migration effects on conditional convergence

Migration effects on conditional

convergence

Joseph Qian

Economics Research Essay 2015

DEPARTMENT OF ECONOMICS RESEARCH ESSAY COVER SHEET

SUBJECT CODE

ECON40016

SUBJECT NAME

Economics Research Essay

STUDENT ID.

588355

STUDENT NAME

Joseph Qian

WORD COUNT

3145

PLAGIARISM

Plagiarism is the presentation by a student of an assignment which has in fact been copied in whole

or in part from another student’s work, or from any other source (E.g. published books or periodicals), without due acknowledgement in the text.

COLLUSION

Collusion is the presentation by a student of an assignment as his or her own which is in fact the

result in whole or part of unauthorised collaboration with another person or persons.

DECLARATION

This essay is the sole work of the author whose name appears on the Title Page and contains no material which the author has previously submitted for assessment at The University of Melbourne or elsewhere. Also, to the best of the author’s knowledge and belief, the essay contains no material

previously published or written by another person except where due reference is made in the text of the essay. I declare that I have read, and in undertaking this research, have complied with the University’s Code of Conduct for Research. I also declare that I understand what is meant by

plagiarism and that this is unacceptable; except where I have expressly indicated otherwise. This essay is my own work and does not contain any plagiarised material in the form of unacknowledged

quotations or mathematical workings or in any other form.

I declare that this assignment is my own work and does not involve plagiarism or collusion.

Signed JOSEPH QIAN Date 10/19/2015

Acknowledgements I acknowledge the assistance of my advisor Shuyun (May) Li for her academic guidance and support

throughout the year. I also acknowledge my Honours in Economics Co-ordinator Neville Norman for

his support throughout the year.

Migration Effects on Conditional Convergence

Abstract

This paper aims to look at the role that migration plays in a country's economic growth. We have

seen over the past century that countries such as Australia and Canada have benefitted greatly from

migration inflows. Whilst on the other side of the story, developing countries as of late have shown

signs of divergence and a persistent migration outflow. I aim to provide insight into this

phenomenon by developing a theoretical model mainly based upon the work of Reichlin and

Rustichini. I will then run numerical simulations using my model to support my theoretical results

and to derive any further results. My main finding is that migration can result in convergence of

wages which would otherwise not be achieved if labour was immobile. I also find that skilled labour

migration plays a significant role in determining whether convergence or divergence will occur in the

long run.

Contents

1. Introduction 2

2. Literature Review 3

3. Theoretical Model 4

3.1 Model Environment 4

3.2 Equilibrium 6

3.3 No Migration Case 6

3.4 Migration Case 7

4. Numerical Simulation 9

5. Results 10

6. Limitations and Extensions 12

7. Conclusion 13

8. References 14

9. Appendices 15

2 | P a g e

Migration Effects on Conditional Convergence

1. Introduction

In the modern world, migration has become more and more prevalent as society becomes more

inclusive and accepting of diversity. However, it remains to be seen what the exact implications are

for countries that primarily have labour inflows or outflows. As many individuals rationally look to

migrate to a country where they can achieve a better life, does that affect the gap between rich and

poor countries?

My paper looks to answer this vital question and determine the whether migration patterns have

any explanatory power in predicting conditional convergence. This area of economics has become a

great concern in recent times, with many countries showing signs of divergence. By gaining a greater

understanding of this field, we can look for ways to close the income gap and achieve convergence.

I will do so by developing a theoretical model based upon the Diamond model and incorporate

migration. The migration element will be drawn from the model developed by Reichlin and

Rustichini (1997). I will then analyse the long run results in terms of wages based on varying initial

parameters.

In Section 2, I will provide an overview of the past literature that looks at areas such as convergence

theory, capital flows and migration. Next I will present my theoretical model in Section 3 and derive

analytical long run outcomes. In Section 4, I will conduct numerical simulations to validate my

theoretical results and explore more outcomes. I will then present the results of my simulation in

Section 5 and discuss the limitations and extensions of my model in Section 6. Finally I will

summarize my findings in Section 7.

3 | P a g e

2. Literature Review

The general consensus on conditional convergence is that it is very plausible. The classic Solow-Swan

model (1956) captures this theory and tests it empirically. The underlying driving force for

convergence comes from capital mobility. Capital flows from rich countries to poor countries allow

the gap between income per capita to close. However, Lucas (1990) noticed that this is not always

the case and looks at reasons why this does not occur. One of the main reasons he contends is that

due to capital market imperfections, convergence is a less likely outcome. Given recent signs of

divergence between rich and poor countries, we are unable to conclude that the neoclassical theory

holds for all cases.

In the absence of perfect capital mobility, it is possible that migration also plays its role in

conditional convergence. Neoclassical models predict that as countries converge, migration should

die away. However, Borjas (1994) extensively studied migration patterns in USA, Canada and

Australia and found that migration inflows to these countries were not diminishing. In this sense,

these models are unable to explain persistent migration flows or the lack of cross country

convergence. The common determinant for migration as supported by the empirical literature is

wage differentials. A contentious result of the Solow-Swan model is that convergence leads to wage

equivalence across countries. Empirical studies have disproved this notion, which possibly explain

the persistence of migration.

Okun and Richardson (1961) and Bhagwati and Rodriguez (1975) share the view that human capital

flows in the form of migration is likely to provide a clearer picture of convergence theory. They argue

that there is a significant externality produced by migration on workers productivity. A study on the

effect of immigrant workers on domestic wages by Greenwood and McDowell (1986) looks at the

possible likelihood that skilled immigrants have a positive effect. Similarly, Lucas (1990) also argues

that differences in human capital and the external benefits of human capital explain the lack of

convergence. Mankiw, Romer and Weil (1990) addresses this common view by developing the

Human Capital Augmented Solow-Swan model. This model proved to perform better empirically

4 | P a g e

than the baseline model, however various assumptions had to be made in establishing what

constituted human capital.

Reichlin and Rustichini (1997) developed a model based on the Arrow model which provides many

conclusions that oppose standard neoclassical model predictions. They define migration flows in

terms of a size and composition effect. They conclude that reduction in the labour force due to

migration outflows are ultimately bad for a country. However, migration that consists mainly of

unskilled workers is beneficial to the labour exporting country as it improves that country’s

composition of labour and hence improves the efficiency of that country’s work force. As these two

effects can operate in the same or opposite directions, it can provide explanations for both

persistent and temporary migration.

Dowrick and Rogers (2002) also point out the unlikelihood of convergence is due to steady state

levels changing over time. They argue that the importance of technological progress allows the

richer country to grow whilst poorer countries are unable to effectively utilize the existing

technology. They emphasize that the lack of skilled workers in poorer countries results in an inability

to take advantage of technological advances that would otherwise be transferable to their country.

3. Theoretical Model

My theoretical model will be based off the model produced by Reichlin in Rustichini (1997). My

model differs to theirs in the specification of productivity function. Given the arguments made by

Dowrick and Rogers (2002), I will make productivity a function of skilled labour, which would

therefore mean that technology may not be perfectly transferable across countries.

3.1 Model Environment

I construct a two country model in discrete time, indexing each country with a superscript i and j. I

follow an overlapping generations model framework, whereby each country consists of young and

old individuals at any given time. A young individual has the option to earn a wage in their own

5 | P a g e

country, or migrate to the other country and earn a wage there. For simplicity, I take the case where

individuals only consume when they are old. Therefore young individuals invest all of their income

and can earn capital income from either country. In the next period, they become an old individual

and consume their initial income plus the accumulated interest. Assuming that there is no disutility

from work, an individual aims to maximise their utility by maximising their consumption when old.

𝑀𝑎𝑥 𝑈(𝑐𝑜𝑡+1) 𝑠. 𝑡 𝑐𝑜𝑡+1 = 𝑤𝑡(1 + 𝑟𝑡+1) (3.1.1)

Each country will follow an identical Cobb-Douglas production function as given by

𝑌𝑡 = (𝐾𝑡)𝛼(𝐴𝑡𝐿𝑡)1−𝛼 , 𝑤ℎ𝑒𝑟𝑒 0 < 𝛼 < 1 (3.1.2)

I assume that each country’s capital and labour markets are perfectly competitive, such that interest

rates and wages are paid according to their marginal products.

𝑟𝑡 = 𝛼(𝑘𝑡)𝛼−1, 𝑤ℎ𝑒𝑟𝑒 𝑘𝑡 =𝐾𝑡

𝐴𝑡𝐿𝑡 (3.1.3)

Furthermore, I decompose labour into skilled and unskilled labour to establish two different wage

rates.

𝐿𝑡 = 𝐿𝑢𝑡 + 𝐿𝑠𝑡 (3.1.4)

I differentiate between the two by allowing total factor productivity to be independent of unskilled

labour and positively related to skilled labour. I do this by assuming that total factor productivity can

be expressed as a function of skilled labour. For simplicity I express this function as

𝐴𝑡 = (𝐿𝑠𝑡)𝜂 , 𝑤ℎ𝑒𝑟𝑒 0 < 𝜂 < 1 (3.1.5)

Hence wages for unskilled and skilled labour will be given by

𝑤𝑢𝑡 =𝛿𝑌𝑡

𝛿𝐿𝑢𝑡= (1 − 𝛼)(𝑘𝑡)𝛼(𝐿𝑠𝑡)𝜂 (3.1.6)

𝑤𝑠𝑡 =𝛿𝑌𝑡

𝛿𝐿𝑠𝑡= (1 − 𝛼)(𝑘𝑡)𝛼(𝜂(𝐿𝑠𝑡)𝜂−1𝐿𝑢𝑡 + (𝜂 + 1)(𝐿𝑠𝑡)𝜂) (3.1.7)

where skilled workers have a strictly larger wage than unskilled workers.

6 | P a g e

3.2 Equilibrium

The total combined capital accumulation between the two countries is given by

𝐾𝑡+1𝑖 + 𝐾𝑡+1

𝑗= 𝐿𝑢𝑡

𝑖 𝑤𝑢𝑡𝑖 + 𝐿𝑠𝑡

𝑖𝑤𝑠𝑡𝑖 + 𝐿𝑢𝑡

𝑗𝑤𝑢𝑡

𝑗+ 𝐿𝑠𝑡

𝑗𝑤𝑠𝑡

𝑗 (3.2.1)

Firstly, if interest rates are the same between the two countries, there is no incentive for individuals

to invest their income in a different country and therefore each countries capital is accumulated

domestically. However, if the interest rate is different between the two countries, young individuals

will prefer to invest in the higher interest rate country. This persists until interest rates are equalized

between the two countries at which point we revert back to the previous result. Therefore, each

countries capital stock will be determined jointly by the above equation, as well as

𝑘𝑡𝑖 = 𝑘𝑡

𝑗 (3.2.2)

Note that this result relies upon the assumption of perfect capital mobility, which we assume

throughout.

A country’s labour force grows at a rate of n, and can differ between countries. I assume that both

skilled labour and unskilled labour grow at the same rate, such that the composition of labour

remains constant when labour is immobile. The equilibrium will therefore be defined such that

equations (3.2.1) and (3.2.2) hold and capital per effective worker is constant. Since interest rates

are always are equal, the long run wages will determine which country’s inhabitants can achieve a

higher utility.

3.3 No Migration Case

I will firstly look at the case when labour is immobile, and determine the long run equilibrium. When

there is no labour migration, and labour grows at the same rate irrespective of quality, we can

express unskilled labour as a function of skilled labour. By defining

𝐿𝑠𝑡 = 𝛾𝐿𝑡 (3.3.1)

𝐿𝑢𝑡 =1−𝛾

𝛾𝐿𝑠𝑡 (3.3.2)

7 | P a g e

we can see that the wage for a skilled worker is

𝑤𝑠𝑡 = (1 − 𝛼)(𝑘𝑡)𝛼(𝜂(1−𝛾)

𝛾+ 𝜂 + 1)(𝐿𝑠𝑡)𝜂 (3.3.3)

With the assumption of perfect capital mobility and identical production functions, we see that the

long run outcomes depend upon the composition and the population growth rates. If we take the

case where both countries have the same labour composition, then we can see that individuals from

the country with the higher skilled labour force will have a higher wage. Therefore the country with

a higher population growth will be better off regardless of their initial population in the long run.

If we then take the case where both countries have the same population growth, then the country

with the higher initial labour force will be better off as they will have a higher wage in every period.

3.4 Migration Case

I will now extend the analysis to allow for labour migration. As mentioned previously, young

individuals are able to migrate to another country and earn a wage that country. I assume that the

incentive to migrate is purely driven by wage differentials, however, unlike capital, labour is not

perfectly mobile.

The migration parameters are characterised by the following function, theta.

𝜃𝑢𝑡𝑖𝑗

= {𝜃𝑢𝑖𝑗 , 𝑖𝑓 𝑤𝑢𝑡

𝑖 < 𝑤𝑢𝑡𝑗

0, 𝑖𝑓 𝑤𝑢𝑡𝑖 ≥ 𝑤𝑢𝑡

𝑗 (3.4.1)

𝜃𝑢𝑡𝑗𝑖

= {𝜃𝑢𝑗𝑖 , 𝑖𝑓 𝑤𝑢𝑡

𝑗< 𝑤𝑢𝑡

𝑖

0 , 𝑖𝑓 𝑤𝑢𝑡𝑗

≥ 𝑤𝑢𝑡𝑖

(3.4.2)

𝜃𝑠𝑡𝑖𝑗

= {𝜃𝑠𝑖𝑗, 𝑖𝑓 𝑤𝑠𝑡

𝑖 < 𝑤𝑠𝑡𝑗

0, 𝑖𝑓 𝑤𝑠𝑡𝑖 ≥ 𝑤𝑠𝑡

𝑗 (3.4.3)

𝜃𝑠𝑡𝑗𝑖

= {𝜃𝑠𝑗𝑖, 𝑖𝑓 𝑤𝑠𝑡

𝑗< 𝑤𝑠𝑡

𝑖

0, 𝑖𝑓 𝑤𝑠𝑡𝑗

≥ 𝑤𝑠𝑡𝑖

(3.4.4)

8 | P a g e

Where 𝜃𝑢𝑖𝑗 , 𝜃𝑢𝑗𝑖 , 𝜃𝑠𝑖𝑗 𝑎𝑛𝑑 𝜃𝑠𝑗𝑖 ∈ (0,1)

For now, I will assume for simplicity that thetas are the same irrespective of skill and direction. This

allows me to express the labour dynamics of each country as follows

𝐿𝑢𝑡+1𝑖 = (1 + 𝑛𝑖)[(1 − 𝜃)𝐿𝑢𝑡

𝑖 + 𝜃𝐿𝑢𝑡𝑗] (3.4.5)

𝐿𝑢𝑡+1𝑗

= (1 + 𝑛𝑗)[(1 − 𝜃)𝐿𝑢𝑡𝑗

+ 𝜃𝐿𝑢𝑡𝑖 ] (3.4.6)

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)[(1 − 𝜃)𝐿𝑠𝑡

𝑖 + 𝜃𝐿𝑠𝑡𝑗] (3.4.7)

𝐿𝑠𝑡+1𝑗

= (1 + 𝑛𝑗)[(1 − 𝜃)𝐿𝑠𝑡𝑗

+ 𝜃𝐿𝑠𝑡𝑖] (3.4.8)

First I will consider the case when both countries have the same labour composition. If both

countries start with the same initial population, the long run outcome will again depend on the

population growth. If one country’s population grows at faster rate, we know that in 2nd period, they

will have a larger skilled labour force, which will create a wage gap that attracts workers to that

country. This trend will continue in the long run, thus making individuals in that country better off in

every period.

However, if they do not have the same initial population, then the long run equilibrium will depend

on the population growth rates of each country and the migration parameters.

The proof for this statement is structured as follows.

Without loss of generality, assume that Country i has a larger initial labour force such that migration

will flow from Country j to Country i. We can than express the labour dynamics of each country as

follows

𝐿𝑢𝑡+1𝑖 = (1 + 𝑛𝑖)[𝐿𝑢𝑡

𝑖 + 𝜃𝐿𝑢𝑡𝑗] (3.4.9)

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)[𝐿𝑠𝑡

𝑖 + 𝜃𝐿𝑠𝑡𝑗] (3.4.10)

𝐿𝑢𝑡+1𝑗

= (1 + 𝑛𝑗)(1 − 𝜃)𝐿𝑢𝑡𝑗 (3.4.11)

9 | P a g e

𝐿𝑠𝑡+1𝑗

= (1 + 𝑛𝑗)(1 − 𝜃)𝐿𝑠𝑡𝑗 (3.4.12)

Migration will continue to persist if Country i’s wage is always higher than Country j’s wage, which is

equivalent to always having a larger skilled labour force. Therefore it is sufficient to look at the

difference in skilled labour between the two countries and determine whether there is convergence.

I find that if the following equation holds, then Country j’s wage will converge to Country i’s wage.

(1 + 𝑛𝑖)(1 + 𝜃) ≤ (1 + 𝑛𝑗)(1 − 𝜃) (3.4.13)

The right hand side of this equation describes the growth rate of skilled labour in Country j, whilst

the left hand side is the long run growth rate of skilled labour in Country i. If the equation holds with

equality, there is an equilibrium where wages will converge in the long run. If the right hand side is

strictly larger than the left hand side, then convergence is achieved after a finite period of time.

From then on, the migration flows in the other direction and wages diverge. If the left hand side is

strictly larger than the right hand side, then wages will not converge, and migration will persist in the

long run.

4. Numerical Simulation

I run several numerical simulations as a robustness check for my theoretical findings, as well as

analyse situations where analytical methods were not used. I conduct these simulations using

different initial values and parameters. For simplicity, I will fix population growth in Country i to be

zero throughout. Furthermore, to ensure consistency, both countries will be endowed with the same

amount of initial capital and Country i will have the same initial labour, both in terms of size and

composition. I then analyse the wages for skilled and unskilled workers and determine the long run

outcomes that are reached.

10 | P a g e

5. Results

First I will look at the case where there is no migration and Country j has a lower initial population

and higher population growth rate. As we can see from figure 1, the result is consistent with the

theoretical results and wages diverge. Although young individuals in Country i born in the first two

periods have a relatively higher wage and therefore utility, in all subsequent periods they are

relatively worse off. I then replicate this analysis to include migration and setting Country j’s

population growth rate to satisfy equation (3.4.13). By including migration into the model I find that

wages converge in the long run as shown in figure 2. Therefore in this scenario, migration is

beneficial to Country i and detrimental to Country j.

Next I will relax the assumption that both countries have the same labour composition. First I will let

Country j have a lower skilled labour composition than Country i. Similar to the previous case, we

can see from figure 3 that wages will converge in the long run. Notably, the labour composition also

Figure 1

Figure 2

11 | P a g e

converges, with Country j’s composition improving with migration whilst Country i’s composition

worsens. The overall long run composition of labour in the economy is lower than in the previous

case. As such, skilled workers a slightly worse off in this scenario, whilst unskilled workers are

marginally better off. I then repeat this process with a higher initial skilled labour composition for

Country j in figure 4 and find that this results in divergence. Although there is an initial migration

flow from Country j to Country i, this flow is reversed after two periods. Subsequently, each

individual in Country j is better off as the wage gap persistently increases.

Figure 3

Figure 4

12 | P a g e

Finally, I will look at the case where only skilled labour can migrate. Using a significantly low

population growth rate for Country j, figure 5 shows that Country j’s wages will eventually diverge

away from Country i’s wage, despite being initially lower. This analysis emphasises the ‘brain drain’

effect, as we can see the significance of skilled labour migration on wage differentials. Compared to

when we allowed for unskilled labour migration, convergence could only be achieved for a

significantly high population growth for Country j. However, when we restrict migration to only be

possible for skilled labour, the same qualitative outcome can be achieved with a lower population

growth rate.

6. Limitations/Extensions

My theoretical models main limitation is its ability to be tested empirically. In particular, it would be

difficult to get values that can capture total factor productivity and the migration parameters. The

model’s framework and the numerical simulation were also significantly simplified by assigning the

same values to each country for many parameters. Additionally, by assuming that the migration

parameters are constant through time and did not vary in direction likely made the model’s results

less realistic.

In this sense the model can be improved by specifying a more precise function for migration

movements. One possible extension could be to lower the migration parameter when the wage gaps

are closer. Currently, my model is specified such that even small wage differences induce the same

Figure 5

13 | P a g e

magnitude of migration. Another way to improve the model is to allow migration to differ across

countries. As there are various other factors which influence migration patterns, it is possible that

these can be added to the model to allow migration to occur even with wage equivalence. Although

the model can be made more accurate by allowing migration to vary between countries and time,

this makes it significantly more complex.

7. Conclusion

The main results are that migration can result in many different outcomes depending on the

population growth rate and the incentives to migrate. From the theoretical model, we can see that

when migration is included in the model, convergence can be achieved even if both countries have

different population growth rates. Therefore, the country with low population growth can be better

off through migration inflows.

However, as discussed in the theoretical findings as well as the numerical simulations, migration may

also be temporary. When migration flows change their direction, the result is that the wage gap will

persistently increase, leaving the poorer country behind. This was shown to be the case when one

country has a significantly higher skilled labour composition. Furthermore, when migration is only

possible for skilled workers, the long run result is the same even for relatively small differences in

population growth.

Therefore, migration can be beneficial in some situations, allowing countries with low population

growth rate to achieve a wage rate similar to a high population growth country. However, migration

can also lead to divergence, especially when the ‘brain drain’ effect causes skilled workers to come

together in a single country, leaving the other country behind.

14 | P a g e

8. References

Bhagwati, J., & Rodriguez, C. (1975). Welfare-theoretical analyses of the brain drain. Journal of

development Economics, 2(3), 195-221.

Borjas, G. J. (1994). The economics of immigration. Journal of economic literature, 1667-1717.

Diamond, P. A. (1965). National debt in a neoclassical growth model. The American Economic

Review, 1126-1150.

Dowrick, S., & Rogers, M. (2002). Classical and technological convergence: beyond the Solow-Swan

growth model. Oxford Economic Papers, 369-385.

Greenwood, M. J., & McDowell, J. M. (1986). The factor market consequences of US immigration.

Journal of Economic Literature, 1738-1772.

Lucas, R. E. (1990). Why doesn't capital flow from rich to poor countries?. The American Economic

Review, 92-96.

Mankiw, N. G., Romer, D., & Weil, D. N. (1990). A contribution to the empirics of economic growth

(No. w3541). National Bureau of Economic Research.

Moody, C. (2006). Migration and economic growth: a 21 st century perspective. Migration, 6, 02.

Okun, B., & Richardson, R. W. (1961). Regional income inequality and internal population migration.

Economic Development and Cultural Change, 9(2), 128-143.

Reichlin, P., & Rustichini, A. (1998). Diverging patterns with endogenous labor migration. Journal of

Economic Dynamics and Control, 22(5), 703-728.

Solow, R. M. (1956). A contribution to the theory of economic growth. The quarterly journal of economics, 65-94. Swan, T. (1956). Economic growth and capital accumulation.

15 | P a g e

9. Appendices Proof of equation (3.4.13)

Given

𝐿𝑠𝑡+1𝑗

= (1 + 𝑛𝑗)(1 − 𝜃)𝐿𝑠𝑡

We can write

𝐿𝑠𝑡𝑗

= (1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡𝐿𝑠0

𝑗

Given

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)[𝐿𝑠𝑡

𝑖 + 𝜃𝐿𝑠𝑡𝑗]

We can write

𝐿𝑠𝑡𝑖 = (1 + 𝑛𝑖)

𝑡(𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗) + 𝜃𝐿𝑠0

𝑗∑(1 + 𝑛𝑖)

𝑘(1 + 𝑛𝑗)

𝑡−𝑘(1 − 𝜃)𝑡−𝑘

𝑡−1

𝑘=1

𝐿𝑠𝑡𝑖 = (1 + 𝑛𝑖)

𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

Proof by Induction

For any t,

𝐿𝑠𝑡𝑖 = (1 + 𝑛𝑖)

𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

For t=1

𝐿𝑠1𝑖 = (1 + 𝑛𝑖)

1𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

1

𝑘=1

𝐿𝑢1𝑖 = (1 + 𝑛𝑖)(𝐿𝑢0

𝑖 + 𝜃𝐿𝑢0𝑗)

Assume for any t,

𝐿𝑠𝑡𝑖 = (1 + 𝑛𝑖)

𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

For t+1,

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)[𝐿𝑠𝑡

𝑖 + 𝜃𝐿𝑠𝑡𝑗]

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖) [(1 + 𝑛𝑖)

𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

+ 𝜃(1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡𝐿𝑠0

𝑗]

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)

𝑡+1𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗(∑(1 + 𝑛𝑖)

𝑘+1(1 + 𝑛𝑗)

𝑡−𝑘(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

+ (1

+ 𝑛𝑖)(1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡)

𝐿𝑠𝑡+1𝑖 = (1 + 𝑛𝑖)

𝑡+1𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡+1

𝑘=1

16 | P a g e

𝐿𝑠𝑡𝑖 − 𝐿𝑠𝑡

𝑗

= [(1 + 𝑛𝑖)𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑(1 + 𝑛𝑖)𝑘

(1 + 𝑛𝑗)𝑡−𝑘

(1 − 𝜃)𝑡−𝑘

𝑡

𝑘=1

] − (1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡𝐿𝑠0

𝑗

= (1 + 𝑛𝑖)𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘

((1 + 𝑛𝑗)(1 − 𝜃))𝑡

𝑡

𝑘=1

− (1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡𝐿𝑠0

𝑗

= (1 + 𝑛𝑖)𝑡𝐿𝑠0

𝑖 + 𝜃𝐿𝑠0𝑗

((1 + 𝑛𝑗)(1 − 𝜃))𝑡

∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘𝑡

𝑘=1

− (1 + 𝑛𝑗)𝑡(1 − 𝜃)𝑡𝐿𝑠0

𝑗

= (1 + 𝑛𝑖)𝑡𝐿𝑠0

𝑖 + 𝐿𝑠0𝑗

((1 + 𝑛𝑗)(1 − 𝜃))𝑡

(𝜃 ∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘

− 1)

𝑡

𝑘=1

For the above distance to converge to zero,

𝜃 ∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘

− 1

𝑡

𝑘=1

< 0

𝜃 ∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘𝑡

𝑘=1

< 1

𝑀𝑎𝑥 ∑ ((1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘𝑡

𝑘=1

𝑤ℎ𝑒𝑛 𝑡 → ∞

∑ (1 + 𝑛𝑖

(1 + 𝑛𝑗)(1 − 𝜃))

𝑘

𝑡

𝑘=1

→1

1 − (1 + 𝑛𝑖

(1 + 𝑛𝑗)(1 − 𝜃))

− 1

→(1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃) − (1 + 𝑛𝑖)

𝜃(1 + 𝑛𝑖)

(1 + 𝑛𝑗)(1 − 𝜃) − (1 + 𝑛𝑖)< 1

1 + 𝜃

1 − 𝜃≤

1 + 𝑛𝑗

1 + 𝑛𝑖

If this equation holds, wages will converge

17 | P a g e

Figure 1

𝑛𝑖 𝑛𝑗 𝛼 𝜃 𝜂

0 0.5 0.3 0 0.5 Country i

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 7 3.00 1.73 11.93 0.29 0.84 2.23 0.3

1 11.26 7 3.00 1.73 15.22 0.65 1.07 2.84 0.3

2 12.02 7 3.00 1.73 15.52 0.69 1.09 2.90 0.3

3 11.20 7 3.00 1.73 15.20 0.65 1.06 2.84 0.3

4 10.13 7 3.00 1.73 14.75 0.58 1.03 2.75 0.3

5 9.27 7 3.00 1.73 14.36 0.54 1.01 2.68 0.3

6 8.68 7 3.00 1.73 14.08 0.50 0.99 2.63 0.3

7 8.30 7 3.00 1.73 13.89 0.48 0.97 2.59 0.3

8 8.08 7 3.00 1.73 13.78 0.47 0.96 2.57 0.3

9 7.95 7 3.00 1.73 13.71 0.46 0.96 2.56 0.3

10 7.88 7 3.00 1.73 13.68 0.45 0.96 2.55 0.3

Country j

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 3.50 1.50 1.22 5.76 0.82 0.81 2.15 0.3

1 7.32 5.25 2.25 1.50 9.89 0.65 0.92 2.46 0.3

2 14.34 7.88 3.38 1.84 18.52 0.69 1.15 3.07 0.3

3 24.55 11.81 5.06 2.25 33.31 0.65 1.38 3.68 0.3

4 40.80 17.72 7.59 2.76 59.39 0.58 1.64 4.38 0.3

5 68.57 26.58 11.39 3.38 106.23 0.54 1.96 5.22 0.3

6 117.94 39.87 17.09 4.13 191.33 0.50 2.35 6.27 0.3

7 207.37 59.80 25.63 5.06 346.90 0.48 2.84 7.58 0.3

8 370.75 89.70 38.44 6.20 632.12 0.47 3.45 9.21 0.3

9 670.25 134.55 57.67 7.59 1155.70 0.46 4.21 11.22 0.3

10 1220.00 201.83 86.50 9.30 2117.27 0.45 5.14 13.71 0.3

Figure 2

𝑛𝑖 𝑛𝑗 𝛼 𝜃 𝜂

0 0.5 0.3 0.2 0.5 Country i

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 7.00 3.00 1.73 11.93 0.29 0.84 2.23 0.3

1 13.24 7.70 3.30 1.82 17.66 0.66 1.12 3.00 0.3

2 17.90 8.54 3.66 1.91 21.55 0.77 1.24 3.30 0.3

3 21.87 9.55 4.09 2.02 25.73 0.79 1.32 3.52 0.3

4 26.15 10.76 4.61 2.15 30.77 0.79 1.40 3.74 0.3

5 31.34 12.21 5.23 2.29 37.11 0.79 1.49 3.97 0.3

6 37.88 13.95 5.98 2.45 45.18 0.78 1.59 4.23 0.3

7 46.24 16.04 6.87 2.62 55.54 0.77 1.70 4.52 0.3

8 56.99 18.55 7.95 2.82 68.89 0.76 1.82 4.85 0.3

9 70.86 21.56 9.24 3.04 86.11 0.76 1.96 5.22 0.3

10 88.80 25.17 10.79 3.28 108.42 0.75 2.11 5.63 0.3

Country j

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾

18 | P a g e

0 5.00 3.50 1.50 1.22 5.76 0.82 0.81 2.15 0.3

1 5.33 4.20 1.80 1.34 7.12 0.66 0.83 2.21 0.3

2 8.12 5.04 2.16 1.47 9.77 0.77 0.95 2.53 0.3

3 11.02 6.05 2.59 1.61 12.97 0.79 1.05 2.80 0.3

4 14.49 7.26 3.11 1.76 17.05 0.79 1.15 3.07 0.3

5 18.88 8.71 3.73 1.93 22.36 0.79 1.26 3.35 0.3

6 24.56 10.45 4.48 2.12 29.30 0.78 1.37 3.66 0.3

7 31.97 12.54 5.37 2.32 38.40 0.77 1.50 4.00 0.3

8 41.65 15.05 6.45 2.54 50.34 0.76 1.64 4.37 0.3

9 54.33 18.06 7.74 2.78 66.02 0.76 1.79 4.78 0.3

10 70.94 21.67 9.29 3.05 86.61 0.75 1.96 5.22 0.3

Figure 3

𝑛𝑖 𝑛𝑗 𝛼 𝜃 𝜂

0 0.5 0.3 0.2 0.5 Country i

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 10.00 7.00 3.00 1.73 11.93 0.29 0.84 0.30

1 10.62 10.30 7.90 2.40 1.55 14.12 0.67 0.96 0.23

2 14.39 11.71 8.98 2.73 1.65 17.70 0.74 1.06 0.23

3 18.08 13.40 10.28 3.13 1.77 21.85 0.76 1.14 0.23

4 22.38 15.43 11.83 3.60 1.90 27.01 0.76 1.23 0.23

5 27.74 17.87 13.70 4.17 2.04 33.61 0.76 1.32 0.23

6 34.60 20.79 15.94 4.86 2.20 42.12 0.76 1.42 0.23

7 43.46 24.30 18.62 5.68 2.38 53.12 0.75 1.53 0.23

8 54.94 28.51 21.85 6.66 2.58 67.41 0.75 1.66 0.23

9 69.86 33.56 25.72 7.84 2.80 86.00 0.74 1.79 0.23

10 89.28 39.63 30.36 9.26 3.04 110.20 0.74 1.95 0.23

Country j

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 4.50 0.50 0.71 3.92 1.41 0.55 3.30 0.10

1 6.03 5.40 1.65 1.28 8.01 0.67 0.80 2.50 0.23

2 8.85 6.48 1.98 1.41 10.89 0.74 0.90 2.83 0.23

3 11.94 7.78 2.38 1.54 14.43 0.76 0.99 3.12 0.23

4 15.72 9.33 2.85 1.69 18.98 0.76 1.09 3.42 0.23

5 20.55 11.20 3.42 1.85 24.90 0.76 1.19 3.74 0.23

6 26.84 13.44 4.11 2.03 32.67 0.76 1.30 4.09 0.23

7 35.07 16.12 4.93 2.22 42.87 0.75 1.43 4.47 0.23

8 45.86 19.35 5.91 2.43 56.27 0.75 1.56 4.89 0.23

9 60.00 23.22 7.09 2.66 73.86 0.74 1.71 5.35 0.23

10 78.57 27.86 8.51 2.92 96.98 0.74 1.87 5.85 0.23

Figure 4

𝑛𝑖 𝑛𝑗 𝛼 𝜃 𝜂

0 0.5 0.3 0.2 0.5 Country i

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 7.00 3.00 1.73 11.93 0.29 0.84 2.23 0.30

1 10.40 5.60 3.60 1.90 14.95 0.60 1.14 2.59 0.39

2 13.84 5.60 4.32 2.08 18.29 0.67 1.29 2.77 0.44

19 | P a g e

3 10.12 5.60 3.46 1.86 14.45 0.60 1.12 2.58 0.38

4 6.14 4.48 2.76 1.66 9.84 0.51 0.95 2.20 0.38

5 4.01 3.58 2.21 1.49 6.85 0.47 0.83 1.91 0.38

6 2.77 2.87 1.77 1.33 4.85 0.45 0.73 1.69 0.38

7 1.97 2.29 1.42 1.19 3.46 0.45 0.65 1.51 0.38

8 1.41 1.84 1.13 1.06 2.48 0.45 0.58 1.35 0.38

9 1.01 1.47 0.91 0.95 1.77 0.45 0.52 1.21 0.38

10 0.72 1.17 0.72 0.85 1.27 0.45 0.47 1.08 0.38

Country j

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 5.00 2.00 3.00 1.73 7.34 0.58 1.03 0.60

1 9.84 8.70 5.10 3.60 1.90 14.13 0.60 1.14 0.41

2 16.70 11.97 7.65 4.32 2.08 22.07 0.67 1.29 0.36

3 32.27 19.25 11.48 7.78 2.79 46.08 0.60 1.68 0.40

4 57.41 31.59 18.89 12.70 3.56 91.99 0.51 2.04 0.40

5 102.91 49.56 29.68 19.88 4.46 175.71 0.47 2.48 0.40

6 188.92 76.08 45.60 30.48 5.52 330.54 0.45 3.04 0.40

7 350.19 115.52 69.26 46.26 6.80 616.53 0.45 3.74 0.40

8 649.59 174.39 104.58 69.81 8.36 1143.47 0.45 4.59 0.40

9 1202.24 262.47 157.42 105.06 10.25 2112.78 0.45 5.63 0.40

10 2219.55 394.42 236.56 157.86 12.56 3894.42 0.45 6.91 0.40

Figure 5

𝑛𝑖 𝑛𝑗 𝛼 𝜃𝑠 𝜂

0 0.5 0.3 0.2 0.5 Country i

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 7.00 3.00 1.73 11.93 0.29 0.84 2.23 0.30

1 14.10 7.00 3.30 1.82 17.19 0.75 1.17 2.99 0.32

2 18.41 7.00 3.56 1.89 19.47 0.92 1.29 3.20 0.34

3 20.74 7.00 3.80 1.95 20.95 0.99 1.36 3.29 0.35

4 22.23 7.00 4.00 2.00 22.07 1.01 1.40 3.34 0.36

5 23.35 7.00 4.18 2.04 23.01 1.02 1.44 3.37 0.37

6 24.28 7.00 4.34 2.08 23.82 1.03 1.47 3.39 0.38

7 19.76 7.00 3.47 1.86 19.58 1.01 1.31 3.28 0.33

8 21.22 7.00 3.82 1.95 21.15 1.00 1.37 3.31 0.35

9 17.20 7.00 3.05 1.75 17.45 0.98 1.22 3.22 0.30

10 14.05 7.00 2.44 1.56 14.54 0.95 1.08 3.16 0.26

Country j

𝑡 𝐾 𝐿𝑢 𝐿𝑠 𝐴 𝑌 𝑘 𝑤𝑢 𝑤𝑠 𝛾 0 5.00 3.50 1.50 1.22 5.76 0.82 0.81 2.15 0.30

1 4.48 3.85 1.32 1.15 5.46 0.75 0.74 2.19 0.26

2 5.37 4.24 1.16 1.08 5.68 0.92 0.74 2.45 0.22

3 5.66 4.66 1.02 1.01 5.72 0.99 0.70 2.66 0.18

4 5.77 5.12 0.90 0.95 5.73 1.01 0.67 2.90 0.15

5 5.84 5.64 0.79 0.89 5.76 1.02 0.63 3.17 0.12

6 5.92 6.20 0.70 0.83 5.80 1.03 0.59 3.51 0.10

7 11.35 6.82 1.72 1.31 11.25 1.01 0.92 3.21 0.20

8 11.15 7.50 1.51 1.23 11.11 1.00 0.86 3.43 0.17

9 16.68 8.25 2.51 1.58 16.92 0.98 1.10 3.47 0.23

20 | P a g e

10 22.04 9.08 3.43 1.85 22.81 0.95 1.28 3.61 0.27