## Abstract

It is widely believed that many migrations are undertaken at least in part for the benefit of future generations. To provide evidence on the effect of intergenerational altruism on migration, I estimate a dynamic residential location choice model of the African American Great Migration in which individuals take the welfare of future generations into account when deciding to remain in the Southern USA or migrate to the North. I measure the influence of altruism on the migration decision as the implied difference between the migration probabilities of altruistic individuals and myopic ones who consider only current-generation utility when making their location decisions. My preferred estimates suggest that intergenerational altruism explains between 24 and 42% of the Northward migration that took place during the period that I study, depending on the generation.

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## Notes

See minute 2:00 of the interview and transcript available at minute 2:00 of the interview available at https://goinnorth.org/ella-lee-interview.

See minute 6:00 of the interview and transcript available at https://goinnorth.org/beulah-collins-interview-1983https://goinnorth.org/beulah-collins-interview-1983.

One explanation for the paucity of evidence on the magnitude of this effect is that, because intergenerational altruism is not directly observable, inferring its effects requires both a behavioral model of the intergenerational migration decision and data sufficiently rich to permit identification of the model.

I code respondents’ current and birth locations as Southern according to the Census Bureau’s definition of the Southern region. According to this definition, Alabama, Arkansas, Delaware, Florida, Georgia, Kentucky, Louisiana, Maryland, Mississippi, North Carolina, Oklahoma, South Carolina, Tennessee, Texas, Virginia, and West Virginia are Southern states. I code all other locations within the USA as belonging to the North. I then classify Southern-born respondents as migrants if they lived in the North at the time of the survey. Because the data do not contain information on the timing of migration, it is unavoidably possible under this scheme that the actual migration decision was made by respondents’ parents.

These modest migration rates are not at odds with the notion that Northward migration was widespread. Although a relatively small fraction of any given generation migrated North, overall about 35% of families eventually left the South; this figure is roughly comparable to outmigration rates documented elsewhere (see, for example, Tolnay 2003).

This is not a strong assumption. The lack of return migration necessarily means that residents of the North either have preferences for living there, or face costs of Southward migration, that approach infinity. Treating the North as absorbing simply obviates the need to estimate these extreme parameters.

Although assumption is clearly stylized, it is also consistent with the observation that widespread Northward migration had come to an end by the mid 1970s. This assumption does not require that there were no regional differences in expected lifetime utility. For example, it allows for the possibility that the North continued to offer greater economic opportunities that were offset by strong family and community ties to the South. It also allows for the possibility that there were realized North-South differences in utility, despite agents’ expectations that utility would equilibrate between the two regions.

Using a calibrated discount factor is standard practice (see Magnac and Thesmar2002; Arcidiacono and Miller 2017). Because the steady-state assumption is equivalent to a finite time horizon, the discount factor can technically be identified under the assumption that the flow utility functions are stable across time, although identification in this case is achieved only tentatively through the nonlinear manner in which the discount factor enters the likelihood function. In contrast, models such as that in Heckman and Raut (2016) recover altruistic preferences directly when parents make investments that can only benefit them through their children, making them much better suited to estimate the altruism parameter. Hence, the results in this paper can be interpreted as answering the question, given a rate of intergenerational altruism consistent with prior research, what is the implied effect of altruism on migration?

The use of the Type I Extreme Value distribution is common because it generates analytical expressions for the conditional valuation functions; in the static case, it is equivalent to using a logit model.

In principle, it is possible to allow the cost of migration to depend on the covariates. In practice, the lack of power of the covariates to explain observed migrations means that these covariate-specific costs are not well identified.

Although they are likely choice variables, I treat education and fertility as exogenously endowed in order to focus on the migration decision. Note however that jointly choosing education, fertility and location is equivalent to choosing education and fertility with the understanding that the optimal location will depend on these choices.

To see this formally, consider a simplified model without covariates in which

*u*_{ng}(*b*_{g}) =*β*_{0}−*c*1(*b*_{g}=*s*) and*u*_{sg}= 0. Since the dynamic decision problem ends after the third generation, the North-South difference in conditional valuation functions for Southern-born members of the third generation is*v*_{n3}(*s*) −*v*_{s3}(*s*) =*β*_{0}−*c*. Since the North is an absorbing state, it follows from properties of the Type I Extreme Value distribution (see, e.g., Rust 1987) that for the second generation this difference is*v*_{n2}(*s*) −*v*_{s2}(*s*) =*u*_{n2}(*s*) +*λ**E*[*V*_{3}(*n*) −*V*_{2}(*s*)] =*β*_{0}−*c*+*λ*{*β*_{0}+*γ*− log[1 + exp(*β*_{0}−*c*)] −*γ*}, where*γ*≈ .5772 is Euler’s constant. Since*β*_{0}appears in the*v*_{l2}independently of the*β*_{0}−*c*term (and since the probability that generation*g*migrates is the same as the probability that*v*_{ng}>*v*_{sg}) the latter can be identified from maximum likelihood on the third generation and the former from maximum likelihood on the second. Note that, by this logic,*λ*is technically identified from maximum likelihood on*v*_{1n}(*s*) −*v*_{1s}(*s*) =*β*_{0}−*c*+ (*λ*+*λ*^{2})(*β*_{0}+*γ*) −*λ*{log[exp(*v*_{n2}(*s*)) + exp(*v*_{s2}(*s*))]}, although only through the nonlinear way that it enters into this function.I include the gender of generation

*g*+ 1 in order to allow for the possibility that transitions from parent to child differ for male and female children. Ideally, the transition structure would allow both the mother’s and father’s state variables to influence those of the child. Unfortunately, this is not possible with my data, which only report education and fertility for either the mother or the father. To test the sensitivity of my results to this modeling choice, I have also estimated versions of the models presented below that either completely omit gender (and hence implicitly average over males and females) or allow for the state transitions to depend on the gender of the parent as well as the child. These extensions do not alter the substantive conclusions of the paper.By design, the estimated transitions to state space elements for generation

*g*+ 1 do not depend on the gender of the parent observed in generation*g*.Because maximizing the expected sum is equivalent to maximizing the expected summand, this is the dynamic analog of a standard logistic regression model, treating the future value terms as observed.

In general, the coefficients on variables in a discrete choice model are only directly informative about the corresponding partial effects (and not the contributions of the variables themselves to the decision). Since all of the variables in my model change in increments of one (the observed covariates are all binary, the moving cost can be viewed as the coefficient on being Southern-born, and time varies from one to three), the relative magnitudes of the coefficients also identify the relative contributions of the corresponding variables to the migration decision.

Rates of northward migration among blacks were significantly larger during the Great Migration than rates of interstate migration in the contemporary USA, despite the fact that the migration cost is estimated to be large in both periods. As an anonymous reviewer has noted, since migration is associated with income gains in both periods (c.f. Kennan and Walker 2011; Collins and Wanamaker 2014), it is unclear whether differences in income gains from migration between these two periods can explain why migration rates were so much larger during the Great Migration. A natural hypothesis is that blacks were also attracted to non-pecuniary amenities such as reduced discrimination in the Great Migration North. However, disentangling these potential motives is a complex empirical problem; since my model does not distinguish between the pecuniary and non-pecuniary components of location preferences, it cannot provide any new evidence on this hypothesis.

This may suggest that larger families prefer the South, that fertility and preferences for the South are positively correlated with an unobserved factor, or since fertility and location decisions may be made jointly, that living in the North discourages fertility.

I compute the dynamic probabilities as \(\exp [v_{gn}(x_{g}, b_{g}=s;\hat \theta )-v_{gs}(x_{g}, b_{g}=s;\hat \theta )]\cdot \{1+\exp [v_{gn}(x_{g}, b_{g}=s;\hat \theta )-v_{gs}(x_{g}, b_{g}=s;\hat \theta )]\}^{-1}\) and the static probabilities as \(\exp [u_{gn}(x_{g};b_{g}=s;\hat \theta )-u_{gs}(x_{g}, b_{g}=s;\hat \theta )]\cdot \{1+\exp [u_{gn}(x_{g}, b_{g}=s;\hat \theta )-u_{gs}(x_{g}, b_{g}=s;\hat \theta )]\}^{-1}\).

The predicted rates for the third generation are considerably higher than the observed rates. However, since some of the migration histories for this generation are likely right censored, this can be viewed as a benefit of using stable utility parameters and allowing for a time trend.

The notional question answered by this difference is the extent to which altruistic parents are more likely to migrate than myopic ones on average across the distribution of observed covariates (which are correlated with the expected benefits to future generations of living in the North) and the idiosyncratic errors. To a first-order approximation, this difference can also be interpreted as an average effect across families of varying intergenerational altruism (as long as the assumed altruism parameter is interpreted as the population average parameter). In principle, such an average could be estimated directly by allowing for unobserved heterogeneity in both altruism and location preferences, although for reasons described above the structure of my data is poorly suited for this kind of identification. Also note that any motive that agents have to benefit future generations who are not their direct descendants will be absorbed by the idiosyncratic preference components, as long as they are not correlated between different generations of the same family (the models with unobserved heterogeneity discussed below allow for the possibility that such motives are correlated across generations).

Nonparametric identification of the model that I estimate below follows from remark 3 of Kasahara and Shimotsu (2009). A model where preferences depended on further lags of the location decision, introducing state dependence, would not be identified through their construction without observations on further generations. However, because the very existence of the black population in the Southern USA was a consequence of slavery, the inclusion of a generational time trend in the model helps to account for state dependence and duration effects.

The assumption of binary heterogeneity is particularly appropriate for small samples such as mine, but can also be viewed as an approximation of a higher-dimensional unobserved state variable.

To see this, suppose that a fraction

*π*of the population migrate with probability*p*_{1}and a fraction (1 −*π*) migrate with probability*p*_{2}<*p*_{1}, so that the observed probability is*p*=*π**p*_{1}+ (1 −*π*)*p*_{2}. Unless*p*_{1}≈*p*_{2}≈*p*(in which case the unobserved heterogeneity is irrelevant),*p*small implies that*p*_{2}≈ 0 and*p*_{1}≈*p*/*π*>*p*.Although some “movers” will remain in the South by chance, they will be missing at random from the limited estimation sample without affecting the consistency of the parameter estimates. Although “stayers”’ preferences are not identified under this approximation, they are also uninteresting, since members of that group never migrate.

These mover-specific altruism effects imply population-average effects of about .16 × .35 = .056, which the model without unobserved heterogeneity approximates reasonably well for first-generation Southerners, but understates for the second generation.

Although it seems counterintuitive that increasing the discount factor decreases the estimated effects of altruism, this is consistent with the changes in the estimated moving cost. The higher the discount factor, the lower the migration cost that parents are willing to migrate to spare future generations, which in a nonlinear model can imply a smaller impact of altruism on migration.

To estimate the model, I assume as before that the

*f*_{g+ 1|g}are independent of*τ*, and maximize the sample analog of$$ E\left[\log\left( \pi \prod\limits_{g=1}^{3} P(l_{g} | x_{g}, b_{g}=s; \beta_{01}, \beta, \hat\rho) + (1-\pi) \prod\limits_{g=1}^{3} P(l_{g} | x_{g}, b_{g}=s; \beta_{02}, \beta, \hat\rho) \right)\right], $$where

*β*_{01}and*β*_{02}are the type-specific constants,*β*are the remaining utility function parameters (excluding a constant), and \(\hat \rho \) are the transition function parameters. Although in principle all of the parameters could be indexed by*τ*, the limited power of observables to explain migration makes these type-specific parameters difficult to identify, particularly for the group with lower migration probabilities.It is of course possible to include time trends in pure location preferences as well as the cost of migrating. However, since the time trends are identified purely by functional form assumptions, the resulting estimates are too imprecise to be of much use.

The literature on the timing of the Great Migration (e.g., Carrington et al. 1996; Collins 1997; Chay and Munshi 2015) asks why Northward migration did not begin earlier, given the apparently large potential gains accruing to migrants. Consistent with the essential stylized fact of this literature, both the estimated time-trend and moving-cost models reflect increasing rates of migration over time. The former attributes this trend to increases in the utility of living in the North over time while the latter attributes it to decreases in the cost of migration. Unfortunately, because both the time-trend and moving-cost parameters are identified from intergenerational differences in migration rates, neither estimated model provides any insight into why migrating North became more attractive, or less costly, over time.

Following the procedure in Wooldridge (2010, Ch. 13), I implement this test by regressing the differences in maximized likelihoods \(\ell _{i}(\hat \theta ^{c})-\ell _{i}(\hat \theta ^{t})\) between the cost- and time-trend models for each family

*i*on a constant to test their difference from zero. The estimated coefficient of −.015 is significant with a*p*value of .016.

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## Acknowledgements

I thank Robert Miller, George-Levi Gayle, four anonymous reviewers, and the editor for their helpful comments and suggestions.

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## Appendices

### Appendix A: Transition functions

### Appendix B: Alternative specifications

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Gardner, J. Intergenerational altruism in the migration decision calculus: evidence from the African American Great Migration.
*J Popul Econ* **33**, 115–154 (2020). https://doi.org/10.1007/s00148-019-00738-5

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DOI: https://doi.org/10.1007/s00148-019-00738-5

### Keywords

- Altruism
- Intergenerational altruism
- Migration
- Immigration
- Great migration
- Dynamic discrete choice

### JEL Classification

- J61
- D64
- R23