A dynamic model of migration

Hatton’s model builds on a microeconomic analysis which treats migration as a decision of a utility maximizing individual. A key dynamic feature of the model is that individual i’s decision to migrate at time t does not only depend on the current utility, V_t, but also on the net present value of all future utility streams, denoted V_t^∗. As such, the total net present value of staying in the origin country o at time t

is given by

U_iot= β (αVot+V_ot^∗) + εot (2.1)

where α > 1 reflects the extra weight given to current conditions and εot is an i.i.d. extreme-value distributed random term. Analogously, the net present value of residing in country d at time t can be written as

U_idt= β (αVdt+V_dt^∗) + εdt. (2.2)

The utility maximizing individual will choose the country that provides the best opportunities among the home country and all potential destination countries. In line with McFadden (1978), the probability that individual i will migrate from o to d at time t (m_idot = 1) can be written as where N_ot denotes the size of the native population in country o. Analogously, the probability that individual i will stay in country o at time t (m_idot = 0) is given by

Pr where N_ootdenotes the number of stayers in country o at time t. Combining equa-tions (2.4) and (2.6), aggregate migration from o to d to the resident population in

country o is given by

M_dot

N_oot = exp βαV_dt+ βV_dt^∗

exp(βαVot+ βV_ot^∗) (2.7) or, taking logs,

ln M_dot N_oot



= βαVdt+ βV_dt^∗− βαVot− βV_ot^∗ (2.8)

= βα (V_dt−V_ot) + β (V_dt^∗−V_ot^∗) (2.9)

= βαd_t+ βd_t^∗ (2.10)

where d_t denotes the current expected utility difference and d_t^∗captures expected future utility differences in year t. As suggested by Hatton (1995), the former can be defined as

d_t = Eu (y_dt) − Eu (y_ot) + z_dot (2.11)

where y denotes income and z_dot captures the average non-pecuniary utility dif-ferences between the two countries as well as the cost of migration. Following Todaro (1969), Hatton defines expected income as the wage (w) times the em-ployment rate (e), with income uncertainty being due to uncertain emem-ployment prospects. To take into account the welfare magnet theory presented in Borjas (1987, 1999), we extend this definition of expected income by adding the pro-vision of public services (ps) in the form of social protection benefits² (see also Pedersen et al., 2008; Warin and Svaton, 2008). Assuming a logarithmic utility function and a binomial distribution to characterize the probability of

employ-2The inclusion of public services might also be linked to the cost of migration, z_dot. In that sense, immigrants are expected to prefer countries with a generous system of public services since the presence of a safety net lowers the psychological cost of migration.

ment, equation (2.11) can be rewritten as

d_t = η1ln w_dt− η2ln w_ot+ η3ln ps_dt+ η4ln e_dt− η5ln e_ot+ z_dot. (2.12)

Assuming that expectations about future utility streams are a geometric series of past utility differences³

d_t^∗= λdt+ λ²d_t−1+ λ³d_t−2+ λ⁴d_t−3+ ... (2.13)

and applying a Koyck transformation gives

ln M_dot

Substituting (2.11) in (2.16) and rearranging results in the following aggregate dynamic migration equation

3As shown by Hatton (1995) this is consistent with rational expectations if d_itfollows an AR(1) process.

Hatton assumes that z_dot is determined by the stock of previous immigrants and a time variable such that

z_dot= γ0+ γ1ln MST_dot+ γ_t+ γ_do (2.17)

where MST_dot is the stock of migrants from origin country o residing in destina-tion country d at the beginning of time t. This stock variable is included to capture network effects: friends and relatives who already live in the host country reduce the monetary and psychological costs of migration. The higher the stock of pre-vious immigrants from the same origin country, the lower the costs of migration and the higher the immigrant flow. Nevertheless, this is not the only cost deter-mining factor. Also decreasing transportation and communication costs lower the cost of migration over time. In our model, these decreasing costs are captured by the year dummies γ_t. The latter might however also represent, among other things, the impact of joint changes in origin and destination countries’ emigration and immigration policies and they also capture the impact of changes in the ori-gin population ln (Noot/N_oot−1).⁴ Furthermore, also distance, common language, similar culture, colonial ties and immigration policy affect the cost of migration.

To the extent that these factors are time invariant, they are captured by the bilateral fixed effect γ_do.

The stock of migrants diminishes at a rate δ_dodue to deaths and return migration

4Given that we are not interested in the role of population growth in shaping bilateral migration, we assume equal population growth across countries to get rid of the population ratio in equation (2.16). Moreover, as argued by Egger and Pfaffermayr (2003), a three-way model that includes both bilateral effects and time effects is identical to a model that includes also country specific effects. As such, the empirical model described above indirectly controls for both time-variant and cross-sectional differences in population growth. Consequently, the population ratio can be disregarded without further implications.

but increases due to the inflow of new migrants such that

MST_dot= (1 − δdo) MST_dot−1+ M_dot−1. (2.18)

where δ_dois allowed to vary across destination and origin country pairs. In a later stage, this relationship will be used to account for the link between immigrant flows and stocks. For the moment, we use this expression to eliminate ln MST_dot−1 from z_dot−1in equation (2.16) by applying a logarithmic expansion of the migrant stock and its components in equation (2.18) about their mean values so that

ln MST_dot= (1 − Ω) ln [(1 − δ_do) MST_dot−1] + Ω ln M_dot−1 (2.19)

where Ω = (1−δ)MST +M^M > 0.⁵

5First, we can write ln {MST_dot/ [(1 − δdo) MST_dot−1]} as ln {1 + exp [ln M_dot−1− (1 − δdo) ln MSTdot−1]}. A first-order Taylor expansion of the latter around the mean values of M_dot−1 and (1 − δdo) MST_dot−1 gives ln {MST_dot/ [(1 − δdo) MST_dot−1]} ≈ Ω [ln M_dot−1− (1 − δdo) ln MSTdot−1] + c where c is an arbitrary constant which we ignore for notational convenience. Now add ln [(1 − δdo) MST_dot−1] to both sides of the equation to approximate ln MST_dot= ln [(1 − δ_do) MST_dot−1+ M_dot−1] which gives (2.19) in the text.

Substituting (2.17) and (2.19) in (2.16) and rearranging gives

A number of key features of this model are worth discussing. First, note that equa-tion (2.20) is of the double log form, which results from the choice of funcequa-tional form for the utility function and from taking migration and the migrant stock in equations (2.14) and (2.17) as logarithmic. Although Hatton’s (1995) origi-nal model is semi-logarithmic, he emphasizes that this model is only one among many different functional forms and also suggests and estimates a double log ver-sion. Given our panel dataset, with countries that greatly differ in size, the double log form has the important advantage that it eliminates the scale of the migrant flows and stocks. As an alternative, some studies divide the immigrant flow by the population in the origin or destination countries (see e.g. Fertig, 2001; Boeri et al., 2002; Pedersen et al., 2008; Mayda, 2010), but this only partly removes problems of scale. Only dividing by the population in both sending and receiving countries or taking the natural logarithm entirely solves the problem (see Lewer

and Van den Berg, 2008; Warin and Svaton, 2008; Ortega and Peri, 2009).

Second, lagged migration flow and migrant stock enter equation (2.20) as two separate determinants. This contradicts the common practice in empirical stud-ies to include either the lagged migration flow (see e.g. Bertocchi and Strozzi, 2008; Mayda, 2010) or the migrant stock (see e.g. Hooghe et al., 2008; Lewer and Van den Berg, 2008; Pedersen et al., 2008; Warin and Svaton, 2008), with both variables typically being argued to capture network effects⁶. Laber (1972) has already highlighted that it is not clear whether these dynamic terms represent network effects or rather capture a partial adjustment mechanism. Dunlevy and Gemery (1977) argue that lagged migration and migrant stock should both be in-cluded as determinants to capture the separate impact of partial adjustment and network effects respectively. This is confirmed by equation (2.20) which shows that a nonzero coefficient on ln M_dot−1 implies partial adjustment (λ 6= 0) stem-ming from the process by which expectations are formed and updated while a nonzero coefficient on ln MST_dot implies network effects (γ₁6= 0).

Third, an additional dynamic feature of the model is that it includes both lagged levels and current changes of the explanatory variables. The latter capture im-mediate responses of the immigrant flow to changes in the explanatory variables.

This stems from the fact that migration decisions can be postponed when eco-nomic conditions are unfavorable such that migration may fluctuate more closely with current conditions than might be expected from individuals that maximize their lifetime utilities.

6A popular motivation for not including both lagged migrant flow and migrant stock is that the latter is, as presented in equation (2.18), the sum of all past immigrant flows less deaths and return migrants. Hence, the migrant stock is itself a function of all those factors which influenced the earlier immigrant flows. Therefore it will be correlated with all the explanatory variables.

However, multicollinearity is no reason to omit the migrant stock variable as this may result in a specification bias as well as in a loss of information regarding the network effect.