The Dynamics of Internal Migration: A New Fact and its Implications

Howard and Shao document a new empirical regularity in U.S. internal migration: the t-year interstate migration rate — defined as the share of people living in a different state than they did t years ago — is approximately proportional to the square root of t. The fact is established using the Gies Consumer and Small Business Credit Panel (GCCP), a 15-year panel (2004–2018) covering approximately 1 percent of all Americans with a credit report, and is corroborated in the Panel Survey of Income Dynamics (PSID, 1969–1997), where the square root pattern holds out to a 25-year horizon. The fact is not an artifact of averaging across origins, destinations, cohorts, or age groups: most of the distribution across these cuts is concentrated close to the square root line. It holds for both people under 45 and over 45, and is robust to the choice of time period and inter-state distance.

The standard moving cost model — in which location choice is a Markov process with i.i.d. extreme-value utility shocks and large bilateral moving costs — is shown (Proposition 1) to imply that the t-year migration rate is approximately proportional to t, not sqrt(t), as moving costs tend to infinity. Simulations confirm the linear pattern persists in calibrated versions of the moving cost model even when adding state variables for prior location, home state, or age.

The paper’s main theoretical contribution is the SPACE model (Spatially and Persistently Autocorrelated Epsilons). Rather than imposing moving costs, the SPACE model assumes that person-location match-specific utility is (i) persistent over time, governed by an autocorrelation parameter rho, and (ii) spatially correlated across locations via a generalized extreme-value (cross-nested logit) structure. The model has no moving costs by default. Proposition 3 proves that as rho approaches 1, the ratio of t-year migration to 1-year migration is bounded below by sqrt(t) and above by sqrt(pi/3) * sqrt(t) — a tight bound, since sqrt(pi/3) is approximately 1.023. The calibrated rho-tilde is 0.892, implying a period-to-period autocorrelation of 1 − (1 − rho-tilde)^2 = 0.988.

The SPACE model replicates bilateral one-year migration flows, matches the decreasing hazard rate of migration conditional on duration of stay, reproduces the distribution of lifetime move counts (including the large fraction who never move and the few percent who move four or more times in 14 years), and outperforms the moving cost model at out-of-sample individual location forecasting: by 2018, the moving cost model’s mean Kullback-Leibler divergence reaches approximately 0.12 log-points per observation above the maximum-possible benchmark, versus only 0.014 log-points for the SPACE model.

Key divergences from the moving cost model arise in four areas. First, moving costs need not be large: the SPACE model rationalizes observed low migration without any moving costs, in contrast to Kennan and Walker’s (2011) estimate of average moving costs of $312,146 (2010 dollars), more than six times median household income; when moving costs are added to the SPACE model, they are roughly two orders of magnitude smaller. Second, long-run population elasticities differ sharply: in the SPACE model they remain proportional to bilateral gross migration rates, while in the moving cost model they converge to a static logit proportional to population shares — and population shares and gross migration rates have little empirical correlation, so the long-run elasticities of the two models are essentially uncorrelated across state pairs. Third, adjustment dynamics differ: in the SPACE model a permanent utility shock to Louisiana produces immediate, full population adjustment; in the moving cost model adjustment takes roughly 200 years, with Mississippi overshooting its new steady-state and New York adjusting implausibly slowly. Fourth, welfare inferences are almost reversed: the correlation between log utility changes implied by the two models using U.S. population data is −0.497, with the SPACE model attributing relative utility gains to the South and West and the moving cost model attributing gains to New York and New England.

Q: What is the square root fact, and which datasets confirm it? A: The t-year interstate migration rate scales approximately as sqrt(t). It is documented in the GCCP (2004–2018, ~1% of Americans with credit reports) and verified in the PSID (1969–1997), where the pattern holds out to a 25-year horizon. It is not driven by averaging across subgroups: the distribution of the fact across origin-destination pairs, age groups, cohorts, and starting years is concentrated close to the square root line.

Q: Why does the standard moving cost model fail to match the square root fact? A: In the moving cost model, location choice is a Markov process with i.i.d. extreme-value shocks. Proposition 1 proves that as the common component of moving costs tends to infinity, the t-year migration rate is proportional to t (linear). Because the model requires large moving costs to rationalize low migration rates, the linear prediction is unavoidable. Simulations of calibrated versions — including variants with home bias, prior-location state variables, or age — confirm the relationship remains approximately linear.

Q: What is the SPACE model, and why does it generate a square root? A: The SPACE model replaces moving costs with persistent and spatially correlated person-location match-specific utility. Utility shocks are drawn from a generalized extreme-value (cross-nested logit) distribution that allows spatial correlation, and they are autocorrelated over time with persistence parameter rho. Proposition 3 shows that as rho → 1, the ratio of t-year to 1-year migration is bounded in [sqrt(t), sqrt(pi/3)*sqrt(t)], a tight interval since sqrt(pi/3) ≈ 1.023. The intuition is that when rho is close to 1, the idiosyncratic utility process resembles a random walk, whose standard deviation grows as sqrt(t), causing migration thresholds to be crossed at a sqrt(t) rate.

Q: What is the calibrated persistence parameter, and what does it imply? A: The calibrated rho-tilde is 0.892, close enough to 1 to generate the square root fact in simulations. The implied period-to-period autocorrelation of match-specific utility is 1 − (1 − 0.892)^2 = 0.988. This calibration is achieved by solving for the largest eigenvalue of an I×I matrix of conditional migration rates.

Q: How do the two models compare on individual-level forecasting accuracy? A: Performance is evaluated using mean Kullback-Leibler divergence from the maximum-achievable log likelihood. Both models perform similarly in 2005, but by 2018 the moving cost model’s KL divergence reaches approximately 0.12 log-points per observation, while the SPACE model’s reaches only 0.014 log-points — roughly an order of magnitude better — leaving little room for improvement.

Q: How large are implied moving costs under each model? A: Kennan and Walker (2011) estimate average moving costs of $312,146 in 2010 dollars, exceeding six times the median household income. The baseline SPACE model requires zero moving costs to match observed migration levels. When an augmented SPACE model with both persistence and moving costs is calibrated to match the one-year and ten-year migration rates, the estimated moving costs are approximately two orders of magnitude smaller than those from a moving-cost-only model.

Q: How do short-run population elasticities compare across models? A: In both models, the short-run cross-elasticity of population in state i with respect to utility in state j is approximately proportional to the gross migration rate between them. Corollary 1 formalizes this for the SPACE model: dp_i/du_j = −(1/(1−rho)) * m_{i→j} for i ≠ j. This means that in the short run, both models deliver similar predictions for how populations respond to local shocks.

Q: How do long-run population elasticities differ? A: In the SPACE model, long-run elasticities remain proportional to bilateral gross migration rates — the same relationship as in the short run. In the moving cost model, Proposition 4 shows that the long-run elasticity converges to the static logit: d(log p_i)/d(v_j) = −2*p_j for i ≠ j, depending only on population shares. Since population shares and gross migration rates are empirically uncorrelated, the long-run elasticities of the two models are essentially uncorrelated across state pairs.

Q: What do the models predict about the speed of regional adjustment? A: In the SPACE model, a permanent utility shock to Louisiana causes full, immediate population adjustment in the first period with no further dynamics. In the moving cost model, the same shock generates adjustment lasting roughly 200 years. Mississippi overshoots its long-run steady state in the moving cost model due to high bilateral migration with Louisiana, while New York adjusts especially slowly due to low bilateral migration — a pattern the authors describe as potentially counterintuitive.

Q: How do the models handle events involving rapid population change, such as Hurricane Katrina? A: The SPACE model accommodates fast adjustments by assuming rapid utility changes, consistent with the observed sharp decline in Louisiana’s population share followed by a small rebound. The moving cost model requires implausible utility assumptions to match these dynamics: it implies that Louisiana utility two years after Katrina was higher than before the hurricane.

Q: What do the two models infer about which U.S. states have gained or lost relative utility over time? A: Using exact-hat algebra applied to observed U.S. population changes, the SPACE model infers that the South and West have the largest relative utility gains, while New England and the Rust Belt have the largest relative declines. The moving cost model produces nearly the opposite inference: New York and New England show relative utility gains, while the South and West show declines. The correlation between the log utility changes implied by the two models is −0.497.

Q: Why do the authors argue that spatially and temporally correlated utility is realistic, not merely a mathematical convenience? A: Surveys (Jia et al., 2023) show that people primarily cite family and employment considerations as reasons for interstate moves — both are persistent and geographically concentrated. Proximity to family is spatially correlated: if state i is close to one’s family, nearby states are also relatively close. Job opportunities in specific industries or skills are geographically clustered. Natural amenities and regional cultures are spatially correlated as well. The authors argue it is harder to defend the i.i.d. assumption of the moving cost model than the SPACE model’s correlated structure.

Q: What is the distinction between moving costs and persistent match-specific utility? A: A moving cost is a one-time irreversible cost paid upon leaving a location. Persistent match-specific utility implies that the utility change from moving is ongoing, partially reversible upon return, and decays with time away from the original location. The authors argue that many factors labeled “moving costs” in the literature — such as distance from friends or amenities — are more accurately characterized as persistent and partially reversible utility losses, a distinction previous models could not draw.

Q: Does the SPACE model replicate the gravity equation for bilateral migration? A: Yes. Proposition 2 shows that migration from i to j in the SPACE model is given by m_{i→j} = (1 − rho) * p_i * p_j * (1 + tau_ij), where tau_ij captures spatial correlation. This resembles a gravity equation: more spatially correlated location pairs have higher bilateral migration, and higher persistence (higher rho) implies lower overall migration levels.

Q: Can the SPACE model be embedded in broader quantitative spatial models? A: Yes. The SPACE model admits closed-form solutions for state populations and bilateral migration flows, is compatible with exact-hat algebra for dynamic counterfactuals, and supports computationally feasible individual-level simulations. Appendix E embeds the SPACE model in a housing model with durable local housing production and shows that slow population adjustment can emerge from housing durability rather than slow migration per se, providing an alternative explanation for regional divergence persistence.

SPACE model: A model of internal migration featuring Spatially and Persistently Autocorrelated Epsilons — person-location match-specific utility that is both autocorrelated over time (with persistence parameter rho) and spatially correlated across locations via a generalized extreme-value (cross-nested logit) distribution. The model contains no moving costs by default.

Square root fact: The empirical regularity that the t-year interstate migration rate (share of people living in a different state than t years ago) is approximately proportional to sqrt(t). Documented in GCCP data (2004–2018) and PSID (1969–1997) up to a 25-year horizon.

Moving cost model: The standard dynamic discrete-choice model of migration in which an agent living in state i chooses location j to maximize u_j − delta_ij + epsilon_j + beta*E[V’], where delta_ij is a bilateral one-time irreversible moving cost and epsilon_j is i.i.d. extreme-value. Low migration rates are rationalized by large moving costs (e.g., $312,146 average in Kennan and Walker 2011).

Persistence parameter (rho): In the SPACE model, rho governs the autocorrelation of match-specific utility over time. The calibrated value is rho-tilde = 0.892, implying period-to-period autocorrelation of 0.988. As rho → 1, the model generates a square root relationship between the t-year migration rate and t.

Population cross-elasticity: The elasticity of population in state i with respect to utility in state j. In both models it is proportional to gross bilateral migration in the short run. In the long run, the SPACE model retains this proportionality to migration rates, while the moving cost model converges to a static logit proportional to population shares.

Exact-hat algebra: A solution method for computing counterfactual equilibria in terms of ratios of new to old values (hats), without requiring knowledge of levels. The SPACE model admits simple exact-hat formulas for population changes; the moving cost model’s exact-hat algebra additionally requires tracking past population changes.

Kullback-Leibler divergence (in this context): The mean divergence between a model’s predicted distribution over future locations and the empirical distribution, used as a measure of forecasting accuracy. By 2018, the SPACE model achieves KL divergence of 0.014 log-points per observation versus approximately 0.12 for the moving cost model.