I briey recap the structural migration gravity system proposed by Anderson (2011).
In a multi-country setting emigration is a discrete choice from the set of countries in the world from the perspective of a single worker. A worker h migrates from country o (origin) to d (destination) only if her utility of choosing d is bigger than for all other possible choices. The utility in country o is given by her wage, wo plus an idiosyncratic part of utility. Migration to country d involves country-pair specic costs of migration, δod > 1 ∀ d 6= o and δoo = 1, which reduce utility in country d in an iceberg type way, wd/δod. Migration additionally involves a worker and country-pair specic factor of utility odh. So a worker decides to migrate from country o to d i (wd/δod)odh ≥ woooh. In line with discrete choice theory, utility of a representative migrant is separated into two parts. One part which is observable and determined by characteristics at the country-pair-level, Vod = ln(wd) − ln(wo) − ln(δod). The second part of the utility, which is worker and country-pair specic,εodh = lnodh, is not observable for the researcher. With distributional assumptions for εodh, one can derive the probability of a randomly drawn worker to migrate.6
From multiplying the number of people in country o with the migration probability of a randomly drawn worker of country o, G(Vod), we gain an aggregate multi-country migration ow equation,
Mod=G(Vod)No, (2.1)
whereNo is the number of natives ino and G(Vod)gives the proportion of migrants from o to d, which is given by
G(Vod) = eVod P
keVok. (2.2)
5Anderson (2011) highlights the general modularity of the gravity equation in more detail and with respect to a sectoral analysis.
6Adopted to the multi-country discrete choice of a representative worker, a derivation of the multinomial-logit probabilities is given in Appendix A.1.
Plugging in the V's yields a multilateral migration ow equation as
The migration ow from country o to d is positively associated with the wage in the destination countryd, bilateral migration barriers to all other potential countries thand, δok, and the number of natives of the source countryo,No.7 Migration is negatively asso-ciated with bilateral migration barriers, captured byδod, and wages in all other countries than d, wk. Note that the idiosyncratic or worker specic part of the utility is captured implicitly by the functional form of Equation (2.3). So the individual probabilities, which are derived in Appendix A.1, already capture the unobserved part of the migrant's utility, εodh.
Using accounting identities and the labor market clearance condition, Anderson (2011) provides the following migration gravity system:8
Mod = LdNo
The masses which drive migration ows in this gravity model are given by No, the pop-ulation of the origin country, and by Ld, the labor force supplied to country d. Both increase migration ows between a bilateral pair of countries and their product goes into the ow equation relatively to the world population Nw. Bilateral migration barriers, δod, decrease migration ows. Ωd and Wo indicate the multilateral resistance terms to migration.
Section 2.4 estimates Equation (2.4) structurally to inferδod, and in Section 2.7 I use this system to conduct the comparative static analysis. This can be done by realizing that multilateral resistance terms can be solved for observed values of No, Ld, and δod.
Before I go on, several things are worth mentioning about this model. First of all, we can observe the hypothetical migration pattern of a frictionless world by the rst part of Equation (2.4). In a world without any friction to migration, we would observe the
mi-7Beine et al. (2015) call the latter the potential of a country for sending migrants.
8For intermediate steps of the derivation see Appendix A.2.
grant share from countryoof the labor force supplied todto be equal to countryo's share of the world population. From this we can nicely observe the general two-way migration nature of the model. The precise two-way migration pattern is additionally shifted by bilateral migration costs and multilateral resistance terms. The frictionless view already points to the second important fact, that the model would only imply a zero migration ow if the frictions between two countriesoanddwere innitely large. Migration frictions are collected in the second part of Equation (2.4). Frictions are a composite of bilateral migration barriers,δod, and multilateral resistance terms. Bilateral migration costs aect bilateral migration ows relative to the multilateral resistance terms. We can already see that multilateral resistance terms depend on bilateral migration barriers. Therefore, a change in the bilateral migration cost vector for one country-pair aects all countries' multilateral resistance terms which has to be accounted for when it comes to a predic-tion of migrapredic-tion ows. Technically multilateral resistance terms are averages of inverse migration frictions weighted by the relative size of a country. The inward multilateral resistance term collects all barriers for migrants to a specic migration destination coun-try, while the outward multilateral resistance term collects all barriers for migrants from a specic migration origin country. Anderson and Yotov (2010) give a nice intuition for these terms for trade ows. They suggest understanding inward multilateral resistance as the uniform markup a buyer pays for a bundle of goods from a hypothetical world market.
Outward multilateral resistance is then understood as the average trade cost which an exporter faces when selling to this world market. Transferring this intuition to migration means that inward multilateral resistance captures migration barriers for every migrant to destination country d for migrants from a hypothetical world origin, i.e. irrespective of her origin country. Then, outward multilateral resistance measures the uniform costs every migrant faces for migration from country o to the hypothetical migrant's country, i.e. irrespective of her actual destination country.9 Put dierently, inward multilateral resistance of a country aggregates unilateral immigration barriers from a hypothetical world origin country and outward multilateral resistance of a country aggregates emi-gration barriers to a hypothetical world destination. Multilateral resistance terms are aggregate concepts. Migration ows at the aggregate (Equation (2.4)) are determined by bilateral migration barriers relative to multilateral resistance terms. Also, multilateral resistance terms vary across countries. A change in bilateral migration barriers results in heterogeneous migration eects. The multilateral resistance terms entail non-trivial, multilateral changes of the migration pattern from bilateral changes in migration barriers,
9How to transfer the incidence intuition to migration is not obvious since for migration it is not clear who is the hypothetical entity which is actually charged.
which can be inferred from the comparative static analysis in Section 2.7.
Also note that there is no term left in Equation (2.4) which explicitly captures wage dierentials, since they were substituted out via the labor market clearance equation (see Appendix A.2). This explains the dierence of the empirical specication of the migration gravity to other RUM based migration approaches like Grogger and Hanson (2011). Furthermore, the theoretical migration gravity model is, in a way, agnostic about the classical dierentiation between push and pull factors and the importance of specic migration barriers. Simply put, δod is not specied by the model. The specication of migration barriers is an empirical question and oftentimes hinges on the availability of bilateral measures and data.10 I leave the presentation of the empirical specication for Section 2.4.