Dynamic Spatial General Equilibrium
动态空间一般均衡

2.6.1 Capital Returns and Accumulation
2.6.1 资本回报与积累

Using capital market clearing (9), the gross return on capital in each location i must satisfy
使用资本市场清算（9），每个位置 i 的资本总回报必须满足

where the price index (4) of each location becomes
每个地点的价格指数（4）变为

(10)

The law of motion for capital is
资本运动定律是

(11)

where is the saving rate defined recursively as in Lemma 1:
其中 是递归定义的储蓄率，定义如下：

2.6.2 Goods Market Clearing
2.6.2 商品市场出清

Using the CES expenditure share, the equilibrium pricing rule (2), and the capital market clearing condition (9) in the goods market clearing condition (8), the requirement that income equals expenditure on the goods produced by a location can be written solely in terms of labor income:
利用 CES 支出份额、均衡定价规则（2）和资本市场清算条件（9）在商品市场清算条件（8）中，收入等于该地区生产商品支出的要求可以仅用劳动收入来表示：

(12)

(13)

where we have used the property that capital income is a constant multiple of labor income, is the expenditure share of importer n on exporter i at time t, we have defined as the corresponding income share of exporter i from importer n at time t, and note that the order of subscripts switches between the expenditure share () and the income share (), because the first and second subscripts will correspond below to rows and columns of a matrix, respectively.
在本文中，我们使用了资本收入是劳动收入常数倍的性质， 表示在时间 t 进口商 n 对出口商 i 的支出份额，我们定义 为在时间 t 出口商 i 从进口商 n 获得的相应收入份额，并注意下标顺序在支出份额（ ）和收入份额（ ）之间切换，因为下面的一、二个下标将分别对应矩阵的行和列。

2.6.3 Worker Value Function
2.6.3 工人价值函数

Using the value function (7), indirect utility function (3) and the properties of the extreme value distribution, the expected value from living in location n at time t after taking expectations with respect to the idiosyncratic mobility shocks (i.e., ), can be written as
利用价值函数（7）、间接效用函数（3）以及极值分布的性质，在考虑针对特定流动性冲击 （即 ）的期望后，居住在位置 n 在时间 t 的期望值可以表示为

(14)

where the expectation is taken over future fundamentals .
在何处对未来基本面 的预期 被考虑。

2.6.4 Population Flow 2.6.4 人口流动

Using the properties of the extreme value distribution, the population flow condition for the evolution of the population distribution over time is given by
利用极值分布的性质，给出随时间演化的人口分布的种群流动条件

(15)

(16)

where is the outmigration probability from location i to location g between time t and ; we have defined as the corresponding inmigration probability to location g from location i between time t and . Again note that the order of subscripts switches between the outmigration probability () and the inmigration probability (), because the first and second subscripts will correspond below to rows and columns of a matrix, respectively.
在位置 i 到位置 g 在时间 t 到 之间的迁出概率 ；我们已将 定义为在时间 t 到 之间从位置 i 到位置 g 的迁入概率。再次注意，下标顺序在迁出概率（ ）和迁入概率（ ）之间切换，因为第一个和第二个下标将分别对应矩阵的行和列。

2.6.5 Properties of General Equilibrium
2.6.5 一般均衡的性质

Given the state variables and the realized location fundamentals , the general equilibrium in each period is determined as in a standard static international trade model. Between periods, the evolution of the stock of capital is determined by the equilibrium saving rate, and the dynamics of the population distribution are determined by the gravity equation for migration. We now formally define equilibrium.
考虑到状态变量 和实际位置基础 ，每个时期的总体均衡被确定为与标准静态国际贸易模型相同。在时期之间，资本存量 的演变由均衡储蓄率决定，人口分布 的动态由移民的引力方程决定。我们现在正式定义均衡。

We define a deterministic steady-state equilibrium as one in which the fundamentals and the endogenous variables are constant over time, where we use an asterisk to denote the steady-state value of variables.
我们定义一个确定性稳态均衡为一种在时间上基本要素 和内生变量 保持恒定的状态，其中我们用星号表示变量的稳态值。

2.6.6 Trade and Migration Share Matrices
2.6.6 贸易与移民份额矩阵

We now introduce the trade and migration share matrices that we use to characterize the model's transition dynamics in response to shocks to fundamentals. Let S be the matrix with the nith element equal to importer n's expenditure on exporter i. Let T be the matrix with the inth element equal to the fraction of income that exporter i derives from selling to importer n. We refer to S as the expenditure share matrix and to T as the income share matrix. Intuitively, captures the importance of i as a supplier to location n, and captures the importance of n as a buyer for country i. Note the order of subscripts: in matrix S, rows are buyers and columns are suppliers, whereas in matrix T, rows are suppliers and columns are buyers.
我们现在引入贸易和移民份额矩阵，这些矩阵用于描述模型对基本面冲击的反应的动态变化。令 S 为 矩阵，其中第 n 个元素等于进口国 n 对出口国 i 的支出。令 T 为 矩阵，其中第 in 个元素等于出口国 i 从向进口国 n 销售中获得的收入比例。我们将 S 称为支出份额矩阵，将 T 称为收入份额矩阵。直观上， 捕捉了 i 作为 n 地供应商的重要性，而 捕捉了 n 作为 i 国买家的重视程度。注意下标的顺序：在矩阵 S 中，行是买家，列是供应商，而在矩阵 T 中，行是供应商，列是买家。

Similarly, let D be the matrix with the nith element equal to the share of outmigrants from origin n to destination i. Let E be the matrix with the inth element equal to the share of inmigrants to destination i from origin n. We refer to D as the outmigration matrix and to E as the inmigration matrix. Intuitively, captures the importance of i as a destination for origin n, and captures the importance of n as an origin for destination i. Again note the order of subscripts: in matrix D, rows are origins and columns are destinations, whereas in matrix E, rows are destinations and columns are origins.⁴
同样，设 D 为第 0#矩阵，其中第 n 个元素等于从起源 n 到目的地 i 的移民份额。设 E 为第 1#矩阵，其中第 i 个元素等于从起源 n 到目的地 i 的移民份额。我们将 D 称为移民矩阵，将 E 称为入境矩阵。直观上，第 2#捕捉了 i 作为起源 n 的目的地的重要性，而第 3#捕捉了 n 作为目的地 i 的起源的重要性。再次注意下标的顺序：在矩阵 D 中，行是起源，列是目的地，而在矩阵 E 中，行是目的地，列是起源。第 4#

2.6.7 Dynamic Exact-Hat Algebra
2.6.7 动态精确帽代数

We now generalize existing dynamic exact-hat algebra results for undertaking counterfactuals under perfect foresight in migration models from Caliendo, Dvorkin, and Parro (2019) to incorporate forward-looking investment decisions. We suppose that we observe the spatial distribution of economic activity somewhere along the transition path toward an unobserved steady state. Given the initial observed endogenous variables of the model, we show that we able to solve for the economy's transition path in time differences () for any anticipated convergent sequence of future changes in fundamentals, without having to solve for the unobserved initial level of fundamentals.
我们现在将 Caliendo、Dvorkin 和 Parro（2019）关于在移民模型中基于完美预见的反事实动态精确帽代数结果进行推广，以纳入前瞻性投资决策。我们假设我们观察到经济活动在向未观察到的稳态过渡路径上的空间分布。鉴于模型中观察到的初始内生变量，我们表明，对于任何预期的基本面的未来变化收敛序列，我们能够求解经济的时间差异（ ）过渡路径，而无需求解未观察到的初始基本面水平。

PROPOSITION 2. (Dynamic Exact-Hat Algebra)Given an initial observed allocation of the economy, , and a convergent sequence of future changes in fundamentals under perfect foresight:
命题 2.（动态精确帽代数）给定经济的一个初始观察到的分配 ，以及在完美预测下，基本面未来变化的收敛序列：

the solution for the sequence of changes in the model's endogenous variables does not require information on the level of fundamentals:
模型内生变量变化序列的解不需要关于基本面水平的信息：

In addition to these dynamic exact-hat algebra results in Proposition 2, we can invert the model to solve for the unobserved changes in productivity, amenities, trade costs, and migration costs that are implied by the observed changes in the endogenous variables of the model under perfect foresight, as shown in Online Supplement S.2.1. Importantly, we can undertake this model inversion along the transition path without making assumptions about the precise sequence of future fundamentals, because the observed changes in migration flows and the capital stock capture agents' expectations about this sequence of future fundamentals.
除了命题 2 中的这些动态精确帽代数结果外，我们还可以将模型反转来解决由模型内生变量观察到的变化所隐含的生产率、便利设施、贸易成本和移民成本的变化，正如在线补充 S.2.1 所示。重要的是，我们可以在转换路径上执行这种模型反转，而不需要对未来基本面的精确顺序做出假设，因为观察到的移民流量和资本存量变化捕捉了代理人对这一未来基本面顺序的预期。

3.2.1 System of Second-Order Difference Equations
3.2.1 二阶差分方程组

In Online Appendices B.4.4–B.4.5, we show that the model's transition dynamics can be reduced to the following linear system of second-order difference equations in the state variables:
在在线附录 B.4.4–B.4.5 中，我们表明该模型的转换动力学可以简化为以下关于状态变量的二阶差分方程的线性系统：

(22)

where is a vector of the state variables; is a vector of the shocks to fundamentals; and Ψ, Γ, Θ, and Π are matrices that only depend on the structural parameters of the model and the observed trade and migration share matrices .
在 中是状态变量的 向量； 是基本面的 向量；Ψ、Γ、Θ和Π是仅依赖于模型 的结构参数和观察到的贸易和移民份额矩阵 的 矩阵。

We solve this matrix system of equations using the method of undetermined coefficients following Uhlig (1999) to obtain a closed-form solution for the economy's transition path in terms of an impact matrix (R), which captures the initial impact of the fundamental shocks, and a transition matrix (P), which governs the evolution of the state variables over time.⁶ Specifically, one can show that the matrix has eigenvectors of the form , where are the corresponding eigenvalues, and are vectors. If the eigenvalues are stable (), the linearized system has a unique stable transition path (see, e.g., Dejong and Dave (2011)).⁷
我们采用 Uhlig（1999）的不定系数法来解这个矩阵方程组，从而获得一个关于冲击矩阵（R）的封闭形式解，该矩阵捕捉了基本冲击的初始影响，以及一个过渡矩阵（P），它控制状态变量随时间的变化。具体来说，可以证明矩阵 具有形式 的特征向量，其中 是相应的特征值， 是 向量。如果特征值是稳定的（ ），则线性化系统具有唯一的稳定过渡路径（例如，参见 Dejong 和 Dave（2011））。 ⁷

PROPOSITION 3. (Transition Path)Suppose that the economy at time is on a convergence path toward an initial steady state with constant fundamentals (z, b, κ, τ). At time , agents learn about one-time, permanent shocks to productivity and amenities () from time onwards. There exists a transition matrix (P) and a impact matrix (R) such that the second-order difference equation system in (22) has a closed-form solution of the form:
命题 3.（过渡路径）假设在时间 时，经济处于向具有恒定基本要素（z, b, κ, τ）的初始稳态收敛的路径上。在时间 时，从时间 开始，代理人了解到一次性的、永久性的生产率和便利性冲击（ ）。存在一个 过渡矩阵（P）和一个 影响矩阵（R），使得（22）中的二阶差分方程组具有以下形式的封闭形式解：

(23)

The transition matrix P satisfies
矩阵 P 满足

where Λ is a diagonal matrix of 2N stable eigenvalues and U is a matrix stacking the corresponding 2N eigenvectors . The impact matrix (R) is given by
在Λ是一个 2N 个稳定特征值 的对角矩阵，U 是一个堆叠相应 2N 个特征向量 的矩阵。影响矩阵（R）由以下给出

where (Ψ, Γ, Θ, Π) are the matrices from the system of second-order difference equations (22).
其中(Ψ, Γ, Θ, Π)是二阶差分方程组(22)中的矩阵。

3.2.2 Convergence Dynamics versus Fundamental Shocks
3.2.2 收敛动力学与基本冲击对比

Using this closed-form solution in Proposition 3, the transition path of the economy's state variables can be additively decomposed into the contributions of convergence dynamics given initial conditions and fundamental shocks. Applying equation (23) across time periods, we obtain
利用命题 3 中的这个闭式解，经济状态变量的转换路径可以加性分解为给定初始条件和基本冲击的收敛动力学贡献。将方程（23）应用于时间区间，我们得到

(24)

In the absence of shocks to fundamentals (), the second term on the right-hand side of equation (24) is zero. In this case, the evolution of the state variables is shaped solely by convergence dynamics given initial conditions, and converges over time to
在基本因素没有受到冲击（ ）的情况下，方程（24）右侧的第二项为零。在这种情况下，状态变量的演变完全由初始条件给出的收敛动力学决定，并且随着时间的推移收敛到

(25)

where is well-defined under the condition that the spectral radius of P is smaller than one.
在 P 的特征半径小于 1 的条件下， 是良好定义的。

In contrast, if the economy is initially in a steady state at time 0, the first term on the right-hand side of equation (24) is zero. In this case, the transition path of the state variables is solely driven by the second term for fundamental shocks, and follows:
与对比，如果经济在时间 0 时处于稳态，则方程（24）右侧的第一个项为零。在这种情况下，状态变量的转换路径仅由基本冲击的第二项驱动，并遵循：

(26)

In the period when the shocks occur, the response of the state variables is . Taking the limit as in equation (26), the comparative steady-state response is
在冲击发生的时期 ，状态变量的响应为 。将方程（26）中的 取极限，比较稳态响应为

(27)

A key implication of this additive separability in equation (24) is that we can examine the economy's dynamic response to fundamental shocks separately from its convergence towards an initial steady state with unchanged fundamentals. Therefore, without loss of generality, we focus in the remainder of this section on an economy that is initially in steady state.
方程（24）中这种加性可分性的一个关键含义是，我们可以分别考察经济对基本冲击的动态反应以及其向初始稳态收敛的过程，而基本条件保持不变。因此，在接下来的部分中，我们不妨专注于一个最初处于稳态的经济体。

3.2.3 Spectral Analysis of the Transition Matrix
3.2.3 转换矩阵的光谱分析

We now provide a further analytical characterization of the roles of capital and labor dynamics in shaping the economy's gradual adjustment to shocks using a spectral analysis of the transition matrix. We show that both the speed of convergence to steady state and the evolution of the state variables along the transition path to steady state can be written solely in terms of the eigenvalues and eigenvectors of this transition matrix.
我们现在使用过渡矩阵的频谱分析，进一步分析了资本和劳动力动态在塑造经济逐渐适应冲击中的作用。我们表明，收敛到稳态的速度以及沿着过渡路径到稳态的状态变量演变，都可以仅用这个过渡矩阵的特征值和特征向量来表示。

3.2.3.1 Eigendecomposition of the Transition Matrix
3.2.3.1 转移矩阵的特征分解

We use the eigendecomposition of the transition matrix, , where Λ is a diagonal matrix of eigenvalues arranged in decreasing order by absolute values, and . For each eigenvalue , the hth column of U () and the hth row of V () are the corresponding right and left eigenvectors of P, respectively, such that
我们使用转移矩阵 的特征分解，其中Λ是一个按绝对值递减排列的特征值对角矩阵，以及 。对于每个特征值 ，U( )的第 h 列和 V( )的第 h 行分别是 P 的对应右特征向量和左特征向量，使得

That is, () is the vector that, when left-multiplied (right-multiplied) by P, is proportional to itself but scaled by the corresponding eigenvalue .⁸ We refer to simply as eigenvectors. Both and are bases that span the 2N-dimensional vector space.
这是指 （ ）是这样一个向量，当它左乘（右乘）P 时，与自身成比例，但按相应的特征值 缩放。 ⁸ 我们简单地将 称为特征向量。 和 都是张成 2N 维向量空间的基。

3.2.3.2 Eigenshock 3.2.3.2 特征冲击

We next introduce a particular type of shock to fundamentals that proves useful for characterizing the model's transition dynamics. We define an eigenshock as a nonzero shock to productivity and amenities () for which the initial impact of these shocks on the state variables () coincides with a real eigenvector of the transition matrix () or the zero vector. Generically, the eigenshocks form a basis that spans the 2N-dimensional shock space. Each eigenvector of P with a nonzero eigenvalue () has a corresponding eigenshock for which . We refer to such as eigenvector with a nonzero eigenvalue as “nontrivial,” because it affects the dynamics of the state variables. Additionally, P has an eigenvector with a zero eigenvalue (), because population shares sum to one, and thus one of the 2N dimensions of the state space is redundant. The corresponding fundamental shock is the vector of a common amenity shock to all locations. Such a common amenity shock affects worker flow utility, but does not affect any prices or quantities in the equilibrium, and thus is trivial in the sense that it does not affect the dynamics of the state variables. We use the index 1 for this trivial eigencomponent.
我们接下来介绍一种对基本面的特定类型冲击，这种冲击对于描述模型的过渡动力学是有用的。我们定义特征冲击为一个非零的针对生产力和便利性的冲击（ ），其中这些冲击对状态变量（ ）的初始影响与过渡矩阵（ ）的实特征向量或零向量相一致。一般而言，特征冲击 构成了一个基，它覆盖了 2N 维冲击空间。P 的每个具有非零特征值（ ）的特征向量都有一个相应的特征冲击，其中 。我们将具有非零特征值的特征向量称为“非平凡”的，因为它会影响状态变量的动力学。此外，P 有一个特征向量 ，其特征值为零（ ），因为人口份额之和为 1，因此状态空间的 2N 个维度中有一个是冗余的。相应的根本冲击 是针对所有位置的共同便利性冲击的向量。这种共同便利性冲击会影响工人流动效用，但不会影响均衡中的任何价格或数量，因此它是平凡的，因为它不影响状态变量的动力学。 我们使用索引 1 表示这个平凡的特征分量。

In general, there is no reason why an empirical shock should correspond to an eigen-shock. But we can use these eigenshocks to characterize the impact of any empirical shock using the following two properties. First, we can solve for these eigenshocks from the observed data, because the impact matrix (R) and the transition matrix (P) depend solely on our observed trade and migration share matrices (S, T, D, E) and the structural parameters of the model . Second, any empirical productivity and amenity shocks () can be expressed as a linear combination of the eigenshocks (), where the weights or loadings in this linear combination can be recovered from a linear projection (regression) of the observed shocks () on the eigenshocks (). Using this property, the transition path of the state variables in response to any empirical productivity and amenity shocks can be expressed solely in terms of the eigenvalues and eigenvectors of the transition matrix, as summarized in the following proposition.
一般来说，没有理由认为一个经验冲击应该对应于本征冲击。但我们可以利用这些本征冲击，通过以下两个特性来表征任何经验冲击的影响。首先，我们可以从观测数据中求解这些本征冲击，因为影响矩阵（R）和转换矩阵（P）仅取决于我们观测到的贸易和移民份额矩阵（S，T，D，E）以及模型的结构性参数 。其次，任何经验生产力和便利性冲击（ ）都可以表示为本征冲击（ ）的线性组合，其中这个线性组合中的权重或负荷可以从对观测冲击（ ）在本征冲击（ ）上的线性投影（回归）中恢复。利用这一特性，可以仅用转换矩阵的本征值和本征向量来表示对任何经验生产力和便利性冲击的响应状态变量的转换路径，如下命题所述。

PROPOSITION 4.Spectral Analysis. Consider an economy that is initially in steady state at time when agents learn about one-time, permanent shocks to productivity and amenities () from time onwards. The transition path of the state variables can be written as a linear combination of the eigenvalues () and eigenvectors () of the transition matrix:
命题 4.光谱分析。考虑一个在时间 时处于稳态的经济体，从时间 开始，代理者了解到一次性的、永久性的生产力和便利性冲击（ ）。状态变量的转换路径可以表示为转换矩阵的特征值（ ）和特征向量（ ）的线性组合：

(28)

where the weights in this linear combination () can be recovered as the coefficients in a linear projection (regression) of the observed shocks () on the eigenshocks ().
在这一点上，这个线性组合（ ）中的权重可以恢复为对观测到的冲击（ ）在特征冲击（ ）上的线性投影（回归）中的系数。

PROOF.The proposition follows from the eigendecomposition of the transition matrix: , as shown in Online Appendix B.4.9. Q.E.D.
证明。该命题可从转移矩阵的特征分解得出： ，如在线附录 B.4.9 所示。证毕。

Additionally, the speed of convergence to steady state for an eigenshock, as measured by the half-life of convergence to steady state, depends solely on the associated eigenvalue of the transition matrix, as summarized in the following proposition.
此外，通过稳定状态收敛的半衰期来衡量的特征激波的收敛速度，仅取决于过渡矩阵相关的特征值，如下命题所述。

PROPOSITION 5. (Speed of Convergence)Consider an economy that is initially in steady state at time when agents learn about one-time, permanent shocks to productivity and amenities () from time onwards. Suppose that these shocks are a nontrivial eigenshock (), for which the initial impact on the state variables at time coincides with a real eigenvector () of the transition matrix (P): . The transition path of the state variables () in response to such an eigenshock () is
命题 5.（收敛速度）考虑一个经济体系，在时间 时处于稳态，从时间 开始，代理者了解到一次性的、永久性的生产率和便利性冲击（ ）。假设这些冲击是一个非平凡的特征冲击（ ），在时间 对状态变量的初始影响与过渡矩阵（P）的实特征向量（ ）一致： 。对于这样的特征冲击（ ），状态变量的过渡路径（ ）是

and the half-life of convergence to steady state is given by
并且达到稳态的半衰期由以下公式给出

for all state variables , where , and is the ceiling function. The trivial eigenshock with an associated eigenvalue of zero has a zero half-life.
对于所有状态变量 ，其中 和 是上取整函数。与零特征值相关的平凡特征震荡具有零半衰期。

PROOF.The proposition follows from the eigendecomposition of the transition matrix (), for the case of a nontrivial eigenshock in which the initial impact of the shocks to productivity and amenities on the state variables at time coincides with a real eigenvector () of the transition matrix (P), as shown in Online Appendix B.4.9. Q.E.D.
证明。该命题可从转移矩阵（ ）的特征分解得出，对于非平凡特征冲击的情况，冲击对生产力和便利设施在时间 对状态变量的初始影响与转移矩阵（P）的实特征向量（ ）相一致，如在线附录 B.4.9 所示。证毕。

From Proposition 5, the impact of a nontrivial eigenshock () on the state variables in each time period is always proportional to the corresponding eigenvector (), and decays exponentially at a rate determined by the associated eigenvalue (), as the economy converges to the new steady state.⁹ These eigenvalues fully summarize the economy's speed of convergence in response to eigenshocks, even in our setting with a high-dimensional state space, a rich geography of trade and migration costs, and multiple sources of dynamics.
从命题 5，非平凡特征冲击（ ）对每个时间段的变量状态的影响始终与相应的特征向量（ ）成比例，并且随着经济收敛到新的稳态，以与相关特征值（ ）确定的速率指数衰减。 ⁹ 这些特征值完全概括了经济对特征冲击的收敛速度，即使在具有高维状态空间、丰富的贸易和移民成本地理以及多个动态来源的设置中也是如此。

In general, each eigenshock () has a different speed of convergence (as captured by the associated eigenvalue ), which reflects the fact that the speed of convergence to steady state does not only depend on the structural parameters , but also on the incidence of the shock on the labor and capital state variables in each location (as captured by ). From Proposition 4, any empirical shock () can be expressed as a linear combination of the eigenshocks. Therefore, the speed of convergence also varies across these empirical shocks, depending on their incidence on the labor and capital state variables in each location.
一般来说，每个特征冲击（ ）具有不同的收敛速度（由相关特征值 所捕捉），这反映了收敛到稳态的速度不仅取决于结构参数 ，还取决于冲击在每个位置的劳动和资本状态变量上的影响（由 所捕捉）。根据命题 4，任何经验冲击（ ）都可以表示为特征冲击的线性组合。因此，收敛速度也因这些经验冲击在每个位置的劳动和资本状态变量上的影响而有所不同。

5.3.1 Initial Conditions versus Fundamental Shocks

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

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.

5.3.2 Capital Accumulation versus Migration Dynamics

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.

5.4.1 Speed of Convergence to Steady State

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 (). 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

As in our symmetric two-region example above, these eigencomponents have an intuitive interpretation. One eigenvector 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 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 in the special case of log preferences (), 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 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 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 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.

5.4.2 Initial Conditions and Convergence

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.

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.

5.4.3 Fundamental Shocks

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.

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.

5.4.4 Impulse Responses

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.

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.

5.4.5 Comparative Statics of the Speed of Convergence

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 . 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.

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).