Liquidity-constrained migrants

Journal of International Economics 93 (2014) 210–224

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Journal of International Economics journal homepage: www.elsevier.com/locate/jie

Liquidity-constrained migrants☆ Slobodan Djajić a,⁎, Alexandra Vinogradova b a b

The Graduate Institute of International and Development Studies, 11A Avenue de la Paix, CH-1211 Geneva, Switzerland Center of Economic Research, CER-ETH, Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 20 March 2012 Received in revised form 4 December 2013 Accepted 25 February 2014 Available online 17 March 2014 JEL classification: F22 J61

a b s t r a c t Liquidity constraints represent a major obstacle for potential migrants trying to meet the high cost of undocumented international migration. Some cover it by borrowing from a smuggling organization with a commitment to repay the loan by working in the destination country as bonded laborers. This paper compares alternative ways of financing migration and shows that debt bondage is optimal only if the international wage differential is sufficiently large in relation to migration costs. Tougher border controls as well as internal enforcement measures can be expected to reduce the incidence of debt-bonded relative to self-financed migration, although they may not necessarily lower the overall inflow of illegal aliens. © 2014 Elsevier B.V. All rights reserved.

Keywords: Liquidity constraints Bonded labor Illegal immigration Human smuggling

1. Introduction In an effort to control immigration over the last couple of decades, the advanced countries have introduced new barriers to international mobility of low-skilled workers. With the increasing complexity of overcoming these barriers, migrants are relying more and more on the services of human smuggling organizations to help them reach their desired destination. As reported by Petros (2005), the fees for smuggling services vary depending on the distance traveled, the means of transport, and the entry strategy, reaching tens of thousands of dollars on certain long-haul routes. Although the amounts paid to smugglers may not be very large in relation to the expected income abroad, from the perspective of low-skilled workers in the poor developing countries, the cost of

☆ We are grateful to Teresa Sobieszczyk for sharing her data and two anonymous referees for very helpful comments on an earlier draft of this paper. ⁎ Corresponding author. Tel.: +41 22 908 5934; fax: +41 22 733 3049. E-mail addresses: [email protected] (S. Djajić), [email protected] (A. Vinogradova).

http://dx.doi.org/10.1016/j.jinteco.2014.02.004 0022-1996/© 2014 Elsevier B.V. All rights reserved.

migration represents a big obstacle that stands in the way of their migration plans.1 A key question is how to pay for the cost of migration. One possibility is to accumulate enough savings out of income earned in the source country. We might expect this “self-finance” solution to be attractive when the cost of migration is low in relation to the source-country wage. When the cost is in the tens of thousands of dollars, as in the case of undocumented migration from China to Western Europe and North America, there may be no scope for accumulating the required amount out of earnings at home. In such cases it would be necessary to borrow in order to migrate. 1 There is a growing empirical literature that offers evidence on the effects of liquidity constraints on international migration. Angelucci (2004) uses data from the Progresa program in Mexico to study the impact of transfers to liquidity-constrained, rural households on both internal and international migration. She finds that unconditional cash transfers are associated with a 60% increase in the average migration rate, while the likelihood of having migrants in the household is a positive function of the amount received through the program. In the case of El Salvador, Halliday (2006) reports that higher household wealth is positively associated with migration to the U.S.A. For internal migration in Russia, Andrienko and Guriev (2004) find evidence that inter-regional migration is constrained by lack of liquidity and that it rises with an increase in income. All these studies point to the importance of liquidity constraints in restricting contemporary international migration, confirming what we already know about the role of such constraints in the 18th and 19th centuries (see, e.g., Hatton and Williamson (1992, p. 7) and Chiswick and Hatton (2006, p. 2). See also Grubb (1985), Galenson (1984), and Hatton and Williamson (1994, 1998).

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

Borrowing can take place from a network of family and friends, part of which may already be located in the host country, or by getting indebted to a human smuggling organization. When borrowing from relatives or friends, the loan agreement is typically informal, with the interest obligations (if any) and the contract-enforcement mechanism varying from one culture to another. By contrast, when a migrant borrows from a smuggling organization, enforcement is very strict and the rates of interest are often 20%, 30% or even 60% per annum.2 These rates reflect not only the risk incurred by the lender but also the high transactions and enforcement costs. As a way of controlling these costs, the smuggler typically obliges the migrant to become a bonded laborer with (a partner of) the smuggling organization until the loan is paid off. While in bondage, the migrant's freedom of movement is limited and the wage earned is usually lower than the free-market wage in the host country.3 The purpose of the present study is to investigate the problem facing liquidity-constrained candidates for migration and to characterize the conditions under which they choose debt bondage as the optimal mode of financing their migration costs. This analysis is essential to an informed debate on what factors contribute to the growing incidence of debt-bonded migration and how immigration policies, including border controls and internal enforcement measures of the host countries, affect migration decisions. The scope of our study is limited to voluntary debt-bondage contracts, which are entered into on the basis of more or less perfect information.4 An analysis of human trafficking, which involves deception, strategic behavior, coercion, kidnapping, and violence, is beyond the scope of our paper.5 The present study is not the first to analyze the behavior of debtbonded agents in a model of international migration. Friebel and Guriev (2006) examine the interaction between wealth-constrained migrants and smugglers, with a focus on the conditions under which the latter are willing to offer credit to the former. They confine their analysis, as we do, to voluntary debt-bondage arrangements and provide a number of important new findings on the effectiveness of border controls and deportation measures in deterring illegal immigration of liquidity-constrained individuals. Friebel and Guriev (2006), however, do not explicitly model saving behavior. Their candidates for migration are endowed with a certain initial stock of assets, which can be either greater or smaller than the cost of migration. If it is smaller, they can migrate only as bonded laborers. By contrast, the focus of the present study is on the optimizing behavior of liquidity-constrained individuals, including their saving behavior. This opens up a wider range of options for a potential migrant, both with respect to the mode of financing and the optimal timing of departure from the source country. Our objective is to determine how a worker's optimal migration strategy is related to the cost of migration, the conditions in the labor

2 See Kwong (1997, p. 38), Gao (2004, p. 11) and Sobieszczyk (2000, p. 412). According to Kwong, in the case of Chinese migrants to the West, interest rate of 2% per month is most common. 3 According to Jordan (2011): “An example of a debt bondage situation is a person who agrees to repay a debt of $5000 for recruitment fees and travel costs allegedly paid by the employer/enforcer. The worker agrees to sew clothes until this ‘debt’ is repaid. The market wage for the work is $50 per day but the employer/enforcer only deducts $20 a day from the debt…”. See Gao and Poisson (2005), Human Rights Watch (2000), Kwong (1997), Salt (2000), Sobieszczyk (2000), Stein (2003), Surtees (2003), and Vayrynen (2003) for informative discussions of the conditions facing migrants in debt bondage. 4 In light of some media reports on the experience of illegal immigrants, it may seem odd that we should think of human smuggling and debt-bonded migration in the context of a perfect-information framework. As we shall see below, whether such a framework is a reasonable approximation depends largely on the characteristics of the market for human smuggling and the role of an operator's reputation in enabling him to attract new clients. 5 The problem of trafficking is analyzed from a theoretical perspective by Tamura (2010, 2013). He examines the equilibrium degree of migrant exploitation by the smugglers in a model where the migrants are not liquidity constrained, but have enough personal savings to pay the smuggling fee on arrival at the destination. A recent empirical study by Mahmoud and Trebesch (2010) examines the factors that influence the incidence of trafficking within a migrant population. Their work, as well, does not touch on the issue of how migration is financed.

211

markets at home and abroad, and the cost of borrowing from a smuggling organization. We find that debt bondage is the preferred option when the international wage differential is sufficiently large in relation to migration costs. More restrictive border-control measures can reduce the incidence of debt-bonded migration. Depending on the wage gap between the host and source countries, however, such measures may merely induce migrants to switch from one mode of financing to another, rather than reduce the total flow of undocumented immigrants. Tougher internal enforcement policies that increase the costs and risks facing employers of bonded laborers are found to reduce the incidence of debt-bonded migration, increase the incidence of self-financed migration and reduce the overall inflow of undocumented workers. Our model suggests that the reduction in the inflow is likely to be from the relatively poorer of the sending countries. The remainder of the paper is organized as follows. Section 2 describes the market for human smuggling and defines the migrant's optimization problem in the debt-bondage and self-finance scenarios. Section 3 compares the utility of remaining at home with the utilities of migrating under these two alternative financing schemes and characterizes the conditions under which one or the other is more attractive. The links between our model and some stylized facts are discussed in Section 4. Section 5 extends the baseline model (i) to include the possibility of optimally combining self finance with a debt-bondage arrangement in order to pay for the cost of migration and (ii) to account for the fixed costs of entering into a loan agreement with a smuggling organization. Section 6 concludes the paper by summarizing its main results and offering suggestions for future research. 2. Self-financed vs debt-bonded migration We compare two alternative ways of paying for migration costs: By accumulating savings out of source-country income (self-financed migration) and by borrowing from a smuggler with a commitment to repay the loan out of income earned in the destination country (debtbonded migration). Either way, once the migration cost is paid, we assume that the smuggling organization guarantees passage to the destination.6 Human smuggling operations take many different shapes and forms. Some are run by genuine travel agents, who gradually entered the smuggling business in the process of trying to help their clients realize their travel plans without proper documentation. Enterprises of this type can be found throughout South, South-East, and East Asia. They charge a fee for providing business or academic credentials, letters of invitation, false or modified stolen passport, and other documentation needed for travel to the desired destination. They seem to operate competitively in areas where their customers live, their track record is well known in the community, and they depend very much on their reputation in attracting new clients. Smuggling of Chinese undocumented migrants into Western Europe and North America has similar features in that the reputation of the service provider is a key asset. Moreover, Chinese smuggling networks “…avoid criminality which is likely to attract sustained law-enforcement activity” (Silvertone, 2011, p. 109). These considerations limit the scope for client abuse and opportunistic behavior on the part of the smugglers.7 6 This is usually the case in the Chinese market for human smuggling. The client is initially required to make a fractional down payment. If a smuggling attempt is unsuccessful, the contract calls on the smuggling organization to try again to bring the client to the destination. Full payment for smuggling services is made only after the client arrives safely at the destination. 7 Chin (1999) reports on the basis of his New York survey that smuggled Chinese nationals often considered their smugglers (or “snakeheads”) as philanthropists. Another survey based on 129 interviews with snakeheads in New York City, Los Angeles, and Fuzhou, conducted by Zhang and Chin (2002), provides details on the structure of Chinese human-smuggling operations into the United States and on the relationship between the smugglers and their clients. There is a clear sense that the smugglers are genuinely concerned about the responsibilities to their clients. See Djajić and Vinogradova (2013) for further discussion.

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By contrast, the situation is very different in the human-smuggling markets of the Balkans, North Africa, and Turkey. In those cases migrants from distant countries, poorly informed and eager to get to their final destination, end up involved in arrangements with opportunistic smugglers who may in fact be exploitative criminals. In such markets, where a solid reputation of the service provider is not essential for getting new clients, because poorly informed migrants arrive spontaneously to the market to be matched almost at random with the smugglers, transport services and criminal abuse are often parts of a single package, as analyzed very carefully and discussed in papers by Tamura (2010, 2013).8 Better informed or more experienced migrants fare better in these markets than the ones who are not (see Gathmann, 2008). Following Friebel and Guriev (2006), however, we focus on purely human-smuggling activities that do not involve exploitation of clients through strategic behavior, deception, and physical abuse. We assume that a competitive smuggling organization offers the migrant a contract and honors it in full. The advantage of self-financed in relation to debt-bonded migration in this setting is not having to pay excessive interest charges and not being subjected to the constraints of bondage on arrival in the host country. The advantage of debt-bondage is that it allows the migrant to reach the host country sooner. This means being able to sell labor services at a higher wage earlier in life, although the bonded wage may be lower than the free-market wage at the destination.

Let us assume that the utility function takes the following CRRA form, u(ct) = c1t − θ/(1 − θ), where 1/θ is the elasticity of intertemporal consumption substitution (EICS). Then the consumption path during the period of asset accumulation [0, ϕ] is given by (all the derivations are relegated to Appendix A.1.1)

2.1. Self-financed migration

  h    i
−δϕ −θ −ρϕ u w −u cϕ e −c0 w−cϕ þ ρK e ¼ 0;

Consider first the problem facing a migrant who pays for migration cost out of accumulated savings in the source country. His objective is to maximize utility of consumption over a planning horizon which is assumed to extend from time 0 to T. During the period [0, ϕ] he earns the source-country wage, w, consumes ct at each instant, and saves the rest of his income to pay for the cost of migration, K, at the optimally-chosen time of departure, ϕ. The rate of interest earned on assets accumulated at home is denoted by ρ and the rate of time preference by δ. From time ϕ until T, he stays in the host country, earns w⁎ N w, consumes at the rate c⁎t , and is able to lend and borrow at the host-country, risk-free interest rate r⁎. The migrant's problem is to choose the consumption rates at home and abroad, ct and c⁎t , respectively, and the duration of the predeparture, asset-accumulation period, ϕ, given δ, w⁎, w, ρ, r⁎, and K, all of which are assumed constant. Migration takes place instantaneously and the migrant has no initial asset holdings. The objective function can be written as Z max

ct ;c
t ;ϕ

ϕ 0

−δt

uðct Þe

Z dt þ

T ϕ


 −δt u ct e dt:

ð1Þ

In maximizing Eq. (1), the migrant faces two budget constraints. First, over the pre-migration period, his accumulated savings must sum up to the cost of migration: Z

ϕ 0

ρðϕ−t Þ

ðw−ct Þe

dt ¼ K:

ð2Þ

Second, his net savings while abroad, discounted at r⁎, must be equal to zero in the absence of a bequest motive: Z

T ϕ



 −r
t w −ct e dt ¼ 0:

ð3Þ

8 Similar conditions prevail in the markets for human smuggling services along the border between the US and Mexico. Migrants arrive there after a long journey from the interior of Mexico or yet another country and often lack knowledge of the market conditions or service providers.

½ðρ−δÞ=θt

c t ¼ c0 e

:

ð4Þ

Substituting Eq. (4) in the budget constraint (2) we get   c  w −ρϕ γϕ −ρϕ ; 1−e − 0 e −1 ¼ Ke γ ρ

ð5Þ

where γ ≡ ρ−δ θ −ρ is the proportional growth rate of the discounted (time 0) value of the consumption rate ct. Eq. (5) sets the migrant's savings in the source country equal to the cost of migration and implies that: c0 ¼

" # −ρϕ γ 1−e −ρϕ : −Ke w ρ eγϕ −1

ð6Þ

Assuming for simplicity that the migrant's rate of time preference, δ, equals the foreign risk-free rate, r⁎, his consumption abroad is constant (c∗t = c∗) and equal to his income, w⁎. With c⁎ = w⁎ the optimality condition with respect to the departure date, ϕ, can be rewritten as

½ð

ð7Þ

Þ=θϕ

where cϕ ¼ c0 e ρ−δ . Thus at the optimal time of departure from the source country, the utility sacrificed by staying at home an instant longer, [u(w∗) − u(cϕ)]e−δϕ, must be equal to the benefit, c−θ 0 (w − cϕ + ρK)e−ρϕ, which is the utility value of the savings accumulated over that unit of time. Note that on arrival in the host country, the migrant's consumption jumps instantaneously from cϕ to w⁎. Eqs. (6) and (7) can be solved for the two key endogenous variables, c0 and ϕ, as functions of the parameters that describe the environment facing the migrant: w, w⁎, ρ and K. The comparative statics results are provided in Appendix A.1.1. Most important to us is the level of discounted lifetime utility, USF, enjoyed by a migrant under the selffinance arrangement: U

SF

" # " # c1−θ eγϕ −1 ðw
Þ1−θ e−δϕ −e−δT 0 ; ¼ þ 1−θ 1−θ δ γ

ð8Þ

where both c0 and ϕ are optimally chosen and thus functions of the exogenous variables. We will subsequently compare this utility with the levels enjoyed under alternative financing arrangements to determine which of the available options is superior. 2.2. Debt-bonded migration One can envisage different forms of debt-for-labor contracts. A prominent example is the one used in colonial America in the 17th and 18th centuries to bring indentured servants across the Atlantic. Migrants who could not afford to pay for their passage to the colonies from Britain or continental Europe would indenture themselves, agreeing to repay the loan after arrival with a number of years of labor. The duration of servitude depended on the productivity of the migrant, as reflected in his or her age and skill profile as well as the market conditions at the destination. Over the repayment period, indentured servants were not paid wages, but they were provided with accommodation, food, clothing, and training by their employer. The contracts were strictly regulated and supported by the colonial governors, eager to facilitate migration and meet labor shortages in the colonies (see Galenson, 1984). A very similar type of contract has been used to bring Chinese migrants to the West over the last three decades. In many cases the

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

migration project is sponsored by an employer in the destination country, who is a member of the migrant's extended social network. The sponsor typically has an established business that hires labor to work in a restaurant, a sewing factory or some other enterprise in the destination country. He would like to assist a relative or a person from his village in China by paying for his migration and letting him work off the fee in say three years. He figures that the cost of a hired worker is some $500 per week or $25,000 a year, so that paying a $50,000 migration fee and getting three years of labor from the migrant will make both of them better off. How the potential benefits are divided by the two parties usually depends on the proximity of the relationship between them within the family or community network. Over the repayment period, the migrant is provided with food and shelter, usually on the premises of the business, and is expected to work 40–60 h per week. Within such arrangements, it is impossible to disentangle the migrant's earnings and interest charges as the contract stipulates the duration of the repayment period rather than the debtor's wage and the interest rate.9 While we examine the behavior of migrants under the Chinese (or Colonial) type of debt-for-labor contract in an earlier version of this paper (available on request from the authors), we consider here a more general contractual framework that enables the migrant to optimally choose the duration of the repayment period.10 We assume that the smuggling organization brings the migrant into the destination country at time 0, where he stays until time T. The loan from the smugglers covers the entire cost of migration, K. The migrant commits to repay the debt by the time τ ∈ (0, T), with the interest rate, r, while working for (a partner of) the smuggling organization at the bonded wage, wb. Let us suppose that r N r⁎ and w b wb b w⁎, which corresponds to most cases of debt-bonded migration.11 We assume in what follows that a candidate for migration takes r and wb as given. Issues related to the optimality of the debt-for-labor contract in a framework with endogenously determined r and wb are outside of the scope of this paper. Once the debt is repaid, the migrant is released from bondage and free to earn w⁎, as well as to lend and borrow at the rate r⁎. As it is rarely the case that a migrant is able to default on a loan from the smuggling organization, we assume that the loan is always paid back. The migrant's objective in this setting is to maximize lifetime utility Z

τ 0

Z T     b −δt b
−δt u ct e dt þ u ct e dt; τ

period, the present value of his savings, discounted at the smuggler's rate of interest, r, must be equal to the size of the debt: Z τ 0

with respect to the duration of the debt-repayment period, τ, his consumption rates while indebted, cbt , and after being released from bondage, cb∗ t , subject to two budget constraints. First, during the bondage

Z T τ

ð10Þ




b


w −ct


−r t dt ¼ 0: e

ð11Þ

Following standard optimization techniques (see Appendix A.1.2), we derive the migrant's optimal consumption path during the period of indebtedness as b

b ½ðr−δÞ=θt

c t ¼ c0 e

;

ð12Þ

so that the consumption rate while in bondage grows at a proportional rate equal to the product of the EICS and the difference between the rate of interest charged by the smuggler and the migrant's rate of time preference. Combining Eq. (12) with Eq. (10) we obtain  wb  cb  gτ −r τ  1−e − 0 e −1 ¼ K; r g

ð13Þ

where g ≡ r−δ θ −r is the proportional growth rate of the discounted (time 0) value of the consumption rate cbt . Having assumed that δ = r⁎, the consumption rate of a debt-free migrant (i.e., after time τ), is constant at cb⁎ = w⁎. The optimality condition with respect to the debt-repayment date can then be written as h    i
 −θ  
b −r τ b b b −r τ − c0 w −cτ e ¼ 0; u w −u cτ e

ð14Þ

which states that when τ is optimally chosen, the cost (in terms of utilh  i

b −r τ ity) of remaining in bondage an instant longer, uðw Þ−u cτ e , must be equal to the benefit, (cb0)−θ(wb − cbτ)e−r τ, which is the utility value of net savings accumulated during this extra instant. Noting that ð Þ cb ¼ cb e½ r−δ =θτ , Eqs. (13) and (14) can be solved for the optimal length 0

of the repayment period, τ, and the initial consumption rate, cb0, as functions of the exogenous variables (see Appendix A.1.2 for comparative statics results). At the time of release from bondage the migrant's consumption jumps instantaneously from cbτ to w⁎. The discounted lifetime utility of a debt-bonded migrant is given by

U 9 Another type of contract is offered to debt-bonded migrants from Thailand who work in the sex industry of Japan and other wealthy countries in East Asia. Workers are transported to the host country without having to make any payments. The cost of migration is recovered by the smuggling organization when the worker's migration debt is sold to an employer at the destination. While the employer pays in the range of $15,000– $20,000, the migrant is expected to provide the employer with, say, $30,000 worth of services (see Human Rights Watch, 2000). Some migrants repay the debt within a few months, others may take a year or more, depending on the number of clients they choose to serve on an average day. Although numerous media reports seem to suggest that debtbonded migrants, especially in the sex industry, are coerced, subjected to violence and sometimes even indefinite slavery, the vast majority of migrants are fully aware of the conditions of employment abroad before entering into their contracts. See Skeldon (2000, p. 19) and Human Rights Watch (2000). 10 Debt-bondage contract with a flexible repayment period are observed, for example, in the sex industry (see footnote 9), in the garment industry, and other activities where compensation is based on piece rates. 11 As noted in the Introduction, the interest rate is often 2% or more per month, reflecting transactions and contract enforcement costs, possible risks to the lender, as well as rents that the lender may be able to extract from the borrower under the arrangement. The gap between wb and w⁎ reflects similar considerations (see also footnote 3).

 b b −rt w −ct e dt ¼ K:

Second, once the debt is repaid, the migrant's savings over the remainder of his planning horizon, discounted at the risk-free rate, must sum up to zero:

τ

ð9Þ

213

DB

¼

 1−θ  gτ  cb0 e −1 1−θ

g

" # ðw
Þ1−θ e−δτ −e−δT ; þ 1−θ δ

ð15Þ

where cb0 and τ are optimally chosen. 2.3. No migration Another option available to a potential migrant is simply to remain permanently in the source country and work for the wage w, earning the interest rate ρ on accumulated savings. His optimal consumption  −ρT   γ ½ðρ−δÞ=θt ¼ cNM , where cNM ¼ w 1−eρ . path is then simply cNM t γT 0 e 0 e −1 The discounted lifetime utility stemming from his optimal consumption program is given by

U

NM

¼

 1−θ " # cNM 0 eγT −1 1−θ

γ

;

where NM stands for “no migration” and γ ≡

ð16Þ ρ−δ θ −ρ.

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S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

Benchmark: δ=r*=ρ=5%, r=30%, σ=1/3, θ=0.95, T=30

3

financing mode gets them sooner to the foreign, high-wage country. For any given σ, getting abroad sooner has a greater impact on welfare, the larger the international wage differential. High interest charges on loans provided by the smuggling organization are, on the other hand, a disadvantage, the weight of which is heavier, the higher the cost of migration. Given r and σ, this implies a positive relationship between the foreign wage and the cost of migration that makes potential migrants indifferent between self financing their migration costs and borrowing from a smuggling organization. For any combination of the foreign wage and migration cost that is above the SF = DB locus, DB is preferred over SF and vice versa. Finally, agents are indifferent between debt-bonded migration and “no migration” along the DB = NM schedule. Above it, UDB N UNM, while below it, debt-bonded migration is less attractive than the NM option.

2

Proposition 1. The schedules SF = NM, SF = DB, and DB = NM are all positively sloped. The magnitudes of the slopes are given by:

8

7

w*/w

6

5

4

0

100

200

300

400

500

600

700

800

900

K/w Fig. 1. Optimal arrangements for financing migration costs.

In the next section we compare the three options with the aim of (i) identifying the conditions under which international migration is optimal and, (ii) when migration does increase lifetime welfare, determining under what conditions migrants prefer debt-bondage over self-finance as a way of meeting migration costs. The possibility of self-financing only part of migration costs and then borrowing the rest from a smuggling organization is examined in Section 5. Here we focus on the three pure strategies: SF, DB, and NM.

 dw
 ¼ dK SF¼NM

12 We have tried a wide range of values for T and θ in our simulations, only to find that the main results of the paper remain unaffected. With a longer time horizon, T, an increase in K requires a smaller increase in w⁎ to keep the utility of SF equal to that of NM, making the SF = NM schedule flatter. By contrast, an increase in the degree of concavity of the utility function makes the SF = NM schedule steeper. That is, for any given increase in K, it requires a larger increase in future income (and hence w⁎) to keep the agent indifferent between SF and NM.

u0 ðw
Þ

 dw
 ¼ dK DB¼NM

 dw
 ¼ dK SF¼DB

3. Comparing the alternatives The choices available to a potential migrant are: (a) no migration (NM), resulting in utility UNM, (b) self-financed migration (SF), resulting in utility USF, and (c) debt-bonded migration (DB), giving rise to a utility level UDB. The relationship among these options is illustrated in Fig. 1, where we have the ratio of the host- to source-country wage on the vertical axis and the ratio of the migration cost to the source-country wage on the horizontal axis. The SF = NM locus shows combinations of w⁎/w and K/w such that a potential migrant is indifferent between selffinanced migration and no migration. The schedule is drawn for T = 30 years, θ = 0.95, and δ = ρ = r⁎ = 5% per annum, while wages w⁎ and w are measured as flows per week, with w normalized to 1. These same values are used in our calculations throughout the paper.12 Anywhere above and to the left of the SF = NM schedule, USF N UNM, so that a worker is better off migrating under the self-finance arrangement rather than staying permanently at home. In the region below and to the right of SF = NM it does not pay to migrate if migration has to be selffinanced. The SF = DB locus shows combinations of w⁎ and K such that a potential migrant is indifferent between self-financed migration and debtbonded migration under the assumptions that the smuggling organization charges r = 30% per annum and offers a bonded wage which is only two thirds of the market wage in the host country (i.e., wb = (1 − σ)w⁎, where σ = 1/3). What can make debt bondage appealing to potential migrants, in spite of the high interest rate charged by the smuggling organization and the prospect of being underpaid abroad, is that this

u0 ðc0 Þe−ρϕ e−δϕ −e−δT δ

N0;

ð17Þ

  u0 cb0 ð1−σ Þ

−δτ 1−e−r τ 0  b  −e−δT 0
 e u c0 þ u w r δ

  u0 cb0 −u0 ðc0 Þe−ρϕ ð1−σ Þ

−δϕ 1−e−r τ 0  b  −e−δτ 0
 e u c0 −u w r δ

N0;

ð18Þ

N0:

ð19Þ

Proof. See Appendix A.2. □ All three schedules intersect at point A, which we shall refer to as the “triple-indifference point.” In fact, the three schedules must always intersect at the same point. Consider a point of intersection between the SF = DB locus and the SF = NM locus. For that combination of w⁎ and K, it must also be the case that DB = NM. Appendix A.2.5 provides a discussion of the conditions for the existence of the intersection point. Corollary. In the neighborhood of the triple-indifference point, the SF = DB schedule is flatter than the DB = NM schedule, which is in turn flatter than the SF = NM schedule, provided that u0 ðc0 Þe−ρϕ u0 ðw
Þ

e−δϕ −e−δT δ

  u0 cb0

N ð1−σ Þ

−δτ 1−e−r τ 0  b  −e−δT 0
 e u c0 þ u w r δ

:

Proof. See Appendix A.2.4, where we also show that this condition is satisfied at the point of intersection of the three schedules. □ This inequality states that, with respect to a migrant's discounted utility, the impact of a change in K relative to that of a change in w⁎, must be greater in the case of SF than it is in the case of DB at the triple-indifference point.13 The schedules serve to identify in Fig. 1 the combinations of w⁎ and K for which each of the three options is optimal. Source-country workers will opt for debt-bonded migration when combinations of w⁎ and K fall 13 Note that the effects on the discounted utility of changes in both K and w⁎ are larger in the case of DB than they are in the case of SF. Since an increase in K has a negative and an increase in w⁎ a positive impact on utility, it follows that anywhere to the right of the SF = DB schedule, USF N UDB and anywhere above it, UDB N USF.

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

14 Although the precise relationship between the intensity of border controls and the value of K depends on the technology of enforcement, the characteristics of the market for human smuggling, and numerous other factors, we are interested here in only the qualitative impact of a policy change on the value of K. There is in fact very little evidence on the quantitative effect of a change in the intensity of border patrols on K, other than in the case of the Mexico–US border. See, e.g., Gathmann (2008) and Hanson and Spilimbergo (1999). 15 We take this increase in r to be exogenous. As pointed out by a referee, in a generalequilibrium framework r may in fact be affected by the cost of migration. See Section 6 for possible extensions on the agenda for future research.

r=30% (dashed) vs r=40% (solid), δ=r*=ρ=5%, σ=1/3, θ=0.95, T=30

10 9 8 7

w*/w

into the dotted area above the SF = DB schedule to the left of point A and above the DB = NM schedule to the right of A. Self-finance is optimal when combinations of w⁎ and K fall into the white, unshaded area between the SF = DB and the SF = NM schedules below and to the left of point A. No migration is optimal in the remaining area shaded by thin diagonal lines. The figure illustrates some obvious points, but it also reveals a number of interesting implications of our analysis. First, it shows that NM is optimal when migration costs are high, while the foreign wage is not attractive enough to warrant moving abroad. By contrast, when migration costs are low and the foreign wage is high, debt-bonded migration is optimal. A low K and a high w⁎ ensure, respectively, that the debt burden is not too heavy and that the loan can be repaid relatively quickly out of earnings abroad, even at an exorbitant rate of interest charged by the smuggler. For somewhat higher migration costs and/or lower foreign wage, the self-finance option dominates debt-bondage in the unshaded region. This is because a higher K imposes a larger debt that must be serviced under DB at a high rate of interest, while a reduction in w⁎ relative to w erodes the advantage of getting abroad sooner as a bonded laborer. Self-finance is then the optimal way to pay for migration costs. An important implication of this analysis is that, by increasing K, tougher border controls can help reduce the incidence of debtbondage.14 This goes against the conventional wisdom that higher migration costs fuel growth of debt-bondage. The conventional view is based on the notion that if a potential migrant's wealth is smaller than K, he will be inclined to borrow from the smuggler. An increase in K can then induce a larger flow of DB migrants. Once we consider the possibility of saving for the purpose of meeting migration costs, however, we find that an increase in K makes DB less attractive relative to SF, but also relative to NM. It is interesting to note that an increase in border controls has different implications depending on the magnitude of the international wage differential. For relatively low values of w⁎/w (i.e., below the line XAY in Fig. 1), a marginal increase in K/w that is effective in reducing debtbonded migration will result in an offsetting increase in self-financed migration. In that range of values of w⁎/w, a higher K/w induces migrants to switch from DB to SF, but does not discourage them from attempting to migrate. It is only for values of w⁎/w above the intersection of the three schedules that tougher border enforcement measures that deter debt-bonded migration are also effective in reducing illegal immigration one for one. In that range it does not pay to switch to SF, but rather to NM. We consider next the effects of an increase in the cost of servicing the loan from a smuggler, i.e., an increase in r from 30% to 40% per year.15 In Fig. 2, the dashed schedules correspond to the benchmark case, while the solid ones are drawn for r = 0.4. Note that the SF = NM schedule is unaffected since an increase in r has no influence on the attractiveness of SF in relation to NM. It does, however, make debt-bonded migration less appealing: The dotted area is now smaller, while the unshaded “self-finance” area is larger by the amount EA′AD. The “no-migration” area also expands at the expense of DB to include the area A′BICA. For combinations of w⁎/w and K/w within this area, it no longer pays to migrate if the rate of interest charged by a smuggler is raised from 0.3 to 0.4 per annum.

215

6 5 4 3 2 0

100

200

300

400

500

600

700

800

900

K/w Fig. 2. Effect of an increase in r.

The implications of an increase in σ, the proportion by which the bonded wage falls short of the foreign free-market wage, are very similar to those of an increase in r: The SF = DB and DB = NM schedules shift up and to the left to intersect the unaffected SF = NM locus at higher levels of both w⁎/w and K/w. The area of “debt-bondage” is thus reduced while the areas of “self-finance” and “no migration” expand in a manner very similar to that illustrated in Fig. 2. This suggests that to the extent that tougher enforcement measures serve to increase the risks and costs of employing debt-bonded labor, thereby contributing to an increase in σ, they are likely to reduce the incidence of debtbonded migration, increase the incidence of self-financed migration and reduce the overall migration flow.16 The reduction in the flow will be from the source countries whose emigrants face an environment characterized by very large values of w⁎/w and K/w, as shown by the area A′BICA in Fig. 2. The switch from debt bondage to self finance, with no reduction in the flow, will be from the economies with intermediate values of w⁎/w and K/w that correspond to the area EA′AD.

4. The model and some stylized facts After a thorough search for empirical evidence on modern-day debtbonded migration, that could possibly be used to test the predictions of our model at either the micro or macro levels, we have found only one useful data set which is based on interviews with return migrants in Thailand in the late 1990s. This data set, gathered for the Sobieszczyk (2000) study, contains observations on 104 migrants (including 13 former debt-bonded) and provides information on marital status, number of children, age at migration, level of education, commission paid to go abroad, salary abroad, destination country, and other variables. The thirteen former debt-bonded migrants worked in Japan (6), Singapore (5), Macao (1), and Taiwan (1). The other 91 migrants, reported to be self-financed, worked in Taiwan (44), Japan (29), Hong Kong (6), Brunei (4), South Korea (4), Malaysia (2), and Singapore (2). 16 By tougher enforcement, we mean harsher penalties for the employers of bonded labor, while continuing to assume that the probability of an illegal alien getting deported or otherwise punished by the authorities is zero. Here we follow the seminal work of Ethier (1986) in assuming that the internal enforcement measures are directed strictly at the employers (in our case, employers of bonded labor) rather than those who work for them. For an analysis of how the prospect of deportation affects the behavior of debt-bonded migrants, see Djajić and Vinogradova (2013) and for the case of debt-free illegal aliens in the context of a dynamic stochastic optimization model see Vinogradova (2011, 2014).

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On the basis of this data set, which admittedly does not allow for rigorous empirical analysis, we are nonetheless able to draw some insights on the relationship between personal characteristics of migrants and their choice between DB and SF. DB migrants tend to be relatively young (23.7 years old, on average, vs 29.6), to have a low level of education (5.6 years of schooling for DB vs 7 for SF), and to migrate to high-income/high-wage countries. Of the thirteen DB migrants in the sample, the majority worked in Japan and Singapore, the two highestincome destinations. By contrast, more than half of the SF migrants worked in Taiwan and Malaysia, the two poorest of the seven destinations. This evidence is consistent with the prediction of our model that the higher the international wage differential, the stronger the incentive to choose DB over SF. With respect to the choice of destination for DB migrants, a similar pattern has been observed centuries ago in colonial America. Many of the migrants in that era chose to meet the cost of passage from Europe by entering into servitude contracts. 17 In relation to the predictions of our model, it is interesting to note that the proportion of immigrants that chose servitude as a means of financing migration varies significantly across colonies. Colonies with a relatively high proportion of servants among their immigrants were Virginia, Maryland and Pennsylvania. The Carolinas and Georgia to the south and the colonies to the north of Pennsylvania had a much lower incidence of servitude.18 Why were some colonies so attractive to debt-bonded migrants while others received mostly self-financed immigrants? Our model predicts that for any given cost of migration (and the cost of passage from England was roughly the same at this time, regardless of which colony was chosen as the destination), a higher ratio of host- to source-country wage makes DB more attractive relative to SF. The colonies that show a high incidence of servitude among their immigrants were precisely those that offered better compensation and working conditions. According to Grubb (1985), the productivity of farm labor in the northern colonies was not high enough to enable recruiters to offer competitive contracts. The highest productivity of labor in agriculture was in the middle colonies, where tobacco and grains were produced for export. Colonies north of Pennsylvania lacked the lucrative export crops that the middle and southern colonies produced. The southern colonies were unattractive for those entering servitude contracts because the working conditions in the rice fields of South Carolina were perceived to be much less favorable than those on tobacco and grain farms of the middle colonies (Grubb, 1985, p. 335). This raises the following question: If the cost of passage was the same to all colonies, but some colonies offered higher wages, why would anyone have migrated to the low-wage colonies? In fact the “low-wage” colonies did attract fewer migrants. Moreover, those who were attracted, often came for reasons other than wealth accumulation, as in the case of Puritan migration to the New England colonies. The point we wish to make on the basis of our model is that a destination with a higher wage is relatively more attractive to a DB migrant than it is to an SF migrant. We should therefore observe, as documented by Grubb (1985, p. 334), that higher-wage

destinations experienced more migration and a higher incidence of DB migration.19 Later in the 19th century, debt-bonded migration from India and China met shortages of labor on sugar plantations of the West Indies and Hawaii, in the mines of California and South America, and on the building of railroads. These were the types of employment avoided by the free white settlers. Coincidental with the bound Asian migration was the primarily self-financed migration of Europeans to the United States. Galenson (1984, p. 25) explains this phenomenon in a way consistent with the predictions of our model, by pointing to migration costs. For Asian migrants to the Western Hemisphere, they were 20 to 40 times higher, when measured in terms of per-capita income of the source country, than they were for migrants from Great Britain, Ireland, Germany, and the Scandinavian countries. Self-finance was then an attractive option for the Europeans, whose home wages were catching up with those in the Americas and migration costs were to the left of point A in Fig. 1, while for Indian and Chinese migrants, facing K/w to the right of point A, DB was the way to go. This holds true regardless of whether migration flows were triggered at the time by an increase in the destination wage or a reduction in transport costs. 5. Extensions of the basic model This section examines two extensions of our model. The first considers the possibility of self-financing part of migration costs and borrowing the rest from a smuggler at the rate of interest r.20 In the second extension we use this more general framework to study the implications of relaxing the assumption that there are no fixed costs of entering into a loan agreement with a smuggler. In reality, such costs can be sufficiently large to deter individuals from becoming a DB migrant when the cost of migration is low in relation to earnings at home. 5.1. The hybrid case Let us consider the possibility of optimally combining the SF and DB options. Suppose that from time 0 until an optimally chosen time ϕ, the migrant accumulates savings at home up to a certain critical level. At that point he borrows from a smuggler the balance needed to meet the cost of migration, K, migrates, and repays the loan from time ϕ until an optimally chosen time τ, by working for a partner of the smuggling organization at the bonded wage wb. After release from bondage at time τ, the migrant remains in the host country and earns the freemarket wage, w⁎. We refer to this arrangement as the “Hybrid” option (H) because it combines SF and DB in an optimal manner to meet the cost of migration. The problem for a migrant is to Z max

ct ;cbt ;c
t ;ϕ;τ

0

−δt

uðct Þe

Z dt þ

τ ϕ

Z T   
 −δt b −δt u ct e dt þ u ct e dt τ

s.t. Z

ϕ

17

The cost of ocean passage from Britain to the American colonies in the 17th and 18th centuries was roughly one half of a year's income for a low-skilled British emigrant and a year's income for someone migrating from Germany (see Grubb (1985) and Galenson (1984)). According to Smith (1947, p. 336), if one excludes Puritan migration of the 1630s, “…not less than half nor more than two thirds of all white immigrants to the colonies were indentured servants, redemptioners or convicts.” For the period from 1785 to 1804, Grubb (1985, p. 319) estimates that the incidence of indentured servitude among the 7837 German immigrants arriving in Philadelphia was 44.8% overall and over 50% for single adults. 18 Between 1773 and 1776, emigration records were kept by English authorities, including the name of the colony of destination and whether the passenger paid the fare in full or entered, instead, into a servitude contract. As reported in Table 6 of Grubb (1985, p. 334), the percentages of English emigrants destined for various colonies as servants varied between 98.33% for Maryland and 1.85% for the New England Colonies.

ϕ

0

ρðϕ−t Þ

ðw−ct Þe

dt þ

Z τ ϕ

 b b −r ðt−ϕÞ w −ct e dt ¼ K;

ð20Þ

19 According to Grubb (1985, p. 334), the destinations of English migrants to Colonial America in the years 1773–1776 were as follows: New England Colonies, just 54 migrants, Middle Colonies, 1162 migrants (of which New York: 303 and Pennsylvania: 859), Chesapeake, 2984 (of which Maryland: 2217 and Virginia: 767), and Lower South Colonies, 307 (of which Carolinas: 106, Georgia: 196, and Florida: 5), while the percentages of English emigrants destined for various colonies as servants are as follows: Maryland, 98.33%; Virginia, 90.35%; Pennsylvania, 78.81%; Carolinas, 23.58%; Georgia, 17.86%; New York, 11.55%; Canada, 9.68%; Nova Scotia, 7.76%; and New England, 1.85%. 20 We are thankful to an anonymous referee for suggesting that we examine this option.

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

Assuming that ρ = δ = r⁎, the consumption rate in the source country, c, is constant and the consumption rate after release from bondage is also constant at c⁎ = w⁎. The consumption rate while in bondage ð Þ is given by cbt ¼ ce½ r−δ =θðt−ϕÞ . Note that there is no jump in consumpb tion at t = ϕ : cϕ = c, as can be seen from the first-order conditions (22) and (23). Substituting for cbt in the budget constraint (20) and integrating we obtain:

Hybrid Case (no fixed cost of borrowing): δ=r*=ρ=5%, r=30%, σ=1/3, θ=0.95, T=30

8

7

A 6

w*/w

217

ð

w

5

c¼

Þ

−r τ−ϕ eρϕ −1 b 1−e −K þw r ρ

eρϕ −1 egðτ−ϕÞ −1 þ ρ g

4

:

Eq. (25) simplifies to: 3

" b

w−w þ δK þ ðr−δÞ w

2 0

100

200

300

400

500

600

700

b

800

# 1−e−rðτ−ϕÞ egðτ−ϕÞ −1 −c ¼ 0; r g

K/w while Eq. (26) becomes: Fig. 3. Introducing the hybrid option.

Z

T τ



 −r
t w −ct e dt ¼ 0;

    
 b δðτ−ϕÞ −θ b b −r ðτ−ϕÞ c w −cτ e ¼ 0: u cτ −u w þ e ð21Þ

where ρ is the interest rate at home and δ is the migrant's rate of time preference. The Lagrangian may be written as: Z T   
 −δt b −δt uðct Þe dt þ u ct e dt þ u ct e dt L¼ 0 ϕ τ Z ϕ  Z τ  −ρϕ ρðϕ−t Þ b b −r ðt−ϕÞ þλe ðw−ct Þe dt þ w −ct e dt−K 0 ϕ Z T 

 −r
t w −ct e dt þμ Z

ϕ

−δt

Z

τ

H

U ¼

τ

and the first-order conditions: ∂L 0 −δt −ρt ¼ u ðct Þe −λe ¼ 0; ∂ct

∂L ∂cbt

  0 b −δt −rt ðr−ρÞϕ ¼ u ct e −λe e ¼ 0;

∂L 0 
 −δt −r
t ¼ u ct e −μe ¼ 0; ∂c
t

The three equations above simultaneously determine the three key endogenous variables, c, ϕ, and τ, with the discounted lifetime utility of an H migrant given by:

ð22Þ

ð23Þ

ð24Þ

 i h    i ∂L h   b −δϕ −ρϕ b b þe λ w−cϕ − w −cϕ þ ρK ¼ u cϕ −u cϕ e ∂ϕ Z τ ð25Þ  −ρϕ b b −rðt−ϕÞ λðr−ρÞ w −ct e dt ¼ 0; þe ϕ

  
i −δτ 

 −r
τ ∂L h  b  −ρϕ b b −r ðτ−ϕÞ ¼ u cτ −u cτ e þe λ w −cτ e −μ w −cτ e ¼ 0: ∂τ

ð26Þ

Eqs. (22)–(24) relate the marginal utilities of consumption in each of the three phases of the planning horizon to the marginal utility of wealth in the corresponding phase. Eq. (25) guarantees that the time of migration, ϕ (which corresponds to the time of entry into a loan agreement with the smuggler) is optimal, while Eq. (26) is satisfied when τ, the time of loan-repayment completion (and release from bondage) is optimally chosen.

" # " # 1−θ −δϕ gðτ−ϕÞ−δϕ −δϕ
1−θ −δτ −δT c 1−e e −e ðw Þ e −e þ þ : 1−θ δ g 1−θ δ

We compare in Fig. 3 this hybrid financing scheme with the pure DB, SF, and NM options. What we observe is that for the values of K/w and w⁎/w in the area above the H = DB locus, DB dominates H. It is then optimal to finance migration by relying on the pure DB option. Anywhere below the schedule, it is better to self-finance at least a part of K. The dashed SF = DB, DB = NM, and SF = NM schedules are the same as in Fig. 1, while the H = NM schedule is to the right of the SF = NM locus. Thus for combinations of w⁎ and K in the light shaded area to the right of SF = NM and above H = NM it does not pay to migrate under either SF or DB, but it does if one can resort to an optimal combination of the two. Perhaps surprisingly, the SF = H schedule does not exist. As explained below, the reason is that the H option always dominates SF. Thus anywhere above the H = DB schedule, migrants optimally choose the pure DB option, between the H = DB and the H = NM schedules they choose the H option, and in the area below the H = NM schedule it is best to stay at home. It is interesting that once the H option becomes available, it is never optimal to finance the entire cost of migration out of earnings at home. It is always better to borrow at least a little bit from a smuggler so as to get abroad relatively sooner, assuming that wb N w. Imagine a candidate for migration who earns 1 unit of output per year at home, has an opportunity to earn a wage of 3 units of output per year while in bondage abroad, and faces r of 40% per year. It is clearly beneficial for that individual to borrow from the smuggler the last unit of output needed to pay for K. Doing so gets him abroad at least a year sooner, enables him to earn 3 units of output in a single year, to pay off the debt amounting to 1.4 units, and still have 1.6 units left over for consumption during the year. This is more than he can possibly consume at home. Even if we consider a very extreme (and unrealistic) case where r is 365% per year, it still pays to borrow at least an amount corresponding to a day's earnings at home, provided that wb is more than one percent higher than w. According to this logic, pure SF is never optimal when wb exceeds w.

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Hybrid case with fixed cost of borrowing NM δ=r*=ρ=5%, r=30%, σ=1/3, θ=0.95, T=30, β=0.02*U

8

7

A

6

w*/w

B 5

C 4

3

2 0

100

200

300

400

500

600

700

800

900

K/w Fig. 4. The hybrid option with a fixed cost of borrowing from a smuggler.

5.2. Fixed cost of borrowing from a smuggler So why don't we observe that all migrants borrow at least a small amount from their smuggler? Entering into a loan agreement with a criminal organization is a daring endeavor. It is likely to impose a significant fixed transactions cost on a borrower, such as that related to the gathering of information needed for an informed decision on whether to enter into a highly constraining loan agreement with a shady partner. This may well discourage migrants from borrowing relatively “small” amounts. It is important to recall that taking a loan from a smuggler implies losing freedom to change employers, being dispossessed of identity documents, and limitations on the freedom of movement during the indebtedness period. The simplest way of introducing a fixed cost of borrowing from a smuggler into our model is by subtracting a constant amount of utility, β, from the migrant's objective function. His optimal consumption path is unaffected if we do so. The implications are illustrated in Fig. 4, where the levels of utility associated with the two options involving borrowing from a smuggler (H and DB) are reduced by an amount equal to 2% of a non-migrant's discounted lifetime utility. Two points are worth noting. First, the SF = DB schedule is now Ushaped. Anywhere above (below) it, the DB option offers a higher (lower) level of discounted utility when compared with SF. To the left of the negatively-sloped portion of SF = DB, we see that for low values of K it does not pay to get credit from a smuggler. The fixed cost of borrowing makes such an arrangement unattractive. As we consider somewhat higher migration costs, the fixed relative to the variable component of the cost of borrowing becomes smaller and, if the international wage differential is large enough, DB dominates SF as soon as we cross the SF = DB schedule into the dotted area. Further increases in K augment the burden of the debt and the SF option becomes once again more attractive than DB. This occurs as we cross from the left the positively-sloped portion of the U-shaped SF = DB locus. Second, the SF = H schedule now does exist. With a fixed cost of borrowing from a smuggler, the SF option dominates H in the region below the SF = H schedule to the right of point C and in the region below and to the left of the SF = DB locus to the left of point C.21 In fact SF is the optimal financing option anywhere in the white, unshaded area of the figure; DB is optimal in the dotted area between the downward21 At point C, DB, SF, and H generate identical levels of discounted lifetime utility. Note that to the left of the negatively-sloped portion of the SF = DB schedule, SF dominates DB and anywhere above the H = DB locus, DB dominates H.

sloping part of SF = DB and the H = DB schedule above point C; H is optimal between H = DB and H = NM in the vertically shaded area above the SF = H locus; and NM is optimal in the area shaded by the thin diagonal lines to the right of H = NM (above point B) and SF = NM (below point B). The main implications of accounting for a fixed cost of borrowing from a smuggler and the possibility of exercising the H option are the following: First, with respect to the choice between migrating and remaining permanently at home, the possibility of combining SF and DB in the form of H makes migration attractive for certain combinations of K/w and w∗/w that would make NM optimal if only the pure DB and SF options were available. Second, with respect to the mode of financing international migration, we note that SF is optimal when both the cost of migration and the international wage differential are relatively low, yet in such combinations that make migration nonetheless attractive. As we consider larger magnitudes of w∗/w, it becomes more urgent for the migrant to reach the foreign labor market relatively sooner. This increases the attractiveness of borrowing from a smuggler. The extent of borrowing for a given r, and thus the choice between the DB and H options, hinges on the magnitude of migration costs. If that magnitude is in an intermediate range and the international wage differential is sufficiently large, as in the dotted area in Fig. 4, the DB option is optimal. For higher values of K, carrying the debt burden under the DB option becomes too heavy and a dose of SF is required to maximize welfare. This makes the H option optimal in the vertically shaded area to the right of the H = DB locus. In relation to our findings in Section 3, note that with a fixed cost of borrowing from a smuggler, it is no longer attractive to opt for DB if K is relatively low. Higher values of K, however, can induce migrants to switch from SF to DB, in line with the conventional view, but only if the international wage differential is sufficiently high. Further increases in K eventually call for at least partial self finance to lower the burden of the debt that can become excessive under a pure DB arrangement. If an H option is available, migrants then switch from DB to H, and with further increases in K to pure SF or NM, depending on the magnitude of w∗/w (i.e., depending on whether w∗/w is above or below that at point B in Fig. 4). On the practical significance of the Hybrid option, the literature on migration to colonial America identifies cases of families arriving from Europe with enough accumulated savings to pay outright for the fare of certain family members (typically the parents and very young children) with older children entering into debt-bondage arrangements to pay for their passage. Grubb (1985, p. 324) in fact finds a positive relationship between the number of children in a migrating family and the number of children from that family that ended up as servants. Thus in the case of German immigrant children arriving in the port of Philadelphia between 1785 and 1804, he finds that while an only child had a 24% chance of entering servitude, having three or more siblings practically doubled this chance. This is a clear indication that the hybrid option has been used in colonial America at the household level, with accumulated savings (SF) used to pay for passage of some family members, while relying on servitude contracts (DB) to pay for passage of other members. Thus the possibility of combining SF and DB options may have enabled some families to migrate under the conditions that would have made NM optimal if only the pure SF and DB options were available at the level of the family. 6. Conclusion The present study characterizes the economic environment in which migration is an attractive option for liquidity-constrained individuals and, when it is, under what conditions they choose debt-bondage as the optimal means of financing migration costs. What makes debtbondage appealing to potential migrants, in spite of the high interest charges and the prospect of being underpaid abroad while repaying the debt, is that this financing mode brings them sooner to the foreign,

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

high-wage economy. Getting abroad sooner is of greater significance when the international wage differential is larger. High interest charges on loans provided by human smugglers are, however, a disadvantage, the weight of which is greater, when the cost of migration is higher. We therefore find that debt bondage is the preferred mode of financing when the international wage differential is large in relation to migration costs. With respect to policy, we show that tougher border control measures may help reduce the incidence of debt-bonded migration. This goes against the conventional wisdom that higher migration costs compel more migrants to become indebted to the smugglers. Tougher internal enforcement measures that increase the risks and costs of operating a human-smuggling organization or employing bonded laborers tend to reduce migration flows and the incidence of debt-bonded relative to self-financed migration. The reductions in the flows are shown to be from the very poor source countries, where the local wage is low in relation to the cost of migration and the host-country wage. From other source countries with sufficiently high local wages, these policies may not deter illegal immigration, but rather induce a switch from debtbonded to self-financed migration. The possibility of self financing a part of migration costs and borrowing the rest from a smuggler is also examined. This hybrid option, which optimally combines SF and DB, is shown to render migration attractive under certain conditions that make NM optimal if only the pure SF and DB options are available. We also consider the case where a fixed cost of entering into a loan agreement with a smuggler deters migrants from borrowing relatively small amounts. A fixed cost of borrowing results in the SF = DB schedule becoming U shaped. SF is then the optimal strategy when the cost of migration is low, with DB becoming attractive only when the international wage differential and the cost of migration are sufficiently high. Our baseline debt-for-labor contract is designed to capture the features of realistic arrangements that we observe in relation to current Chinese migration to the West or in the case of South-East Asian migrants seeking access to the lucrative labor markets of East Asia and North America. This leaves a number of interesting problems on the agenda for future research. One such problem is to characterize the optimal debt-for-labor contract in the policy environment that potential migrants and human smugglers face today. What determines the interest rate on the debt or the degree to which indebted migrants are underpaid by their creditors/employers? How do the various policies of the host country influence these variables in general equilibrium where the flows of migrants, their compensation and interest charges are endogenously determined? Just these few important questions leave us with a very rich agenda for future research.

Appendix A A.1. Analytic solutions for self-finance and debt-bondage cases This Appendix examines in more detail the optimization problem facing self-financed and debt-bonded migrants.

A.1.1. Self-financed migration A.1.1.1. Derivation of the solution. The Lagrangian function is given by

219

The first-order conditions, ∂L 0 −δt −ρt ¼ u ðct Þe −λe ¼ 0; ∂ct

ð27Þ

∂L 0 
 −δt −r
t −μe ¼ 0;
¼ u ct e ∂ct

ð28Þ

     
  ∂L −δϕ
−δϕ −ρϕ

−r ϕ −u cϕ e þ λ w−cϕ þ ρK e −μ w −cϕ e ¼ 0; ¼ u cϕ e ∂ϕ

ð29Þ and the budget constraints (2) and (3) determine the five endogenous variables ct, c∗t , ϕ, λ, and μ. Eqs. (27)–(28) relate the marginal utilities of consumption before and after ϕ to the utility values of wealth while in the source country (λ) and after migration (μ), respectively. Eq. (29) states that, at the optimal time of departure, ϕ, the cost of remaining in the source country for an extra instant, [u(c∗ϕ) − u(cϕ)]e−δϕ, must be  
  equal to the benefit, λ w−cϕ þ ρK e−ρϕ −μ w
−c
ϕ e−r ϕ , which is the utility value of the increase in wealth that results from staying in the source country an instant longer. A.1.1.2. Comparative statics. Total differentiation of Eqs. (5) and (7) of the main text yields the following comparative statics results γϕ  θ e −1 −ρϕ w−cϕ þ ρK e þ ρ dϕ c0 γ   ¼ N0 dK w−cϕ þ ρK Δ

θ eγϕ −1 1−e−ρϕ  w−cϕ þ ρK − dϕ ρ γ c0   ¼ ≷0 dw w−cϕ þ ρK Δ

γϕ 0
−δϕ e −1 u ðw Þe dϕ γ   b0 ¼− dw
c−θ e−ρϕ w−c þ ρK Δ 0

ð30Þ

ð31Þ

ð32Þ

ϕ

dc0 δe−ρϕ ¼ N0 dK Δ

ð33Þ

−ρϕ dc0 1−eρ ðρ−δÞ þ e ¼ ≷0 dw Δ

ð34Þ

dc0 u0 ðw
Þe−δϕ b0; ¼− dw
u0 ðc0 ÞΔ

ð35Þ

−ρϕ

  γϕ −ρϕ þ e γ−1 ðρ−δÞ. Note that Δ is unambigwhere Δ ¼ w−cϕ þ ρK cθ e 0

Z ϕ  Z T 
 −δt −δt −ρt −ρϕ uðct Þe dt þ u ct e dt þ λ ðw−ct Þe dt−Ke 0Z ϕ 0 T

 −r
t w −ct e dt; þμ Z

L¼

ϕ

ϕ

where λ and μ are the multipliers attached to the constraints (2) and (3) of the main text, respectively.

uously positive when ρ ≥ δ. These results can be summarized as follows: An increase in migration costs prolongs the period of saving at home prior to emigration and increases the initial consumption rate, c0. An increase in w has an ambiguous effect on both ϕ and c0, as shown in Eqs. (31) and (34). This reflects the opposing forces of the income and the substitution effects of an increase in w. By contrast, an increase in w⁎ makes it more urgent to emigrate earlier, encouraging the migrant to save at a higher rate (dc0/dw⁎ b 0 in Eq. (35)) and leave the source country sooner (dϕ/dw⁎ b 0 in Eq. (32)).

220

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

A.1.2. Debt-bonded migration A.1.2.1. Derivation of the solution. The Lagrangian function is given by Z τ   Z T      b −δt b
−δt b b b −rt u ct e dt þ u ct e dt þ λ w −ct e dt−K 0 Z 0  τ
T b
b
−r t w −ct e dt; þμ Z

b

L ¼

τ

τ

with the first-order conditions consisting of

 −θ cb0 eðδ−rÞτ θ dτ N0 ¼ b dK Δ cb0  −θ " # b ðδ−r Þτ b gτ c0 e dτ θ τc0 e −1 − ¼ −B r ≷0; θ dr g Δb cb0

ð36Þ

  ∂L 0 b
−δt b −r
t ¼ u ct e −μ e ¼ 0; b
∂ct

ð37Þ

b

    ∂Lb b −δτ b
−δτ ¼ u cτ e −u cτ e ∂τ    
b b b −r τ b
b
−r τ þ λ w −cτ e −μ w −cτ e ¼ 0;

ð38Þ

and the budget constraints (10) and (11). These five equations b b determine the five endogenous variables cbt , cb∗ t , τ, λ , and μ . Eqs. (36)–(37) are the usual Euler equations, while Eq. (38) states that when τ is optimally chosen, the cost (in terms of utility) of remaining ) − u(cbτ)]e−δτ, must be equal to in bondage an instant longer, [u(cb∗   τ 
−r τ the benefit, λb wb −cbτ e−r τ −μ b w
−cb
, which is the utility τ e value of net savings accumulated during this extra instant. A.1.2.2. Comparative statics. We totally differentiate the system of Eqs. (13) and (14) of the main text to obtain the following comparative statics results on the assumption that r N δ:

ð1−σ Þ dc0
¼ dw b

h    i 1−e−r τ 0  b  −r τ 0
0 b u cτ ðr−δÞ−e u w −ð1−σ Þu cτ r ≷0 Δb ð39Þ

 −θ   w
cbτ dcb0 1−e−r τ −r τ b0 ¼− ð r−δ Þ þ e dσ r Δb  −θ b ðδ−r Þτ b c0 e dc0 ¼− ðr−δÞb0 dK Δb  −θ ð Þ b  i cb0 e δ−r h dc0 b −r τ b gτ b0; ¼ Br ðr−δÞ−τ w e −c0 e b dr Δ

−r τ

τe

−r τ

− 1−er

both in bondage and after release. This gives rise to both income and substitution effects that pull in opposite directions so that dcb0/dw∗ and dτ/dw⁎ are in general ambiguous. Note, however, that if we decouple wb and w⁎ and consider the effects of an increase in w⁎, holding wb constant, the terms involving 1 − σ in expressions (39) and (43) vanish. Both dcb0/dw∗ and dτ/dw⁎ are then unambiguously negative. In that context, an increase in w⁎ has only the effect of making the post-bondage phase more attractive to be in, inducing a migrant in debt to save at a higher rate so as to gain release from bondage sooner. Alternatively, if one sets σ = 0 in Eq. (39), an increase in w⁎ raises the bonded wage by the same amount, in which case the optimal response is to increase consumption (but also the saving rate) while in bondage. An increase in σ lowers the migrant's optimal consumption rate and has an ambiguous effect on the duration of the repayment period, as shown in Eqs. (40) and (44). An increase in K tightens the migrant's budget constraint while in bondage, causing cb0 to fall and τ to increase in Eqs. (41) and (45), respectively. Finally, an increase in r encourages the migrant to save more in an effort to repay the debt more quickly (i.e., dcb0/dr b 0, as shown in Eq. (42)). At the same time it lowers the present value of savings generated in bondage, requiring a longer repayment period. When the optimal saving rate is already relatively high, either because of a large r or a large gap between w⁎ and wb, dw/dτ N 0. Otherwise τ decreases with an increase in r. Thus in general, dτ/dr is of ambiguous sign, as shown in Eq. (46). For a more extensive analysis of the behavior of debt-bonded migrants, see Djajić and Vinogradova (2013). A.2. Derivation of the three schedules

ð40Þ

ð41Þ

ð42Þ

This appendix derives and compares the slopes of the SF = NM, DB = NM and SF = DB schedules. The derivations are based on the assumption that r N r⁎ = δ = ρ. A.2.1. The SF = NM schedule and it's slope Along the SF = NM schedule the welfare of an SF migrant is identical to that of a non-migrant, so that DSF ≡ USF − UNM = 0 or " # " # " # c1−θ 1−e−δϕ ðw
Þ1−θ e−δϕ −e−δT w1−θ 1−e−δT 0 þ − ¼ 0: ¼ 1−θ δ 1−θ δ 1−θ δ

SF

D

  2 3 0
0 b gτ −r τ dτ 1 4e −1 u ðw Þ−ð1−σ Þu cτ 1−e θ 0  b 5 ¼− b þ ð1−σ Þ u cτ ≷0 r dw
g wb −cbτ Δ cb0

ð43Þ  −θ " #
b dτ w cτ θ 1−e−r τ egτ −1 −   ≷0 ¼ r dσ Δb cb0 g wb −cbτ



ð46Þ

 b   gτ gτ τe − e g−1 b 0 represents the − cg0 1−θ θ  −θ b b effect of r on the migrant's budget while in bondage and Δ ¼ cτ h gτ   i b b θ gτ e −1 N0. g ðr−δÞ þ w −cτ cbτ e b ⁎ Since w = (1 − σ)w , an increase in w⁎ raises the migrant's wage b

where Br ¼ wr

  ∂L 0 b −δt b −r t ¼ u ct e −λ e ¼ 0; b ∂ct b

ð45Þ

The slope of this schedule in w⁎ and K space is given by

SF

ð44Þ

SF

∂D dc0 ∂D dϕ  þ dw  dDSF =dK ∂c dK ∂ϕ dK ¼ − SF 0 N0; DSF ¼0 ¼ − SF
dK  dD =dw ∂D dc0 ∂DSF dϕ ∂DSF þ þ ∂c0 dw
∂ϕ dw
∂w



S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

where SF

∂D 1−e−δϕ 0 N0; ¼ u ðc0 Þ δ ∂c0 dc0 δ ¼ N0; θ dK ðw−c0 þ ρK Þ c0 
 −δϕ ∂DSF 0 −ρϕ ¼ −u ðc0 Þðw−c0 þ ρK Þe b0; ¼ uðc0 Þ−u w e ∂ϕ   θ −ρϕ −ρϕ ðw−c0 þ ρK Þ e þ 1−e dϕ c0 N0; ¼ θ −ρϕ dK ðw−c0 þ ρK Þ2 e c0 dc0 u0 ðw
Þ b0;
¼− θ dw u0 ðc0 Þðw−c0 þ ρK Þ c0 −ρϕ 0
1−e u ðw Þ dϕ ρ b0;
¼− θ −ρϕ dw u0 ðc0 Þðw−c0 þ ρK Þ2 e " # c0 −δϕ ∂DSF −e−δT 0
 e N0: ¼u w δ ∂w
Hence, the numerator of the slope can be written as SF

  0 −δϕ δ u ðc0 Þ 1−e

SF

∂D dc0 ∂D dϕ þ ¼ ∂c0 dK ∂ϕ dK

θ δðw−c0 þ ρK Þ  c0  θ −ρϕ  −ρϕ u0 ðc0 Þðw−c0 þ ρK Þe−ρϕ ðw−c0 þ ρK Þ e þ 1−e c0 − 2 θ −ρϕ ðw−c0 þ ρK Þ e c0  0   u ðc0 Þ θ −ρϕ  −δϕ −ρϕ −ðw−c0 þ ρK Þ e ¼ 1−e − 1−e θ c0 ðw−c0 þ ρK Þ c0   0 u ðc0 Þ θ −ρϕ 0 −ρϕ ¼ −ðw−c0 þ ρK Þ e ; ¼ −u ðc0 Þe θ c0 ðw−c0 þ ρK Þ c0

where the last equality follows from our assumption that ρ = δ. The denominator can be written as ∂DSF dc0 ∂DSF dϕ ∂DSF þ þ ∂c0 dw
∂ϕ dw
∂w
1−e−δϕ 0 
 −δϕ u w e δ ¼− θ u0 ðc0 Þðw−c0 þ ρK Þ c0 −ρϕ 1−e −δϕ e u0 ðc0 Þðw−c0 þ ρK Þe−ρϕ u0 ðw
Þ ρ þ θ e−ρϕ u0 ðc0 Þðw−c0 þ ρK Þ2 c 0 1−e−δϕ 0 
 −δϕ −δϕ −δT u w e −e 0
 e δ ¼− þu w θ δ ðw−c0 þ ρK Þ c0 −ρϕ −δϕ 0
1−e e u ðw Þ −δϕ −δT −δϕ −δT −e −e ρ 0
 e 0
 e ¼u w : þ þu w θ δ δ ðw−c0 þ ρK Þ c0

221

A.2.2. The DB = NM schedule and its slope Along the DB = NM schedule the welfare of a debt-bonded migrant is identical to that of a non-migrant. It follows that DDB ≡ UDB − UNM = 0 or

DB

D

 1−θ  gτ  cb0 e −1

¼

1−θ

g

" # " # ðw
Þ1−θ e−δτ −e−δT w1−θ 1−e−δT − þ 1−θ δ 1−θ δ

¼ 0: The slope of this schedule in w⁎ and K space is given by ∂DDB dcb0 ∂DDB dτ þ  DB  ∂τ dK dw  dD =dK ∂cb0 dK ¼ − ¼ − N0; DB b DB DB dK DDB ¼0 dDDB =dw
∂D dc0 ∂D dτ ∂D þ þ
∂τ dw
∂w
∂cb0 dw


where   egτ −1 ∂DDB 0 b ¼ u c0 N0; g ∂cb0 b

dc0 ¼ − " gτ dK e −1 g

ðr−δÞ # b 0;   b b −r τ θ ðr−δÞ þ w −cτ e cb0

h      
i −δτ ∂D b 0 b b b ¼ −u cτ w −cτ b 0; ¼ u cτ −u w e ∂τ DB

b

dτ θ=c0 N 0; ¼   dK egτ −1 b b −r τ θ ðr−δÞ þ w −cτ e b g c0 ð1−σ Þ dcb0 ¼ dw


−r τ   h    i 1−e 0 b 0
0 b −δτ u c0 ðr−δÞ− u w −ð1−σ Þu cτ e r " # ≷0; gτ     e −1 b b −r τ θ u0 cb0 ðr−δÞ þ w −cτ e b g c0

 i egτ −1 h 0 
 1−e−r τ θ 0  b  b b  0 b u cτ w −cτ u w −ð1−σ Þu cτ −ð1−σ Þ r g cb0 " # b 0; gτ      e −1 b b −r τ θ u0 cbτ wb −cbτ ðr−δÞ þ w −cτ e g cb0 " # DB −δτ −δT ∂D −e 0
 e N 0: ¼u w δ ∂w


dτ ¼ dw


−

0

u ðc0 Þ

The slope of the SF = NM schedule can thus be simply written as:
 dw  ¼ dK DSF ¼0

0

−ρϕ

u ðc0 Þe u0 ðw
Þ

−δϕ

e

−δT

−e δ

N0:

After some simplifications (see the longer version of this paper, which is available from the authors on request), the slope of the DB = NM schedule can be written as  dw
  DB ¼ dK D ¼0

  u0 cb0 ð1−σ Þ

−δτ 1−e−r τ 0  b  −e−δT 0
 e u c0 þ u w r δ

N0:

A.2.3. The SF = DB schedule and its slope Along the SF = DB schedule, a migrant's welfare under the selffinance scenario is identical to that under debt-bondage. We thus have D ≡ USF − UDB = 0 or " #  b 1−θ  " #  c0 c1−θ 1−e−δϕ egτ −1 ðw
Þ1−θ e−δϕ −e−δτ 0 − ¼ 0: D¼ þ 1−θ δ 1−θ 1−θ δ g

222

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224 −δϕ   egτ −1   1−e b b 0 b −uðc0 Þ 
 u c0 1−e−r τ c0 u c0 δ g ð1−σ Þw ∼u w uðw
Þ r ð1−θÞ cb0   b −r τ u c   egτ −1 0 1−e−δϕ b 1−e b ∼u c0 w −uðc0 Þ b r δ g c0   b  gτ −δϕ u c0  b 1−e−r τ 1−e b e −1 −c0 w ∼−uðc0 Þ b r δ g c0

The slope of this schedule in w⁎ and K space is given by




 dw
 dD=dK ¼− dK D¼0 dD=dw
∂D dc0 ∂D dcb0 ∂D dϕ ∂D dτ þ þ þ ∂c0 dK ∂cb0 dK ∂ϕ dK ∂τ dK ¼− N0; ∂D dc0 ∂D dcb0 ∂D dϕ ∂D dτ ∂D þ þ þ þ ∂c0 dw
∂cb0 dw
∂ϕ dw
∂τ dw
∂w
∂D ∂c0

where −δϕ

u0 ðw
Þ e

SF

¼ ∂D ; ∂c 0

−e−δτ δ

 dw  ¼ dK D¼0

∂D ∂cb0

SF

DB

¼ − ∂D ; ∂cb 0

∂D ∂ϕ

SF

¼ ∂D ; ∂ϕ

∂D ∂τ

DB

¼ − ∂D∂τ ;

∂D ∂w


recognizing that the expression in the brackets is equal to K (see Eq. (13) in the text), we finally obtain

¼

  b u c0

DB

¼ ∂D − ∂D , so we have ∂w
∂w


cb0

  b 0 −ρϕ u c0 −u ðc0 Þe 0




ð1−σ Þ

−r τ

1−e r

: −δϕ −δτ   −e 0 b 0
 e u c0 −u w δ

    e−δϕ −e−δτ 0 b 0
u c0 ∼u w δ w
multiplying both sides by yields 1−θ −r τ     e−δϕ −e−δτ
1−e 0 b
ð1−σ Þw u c0 ∼u w r ð1−θÞ δ e−δϕ −e−δτ substituting for from D ¼ 0 condition δ −r τ

y þ u0 ðw
Þ

12

12

10

10

8

6

4

4

2 0.2

0.3

0.4

0.5

2 0.5

0.6

−δT

−e δ

  u0 cb0 −u0 ðc0 Þe−ρϕ

∼

y−u0 ðw
Þ

−δϕ

e

−e δ

−δτ

:

Equilibrium points (circle), threshold points (x) δ=r*=ρ=5%, r=30%, σ=1/3, T=30

8

6

0.1

−δτ

e

14

w*/w

w*/w

  u0 cb0

Equilibrium points (circle), threshold points (x) δ=r*=ρ=5%, r=30%, θ=0.95, T=30

0

:

A.2.4. Comparing the slopes We first show that when the SF = NM schedule is steeper than the DB = NM schedule, DB = NM is necessarily steeper than SF = DB.   −r τ Proof. Let us define y ¼ ð1−σ Þ 1−er u0 cb0 for notational convenience. Then compare the following expressions, which represent the slope of DB = NM (LHS) and the slope of SF = DB (RHS):

1−e r

14

−δϕ

1−e δ

Hence, the denominator is also positive and SF = DB is therefore positively sloped.

The numerator is unambiguously positive, while the sign of the denominator is determined by comparing the two terms:

0.6

0.7

0.8

σ

0.9

1

1.1

1.2

1.3

1.4

1.5

θ

b) Elasticity of marginal utility, θ.

a) Wage penalty, σ . Equilibrium points (circle), threshold points (x) δ=r*=ρ=5%, σ=1/3, θ=0.95, T=30

Equilibrium points (circle), threshold points (x) σ=0, θ=1.5, T=30, r*=ρ=0.05

8

25

7 20 6 15

w*/w

w*/w

ð1−σ Þ

K N−uðc0 Þ

10

4 3 2

5 0.2

5

0.3

0.4

0.5

0.6

0.7

0.8

1 0.2

r

c) Smuggler’s interest rate, .

0.3

0.4

0.5

0.6

0.7

0.8

r

d) Smuggler’s interest rate, , for σ = 0 and θ = 1.5. Fig. 5. Robustness checks.

S. Djajić, A. Vinogradova / Journal of International Economics 93 (2014) 210–224

After diagonal multiplication we obtain:       e−δϕ −e−δτ     e−δτ −e−δT 0 b 0 b 0
0 b 0
∼u c0 u w u c0 y−u c0 u w δ δ −δτ −δT   −e 0 b 0 −ρϕ 0 
 e 0 −ρϕ −u ðc0 Þe þu c0 y−u ðc0 Þe u w y: δ Eliminating identical terms on both sides and collecting remaining terms: " # −δτ −δT     e−δϕ −e−δT −e 0 b 0
0 −ρϕ 0
 e ∼−u ðc0 Þe þy ; u w −u c0 u w δ δ multiplying both sides by −1 and keeping in mind that the inequality sign will change: " # −δτ     e−δϕ −e−δT −e−δT 0 b 0
0 −ρϕ 0
 e ∼u ðc0 Þe þy ; u w u c0 u w δ δ which can be rewritten as:   0 b u c0 u0 ðw
Þ

e

−δτ

−δT

−e δ

0

u ðc0 Þe

b

þ y u0 ðw
Þ

e

−ρϕ

−δϕ

−e−δT δ

;

where the inequality sign follows from the fact that the LHS is the slope of DB = NM and the RHS is the slope of SF = NM. □ Next, we show with the aid of Fig. 5 that, for realistic parameter values, the SF = NM schedule is in fact steeper than the DB = NM schedule in a large neighborhood of the triple-indifference point. Consider Fig. 5a, for example, where we hold the key parameters of the model at their benchmark levels, including θ = 0.95 and r = 0.3. On the horizontal axes we plot σ (the wage penalty faced by bonded laborers) covering a wide range from 0 to 0.6, while on the vertical axis we have w⁎/w. For each σ, a black dot plots the wage differential corresponding to the triple-indifference point, such as A in Fig. 1. Although each point corresponds to the equilibrium combination of w⁎/w and K/w, we show only the wage differential in order to keep the figure in just two dimensions. The corresponding value of K/w is taken into account in all the calculations. For that value of K/w, an x plots the threshold value of w⁎/w for each σ, such that the slope of SF = NM is exactly equal to that of DB = NM. For any wage differentials above (below) an x, SF = NM is in fact steeper (flatter) than DB = NM. As the figure demonstrates, the equilibrium wage differential (black dot) for every value of σ 4

x 10 1.68 1.67

discounted utility

1.66 DB

U

1.65

, r = 10% U

DB

, r = 20% DB

1.64

U

, r = 30%

SF

U

1.63

UNM

1.62

K

K1

0

1.61 1.6

0

50

100

150

200

250

300

350

400

450

migration cost, K Fig. 6. Welfare of SF, DB and NM as functions of migration cost.

500

223

is largely above the threshold wage differential (x), with the distance increasing with the value of σ. We conduct a similar robustness check with respect to the elasticity of marginal utility, θ, in Fig. 5b. The range of θ is from 0.5 to 1.5 to cover the empirically relevant values for this parameter. The other parameters are kept at their benchmark levels. Clearly, the triple-indifference points that correspond to the intersection of the three schedules are largely above the threshold wage differentials, with the gap between the two diminishing in θ. In Fig. 5c we have the smuggler's interest rate, r, on the horizontal axis, ranging from an unrealistically low value of 0.2 up to 0.8, with other parameters at their benchmark values. We note that the gap between the equilibrium and the threshold value of the wage differential is increasing in r. Finally, in Fig. 5d, we stack the cards against the case that the SF = NM schedule is steeper than the DB = NM locus by setting σ = 0 and θ = 1.5, while allowing r to vary again between 0.2 and 0.8. We find that the necessary and sufficient condition for SF = NM to be steeper than DB = NM is still satisfied at the triple-indifference point. Note, however, that as r is reduced further, the SF option eventually becomes irrelevant. As we explain below in Appendix A.2.5, it is dominated by DB for a sufficiently low cost of borrowing from the smuggler. A.2.5. Existence Existence of the triple-indifference point essentially requires that the economic environment is not too “favorable” for a DB migrant, so that SF can in fact be optimal for at least some values of migration cost, K, and the foreign free-market wage, w⁎. That is to say, the interest rate, r, charged by the smuggler and the wage discount, σ, should be sufficiently large, so that SF dominates DB over a certain range of K and w⁎. If r is close enough to the rate of time preference and the bonded wage is similar to the market wage abroad, SF is dominated by the other two options and the triple-indifference point does not exist. In Fig. 6 we plot the discounted utility of a DB migrant (UDB) and that of an SF migrant (USF) as functions of K for a given foreign wage, w⁎ = 2. This represents a 100% premium over the home-country wage and we can think of it as a lower bound for w⁎ in a model of illegal immigration. In this illustration, the other parameter values are set as in our benchmark case, except that σ = 0 (i.e., the bonded wage is the same as the market wage), and we focus solely on the role of the interest rate, which takes on three different values: 10, 20 and 30% (r = 30% is our benchmark). Since the welfare of NM (UNM) is independent of K, it is just a horizontal line. USF is the convex schedule and UDB is the concave schedule (the convexity and concavity can be shown mathematically). Consider a value of K, call it K0, such that USF intersects UNM. The necessary and sufficient condition for the triple-indifference point to exist, is that the UDB schedule intersects the UNM schedule at a value of K ⩽ K0 when w⁎ is at its lower bound. In other words, UDB must be “concave enough” in K, as shown by the solid and the dotted schedules in the graph. This strong concavity in K (which is also the stock of debt for a DB migrant) is generated by either a high enough r or a large enough σ or some combination of both. If the UDB schedule is “insufficiently” concave, as shown by the dashed line corresponding to r = 10%, then the DB option dominates SF for all K which are smaller than the value corresponding to the intersection of UDB and UNM (call it K1). On the other hand, for migration costs above K1, the NM option dominates both DB and SF, and so the SF option is not optimal for any value of K. We note that if this is the case for some w⁎, it is also true for any higher value of w⁎, because an increase in the foreign wage has a stronger positive impact, ceteris paribus, on UDB relative to USF. In sum, the triple-indifference point exists if and only if K1 ≤ K0, where K1 is such that UDB = UNM and K0 is such that USF = UNM. For the particular calibration in Fig. 6, with σ = 0, it turns out that the interest rate must be at least 20% or higher in order to guarantee the existence of the triple-indifference point. This critical value of r is smaller, the smaller is w⁎ and the larger is σ. We should note, however,

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that if the triple-indifference point does not exist, it merely implies that one of the migration options — SF in the example above with r = 10% — is not optimal regardless of what it costs to migrate. An individual simply chooses in that case between DB and NM. Fig. 1 of the main text is then split into two (instead of three) relevant regions, with the DB = NM schedule serving as the border between them. For any combination of w⁎ and K above the schedule, migrants will choose DB and for any combination below the schedule, NM is optimal. The SF = NM locus is then entirely above the DB = NM schedule, while the SF = DB locus is entirely to the right of it.22 References Andrienko, Y., Guriev, S., 2004. Determinants of interregional mobility in Russia. Evidence from panel data. Econ. Transit. 12 (1), 1–27. Angelucci, M., 2004. Aid and migration: an analysis of the impact of Progressa on the timing and size of labour migration. IZA Discussion Paper No. 1187. Chin, K.-l., 1999. Smuggled Chinese. Temple University Press, Philadelphia. Chiswick, B.R., Hatton, T.J., 2006. International migration and the integration of labor markets. IZA Discussion Paper No. 559. Djajić, S., Vinogradova, A., 2013. Undocumented migrants in debt. Labour Econ. 21, 15–24. Ethier, W., 1986. Illegal immigration: the host country problem. Am. Econ. Rev. 76 (1), 56–71. Friebel, G., Guriev, S., 2006. Smuggling humans: a theory of debt-financed migration. J. Eur. Econ. Assoc. 4, 1085–1111. Galenson, D.W., 1984. The rise and fall of indentured servitude in the Americas: an economic analysis. J. Econ. Hist. 44, 1–26. Gao, Y., 2004. Chinese migrants and forced labour in Europe. ILO WP 32, Geneva. Gao, Y., Poisson, V., 2005. Le traffic et l'exploitation des immigrants chinois en France. ILO, Geneva. Gathmann, C., 2008. Effects of enforcement on illegal markets: evidence from migrant smuggling along the southwestern border. J. Public Econ. 92, 1926–1941. Grubb, F., 1985. The incidence of servitude in trans-Atlantic migration, 1771–1804. Explor. Econ. Hist. 22, 316–339. Halliday, T., 2006. Migration, risk, and liquidity constraints in El Salvador. Econ. Dev. Cult. Chang. 54 (4), 893–925. Hanson, G.H., Spilimbergo, A., 1999. Illegal immigration, border enforcement, and relative wages: evidence from apprehensions at the U.S.–Mexico border. Am. Econ. Rev. 89, 1337–1357.

22 This discussion takes for granted that wb N w, as assumed throughout the paper, otherwise it never pays to go abroad as a bonded laborer.

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