J|

∂Γ∗J
dθ∗
= ∗
db̄∗
∂ b̄



29

∂ΓK
∂ΓJ
−
∂θ
∂θ



1
> 0,
|J|

∂ΓK
∂ΓJ
−
∂θ
∂θ



1
< 0,
|J|

where∂Γ∗J(1 − α)(1 − β)(1 − b̄∗ )m(θ∗ )
> 0,
=
∂△∗
(1 − σ)(1 − (1 − β)b̄∗ )(1 − △∗ )2 h̄∗∂Γ∗J(1 − β)β(1 − α + (1 − △∗ )h̄∗ θ)m(θ∗ )
=
−
< 0.
∂ b̄∗
(1 − σ)(1 − (1 − β)b̄∗ )2 (1 − △∗ )h̄∗Appendix F: Initial Consumption
First note that the initial asset position of the home household is

A0 = κ (J(1 − u)L + J ∗ (1 − u∗ )L∗ ) .

The time derivative of this expression gives
˙ − u)L − J u̇L + J˙∗ (1 − u∗)L∗ − J ∗ u˙∗ L∗ .
Ȧ = κ J(1Then, using v = uθ with the above two equations and (22), (23), and (25), we obtainrA− Ȧ = κ ((πY −w)(1 − u)L + (πY∗ −w ∗ )(1 − u∗ )L∗− (1 − △)hvL − (1 − △∗ )h∗ v ∗ L∗ ).
Substituting this expression into (2) yields C0 = w(1 − u)L − △hvL + κΘ.

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