留學與移民決策內生化：政策分析

國立臺灣大學社會科學院經濟學系 碩士論文

Department of Economics College of Social Sciences National Taiwan University

Master Thesis

留學與移民決策內生化 ^: 政策分析

Endogenous Determination of the Location of Human Capital Accumulation and Migration: A Policy Analysis

Advisor: Hung-Ju Chen Ph.D 指導教授 ^: 陳虹如 博士

Chung-Han Yang 楊宗翰

July 2012

中華民國 ¹⁰¹ 年 ⁷ 月

誌謝

本篇論文完成首先要感謝陳虹如老師的指導。 老師讓我自由地發揮我的想法, 不予以限 制, 並且適時的給予建議, 才讓本篇論文得以完成。 同時也要感謝張俊仁與陳明郎兩位 口試委員所給予寶貴又實用的建議跟方向, 讓我能夠更進一步思考這篇論文未來的可能 性。 另外要感謝台大經濟博士班的李文基學長對於論文給予的意見, 以及在我大學時期 課業的指導與討論。

台大經濟系如此卓越的研究環境, 實在超乎我原有想像的界限。 幽靜的環境讓我更專 心; 豐富的藏書與資料庫讓我任意遨遊浩瀚文獻; 慘烈的食物徹底了滅絕我對口腹之慾 的追尋, 讓我可以更認真; 社科院的超高速網路更是粉碎了使用 Youtube、 觀看新聞或 是收聽網路廣播打混的任何機會; 每天計算使得熱量極大 (給定放棄追求任何風味) 的 三餐組合, 使我在計算最適化問題時更得心應手。

要感謝217台大統計與 TEX研究中心的各位夥伴們。 感謝曾中信時常提供排版上的 幫助; 與許誠哲、 李國恆和劉彥汝的討論常使我獲益良多。 私底下在研究室一起婊人更 提供了許多樂趣。

最後要感謝阿婕一直以來的幫助、 支持和照顧。

Abstract

I developed a model that endogenizes decisions regarding the location of human capital accumulation and migration, and I identified the factors that affect peo- ple’s decisions on locations of working and human capital accumulation. Based on the welfare comparison, the model shows that government of a developing country subsidizing wages for citizens who accumulate their human capital in a developed country is better than subsidizing over the cost of accumulating their human capital in a developed country.

摘要

本文發展了一個可將人力資本累積地點與移民決策內生化的模型, 同時也討論了會影響 人力資本累積地點與移民決策的因素。 且基於福利分析, 此模型指出開發中國家對國外 累積人力資本者回國工資予以補貼會比補貼於已開發國家中求學之成本較佳。

Contents

1 Introduction 1

2 The model 3

2.1 Human capital accumulation . . . 3

2.2 Decisions on working location . . . 4

2.3 Individual budget constraint . . . 5

2.4 Preference . . . 6

2.5 Bellman’s equation . . . 6

2.6 First-period decisions . . . 7

2.7 Functional form . . . 7

3 Equilibrium 9 3.1 Second-period decisions . . . 10

3.2 First-period decisions . . . 13

4 Welfare 19 5 Policy instruments 20 5.1 Government’s budget constraint . . . 20

5.2 Individual’s budget constraint . . . 21

6 Equilibrium under the policy 23 6.1 Second-period decisions under the policies . . . 23

6.2 First-period decisions under the policies . . . 25

7 Policy analysis 28 7.1 Subsidizing the cost of studying in a foreign country . . . 28

7.2 Subsidizing the home country income for an agent studying in a foreign country 29 7.3 Policy comparison . . . 30

8 Conclusions 33

Chapter 1 Introduction

Human capital accumulation and migration have long interested economists. Previous studies on the wage differential between countries as a key determinant for migration, include Sjaastad (1962), Harris and Todaro (1970), Davies and Wooton (1992) and Engerman and Jones (1997).

These studies have shown that migration can cause exodus of trained workers (brain drain) from developing countries ¹. However, recent studies of Stark and Wang (2002), Beine et al.

(2001) and Mountford (1997), treat migration as an instrument which induces human capital accumulation. There is a strong consensus that the location of human capital accumulation and migration are related, but brain drain literatures have neglected the connection between them.

Baruch et al. (2007), Scheffel (1990) and King and Ruiz-Gelices (2003) analyzed the fac- tors that affect foreign students’ decision of returning to the home country or not, but these studies can not provide a structure to analyze the welfare for a policy implementation. In this paper, I consider human capital accumulation from a different point of view. When people are accumulating their human capital in their early life in a developing country, they are provided with two choices. They can choose to accumulate their human capital in the home country, or they can choose to accumulate their human capital in a foreign country (developed country).

In other words, they do not choose the level of human capital, they merely choose the location to accumulate their human capital. Persons also choose whether to migrate or not. Human capital of individuals and migration are decisions made by individuals, and these two decisions affect a country’s long-run economic performance. Many developing countries encourage their nationals to study in developed countries by subsidizing the cost. For example, the government of Taiwan has implemented a policy like this ². The main reason why people prefer studying in developed countries is that developed countries can provide a better quality of education.

A better quality of education helps people to accumulate more human capital. Higher human capital induces higher wage, which would improve welfare.

When people finish accumulating their human capital, they must decide whether to work

1See Bhagwati and Wilson (2008) and Giannoccolo (2009), and Commander et al. (2004) on this topic.

2The detail of the policy can be seen at the website of Bureau of International Cultural and Educational Relations of the R.O.C

in the home country or to work in the foreign country. People with more human capital are more apt to work in the foreign country. A developed country usually provides higher wages, so people in the developing country prefer to work in the developed country. As a result, people will study and work in the developed country. However, we know that different people make different decisions regarding the locations of their study and work.

In this study, I develop a simple model to study how different kinds of people make those decisions. I argue that the decisions they make are influenced by their innate ability. I show how government policies can affect their decisions and welfare under this simple structure.

Chapter 2 The model

There are two countries, a developing country and a developed country. I refer the developing country as the home country and the developed country as the foreign country. Individuals accumulate their human capital in the early stage. I refer to the accumulation of human capital as studying.

The approach used is to introduce heterogeneous agents in the model to enable all possible outcomes. Let q be agent’s quality endowment. q can be interpreted as IQ, potential or diligence. q is randomly distributed between agents with the following distribution: q ∈ [q, q], where Ψ(q) is cumulative distribution of q. And Ψ satisfiesq

q dΨ(q) = 1.

We have an overlapping generation structure in the model. An agent lives for two periods.

In the first period, he has no income, so he must finance his consumption by borrowing. The agent also decides whether to study in the foreign country or not. In the second period, after he finish studying, he has to decide whether to work in the home country or to work in the foreign country. Besides, The agent must repay the debt that he owes in the first period by using his second-period income.

2.1 Human capital accumulation

The location of human capital accumulation is not the only factor that affects the level of human capital. The difference of efficiency on human capital accumulation between the foreign country and the home country is also crucial. We assume that the home country and the foreign country have different human capital accumulation function. Let h = g(q) be the human capital accumulation function of the home country and let h = f (q) be the human capital accumulation function of the foreign country.

For simplification, capital does not enter the human capital accumulation function. This specification assumes that agents devote all their time to studying, and the only factor allowing agents sharing the same human capital accumulation technology to have different stocks of human capital is their innate ability. f (q), g(q) satisfy f (q) > g(q), ∀q and, g^(q) > 0, g^(q) <

0, f^(q) > 0, f^(q) < 0. f > g, ∀q implies that the foreign country human capital accumu- lation can always produce more human capital then the home country for people who have

2.2 Decisions on working location

the same quality. g^(q) > 0, g^(q) < 0, f^(q) > 0, f^(q) < 0 states that the higher quality the agent endowed, the more human capital he can accumulate, and the marginal human capital accumulated is decreasing in quality.

2.2 Decisions on working location

Agents studying in the foreign country

In the second period, the agent who accumulates human capital h, decides whether to go back working in the home country or to emigrate to the foreign country. The income structure is different in two countries. The income functions in the home country and the foreign country are

Income function of the foreign country: y_f^f = y^f(h) Income function of the home country: y_d^d= y^d(h) + IS.

The subscript f indicates agent who studying in the foreign country in the first period, the superscript shows where the agent works in the second period, d for going back to the home country and f for emigrating to the foreign country. IS is a premium, I assume that home country employer would pay a premium IS to hire an agent who has accumulated his human capital in the foreign country. ¹ The income an agent receives has two parts. The first part is the basic wage, y^f or y^d, this part of the wage is the wage evaluated purely by the working performance, and the second part is the premium. The wage functions satisfying y^f(h) > y^d(h),∀h, and, y^f(h) > 0, y^f(h) < 0, y^d(h) > 0, y^d(h) < 0. y^f(h) > y^d(h),∀h indicate that for agents who possess the same amount of human capital, the foreign country always offers a higher wage. y^f(h) > 0, y^f(h) < 0, y^d(h) > 0, y^d(h) < 0 state that marginal wage is positive and decreasing.

Suppose that IS does not exist, then a foreign job offer is always more attractive to agents.

Thus no one studying abroad will come back to the home country. The existence of IS reduces the attractiveness of foreign job offer, thus only the agent with relative high human capital will stay in the foreign country.

Agents studying in the home country

The decisions for agents studying in the home country in the first period are similar to agents studying in the foreign country in the first period. After they accumulate human capital h at the home country, agents have to decide whether to work in the home country or work in the

1Since agents differ, and the employer does not have the full information to all people, the rest of the paper will show that the agents with higher quality are more likely to accumulate human capital in the foreign country.

Thus studying in the foreign country in the first period can be a signal that an agent is more likely to be a better worker. This can justify the existence of premiumIS.

2.3 Individual budget constraint

foreign country. The income functions in the home country and the foreign country are:

Income function in the foreign country: y_d^f = y^f(h)− MC Income function in the home country: y_d^d= y^d(h)

where M C represent the cost for agents work in the foreign country. This cost includes the cost of emigration, wage discrimination or maladaptation of emigration. Suppose M C does not exist, since the foreign country offer higher wage rate, then everyone in the home country would prefer working in the foreign country. The existence of M C reduce the attractiveness of emigration. Only people with relative high human capital will emigrate.

2.3 Individual budget constraint

Not only the human capital accumulating functions are different between the home country and the foreign country, but the foreign technology is also much more efficient than the home country. Suppose the cost of staying in the home country and go to the foreign country are the same, no one would stay in the home country. Eventually, everyone will go studying in the foreign country. In reality, studying in the foreign country requires some fixed costs and tuition.

Let the fixed cost and tuition of studying in the foreign country be SC, agent’s first-period consumption be c₁. Thus the total expenditure for agents who study in the foreign country in the first period is c₁ + SC. The total consumption for the agents who study in the home country in the first period is c₁.

Since an agent has no income at all in the first period, the consumption becomes the debts he owes in the second period. SC can be considerably high, which can reduce the payoff of studying in the foreign country for an agent with relative low quality. Thus this cost discourages the agents with lower quality to study in the foreign country. In first period, agents finance their consumption by borrowing. Assume that the market is complete and open, and the home country is a small country. This implies that the interest rate satisfies r_t = r, where r is the international interest rate.

Studying in the home country

If an agent chooses to study in the home country, his first-period consumption is c₁. At the second period, the agent has to decide where to work at. Suppose he works at the home country, then his second-period disposable income would be y^d(h). His life-time budget constraint is:

y^d(h) = (1 + r)c₁+ c₂.

Suppose the agent chooses to emigrate, his second period disposable income would be y^f(h)− M C. His life-time budget constraint would be

y^f(h)− MC = (1 + r)c₁+ c₂

2.4 Preference

Studying in the foreign country

If an agent chooses to study in the foreign country, his first-period consumption would be c₁+ SC. In the second period, he has to decide where to work. Suppose the agent chooses to work in his home country, his second-period disposable income will be y^d(h) + IS. His life-time budget constraint is:

y^d(h) + IS = (1 + r)(c₁ + SC) + c₂.

Suppose the agent choose to work in the foreign country, his second-period disposable income is y^f(h). His life-time budget constraint is

y^f(h) = (1 + r)(c₁+ SC) + c₂

2.4 Preference

The agent’s life-time utility function is

U = u(c₁) + βu(c₂).

The instantaneous utility function satisfies u^ > 0, u^ < 0. The parameter β is a discount factor, and β ∈ (0, 1)

2.5 Bellman’s equation

the following model is solved recursively. Let D be the debt an agent owes in the first period.

In the second stage, the agent possesses h of human capital and D of debt, and he has chosen whether to study in the home country or in the foreign country. Agents decide whether to work in the home or the foreign country based on these given conditions.

Studying in the home country

Since the marginal utility of consumption is positive, the optimal decision is to consume all the income left after paying all debts. The decision problem is equivalent to maximizing remaining income. Thus, the second-period Bellman’s equation for agents who study in the home country is

V^d(h, D) = V^d(h, c₁) = max



y^f(h)− MC − (1 + r)c1, y^d(h)− (1 + r)c1



. (2.1) For given human capital h and debts D, suppose the agent decides to emigrate to the foreign country. The earned income is y^f(h)− MC − (1 + r)c₁, or if he chooses to work in the home country, the remaining income he will earn is y^d(h)− (1 + r)c₁. The agent will pick the option which brings the highest remaining income.

2.6 First-period decisions

Studying in the foreign country

Decisions made by agents who study in the foreign country are similar to those who choose to study in the home country. The Bellman’s equation for agents who study in the foreign country is:

V^f(h, D) = V^f(h, c₁+ SC)

= max



y^f(h)− (1 + r)(c1+ SC), y^d(h)− (1 + r)(c1+ SC) + IS



(2.2) For given human capital h and debts D, suppose an agent is working in the foreign country The earned income is y^f(h)− (1 + r)(c₁+ SC). If he chooses to work in the home country, the remaining income he will earn is y^d(h)− (1 + r)(c₁ + SC) + IS. The option which brings the highest remaining income will be selected.

2.6 First-period decisions

Given the agent’s quality q at stage one, agents have to decide whether to study in the home country or in the foreign country.

Combining the Bellman’s equation in equations (2.1) and (2.2), Bellman’s equation for the first period is:

V (q) = max



u(c₁) + βu(V^f(h, c₁+ SC)), u(c₁) + βu(V^d(h, c₁))



where u(c₁) + βu(V^f(h, c₁+ SC)) represents the maximized life-time utility to agents who choose to study in the foreign country and u(c₁) + βu(V^d(h, c₁)) represents the maximized life-time utility for agents who choose to study in the home country. Agents would choose the option which offers the greatest life-time utility.

Plugging the human capital production function into above Bellman’s equation, we obtain V (q) = max



u(c₁) + βu(V^f(f (q), c₁+ SC)), u(c₁) + βu(V^d(g(q), c₁))



2.7 Functional form

Here the form functions of the preference and technologies that satisfy the condition of the assumptions are specified.

Instantaneous utility function

u(c) = ln c

2.7 Functional form

Human capital production functions

Home country: g(q) = A₁q^α

Foreign country: f (q) = A₂q^α , and A₂ > A₁ Income functions

Home country: y^d(h) = A₃h^γ

Foreign country: y^f(h) = A₄h^γ , and A₄ > A₃

Here an assumption is made on the technology parameters to ensure that the model is solvable

Assumption 1. Parameters satisfy

A₃A^γ₂ − A4A^γ₁ > 0.

Chapter 3 Equilibrium

Figure 1 illustrates the decision nodes of the model.

I.V (q)

II.V^f(h, D) Study abroad

Work at home Work abroad

III.V^d(h, D) Study at home

Work at home Work abroad

Figure 3.1: Decision nodes of the model.

At each node, individuals make different decisions

I. Decide how much to consume in the first period, and whether to study in the home country or the foreign country.

II. Decide whether working in the home country or the foreign country for individuals who study abroad.

III. Decide whether working in the home country or the foreign country for individuals who study in the home country.

The model’s solving procedures are listed below

1. We begin with agents who study in the home country. First human capital h^∗(c₁) which makes an individual feels indifferent between working in the home country or the foreign

3.1 Second-period decisions

country for any given c₁. For people possessing human capital higher than h^∗(c₁), they will work in the foreign country. For people possessing human capital less than h^∗(c₁), they will work in the home country.

2. For agents who study in the foreign country, we solve for the human capital h^∗∗(c₁) which makes an individual feels indifferent between working in the home country or the foreign country for any given c₁. For people possessing human capital higher than h^∗∗(c₁), they will work in the foreign country. For people possessing human capital less than h^∗∗(c₁), they will work in the home country.

3. Use the human capital accumulation function to solve the innate ability level q^∗ and q^∗∗

that correspond to human capital level h^∗ and h^∗∗.

4. Use q^∗ and q^∗∗ to solve the first-period decisions, then q can be found. An agent with quality q will be indifferent to accumulating human capital in the home country or the foreign country.

3.1 Second-period decisions

Decision rule on node III.

The decision on node III is based on the optimization of the following Bellman’s equation V^d(h, D) = V^d(h, c₁) = max



y^f(h)− MC − (1 + r)c1, y^d(h)− (1 + r)c1}

= max{A₄h^γ− MC − (1 + r)c₁, A₃h^γ− (1 + r)c₁ .

The first term in the brackets is the utility for agents who choose to emigrate to the foreign country, and the second term in the brackets is the utility for agents who choose working in the home country. Equating these two terms, we can solve for the value of human capital which cause an agent who possess it feel indifferent between working in the home country or the foreign country. Let that value be h^∗. Then

h^∗ = [ M C A₄ − A₃]^γ1.

Note that h^∗ can be affected by M C, A₄ and A₃. With h^∗ in hand, the Bellman’s equation can be used as: which is

V^d(h, c₁) =



A₄h^γ− MC − (1 + r)c1, if h > h^∗

A₃h^γ− (1 + r)c1, if h≤ h^∗. (3.1) And we have the decision rule on node III

3.1 Second-period decisions

Lemma 1. Given an agent studying in the home county in the first period, his second-period decision rule is:

This agent will work in the foreign country if his human capital h satisfies h > h^∗ = [ M C

A₄− A3

]^1γ

This agent will work in the home country if his human capital h satisfies h < h^∗ = [ M C

A₄− A3

]^1γ

Representing h^∗as a function of M C, A₃ and A₄, h^∗(M C, A₃, A₄), then we have the following proposition

Proposition 1. The h^∗ represented by h^∗(M C, A₃, A₄) satisfies

∂h^∗

∂M C = 1

γ[ M C

A₄− A3]^1γ−1 1

A₄− A3 > 0

∂h^∗

∂A₃ = 1

γ[ M C

A₄− A₃]^γ1−1 M C

(A₄− A₃)² > 0

∂h^∗

∂A₄ =−1

γ[ M C

A₄ − A₃]^γ1−1 M C

(A₄− A₃)² < 0

The results are also listed in Table 1. Proposition 1 illustrates that a rise in the cost of adaptation M C will decrease the number of people working in the foreign country. A rise in the wage rate in foreign country A₄ will increase the number of people working in the foreign country. A rise in the wage rate in the home country A₃ will decrease the number of people working in the foreign country.

Decision rule on node II.

The decision on node II is based on the optimization of the following Bellman’s equation V^f(h, D) = V^f(h, c₁+ SC)

= max



y^f(h)− (1 + r)(c₁+ SC), y^d(h)− (1 + r)(c₁+ SC) + IS



= max



A₄h^γ− (1 + r)(c1+ SC), A₃h^γ− (1 + r)(c1+ SC) + IS



The first term in the brackets is the utility for agents who choose working in the foreign country, and the second term in the bracket is the utility for agents who choose working in the home country. Equating these two terms, we can solve for the value of human capital which cause agents who possess it feel indifferent between working in the home country or the foreign country. Let that value be h^∗∗.

3.1 Second-period decisions

h^∗∗ = [ IS A₄− A3

]^1γ

Note that h^∗∗ can be affected by IS, A₃ and A₄. With h^∗∗ in hand, we can then solve the Bellman’s equation, which is

V^f(h, c₁+ SC) =



A₄h^γ− (1 + r)(c₁+ SC), if h > h^∗∗

y^d(h)− (1 + r)(c₁+ SC) + IS, if h≤ h^∗∗. (3.2) And we have the decision rule on node II.

Lemma 2. Given an agent studying in the foreign county in the first period, his second period decision rule is:

This agent will work in the foreign country if his human capital h satisfies h > h^∗∗= [ IS

A₄− A3]^γ1.

This agent will work in the home country if his human capital h satisfies h < h^∗∗= [ IS

A₄− A₃]^γ1.

Representing h^∗∗ as a function of M C, A₃ and A₄, h^∗∗(IS, A₃, A₄), then we have the following proposition

Proposition 2. The h^∗∗ represented by h^∗∗(IS, A₃, A₄) satisfies

∂h^∗∗

∂IS = 1 γ[ IS

A₄− A3]^1γ−1 1

A₄− A3 > 0

∂h^∗∗

∂A₃ = 1 γ[ IS

A₄− A₃]^1γ−1 IS

(A₄− A₃)² > 0

∂h^∗∗

∂A₄ =−1 γ[ IS

A₄− A3

]^1γ−1 IS

(A₄− A3)² < 0

The results are also listed in Table 1. Proposition 2 shows that a rise in the premium of the home country income IS will decrease the number of people working in the foreign country. An increase in the wage rate in the foreign country A₄ will increase the number of people working in the foreign country. An increase in the wage rate in the home country A₃ will decrease the number of people working in the foreign country.

Combining the above results, as the adaptation cost M C of working in the foreign country gets higher, more people will choose to work in the foreign country for people who study in the home country. As the premium for the home country income IS gets higher, more people will choose to work in the home country for people who study in the foreign country.

3.2 First-period decisions Table 3.1

h^∗

exogenous variables definition

MC Migration cost +

A₃ Home country productivity + A4 Foreign country productivity − h^∗∗

exogenous variables definition

IS Income subsidy +

A₃ Home country productivity + A4 Foreign country productivity −

3.2 First-period decisions

To solve the decision rule for node I, The utility function is used with the first-period Bellman’s equation

V (q) = max



ln c₁+ β ln[V^f(f (q), c₁+ SC)], ln c₁+ β ln(V^d(g(q), c₁)]

 . At first stage, there are two things for agents to decide

1. first-period consumption

2. whether to study in the home country or the foreign country.

To solve this problem, we first solve the optimal consumption in the first period for agents who are going to study in the home country for all quality q. Then the optimal consumption in the first period for agents who are going to study in the foreign country for all quality q is solved. Then these solutions are combined to solve agent’s first-period location decision of studying.

First-period consumption c^d₁^∗ for agents studying in the home country From equation (3.1), the optimization problem can be reduced to the problem below

max{c^d₁} ln c^d₁+ β ln[V^d(A₁q^α, c^d₁)].

By observing equation (3.1), V^d(·) is differentiable over c^d₁^∗. Thus we have the first order condition for finding c^d₁^∗

3.2 First-period decisions

1

c^d₁^∗ = β(1 + r) V^d(A₁q^α, c^d₁^∗)

=

⎧⎪

⎪⎨

⎪⎪

⎩

β(1 + r)

A₄[A₁q^α]^γ− MC − (1 + r)c^d₁^∗ if A₁q^α > h^∗ β(1 + r)

A₃[A₁q^α]^γ− (1 + r)c^d₁^∗ if A₁q^α ≤ h^∗

=

⎧⎪

⎪⎨

⎪⎪

⎩

β(1 + r)

A₄[A₁q^α]^γ− MC − (1 + r)c^d₁^∗ if q > A⁻₁ ^1α[ M C A₄− A3]^αγ1 β(1 + r)

A₃[A₁q^α]^γ− (1 + r)c^d₁^∗ if q≤ A⁻₁ ^α1[ M C A₄ − A₃]^αγ1 . Let

q^∗ = A⁻₁ ^α1[ M C A₄ − A3]^αγ1 .

Rearranging the first order condition, the optimal consumption in the first period c^d₁^∗ for agent who are going to study in the home country is:

c^d₁^∗ =

⎧⎪

⎨

⎪⎩

β

(1 + β)(1 + r)[A₄A^γ₁q^αγ− MC] if q > q^∗ β

(1 + β)(1 + r)[A₃A^γ₁q^αγ] if q≤ q^∗ . (3.3) First-period consumption c^f₁^∗ for agents studying in the foreign country

From equation (3.2), the optimization problem can be reduced to the problem below max

{c^f₁} ln c^f₁ + β ln[V^f(A₂q^α, c^f₁ + SC)]

By observing equation (3.2), V^f(·) is differentiable over c^f₁^∗ . Thus we have the first order condition for finding c^f₁^∗

1

c^f₁^∗ = β(1 + r) V^f(A₂q^α, c^f₁^∗+ SC)

=

⎧⎪

⎪⎨

⎪⎪

⎩

β(1 + r)

A₄[A₂q^α]^γ− (1 + r)(c^f₁^∗+ SC) if A₂q^α > h^∗∗

β(1 + r)

A₃[A₂q^α]^γ− (1 + r)(c^f₁^∗+ SC) + IS if A₂q^α ≤ h^∗∗

=

⎧⎪

⎪⎨

⎪⎪

⎩

β(1 + r)

A₄[A₂q^α]^γ− (1 + r)(c^f₁^∗+ SC) if q > A⁻₂^α1[ IS A₄− A3]^αγ1 β(1 + r)

A₃[A₂q^α]^γ− (1 + r)(c^f₁^∗+ SC) + IS if q ≤ A⁻₂ ^1α[ IS A₄− A3]^αγ1

3.2 First-period decisions

Let

q^∗∗ = A⁻₂ ^α1[ IS

A₄ − A3]^αγ1 .

Rearranging the first order condition, we can solve the optimal consumption in the first period c^f₁^∗ for agents who study in the foreign country.

c^f₁^∗ =

⎧⎪

⎨

⎪⎩

β

(1 + β)(1 + r)[A₄A^γ₂q^αγ− (1 + r)SC] if q > q^∗∗

β

(1 + β)(1 + r)[A₃A^γ₂q^αγ− (1 + r)SC + IS], if q ≤ q^∗∗ (3.4) Decision on location of studying

Assume that for an agent who feels indifferent between studying in the home country or the foreign country possesses q quality. We assume that q ∈ [q^∗, q^∗∗], that is

Assumption 2. q^∗, q^∗∗ and q satisfy q^∗ < q < q^∗∗,which is A⁻₁ ^α1[ M C

A₄− A3]^αγ1 <q < A⁻₂ ^1α[ IS A₄− A3]^αγ1

An agent with qualityqfeels indifferent between studying in the home country or the foreign country. By Assumption 2 and the decision rule for the location of work, we know that

If the agent with quality q decides to study in the home country, he will emigrate to the foreign country in the next period.

If the agent with quality q decides to study at the foreign country, he will work in the home country in the next period.

Using equations (3.1) and (3.2) in the first-period Bellman’s equation for an agent who possesses quality q is:

V (q) = max



ln c^f₁ + β ln 

A₃(A₂q^α)^γ− (1 + r)(c^f₁ + SC) + IS 

,

ln c^d₁+ β ln 

A₄(A₁q^α)^γ− MC − (1 + r)c^d₁ .

Once the second-period decision of agents with quality q is known, we can substitute the c^d₁, c^f₁ into equations (3.3) and (3.4) with the optimal first-period consumptions

c^f₁ = β

(1 + β)(1 + r)[A₃A^γ₂q^αγ− (1 + r)SC + IS]

c^d₁ = β

(1 + β)(1 + r)[A₄A^γ₁q^αγ− MC]

3.2 First-period decisions

Using above equations in the first-period Bellman’s equation, we have

V (q) = max



(1 + β) ln[A₃A^γ₂q^αγ− (1 + r)SC + IS] + ln β

(1 + β)^1+β(1 + r) 

,

(1 + β) ln 

A₄A^γ₁q^αγ− MC] + ln β

(1 + β)^1+β(1 + r) 

.

The first term in the brackets is the utility of studying in the foreign country for an agent who possesses of quality q, the second term in the brackets is the utility of studying in the home country for an agent who possesses quality q. Since the agent who possesses quality q feels indifferent between studying in the home country or the foreign country, q satisfies

(1 + β) ln[A₃A^γ₂q^αγ− (1 + r)SC + IS] + ln β

(1 + β)^1+β(1 + r) 

=(1 + β) ln[A₄A^γ₁q^αγ− MC] + ln β

(1 + β)^1+β(1 + r) 

. Rearranging the above equation, we have

q =



(1 + r)SC− IS − MC A₃A^γ₂ − A₄A^γ₁

 ¹

αγ

(3.5) Thus we have the decision rule in the first period.

Lemma 3. If the quality q that an agent possesses is higher than q (q > q), he will choose to accumulate human capital in the foreign country, and his first-period consumption is

c^f₁^∗ =

⎧⎪

⎨

⎪⎩

β

(1 + β)(1 + r)[A₄A^γ₂q^αγ − (1 + r)SC] if q > q^∗∗

β

(1 + β)(1 + r)[A₃A^γ₂q^αγ − (1 + r)SC + IS], if q ≤ q^∗∗

If the quality q that an agent possesses is smaller than q (q < q), he will choose to accumulate human capital in the home country, and his first-period consumption is

c^d₁^∗ =

⎧⎪

⎨

⎪⎩

β

(1 + β)(1 + r)[A₄A^γ₁q^αγ− MC] if q > q^∗ β

(1 + β)(1 + r)[A₃A^γ₁q^αγ] if q ≤ q^∗

To examine what factors might affect q, q is a function of SC, MC, IS, A₁, A₂, A₃ and A₄, which is q(SC, MC, IS, A₁, A₂, A₃, A₄), and let

Γ = A₃A^γ₂ − A₄A^γ₁ By assumption 1, Γ > 0. We have the following proposition

3.2 First-period decisions

Proposition 3. The q represented by q(SC, MC, IS, A₁, A₂, A₃) satisfies

∂q

∂SC > 0; ∂q

∂M C < 0; ∂q

∂IS < 0; ∂q

∂A₁ > 0;

∂q

∂A₂ < 0; ∂q

∂A₃ < 0; ∂q

∂A₄ > 0

The results are also listed in Table 2. The above result is intuitive.¹ A rise in q indicates that people are less willing to study in the foreign country, because more people will study in the home country. Proposition 3 demonstrates that a rise in the cost of studying in the foreign country SC, will lower the number of people studying in the foreign country. A rise in the premium of the home country income IS, will increase the number of people studying in the foreign country. A rise in the cost of emigration to the foreign country M C, will increase the number of people studying in the foreign country. A rise in the efficiency of human capital accumulation in the home country A₁, will lower the number of people studying in the foreign country. A rise in the efficiency of human capital accumulation in the foreign country A₂, will increase the number of people studying in the foreign country. A rise in the efficiency of human capital accumulation in the home country A₁, will lower the number of people studying in the foreign country. A rise in the wage of home country A₃, will increase the number of people studying in the foreign country. A rise in the wage of the foreign country A₄, will reduce the number of people studying in the foreign country.

Since the agent possesses the quality q will work in the home country if he has chosen to study in the foreign country, he will work in the foreign country if he has chosen to study in the home country. A rise in home country wage A₃ will make an agent with quality q become more willing to study in the foreign country, because he would expect a higher income when he comes back to work in the home country. A rise in the home foreign country wages A₄ will make an agent with quality q become less willing to study in the foreign country, because he would expect a higher income when he emigrates to the foreign country.

Table 3.2

q

exogenous variables definition

SC Cost of studying abroad +

MC Migration cost −

IS Income subsidy −

A₁ Productivity on domestic human capital accumulation + A2 Productivity on foreign human capital accumulation −

A₃ Home country productivity −

A₄ Foreign country productivity +

1See the proof of Proposition 3 in appendix A.

3.2 First-period decisions

By the decision rule, and defining Ω = ln

 β

(1 + β)^1+β(1 + r) 

The first-period value function is

V (q) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

(1 + β) ln[A₃A^γ₁q^αγ] + Ω, if q < q < q^∗ (1 + β) ln

A₄A^γ₁q^αγ− MC] + Ω, if q^∗ < q < q (1 + β) ln[A₃A^γ₂q^αγ− (1 + r)SC + IS] + Ω, if q < q < q^∗∗

(1 + β) ln 

A₄A^γ₂q^αγ− (1 + r)SC] + Ω, if q^∗∗< q < q

(3.6)

Thus, based on their quality, native-born agents can be classified into four groups:

1. if q ∈ [q, q^∗]: they will study in the home country and work in the home country.

2. if q ∈ [q^∗,q]: they will study in the home country and work in the foreign country.

3. if q ∈ [q, q^∗∗]: they will study in the foreign country and work in the home country.

4. if q ∈ [q^∗∗, q]: they will study in the foreign country and work in the foreign country.

The first-period consumption, the decision on locations of studying and working are different for each kind of agents, thus their value functions are different, so, their welfare is calculated separately.

Chapter 4 Welfare

The welfare of all native-born people W is W =

 q q

V (q)ψ(q)dq =

 q^∗ q

V (q)ψ(q)dq +

 q_ q^∗

V (q)ψ(q)dq

+

 q^∗∗

q

V (q)ψ(q)dq +

 q q^∗∗

V (q)ψ(q)dq.

Even though we can calculate the welfare of all native-born persons using the above equation, when a government implements a policy, it does not concern the welfare of all native-born persons. Instead, it concerns the welfare of all residents, which are the people who eventually live in the home country.

Let W^g represent the welfare of all residents, that is W^g =

 q^∗ q

V (q)ψ(q)dq +

 q^∗∗

q

V (q)ψ(q)dq.

Chapter 5

Policy instruments

A Government has two policy instruments available:

1. Subsidizing the cost of studying in the foreign country, SC.

2. Subsidizing the income to individuals who study in foreign country, IS.

These policies are financed through lump sum tax.

5.1 Government’s budget constraint

Let T be government’s total tax revenue, G be government’s total spending. The government runs a balanced budget. That is

T = G

Assume the government levies lump sum tax on residents. Let the tax rate be t. We can write the tax revenue as

T = t×

 q^∗ q

ψ(q)dq + t×

 q^∗∗

q

ψ(q)dq.

We will obtain the government’s budget constraint after we calculate the total spending.

Subsidizing the cost of studying in the foreign country

We first assume that government implements the policy of subsidizing the cost of studying in the foreign country. Let the subsidy be s₁. Since only the people who study in the foreign country will receive the subsidy, the total government spending is

G = s₁×

 _q

q

ψ(q)dq.

The government’s budget constraint under this policy is T = t.×

 q^∗ q

ψ(q)dq + t×

 q^∗∗

 q

ψ(q)dq = s₁×

 q_ q

ψ(q)dq = G.

5.2 Individual’s budget constraint

Subsidizing the wage of the home country income

We assume that the government only implements the policy of subsidizing the income of the home country for people who study in the foreign country. Let the subsidy be s₂. Since only the people who study in foreign country will receive the subsidy, the total government spending is

G = s₂×

 _q^∗∗

q

ψ(q)dq.

The government’s budget constraint under this policy is T = t×

 q^∗ q

ψ(q)dq + t×

 q^∗∗

q

ψ(q)dq = s₂ ×

 q^∗∗

q

ψ(q)dq = G.

Mixing the policies

If the government subsidizes both the cost of studying in the foreign country and the home country income for people who study in the foreign country, the government’s budget constraint will become

T = t×

 q^∗ q

ψ(q)dq + t×

 q^∗∗

q

ψ(q)dq = s₁×

 q_ q

ψ(q)dq + s₂×

 q^∗∗

q

ψ(q)dq = G.

5.2 Individual’s budget constraint

For simplicity, assume that the government implements both policies, then we derive the agent’s individual budget constraint. When we analyze the effect of one of the policies, we only have to set the other policy variable to zero.

Accumulating human capital in the home country

Suppose that the agent chooses to study in the home country, and his first-period consumption is c₁. If the agent chooses to work in the home country, his second-period disposable income is y^d(h)− t = A₃h^γ− t. Then his life-time budget constraint is

A₃h^γ− t = (1 + r)c₁+ c₂.

If he choose to work in the foreign country, his second-period income is y^f(h)− MC = A4h^γ− M C. Then his life-time budget constraint is

A₄h^γ− MC = (1 + r)c1+ c₂.

5.2 Individual’s budget constraint

Accumulating human capital in the foreign country

Suppose that the agent chooses to study in the foreign country, and his first-period consumption is c₁+ (SC−s1). If the agent chooses to work in the home country, his second-period disposable income is y^d(h) + (IS + s₂)− t = A₃h^γ+ (IS + s₂)− t. Then his life-time budget constraint is

A₃h^γ + (IS + s₂)− t = (1 + r)[c₁+ (SC− s₁)] + c₂.

If he chooses to work in the foreign country, his second-period income is y^f(h) = A₄h^γ. Then his life-time budget constraint is

A₄h^γ = (1 + r)[c₁+ (SC− s1)] + c₂.

Chapter 6

Equilibrium under the policy

Since the model’s solution is similar to an equilibrium without policy, I only show the result in this section¹.

6.1 Second-period decisions under the policies

Agents studying in the home country

Suppose an agent chooses to study in the home country in the first period, and the level of human capital which he possesses that makes him feel indifferent between working in the home country or the foreign country is

h^∗_p = [M C − t A₄ − A₃]^γ1.

Writing h^∗_p as a function of M C, t, IS, A₄ and A₃, we can state the following proposition Proposition 4. The h^∗_p represented by h^∗_p(M C, A₃, A₄, t) satisfies

∂h^∗_p

∂M C = 1

γ[M C− t

A₄− A3]^1γ−1 1

A₄− A3 > 0

∂h^∗_p

∂A₃ = 1

γ[M C − t

A₄− A3]^γ1−1 M C

(A₄− A3)² > 0

∂h^∗_p

∂A₄ =−1

γ[M C − t

A₄ − A3]^γ1−1 M C

(A₄− A3)² < 0

∂h^∗_p

∂t =−1

γ[M C − t

A₄− A3]^γ1−1 1

A₄− A3 < 0

The results are also listed in Table 3. The implications of proposition 4 are similar to proposition 1. Proposition 4 further predicts that an increase in tax rate t will discourage working in the home country for individuals who study in the home country. The second- period decision rule is

1We solve the equilibrium in appendix B.

6.1 Second-period decisions under the policies

Lemma 4. Given an agent studying in the home county in the first period, his second-period decision rule is:

The agent will work in the foreign country if his human capital h satisfies h > h^∗_p = [M C − t

A₄− A3

]^γ1.

The agent will work in the home country if his human capital h satisfies h < h^∗_p = [M C − t

A₄− A3]^1γ Agents studying in the foreign country

Suppose the agent chooses to study in the foreign country in the first period, the level of human capital which the agent possesses that makes him feel indifferent between working in the home country or the foreign country is

h^∗∗_p = [(IS + s₂)− t A₄− A3 ]^γ1.

With h^∗∗_p as a function of s₂, t, IS, A₃ and A₄, we have the following proposition Proposition 5. The h^∗∗_p represented by h^∗∗_p (IS, A₃, A₄, t, s₂) satisfies

∂h^∗∗_p

∂IS = 1

γ[(IS + s₂)− t

A₄− A3 ]^γ1−1 1

A₄− A3 > 0

∂h^∗∗_p

∂A₃ = 1

γ[(IS + s₂)− t

A₄− A3 ]^γ1−1(IS− s₂)− t (A₄− A3)² > 0

∂h^∗∗_p

∂A₄ =−1

γ[(IS + s₂)− t

A₄− A3 ]^γ1−1(IS− s₂)− t (A₄− A3)² < 0

∂h^∗∗_p

∂t =−1

γ[(IS + s₂)− t

A₄− A3 ]^γ1−1 1

A₄− A3 < 0

∂h^∗∗_p

∂s₂ = 1

γ[(IS + s₂)− t

A₄− A3 ]^γ1−1 1

A₄− A3 > 0

The results are also listed in Table 3. The implications of proposition 5 are similar to proposition 2. Proposition 5 further predicts that an increase in tax rate t and a reduction of the income subsidy s₂ will discourage working in the home country for individuals who study in the foreign country. The second-period decision rule is

Lemma 5. Given the agent studying in the home county in the first period, his second-period decision rule is:

The agent will work in the foreign country if his human capital h satisfies h > h^∗∗_p = [(IS + s₂)− t

A₄− A₃ ]^γ1.