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The Effects of Trade and Endogenous Technological Change on the Wage Gap and Unemployment

Benedikt Rydzek

The Effects of Trade and Endogenous Technological

Change on the Wage Gap and Unemployment

Benedikt Rydzek ∗

This version 09.05.2012

Abstract

Skill biased technological change and international trade have emerged as

alternative explanation for the increasing wage gap in the US since the 1980s.

This paper investigates the relationship between trade liberalization, techno-

logical change, unemployment and the wage gap. In the model a high and a

low skilled sector produce high and low skilled intermediate goods, respectively.

Profit maximizing R&D firms develop the technology used in each sector. Tech-

nological responds to the prices of the intermediate goods and the market size.

In the model an increasing supply of skilled workers will always induce skill

biased technological change, but the wage gap will only increase for unreason-

ably high elasticities of substitution between the intermediate goods or strong

changes of the high skilled labor supply.

On the other hand, trade in intermediate goods will increases the wage gap, e.g.

trade opening of the US to Japan in the 1980s increased the wage gap by about

14%. If trade effects are sufficiently strong to outweigh the negative supply side

effect of skilled workers, the increasing trade liberalization since the 1980s can

explain the increasing wage gap.

Furthermore, trade with developed countries can explain the declining unem-

ployment rates of high and low skilled workers as well as the disproportional

increases of high skilled wages.

JEL classification : F16, J2, J31, O31

Key Words : Trade Liberalization, R&D, Endogenous Technological

Change, Skill Biased Technological Change, Search Unemployment,

Wage Gap

∗ETH Zurich - KOF, Weinbergstrasse 35, 8092 Zurich, Switzerland. Acknowledgment what else to

one I would like to thank Carolina Silva, Mariano Bosch and Gonzague Vannoorenberghe for theirvaluable comments during the development of this paper.

1

1 Introduction

Unemployment and wage inequality are two important variables for every economy. A

large literature documents a rise of wage differentials by education, by occupation, by

age, and by experience groups in the United States (US) since the late 1970s. While

the wage gap increased, the unemployment declined.1 These developments coincide

with a strong increase of US trade, an increase of the share of skilled workers in the

population (see Figure 1) as well as technological progress.

Trade liberalization and skill biased technological change (SBTC) have emerged

as alternative explanations for the increasing wage gap in the US. An remarkable ex-

ception is Acemoglu (2003, [2]) who considers trade induced skill biased technological

change.

Already in the early 1990s Katz and Murphy (1992, [6]) analyzed the increasing wage

inequality in the United States. Since the 1960s the supply of skilled workers has in-

creased steadily in the US. Common supply and demand theory predicts a decreasing

wage gap in this case. Autor, Katz and Kearny (2008, [3]) identify skill biased techno-

logical change as the reason for this counterintuitive development. In their opinion the

productivity of high skilled workers increased strongly and out weighted the negative

supply effect.2

Most trade models strongly rely on changes in the relative price to explain changes in

the relative wage. Empirically Lawrence and Slaughter (1993, [7]) find a small decline

of the relative price while Sachs and Shatz (1996, [13]) find that the relative price

increased slightly after trade opening in the 1980s. They conclude that trade liber-

alization cannot explain the increasing wage gap as the relative price change was too

small.

Moore and Ranjan (2005, [8]) compare the effects of SBTC and trade on the wage

gap and unemployment and find that a trade liberalization increases the unemploy-

ment among low skilled workers. They argue that trade opening can increase the wage

gap, but it is associated with an increasing low skilled unemployment rate. This does

not reconcile with the decline of the overall and low skilled unemployment rates since

the 1980s. In the Moore and Ranjan model technology is asumped to be constant after

trade opening. On the other hand Acemoglu (2003, [2]) shows that trade can induce

technological progress by a price and market size effect. Furthermore, Pissarides and

1Throughout this paper the notion wage gap refers to the relative wage of workers with at least acollege degree and workers with only a high school diploma, wh/wl.

2Card and DiNardo (2002, [4]) critically remark that whenever the changes in the relative wagesare not fully explained by changes in the relative skill supply, it is claimed that technological changehave caused the opposing wage development. Hence, SBTC is a residual explanation for the increasingwage gap, which is hard to verify empirically.

2

Figure 1: Wage Gap (Autor, Katz and Kearny, 2008), Unemployment Rate (U.S.Bureau of Labor Statistics, 2010), Skill Ratio - Supply of workers with at least acollege degree over supply of workers without a college degree (US Census Bureau,Current Population Survey, 2010), Trade Volume (WTO Statistical Database, 2010).

Vallanti (2007, [12]) find that total factor productivity growth leads to a higher job

creation and hence lower unemployment rates. Thus, combining the concepts of the

above authors I argue that trade liberalization can be a consistent explanation for the

increasing wage gap and the declining unemployment rate since the 1980s.

This paper explores the relationship between the labor markets, technological

change and trade. The model is based on the concept of trade induced technolog-

ical change by Acemoglu (2003, [2]) and integrates imperfect labor markets as given

by Moore and Ranjan (2005, [8]). To the best knowledge of the author there is no

theoretical model that investigates the effect of endogenous technological change on

labor markets and unemployment under various trade scenarios.

In the model technological progress is skill biased if the supply of skilled labor in-

creases faster than for low skilled labor or if the relative price (high skilled to low skilled

goods) increases. A monopolistic R&D (research and development) sector responds

by increasing the research effort of high skilled complementary technology relative to

low skilled technology. Hence, the model has a medium run perspective as R&D firms

are able to respond to changes in the relative skill supply while the education decision

of the population is fixed.

3

As in the model of Moore and Ranjan (2005, [8]) search frictions exists in the labor

markets for high and low skilled workers. In contrast to Moore and Ranjan technolog-

ical change is endogenous allowing a more sophisticated analysis of trade and SBTC.

With endogenous technological change the unemployment rate of low skilled workers

depends on the (relative) labor supply and the intellectual property rights agreements

of the trading partners. For instance, trading with a symmetric developed country

increases the wage gap and decreases the low skilled unemployment rate. Hence, the

identification of trade effects through a decline of the low skilled unemployment rate

is not sufficient.

The trade effects found in the model are similar to those found in Acemoglu (1998,

[1] and 2003, [2]). He focuses on trade with developing countries, while here trade with

developed countries is considered explicitly. A further key difference is the imperfect

labor market, hence the wages of workers are not directly linked to their marginal

productivity. In the model an increasing labor supply leads to a decreasing unem-

ployment rate in the corresponding sector. Workers do not only profit from a higher

productivity due to a higher technology level, but as well from better labor market

conditions. The driving parameter in the model is the (relative) skill supply as it

changes the relative price and the technology level in the economy. Adding imperfect

labor markets to the model diminishes the magnitude of price changes necessary to

cause a significant increase of the wage gap. Trade liberalization changes the relative

price and increases the market size and hence trade opening can be consistent with

the small changes of the relative price found by Sachs and Schatz (1996, [13]) and

Lawrence and Slaughter (1993, [7]). In contrast to Acemoglu (2003, [2]) I show that

the wage gap might decrease although relative technology develops in favor of high

skilled workers, hence SBTC is not sufficient to explain an increasing wage gap.

The outline of the paper is as follows. Section 2 describes the basic model. The

equilibrium is discussed in detail in Section 3. Section 4 introduces trade with devel-

oping and developed countries. Section 5 presents numerical solutions for the basic

model and various trade liberalization scenarios. Finally, Section 6 discusses the main

findings and concludes.

4

2 Model

2.1 Production

All individuals have equal lifetime preferences that depend on their consumption of a

final good c. The utility function is given by

uj =

∫ ∞τ=0

cjτ exp−ρτ ∀ j = D,F (1)

where ρ is the discount factor, τ is a time index and j = D,F defines the country

domestic, D, and the foreign, F , country, both indices will be omitted when ever it

does not lead to any confusion. In each period the final good c is produced with CES

(constant elasticity of substitution) production using a high skilled intermediate good,

ch, and a low skilled intermediate good, cl.

c =(ωc

ε−1ε

l + (1− ω)cε−1ε

h

) εε−1

(2)

Labor is not necessary for the production of the final good. The parameter ε defines the

elasticity of substitution between the two goods and ω is a share parameter between the

two intermediate goods. The subscript i = h, l indicates high and low skilled variables.

In each country the two intermediate goods are produced separately by a repre-

sentative firm using the corresponding kind of local labor and technology, Ah and Al,

respectively. Each intermediate good has its own specific machines, xi, which are com-

plementary to the sector specific labor, i.e., high skilled machines can only be used

by high skilled workers and vice versa. The intermediate good firm decides about the

optimal machine and labor demand and takes prices and technology level as given.

The intermediate goods are produced with a Cobb-Douglas production function

yh = Aβhx1−βh Nβ

h (3)

yl = Aβl x1−βl Nβ

l (4)

where β ∈ (0, 1) is a common production coefficient. Ni gives the high and low skilled

labor employed in the corresponding sector, which is different from the total labor

supply. The total labor supply will be indicate by N i and hence the employed labor

is given by Ni = (1− ui)N i, where ui is the sector specific unemployment rate.

5

2.2 Prices

With competitive intermediate goods markets the relative price, p, follows from the

equation (2). In the optimum the marginal rate of substitution between the two

intermediate goods have to be equal to the relative price.

p =phpl

=

(∂c∂ch

)(∂c∂cl

) =1− ωω

(chcl

)− 1ε

(5)

The price of the final consumption good, pc, depends on the prices of both inputs. For

a CES production function the price is given by

pc = (ωεp1−εl + (1− ω)εp1−εh )1

1−ε (6)

The real prices of the intermediate goods in terms of the relative price are derived

using the above expression for the final good price

ph =phpc

=p

(ωε + (1− ω)εp1−ε))1

1−ε(7)

pl =phpc

=1

(ωε + (1− ω)εp1−ε))1

1−ε(8)

The price of the high skilled good increases with the relative price while the price of

the low skilled good decreases, ∂ph∂p

> 0 and ∂pl∂p< 0.

2.3 Machine Demand and R&D

High and low skilled R&D firms make technological innovations and are the monopo-

listic producer of machines. Intermediate good firms rent the machines in each period.

The R&D firm decides about the technology level and the price of the machines. The

technology is embedded in the machine, when an intermediate good firm rents a ma-

chine it produces with the corresponding technology level. For simplicity only the

newest vintage can be rented.3

The intermediate good firms take the rental price, χi, and the technology level of the

machines as given and maximize their profits with respect to the machines, xi.4

3If the technology level drops for some reasons, intermediate good firms want to rent older ma-chines, as they have a higher technology level and productivity. This assumption excludes thispossibility.

4The labor demand is not derived straightforward as search frictions exists, hence the firm considersNi as the given employment level. The optimal employment level is determined later in context of asearch model.

6

maximizexi

πi = yipi − χixi −Niwi

subject to yi = Aβi x1−βi Nβ

i

(9)

Wages and capital costs are in final good prices. This yields the machine demand as a

function of the rental price, the real intermediate good price, the technology level and

the employed labor.

xi =

((1− β)pi

χi

) 1β

AiNi (10)

The machines are produced by the R&D firm that will gain monopoly rents. As the

demand is iso-elastic, the common expression of the Lerner index is used to determine

the optimal monopoly mark up of the firm.

− 1

εxi=χi −MC

χi⇒ β =

χi − (1− β)2

χi

The elasticity of demand is εxi = − 1β. The R&D firm’s marginal costs (MC) are

constant and fixed to (1− β)2, hence the monopoly price for each type of machine is

constant.

χi = (1− β) (11)

Replace the monopoly price, χi, in equation (10) to obtain the machine demand for

an intermediate producer.

xi = p1β

i AiNi (12)

The machine demand increases with the employed labor, the intermediate good price

and the technology level. Substituting the demand in the intermediate good produc-

tion function yields

yh = p1−ββ

h AhNh (13)

yl = p1−ββ

l AlNl (14)

Research firms decide about the technology level, Ai, which partly determines the

machine demand. The technology production function for each R&D firm is given by

Ai = zµi q1−µi (15)

where zi is the research effort in final good units, qi > 0 is a scale parameter and

7

µ ∈ (0, 1) is a production coefficient. Technology production has diminishing returns

to research effort, hence higher levels of technology are more costly to achieve and to

maintain.

The research firms maximize their profits with respect to their research effort consider-

ing the constant markup price from equation (11) and the iso-elastic demand function

in equation (12). On the other hand, R&D firms take the intermediate good price and

the employment level as given.

maximizezi

πRDi = χixi −MCxi − ziµβ(1− β) = β(1− β)p1β

i AiNi − ziµβ(1− β)

subject to Ai = zµi q1−µi

Research effort costs, zi, are scaled by the expression µβ(1− β) to simplify notation.

The optimal research effort is

zi = qi(p1β

i Ni)1

1−µ (16)

which implies a technology level of

Ai = qi(p1β

i Ni)µ

1−µ (17)

The higher is the intermediate good price the more profitable is the production for the

intermediate good firm and hence the higher is the machine demand. Similar for the

number of workers employed. The higher machine demand increases the profits of the

R&D firm and the technology level.

For a country in autarky the consumption of high and low skilled goods equals the

local production. Considering the technology level the relative price can be written as

a function of the labor supply ratio.

p =

((1− ωω

)−εyhyl

)− 1ε

=

((1− ωω

)−εp

1−ββAhNh

AlNl

)− 1ε

=

((1− ωω

)−εp

(1−µ)(1−β)+µ(1−µ)β

(qhql

)(Nh

Nl

) 11−µ)− 1

ε

(18)

Solving for p yields

p =

((1− ωω

)−ε(qhql

)(Nh

Nl

) 11−µ)− β(1−µ)

(1−µ)(βε+(1−β))+µ

(19)

A higher employment in the skilled sector increases the supply of skilled interme-

diate goods and hence decreases its price.

8

2.4 Labor Markets

This section introduces search model as given by Pissarides (2000, [9]). The model

allows determining wages and unemployment rates in each sector. Firms will only cre-

ate a vacant position if it is profitable. Workers will only accept a job offer if the wage

paid is higher than their reservation wage. Exogenous shocks destroy filled positions.

Unemployment exists as it takes time for the firm and the worker to form a match.

I use two symmetric matching function for the high and low skilled sector.

M(vhNh;uhNh) = kvγhu1−γh Nh = kθγhuhNh (20)

M(vlN l;ulN l) = kvγl u1−γl N l = kθγl ulN l (21)

The exogenous parameters, such as recruitment costs or social benefits, are the same

for high and low skilled workers and firms.5 θi = viui

gives the labor market tightness,

where vi is the vacancy rate and ui is the unemployment rate in sector i, γ ∈ (0, 1) is a

matching coefficient and k is a scale parameter. Recall that N i gives the total supply

of skilled and unskilled labot.

Following the standard Pissarides model the equilibrium unemployment rate is given

by

ui =ψ

ψ + kθγi(22)

where ψ is the exogenous job destruction rate.

2.4.1 Firms

For the wage determination firms consider the value of a filled and a vacant position.

Fi represents the discounted value of a vacancy in a firm and Ji the present discounted

value of a filled position in a firm.

A vacant position is an asset for the firm. If the capital markets are perfect, the

valuation of this asset will be ρFi and equal to the expect gains from filling a position,

5Different values for the high and low skilled sector do not change the qualitative results of themodel. For example higher recruitment costs for high skilled workers increase the high skilled unem-ployment rate and decreases the high skilled wages and consequently the wage gap. This reconcileswith the behavior of unemployment and wages we would expect in a search unemployment model.Davidson et al. (1999, [5]) note that different labor market institutions are important for the evalu-ation of trade. Nevertheless, this paper focuses less on the labor market institutions, but rather onthe interaction of trade, labor markets and technological progress.

9

which gives the right hand side of the following equation.

ρFi = kθγ−1i (Ji − Fi)− δ (23)

Similarly, a filled position has a value for the firm, which is equal to the marginal

profits. This is calculated by subtracting the wage, wi, and the marginal costs of

machines, ri, from the marginal revenues of a worker.6

ρJi = ti − wi − ri + ψ(Fi − Ji) (24)

The last term on the right hand side of equation (24) represents the expected change

of the value if a match is resolved in the future.7

The intermediate firm takes the technology level as given and hence the marginal

revenues are given by equation (13) and (14).

ti =

(∂yi∂Ni

)pi = p

1β

i Ai (25)

The capital costs are calculated as χhxh using equations (11) and (12), hence the

marginal rental costs are

ri =∂(χixi)

∂Ni

= p1β

i Ai(1− β) (26)

Using equation (23) and (24) with the fact that the value of a vacancy have to be zero

in the equilibrium, Fi = 0.

kθγ−1i (ti − ri − wi) = kθγ−1i (βp1β

i Ai − wi) = δ(ψ + ρ) (27)

where the equations (25) and (26) are substituted in equation (24) to receive the second

equality.

2.4.2 Workers

Workers accept any job that pays a higher wage than their reservation wage. The

present discounted value of unemployment is equal to the social benefits plus the

expected gain from finding a job. On the other hand, the present discounted value of

6Employing an additional worker increases the machine demand and by this the marginal costs ofan additional worker.

7Note that if the job destruction rate is zero, no unemployment exists in the model. Hence,equation (24) would simplify to ρJ = ti − wi − ri. In this setting the value of a filled position, J ,would be zero and the above equation would correspond to the first order condition for labor of thecommon firm’s maximization problem in expression (9).

10

being employed is equal to the wage plus the expected loss when a match is destroyed.

These considerations lead to the following two standard Bellman equations in the

Pissarides model.

ρUi = b+ kθγi (Wi − Ui) (28)

ρWi = wi + ψ(Ui −Wi) (29)

where Ui is the present discounted value of unemployment and Wi is the present dis-

counted value of employment. The worker receives social benefits b if unemployed. kθγi

gives the rate at which workers find a job and ψ the rate at which workers lose their job.

2.4.3 Wage Bargaining

Wages are determined by Nash bargaining over the profits of a filled position. The

parameter η defines the bargaining power of workers and firms. A higher η gives more

weight to the workers, η = 0.5 implies symmetric bargaining.

wi = arg max(Wi − Ui)η(Ji − Fi)1−η (30)

The first order condition for equation (30) is

Wi − Ui =η

1− η(Ji − Fi) (31)

From this expression the wage equation can be derived.

wi = (1− η)b+ η(βp1/βi Ai + δθi) (32)

The key parameter for the equilibrium is the labor market tightness, which is given

by an implicit function, emanating from wage equation and the equation (27). The

mathematical derivation is shown in the Appendix.

(1− η)[βp1/βi Ai − b]− ηδθi −

δ(ρ+ ψ)

kθγ−1i

= 0 (33)

The labor market tightness is an increasing function of technology, Ai, and the real

intermediate good prices, pi

∂θi∂Ai

> 0∂θi∂pi

> 0

By equation (22) a higher θi decreases the unemployment rate. Thus, higher inter-

11

mediate good prices and technological progress decrease the unemployment as both

factors make the employment of more workers more profitable for the intermediate

firm.

3 Equilibrium

Equation (33) gives the equilibrium condition for each sector in the model. After

substituting equations (7), (8), (5) and (17) the labor market tightness is a function of

the labor supply. Note, that the relative price depends on the labor market tightness

of both sectors, thus the implicit function in equation (33) has to hold simultaneously

for both sectors. Once the labor market tightness for each sector is determined for a

given labor supply, all other parameters can be recovered.

For a country in autarky the equilibrium is defined by

(1− η)[βp1β

hAh − b]− ηδθh −δ(ρ+ ψ)

kθγ−1h

= 0

(1− η)[βp1β

l Al − b]− ηδθl −δ(ρ+ ψ)

kθγ−1l

= 0

(34)

where

p =

((1− ωω

)−ε(qhql

)((1− uh)Nh

(1− ul)N l

) 11−µ)− β(1−µ)

(1−µ)(βε+(1−β))+µ

ph =p

(ωε + (1− ω)εp1−ε))1

1−ε

pl =1

(ωε + (1− ω)εp1−ε))1

1−ε

Ai = qi(p1β

i (1− ui)N i)µ

1−µ

ui =ψ

ψ + kθγi

Although the system has no closed form solution, comparative statics help to under-

stand the mechanism of the model. Assuming that the labor market tightness of the

other sector is constant, the labor market tightness of each sector is increasing with

its labor supply. This implies a decreasing unemployment rate in each sector.

12

∂θi

∂N i

> 0 i = h, l

The decreasing unemployment rate is driven by the increasing technology, which

can be interpreted as ”capitalization” effect, i.e., recruitment costs become less and

less important and new matches are formed easier. This is consistent with the em-

pirical evidence for a negative relationship between the unemployment rate and labor

productivity, see Pissarides & Vallanti (2007, [12]).

Proposition 1: For a given θl (θh), an increase in the labor supply Nh (N l), will

increase the labor market tightness θh (θl) and decrease the unemployment rate uh (ul).

A high skilled labor supply shock will have two opposing effects on the technology

level. First, it decreases the relative price by increasing the production of the high

skilled good. A decreasing intermediate good price will diminish the profit incentives

of the research firm, which will lead to a lower technology level. On the other hand,

the demand for high skilled machines will increase as more high skilled workers are

employed. Which in turn increases the profits of the R&D firm and hence the tech-

nology level. The net effect of the negative price change and the positive labor market

size effect on the relative technology level can be evaluated using equation (17) and

replacing the relative price by the equation (19).

AhAl

=

(qhql

)(p

1βNh

Nl

) µ1−µ

= κ

(qhql

)((1− uh)Nh

(1− ul)N l

) µβ(ε−1)(1−µ)(βε+1−β)+µ

(35)

where κ =

((1−ωω

)−ε ( qhql

)1−µ) −(1−µ)(1−µ)(εβ+1−β)+µ

The relative technology increases in Nh and decreases in N l for given unemployment

rates if ε > 1, hence the technological change will be skill biased if the skill ratio

increases.

Proposition 2: For a given labor market tightness in the two sectors an increase in

the labor supply Nh (N l) increases (decreases) the relative technology if ε > 1. An

increasing skill ratio implies skill biased technological change.

The effect on wages are ambiguous. The wage in each sector depends on the sectoral

technology level, the intermediate good price and the labor market tightness, which

can be seen from equation (32). An increase in the skill supply will have a positive

impact on the labor market tightness and the technology, but at the same time the

13

intermediate good price will decrease. Which effect is stronger is not clear a priori.

4 Trade Models

Trade occurs in intermediate goods between two countries countries. Trade in inter-

mediate goods equalizes the relative prices in the two economies. As all individuals

have the same preferences and face the same prices after trade opening, they have the

same relative demand of high and low skilled goods. Hence, the post-trade relative

price p satisfies

p =phpl

=1− ωω

(yDh + yFhyDl + yFl

)− 1ε

(36)

the post-trade prices for high and low skilled goods are still given by equations (7)

and (8) using the above post-trade relative price.

For the R&D firms the intellectual property rights (IPR) are crucial. I distinguish

two cases. First, intellectual property rights are only enforced in the domestic country

and not in the foreign country. This implies that R&D activities only takes place

in the domestic country and the foreign country copies the technology and machines

developed in the domestic country. This case can be interpreted as trade with a

developing country. In the second case, IPRs are enforced in the foreign country after

trade opening. As the R&D firm has a monopoly, only one R&D firm for each sector

rents machines to intermediate good firms in both countries. I will refer to this case

as trade with a developed country.

4.1 Developing Country

A foreign developing country has no research sector and copies the existing technology

of the domestic country. Consequently, the domestic R&D sector does not take into

account the machine demand of the foreign country. The technology in both countries

will be given by equation (17), but the domestic R&D firm considers the post-trade

price. Trade liberalization with a developing country has a price effect, but no market

size effect for technology.

Ai = qi(p1β

i NDi )

µ1−µ (37)

Proposition 2 still holds after trade liberalization, i.e. if the skill ratio in the

domestic country increases, technological progress is skill biased.

14

Ah

Al=

(qhql

)(p

1β

(1− uh)Nh

(1− ul)N l

) µ1−µ

(38)

Furthermore, trade liberalization induces skill biased technological change if the

skill ratio of the domestic country is higher than of the foreign country. To see this,

substitute the post-trade technology level from equation (37) in the production func-

tion of the domestic and foreign countries. Considering the relationship for the relative

price p as given in equation (36) and using that the trade in intermediate goods equal-

izes the prices in the two economies, the relative world market price is a function of

the labor supply of the trading countries.

p =

((1− ωω

)−εp

(1−µ)(1−β)+µ(1−µ)β

(qhql

)(NDh

NDl

) 11−µ)− 1

ε

︸ ︷︷ ︸p∗

1 +NFh

NDh

1 +NFl

NDl

−1ε

(39)

The first parenthesis in equation (39) would be the the relative price in the domestic

country without trade and the non-trade technology level, compare equation (18). The

relative price will increase in the domestic country if it starts trading with a country

that is scarcer in high skilled labor, i.e.,NDh

NDl>

NFh

NFl

. The post-trade relative price

increase will be greater, the bigger the foreign country and the higher the difference

between the skill ratios of the two countries. A higher relative price leads to skill

biased technological change by equation (38).

Intuitively trade liberalization with a developing country with a lower skill ratio leads

to an increasing wage gap. First, the price for high skilled goods increases and the

price for low skilled goods decreases. Second, technological progress is biased towards

skilled workers. Both developments favor the wages of high skilled workers.

Proposition 3: For given high and low skilled unemployment rates trade opening to a

country with a lower skill ratio will increase the relative price in the domestic country

and induces skill biased technological change. The wage gap will increase.

Solving the above equation for p yields an analogous expression to equation (19).

p = κ1β

((NDh

NDl

) µ1−µ(NDh +NF

h

NDl +NF

l

))− β(1−µ)(1−µ)(εβ+1−β)+µ

(40)

As the technology level and the relative price are changing after trade liberalization

with a developing country, the expression for the labor market tightness have to be

adapted. Furthermore, in the post-trade relative price depends on the employment

15

levels in both countries and hence the implicit functions for the labor market tightness

for the two sectors in the two countries have to be satisfied simultaneously in the

equilibrium.

(1− η)[βp1β

h Ah − b]− ηδθjh −

δ(ρ+ ψ)

k(θjh)γ−1

= 0 for j = D,F

(1− η)[βp1β

l Al − b]− ηδθjl −

δ(ρ+ ψ)

k(θjl )γ−1

= 0 for j = D,F

(41)

where pi is given by equation (7) and (8) substituting the relative world market price

p. The expression for the unemployment rate, ui, is unchanged and given by equation

(22). Furthermore, the labor market tightness have to be the same in both countries

after trade liberalization as all exogenous parameters in the matching function, the

prices and the technology levels are the same in both countries.

4.2 Developed Country

Trading with a developed country is very similar to a labor supply shock. Property

rights are respected in developed countries, consequently the domestic R&D firm con-

siders the demand of foreign intermediate good firms for machines. In this setting trade

opening implies that only one R&D firm exists in both countries. The monopolistic

R&D firm considers the demand of both countries for machines.

maximizezi

πRDi = χi(xDi + xFi )−MC(xDi + xFi )− ziµβ(1− β)

subject to Ai = zµi q1−µi

Using that the technology levels and prices are the same in both countries after

trade liberalization, the above maximization yields the optimal technology level after

trade liberalization with a developed country.

Ai = qi(p1β

i (NDi +NF

i ))µ

1−µ = qi(p1β

i ((1− uDi )ND

i + (1− uFi )NF

i ))µ

1−µ (42)

The expression for the technology level have to be substituted in the equilibrium

condition in equation (41), which again have to hold for the two sectors in both coun-

tries.

Proposition 3 holds as well for trade liberalization with a developed country that

has a lower skill ratio than the domestic country. If the combined skill ratios of the

domestic and foreign country lowers, technology will increases more for low skilled

workers as for high skilled workers. This follows from Proposition 2.

16

A special and simple case is noteworthy. If the foreign developed country and the

domestic country are completely symmetric, trade opening will not change the relative

price for given unemployment rates, only a market size effect for technology can be

observed.

5 Numerical Results

The equilibrium conditions for the basic model and the trade model, equation (34) and

(41) respectively, have no closed form solution, but they can be solved numerically.

I present three different scenarios. First, a country in autarky with a changing skill

ratio. Second, I show the effects of trade liberalization to a developed country that

respects IPR. Finally, the effects of trade liberalization to a less developed country,

i.e. without IPR.

5.1 Calibration

I calibrate the model to match the US economy, i.e. the wage gap and unemployment

rates. The values for the exogenous variables presented in Table 1 are in line with

the common values taken by Pissarides (2007, [10] and 2009, [11]). For the supply of

skilled and unskilled workers I use data from the US Census of Population Educational

Attainment from 1963 to 2003, where skilled workers have at least a college degree.

The skill ratio increases because the supply of high skilled workers increases stronger

than the supply of low skilled workers. The common estimate for the elasticity of

substitution in the final good production function is 1.5

5.2 Robustness

A key parameter is the elasticity of substitution, ε, in the production function of

the final good. I solve the models for ε between 1.5 and 2.5 and discuss the effects

when needed. The parameter b for social benefits and the recruitments costs δ have

to be sufficiently low relative to the wage to ensure a solution. A higher technology

production coefficient µ increases the spread between pre- and post-trade wage levels.

The technology scale parameter qi is set to 3.5 for the high and low skilled technology

to give a reasonable wage gap.

A system of non-linear implicit functions have to be solved simultaneously, there-

fore I use a Levenberg-Marquard algorithm to solve the system of equation which is

more robust than a Gauss-Newton algorithm. As a guess for the solution I use θs so

that the unemployment rate for high skilled workers is around 3% and for low skilled

17

Parameter Value Descriptionβ 0.66 Cobb-Douglas production coefficientγ 0.5 Matching parameterδ ∈(0.1,0.5) Recruitment costsε ∈(1.5,2.5) Elasticity of substitution for the final good productionη 0.5 Bargaining power parameterµ ∈(0.25,0.6) Technology production parameterρ 0.004 Discount factorψ 0.019 Job destruction rateb free Social benefitsk free Unemployment scale parameterqh free Skilled technology scale parameterql free Low skilled technology scale parameter

Table 1: Value of the exogenous variables following Pissarides (2007 and 2009) andAcemoglu (2003).

workers around 6%. The results for a country in autarky are robust for changes of

the initial guess to solve the system of equations, indicating that a unique solution

is more likely. As all exogenous parameters determining the labor market tightness

are the same for the two countries and the post-trade prices and the technology levels

are the same, the labor market tightness have to be the same in both countries after

trade liberalization. This assumptions make the numerical results stable to changes in

the initial starting values for the algorithm. The results differ once labor market pa-

rameters are different in the two countries. Nevertheless, results change only slightly,

as long as the initial guess is associated with a reasonable low unemployment rate in

both countries.

5.3 US in Autarky

In this section I show the results for the wages, unemployment rates and technology

of both skill groups for a country in autarky. I use the labor supply of the US, which

implies an increase of the skill ratio from 0.09 to 0.37 between 1963 and 2003. Figure

2 presents the solution graphically.

The wage gap in general decreases for even very high elasticities of substitution.

This is contradicts the findings of Acemoglu (2003, [2]) and Moore and Ranjan (2005,

[8]) who find that ε > 1 is sufficient for an increasing wage gap. A elasticity of 2.5

is needed to explain a slight increase of the wage gap, by 4 percentage points since

the mid 1990s. Still the increasing skill ratio will lead to skill biased technological

progress for any ε > 1. This models is able to consider the supply of high and low

18

Figure 2: Numerical solution for the US labor supply of skilled and unskilled workersusing various ε.

skilled workers separately. Most previous models focus just on the skill ratio, which

steadily was increasing in the US. As a counterfactual I can solve the model for changes

in the labor supply only occurring in the high skilled sector, the skill ratio would be

identical, but an elasticities of substitution greater than one is sufficient to explain an

increasing wage gap.

The observed changes in the labor supply of high skilled wages are to small to explain

the increasing wage gap in combination with a low elasticity of substitution.

5.4 Trade with a Developed Country

The case of trade with a developed country that does respect IPR is shown in Figure

3. The supply of low skilled workers is 70% of the supply in the domestic country

(US) and the supply of skilled workers is 60% of the US supply. Hence the foreign

country is smaller than the domestic country and has a slightly lower relative labor

supply.8 The values for the foreign are chosen to match the skill supply and the size

of the Japanese population in the 1980 census.

The solid line gives the post-trade results and the dashed line the pre-trade results.

The wage gap increased significantly, technology became more skill biased, unemploy-

ment rates decreased and wages increased for both skill groups. Two points are very

important. First, the wage inequality rises because the high skilled wages rises much

faster than low skilled wages. Second, the unemployment rates for both skill groups are

8This ensures that the relative price increases after trade liberalization.

19

lower after trade opening. This is considerably different from the findings of Moore

and Ranjan (2005), who argue that trade liberalization leads to an increasing low

skilled unemployment rate and decreasing low skilled wages. Theses results reconcile

much better with the empirical findings.

The results are stronger if the foreign country has a even lower skill ratio than the

domestic country.

Figure 3: Numerical solution for trade with smaller and less skilled developed country.ε = 1.5.

The numerical result can be seen as evidence that the increasing trade, especially

with Japan, in the 1980s contributed to the development of the wage gap in the US.

In this setup a relative small price change is able to explain a significant increase of

the wage gap, i.e. the mean relative price increases by less than 4% while the mean

wage gap increases by about 14%. The results seem more consistent with the empirical

findings of Sachs and Schatz (1996, [13]) and Lawrence and Slaughter (1993, [7]).

5.5 Trade with Developing Countries

Figure 4 gives the numerical results for trade with a developing country that does not

respect IPR. The supply of skilled workers in the domestic country is only 25% of the

supply in the domestic country and 50% for low skilled workers. This implies that

the price change is considerably greater than in the case of trade with a developed

country.

Again, trade liberalization leads to an increasing wage gap and skill biased techno-

logical change in the domestic country. The high skilled unemployment rate decreases

20

Figure 4: Numerical solution for trade with a small developing country (no IPR) witha low skill ratio. ε = 1.5

while the low skilled unemployment rate increases. This behavior is consistent with

the findings of Moore & Ranjan (2005, [8]) who use the increasing unemployment rate

of low skilled workers to identify the effect of trade on the wage gap. Similar to Moore

& Ranjan trade causes the low skilled wages to decrease and the high skilled wages to

increase.

6 Concluding Remarks

In this paper I investigate the increasing wage gap in the US and its relationship with

unemployment rates of high and low skilled workers in the context of trade liberaliza-

tion and endogenous technological change. In contrast to previous studies, the level

of technology is an equilibrium outcome of the model. Furthermore, I consider the

supply of high and low skilled workers separately and do not focus exclusively on the

skill ratio. With the help of the search frictions in the labor market and I distinguish

between the labor employed and the labor supply of the population. The first is an

equilibrium outcome of the model while the second is exogenously given.

The model allows me to evaluate various economic scenarios: an increasing labor sup-

ply, trade with developing countries and trade with developed countries.

I find that an increasing skill ratio, due to a higher supply of skilled labor, always

leads to classical skill biased technological change. In the model the unemployment

21

rates decrease with an increasing labor supply for both skill groups. A higher supply of

a certain type workers increases the employment level of theses workers. This increases

the demand for the machines complementary to theses workers and the incentives for

R&D firms to develop the complementary technology. This can be interpreted as a

capitalization effect.

For a country in autarky the wage gap increases only if the elasticity of substitution

between the two intermediate goods in the production of the final good is sufficiently

high or the high skilled labor supply rises strongly relative to low skilled labor supply.

The numerical results show that for the labor supply of the US the elasticity of substi-

tution of at least 2.5 is needed to ensure an increasing wage gap.9 For low elasticities

the technological progress is skill biased, but the negative price effect is stronger than

the positive technology effect on wages.

The observed changes in the labor supply of high skilled wages are to small to explain

the increasing wage gap in the US in combination with the common estimates for the

elasticity of substitution.

Trade liberalization to a developed country that respects intellectual property

rights can explain the increasing wage gap. The wage gap increases as the high skilled

wages increases stronger than the wages of low skilled workers, which is consistent

with the empirical findings of Autor, Katz and Kearny (2008, [3]) who show that the

greater dispersion of the wage distribution comes from increasing wages in the upper

tail of the income distribution. This stands in strong contrast to the finding of Moore

and Ranjan (2005, [8]) that the increasing wage inequality after trade liberalization

arises from a decreasing low skilled wage.

Furthermore, as observed in the US the unemployment rates of both skill groups de-

crease after trade liberalization with a developed country. Trade with a symmetric

developed country will have no direct price effect, as the relative and absolute en-

dowment with skilled labor is the same, but only a market size effect for technology.

It increases the technology level for high and low skilled machines and consequently

decreases the unemployment rates of both skill groups.

I find that the decline in the low skilled unemployment rate is stronger, causing the

employed skill ratio to change, which in turn leads to an increase in the relative price.10

This combination leads to an increasing wage gap after trade liberalization. Further-

more, the price change in the case of trade between two developed countries with

similar skill ratio is relatively small while the wage gap increases significantly. The be-

9The literature states typical ε around 1.5.10The rise of the relative price would be even greater if the foreign developed country has a lower

skill ratio.

22

havior of the relative price reconcile with the empirical findings of Sachs-Schatz (1996,

[13]) or Lawrence-Slaugher (1993, [7]).

The ongoing trade liberalization in the US with other developed countries, namely

Japan and the European countries, can explain the increasing wage gap and the de-

creasing unemployment rates for both skill groups.

In this paper I argue that just considering the decreasing low skilled unemployment

rate to identify trade effects and skill biased technological change effects (induced by

an increasing skill ratio) on the wage gap is not sufficient. First, endogenous skill

biased technological change does not necessarily lead to an increasing wage gap, only

if the elasticity of substitution is sufficiently high. Second, trade liberalization to a

country that respects intellectual property rights leads to skill biased technological

change and a decreasing low skilled unemployment rate. Consequently, one should be

cautious when referring to SBTC to explain an increasing wage gap whenever skill

supply and the wage development are not inversely related. Furthermore, I showed

that wages for both skill groups increases after trade opening and that the increasing

wage gap is due to the stronger increase of high skilled wages.

23

References

[1] Daron Acemoglu. Why do new technologies complement skills? directed technical

change and wage inequality. The Quarterly Journal of Economics, 113(4):1055–

1089, 1998.

[2] Daron Acemoglu. Patterns of skill premia. Review of Economic Studies,

70(2):199–230, 2003.

[3] David H. Autor, Lawrence F. Katz, and Melissa S. Kearney. Trends in u.s.

wage inequality: Revising the revisionists. Review of Economics and Statistics,

90(2):300–323, 2008.

[4] David Card and John Dinardo. Skill-Biased Technological Change and Rising

Wage Inequality: Some Problems and Puzzles. Journal of Labor Economics,

20(4):733–783, 2002.

[5] Carl Davidson, Lawrence Martin, and Steven Matusz. Trade and Search generated

Unemployment. Journal of International Economics, 48(2):271–299, 8 1999.

[6] Lawrence F. Katz and Kevin M. Murphy. Changes in relative Wages, 1963 - 1987:

Supply and Demand Factors. The Quarterly Journal of Economics, 107(1):35–78,

1992.

[7] Robert Lawrence, Matthew Slaughter, Robert Hall, Steven Davis, and Robert

Topel. International trade and american wages in the 1980s: Giant sucking

sound or small hiccup? Brookings Papers on Economic Activity. Microeconomics,

1993(2):161–226, 1993.

[8] Mark P. Moore and Priya Ranjan. Globalisation vs skill-biased technological

change: Implications for unemployment and wage inequality. The Economic Jour-

nal, 115(503):391–422, 2005.

[9] Christopher A. Pissarides. Equilibrium Unemployment Theory. MIT Press, 2nd

edition, 2000.

[10] Christopher A. Pissarides. The Unemployment Volatility Puzzle: Is Wage Sticki-

ness the Answer? Open access publications from london school of economics and

political science, London School of Economics and Political Science, 2007.

[11] Christopher A. Pissarides. The Unemployment Volatility Puzzle: Is Wage Stick-

iness the Answer? Econometrica, 77(5):1339–1369, 2009.

24

[12] Christopher A. Pissarides and Giovanna Vallanti. The Impact of TFP Growth

on steady-state Unemployment. International Economic Review, 48(2):607–640,

2007.

[13] Jeffrey D. Sachs and Howard J. Shatz. U.S. Trade with Developing Countries and

Wage Inequality. American Economic Review, 82(2):234–239, 1996.

25

A Search Unemployment

This section derives the basic equations for the search model used in Section 2.4.

Matching functions

M(vhNh;uhNh) = kvγhu1−γh Nh = kθγhuhNh (43)

M(vlN l;ulN l) = kvγl u1−γl N l = kθγl ulN l (44)

The rate at which a worker finds a job is defined as

M(viN i;uiN i)

uiN i

= kθγi (45)

which is increasing in the labor market tightness. The rate at which a vacant position

is filled is given by

M(viN i;uiN i)

viN i

= kθγ−1i (46)

which is decreasing in the labor market tightness.

ui = ψ(1− ui)− kθγi ui (47)

The flow rate into unemployment per unit of time is given as the rate of exogenously

broken matches less the rate of workers newly employed. ψ gives an exogenous break

up rate for filled positions. In the equilibrium ui = 0 will be satisfied. This gives the

steady state unemployment rate.

ui =ψ

ψ + kθγi(48)

A vacant position is an asset for the firm. If the capital markets are perfect, the capital

costs ρFi have to be equal to the rate of return on the asset. The later is given as the

expect gains from filling a position less the recruitment costs.

26

ρFi = kθγ−1i (Ji − Fi)− δ (49)

ρJi = ti − wi − ri + ψ(Fi − Ji) (50)

Similarly, a filled position has a capital costs ρJi for the firm. This has to be equal

to the current marginal revenues of a worker less the wage and the (marginal) rental

costs of machines less the expected loss if the match is destroyed at some point in time.

In the equilibrium all profit opportunities from new jobs are exploited, driving rents

from a vacant position to zero. Therefore the equilibrium condition for the supply of

vacant jobs is zero. Given a non-zero discount factor ρ, this is satisfied if Fi = 0. This

implies that firms can enter and exit the market freely.

Ji =ti − wi − riρ+ ψ

(51)

After substituting Ji from equation (59) the above equation can be written as

kθγ−1i (ti − wi − ri) = δ(ψ + ρ) (52)

Workers face a similar problem as firms. Ui is the present discounted value of unem-

ployment and Wi is the present discounted value of employment. The worker receives

social benefits b if unemployed. kθγi gives the rate at which workers are employed and

ψ the rate at which workers lose their job.

These considerations lead to the following two Bellman equations:

ρUi = b+ kθγi (Wi − Ui) (53)

ρWi = wi + ψ(Ui −Wi) (54)

Note that the permanent income Wi is different from the actual wage rate wi. This is

caused by the risk of unemployment and hence a lower income. It is assumed that the

wage will be higher than social benefits, i.e. wi > b, thus an incentive to work exists.

27

Solving equation above equation for Wi yields

Wi =wi + ψUiρ+ ψ

(55)

To derive the wage equation as given in equation (32) a common Nash bargaining

model is used.

wi = arg max(Wi − Ui)η(Ji − Fi)1−η (56)

The first order condition is

Wi − Ui =η

1− η(Ji − Fi) (57)

First substitute (51) and (55) in (57) and use Fi = 0.

wi = ρUi + η(ti − ri − ρUi) (58)

From equation (59) and Fi = 0 follows

Ji =δ

kθγ−1i

(59)

Replace Ji in equation (57) by equation (59)

Wi − Ui =η

1− η

(δ

kθγ−1i

)(60)

Now substitute equation (60) in (53)

ρUi = b+η

1− ηδθi (61)

Use (61) with (58)

wi = b+ η1−ηδθi + η(ti + ri − b− η

1−ηδθi)

wi = (1− η)b+ η1−ηδθi + ηti − ηri − η2

1−ηδθi(62)

which then simplifies to the wage equation as given by equation (32).

wi = (1− η)b+ η(ti − ri + δθi) (63)

28

The wage depends on three endogenous parameters: marginal revenues, ti, (marginal)

rental costs, ri, and labor market tightness, θi. Substituting the expression for the ti

and ri in the above equation yields.

wi = (1− η)b+ η(βp1/βi qiAi + δθi) (64)

θi can be defined as an implicit function using equation (63) and (52).

The labor market tightness as well depends on the marginal revenues, ti, and the

marginal rental costs, ri.

ti − ri − [(1− η)b+ η(ti − ri + δθi)]−δ(ρ+ ψ)

kθγ−1= 0 (65)

After substituting ti and ri from equation (25) and (26) this simplifies to

(1− η)[βp1/βi qiAi − b]− ηδθi −

δ(ρ+ ψ)

kθγ−1i

= 0 (66)

29