Assumptions about preferences play a central role in shaping consumers’ location decisions. Five main sets of assumptions about preferences can be distinguished.
假设关于偏好的观点在塑造消费者位置决策中扮演着核心角色。可以区分出关于偏好的五组主要假设。

Assumptions about production technology critically influence firms’ location decisions. Four main sets of assumptions concerning production technology can be distinguished.
生产技术的假设对企业的区位决策产生关键影响。可以区分出关于生产技术的四组主要假设。

Several mechanisms can explain the importance of the location of agents relative to one another in quantitative spatial models. The first of these mechanisms is the cost of trading goods. Four main sets of assumptions concerning the costs of trading goods can be delineated.
几种机制可以解释在定量空间模型中，代理相对于彼此的位置的重要性。这些机制中的第一种是交易商品的成本。可以区分出关于交易商品成本的四个主要假设集。

The second mechanism explaining the importance of the location of agents relative to one another in quantitative spatial models is friction in idea flows. Three main sets of assumptions concerning the technology of idea flows are considered.
第二机制解释了在定量空间模型中，代理相对位置的重要性，即思想流动的摩擦。关于思想流动技术的三个主要假设集被考虑。

The third mechanism that explains the importance of the spatial location of agents relative to one another in quantitative spatial models is friction in the movement of people. In this section, four main sets of assumptions are described.
第三种解释在定量空间模型中代理相对空间位置重要性的机制是人们在运动中的摩擦。在本节中，描述了四组主要假设。

The above choices about preferences; production technology; and the costs of the movement of goods, ideas, and people need to be combined with choices on the endowments of the economy.
上述关于偏好、生产技术和商品、思想及人员流动成本的抉择，需要与关于经济禀赋的抉择相结合。

Given the above assumptions about preferences; production technology; endowments; and the technologies for the movement of goods, ideas, and people, a final set of assumptions concerning the equilibrium conditions of the model must be considered.
考虑到上述关于偏好的假设、生产技术、禀赋以及商品、思想和人员流动的技术，必须考虑关于模型均衡条件的最终假设集。

Having outlined the modules or building blocks that can be used to create quantitative spatial models, we now discuss some of the possible criteria for choosing among these modules.
概述了可用于创建定量空间模型的模块或构建块后，我们现在讨论一些在选择这些模块时可能适用的标准。

Preferences are defined over goods consumption (Cn ) and residential land use (hn ) and are assumed to take the Cobb-Douglas form: ⁶
偏好定义在商品消费（Cn）和住宅土地利用（hn）上，并假定其采用 Cobb-Douglas 形式： ⁶

1. 

The goods consumption index Cn is defined over consumption cni (j) of each horizontally differentiated variety j from the endogenous measures Mi supplied by each region. This consumption index and its dual price index, Pn , are given by
商品消费指数 Cn 是在每个地区提供的内生度量 M 的基础上，定义了每个水平差异化的品种 j 的消费 cni(j)。该消费指数及其对偶价格指数 Pn 由以下公式给出

2. 

Varieties are produced under conditions of monopolistic competition and increasing returns to scale. To produce a variety, a firm must incur a fixed cost of F units of labor and a constant variable cost in terms of labor that depends on a location's productivity Ai . Therefore, the total amount of labor, li (j), required to produce xi (j) units of a variety j in location i is
品种在垄断竞争和规模报酬递增的条件下产生。为了生产一个品种，企业必须承担 F 单位劳动力的固定成本以及与地点 i 的生产力 A 相关的劳动力的常数变动成本。因此，在地点 i 生产品种 j 的 x(j)单位所需的劳动力总量 l(j)是

3. 

Profit maximization and zero profits imply that equilibrium prices are a constant markup over the marginal cost of supplying a variety to a market,
利润最大化与零利润意味着均衡价格是市场供应某种商品边际成本的一个常数加成

4. 

and equilibrium output of each variety is equal to a constant that depends on location productivity, namely,
并且每种品种的均衡产出等于一个依赖于地理位置生产力的常数，即，

5. 

which implies that equilibrium employment for each variety is the same for all locations, so
这表明每种品种的均衡就业在所有地点都是相同的，因此

6. 

Given this constant equilibrium employment for each variety, labor market clearing implies that the total measure of varieties supplied by each location is proportional to the endogenous supply of workers choosing to locate there:
考虑到每种品种的恒定均衡就业，劳动力市场出清意味着每个地点提供的品种总供应量与选择在该地定居的工人的内生供应量成比例：

7. 

Using equilibrium prices (Equation 4) and labor market clearing (Equation 7), one can express the price index dual to the consumption index (Equation 2) as
使用均衡价格（方程式 4）和劳动力市场均衡（方程式 7），可以将与消费指数（方程式 2）相对应的价格指数表示为

8. 

Using the constant elasticity of substitution (CES) expenditure function, equilibrium prices (Equation 4), and labor market clearing (Equation 7), the share of location N’s expenditure on goods produced in location i is
利用常弹性替代（CES）支出函数、均衡价格（方程 4）和劳动力市场均衡（方程 7），位置 N 在位置 i 生产的商品上的支出份额是

9. 

The model therefore implies a gravity equation for goods trade, where the bilateral trade between locations N and i depends on both bilateral resistance (bilateral trade costs dni ) and multilateral resistance (trade costs to all other locations k dnk ), as in the work of Anderson & van Wincoop (2003). Together, Equations 8 and 9 imply that each location's price index can be written in terms of its trade share with itself, so
该模型因此暗示了一个关于货物贸易的重力方程，其中 N 和 i 之间的双边贸易取决于双边阻力（双边贸易成本 dni）和多边阻力（对所有其他地点的贸易成本 k dnk），正如 Anderson 和 van Wincoop（2003）的研究所示。8 号和 9 号方程共同意味着每个地点的价格指数可以用其与自身的贸易份额来表示，因此

10. 

Expenditure on land in each location is redistributed in a lump sum to the workers residing in that location. Therefore, trade balance at each location implies that per capita income vn in each location equals labor income wn plus per capita expenditure on residential land, (1−α)vn , namely,
各地点的土地支出以总额形式重新分配给该地点居住的工人。因此，每个地点的贸易平衡意味着每个地点的人均收入 vn 等于劳动收入 wn 加上人均住宅土地支出(1−α)vn，即，

11. 

Land market clearing implies that the supply of quality-adjusted land, Hn , equals the demand for land, Lnhn . By combining this market clearing condition with the first-order condition of the consumer problem, we obtain the result that land rents, rn , are given by
土地市场出清意味着质量调整后的土地供应量 Hn 等于土地需求量 Lnhn。将这一市场出清条件与消费者问题的第一阶条件相结合，我们得到土地租金 rn 的表达式为

12. 

Population mobility implies that workers receive the same real income in all populated locations, thus
人口流动性意味着工人在所有人口密集地区都能获得相同实际收入，因此

13. 

By using the price index (Equation 10), the assumption that trade is balanced at each location such that income equals expenditure (Equation 11), and land market clearing (Equation 12) in the population mobility condition (Equation 13), one determines that real wage equalization implies that the population Ln and domestic trade share π nn of each location must satisfy
通过使用价格指数（公式 10），在人口流动性条件（公式 13）下，贸易在每个地点平衡的假设（收入等于支出，公式 11）和土地市场出清（公式 12），可以确定实际工资均等化意味着每个地点的人口 Ln 和国内贸易份额πnn 必须满足

14. 

Therefore, the population share of each location ( ) depends on its productivity An , supply of land Hn , and domestic trade share π nn relative to those of all other locations,
因此，每个地点的人口份额（ ）取决于其生产率 An、土地供应 Hn 以及相对于所有其他地点的国内贸易份额πnn

15. 

where each location's domestic trade share π nn summarizes its market access to other locations.
每个位置的国内贸易份额πnn 概括了其对其他地点的市场准入。

The properties of the general equilibrium of the model can be characterized analytically by combining the trade share (Equation 9), price index (Equation 8), and population mobility condition (Equation 13). Under the assumption that trade costs are symmetric (dni =din ), one can follow the arguments of Allen & Arkolakis (2014) to show that these three sets of relationships reduce to the following system of N equations in the N populations of each location:
模型的总体均衡性质可以通过结合贸易份额（方程 9）、价格指数（方程 8）和人口流动性条件（方程 13）进行解析性描述。在贸易成本对称（dni = din）的假设下，可以遵循 Allen & Arkolakis（2014）的论证，表明这三组关系简化为每个地点 N 个人口中的 N 个方程组：

16. 

where the scalar is determined by the requirement that the labor market clears ( ) and
在满足劳动力市场出清（ ）的要求下确定标量

Wages in turn are implicitly determined by
工资反过来是由...隐含决定的

17. 

where ξ is a scalar that normalizes wages. Allen & Arkolakis (2014) use this argument, together with the mathematical results for fixed points of systems of equations of the form given by Equation 16 (Fujimoto & Krause 1985), to show that there exists a unique Ln for each n that satisfies Equation 16 as long as γ2/γ1∈(0, 1). Thus, given the land area and productivity parameters {Hn , An } and symmetric bilateral trade frictions {dni } for all locations n, i∈N, there exists a unique equilibrium as long as this parametric restriction is satisfied. Furthermore, if γ2/γ1∈(0, 1), then one can also guarantee that a solution to Equation 16 can be found by iteration from any initial distribution of populations. ⁷
ξ是一个标准化工资的标量。Allen & Arkolakis（2014）使用这个论点，结合方程 16（Fujimoto & Krause 1985）给出的方程组不动点数学结果，来证明对于每个 n 存在唯一的 Ln，只要γ2/γ1∈(0, 1)，它就满足方程 16。因此，给定所有位置 n，i∈N 的土地面积和生产率参数{Hn, An}以及对称的双边贸易摩擦{dni}，只要满足这个参数限制，就存在唯一的均衡。此外，如果γ2/γ1∈(0, 1)，还可以保证可以从任何初始人口分布通过迭代找到方程 16 的解。 ⁷

The parameter restrictions that guarantee that an equilibrium exists and is unique amount to imposing conditions that guarantee that congestion forces always dominate agglomeration forces. In our simple model, a sufficient condition for γ2/γ1∈(0, 1) is σ(1−α)>1. Intuitively, as population concentrates in a location, the measure of varieties produced there expands, which, in the presence of trade costs, makes that location a more attractive residence (an agglomeration force). However, as population concentrates in a location, this also bids up land prices (a dispersion force). The higher the elasticity of substitution (σ), the weaker is the agglomeration force. The higher the share of land (1−α), the stronger is the dispersion force. For parameter values for which σ(1−α)>1, the dispersion force dominates the agglomeration force, and a unique equilibrium distribution of economic activity exists.
参数限制保证存在且唯一的均衡，相当于施加条件以保证拥堵力量始终支配聚集力量。在我们的简单模型中，γ2/γ1∈(0, 1)的充分条件是σ(1−α)>1。直观地说，随着人口在某一地点集中，该地点生产的品种数量增加，这在存在贸易成本的情况下，使该地点成为更吸引人的居住地（聚集力量）。然而，随着人口在某一地点集中，这也推高了土地价格（分散力量）。替代弹性的弹性（σ）越高，聚集力量越弱。土地份额（1−α）越高，分散力量越强。对于σ(1−α)>1 的参数值，分散力量支配聚集力量，存在唯一的经济活动均衡分布。

The existence of such a unique equilibrium is important because it ensures that counterfactuals for transport infrastructure improvements or other public policy interventions have determinate implications for the spatial distribution of economic activity. Although this is a convenient property of the model for quantitative empirical work, a central feature of the theoretical literature on new economic geography was the presence of multiple equilibria [as in the original core–periphery model of Krugman (1991b)], and assuming σ(1−α)>1 excludes this possibility. Thus, for the range of parameters where σ(1−α)>1, the model cannot generate agglomerations when space is perfectly homogeneous; it can generate agglomerations only as a result of initial differences across locations. Of course, in this general class of models, the vector of initial differences can be multidimensional and quite rich, as discussed in the context of a richer model with many sources of heterogeneity across locations by Desmet et al. (2016).
该独特均衡的存在具有重要意义，因为它确保了关于交通基础设施改善或其他公共政策干预的假设情景对经济活动空间分布的确定影响。尽管这是模型进行定量实证研究的一个便利属性，但新经济地理学理论文献的一个中心特征是存在多个均衡[如克鲁格曼（1991b）的原核-外围模型]，而假设σ(1−α)>1 排除了这种可能性。因此，在σ(1−α)>1 的参数范围内，模型在空间完全同质的情况下不能产生集聚；它只能通过不同地点的初始差异产生集聚。当然，在这个一般模型类中，初始差异向量可以是多维的，相当丰富，正如 Desmet 等人（2016）在讨论具有许多地点异质性来源的更丰富模型时所指出的。

At a more philosophical level, whether a model is characterized by multiple equilibria may depend on its level of abstraction. On the one hand, a model may exhibit multiple equilibria because it omits the relevant idiosyncratic factors that determine one allocation rather than another in the data. On the other hand, if a model includes all such idiosyncratic factors, it ceases to be a model and becomes a descriptive representation of reality. One possible interpretation of such idiosyncratic factors is as random realizations of a stochastic error that is drawn from a parameterized distribution. At smaller spatial scales (e.g., blocks within cities), one might expect such random idiosyncratic factors to be more important relative to the systematic deterministic components of a model (e.g., natural resource abundance) than at larger spatial scales (e.g., across regions or countries).
在更哲学的层面上，一个模型是否具有多重均衡可能取决于其抽象程度。一方面，一个模型可能表现出多重均衡，因为它省略了决定数据中某一分配而非另一分配的相关独特因素。另一方面，如果一个模型包括了所有这样的独特因素，它就不再是模型，而变成了对现实的描述性表征。对这种独特因素的一种可能的解释是它们是随机实现，这些随机实现是从参数化分布中抽取的随机误差。在较小的空间尺度上（例如，城市内的街区），人们可能会预期这种随机的独特因素相对于模型中系统的确定性成分（例如，自然资源丰富度）在较大空间尺度上（例如，跨区域或国家）更为重要。

Throughout this section, we focus on a canonical new economic geography model with increasing returns to scale and monopolistic competition. However, similar properties hold in a wider class of models. Allen & Arkolakis (2014) demonstrate an isomorphism to a perfectly competitive Armington trade model (in which goods are differentiated by location of origin), extended to incorporate labor mobility and external economies. Allen & Arkolakis (2014) and Redding (2016) show that similar properties also hold in a perfectly competitive Ricardian trade model following Eaton & Kortum (2002) with labor mobility and external economies of scale.
在整个本节中，我们关注一个具有规模报酬递增和垄断竞争的典型新经济地理模型。然而，类似的性质也存在于更广泛的模型类别中。Allen & Arkolakis（2014）证明了与一个完全竞争的 Armington 贸易模型（其中商品通过原产地进行区分）的同构性，并将其扩展到包含劳动力流动和外部经济。Allen & Arkolakis（2014）和 Redding（2016）表明，在遵循 Eaton & Kortum（2002）的完全竞争 Ricardian 贸易模型中，也具有类似的性质，该模型具有劳动力流动和规模经济的外部性。

In this section, we describe how the quantitative spatial model can be used to rationalize observed data. We suppose that a researcher has estimates of the model's two key parameters: the share of residential land in consumer expenditure (α) and the elasticity of substitution between varieties (σ). The researcher is also assumed to have parameterized symmetric bilateral trade costs (dni ), for example, by assuming that they are a constant elasticity function of distance, and to observe an endogenous population, {Ln }, and nominal wages, {wn }. One can show that there is a one-to-one mapping from the model's parameters and the observed data to the unobserved values of quality-adjusted land {Hn } and productivities {An } (up to a normalization constant). That is, the model can be inverted to recover the unique values of unobserved quality-adjusted land and productivities that rationalize the observed data as an equilibrium outcome of the model.
在这一节中，我们描述了如何使用定量空间模型来合理化观察到的数据。我们假设研究者已经估计了模型的两个关键参数：住宅用地在消费支出中的份额（α）和品种之间的替代弹性（σ）。研究者还被假设已经对对称的双边贸易成本（dni）进行了参数化，例如，假设它们是距离的常数弹性函数，并观察到内生人口{Ln}和名义工资{wn}。可以证明，从模型的参数和观察到的数据到未观察到的质量调整后土地{Hn}和生产力{An}的值之间存在一一对应的关系（直到归一化常数）。也就是说，可以通过逆模型来恢复未观察到的质量调整后土地和生产力唯一值，这些值可以合理化观察到的数据，作为模型均衡结果的体现。

Inverting the model amounts to using Equations 16 and 17 to solve for {An , Hn } given {Ln , wn }. This is exactly the opposite of what we do when we solve for an equilibrium of the model, where we solve for {Ln , wn } given {An , Hn }. To guarantee that there exists a unique set of values {An , Hn } that rationalize the observed data (up to a normalization or choice of units), we can proceed as follows. Using Equation 17, we can solve for Hn and substitute this solution in Equation 16. The resulting equation can then be solved for {An } using information on {Ln , wn }. To show that such a solution exists and is unique, we can again use the mathematical theorems that guarantee solutions to these type of equations [e.g., those of Fujimoto & Krause (1985) for discrete space or Zabreyko et al. (1975) for continuous space]. As in the previous section, a solution exists if, after substituting for Hn , the exponent of the Ai term inside the sum is smaller than the exponent of the An term outside the sum. In our example, this is guaranteed if σ(1−α)>1. We can then recover {Hn } using Equation 17.
将模型进行逆运算相当于使用方程 16 和 17 求解{An, Hn}，给定{Ln, wn}。这与我们在求解模型平衡时的做法正好相反，在求解模型平衡时，我们给定{An, Hn}求解{Ln, wn}。为了保证存在一组唯一的值{An, Hn}来合理化观察到的数据（直到归一化或选择单位），我们可以按以下步骤进行。使用方程 17，我们可以求解 Hn，并将此解代入方程 16。然后，可以使用{Ln, wn}的信息求解{An}。为了证明这样的解存在且唯一，我们可以再次使用保证这类方程解的数学定理[例如，Fujimoto & Krause（1985）的离散空间定理或 Zabreyko 等（1975）的连续空间定理]。与上一节一样，如果将 Hn 代入后，求和式中 A 项的指数小于求和式外的 An 项的指数，则存在解。在我们的例子中，这由σ(1−α)>1 保证。然后，我们可以使用方程 17 恢复{Hn}。

Having recovered the unobserved productivities {An }, these can be used, together with the parameterization of trade costs (dni ) and observed wages (wn ) in the trade shares (Equation 9), to generate predictions for unobserved bilateral trade shares (π ni ) in the equilibrium observed in the data. Related arguments hold if the researcher directly observes bilateral trade shares (π ni ) instead of having to assume values for bilateral trade costs (dni ). In this case, unobserved quality-adjusted land supplies (Hn ), productivities (An ), and bilateral trade costs (dni ) can be recovered from the observed data (up to a normalization or choice of units).
已恢复未观察到的生产率{An}，这些可以与贸易成本的参数化（dni）和观察到的工资（wn）在贸易份额（方程 9）中一起使用，以生成对数据中观察到的均衡状态下未观察到的双边贸易份额（πni）的预测。如果研究者直接观察到双边贸易份额（πni）而不是必须假设双边贸易成本（dni）的值，相关的论点仍然成立。在这种情况下，可以从观察到的数据中恢复未观察到的质量调整后的土地供应（Hn）、生产率（An）和双边贸易成本（dni）（直至归一化或单位选择）。

These arguments imply that the model is exactly identified, in the sense that it has the same number of degrees of freedom (unobserved location characteristics in the form of the quality-adjusted land supplies Hn and productivities An for each location) as observed endogenous variables (population Ln and wages wn ). Therefore, the model's ability to explain the observed data cannot be used as a test of the model because the unobserved quality-adjusted land supplies and productivities are free parameters that can be adjusted so as to ensure that the model exactly matches the data. Note also that, because Hn denotes quality-adjusted land, it cannot be directly compared to observations on total land supply in a location. Doing so would simply yield a measure of the average quality of land in that region (which is related to its residential amenities). Furthermore, the observed data on wages and populations cannot be used to estimate the model's structural parameters, α and σ. Given any value for these parameters, quality-adjusted land supplies and productivities can be adjusted so as to ensure that the model exactly rationalizes the data. Therefore, any change in the value of the structural parameters can be offset by changes in these unobserved location characteristics such that the model continues to explain the data.
这些论点意味着模型在精确识别的意义上，具有与观察到的内生变量（人口 Ln 和工资 wn）相同的自由度（以每个位置的质量调整后的土地供应 Hn 和生产率 An 形式存在的未观察到的位置特征）。因此，模型解释观察到的数据的能力不能作为对模型的检验，因为未观察到的质量调整后的土地供应和生产率是自由参数，可以调整以确保模型与数据完全匹配。注意，由于 Hn 表示质量调整后的土地，它不能直接与某个位置的总体土地供应的观察值进行比较。这样做只会得到该地区土地平均质量的度量（这与它的居住便利性有关）。此外，观察到的工资和人口数据不能用来估计模型的结构参数α和σ。对于这些参数的任何值，都可以调整质量调整后的土地供应和生产率，以确保模型能够完全合理化数据。 因此，结构参数值的任何变化都可以通过这些未观察到的位置特征的变化来抵消，从而使模型继续解释数据。

Nonetheless, the model's ability to exactly explain the data implies that it provides a framework that can be used to decompose the observed variation in endogenous variables (e.g., population and wages) into the contribution of different exogenous determinants (e.g., trade costs and exogenous determinants of productivity and quality-adjusted land). Note that the recovered exogenous fundamentals do not explain the variation in endogenous variables but simply account for it according to the model. Furthermore, additional data not used for the quantification of the model or exogenous shocks to the economy (e.g., natural experiments from history) can be used to provide overidentification checks (tests of the model's external validity) or to estimate the model's structural parameters. We consider several empirical studies in Section 5 that have sought to provide such overidentification checks or undertake such structural estimation.
尽管如此，模型精确解释数据的能力意味着它提供了一个框架，可以用来将内生变量（例如人口和工资）观察到的变化分解为不同外生决定因素（例如贸易成本和生产力及质量调整后的土地的外生决定因素）的贡献。请注意，恢复的外生基本因素并不能解释内生变量的变化，而只是根据模型对其进行解释。此外，未用于模型量化或对经济的外生冲击（例如历史中的自然实验）的额外数据可以用来提供过度识别检验（对模型外部有效性的检验）或估计模型的结构参数。我们在第 5 节中考虑了几项实证研究，这些研究试图提供这样的过度识别检验或进行这样的结构估计。

In this section, we show how our quantitative spatial model can be used to undertake counterfactuals for the effects of public policy interventions, such as transport infrastructure improvements. We show that these counterfactuals can be undertaken using the observed values of the endogenous variables of the model in an initial equilibrium without having to solve for the unobserved location characteristics, as in the work of Dekle et al. (2007). We denote the (unknown) value of variables in the counterfactual equilibrium with a prime (x′) and the relative value of variables in the counterfactual and observed equilibria by a hat ( ). We suppose that the researcher observes population (Ln ), wages (wn ), and trade shares (π ni ) in the initial equilibrium and can parameterize the change in bilateral trade costs as a result of the transport infrastructure improvement ( ). From the trade share (Equation 9), price index (Equation 10), income equal to expenditure (Equation 11), land market clearing (Equation 12), and population mobility (Equation 13), we obtain the following system of equations that can be used to solve for the counterfactual changes in wages, trade shares, and population shares ({ , , }) given only the observed wages, trade shares, and population shares in the initial equilibrium ({w, π ni , λ n }):
在这一节中，我们展示了如何使用我们的定量空间模型来对公共政策干预的影响进行反事实分析，例如交通基础设施的改善。我们表明，这些反事实分析可以使用模型内生变量的初始均衡观测值进行，而无需解决未观测到的位置特征，正如 Dekle 等人（2007 年）的研究所示。我们用撇号（x′）表示反事实均衡中变量的（未知）值，用帽子（ ）表示反事实和观测均衡中变量的相对值。我们假设研究者观察到初始均衡中的人口（Ln）、工资（wn）和贸易份额（πni），并且可以将双边贸易成本的变动参数化为运输基础设施改善的结果（ ）。从贸易份额（方程 9）、价格指数（方程 10）、收入等于支出（方程 11）、土地市场出清（方程 12）和人口流动性（方程 13），我们得到以下方程组，可用于求解在初始均衡中仅给定观察到的工资、贸易份额和人口份额时，工资、贸易份额和人口份额的逆事实变化（{ ， ， }）：

18. 

19. 

20. 

Given our assumptions of σ(1−α)>1 and symmetric trade costs, there exists a unique general equilibrium in the model, which ensures that these counterfactuals yield determinate predictions for the impact of the transport infrastructure improvement or another public policy intervention on the spatial equilibrium distribution of economic activity.
鉴于我们假设σ(1−α)>1 和对称的贸易成本，模型中存在一个唯一的一般均衡，这确保了这些反事实能够对运输基础设施改善或另一项公共政策干预对经济活动空间均衡分布的影响做出确定的预测。

A further implication of this class of quantitative spatial models is that the welfare effects of public policy interventions that change trade costs can be expressed solely in terms of empirically observable sufficient statistics. Consider a transport infrastructure improvement that reduces trade costs between an initial equilibrium (indexed by 0) and a subsequent equilibrium (indexed by 1). Perfect population mobility implies that the transport infrastructure improvement leads to reallocations of population across locations until real wages are equalized. Using the population mobility condition (Equation 14), the changes in the domestic trade share (π nn ) and population (Ln ) for any one location are sufficient statistics for the welfare impact of the transport infrastructure improvement on all locations:
这一类定量空间模型的一个进一步含义是，改变贸易成本的公共政策干预的福利效应可以仅用经验上可观察的充分统计量来表示。考虑一种运输基础设施的改善，它降低了初始均衡（以 0 为指数）和随后均衡（以 1 为指数）之间的贸易成本。完美的人口流动性意味着运输基础设施的改善导致人口在不同地点之间的重新分配，直到实际工资均等化。使用人口流动性条件（方程 14），任何地点国内贸易份额（πnn）和人口（Ln）的变化是运输基础设施改善对所有地点福利影响的充分统计量：

21. 

Given our assumption of σ(1−α)>1, a larger reduction in a location's domestic trade share must be offset by a larger increase in its population to preserve real wage equalization. Intuitively, if the transport infrastructure improvement decreases trade costs for one location more than for other locations (and thus reduces the domestic trade share of this location), the resulting upwards pressure on its real wage induces a population inflow until the price of the immobile factor land is bid up to restore real wage equalization. This implication is a direct analog of the result found by the international trade literature that the domestic trade share is a sufficient statistic for the welfare gains from trade in a class of trade models (see Arkolakis et al. 2012). In an economic geography model in which an immobile factor of production, such as land, is used residentially or commercially, changes in the distribution of mobile factors of production across locations must also be taken into account (see Caliendo et al. 2014, Redding 2016).
鉴于我们假设σ(1−α)>1，一个地区国内贸易份额的更大减少必须通过其人口的更大增加来抵消，以保持实际工资均等化。直观地说，如果交通基础设施的改善使一个地区的贸易成本比其他地区降低更多（从而减少了该地区的国内贸易份额），那么对其实际工资产生的向上压力将导致人口流入，直到不可移动的生产要素土地的价格被推高以恢复实际工资均等化。这一推论是国际贸易文献中发现的直接类比结果，即国内贸易份额是一类贸易模型中贸易福利收益的充分统计量（参见 Arkolakis 等人，2012 年）。在一个生产要素如土地等不可移动的生产要素被用于居住或商业用途的经济地理模型中，还必须考虑生产要素在地区间的分布变化（参见 Caliendo 等人，2014 年，Redding，2016 年）。

In this section, we outline a quantitative illustration of the model for which accompanying MATLAB code is available. We show how the quantitative model can be used to evaluate the impact of trade frictions (both between countries and between regions within countries) on the spatial distribution of economic activity and welfare. We consider a model economy on a 30×30 latitude and longitude grid. We assume that this economy consists of two countries, one occupying the western half of the grid (West) and the other occupying the eastern half (East). We assume that labor is perfectly mobile across locations within each country but perfectly immobile across countries. Following Donaldson (2016), we compute a measure of lowest-cost-route effective distance, which minimizes the cumulative cost of traveling between each origin and destination. In particular, this measure assigns a cost (or weight) to traveling across each point N on the grid of δ n . For a pair of neighboring points N and i, the effective distance for an orthogonal link between these locations (moving vertically or horizontally across the grid) is . In contrast, the effective distance for a diagonal link between these locations (moving diagonally across the grid) is . For any origin and destination, the effective distance is the sum across all of the links between neighboring locations along the lowest-cost route. We assume that the cost (weight) is the same for each location on the grid (δ n =δ i =δ) and normalize this common distance weight to one.
在这一节中，我们概述了一个模型的定量说明，并提供了相应的 MATLAB 代码。我们展示了如何使用定量模型来评估贸易摩擦（国家间和国内地区间）对经济活动和福利空间分布的影响。我们考虑了一个在 30×30 纬度和经度网格上的模型经济。我们假设这个经济由两个国家组成，一个占据网格的西部一半（西部），另一个占据东部一半（东部）。我们假设劳动力在每个国家内部的位置之间可以完全流动，但在国家之间完全不可流动。遵循 Donaldson（2016）的方法，我们计算了一个最低成本路线有效距离的度量，该度量最小化了每个起点和目的地之间旅行的累积成本。特别是，这个度量将成本（或权重）分配给网格上每个点 N 的δn。对于相邻的点 N 和 i，这两个位置之间正交链接的有效距离（在网格上垂直或水平移动）是 。 与这些地点之间的对角线链接的有效距离（在网格上斜向移动）为 。对于任何起点和终点，有效距离是沿着最低成本路线，在相邻位置之间的所有链接上的总和。我们假设网格上每个位置的成本（权重）相同（δn = δ = δ），并将这个共同距离权重归一化为一。

We allow productivity to differ randomly across locations. For each location, we draw a realization for productivity {An } from an independent standard log normal distribution. Figure 1 displays the realization of productivities for the 900 locations in our grid. In this realization, there are two clusters of high-productivity areas, one northwest of the border and another southeast of it. Other, more isolated, high-productivity areas are also evident. Of course, in this example the location of high-productivity areas is purely random.
我们允许生产率在不同地区随机差异。对于每个地区，我们从独立的标准对数正态分布中抽取一个生产率{An}的实现值。图 1 显示了我们网格中 900 个地点的生产率实现值。在这个实现中，有两个高生产率区域的集群，一个位于边界西北部，另一个位于东南部。其他更孤立的高生产率区域也很明显。当然，在这个例子中，高生产率区域的地理位置完全是随机的。

Figure 1 图 1  Realization of random local productivity for the 900 locations in our grid. Cold (blue) colors correspond to lower values, and hot (yellow) colors correspond to higher values.
实现网格中 900 个位置的随机局部生产力。冷（蓝色）颜色对应较低值，热（黄色）颜色对应较高值。

Figure 1 

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For simplicity, we assume that each location has the same quality-adjusted land area (Hn ) of 100 km². We choose central values for the model's parameters based on the existing empirical literature. First, we set the share of land in residential consumption expenditure (1−α) to 25%, which is in line with the housing expenditure share of Davis & Ortalo-Magne (2011). Second, we set the elasticity of substitution (σ) equal to 5, which implies an elasticity of trade flows with respect to trade costs of σ−1=4, in line with the estimates of Simonovska & Waugh (2014). Third, we assume that trade costs are a constant elasticity function of effective distance ( ), which implies an elasticity of trade flows with respect to effective distance of −(σ−1)ϕ [because trade flows depend on ]. We choose the parameter ϕ to match the elasticity of trade flows with respect to distance in gravity equations using interregional trade data of −(σ−1)ϕ=−1.5, which, for our assumed value for σ−1, implies ϕ=0.375.
为简化起见，我们假设每个地点具有相同的质量调整后土地面积（Hn）为 100 平方公里 ² 。我们根据现有的实证文献选择模型参数的中央值。首先，我们将土地在住宅消费支出中的份额（1−α）设定为 25%，这与 Davis & Ortalo-Magne（2011）的住房支出份额一致。其次，我们将替代弹性（σ）设定为 5，这意味着贸易流量对贸易成本的弹性为σ−1=4，与 Simonovska & Waugh（2014）的估计一致。第三，我们假设贸易成本是有效距离（ ）的常数弹性函数，这意味着贸易流量对有效距离的弹性为−(σ−1)ϕ[因为贸易流量取决于 ]。我们选择参数ϕ以匹配重力方程中贸易流量对距离的弹性，使用区域间贸易数据得到−(σ−1)ϕ=−1.5，对于我们的σ−1 假设值，这意味着ϕ=0.375。

In addition to these geographical frictions from transport costs, we consider two forms of economic frictions to trade between locations. First, we assume a proportional internal tax on trade with other locations of 100% (τⁱⁿ=2), which is paid whenever a good flows from one location to another. Second, we assume a proportional external tax on trade between the two countries of 100% (τ^out=2), which is paid whenever a good crosses the border between the two countries (between latitude 15 and 16). For simplicity, we assume that the revenue from both taxes is wasted, and thus both correspond to real resource costs. In Figure 2 , we display the log level of economic activity across locations in the initial equilibrium with both taxes. Areas of high productivity have large population concentrations ( Figure 2a ), high wages ( Figure 2b ), and high land prices ( Figure 2c ). As expected, the log of the price index ( Figure 2d ) is a smooth surface with the gradient governed by trade costs. Prices are lower in areas that produce a large variety of goods, for example, at the two large cities close to the border. The largest agglomerations in this economy are most clearly appreciated in this panel. Panel d also clearly exhibits the border effect created by the tariff between the two countries.
除了这些由运输成本引起的地理摩擦外，我们还考虑两种形式的经济摩擦来分析不同地点之间的贸易。首先，我们假设对与其他地点的贸易征收 100%比例的内部税（τ ⁱⁿ =2），每当商品从一个地点流向另一个地点时支付。其次，我们假设对两国之间的贸易征收 100%比例的外部税（τ ^out =2），每当商品跨越两国边境（在纬度 15 和 16 之间）时支付。为了简化，我们假设这两种税收的收入都被浪费了，因此两者都对应于实际资源成本。在图 2 中，我们展示了在两种税收的初始均衡状态下，不同地点的经济活动对数水平。高生产力的地区有大量人口聚集（图 2a），高工资（图 2b）和高土地价格（图 2c）。正如预期的那样，价格指数的对数（图 2d）是一个平滑的表面，其梯度由贸易成本控制。在生产和多种商品的地区，价格较低，例如在靠近边境的两个大城市。 在这个经济体的最大聚集地在这张面板中表现得最为明显。面板 d 也清楚地展示了两个国家之间关税产生的边界效应。

Figure 2 图 2  Initial equilibrium, showing (a) log population, (b) log wages, (c) log land prices, and (d) log price index. Cold (blue) colors correspond to lower values, and hot (yellow) colors correspond to higher values.
初始均衡，显示（a）对数人口，（b）对数工资，（c）对数地价和（d）对数价格指数。冷色（蓝色）对应较低值，暖色（黄色）对应较高值。

Figure 2 

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In Figure 3 , we display the log relative changes in population ( ), wages ( ), land prices ( ), and price indices ( ) as a result of the removal of the proportional tax on trade between the two countries. As trade costs between the two countries fall, economic activity reallocates toward the border between them. The areas that benefit the most are the ones close to but on the opposite side of the border from the large cities. These locations can now trade more cheaply with the large market in those cities and thus experience the largest increases in population ( Figure 3a ), wages ( Figure 3b ), and land rents ( Figure 3c ) and the largest reductions in the price index ( Figure 3d ). In contrast, the largest agglomerations lose relative to these up and coming locations. In the first row of Table 1 , we report the resulting impact on the common level of welfare across locations within each country. We find that this external trade liberalization raises welfare in West and East by around 0.2% and 0.3%, respectively.
在图 3 中，我们展示了由于两国间贸易比例税的取消，人口（ ）、工资（ ）、土地价格（ ）和价格指数（ ）的相对变化。随着两国间贸易成本的降低，经济活动重新分配到两国边境。受益最大的地区是靠近但位于大城市对面的地区。现在，这些地区可以以更低的成本与那些城市的大市场进行贸易，因此人口（图 3a）、工资（图 3b）和土地租金（图 3c）的增长最大，价格指数（图 3d）的降幅也最大。相比之下，最大的聚集地相对于这些新兴地区有所下降。在表 1 的第一行中，我们报告了这种外部贸易自由化对每个国家内部各地点的福利水平的影响。我们发现，这种外部贸易自由化使西部和东部的福利分别提高了约 0.2%和 0.3%。

Figure 3 图 3  External liberalization, showing the ratio of counterfactual to initial values, : (a) log relative population, (b) log relative wages, (c) log relative land rents, and (d) log relative price index. Cold (blue) colors correspond to lower values, and hot (yellow) colors correspond to higher values.
外部自由化，显示反事实值与初始值的比率， ：（a）相对人口的对数，（b）相对工资的对数，（c）相对地租的对数，以及（d）相对价格指数的对数。冷（蓝色）颜色对应较低值，热（黄色）颜色对应较高值。

Figure 3 

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	Welfare gains () 福利收益 ()
	West 西方	East 东方
External liberalization 外部自由化	0.2%	0.3%
Internal liberalization 内部自由化	1.4%	2.3%

In Figure 4 , we present an alternative counterfactual experiment where we remove all internal trade costs but leave international trade costs as in the initial equilibrium. The figure presents relative changes with respect to the initial equilibrium. The implications of an internal reduction in trade costs are clearly quite different than the ones from an external trade cost reduction. The main effect of the internal liberalization is to reduce the size of the two large cities in favor of rural areas, thereby making economic activity more dispersed ( Figure 4a ). As trade costs decline, the home market effect reducing local price indexes in large cities weakens, so that prices fall everywhere but less so in areas with larger populations ( Figure 4d ). Wages and land rents also fall in large agglomerations, whereas they increase in all other regions ( Figure 4b,c ). The second row of Table 1 reports the welfare impact of the removal of the internal tax on trade with other locations. We find that this internal trade liberalization raises welfare in West and East by around 1.4% and 2.3%, respectively, more than five times the effects of the external trade liberalization.
在图 4 中，我们提出了一种替代的逆事实实验，其中我们消除了所有内部贸易成本，但保留与国际贸易成本相同的初始均衡。该图展示了相对于初始均衡的相对变化。内部贸易成本降低的影响与外部贸易成本降低的影响明显不同。内部自由化的主要影响是减少两个大城市规模，有利于农村地区，从而使经济活动更加分散（图 4a）。随着贸易成本的下降，减少大城市本地价格指数的国内市场效应减弱，因此价格在所有地方都下降，但在人口较多的地区下降较少（图 4d）。工资和土地租金在大城市群中也会下降，而在其他所有地区则会上升（图 4b,c）。表 1 的第二行报告了取消与其他地点贸易的内部税收对福利的影响。我们发现，这种内部贸易自由化使西部和东部的福利分别提高了约 1.4%和 2.3%，超过外部贸易自由化效果的五倍。

Figure 4 图 4  Internal liberalization, showing the ratio of counterfactual to initial values, : (a) log relative population, (b) log relative wages (truncated), (c) log relative land rents, and (d) log relative price index (truncated). Cold (blue) colors correspond to lower values, and hot (yellow) colors correspond to higher values.
内部自由化，显示反事实值与初始值的比率， ：（a）相对人口的对数，（b）相对工资（截断），（c）相对地租的对数，和（d）相对价格指数（截断）。冷（蓝色）颜色对应较低值，热（黄色）颜色对应较高值。

Figure 4 

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Intuitively, trade is much larger between regions than between countries, highlighting the greater importance of internal trade frictions relative to external trade frictions.
直观上看，地区间的贸易规模远大于国家间的贸易，突显了内部贸易摩擦相对于外部贸易摩擦的重要性。

Although our quantitative analysis in this section is inevitably stylized, particularly because we started from a random productivity distribution, it highlights the power and flexibility of this class of quantitative spatial models. Simply adding population data for a region would allow us to obtain real-world productivity estimates and would make this exercise quite informative about the region's economy and policy options. Although the particular quantitative model considered in this section abstracts from a number of real-world features, such as multiple traded sectors and input–output linkages, it could be further enriched to incorporate these and other salient features of the economy.
尽管本节中的定量分析不可避免地具有风格化，尤其是因为我们从随机生产力分布开始，但它突出了这类定量空间模型的力量和灵活性。简单地为一个地区添加人口数据就能使我们获得现实世界的生产力估计，这将使这项练习对该地区的经济和政策选择具有很高的信息量。尽管本节考虑的特定定量模型抽象了许多现实世界的特征，如多个贸易部门和投入产出联系，但它可以进一步丰富，以纳入这些以及其他显著的经济特征。

One line of empirical research has examined a key implication of quantitative spatial models, namely, that both wages and population depend on market access. ⁹ We illustrate this prediction in the context of the quantitative spatial model developed in Section 3. Using CES demand, one finds that profit maximization (Equation 4) and zero profits (Equation 6) imply that the free-on-board price (pi ) charged for each variety by a firm in each location i must be low enough to sell the quantity and cover the firm's fixed production costs, so that
一条实证研究线考察了定量空间模型的一个关键含义，即工资和人口都取决于市场准入。我们在第 3 节开发的定量空间模型背景下说明了这一预测。使用 CES 需求，可以发现利润最大化（方程 4）和零利润（方程 6）意味着每个地点 i 的厂商对每种品种收取的离岸价格（p）必须足够低，以便销售数量 并覆盖公司的固定生产成本，从而

31. 

We define the weighted sum of market demands faced by firms as firm market access (FMA i ), following Redding & Venables (2004), such that
我们定义企业面临的市场需求加权总和为企业市场准入（FMA），遵循 Redding & Venables（2004）的研究，即

32. 

where ξ≡[F(σ−1)]^−1/σ(σ−1)/σ collects together earlier constants. Thus, wages are increasing in both productivity (Ai ) and firm market access (FMA i ).
在ξ≡[F(σ−1)] ^−1/σ (σ−1)/σ汇集了早期常数。因此，工资在生产力（A）和企业市场准入（FMA）方面都在增加。

Market access also affects the price index (Equation 8), which depends on consumers’ access to tradeable varieties, as captured by the measure of varieties (Mi ) and their free-on-board prices (pi ) in each location i, together with the trade costs of shipping the varieties from locations i to N (dni ). We summarize this access to tradeable varieties using the concept of consumer market access (CMA n ):
市场准入也影响价格指数（公式 8），该指数取决于消费者对可交易品种的获取，这通过品种（M）及其在每个位置 i 的离岸价格（p）来衡量，以及将品种从位置 i 运送到 N 的贸易成本（dni）。我们使用消费者市场准入（CMAn）的概念来总结这种对可交易品种的获取。

33. 

Using data on a cross section of countries, Redding & Venables (2004) find a strong correlation between wages and these measures of market access. Using data on counties within the United States, Hanson (2005) finds a similarly strong relationship between wages and market access. However, establishing that these relationships are causal is more challenging. For example, Redding & Venables (2004), Hanson (2005), and Bartelme (2016) all report instrumental variable specifications, but it is difficult to definitively establish that the exclusion restriction of the instruments only affecting wages through market access is satisfied. One line of research has used trade liberalization as a natural experiment that changes the relative market access of regions (see, e.g., Hanson 1996, 1997). ¹⁰ Although this evidence from trade liberalization has strengthened the case for a causal interpretation of the role of market access, a remaining concern is that trade liberalization reforms could be endogenous to domestic political economy concerns.
使用国家横截面数据，Redding & Venables（2004）发现工资与市场准入的这些指标之间存在强烈的关联。使用美国县级行政区数据，Hanson（2005）发现工资与市场准入之间存在类似强烈的关联。然而，确立这些关系是因果的更具挑战性。例如，Redding & Venables（2004）、Hanson（2005）和 Bartelme（2016）都报告了工具变量设定，但很难明确地确立工具变量的排除限制仅通过市场准入影响工资得到满足。一条研究线已经使用贸易自由化作为改变地区相对市场准入的自然实验（参见，例如，Hanson 1996，1997）。尽管贸易自由化的证据已经加强了市场准入作用因果解释的论据，但一个持续的担忧是贸易自由化改革可能是国内政治经济问题的内生性。

Another line of research has sought to use transport infrastructure as a source of variation in market access. In this case, the key endogeneity concern is that transport infrastructure is unlikely to be randomly assigned. Therefore, a growing reduced-form literature has sought exogenous sources of variation in transport infrastructure, including from routes planned for strategic reasons, historical exploration routes, and inconsequential places that are connected to transport infrastructure merely because they lie along the route between two other locations (for a review, see Redding & Turner 2015). A smaller number of studies have sought to estimate structurally the impact of transport infrastructure improvements within a quantitative spatial model. Donaldson (2016) undertakes a quantitative evaluation of the construction of India's vast railroad network and finds that there is a strong and statistically significant estimated effect of railroads on real income levels but that this effect becomes statistically insignificant after controlling for the model's sufficient statistic of a region's own trade share. This pattern of results is consistent with the view that the estimated effects of railroads are in fact operating through the market access mechanism in the model. In a similar vein, Donaldson & Hornbeck (2016) investigate the impact of the expansion of the railroad network on the agricultural sector and show that the overall impact on each location can be captured in terms of its market access. Increases in market access from the expansion in the railroad network from 1870 to 1890 are found to substantially increase county agricultural land values.
另一条研究线试图将交通基础设施作为市场接入差异的来源。在这种情况下，关键的内生性问题在于交通基础设施不太可能被随机分配。因此，越来越多的简化形式文献寻求在交通基础设施中的外生变异来源，包括出于战略原因规划的路线、历史勘探路线以及仅仅因为它们位于两个其他地点之间路线上的无关紧要的地方（参见 Redding & Turner 2015 年的综述）。少数研究试图在定量空间模型中结构性地估计交通基础设施改善的影响。Donaldson（2016）对印度庞大的铁路网络建设进行了定量评估，发现铁路对实际收入水平有强烈且统计上显著的估计效应，但这一效应在控制了模型中一个地区的贸易份额这一充分统计量后变得统计上不显著。 这种结果模式与观点一致，即铁路的估计效应实际上是通过模型中的市场准入机制发挥作用的。在这方面，Donaldson & Hornbeck（2016）研究了铁路网络扩张对农业部门的影响，并表明每个地点的总影响可以用其市场准入来衡量。发现从 1870 年到 1890 年铁路网络扩张带来的市场准入增加，显著提高了县农业土地价值。

To provide further evidence of a causal role for market access, Redding & Sturm (2008) use the division of Germany after World War II as a natural experiment that provides plausibly exogenous variation in market access. The division of Germany has a number of attractive features for isolating the role played by market access across West German cities depending on their proximity to the new border with East Germany. First, in contrast to cross-country studies, there is no obvious variation in institutions across cities within West Germany. Second, there are no obvious changes in natural advantage, such as access to navigable rivers or coasts, climatic conditions, or disease environment. Third, the change in market access following German division is much larger than what is typically observed in other contexts, and the effects can be observed over a long period of time. Fourth, the drawing of the border dividing Germany into East and West Germany was based on military considerations that are unlikely to be correlated with the predivision characteristics of cities. ¹¹
为了进一步证明市场准入的因果作用，Redding & Sturm（2008）利用第二次世界大战后德国的分裂作为自然实验，该实验提供了市场准入的合理外生变化。德国的分裂具有许多吸引人的特征，可以隔离市场准入在靠近新边界与东德的城市中发挥的作用。首先，与跨国研究相比，在西德城市之间没有明显的制度差异。其次，没有明显的自然优势变化，如可通航河流或海岸的接入、气候条件或疾病环境。第三，德国分裂后市场准入的变化远大于其他环境中通常观察到的变化，其影响可以在长时间内观察到。第四，将德国划分为东西德边界的划定是基于军事考虑，这些考虑与城市在分裂前的特征不太可能相关。 ¹¹

In the quantitative model from Section 3, the treatment effect of division on border cities relative to those further from the new border depends on two parameter combinations that capture (a) the strength of agglomeration and dispersion forces [σ(1−α)] and (b) the elasticity of trade with respect to distance [(σ−1)ϕ, where ϕ is the elasticity of trade costs with respect to distance]. Redding & Sturm (2008) undertake a quantitative analysis of the model and show that, for plausible values of these parameter combinations that satisfy the condition σ(1−α)>1 for a unique equilibrium, the model can account quantitatively for both the average treatment effect of division and the larger treatment effect for small than for large cities.
在第三节的定量模型中，相对于远离新边界的城市，分割对边境城市的影响效果取决于两个参数组合，这两个参数组合捕捉了（a）聚集和分散力的强度[σ(1−α)]和（b）贸易相对于距离的弹性[(σ−1)ϕ，其中ϕ是贸易成本相对于距离的弹性]。Redding & Sturm（2008）对模型进行了定量分析，并表明，对于满足σ(1−α)>1 以实现唯一均衡的这些参数组合的合理值，模型可以定量解释分割的平均影响效果以及相对于大城市，分割对小城市的影响效果更大。

A large body of empirical literature finds that wages, land prices, productivity, employment, and employment growth are positively correlated with population density (for a survey, see Moretti 2011a). In their survey of this empirical literature, Rosenthal & Strange (2004) report that the elasticity of productivity with respect to the density of economic activity is typically estimated to lie within the range of 3–8%. ¹² However, establishing that this correlation is indeed causal remains challenging, and a relatively small number of studies have sought exogenous sources of variation in the surrounding concentration of economic activity. For example, Rosenthal & Strange (2008) and Combes et al. (2010) use geology as an instrument for population density, exploiting the idea that tall buildings are easy to construct where solid bedrock is accessible. Greenstone et al. (2010) provide evidence on agglomeration spillovers by comparing changes in total factor productivity among incumbent plants in winning counties that attracted a large manufacturing plant and losing counties that were the new plant's runner-up choice.
大量实证文献发现，工资、土地价格、生产力、就业和就业增长与人口密度呈正相关（参见 Moretti 2011a 中的综述）。在他们对这一实证文献的综述中，Rosenthal & Strange（2004）报告称，通常估计生产率对经济活动密度的弹性值在 3-8%之间。然而，确立这种相关性确实是因果关系仍然具有挑战性，并且相对较少的研究寻求在经济活动集中度周围的变异的外生来源。例如，Rosenthal & Strange（2008）和 Combes 等人（2010）将地质学作为人口密度的工具变量，利用了在坚固基岩易于建造高层建筑的观点。Greenstone 等人（2010）通过比较吸引大型制造工厂的获胜县和作为新工厂备选选择的新败县的总要素生产率的变化，提供了关于集聚溢出的证据。

Several recent studies have used exogenous variation from natural experiments to examine whether estimated agglomeration economies are consistent with the predictions of quantitative spatial models. Combining data from an urban revitalization program in Richmond, Virginia, between 1999 and 2004 with a structural model of residential externalities, Rossi-Hansberg et al. (2010) estimate substantial and highly localized housing externalities. Land prices in neighborhoods targeted for revitalization rose by 2–5% more per annum than those in a control neighborhood. With every 1,000 ft of distance from the subsidized area, housing externalities are estimated to decline by around one-half. ¹³
一些近期研究利用自然实验的外生变化来检验估计的集聚经济是否与定量空间模型的预测一致。将弗吉尼亚州里士满 1999 年至 2004 年的一项城市复兴计划的数据与住宅外部性的结构模型相结合，罗西-汉斯伯格等人（2010 年）估计了显著且高度局部化的住房外部性。目标复兴地区的土地价格每年比控制地区的土地价格高出 2-5%。从补贴区域每增加 1,000 英尺距离，住房外部性预计将减少约一半。 ¹³

Using the Tennessee Valley Authority (TVA) as a natural experiment and a structural model, Kline & Moretti (2014a) provide evidence on the long-run effects of one of the most ambitious regional development programs in US history. Using as controls authorities that were proposed but never approved by Congress, Kline & Moretti (2014a) found that the TVA led to large gains in agricultural employment that were eventually reversed when the program's subsidies ended. In contrast, gains in manufacturing employment were found to intensify well after federal transfers had lapsed, consistent with agglomeration economies in manufacturing.
以田纳西河谷管理局（TVA）作为自然实验和结构模型，克莱因和莫雷蒂（2014a）提供了关于美国历史上最雄心勃勃的区域发展计划之一的长期影响的证据。使用国会提出但从未获得批准的机构作为控制变量，克莱因和莫雷蒂（2014a）发现，TVA 导致了农业就业的大幅增长，当该计划的补贴结束时，这些增长最终被逆转。相反，制造业就业的增长被发现是在联邦转移支付结束后才加剧的，这与制造业的集聚经济一致。

Using the division of Berlin following World War II and its reunification after the fall of the Iron Curtain as an exogenous source of variation in the surrounding density of economic activity, Ahlfeldt et al. (2015) structurally estimate the parameters of the quantitative spatial model explained in Section 4. Following the city's division, there was a reorientation of the gradient in land prices and employment in West Berlin away from the main pre-war concentration of economic activity in East Berlin, whereas the city's reunification led to a reemergence of these gradients. The model's parameters are identified from the assumption that the systematic change in the pattern of economic activity in West Berlin following division and reunification is explained by the mechanisms of the model (changes in commuting access and production and residential externalities) rather than by systematic changes in the pattern of structural residuals (production and residential fundamentals).
利用第二次世界大战后柏林的分裂及其在铁幕倒塌后的重新统一作为周围经济活动密度变化的内生来源，Ahlfeldt 等人（2015）对第 4 节中解释的定量空间模型的参数进行了结构估计。在城市的分裂之后，西柏林的土地价格和就业梯度发生了重新定位，远离了东柏林战前经济活动的主要集中区域，而城市的重新统一导致了这些梯度的再次出现。模型的参数是通过以下假设来确定的：西柏林在分裂和重新统一后经济活动模式系统性的变化是由模型机制（通勤可达性、生产和居住外部性变化）解释的，而不是由结构残差（生产和居住基本面）模式系统性的变化解释的。

Both productivity and amenities are found to exhibit substantial and highly localized agglomeration externalities. The estimated elasticities of productivity and amenities with respect to the surrounding densities of workplace employment and residence employment are μ=0.07 and η=0.15, respectively. Undertaking counterfactuals for the impact of division and reunification, one finds that the special case of the model without agglomeration forces (μ=η=0) is unable to account quantitatively for the observed reallocations of economic activity in the data. In contrast, for the estimated values of production and residential externalities, the model is successful in matching the observed impacts of division and reunification, both qualitatively and quantitatively.
生产力与便利设施均被发现存在显著且高度局部化的聚集外部性。生产力和便利设施相对于工作场所就业密度和居住就业密度的弹性估计值分别为μ=0.07 和η=0.15。对分割和重聚的影响进行反事实分析，发现模型在没有聚集力（μ=η=0）的特殊情况下，无法定量解释数据中观察到的经济活动重新分配。相比之下，对于生产和居住外部性的估计值，模型在定性和定量上均成功匹配了分割和重聚的观察影响。

Other research, such as that of Bernard et al. (2016), has examined the role of urban property rights in influencing internal city structure in a developing-country context. Further quantitative evidence on the role of residential externalities in influencing the spatial distribution of economic activity is provided by Diamond (2015). From 1980 to 2000, the rise in the US wage gap between college and high-school graduates coincided with increased geographic sorting as college graduates concentrated in high-wage, high-rent cities. A structural spatial equilibrium model is used to evaluate the causes and welfare consequences of this increased skill sorting. Although local labor demand changes fundamentally caused the increased skill sorting, it was further fueled by endogenous increases in amenities (residential externalities) within higher-skill cities. Changes in cities’ wages, rents, and endogenous amenities increased inequality between high-school and college graduates by more than would be suggested by the increase in the college wage gap alone.
其他研究，如 Bernard 等人（2016）的研究，探讨了在发展中国家背景下城市产权在影响城市内部结构中的作用。Diamond（2015）提供了更多关于住宅外部性在影响经济活动空间分布中作用的定量证据。从 1980 年到 2000 年，美国大学毕业生与高中毕业生之间的工资差距扩大与地理上的排序增加相一致，大学毕业生集中在高工资、高租金的城市。使用结构空间均衡模型来评估这种技能排序增加的原因和福利后果。尽管地方劳动力需求的基本变化是导致技能排序增加的根本原因，但它进一步受到高技能城市内部便利性（住宅外部性）内生增加的推动。城市工资、租金和内生便利性的变化使得高中毕业生与大学毕业生之间的不平等程度超过了仅由大学工资差距增加所暗示的程度。

Several of the above studies find evidence of residential externalities; the microeconomic determinants of these externalities and their implications remain interesting areas for further research.
一些上述研究发现了居住外部性的证据；这些外部性的微观经济决定因素及其影响仍然是进一步研究的有趣领域。

Motivated in part by models with multiple equilibria, one line of empirical research on changes in the spatial distribution of economic activity over time has set out to look for evidence of path dependence, defined as temporary shocks permanently affecting the distribution of economic activity. Davis & Weinstein (2002, 2008) use Japanese war-time bombing as such an exogenous temporary shock and find little evidence of path dependence for either the distribution of population as a whole or employment in individual industries (for other research exploiting war-time bombing as an exogenous shock, see, e.g., Bosker et al. 2007, 2008; Miguel & Roland 2011). Subsequent studies have provided a number of apparent examples of path dependence using a variety of alternative sources of exogenous variation. Redding et al. (2011) find path dependence for the location of Germany's air hub using the natural experiment of Germany's division and reunification. ¹⁴ Bleakley & Lin (2012) find, using variation from portage sites in the United States, that a temporary historical advantage had permanent effects on the spatial distribution of population. Hornbeck & Keniston (2014) find long-lived effects of the Boston fire through the potential for large-scale rebuilding using plot-level data on land values. Michaels & Rauch (2016) report results consistent with path dependence using data on Roman cities.
部分受多均衡模型启发，一条关于经济活动空间分布随时间变化的经验研究路线试图寻找路径依赖的证据，定义为暂时性冲击永久性地影响经济活动的分布。戴维斯和韦恩斯坦（2002，2008）将日本战争时期的轰炸视为此类外生暂时性冲击，并发现对于整体人口分布或单个产业就业分布几乎没有路径依赖的证据（关于利用战争时期轰炸作为外生冲击的其他研究，参见，例如，Bosker 等人 2007，2008；Miguel 和 Roland 2011）。后续研究通过多种外生变化来源提供了许多路径依赖的明显例子。雷丁等人（2011）利用德国分裂和统一的自然实验发现德国航空枢纽位置的路径依赖。布莱克利和林（2012）发现，利用美国港口遗址的变化，一个暂时性的历史优势对人口空间分布产生了永久性影响。 霍恩贝克和肯尼斯顿（2014）通过土地价值的地块级数据发现波士顿火灾的长期影响，通过大规模重建的可能性。迈克尔和劳奇（2016）报告了与路径依赖一致的结果，使用罗马城市的数据。

Although we now have strong evidence of path dependence in the distribution of economic activity, the interpretation of these empirical results remains open to question. In particular, findings of path dependence do not necessarily imply multiple equilibria. If historical advantages lead to investments in local technology and infrastructure and better institutions, dynamic agglomeration effects can lead to maintained investments in these locations. These investments can serve as new sources of local advantages even after the original historical advantage has become obsolete or irrelevant. This is the case in the dynamic models of Desmet & Rossi-Hansberg (2015) and Desmet et al. (2016). The latter article shows that these dynamic agglomeration effects have significant predictive power over long periods. In particular, using only information for the year 2000 and running the dynamics backwards, Desmet et al. (2016) find a correlation between actual and predicted country population levels in 1950 as high as 0.74. In these models, the equilibrium is unique, and so the evolution of the distribution of economic activity is uniquely determined by initial conditions. In that sense, any shock or change in fundamentals in a given period will affect the future evolution and the balanced-growth path of the world economy. Empirical and theoretical research on the dynamic evolution of the distribution of economic activity is reviewed by Desmet & Henderson (2015).
尽管我们现在有关于经济活动分布路径依赖的强烈证据，但这些经验结果的解释仍然存在疑问。特别是，路径依赖的发现并不一定意味着存在多重均衡。如果历史优势导致对当地技术和基础设施的投资以及更好的制度，动态集聚效应可以导致对这些地点的投资持续。这些投资甚至可以在原始历史优势变得过时或不相关后，成为新的地方优势来源。这是 Desmet 和 Rossi-Hansberg（2015 年）以及 Desmet 等人（2016 年）的动态模型中的情况。后一篇文章表明，这些动态集聚效应在长期内具有显著的预测能力。特别是，Desmet 等人（2016 年）仅使用 2000 年的信息，逆向运行动态，发现实际和预测的 1950 年国家人口水平之间的相关性高达 0.74。在这些模型中，均衡是唯一的，因此经济活动分布的演变由初始条件唯一确定。 在这个意义上，任何给定时期的冲击或基本面的变化都将影响世界经济未来的演变和平衡增长路径。Desmet 和 Henderson（2015）回顾了关于经济活动分布动态演变的实证和理论研究。

Clearly, more theoretical and empirical work on the role of shocks and initial conditions in determining the evolution of economic activity is needed. Perhaps this will be one of the more fruitful areas for future research on quantitative spatial economics.
显然，需要更多关于冲击和初始条件在决定经济活动演变中作用的理论和实证研究。或许这将是未来定量空间经济学研究较为富有成效的领域之一。