## Abstract

This paper presents a two-period human capital investment model, which is used to study the optimal investment decisions of credit-constrained married immigrants relative to single immigrants and native couples facing a perfect capital market. The model predicts that: (1) The comparative advantage in investment in local skills of one of the spouses may emerge from his/her higher growth rate of imported human capital. (2) The optimal investment of each spouse is non-increasing in the level of imported human capital of the spouse with the comparative advantage in investment, while it is non-decreasing in the level of imported human capital of the other spouse. (3) When two immigrants get married, the spouse with the comparative advantage in investment invests more than when he/she was single whereas the other spouse invests less.

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## Notes

- 1.
Eckstein and Weiss (2004) included this feature in their model in the context of a single immigrant’s investment in human capital when facing a perfect capital market.

- 2.
In the literature on marriage, Weiss (1997) explored the benefit of marriage under binding liquidity constraints when the human capital investment technology and the utility function are linear. In his model, the comparative advantage in investment in, for example, schooling emerges from the direct costs of the investment. However, due to the linearity of the technology, the model leads to full specialization. In the model presented here, it may be optimal for both spouses to partially invest.

- 3.
- 4.
Cobb-Clark and Crossley (2004) represent an exception in the literature since they do not make the a priori assumption that the wife is the secondary earner, but rather assume (ad hoc) that the principal applicant for a visa to Australia is the spouse who is meant to invest in human capital.

- 5.
For a more detailed discussion, see Eckstein and Weiss (2004).

- 6.
Recent studies have found empirical evidence that locally acquired and imported human capital are complements in “producing earnings”. For example, Duleep and Regets (1999) found that learning the host-country language increases the transferability of imported skills.

- 7.
Assuming that investment in any skill requires the same sacrifice of earnings, and since the prices of local skills are fixed (over time), each immigrant will choose to invest only in the skill with the highest rate of return (

*θ*_{ s }). - 8.
Bernhardt and Backus (1990) use a different approach to study the investment decisions of married couples. In their study, human capital is accumulated on the job and does not lead to a loss of earnings.

- 9.
In terms of the rate of return to skills,

*b*_{ j }is defined by the following equality: \(Ln\left( {1 + \beta _j } \right) = \sum\limits_{{\text{s}} \in {\text{S}}_0 } {\left( {\theta _{s2} - \theta _{s1} } \right)z_{s{\kern 1pt} j} } \). - 10.
*N*otice that*β*_{ j }= 0 reflects cases where either the prices of imported skills faced by the immigrant upon arrival are equal to those of native-born individuals or imported skills are specific to the source country and thus cannot be adapted to the host country’s labor market with time spent in the new country. - 11.
The model produces opposite results when

*β*_{ h }<*β*_{ w }. - 12.
Given the specification of the earnings production function,

*x*_{ th }and*x*_{ tw }also represent the share of time spent on investment at time*t*by the husband and wife, respectively. - 13.
By definition, a credit-constrained family cannot borrow but can transfer money to the next period through the capital market i.e., it is able to save.

- 14.
This paper focuses on immigrant families that would like to borrow but cannot. Therefore, their Earnings Possibilities Frontier, in the relevant range, is also their Consumption Possibilities Frontier.

- 15.
One of the elements that distinguish immigrants from native individuals is that immigrants have skills that were acquired abroad and are being adapted to the host country’s labor market. Thus, the present discussion focuses on the effect of the degree of adjustment on investment in human capital and assumes that the spouses differ in the exogenous growth rate of imported human capital,

*β*_{ j }. However, spouses may also differ in*g*(*x*_{ tj }) and/or the rate of depreciation of local human capital. In that case, the comparative advantage in investment is determined by the product \(\left( {1 + \beta _j } \right)\frac{{dg^j }}{{dx_{1j} }}\). - 16.
The concavity of

*g*(*x*_{ tj }) implies that an iso-*y*_{2}is strictly convex whereas the slope of an iso-*y*_{1}, which is equal to \( - \frac{{K_{01h} }}{{K_{01w} }}\), is constant. Thus, the tangency condition is a sufficient condition for the maximization of*y*_{2}for a given*y*_{1}. - 17.
- 18.
Note that in Fig. 1b, from point

*B*_{2}leftward, the family’s optimal strategy is to transfer money to the second period via the capital market (i.e., savings) rather than through investment in local skills. Thus, immigrants who face binding liquidity constraints will not choose to be from point*B*_{2}leftward. - 19.
Browning and Chiappori (1998) show that when deriving household demand, identical preferences make it possible to use the conventional “unitary” model in which the family maximizes the utility of one of the spouses within the household budget constraint.

- 20.
The effect of an increase in

*β*can be decomposed into a substitution effect and an income effect that work in opposite directions. The resulting increase in the slope of the EPF, for each level of investment, reflects the substitution effect which induces lower current consumption, i.e., higher investment, whereas the increase in potential earnings reflects the income effect which induces higher current consumption (due to the normality of current consumption), or, in other words, lower investment. Given the assumed specific preferences, i.e., \(MRS = \frac{{y_2 }}{{y_1 }}\), the two opposing effects offset each other and thus the investment by singles is independent of their*β*. The same applies to an increase in*β*for married immigrants who have the same*β*(see Lemma 5 in Appendix B). - 21.
Investing less is equivalent to working more. Thus, in case that\(\frac{{1 + r * }}{{1 + \beta _w }} > 1 + r\) immigrant wives’ relative (to natives) labor supply schedule is downward sloping with respect to time, and vise versa if \(\frac{{1 + r*}}{{1 + \beta _w }} < 1 + r\) (recall that in the second period both immigrants only work).

- 22.
From Eq. 1, it follows that when estimating a wage regression where the dependent variable is the logged wage, the ratio

*K*_{01h }/*K*_{01w }is represented by the difference between the levels of the spouses’ imported skills (e.g., years of schooling). - 23.
In principle, the same proof holds for each MRS that is a positive monotonic transformation of the above MRS.

## References

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## Acknowledgment

The authors are grateful to three anonymous referees and to Yoram Weiss for their valuable comments. We also benefited from comments by participants at the Tel-Aviv University labor economics seminar. Sarit Cohen-Goldner acknowledges the generous support from GIF, the German-Israeli Foundation for scientific research and development.

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## Additional information

Responsible editor: Klaus F. Zimmermann

## Appendices

### Appendix A

### Proof of Lemma 3

*Under the assumptions of Lemma 1, the slopes of each spouse’s indifference curve and of the corresponding family’s indifference curve will be equal to *
\(\frac{{U_{y_1 } \left( {1 + \rho } \right)}}{{U_{y_2 } }} = \frac{{y_2 }}{{y_1 }}\left( {1 + \rho } \right)\).^{Footnote 23}
*Totally differentiating the first-order conditions for maximization, Eq. *
14
*, results in:*

where

and

Substituting *y*
_{1} and *y*
_{2} into *E*
_{
w
} and *E*
_{
h
} results in:

and

Assuming that *β*
_{
h
} > *β*
_{
w
}, one obtains *x*
_{1h
} > *x*
_{1w
} and thus:

Solving for \(\frac{{\partial x_{1h} }}{{\partial K_{01h} }}\), \(\frac{{\partial x_{1h} }}{{\partial K_{01w} }}\), \(\frac{{\partial x_{1w} }}{{\partial K_{01h} }}\) and \(\frac{{\partial x_{1w} }}{{\partial K_{01w} }}\) one obtains:

where \(\left| A \right| = a_{11} a_{22} - a_{12} a_{21} >0\). QED

### Appendix B

### Lemma 5

*If both spouses have identical exogenous growth rate of imported human capital, *
*β*
*, then their on-the-job proportion of investment will be identical and will be independent of *
*β*.

###
*Proof*

The family’s optimal investment policy is characterized by Eq. 14. That is:

where

Since *β*
_{
w
} = *β*
_{
h
} = *β*, one obtains that Eq. (27) is equivalent to:

Equation 28 implies that the proportion of investment by each spouse will be identical and independent of *β*. QED

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Cohen-Goldner, S., Gotlibovski, C. & Kahana, N. The role of marriage in immigrants’ human capital investment under liquidity constraints.
*J Popul Econ* **22, **983 (2009). https://doi.org/10.1007/s00148-008-0196-5

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### Keywords

- Human capital investment
- Immigrants
- Marriage
- Binding liquidity constraints

### JEL Classification

- D10
- J24
- J61