## Abstract

We incorporate forward-looking capital accumulation into a dynamic discrete choice model of migration. We characterize the steady-state equilibrium; generalize existing dynamic exact-hat algebra techniques to incorporate investment; and linearize the model to provide an analytical characterization of the economy's transition path using spectral analysis. We show that capital and labor dynamics interact to shape the economy's speed of adjustment toward steady state. We implement our quantitative analysis using data on capital stocks, populations, and bilateral trade and migration flows for U.S. states from 1965–2015. We show that this interaction between capital and labor dynamics plays a central role in explaining the observed decline in the rate of income convergence across U.S. states and the persistent and heterogeneous impact of local shocks.

Original language | English (US) |
---|---|

Pages (from-to) | 385-424 |

Number of pages | 40 |

Journal | Econometrica |

Volume | 91 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2023 |

## All Science Journal Classification (ASJC) codes

- Economics and Econometrics

## Keywords

- economic geography
- migration
- Spatial dynamics
- trade

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*Econometrica*,

*91*(2), 385-424. https://doi.org/10.3982/ECTA20273

**Dynamic Spatial General Equilibrium**. In: Econometrica. 2023 ; Vol. 91, No. 2. pp. 385-424.

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*Econometrica*, vol. 91, no. 2, pp. 385-424. https://doi.org/10.3982/ECTA20273

**Dynamic Spatial General Equilibrium.**/ Kleinman, Benny; Liu, Ernest; Redding, Stephen J.

In: Econometrica, Vol. 91, No. 2, 03.2023, p. 385-424.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - Dynamic Spatial General Equilibrium

AU - Kleinman, Benny

AU - Liu, Ernest

AU - Redding, Stephen J.

N1 - Funding Information: Our main source of data for our baseline quantitative analysis from 1965–2015 is the national economic accounts of the Bureau of Economic Analysis (BEA), which report population, gross domestic product (GDP), and the capital stock for each U.S. state.13 We focus on the 48 contiguous U.S. states plus the District of Columbia, excluding Alaska and Hawaii, because they only became U.S. states in 1959 close to the beginning of our sample period, and could be affected by idiosyncratic factors as a result of their geographical separation. We distinguish four broad geographical groupings of states: Rust Belt, Sun Belt, Other Northern, and Other Southern states.14 We deflate GDP and the capital stock to express them in constant (2012) prices. We use data on bilateral 5-year migration flows between U.S. states from the U.S. population census from 1960–2000 and from the American Community Survey (ACS) after 2000. We define a period in the model as equal to 5 years to match these observed data. We interpolate between census decades to obtain 5-year migration flows for each year of our sample period. To take account of international migration to each state and fertility/mortality differences across states, we adjust these migration flows by a scalar for each origin and destination state, such that origin population in year t premultiplied by the migration matrix equals destination population in year t+1, as required for internal consistency. We construct the value of bilateral shipments between U.S. states from the Commodity Flow Survey (CFS) from 1993–2017 and its predecessor the Commodity Transportation Survey (CTS) for 1977. We again interpolate between reporting years and extrapolate the data backwards in time before 1977 using relative changes in the income of origin and destination states, as discussed in further detail in Online Supplement S.7. For our baseline quantitative analysis with a single traded and nontraded sector, we abstract from direct shipments to and from foreign countries, because of the relatively low level of U.S. trade openness, particularly toward the beginning of our sample period. In our multisector extension, we incorporate foreign trade, using data on exports by origin of movement and imports by destination of shipment. To focus on the impact of incorporating forward-looking investment decisions, we assume standard values of the model's structural parameters from the existing empirical literature in our baseline specification. We assume a trade elasticity of θ=5, as in Costinot and Rodríguez-Clare (2014). We set the 5-year discount rate equal to the conventional value of β=(0.95)5. We assume an intertemporal elasticity of substitution of ψ=1, which corresponds to logarithmic intertemporal utility. We assume a value for the migration elasticity of ρ=3β, which is in line with the value in Caliendo, Dvorkin, and Parro (2019). We set the share of labor in value added to μ=0.65, as a central value in the macro literature. We assume a 5% annual depreciation rate, such that the 5-year depreciation rate is δ=1−(0.95)5, which is again a conventional value in the macro and productivity literatures. We later report comparative statics for how changes in each of these model parameters affect the speed of convergence to steady state using our closed-form solutions for the economy's transition path. We now use our theoretical framework to provide new evidence on the process of income convergence and the persistent and heterogeneous impact of local shocks in the United States. Both issues are the subject of large empirical literatures in economics. However, the existing literature on income convergence typically abstracts from migration and trade between locations, both of which are central features of the data on U.S. states. In contrast, the literature on the persistent and heterogeneous impact of local shocks allows for migration, but typically abstracts from forward-looking investment in local buildings and structures, even though these buildings and structures are central features of the world around us, and there is a large literature on capital accumulation in macroeconomics. In our baseline specification, we consider a version of our single-sector model, augmented to take account of the empirically-relevant distinction between traded and non-traded goods.12 In Section 5.1, we discuss our data sources and the parameterization of the model. In Section 5.2, we provide evidence of a decline in rates of convergence in income per capita across U.S. states since the early 1960s. In Section 5.3, we examine the extent to which this observed decline in income convergence is explained by initial conditions versus fundamental shocks, and quantify the respective contributions of capital and labor dynamics. In Section 5.4, we use our spectral analysis to provide evidence on the speed of convergence to steady state and the role of the interaction between capital and labor dynamics in shaping the persistent and heterogeneous impact of local shocks. In Section 5.5, we summarize the results of implementing our multisector extension for the shorter period from 1999–2015 for which data by sector and region are available, as discussed further in Online Supplement S.6.8. Our main source of data for our baseline quantitative analysis from 1965–2015 is the national economic accounts of the Bureau of Economic Analysis (BEA), which report population, gross domestic product (GDP), and the capital stock for each U.S. state.13 We focus on the 48 contiguous U.S. states plus the District of Columbia, excluding Alaska and Hawaii, because they only became U.S. states in 1959 close to the beginning of our sample period, and could be affected by idiosyncratic factors as a result of their geographical separation. We distinguish four broad geographical groupings of states: Rust Belt, Sun Belt, Other Northern, and Other Southern states.14 We deflate GDP and the capital stock to express them in constant (2012) prices. We use data on bilateral 5-year migration flows between U.S. states from the U.S. population census from 1960–2000 and from the American Community Survey (ACS) after 2000. We define a period in the model as equal to 5 years to match these observed data. We interpolate between census decades to obtain 5-year migration flows for each year of our sample period. To take account of international migration to each state and fertility/mortality differences across states, we adjust these migration flows by a scalar for each origin and destination state, such that origin population in year t premultiplied by the migration matrix equals destination population in year t+1, as required for internal consistency. We construct the value of bilateral shipments between U.S. states from the Commodity Flow Survey (CFS) from 1993–2017 and its predecessor the Commodity Transportation Survey (CTS) for 1977. We again interpolate between reporting years and extrapolate the data backwards in time before 1977 using relative changes in the income of origin and destination states, as discussed in further detail in Online Supplement S.7. For our baseline quantitative analysis with a single traded and nontraded sector, we abstract from direct shipments to and from foreign countries, because of the relatively low level of U.S. trade openness, particularly toward the beginning of our sample period. In our multisector extension, we incorporate foreign trade, using data on exports by origin of movement and imports by destination of shipment. To focus on the impact of incorporating forward-looking investment decisions, we assume standard values of the model's structural parameters from the existing empirical literature in our baseline specification. We assume a trade elasticity of θ=5, as in Costinot and Rodríguez-Clare (2014). We set the 5-year discount rate equal to the conventional value of β=(0.95)5. We assume an intertemporal elasticity of substitution of ψ=1, which corresponds to logarithmic intertemporal utility. We assume a value for the migration elasticity of ρ=3β, which is in line with the value in Caliendo, Dvorkin, and Parro (2019). We set the share of labor in value added to μ=0.65, as a central value in the macro literature. We assume a 5% annual depreciation rate, such that the 5-year depreciation rate is δ=1−(0.95)5, which is again a conventional value in the macro and productivity literatures. We later report comparative statics for how changes in each of these model parameters affect the speed of convergence to steady state using our closed-form solutions for the economy's transition path. We begin by providing evidence of a substantial decline over time in the rate of convergence in income per capita across U.S. states. In Figure 1, we display the annualized rate of growth of income per capita against its initial log level for each U.S. state for different subperiods, which corresponds to a conventional β-convergence specification from the growth literature. The size of the circles is proportional to initial state employment. We also show the regression relationship between these variables as the red solid line. In the opening subperiod from 1963–1980 (the left panel), we find substantial income convergence, with a negative and statistically significant coefficient of −0.0257 (standard error 0.0046), and a regression R-Squared of 0.367. This estimated coefficient is close to the −0.02 estimated by Barro and Sala-i-Martin (1992) for the longer time period from 1880–1988. By the middle subperiod from 1980–2000, we find that this relationship substantially weakens, with the slope coefficient falling by nearly one-half to −0.0148 (standard error 0.0059), and a smaller regression R-squared of 0.153. By the closing subperiod from 2000–2017, we find income divergence rather than income convergence, with a positive but not statistically significant coefficient of 0.0076 (standard error 0.0051), and a regression R-squared of 0.055. Growth and Initial Level of Income Per Capita. Note: Vertical axis shows the annualized rate of growth of income per capita for the relevant subperiod; horizontal axis displays the initial level of log income per capita at the beginning of the relevant subperiod; circles correspond to U.S. states; the size of each circle is proportional to state employment; the solid red line shows the linear regression relationship between the two variables. Within our framework, the rate of income convergence is shaped by two sets of forces: initial conditions (the initial deviation of the state variables from steady state) and shocks to fundamentals (productivity, amenities, trade costs, and migration frictions). For each of these two sets of forces, the rate of income convergence is shaped by both capital accumulation and migration. We now use our framework to provide evidence on the relative importance of each of these determinants in shaping the observed decline in income convergence over time. We now use our generalization of dynamic exact-hat algebra in Proposition 2 to examine the relative importance of initial conditions versus fundamental shocks. Starting from the observed equilibrium in the data at the beginning of our sample period, we solve for the economy's transition path to steady state in the absence of any further changes in fundamentals. We thus obtain counterfactual values for income per capita in each year implied by initial conditions alone. For both the actual and counterfactual values of income per capita, we correlate the 10-year ahead log growth in income per capita with its initial level in each year from 1970–2010. In Figure 2(a), we display these correlation coefficients over time, which summarize the strength of regional convergence for actual income per capita (dashed black line) and counterfactual income per capita in the absence of any further fundamental shocks (solid red line). We find that the decline in the rate of regional convergence is around the same magnitude for both counterfactual and actual income per capita, suggesting that much of the observed decline in the rate of income convergence is explained by initial conditions at the beginning of our sample period rather than by any subsequent fundamental shocks.15 Initial Conditions and Income Convergence. Note: Correlation coefficients between the 10-year ahead log growth in income per capita and its initial log level in each year from 1970–2010; in the left panel, the dashed black line show these correlations in the data; in both panels, the red solid line shows the correlation coefficients for counterfactual income per capita, based on starting at the observed equilibrium in the data at the beginning of our sample period, and solving for the economy's transition path to steady state in the absence of any further shocks to fundamentals; in the right panel, the black dashed line shows results for the special case with no capital accumulation, and the black dashed-dotted line shows results for the special case with no migration. To provide further evidence on the role of initial conditions in explaining the observed decline in income convergence, we regress actual log population growth on its predicted value based on convergence toward an initial steady state with unchanged fundamentals, as discussed further in Online Supplement S.6.5. Predicted population growth is calculated using only the initial values of the labor and capital state variables and the initial trade and migration share matrices, and uses no information about subsequent population growth. Nevertheless, we find a positive and statistically significant relationship, with predicted population growth explaining much of the observed population growth. This relationship is particularly strong from 1975 onwards, because the fundamental shocks from 1965–1975 move states on average further from steady state. Estimating this regression for the period 1975–2015, we find a regression slope of 0.99 (standard error of 0.095) and R-squared of 0.82. We show that this explanatory power of predicted population growth is not driven by mean reversion. Controlling for initial log population and the initial log capital stock, as well as initial log population growth, has little impact on the estimated coefficient on predicted population growth or the regression R-squared. Taken together, these results provide a first key piece of evidence that much of the observed decline in the rate of income convergence is explained by initial conditions at the beginning of our sample period rather than by fundamental shocks. Additionally, the fact that it takes decades for the decline in both actual and counterfactual income convergence to occur provides some first evidence of slow convergence to steady state. We next provide evidence on the role of capital accumulation versus migration dynamics in this impact of initial conditions. We use our generalization of dynamic exact-hat algebra from Proposition 2 for the special cases of the model with no investment (in which case our framework reduces to a dynamic discrete choice migration model following Caliendo, Dvorkin, and Parro (2019)) and no migration (in which case the population share of each state is exogenous at its 1965 level). Again, we start at the observed equilibrium in the data at the beginning of our sample period and solve for the transition to steady state in the absence of any further changes to fundamentals. We thus obtain counterfactual values for income per capita in each year implied by initial conditions alone for these two special cases of the model with no investment and no migration. Using these counterfactual predictions, we again correlate the 10-year ahead log growth in income per capita with its initial level for each year from 1970–2010. In Figure 2(b), we display these correlation coefficients over time for the full model (replicating the results from Figure 2(a), as shown by the solid red line), the model with no investment (dashed line), and the model with no migration (dotted-dashed line). We find substantial contributions to the observed decline in income convergence over time from both investment and migration dynamics. Capital accumulation is more important than migration for these dynamics of income per capita, highlighting the relevance of incorporating investment decisions into dynamic spatial models. Nevertheless, even in the model with no capital, we find a decline in the correlation coefficient for income convergence of around 20 percentage points. More generally, allowing for migration is central to matching the observed changes in population shares across U.S. states over time. We now use our linearization of the model and our spectral analysis to provide further evidence on the role of capital accumulation and migration dynamics in shaping both the impact of initial conditions and fundamental shocks. First, we analyze the determinants of the speed of convergence to steady state. Second, we examine the role of capital and labor dynamics in influencing the convergence process. Third, we evaluate the role of these two sources of dynamics in shaping the persistent and heterogeneous impact of local shocks. Fourth, we evaluate the comparative statics of the speed of convergence to steady state with respect to model parameters. Using Propositions 3–5, we compute half-lives of convergence to steady state as determined by the eigenvalues of the transition matrix. In Figure 3, we show these half-lives (solid black line with circle markers) for the entire spectrum of 2N eigencomponents, sorted by increasing half-life. Each nontrivial eigencomponent corresponds to an eigenshock for which the initial impact of the shock on the state variables is equal to an eigenvector of the transition matrix (uh=Rf˜(h)). We display results based on the transition matrix (P) computed using the implied steady-state trade and migration share matrices (S, T, D, E) for 1975. We compute these implied steady-state matrices using using our dynamic exact-hat algebra results from Proposition 2. We focus on 1975, because states are on average furthest from steady state in this year, but we find a similar pattern of results for other years.16 Spectrum of Eigencomponents for 1975. Note: Spectrum of eigencomponents for the steady-state transition (P) matrix for 1975 recovered using our dynamic exact-hat algebra results from Proposition 2; eigencomponents are sorted in increasing order of half-life of convergence to steady state; black solid line with circle markers shows half-life of convergence to steady state; red dotted vertical line shows the eigenvector [1,…,1,0,…,0]′ with eigenvalue 0; blue solid vertical line shows the eigenvector [0,…,0,1,…,1]′, which with log preferences (ψ = 1) has eigenvalue [1 − μ(1 − β(1 − δ))], as shown by the blue dashed horizontal line; purple solid line with square markers shows the loadings of the 1975 gaps of the state variables from steady state on the eigencomponents; green solid line with diamond markers shows the loadings of the 1975–2015 productivity and amenity shocks on the eigencomponents. As in our symmetric two-region example above, these eigencomponents have an intuitive interpretation. One eigenvector [1,…,1,0,…,0]′ captures a common amenity shock to all locations that leaves population shares and capital stocks unchanged, as shown by the red dotted vertical line. This trivial eigencomponent has an associated eigenvalue of 0, since the initial and new steady state coincide, such that there are no transition dynamics. Another eigenvector [0,…,0,1,…,1]′ captures a common productivity shock to all locations that leaves population shares unchanged, but increases the capital stock in all locations, as shown by the blue solid vertical line. This eigencomponent has an associated eigenvalue of [1−μ(1−β(1−δ))] in the special case of log preferences (ψ=1), as shown by the horizontal blue dashed line, and induces the same capital dynamics as in the closed economy. In between the red dotted and blue solid vertical lines, we have N−1 eigencomponents with a negative correlation between the gaps of the labor and capital state variables from steady state, for which convergence to steady state is relatively rapid. To the right of the blue solid vertical line, we have N−1 eigencomponents with a positive correlation between the gaps of the labor and capital state variables from steady state, for which convergence to steady state is relatively slow. Three features are particularly noteworthy. First, the speed of convergence to steady state is typically slow, with an average half-life across the entire spectrum of eigenshocks of around 20 years. Therefore, our theoretical framework is consistent with reduced-form empirical findings of persistent impacts of local labor market shocks, as found, for example, for the China shock in the United States in Autor, Dorn, and Hanson (2013, 2023) and Brazil's trade liberalization in Dix-Carneiro and Kovak (2017). Second, there is substantial heterogeneity in the speed of convergence across eigenshocks, with the half-life of convergence varying from instantaneous convergence for the trivial eigenshock [0,…0,1,…,1]′ to around 80 years. Hence, our theoretical framework also rationalizes heterogeneous effects of local labor market shocks, as emphasized for example in Eriksson, Russ, Shambaugh, and Xu (2019). Third, the higher the correlation between the gaps of the labor and capital state variables from steady state across locations, the slower the speed of convergence to steady state (the larger the half-life of convergence to steady state). We provide further evidence on the strength of this relationship in Figure S.6.9 in Online Supplement S.6.6.2. This finding that capital and labor dynamics interact to shape the speed of convergence to steady state reflects the interplay between the marginal products of capital and labor in the production technology, as discussed above. If a region experiences a negative shock that reduces the steady-state values of both the labor and capital variables, the gradual process of migration away from declining regions is slowed by the gradual downward adjustment of existing stocks of buildings and structures, and vice versa. Therefore, our framework explains persistent and heterogeneous effects of local shocks through this interaction between capital and labor dynamics. In Figure 3, we also relate both the initial gaps of the state variables from steady state in 1975 and the empirical shocks to productivity and amenities from 1975–2015 to these eigenshocks. We use the property that any deviations of the state variables from steady state or any empirical fundamental shocks can be expressed as a linear combination of the eigencomponents. For the deviations of the state variables from steady state, these loadings can be recovered from a regression of these steady-state deviations on the eigenvectors of the transition matrix. The purple line with square markers shows these loadings for the 1975 gaps from steady state. For the empirical fundamental shocks, these loadings can be recovered from a regression of the empirical fundamental shocks on the eigenshocks corresponding to the eigenvectors of the transition matrix. The green line with diamond markers shows these loadings for the empirical shocks to productivity and amenities from 1975–2015. We recover both the steady-state gaps and the empirical productivity and amenity shocks from the full nonlinear model, as discussed in Online Supplements S.2.2 and S.6.7, respectively. Figure 3 shows the absolute value of these loadings, normalized such that the sum of these absolute values is equal to one. Comparing the two sets of loadings, we find that the steady-state gaps in 1975 typically load more heavily on the upper part of the spectrum of eigencomponents with slow rates of convergence to steady state (the purple line with square markers typically lies above the green line with diamond markers for the upper part of the spectrum to the right of the blue solid vertical line). In contrast, the empirical shocks to productivity and amenities from 1975–2015 generally load more heavily on the lower part of the spectrum of eigencomponents with fast rates of convergence to steady state (the green line with diamond markers typically lies above the purple line with square markers for the lower part of the spectrum to the left of the blue solid vertical line). This pattern of results is consistent with the evidence above that initial conditions explain much of the observed decline in income convergence over our sample period. We find loadings-weighted average half-lives of convergence to steady state of 38 years for the 1975 steady-state gaps and 20 years for the productivity and amenity shocks from 1975–2015. Therefore, while the economy adjusts relatively rapidly to the observed productivity and amenity shocks during our sample period, it takes longer to adjust to the initial gaps of the state variables from steady state. We now use our spectral analysis to probe further the role of capital and labor dynamics in shaping the impact of initial conditions. In Figure 4, we use Proposition 4 to decompose the initial gap of the labor and capital state variables from steady state in 1975 into the contributions of the different eigencomponents. In the left panel, we display the overall log deviations of capital from steady state (vertical axis) against the overall log deviation of labor from steady state (horizontal axis). In the middle panel, we show these log deviations for the top-10 eigencomponents with the slowest convergence to steady state. In the right panel, we show these log deviations for the remaining 88 eigencomponents with faster convergence to steady state. By construction, the overall log deviations in the left panel equal the sum of those in the middle and right panels. We preserve the same scale on the horizontal axis across the three panels, but allow the scale on the vertical axis to differ. We show Rust Belt states in gray, Sun Belt states in red, Other North states in blue, and Other South states in brown. The size of the marker for each state is proportional to the size of its population. Decomposition of Gaps of State Variables from Steady State in 1975. Note: Left panel shows the 1975 log deviations of capital and labor from steady state for each U.S. state; middle and right panels decompose these 1975 steady-state gaps into the contributions of the top 10 eigencomponents with the slowest convergence to steady state (middle panel) and the remaining 88 eigencomponents (right panel). From the left panel, the overall capital and labor gaps are positively correlated across U.S. states, consistent with the slow convergence to steady state established above. Rust Belt states (in gray) appear systematically toward the right with populations above steady state, while Sun Belt states (in red) appear systematically toward the left with populations below steady state. From the vertical axis of the left panel, all states have capital stocks below steady state, again highlighting the relevance of capital dynamics. In general, Rust Belt states have smaller deviations of capital from steady state than Sun Belt states. From the middle panel, much of the positive correlation between the steady-state gaps is driven by the top-10 eigencomponents with the slowest convergence to steady state. For these top-10 eigencomponents, the positive correlation is particularly strong, and there is clear geographical separation between the Rust Belt states (toward the top right) and the Sun Belt states (toward the middle and bottom left). Given the role of geography in shaping migration through the gravity equation for migration flows, this geographical separation contributes to slow convergence to steady state. In contrast, from the right panel, the remaining 88 eigencomponents show a weaker positive correlation between the steady-state gaps, with smaller variation in the absolute magnitude of the labor steady-state gap on the horizontal axis, and a less clear geographical separation between Rust Belt and Sun Belt states. Therefore, our findings of slow convergence toward steady state based on initial conditions are driven by the initial steady-state gaps loading heavily on eigencomponents with strong positive correlations between the capital and labor steady-state gaps, and the clear geographical separation between Rust Belt states with populations above steady state and Sun Belt states with population closer to or below steady state. We next use our spectral analysis to explore further the role of capital and labor dynamics in shaping the impact of fundamental shocks. In Figure 5, we use Proposition 4 to decompose the empirical productivity and amenity shocks from 1975–2015 into the contributions of the different eigencomponents. In the left panel, we display the empirical amenity shocks (vertical axis) against the empirical productivity shocks (horizontal axis) over this time period. In the middle panel, we show the components of these empirical shocks accounted for the top-10 eigencomponents with the slowest convergence to steady state. In the right panel, we show the corresponding components accounted for by the remaining 88 eigencomponents with faster convergence to steady state. By construction, the empirical shocks in the left panel equal the sum of the components in the middle and right panels. We again preserve the same scale on the horizontal axis across the three panels, but allow the scale on the vertical axis to differ. We use the same coloring for the four groups of states as above, and the size of the marker for each state is again proportional to the size of its population. Decomposition of Productivity and Amenity Shocks from 1975–2015. Note: Left panel shows log productivity and amenity shocks from 1975–2015 for each U.S. state; middle and right panels decompose these productivity and amenity shocks into the contributions of the top 10 eigencomponents with the slowest convergence to steady state (middle panel) and the remaining 88 eigencomponents (right panel). From the left panel, we find a negative correlation between the empirical productivity and amenity shocks from 1975–2015. Note that higher productivity raises the marginal productivity of both labor and capital, which increases both state variables. In contrast, higher amenities only directly raise worker utility, which increases the labor state variable. Therefore, this negative correlation between productivity and amenity shocks implies a negative correlation between changes in the labor and capital steady-state gaps, and hence implies relatively rapid convergence, in contrast to our results for initial conditions above.17 From the middle panel, we find a strong positive relationship between the components of the amenity and productivity shocks that are accounted for by the top-10 eigencomponents with the slowest convergence to steady state. But there is much less variation in the absolute magnitude of this component on the horizontal axis than for the overall empirical productivity and amenity shocks in the left panel. Therefore, the top-10 eigencomponents again imply slow convergence to steady state, but they account for a relatively small amount of the empirical amenity and productivity shocks. From the right panel, we find a strong negative relationship between the components of the amenity and productivity shocks that are accounted for by the remaining 88 eigencomponents, with greater variation in the absolute magnitude of the productivity shocks on the horizontal axis. Hence, the negative correlation between the empirical amenity and productivity shocks in the left panel is driven by these remaining 88 eigencomponents with relatively fast convergence to steady state. In contrast to our results for initial conditions above, we observe no clear geographical separation between Rust Belt and Sun Belt states. We thus find that the relatively small contribution from fundamental shocks relative to initial conditions toward the decline in income convergence is explained by these fundamental shocks loading more on eigencomponents characterized by fast convergence to steady state. To provide further evidence on the role of capital and labor dynamics in shaping the persistent and heterogeneous impact of local shocks, we now consider individual empirical shocks to productivity and amenities. We examine impulse response functions for the labor and capital state variables in each U.S. state following a local shock, starting from the steady state implied by 1975 fundamentals. Motivated by the observed secular reallocation of economic activity from the Rust Belt to the Sun Belt, we report results for the empirical shock to relative productivity in Michigan from 1975–2015 (a 15% decline) and the empirical shock to relative amenities in Arizona over this same period (a 34% rise). In Figure 6, we display the impulse response of population shares in each U.S. state in response to the empirical 15% decline in relative productivity in Michigan. In the top-left panel, we show the log deviation of Michigan's population share from the initial steady state along the transition path to the new steady state. We find an intuitive pattern where the decline in Michigan's relative productivity leads to a population outflow, which occurs gradually over time, because of migration frictions and gradual adjustment to capital. Impulse Response of Population Shares for a 15% Decline in Productivity in Michigan. Note: Top-left panel shows overall log deviation of Michigan's population share from steady state (vertical axis) against time in years (horizontal axis) for a 15% decline in Michigan's productivity (its empirical relative decline in productivity from 1975–2015); Top-right panel shows overall log deviation of other states' population shares from steady state (vertical axis) against time in years (horizontal axis) for this shock to Michigan's productivity; blue lines show Michigan's neighbors; gray lines show other states; Middle and bottom panels decompose this overall impulse response into the contribution of eigencomponents 1–88 (fast convergence) and 88–98 (slow convergence), respectively. In the top-right panel, we show the corresponding log deviations of population shares from the initial steady state for all other states. We indicate Michigan's neighbors using the blue lines with circle markers and all other states using the gray lines. We find that the model can generate rich nonmonotonic dynamics for individual states. Initially, the decline in Michigan's productivity raises the population share of its neighbors, since workers face lower migration costs in moving to nearby states. However, as the economy gradually adjusts toward the new steady state, the population share in Michigan's neighbors begins to decline, and can even fall below its value in the initial steady state. Intuitively, workers gradually experience favorable idiosyncratic mobility shocks for states further away from Michigan, and the decline in Michigan's productivity reduces the size of its market for neighboring locations, which can make those neighboring locations less attractive in the new steady state. Population shares in all other states increase in the new steady state relative to the initial steady state. In the middle two panels, we show the log deviations from steady state for the component of population shares attributed to bottom-88 eigencomponents with relatively fast convergence to steady state. In the middle-left panel, the solid black line shows the overall log deviation of Michigan's population share from steady state (the same as in the top-left left panel), while the dashed black line indicates the component due to the bottom-88 eigencomponents. In the middle-right panel, the solid blue line with circle markers shows the overall log deviation from steady state of the population shares of Michigan's neighbors (same as in the top-right panel); the dashed blue line with circle markers indicates the component of these neighbors' population shares due to the bottom-88 eigencomponents; the gray lines represent the population shares of all other states (the same as in the top-right panel). Comparing the two sets of blue lines in the middle-right panel, these eigencomponents featuring fast convergence toward steady-state drive the initial rise in the population shares of Michigan's neighbors. In the bottom two panels, we show the log deviations from steady state for the component of population shares attributed to the top-10 eigencomponents with relatively slow convergence to steady state. In the bottom-left panel, the solid black line shows the overall log deviation of Michigan's population share from steady state (the same as in the top-left panel), while the dashed black line indicates the component due to the top-10 eigencomponents. In the bottom-right panel, the solid blue line with circle markers shows the overall log deviation from steady state of the population shares of Michigan's neighbors (same as in the top-right panel); the dashed blue line with circle markers indicates the component of these neighbors' population shares due to the top-10 eigencomponents; the gray lines represent the population shares of all other states (the same as in the top-right panel). Comparing the two sets of blue lines in the bottom-right panel, these eigencomponents featuring slow convergence toward steady state drive the ultimate reduction in the population shares of Michigan's neighbors. Therefore, the nonmonotonic dynamics for Michigan's neighbors in the top-right panel reflect the changing importance over time of the slow and fast-moving components of the economy's adjustment to the productivity shock in the middle-right and bottom-right panels.18 In Online Supplement S.6.6.6, we report analogous results for the empirical shock to relative amenities in Arizona from 1975–2015 (a 34% rise). Whereas the decline in relative productivity in Michigan decreases its population share above, this increase in relative amenities in Arizona increases its population share. We again find persistent and heterogeneous effects of the shock across states. Individual states can again experience rich dynamics, because of the changing importance over time of the slow and fast-moving components of the economy's adjustment to the shock, although there is less evidence of nonmonotonic dynamics for individual states for this amenities shock than for the productivity shock above. Finally, we show that our spectral analysis permits an analytical characterization of the comparative statics of the speed of convergence to steady state with respect to changes in model parameters. Undertaking these comparative statics in the nonlinear model is challenging, because the speed of convergence to steady state depends on the incidence of the productivity and amenity shocks across the labor and capital state variables in each location. As a result, to fully characterize the impact of changes in model parameters on the speed of convergence in the nonlinear model, one needs to undertake counterfactuals for the economy's transition path in response to the set of all possible productivity and amenity shocks, which is not well-defined. In contrast, our spectral analysis has two key properties. First, the set of all possible productivity and amenity shocks is spanned by the set of eigenshocks, which is well-defined. Second, we have a closed-form solution for the impact matrix (R) and transition matrix (P) in terms of the observed data (S, T, D, E) and structural parameters {ψ,θ,β,ρ,μ,δ}. Therefore, for any alternative model parameters, we can immediately solve for the entire spectrum of eigenvalues (and corresponding half-lives) associated with the eigenshocks using the observed data. Because the eigenshocks span all possible empirical productivity and amenity shocks, understanding how parameters affect the entire spectrum of half-lives translates into an analytically sharp understanding of how convergence rates are affected by model parameters. In Figure 7, we display the half-lives of convergence to steady state across the entire spectrum of eigenshocks for different values of model parameters. Each panel varies the noted parameter, holding constant the other parameters at their baseline values. On the vertical axis, we display the half-life of convergence to steady state. On the horizontal axis, we rank the eigenshocks in terms of increasing half-lives of convergence to steady state for our baseline parameter values. Half-lives of Convergence to Steady State for Alternative Parameter Values. Note: Half-lives of convergence to steady state for each eigenshock for alternative parameter values in the 1975 steady state; vertical axis shows half-life in years; horizontal axis shows the rank of the eigenshocks in terms of their half-lives for our baseline parameter values (with one the lowest half-life); each panel varies the noted parameter, holding constant the other parameters at their baseline values; the blue and red solid lines denote the lower and upper range of the parameter values considered, respectively; each of the other eight lines in between varies the parameters uniformly within the stated range; the thick black dotted line in the bottom-left panel displays half-lives for the special case of our model without capital, which corresponds to the limiting case in which the labor share (μ) converges to one. In the top-left panel, a lower intertemporal elasticity of substitution (ψ) implies a longer half-life (slower convergence), because consumption becomes less substitutable across time for landlords, which reduces their willingness to respond to investment opportunities. In the top-middle panel, a higher migration elasticity (lower ρ) has an ambiguous effect that depends on the interaction of capital and migration dynamics: a higher migration elasticity increases the responsiveness of labor flows to capital accumulation, leading to a longer half-life when the labor and capital gaps from steady state are positively correlated, and the converse when they are negatively correlated. In the top-right panel, a higher discount factor (β) also implies a longer half-life (slower convergence), because landlords have a higher saving rate, which implies a greater role for endogenous capital accumulation, thereby magnifying the impact of productivity and amenity shocks, and implying a longer length of time for adjustment to occur. In the bottom-right panel, we vary the share of expenditure on the single tradable sector relative to the single nontradable sector. A lower share of tradables (γ) implies a longer half- life (slower convergence), because it makes the impact of shocks more concentrated locally, which requires greater labor and capital reallocation between locations. In the bottom-middle panel, a higher trade elasticity (θ) implies a longer half-life (slower convergence), because it increases the responsiveness of production and consumption in the static trade model, and hence requires greater reallocation of capital and labor across locations. In the bottom-left panel, we find that a lower labor share (μ) implies a longer half-life (slower convergence), because it implies a greater role for endogenous capital accumulation, which again magnifies the impact of productivity and amenity shocks, and hence requires a greater length of time for adjustment to occur. In this bottom-left panel, we also show the half-lives of convergence to steady state for the special case of our model with no capital using the black dotted line, which corresponds to the limiting case in which the labor share converges to one. In this special case, we have only N state variables and eigenshocks, compared to 2N state variables and eigenshocks in the general model. We again find that capital accumulation and migration dynamics interact with one another. As we introduce capital (raise the labor share above zero), we find slower convergence to steady state in the configurations of the state space where the gaps of the labor and capital state variables are positively correlated across locations (the largest N eigenvalues become larger). In contrast, we find faster convergence to steady state in the configurations of the state-space where the gaps of the labor and capital state variables are negatively correlated across locations (we add an additional N eigenvalues smaller than those represented by the dotted line). In a final empirical exercise, we implement our multisector extension with region-sector specific capital, as discussed in further detail in Online Supplement S.6.8. We again find slow rates of convergence to steady state in this multisector extension, although the rate of convergence is higher than in our baseline single-sector specification, with an average half-life of 7 years and a maximum half-life of 35 years. This finding is driven by the property of the region-sector migration matrices that flows of people between sectors within states are larger than those between states. An implication is that the persistence of local labor market shocks depends on whether they induce reallocation across industries within the same location or reallocation across different locations. Again we find a strong positive relationship between the half-life of convergence to steady state and the correlation between the gaps from steady state for the labor and capital state variables. Therefore, in the multisector model as for the single-sector model above, we find that the interaction between capital accumulation and migration dynamics shapes the persistent and heterogeneous impact of local shocks. Publisher Copyright: © 2023 The Authors. Econometrica published by John Wiley & Sons Ltd on behalf of The Econometric Society.

PY - 2023/3

Y1 - 2023/3

N2 - We incorporate forward-looking capital accumulation into a dynamic discrete choice model of migration. We characterize the steady-state equilibrium; generalize existing dynamic exact-hat algebra techniques to incorporate investment; and linearize the model to provide an analytical characterization of the economy's transition path using spectral analysis. We show that capital and labor dynamics interact to shape the economy's speed of adjustment toward steady state. We implement our quantitative analysis using data on capital stocks, populations, and bilateral trade and migration flows for U.S. states from 1965–2015. We show that this interaction between capital and labor dynamics plays a central role in explaining the observed decline in the rate of income convergence across U.S. states and the persistent and heterogeneous impact of local shocks.

AB - We incorporate forward-looking capital accumulation into a dynamic discrete choice model of migration. We characterize the steady-state equilibrium; generalize existing dynamic exact-hat algebra techniques to incorporate investment; and linearize the model to provide an analytical characterization of the economy's transition path using spectral analysis. We show that capital and labor dynamics interact to shape the economy's speed of adjustment toward steady state. We implement our quantitative analysis using data on capital stocks, populations, and bilateral trade and migration flows for U.S. states from 1965–2015. We show that this interaction between capital and labor dynamics plays a central role in explaining the observed decline in the rate of income convergence across U.S. states and the persistent and heterogeneous impact of local shocks.

KW - economic geography

KW - migration

KW - Spatial dynamics

KW - trade

UR - http://www.scopus.com/inward/record.url?scp=85150360752&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85150360752&partnerID=8YFLogxK

U2 - 10.3982/ECTA20273

DO - 10.3982/ECTA20273

M3 - Article

AN - SCOPUS:85150360752

SN - 0012-9682

VL - 91

SP - 385

EP - 424

JO - Econometrica

JF - Econometrica

IS - 2

ER -