### 3.1 Modeling the effect of retirement on return migration

We begin by developing a simple model of the net benefit of
return migration concentrating on immigrants’ decisions about where (rather than
how much) to work^{7}. Immigrants decide whether or not to
return to their country of origin on the basis of the total future consumption
achievable in the two countries until the end of life. The model is static and we
do not account for either uncertainty in—or the trajectory of—wages, prices, or
consumption over time. This simple approach allows us to abstract from unnecessary
complexity.

Our main interest is in understanding how retirement affects
the incentives for return migration. An individual’s retirement date is assumed to
be determined outside the model, perhaps as a result of institutional arrangements
that define the age at which he or she may access either public or
employer-provided pension benefits. Immigrants save throughout their working lives
to fund consumption in retirement. We assume that there are fixed costs associated
with return migration, for example retirement savings may only be partially
portable and thus transferring them to the origin country may involve a loss of
benefits. Finally, we assume that immigrants’ preferred bundle of consumption
goods is constant across countries, but that it is less costly in the origin than
in the host country (see Stark et al. 1997; Dustmann and Kirchkamp 2002).^{8}

Time is continuous in the model. Immigrants begin their lives
in at *t* = 0 in the host country and die at
*t* = 1. Retirement occurs at time τ with
0 < τ < 1. Consider first the savings process. In the period prior to
retirement, immigrants save a portion of their earnings to fund post-retirement
consumption. Accumulated retirement savings at time *t* are then given by:

{S}_{t}=\left(1-R\right)\left[t\left({w}^{H}-{c}^{H}\right)\right]+R\left[\tau \left({w}^{H}-{c}^{H}\right)-\left(t-\tau \right){c}^{H}\right]

(1)

where *w*^{H}
denotes host-country wages, *c*^{H}is
the consumption level in the host country, and R is an indicator variable that
takes the value 0 in the pre-retirement period (*t < τ*) and 1 in the post-retirement period (*t* ≥ *τ*). In the
pre-retirement period (*R* = 0), savings are
equal to total earnings minus total consumption to date. In the post-retirement
period (*R* = 1), savings equal the total savings
accumulated at retirement minus any post-retirement consumption. Consumption
levels are chosen so as to exhaust any savings at the end of life.

Following others in the literature (e.g. Dustmann 1997a, 2008; Dustmann and Kirchkamp 2002), we assume that at each period *t* = *t*^{*}
immigrants make a decision whether or not to leave the host country and return
home. Immigrants benefit from return migration if their accumulated retirement
savings and future earnings afford a higher standard of living in the origin
country than in the host country. Specifically, the net benefit to return
migration at time *t*^{*}
is given by the difference in future total consumption achievable in the two
countries. Given that we assume that there are no bequests and all resources are
exhausted at death, this implies that future consumption over one’s remaining life
time is equivalent to future resources. Immigrants are assumed to emigrate
whenever the net benefits from doing so are positive. Return migration occurs,
therefore, if and only if

N{B}_{{t}^{*}}={\mathrm{\Pi}}_{{t}^{*}}^{O}-{\mathrm{\Pi}}_{{t}^{*}}^{H}>0

(2)

where {\mathrm{\Pi}}_{{t}^{*}}^{O} and {\mathrm{\Pi}}_{{t}^{*}}^{H} are the future resources available at time *t*^{*} if immigrants do and do
not choose to return migrate, respectively. More specifically, the net benefit to
return migration at time *t*^{*} can be written in terms of accumulated
savings and any future earnings over one’s remaining career as follows

N{B}_{{t}^{*}}=\frac{1}{p}\left\{{S}_{{t}^{*}}+\left(\tau -{t}^{*}\right){w}^{O}\left(1-R\right)-\overline{C}\right\}-\left\{{S}_{{t}^{*}}+\left(\tau -{t}^{*}\right){w}^{H}\left(1-R\right)\right\}

(3)

where *w*^{O}captures origin-country wages, \overline{C} represents fixed costs (e.g. the loss of pension benefits,
travel costs, etc.) associated with return migration^{9}.
The host-country price level is normalized to 1 and relative origin-country prices
are given by *p*. We assume *w*^{O} < *w*^{H}
and *p* < 1 implying that although economic
opportunities are better in the host country than in the origin country,
immigrants’ preferred consumption bundle is less expensive at home.

The net benefit to return migration will be positive at time
*t*^{*} if the
resources available for consumption over an immigrant’s remaining life time are
higher in the origin country than in the host country. The last term in equation
(3) reflects the total resources
available if an immigrant decides to remain in the host country. Total resources
include retirement savings accumulated to time *t*^{*} while working in the host country as
well as an immigrant’s earnings over his or her remaining working life in the host
country. Post-return resources levels are given by the first term on the
right-hand side of equation (3). Although
accumulated savings are the same ({S}_{{t}^{*}}), future resources will be lower in the origin country because
*w*^{O} < *w*^{H}
and because return migrants must also pay the fixed costs associated with return
migration (\overline{C}). At the same time, each dollar of resources funds more
consumption in the origin country because prices (*p*) are lower. Consistent with other models in the literature (Djajić
1989; Dustmann 1997b; Stark et al. 1997), remigration may occur despite
persistently higher host-country wages because consumption is less expensive in
the origin country.

How does retirement affect the probability of return migration?
To address this question, we consider the way in which the incentives for return
migration change over time both before and after retirement. In the
post-retirement period (*t*^{*} ≥ *τ*), immigrants choose to return to their country of origin if and only
if

{M}_{{t}^{*}}=I\left(\frac{\left(1-p\right){S}_{{t}^{*}}-\overline{C}}{p}>0\right)

(4)

where *I* denotes a simple
indicator function and *M* reflects the return
migration decision. Substituting accumulated savings as given by Equation
(1) and rearranging implies that

\begin{array}{ll}\phantom{\rule{1em}{0ex}}{M}_{{t}^{*}}& =I\left(\left(1-p\right){S}_{{t}^{*}}>\overline{C}\right)\\ =I\left(\left(1-p\right)\left[{S}_{R}-\left({t}^{*}-\tau \right){c}^{H}\right]>\overline{C}\right)\end{array}

(5)

Hence, after retirement, return migration occurs if the costs
of return migration (\overline{C}) are less than the additional consumption made possible by
consuming one’s remaining savings in the origin country where prices are lower.
Equation (5) implies that the change in
the probability of return migration over time in the post-retirement period is
given by:

\frac{\partial \mathrm{Pr}\left(M=1\right)}{\partial {t}^{*}}=-\left(1-p\right){c}^{H}

(6)

Before retirement (i.e. in periods *t*^{'} < *τ*), however, immigrants also take into account the effect that
return migration will have on their future earnings. Given the net benefit to
return migration shown in Equation (3),
immigrants choose to return migrate in the pre-retirement period if and only if

{M}_{{t}^{\text{'}}}=I\left(\frac{{S}_{{t}^{\text{'}}}+\left(\tau -{t}^{\text{'}}\right){w}^{O}-\overline{C}-p\left[{S}_{{t}^{\text{'}}}+\left(\tau -{t}^{\text{'}}\right){w}^{H}\right]}{p}>0\right)

(7)

Substituting accumulated savings and rewriting implies that
immigrants choose to return migrate in the pre-retirement period whenever:

\begin{array}{ll}\phantom{\rule{1em}{0ex}}{M}_{{t}^{\text{'}}}& =I\left({S}_{{t}^{\text{'}}}+\left(\tau -{t}^{\text{'}}\right){w}^{O}-p{S}_{{t}^{\text{'}}}+p\left(\tau -{t}^{\text{'}}\right){w}^{H}>\overline{C}\right)\\ =I\left(\left(1-p\right){S}_{{t}^{\text{'}}}-\left(\tau -{t}^{\text{'}}\right)\left(p{w}^{H}-{w}^{O}\right)>\overline{C}\right)\\ =I\left(\left(1-p\right){t}^{\text{'}}\left({w}^{H}-{c}^{H}\right)-\left(\tau -{t}^{\text{'}}\right)\left(p{w}^{H}-{w}^{O}\right)>\overline{C}\right)\end{array}

(8)

Immigrants return migrate before retirement only if the
advantages of consuming one’s accumulated savings in the origin country outweigh
both the cost of return migration and the earnings loss associated with returning
to a low-wage labor market. Thus, the change in the probability of return
migration over the pre-retirement period is given by:

\frac{\partial \mathrm{Pr}\left(M=1\right)}{\partial t}=\left({w}^{H}-{w}^{O}\right)-\left(1-p\right){c}^{H}

(9)

There are several things to note about these changes over time.
First, the probability of remigration declines over the post-retirement period so
long as consumption in the origin country is less expensive than in the host
country (i.e. *p <* 1) (see Equation
(6)). Every year that return migration
is delayed involves a loss associated with consuming in the higher price market
which is no longer being compensated by higher wages. In the pre-retirement
period, the probability of return migration increases every year so long as the
wage advantage afforded by the host country dominates the higher living costs.
This will be true whenever there is a positive economic return to immigration to
the host country in the first place. Together these relationships imply that the
probability of return migration is maximized at the point of retirement when the
wage advantage of the host country relative to the origin country is no longer
relevant and the consumption benefits of moving one’s retirement savings to the
lower cost country are maximized.

### 3.2 Return migration rates and the retirement status of immigrant
populations

The simple model discussed above is useful in highlighting how
the incentives for return migration change when retirement occurs and higher
relative wages are no longer a factor in immigrants’ decisions about whether to
stay or to return home. We now show how this interdependence between emigration
and retirement affects the aggregate retirement status of the remaining immigrant
population and then link this directly to the empirical models that we
estimate.

Note that the probability that an immigrant *i* from sending country *j* retires in the host country is given by the joint probability:

\begin{array}{ll}\phantom{\rule{1em}{0ex}}\mathsf{Prob}\left({R}_{\mathit{ij}}=1,{M}_{\mathit{ij}}=0\right)& =\mathsf{Prob}\left({M}_{\mathit{ij}}=0|{R}_{\mathit{ij}}=1\right)\mathsf{Prob}\left({R}_{\mathit{ij}}=1\right)\\ =\left[1-\mathsf{Prob}\left({M}_{\mathit{ij}}=1|{R}_{\mathit{ij}}=1\right)\right]\mathsf{Prob}\left({R}_{\mathit{ij}}=1\right)\end{array}

(10)

where, as before, *R* and
*M* are indicator variables for being in the
post-retirement period and having return migrated, respectively. Equation
(10) demonstrates that there is a
negative relationship between the probability that remaining immigrants are
retired and the probability of emigrating in the post-retirement period. In the
limit, when return migration to country *j* is
nearly universal, none of the immigrants from country *j* remaining in the host country will be retired. This implies that
different immigrant populations will have different retirement profiles, not only
because individual retirement behavior differs, but also because variation in
sending-country wages or price levels lead to differing propensities of return
migration.

### 3.3 Estimation model

To empirically analyze the relationship between
country-specific return migration rates and the pattern of retirement, we estimate
reduced-form models of retirement status controlling for country of birth-specific
emigration rates, which proxy for the net benefits of emigration for immigrants in
HILDA from different countries since they reflect the prior migration decisions
made by one’s countrymen. In some models, we also control for individuals’
demographic and human capital characteristics. Including controls for
characteristics that are potentially related to retirement status, such as age,
years in Australia, education, and work experience, allows us to account for the
effect that differences in the composition of immigrant populations from different
countries of origin plays in explaining the relationship between country of
birth-specific emigration rates and retirement status. Since our objective is not
to estimate a behavioral model of the retirement decision, but rather to
understand the way that the propensity to be retired at a point in time (i.e.
retirement status) differs among individuals from different countries of birth, we
adopt a cross-sectional estimator, pooling data from multiple survey waves to
improve efficiency.

We assume that an individual’s propensity to be retired
({R}_{i}^{*}) can be expressed as:

{R}_{\mathit{ij}}^{*}={X}_{\mathit{ij}}\beta +{Z}_{j}\varphi +{\mathit{\u03f5}}_{\mathit{ij}}

(11)

where *X*
_{ij} captures demographic
and human capital characteristics, *Z*
_{j}, is the aggregate
emigration rate over the previous five years for each sending country (see Section
4.2) and *ϵ*
_{ij} is a random error
term. Emigration rates are calculated using administrative data and capture the
cross-national variation in institutional arrangements, price levels, etc. that
underlie the aggregate costs and benefits of emigration for individuals from each
specific origin country. The simple theoretical model discussed above shows that
we should expect to find a negative relationship between country-specific
emigration rates and the propensity for any individual immigrant to report being
retired.

The propensity to be retired is unobserved, so we create an
indicator variable reflecting actual retirement status. Specifically,

\mathrm{Pr}\left({R}_{\mathit{ij}}=1\right)=\mathrm{Pr}\left({X}_{\mathit{ij}}\beta +{Z}_{j}\varphi +{\mathit{\u03f5}}_{\mathit{ij}}>0\right)=\mathrm{\Phi}\left(\mathit{Q\gamma}\right)

(12)

where \mathcal{Q}=\left({X}_{ij}\text{,}{Z}_{j}\right), *γ* = (*β*,*ϕ*), and Φ is the standard
normal cumulative density function. Finally, we assume that *ϵ*
_{ij} *~ N*(0,1), is independent of the explanatory variables
in equation (12) and is
potentially clustered for individuals from the same country of birth *j*.^{10}