This post is part 3 of a series outlining a conceptual framework for the empirical analysis of migration. Read the introductory post to the series here, part 1 here, and part 2 here.

The model in part 2 for comparative statics was extremely complicated. The problem was that there were too many moving parts: there was a selection effect arising from differences in grouping, and there was a treatment effect arising from changes in migration patterns affecting both the marginal migrants and others. This part of the series considers a simpler model. Our first pass at exposition will lay out a very simple toy case, and we’ll then discuss variants and possible ways of ramping up the complexity.

We’ll keep focus on two countries: country *A* (a source country for migrants) and country *B* (a target country for migrants). We’ll assume there are no reverse migration flows, and no other countries to compete as sources and targets for migrants. We are considering a migration policy that would allow a subpopulation of country *A* to migrate. We have three relevant subpopulations:

- A subpopulation of the population of country
*A*that comprises the would-be migrants under the migration policy of interest. We’ll call these people the*potential migrants*. - The remaining population of country
*A*, that would not migrate with or without the migration policy of interest. - The resident population of country
*B*.

We are thus comparing the no-migration scenario with the scenario where an identified subpopulation of country *A* is allowed to migrate to country *B*, and takes advantage of the opportunity.

We are interested in providing a rank-ordering and quantitative comparison for the following four quantities:

- Performance of natives of target country
*B*on indicator*X*. - Performance of natives of source country
*A*(who would not move under either policy) on indicator*X*. - Performance of potential migrants on indicator
*X*if they were allowed to migrate (i.e., in the migration scenario). - Performance of potential migrants on indicator
*X*if they were not allowed to migrate (i.e., in the no-migration scenario).

Note that:

- The difference between (2) and (4) measures the selectivity of migration relative to the source country. In other words, it measures
*emigrant selectivity*. - The difference between (1) and (3) measures the selectivity of migrants relative to the target country, or equivalently, to their failure of assimilation (the assimilation may be “upward” or “downward” depending on how the migrants compare with the target country natives). In other words, it measures
*immigrant selectivity*. - The difference between (3) and (4) refers to the treatment effect of migration on migrants. In other words, it measures the premium of migrating.

#### Aspects of the rank ordering that matter from various normative perspectives

The following hold *prima facie* (here, weighted averages refer to averages weighted by population size):

- The individualist universalist cares mainly about the comparison between (3) and (4), because that’s the main source of change
*to individuals*. - The universalistically inclined analytical nationalist, who cares about how national averages change rather than how individuals do, cares about how (1) compares with the weighted average of (1) and (3) and how the weighted average of (2) and (4) compares with (2). Due to compositional effects, this is not always in agreement with the individualist universalist perspective. In particular, compositional effect paradoxes arise under rank orderings (1) > (3) > (4) > (2) and the reverse ordering (2) > (4) > (3) > (1). In words, (1) > (3) > (4) > (2) means that the migrant subpopulation is better at the indicator than the source country, that migration improves it further, but that even after that improvement, they still fall short of the natives of the target country.
- A person driven by local inequality aversion cares about how the gap between (1) and (3) compares with the gap between (2) and (4) (and also the magnitude of migration relative to source and target country populations).
- Assuming that the performance of people on indicator
*X*affects the well-being of others in the territory (this is true for indicators such as crime), the citizenists and territorialists for country*B*care about how (3) compares with (1). - Assuming that the performance of people on indicator
*X*affects the well-being of others in the territory (this is true for indicators such as crime), the citizenists and territorialists for country*A*care about how (4) compares with (2).

Recall that there are some rare indicators (such as resource use) where we care about the *total* rather than the average. For instance, in a country where water is scarce, we may care about total water use rather than per capita water use. In this case, we also need to know the relevant population sizes, and some of the *prima facie* claims above do not apply.

**Mathematical digression: matrix description**

Suppose country 1 is country *A* and country 2 is country *B*. Following the notation for part 2, our matrix for indicator *X* under the no-migration scenario, which we call the *old* scenario, is:

$latex \begin{pmatrix} x_{11}^o & \text{undefined} \\ \text{undefined} & x_{22}^o \\\end{pmatrix}$

Our matrix for indicator *X* under the migration scenario, which we call the *new* scenario, is:

$latex \begin{pmatrix} x_{11}^n & x_{12}^n \\ \text{undefined} & x_{22}^n \\\end{pmatrix}$

Note that we *do* have $latex x_{22}^n = x_{22}^o$, because the set of people is the same in both cases and the individual indicator values are the same for each of them. On the other hand, we do *not* necessarily have $latex x_{11}^n = x_{11}^o$. This is because even though all the individuals who stay put in country *A* under both scenarios fare the same under both scenarios, the no-migration scenario also sees the potential migrants stay put, thereby affecting the average value of the indicator (the compositional selection effect).

What we’re really interested in is the matrix:

$latex \begin{pmatrix} x_{11}^{n,o} & x_{12}^{n,o} \\ \text{undefined} & x_{22}^{n,o} \\\end{pmatrix}$

This uses the groupings from the migration scenario (i.e., it separates the source country population into those who stay put and those who migrate in the migration scenario) but using indicator values from the no-migration scenario. We want to compare this with the migration scenario matrix:

$latex \begin{pmatrix} x_{11}^n & x_{12}^n \\ \text{undefined} & x_{22}^n \\\end{pmatrix}$

Our conditions now tell us that $latex x_{11}^{n,o} = x_{11}^n$ and $latex x_{22}^{n,o} = x_{22}^n = x_{22}^o$. Therefore, the two matrices above coincide except in the top right entry. In other words, we’re looking at these two matrices:

$latex \begin{pmatrix} x_{11}^n & x_{12}^{n,o} \\ \text{undefined} & x_{22}^n \\\end{pmatrix}, \qquad \begin{pmatrix} x_{11}^n & x_{12}^n \\ \text{undefined} & x_{22}^n \\\end{pmatrix}$

We can now see the four numbers that we were attempting to rank-order and compare quantitatively: $latex x_{11}^n$ (this is (2) in the list), $latex x_{12}^{n,o}$ (this is (4) in the list), $latex x_{12}^n$ (this is (3) in the list), and $latex x_{22}^n$ (this is (1) in the list).

#### Adaptation to migration liberalization and marginal migration

For simplicity, the above analysis considers the no-migration scenario as one extreme. It can be adapted to a comparison of liberalizing an existing migration policy towards a subpopulation. We would then be interested in the question of *marginal migrants*: the additional people who can migrate under liberalization. However, there are complications introduced by the distinction between marginal and average: the average performance for the existing set of migrants who migrate under the less liberalized policy may differ from that of the performance for set of migrants who migrate under the more liberalized policy. The difference could be a difference in composition (selection effect) or a difference in treatment.

#### The place premium: an example

The place premium measures the gap (in proportional terms) between (3) and (4), i.e., the treatment effect of migration on migrants. It is one of the few such measures that people have attempted to compute for a wide range of source and target countries; see for instance this working paper by Clemens, Montenegro, and Pritchett that has computed place premium tables on Page 11.