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CPB Discussion Paper

No

23-August-2010

Wage inequality in trade-in-tasks models

Hugo Rojas-Romagosa

The responsibility for the contents of this CPB Discussion Paper remains with the author(s)

CPB Netherlands Bureau for Economic Policy Analysis

Van Stolkweg 14

P.O. Box 80510

2508 GM The Hague, the Netherlands

Telephone +31 70 338 33 80

Telefax +31 70 338 33 50

Internet www.cpb.nl

ISBN

Abstract in English

3

1 Introduction

Traditional trade theory is based on final goods trade. However, the growing importance of trade

in intermediate goods and services –associated with the fragmentation of production into

different countries– is a well known empirical fact. Accordingly, several papers already

incoporated trade in intermediate goods in the Heckscher-Ohlin (HO) theoretical framework.1

The latest contribution in this topic is the trade-in-tasks model by Grossman and Rossi-Hansberg

(2006, 2008), henceforth GRH model, who introduce tasks directly into a HO framework.

The complexity of the recent GRH trade-in-tasks model does not allow for analytical

solutions that allow clear-cut predictions. 2 Therefore, these predictions have been obtained by

numerically simulating the theoretical models. In this regards, the trade volumes and welfare

implications of these recent trade-in-tasks models have been already evaluated by Markusen

(2010) and Baldwin and Robert-Nicoud (2010). However, the implications for factor prices have

not yet been fully analyzed.

The first objective of this paper is to use numerical simulations using GAMS to evaluate the

relative price effects of the GRH model for different specifications. We use a four different

offshoring cost schedules, factor productions shares, different elasticities of substitution for each

good. Furthermore, the GRH model uses a simple offshoring cost schedule, i.e. there is a general

increase in all skill-specific task costs, regardless of differences between skill-specific tasks, and

previous costs levels. We use recent empirical findings relating offshoring costs, tasks and skills

to introduce more complex offshoring scenarios. This allows a richer variety of offshoring costs

changes that inform about the effects of different offshoring phases on wage inequality. For

example, offshoring costs can change relative to a given level (reflecting existing offshoring)

instead of the absolute cost shift implemented in GRH. We start with two offshoring

possibilities: only offshoring low-skilled tasks (L-tasks) and a sequential offshoring of L-tasks

and H-tasks (high-skill tasks).

These combinations are analyzed for both the one-country model (small country with fixed

international prices) and the two-country model (variable international prices). We then compare

these results with the numerical simulations done by Markusen (2010) using a simplified version

of the GRH model and their simulations on the Markusen and Venables (2007) model.

The second objective of this paper is to create a theoretical expansion of current

trade-in-tasks models that can capture more complex wage inequality dynamics by introducing

three or more factors. In particular, we build upon the trade in tasks model by Grossman and

1 Among others, Dixit and Grossman (1982), Helpman (1984), Jones and Kierzkowski (1990), Deardorff (1998b,a),

Venables (1999), Kohler (2004), Antràs et al. (2006), Markusen (2006), Markusen and Venables (2007).

2 The GRH models does have, however, a special case result, where the relative price of low-skill labour increases with

offshoring when the elasticity of factor substitution between sectors is identical and the country is an international-price

taker. This result is fully analyzed later on.

5

Rossi-Hansberg (GRH) to allow more than two skill-groups and the overlapping of some tasks

between skill groups. With these model extensions we have a richer interplay between skill

groups, task offshoring and wage dynamics that are used to assess the effect of offshoring on

wage inequality. The recent trade-in-tasks models are based on a HOS framework with 2-factors.

However, following Autor et al. (2006), 2008) and (Goos and Manning, 2007) recent wage

inequality data shows a polarization pattern where low and high skilled wages are increasing

relative to medium skilled wages. Therefore, to capture these type of wage inequality dynamics

at least three skill types are needed.

6

2 Models with two production factors

2.1 The GRH model

The Grossman and Rossi-Hansberg paper introduces trade-in-tasks directly into a HOS

modeling framework. It does not deal explicitly with intermediate goods, but they are implicit in

their analysis.3 They assume the standard 2x2x2 conditions. Two countries c: domestic (d) and

foreign ( f ), two sectors j: x and y, and two production factors s: low-skill (L) and high-skill

(H). Furthermore, there is a perfect mapping between skills and tasks: low-skill workers do

L-tasks and high-skill workers H-tasks. 4 They also assume that technologies (factor/tasks

requirements) are the same in both countries.

Their main component is the introduction of a task offshoring technology. Firms can perform

production tasks either at home or offshore them to the foreign country. Offshoring is prefered

when one or both factors are cheaper in the foreign country. However, to offshore a task, the firm

must pay not only the local wages but also additional costs related to the monitoring and

coordinating of remote workers.5

In their setting, t j(i) captures heterogeneous offshoring costs for the various tasks i in

industry j. All tasks are indexed by i ∈ [0,1] and order so that the costs of offshoring are

nondecreasing. t j (i) is continously differentiable, while the task ordering implies: ∂ t j(i)∂ i ≥ 0.

Finally, tasks have the same offshorability regardless of sector, i.e., tx (i) = ty (i) = t (i). There is

a general offshoring cost parameter β , that can be associated with communication and

transportation improvements that proportionally reduce the cost of offshoring for all tasks.

Combining these elements they obtain the offshoring zero-profit condition:

β t (i) pL f ≥ pLd (2.1)

where pL f is the foreign low-skill wage and pLd is the domestic low-skill wage, β is a general

offshoring cost, t (i) is the task-specific offshoring cost function, and I is the equilibrium

marginal task, for which there are equal costs in both locations.

Equivalently to the case of L-task offshoring, the equilibrium condition for H-task offshoring

is given by:

γ z(IH)pH f ≥ pHd (2.2)

3 This is made clear when we introduce the two-country equilibrium and the constraints to the balance of payments.

4 There are substitution possibilities between both set of tasks. They begin exposition with no substitution (σ = 0) but then

continue the model with imperfect substitution (σ > 0).

5 This additional costs can also be associated with the wage premium that multinational pay in order to hire the best local

workers. This fact is corroborated by the heterogeneous firm literature, where MNE firms pay higher wages even than

local exporters.

7

where γ is the general offshoring cost of H-tasks, z (iH) is the offshoring cost function for

H-tasks indexed by iH ∈ [0,1], and IH is the value of iH for which there is a H-task offshoring

equilibrium.

We first employ a Cobb-Douglas production function: j = ALα

j H1−α j where A is a

productivity parameter and α j provides the factor shares in production of sector j. Then the

unit-cost minimization problem of the firm, when there are offshoring possibilities in both L and

H-tasks, results in the following unit-cost function:

c j (p, I, IH) = (pLdΩ(I))α j (pKdΩH (IH))1−α j (2.3)

(2.4)

where Ω≤ 1 and ΩH ≤ 1, are the cost-saving variables derived in GRH associated with L and

H-task offshoring, respectivelty.

The expenditure function is given by:

e (p j) = pα

x p1−α

y (2.5)

The factor demand, is then given by:

Lx =∂cx(·)∂ pLd

=1A

αxΩ(pLdΩ)αx−1(pHdΩH)1−αx (2.6)

Ly =∂cy(·)∂ pLd

=1A

αyΩ(pLdΩ)αy−1(pHdΩH)1−αy (2.7)

Hx =∂cx(·)∂ pHd

=1A(1−αx)ΩH(pLdΩ)αx (pHdΩH)

−αx (2.8)

Hy =∂cy(·)∂ pHd

=1A(1−αy)ΩH(pLdΩ)αy (pHdΩH)

−αy (2.9)

And the factor endowment equilibrium conditions are:

L(1− I)

= Lxx +Lyy (2.10)

H(1− IH)

= Hxx +Hyy (2.11)

2.2 Offshoring costs

To run numerical simulations on the GRH model we first need to specify the offshoring cost

function. In the GRH model t(i) is only constrained to be non-decreasing in i. This leaves

several modeling possibilities. We use the general function:

t(i) = t1 + t2it3 (2.12)

Combining equations (2.1) and (2.12) , and taken I to be the task level at which equation

(2.1) holds as a strict equality, we have that the β0 upper bound limit, at which there is no

offshoring (I = 0) is:

β0 =

pLd

pL f t1(2.13)

8

Accordingly, a positive initial offshoring value (I > 0) requires that t1 + t2It3 ≥ pLdβ pL f

, then:

I =[

1t2

(pLd

β pL f− t1

)] 1t3

(2.14)

Given that at the equilibrium offshoring task I, we have that the total offshoring costs

T (I) =∫ I

0 β t (i)di, then we have:

T (I) = t1I +t2It3+1

t3 +1(2.15)

Finally, the offshoring cost-saving parameter (Ω) is:

Ω = 1− I +T (I)

t (i)≤ 1(2.16)

We use four different combinations for the offshoring cost function. We begin with a linear

specification where t3 = 1. Then we calibrate t1 = 1 so at β = pLd/pL f we have no offshoring

activity. Then we use two different values for t2 = 1,4. This provides two relatively extreme

functions. For t2 = 4 offshoring slowly increases with reductions in β , while for t2 = 1 we have

that offshoring activity is relatively sensitive to changes in β . The second set of functions are

non-linear, with t3 = 1.5,2.5 and t2 = 7. With t3 = 1.5 offshoring is relatively inelastic to β

changes. With t3 = 2.5 we have that offshoring is initially very sensitive for changes in β , but

later on only large reductions in offshoring costs increases its activity. The four offshoring cost

functions are depicted in Figure 2.1.

Figure 2.1 Linear and non-linear offshoring costs

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Offshoring index

Offs

horin

g C

osts

PL/ beta1 * PK

PL/ beta2 * PK

t1=1, t2=1, t3=1

t1=1, t2=4, t3=1

t1=1, t2=7, t3=2.5

t1=1, t2=7, t3=1.5

9

2.3 GRH one-country model

We first assume that the domestic country (d) is a small international-price taking country. In

addition, since we want to analyze the effects of offshoring on developed countries, we assume

that the domestic country is high-skill labour abundant and exports the high-skill intensive good,

which we assume is x , since we calibrate our numerical simulations using αx < αy .

We solve the general equilibrium conditions as a mixed complementarity problem (MCP).

The MCP equations and associated (complementary) variables are specified as follows:

Inequality Equation Comp. var.

L-tasHs offshoring equation 1 β pl f t (I)≥ pLd I

L-tasHs offshoring equation 2 t (I)≥ t1 + t2It3 t (I)

L-tasHs offshoring equation 3∫ I

0 t (i)di = T (I)≥ t1I + t2It3+1

t3+1 T (I)

L-tasHs offshoring equation 4 Ω≥ (1− I)+ T (I)t(I) Ω

H-tasts offshoring equation 1 γ pH f z (IH)≥ pHd IH

H-tasts offshoring equation 2 z (IH)≥ z1 + z3Iz2H z (IH)

H-tasts offshoring equation 3∫ IH

0 t (iH)diH = Z (IH)≥ z1IH +z3I

z2+1H

z2+1 Z (IH)

H-tasts offshoring equation 4 ΩH ≥ (1− IH)+Z(IH )z(IH ) ΩH

Non-positive profits for x cx (·) = 1A [ΩpLd ]

αx (ΩH pHd)1−αx ≥ pxd x

Non-positive profits for y cy (·) = 1A [ΩpLd ]

αy (ΩH pHd)1−αx ≥ pyd y

Non-positive “profits” for w e (p,1) = pλ

xd p1−λ

yd ≥ pw w

Supply ≥ Demand x x +mx ≥ ∂e(p,w)∂ pxd

= wλ pλ−1xd p1−λ

yd mx

Supply ≥ Demand y y +my ≥ ∂e(p,w)∂ pyd

= w (1−λ) pλ

xd p−λ

yd my

Supply ≥ Demand w w ≥ Cpw

pw

Supply ≥ Demand L L1−I ≥ x αx Ω

A (ΩpLd)αx−1 (ΩH pHd)

1−αx

+y αy Ω

A (ΩpLd)αy−1 (ΩH pHd)

1−αy pLd

Supply ≥ Demand H H1−IH

≥ x (1−αx )ΩHA (ΩpLd)

αx (ΩH pHd)−αx

+y (1−αy)ΩH

A (ΩpLd)αy (ΩH pHd)

−αy pHd

Income balance consumer C = pLdL

(1−I) + pHdH

(1−IH ) −ρ C

Foreign price for x pxd (1+ τxd) = px f pxd

Foreign price for y pyd(1+ τyd

)= py f pyd

Offshoring payment balance ρ = pLdL I(1−I) + pHdH IH

(1−IH ) ρ

where m j is the import (export if negative) of good j, ρ is the offshoring payment made by

the domestic country d to foreign ( f ) labor, and τ jc is a tariff levied on good j in country c. In

our setting we define welfare (w) as the representative consumer’s utility.

The MCP formulation means that when an equation holds as an equality the complementary

variable is positive, and if the equation holds as an strict inequality then the complementary

variable is zero.

10

The first eight equations are the offshoring equilibrium conditions taken directly from the

GRH model. The other twelve equations are standard general equilibrium conditions. In this

particular one-country model, we assume that international prices(

p j f)

are constant, and they

determine prices in country d . Also note the if the payment to offshored labor (ρ) is zero, we

have balance trade in final goods: mx +my = 0. However, if there is offshoring activities then

ρ > 0 and this produces a wedge between the initial trade balance, such that mx +my =−ρ.

Then ρ can be associated with imported intermediate inputs, which in turn create that

−mx > my : exports of final goods must exceed imports in order to pay for the imported

intermediate inputs ρ.

2.4 Simulation results for the one-country model

We run a series of simulations, in which we use our four offshoring cost functions, three sets of

factor shares of production: αx = 0.2,αy = 0.7, αx = 0.3,αy = 0.6, and αx = 0.4,αy = 0.5. First

we allow only L-task offshoring, and later both types, but in a way that L-task offshoring always

precedes and is larger than H-task offshoring (i.e. I ≥ IH ).

To allow incentives to offshore for the domestic country, we assume that the domestic wages

for each factor s are higher than the foreign wage, in particular psd/ps f = 2.

In each simulation we vary beta and gamma to increase offshoring. We also vary the relative

factor endowment (L/H) to check if the results are sensitive to relative factor abundance.

However, L/H is kept in limits such that the domestic country is always exporting the

H-intensive good.

In Figure 2.2 we present the relative changes in the main variables of interest. First,

offshoring costs are increasing along with decreases in β , welfare is slightly increasing when β

decreases as well6 We also present changes in sectoral production. Note that increased

offshoring (reductions in β ) increases the production of the L-abundant good y and reduces x .

This in turn, means that offshoring by a small-country erodes is endowment-abundancy

differential with the rest of the world, and consequently it trades less.

Figure 2.3 presents the changes in relative wages (pLd/pHd) when we only allow for L-task

offshoring. In each graph the offshoring costs are changing. In the Appendix, Figures 3.1 and

3.2 show the results when using other production shares (α’s).

In all cases relative wages are increasing. Unless there are corner solutions where the

domestic country is only producing one of the goods. These corner solutions are more common

when αx = 0.4 and αy = 0.5, and the factor-abundancy boundaries are narrower. This result is

the special case commented in ?, which is the only specific relative wage result they can derive

6 Welfare is increasing in relative L (Lr el = L/H) since we simulate increases in Lr el by increasing the endowment of

low-skill labour.

11

Figure 2.2 GRH model, changes in offshoring levels, welfare and production, for αx = 0.3 and αy = 0.6, and

t1 = 1, t2 = 7 and t3 = 1.5

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.870

0.753

0.635

0.517

0.400

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Offshoring index

beta

L_rel

Offshoring index

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.870

0.753

0.635

0.517

0.400

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

welfare

beta

L_rel

Welfare

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.870

0.753

0.635

0.517

0.400

0

0.2

0.4

0.6

0.8

1

1.2

X

beta

L_rel

Production of X

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.870

0.753

0.635

0.517

0.400

0

1

2

3

4

5

6

7

Y

beta

L_rel

Production of Y

at the analytical level.

When we allow for the offshoring of H-tasks as a sequence to the initial offshoring of L-tasks

(i.e. I > IH ), then we still obtain similar results are before. In the case of relative wages the

results are depicted in Figure 2.4.

Comparing these results with the one for only L-task offshoring (Figure 2.3), we observe that

the increase in the relative wage of low-skill workers is still positive, but of a smaller magnitude.

Specially in later stages of offshoring (i.e. for low β values), when the sequential offshoring of

H-tasks is higher.

12

Figure 2.3 GRH model, changes in relative wages with L-tasks offshoring (αx = 0.3 and αy = 0.6)

2.00

1.95

1.90

1.85

1.80

1.75

1.70

1.65

1.60

1.55

1.50

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)

Relative wages (PL/PK)

α1=0.3 α2=0.6

t1=1 t2=1 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.09

0.97

0.86

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=1 t3=12.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=1 t3=1

2.00

1.94

1.88

1.81

1.75

1.69

1.63

1.57

1.50

1.44

1.38

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=1 t3=1

2.5 GRH two-country model

When we model both countries, we assume that only the high-skill abundant domestic country

offshores labour to the other country. Thus, all our offshoring equations from the one-country

MCP remain the same. This also implies that only the domestic country benefits from the

cost-reducing parameters Ω and ΩH . International prices are now determined as part of the

general equilibrium system, and they depend on the relative sizes of each country and their trade

volumes. Finally, the foreign country has less low-skill and high-skill labour when one or both

of these workers are producing tasks for the domestic economy, but it receives the offshoring

payments ρ.

The new MCP equations (excluding the offshoring equations) are the following:

13

Figure 2.4 GRH model, changes in relative wages with both L and H-tasks offshoring (αx = 0.3 and αy = 0.6)

2.00

1.95

1.90

1.85

1.80

1.75

1.70

1.65

1.60

1.55

1.50

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=1 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.09

0.97

0.86

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=4 t3=12.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=7 t3=1.5

2.00

1.94

1.88

1.81

1.75

1.69

1.63

1.57

1.50

1.44

1.38

0.87

0.75

0.63

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.3 α2=0.6

t1=1 t2=7 t3=2.5

Inequality Equation Comp. var.

Non-positive profits for xdd1

Ad[ΩpLd ]

αx (ΩH pHd)1−αx ≥ pxd xdd

Non-positive profits for ydd1

Ad[ΩpLd ]

αy (ΩH pHd)1−αx ≥ pyd ydd

Non-positive profits for x f f1

A fpαx

L f p1−αxH f ≥ px f x f f

Non-positive profits for y f f1

A fpαy

L f p1−αxH f ≥ py f y f f

Non-positive profits for xc1c2 pxc1 (1+ τxc1)≥ pxc2 xc1c2

Non-positive profits for yc1c2 pyc1 (1+ τyc1)≥ pyc2 yc1c2

Non-positive “profits” for wc pλ

xc p1−λ

yc ≥ pwc wc

Supply ≥ Demand xc xc1c1 + xc1c2 + xc2c1 ≥ wλ pλ−1xc p1−λ

yc pxc

Supply ≥ Demand yc yc1c1 + yc1c2 + yc2c1 ≥ w (1−λ) pλ

xc p−λ

yc pyc

Supply ≥ Demand wc wc ≥ Ccpwc

pwc

Supply ≥ Demand LdL

1−I ≥ xddαx Ω

Ad(ΩpLd)

αx−1 (ΩH pHd)1−αx

+yddαy Ω

Ad(ΩpLd)

αy−1 (ΩH pHd)1−αy pLd

Supply ≥ Demand HdH

1−IH≥ xdd

(1−αx )ΩHAd

(ΩpLd)αx (ΩH pHd)

−αx

+ydd(1−αy)ΩH

Ad(ΩpLd)

αy (ΩH pHd)−αy pHd

Supply ≥ Demand L f L f ≥ x f fαxA f

pαx−1L f p1−αx

H f + y f fαyA f

pαy−1L f p1−αy

H f pL f

Supply ≥ Demand H f H f ≥ x f f(1−αx )

A fpαx

L f p−αxH f + y f f

(1−αy)A f

pαyL f p−αy

H f pH f

Income balance consumer d Cd = pLdL

(1−I) + pHdH

(1−IH ) −ρ Cd

Income balance consumer f C f = pL f

(L f −Ld

I(1−I)

)+ pH f

(H f −Hd

IH(1−IH )

)+ρ C f

Offshoring payment balance ρ = pLdL I(1−I) + pHdH IH

(1−IH ) ρ

14

where c1 6= c2, and c1 is the producing country and c2 is the consuming country (e.g. y f d are

exports of y from f to d).

In addition, we calibrate the productivity parameters Ac such that the domestic country is

twice as productive as the foreign country, and thus, wages in d are two times larger than in f .

This inter-country wage differential is the main driving force in the offshoring process, only

hindered by the β costs.

For the two-country simulations we simulate different divisions of the total endowments

between both countries. We do keep the restriction that the domestic country is always relatively

high-skill abundant with respect to the foreign country. Using the total endowment sharing

mechanism we obtain different relative country sizes and thus, different effects on trade volumes

and terms-of-trade effects.

As before, we reduce the value of beta to increase offshoring activity in L-tasks. However,

since we are also changing the relative endowments, factor prices are different for each

endowment point and thus, β0 –the value of β for which there is no offshoring– is also changing.

Therefore, we first find the equilibrium for each endowment point, then estimate β0 and from this

value, we apply 5% reductions. We run four different simulations, in which the general

offshoring costs are changing. For b = 0 we have no offshoring activities, with b = 1, β0 is

reduced by 5%, in b = 2 by 10% and in b = 3 by 15%.

In the Appendix we present the figures with the main results. In Figure 3.3 we have the

changes in offshoring when different reductions are applied to β . The left-hand side graphs use

the linear offshoring cost function t1 = 1, t2 = 1, t3 = 1, while the right-hand side graphs use:

t1 = 1, t2 = 4, t3 = 1. As shown in this figure, and following the cost function shape presented in

Figure 2.3, when t2 is higher, then costs are higher and be equal reductions in β the left-hand

side graphs, with lower offshoring costs have higher offshoring activity increases. We have

chosen these values so they represent extreme situations. With t2 = 1 offshoring is very sensitive

to β changes, and with t2 = 4 offshoring changes slowly with β reductions. We also observe that

offshoring is more sensitive to endowment sets that are closer to the diagonal of equal relative

endowments between countries. The closer to this diagonal, offshoring is the highest, and it is

reduced as endowments go toward the extreme where the domestic country has almost all the

total high-skill workers and relatively few low-skill workers.

Figure 3.4 shows the welfare effects for each shedule of β decreases (b). In the following

graphs, we use the no-offshoring scenario (b = 0) as our baseline, and we graph the results of

each b scenario with respect to b = 0. In other words, we compare how reductions in the general

offshoring cost β affects each variable in relation with the case where there was no offshoring.

As shown in Figure 3.4, welfare is almost unchanged (it is equal to one) for most endowment

sets, and there is only a welfare reduction in the area where the domestic country has relatively

low endowments of L (less than 20% of the total).

This welfare changes are influenced by the changes in the terms-of-trade (TOT). In Figure

15

3.5 we observe an almost identical pattern of change in terms-of-trade, as in welfare.

The results for offshoring, welfare and TOT are very similar when we use the non-linear

offshoring cost function z(iH) and when we use other factor share parameters (α’s).

Finally, in Figure 2.5 we show the changes in relative wages. For the endowment sets in all

scenarios we observe a decline in the relative wage of low-skill with respect to high-skill. The

decline becomes more pronounced as offshoring increases. The only exception is in the small

area where the domestic country has very few low-skill workers and a medium endowment of

high-skill. Here there is a relative increase in low-skill wages. This endowment set corresponds

to that of a relative small country, and thus, is similar to the GRH one-country model results.

This general pattern is repeated for both the linear and non-linear offshoring costs, and for

different combinations of α. In Figure 2.6 we present the case with non-linear costs and

al phax = 0.2,αy = 0.7. Here we see that the only qualitative difference with respect to Figure

2.5 is that These results are in strong contrast to the general relative low-skill wage increases

present in the GRH one-country model.

16

Figure 2.5 GRH two-country model, changes in relative wages with respect to no-offshoring scenario,(αx = 0.3

and αy = 0.6), linear offshoring costs (left-hand side t2 = 1, right-hand side t2 = 4

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share

Relative wages with respect to no offshoring case,

b=1

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=1

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=2

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=2

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=3

0.90

0.70

0.50

0.30

0.10

0.9

0.7

0.5

0.3

0.1

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=3

17

Figure 2.6 GRH two-country model, changes in relative wages with respect to no-offshoring scenario,(αx = 0.2

and αy = 0.7), non-linear offshoring costs (left-hand side t3 = 1.5, right-hand side t3 = 2.5

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=1

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=1

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=2

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=2

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=3

0.90

0.70

0.50

0.30

0.10

0.900

0.700

0.500

0.300

0.100

0.80

0.85

0.90

0.95

1.00

1.05

1.10

plh/pkh

Home H share

Home L share


b=3

18

References

Antràs, P., L. Garicano and E. Rossi-Hansberg, 2006, Offshoring in a knowledge economy,

Quarterly Journal of Economics, vol. 121, no. 1, pp. 31–77.

Autor, D.H., L.F. Katz and M.S. Kearney, 2006, The polarization of the U.S. labor market,

American Economic Review, vol. 96, no. 2, pp. 189–194.

Baldwin, R.E. and F. Robert-Nicoud, 2010, Trade-in-goods and trade-in-tasks: An integrating

framework, NBER Working Paper 15882.

Deardorff, A.V., 1998a, Fragmentation across cones, RSIE Discussion Paper 427, University of

Michigan.

Deardorff, A.V., 1998b, Fragmentation in simple trade models, RSIE Discussion Paper 422,

University of Michigan.

Dixit, A. and G.M. Grossman, 1982, Trade and protection with multi-stage production, Review

of Economic Studies, vol. 49, no. 4, pp. 583–594.

Grossman, G.M. and E. Rossi-Hansberg, 2006, The rise of offshoring: It’s not wine for cloth

anymore, Proceedings: Federal Reserve Bank of Kansas City, pp. 59–102.

Grossman, G.M. and E. Rossi-Hansberg, 2008, Trading tasks: A simple theory of offshoring,

American Economic Review, vol. 98, no. 5, pp. 1978–1997.

Helpman, E., 1984, A simple theory of international trade with multinational corporations,

Journal of Political Economy, vol. 92, no. 3, pp. 451–471.

Jones, R.W. and H. Kierzkowski, 1990, The role of services in production and international

trade: A theoretical framework, in R.W. Jones and A. Krueger, eds., The Political Economy of

International Trade, Basil Blackwell, Oxford.

Kohler, W., 2004, Aspects of international fragmentation, Review of International Economics,

vol. 12, no. 5, pp. 793–816.

Markusen, J., 2006, Modeling the offshoring of white-collar services: From comparative

advantage to the new theories of trade and FDI, in S. Brainard and S. Collins, eds., Brooking

19

Trade Forum 2005: Offshoring White-Collar Work, The Brookings Institution, Washington D.C.

Markusen, J.R., 2010, Expansion of trade at the extensive margin: A general gains-from-trade

result and illustrative examples, CEPR Discussion Paper 7802.

Markusen, J.R. and A.J. Venables, 2007, Interacting factor endowments and trade costs: A

multi-country, multi-good approach to trade theory, Journal of International Economics, vol. 73,

no. 2, pp. 333–354.

Venables, A.J., 1999, Fragmentation and multinational production, European Economic Review,

vol. 43, no. 4-6, pp. 935–945.

20

3 Appendix

Figure 3.1 GRH model, changes in relative wages with L-tasks offshoring, with αx = 0.2 and αy = 0.7

2.00

1.95

1.90

1.85

1.80

1.75

1.70

1.65

1.60

1.55

1.50

0.84

0.73

0.62

0.51

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.2 α2=0.7

t1=1 t2=1 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.09

0.97

0.86

0.84

0.73

0.62

0.51

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.2 α2=0.7

t1=1 t2=4 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.84

0.73

0.62

0.51

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.2 α2=0.7

t1=1 t2=7 t3=1.5

2.00

1.94

1.88

1.81

1.75

1.69

1.63

1.57

1.50

1.44

1.38

0.84

0.73

0.62

0.51

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.2 α2=0.7

t1=1 t2=7 t3=2.5

Figure 3.2 GRH model, changes in relative wages with L-tasks offshoring, with αx = 0.4 and αy = 0.5

2.00

1.95

1.90

1.85

1.80

1.75

1.70

1.65

1.60

1.55

1.50

0.89

0.77

0.64

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.4 α2=0.5

t1=1 t2=1 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.09

0.97

0.86

0.89

0.77

0.64

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.4 α2=0.5

t1=1 t2=4 t3=1

2.00

1.89

1.77

1.66

1.54

1.43

1.31

1.20

1.08

0.97

0.85

0.89

0.77

0.64

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.4 α2=0.5

t1=1 t2=7 t3=1.5

2.00

1.94

1.88

1.81

1.75

1.69

1.63

1.57

1.50

1.44

1.38

0.89

0.77

0.64

0.52

0.40

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

1.175

1.200

PL/PK

beta

Relative labour

(L/K)


α1=0.4 α2=0.5

t1=1 t2=7 t3=2.5

21

Figure 3.3 GRH two-country model, changes in L-task offshoring (αx = 0.3 and αy = 0.6) left hand t2=1, right

hand t2=4

0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share

Offshoring index, b=1

0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share


0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share


0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share


0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share


0.90

0.70

0.50

0.30

0.10

0.90

0.70

0.50

0.30

0.10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Offshoring

index

Home K share

Home L share


22

Figure 3.4 GRH two-country model, welfare changes with respect to no-offshoring scenario, (αx = 0.3 and αy =

0.6) left hand t2=1, right hand t2=4

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare

Home K shareHome L share

Welfare with respect to no offshoring case, b=1

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare


Welfare with respect to no offshoring case, b=10.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.95

1.00

1.05

1.10

Welfare



23

Figure 3.5 GRH two-country model, terms-of-trade changes with respect to no-offshoring scenario,(αx = 0.3 and

αy = 0.6) left hand t2=1, right hand t2=4

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)


Terms of trade with respect to no offshoring case, b=1

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)


Terms of trade with respect to no offshoring case, b=10.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)



0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.90

0.70

0.50

0.30

0.10

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Terms of trade

(pxh/pyh)



24

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