The Effect of Financial Development on Convergence: Theory and Evidence* 金融发展对收敛的影响：理论与实证研究*
Philippe Aghion ^(†)quad{ }^{\dagger} \quad Peter Howitt ^(‡)quad{ }^{\ddagger} \quad David Mayer-Foulkes ^(§){ }^{\S} 菲利普·阿吉翁 ^(†)quad{ }^{\dagger} \quad 彼得·霍维特 ^(‡)quad{ }^{\ddagger} \quad 大卫·梅耶-福尔克斯 ^(§){ }^{\S}
April 8, 2004 2004 年 4 月 8 日
Abstract 摘要
We introduce imperfect creditor protection in a multi-country version of Schumpeterian growth theory with technology transfer. The theory predicts that the growth rate of any country with more than some critical level of financial development will converge to the growth rate of the world technology frontier, and that all other countries will have a strictly lower long-run growth rate. The theory also predicts that in a country that converges to the frontier growth rate, financial development has a positive but eventually vanishing effect on steady-state per-capita GDP relative to the frontier. We present cross-country evidence supporting these two implications. In particular, we find a significant and sizeable effect of an interaction term between the log of initial per-capita GDP (relative to the United States) and a financial intermediation measure, in an otherwise standard growth regression, implying that the likelihood of converging to the U.S. growth rate increases with financial development. We also find that, as predicted by the theory, the direct effect of financial intermediation in this regression is not significantly different from zero. In addition, we find that other variables representing schooling, geography, health, policy, politics and institutions do not affect the significance of the interaction between financial intermediation and initial per capita GDP, and do not show any independent effect on convergence in our cross-country regressions. Our findings are robust to removal of outliers and to alternative conditioning sets, estimation procedures and measures of financial development. 我们提出了一种在包含技术转移的多国版本的熊彼特增长理论中引入不完美债权人保护的模型。该理论预测，任何金融发展水平超过某个临界阈值的国家，其增长率将收敛于世界技术前沿的增长率，而所有其他国家将具有严格较低的长期增长率。该理论还预测，在一个收敛于前沿增长率的国家中，金融发展对稳态人均 GDP 相对于前沿的增长具有正向但最终消失的影响。我们提供了支持这两个推论的跨国证据。具体而言，我们在一个标准的增长回归中发现，初始人均 GDP（相对于美国）的对数与金融中介化指标的交互项存在显著且较大的效应，这表明金融发展越高，趋同于美国增长率的可能性越大。此外，正如理论所预测的，金融中介化在该回归中的直接效应与零无显著差异。此外，我们发现，代表教育、地理、健康、政策、政治和制度的其他变量既不影响金融中介与初始人均 GDP 之间交互作用的显著性，也不在跨国回归中对收敛显示出独立影响。我们的发现对剔除异常值、采用替代条件集、估计方法及金融发展指标均具有稳健性。
1 Introduction 1 引言
Most current theories of the cross-country distribution of per-capita income imply that all countries share the same long-run growth rate (of TFP or per-capita GDP). Yet the historical record shows that growth rates can differ substantially across countries for long periods of time. For example, Pritchett (1997) estimates that the proportional gap in percapita GDP between the richest and poorest countries grew more than five-fold from 1870 to 1990, and according to the tables in Maddison (2001) the proportional gap between the richest group of countries and the poorest ^(1){ }^{1} grew from 3 in 1820 to 19 in 1998. 大多数关于各国人均收入跨国分布的理论假设所有国家在长期内拥有相同的增长率（无论是全要素生产率还是人均国内生产总值）。然而，历史数据表明，不同国家在较长时期内的增长率可能存在显著差异。例如，普里切特（1997）估计，1870 年至 1990 年间，最富裕国家和最贫困国家之间的人均 GDP 比例差距增长了五倍多。根据马德森（2001）的表格，最富裕国家与最贫困国家之间的比例差距从 1820 年的 3 扩大到 1998 年的 19。
The “great divergence” between rich and poor countries continued through the end of the twentieth century. Although many studies ^(2){ }^{2} show that a large group of rich and middleincome countries have been converging to parallel growth paths over the past 50 years or so, the gap between these countries as a whole and the very poorest countries as a whole has continued to widen. For example, the proportional gap in per-capita GDP between MayerFoulkes’s (2002) richest and poorest convergence groups grew by a factor of 2.6 between 1960 and 1995, and the proportional gap between Maddison’s richest and poorest groups grew by a factor of 1.75 between 1950 and 1998 . “贫富国家之间的‘大分化’在 20 世纪末期持续存在。尽管许多研究 ^(2){ }^{2} 表明，过去 50 年左右，一大批富裕国家和中等收入国家已逐渐趋向于平行增长路径，但这些国家整体与最贫困国家整体之间的差距仍在持续扩大。例如，根据梅耶-福尔克斯（Mayer-Foulkes，2002）的研究，最富裕与最贫困国家集团的人均 GDP 比例差距在 1960 年至 1995 年间扩大了 2.6 倍，而根据马德森（Maddison）的研究，最富裕与最贫困国家集团的人均 GDP 比例差距在 1950 年至 1998 年间扩大了 1.75 倍。
Technology appears to be the central factor underlying divergence. Easterly and Levine (2001) estimate that about 60%60 \% of the cross-country variation in growth rates of per-capita GDP is attributable to differences in productivity growth, while Klenow and RodríguezClare (1997) estimate that in their sample about 90%90 \% of the variation is attributable to differences in productivity growth. Although the level of productivity can be affected by many factors other than technology, such as geography and institutions that affect the efficiency of resource allocation, it is hard to see how substantial differences in the growth rate of productivity persisting for such long periods of time can be accounted for by these non-technological factors, which are themselves highly persistent over time. Instead it seems more likely that divergence reflects long-lasting cross-country differences in rates of technological progress. 技术似乎是导致分化的核心因素。伊斯特利和莱文（2001）估计，约 60%60 \% 的各国人均 GDP 增长率差异可归因于生产率增长的差异，而克莱诺和罗德里格斯-克莱尔（1997）在其样本中估计，约 90%90 \% 的差异可归因于生产率增长的差异。尽管生产率水平可能受到技术以外的许多因素影响，如影响资源配置效率的地理和制度因素，但很难解释为何生产率增长率在如此长时期内存在的显著差异，仅由这些非技术因素（其本身在时间上也高度持久）来解释。相反，分化似乎更可能反映各国在技术进步速度上的长期差异。
These facts are especially puzzling when one takes into account the possibility of international technology transfer and the “advantage of backwardness” (Gerschenkron 1952) that it confers on technological laggards. That is, the further a country falls behind the world’s technology leaders the easier it is for that country to progress technologically simply by implementing new technologies that have been discovered elsewhere. Eventually this advantage should be enough to stabilize the proportional gap that separates it from the leaders. This is what happens in neoclassical models where technology transfer is instantaneous (Mankiw, Romer and Weil, 1992), where technologies developed on the frontier are not “appropriate” for poorer countries (Basu and Weil, 1998; Acemoglu and Zilibotti, 2001), where technology transfer can be blocked by special interests (Parente and Prescott, 1994, 1999) and where a country adopts institutions that impede technology transfer (Acemoglu, Aghion and Zilibotti, 2002). 这些事实尤其令人费解，因为考虑到国际技术转移的可能性以及它赋予技术落后者“落后优势”（Gerschenkron 1952）的效应。也就是说，一个国家与世界技术领先者之间的差距越大，该国通过采用在其他地方发现的新技术来实现技术进步就越容易。最终，这种优势应足以稳定该国与领先国家之间的技术差距比例。这是新古典模型中的情况，其中技术转移是即时的（Mankiw、Romer 和 Weil，1992），前沿技术对较贫困国家而言“不合适”（Basu 和 Weil，1998；阿西莫格鲁和齐利博蒂，2001），技术转移可能被特殊利益集团阻碍（帕伦特和普雷斯科特，1994，1999），以及国家采用阻碍技术转移的制度（阿西莫格鲁、阿吉翁和齐利博蒂，2002）。
This paper explores the hypothesis that financial constraints prevent poor countries from taking full advantage of technology transfer and that this is what causes some of them 本文探讨了这样一种假设：金融约束阻碍了贫困国家充分利用技术转移，而这正是导致其中一些国家
to diverge from the growth rate of the world frontier. It introduces credit constraints into a multi-country version of Schumpeterian growth theory with technology transfer, ^(3){ }^{3} and shows that the model implies a form of club convergence consistent with the broad facts outlined above. In the theory, countries above some threshold level of financial development will all converge to the same long-run growth rate and all other countries will have strictly lower long-run growth rates. 偏离世界前沿的增长率。该理论在包含技术转移的多国版本的熊彼特增长理论中引入了信贷约束， ^(3){ }^{3} 并表明该模型蕴含一种与上述基本事实相一致的俱乐部收敛形式。在该理论中，金融发展水平超过某一阈值的国家将全部收敛至相同的长期增长率，而其他所有国家将具有严格较低的长期增长率。
There are three key components to the theory. The first is that because technological knowledge is often tacit and circumstantially specific, ^(4){ }^{4} technology transfer requires the receiving country to invest resources in order to master foreign technologies and adapt them to the local environment. Although these investments may not fit the conventional definition of R&D\mathrm{R} \& \mathrm{D}, they play the same role as R&D\mathrm{R} \& \mathrm{D} in an innovation-based growth model; that is, they generate new technological possibilities where they are conducted, building on previous knowledge. ^(5){ }^{5} Accordingly our theory assigns to R&D the role that Nelson and Phelps (1966) assumed was played by human capital, namely that of determining a country’s “absorptive capacity”. ^(6){ }^{6} 该理论包含三个核心要素。首先，由于技术知识往往具有隐性且情境特定的特性， ^(4){ }^{4} 技术转移要求接受国投入资源以掌握外国技术并将其适应本地环境。尽管这些投资可能不符合传统意义上的 R&D\mathrm{R} \& \mathrm{D} ，但在以创新为基础的增长模型中，它们与 R&D\mathrm{R} \& \mathrm{D} 发挥相同的作用，即在实施地点创造新的技术可能性，并基于现有知识进行拓展。 ^(5){ }^{5} 因此，我们的理论将研发赋予了纳尔逊和菲尔普斯（1966）认为由人力资本所扮演的角色，即决定一个国家“吸收能力”的作用。 ^(6){ }^{6}
The second key component is the assumption that as the global technology frontier advances, the size of investment required in order to keep innovating at the same pace as before rises in proportion. This assumption recognizes the force of increasing complexity, which makes technologies increasingly difficult to master and to adapt to local circumstances. ^(7){ }^{7} 第二个关键要素是假设随着全球技术前沿的不断推进，为了保持与以往相同的创新速度，所需的投资规模也会相应增加。这一假设承认了复杂性不断增加的趋势，这种趋势使得技术越来越难以掌握，也越来越难以适应当地的具体情况。 ^(7){ }^{7}
The third key component is an agency problem that limits an innovator’s access to external finance. Specifically we assume that an innovator can defraud her creditors by hiding the results of a successful innovation, at a cost that depends positively on the level of financial development. Because of this, in equilibrium the innovator’s access to external finance will be limited to some multiple of her own wage income. Since wages are limited by domestic productivity, therefore a technological laggard can face a disadvantage of backwardness that counteracts Gerschenkron’s advantage; that is, the further behind the frontier it falls the less its innovators will be able to invest relative to what is required in order to keep innovating at a given rate. The lower the level of financial development in the country the greater will be this disadvantage. 第三个关键因素是代理问题，它限制了创新者获取外部融资的能力。具体来说，我们假设创新者可以通过隐瞒成功创新的结果来欺骗其债权人，而这种欺骗的成本与金融发展水平成正比。因此，在均衡状态下，创新者获取外部融资的能力将被限制在其自身工资收入的若干倍之内。由于工资受国内生产率限制，因此技术落后者将面临一种落后劣势，该劣势会抵消格申克龙优势；即落后于技术前沿的程度越深，其创新者相对于维持给定创新速度所需的投资能力就越弱。国家金融发展水平越低，这种劣势越显著。
Our paper relates to several important strands of theory relating growth, convergence and financial-market development. There is first the literature on poverty traps and interpersonal convergence or divergence in economies with credit market imperfections, in particular Banerjee and Newman (1993), Galor and Zeira (1993), Aghion and Bolton (1997) 本文涉及与增长、收敛及金融市场发展相关的多个重要理论领域。首先是关于信用市场不完善经济体中贫困陷阱及个人间收敛或分化的文献，其中包括班纳吉和纽曼（1993）、加洛尔和泽拉（1993）、阿吉翁和博尔顿（1997）等研究。
and Piketty (1997). In these models, ^(8){ }^{8} all agents face the same production technology and, unlike in our model, the same (productivity-adjusted) investment costs, ^(9){ }^{9} and what generates poverty traps are either non-convexities in production or monitoring, or pecuniary externalities working through factor prices. However, there is no technical progress and therefore no positive long-run growth in these models, which therefore cannot analyze the issue of long-term convergence in growth rates. 以及皮凯蒂（1997）。在这些模型中，所有经济主体面临相同的生產技術，且與我們的模型不同，其投資成本（經生產力調整後）亦相同，而導致貧困陷阱的因素要么是生產過程或監管中的非凸性，要么是通過要素價格傳導的貨幣外部性。然而，这些模型中不存在技术进步，因此不存在正向长期增长，因此无法分析增长率长期收敛的问题。
Another literature analyzes the effects of financial constraints and/or financial intermediation on long-term growth. Thus, Greenwood and Jovanovic (1990), Levine (1991), Bencivenga and Smith (1991, 1993), Saint-Paul (1992), Sussman (1993), Harrison, Sussman and Zeira (1999) and Kahn (2001) analyze the effects of financial intermediation on growth in an AK-style model with no distinction being made between investing in technology and investing in physical or human capital accumulation. King and Levine (1993b), de la Fuente and Marin (1996), Galetovic (1996), Blackburn and Hung (1998) and Morales (2003) consider the relationship between finance and growth in the context of innovation-based growth models. De Gregorio (1996) studies the effects on growth of financial constraints that inhibit human capital accumulation. Krebs (2003) shows how imperfect sharing of individual human-capital risk can depress long-run growth. However, none of these models analyzes the process of technology transfer that we are focusing on, and therefore none of them is capable of addressing the question of why technology transfer is not sufficient to put all countries on parallel long-run growth paths. 另一项文献分析了金融约束和/或金融中介对长期增长的影响。因此，格林伍德和约瓦诺维奇（1990）、莱文（1991）、本西文加和史密斯（1991、1993）、圣保罗（1992）、苏斯曼（1993）、哈里森、苏斯曼和泽拉（1999）以及卡恩（2001）在 AK 型模型中分析了金融中介对增长的影响，该模型未区分技术投资与实物或人力资本积累。金和莱文（1993b）、德拉富恩特和马林（1996）、加莱托维奇（1996）、布莱克本和洪（1998）以及莫拉莱斯（2003）探讨了金融与增长在创新驱动增长模型中的关系。德格雷戈里奥（1996）研究了阻碍人力资本积累的金融约束对增长的影响。克雷布斯（Krebs，2003）展示了个人人力资本风险分配不完善如何抑制长期增长。然而，这些模型均未分析我们关注的技术转移过程，因此无法解答为何技术转移不足以使所有国家走上平行长期增长路径的问题。
The paper also produces evidence to support its main implications. There is already a substantial body of evidence ^(10){ }^{10} to the effect that financial development is an important determinant of a country’s short-run growth rate, almost all of which is predicated on the assumption of long-run convergence in growth rates. We extend this analysis to allow for the possibility of different long-run growth rates, using a cross section of 71 countries over the period 1960-1995. Specifically, we estimate the effect of an interaction term between the log of initial per-capita GDP (relative to the United States) and financial development in an otherwise standard cross-country growth regression. We interpret a negative coefficient as evidence that low financial development makes convergence less likely. Using a measure of financial development first introduced by Levine, Loayza and Beck (2000) we find that the coefficient is indeed negative, and is large both statistically and economically. 本文还提供了支持其主要结论的证据。已有大量研究 ^(10){ }^{10} 表明，金融发展是决定一国短期增长率的重要因素，而这些研究几乎都基于长期增长率趋同的假设。我们扩展了这一分析，以考虑长期增长率可能存在差异的情况，使用了 1960 年至 1995 年期间 71 个国家的横截面数据。具体而言，我们在一个标准的跨国增长回归中估计了初始人均 GDP（相对于美国）的对数与金融发展之间的交互项的影响。我们将负系数解释为金融发展水平较低会降低收敛可能性的证据。采用莱文、洛亚扎和贝克（2000）首次提出的金融发展指标，我们发现该系数确实为负，且在统计上和经济上均显著。
Our empirical methodology is similar to that of Benhabib and Spiegel (2002), who found a negative interaction term between initial TFP and schooling and concluded that schooling was a key determinant of whether or not a country will converge to the frontier growth rate. We test the robustness of our results by including both schooling and an interaction term between the initial GDP gap and schooling as additional regressors in our equation. In addition, we repeat this robustness test using instead of schooling a large number of different variables suggested by other growth theories. In all cases the main implications of our theory pass the test. We also present evidence to the effect that the main channel through which financial development affects convergence is productivity growth, as implied 我们的实证方法与 Benhabib 和 Spiegel（2002）相似，他们发现初始全要素生产率（TFP）与教育水平之间存在负相关关系，并得出结论认为教育水平是决定一个国家是否能收敛到前沿增长率的关键因素。我们通过在方程中加入教育水平以及初始 GDP 差距与教育水平的交互项作为额外自变量，来检验结果的稳健性。此外，我们还使用其他增长理论提出的众多不同变量替代教育水平，重复了这一稳健性检验。在所有情况下，我们理论的主要结论均经受住了检验。我们还提供了证据，表明金融发展影响收敛的主要渠道是生产率增长，这与理论推论一致。
by the theory, rather than capital accumulation, and show that our results are robust to elimination of outliers, to alternative conditioning sets, to alternative estimation procedures and to alternative measures of financial development. 根据理论，而非资本积累，并证明我们的结果在剔除异常值、采用替代条件集、替代估计方法以及替代金融发展指标时均具有稳健性。
2 Theoretical framework 2 理论框架
We follow Acemoglu, Aghion and Zilibotti (2002) in casting Schumpeterian growth theory in a simple discrete-time framework. There are mm countries, who do not exchange goods or factors, but do make use of each others’ technological ideas. There is a continuum of individuals in each country. Each country has a fixed population PP, which for notational convenience we normalize to unity. Thus aggregate and per-capita quantities are identical. Everyone lives for two periods, being endowed with two units of labor services in the first period and none in the second, with a utility function linear ^(11){ }^{11} in consumption: U=c_(1)+betac_(2)U=c_{1}+\beta c_{2}, where 0 < beta < 10<\beta<1. Within each country the growth path is determined as follows. 我们遵循阿西莫格鲁、阿吉翁和齐利博蒂（2002）的思路，将熊彼特增长理论置于一个简单的离散时间框架中。存在 mm 个国家，这些国家之间不进行商品或要素的交换，但会利用彼此的技术理念。每个国家存在一个连续的个体群体。每个国家的人口为固定值 PP ，为便于记号，我们将其归一化为 1。因此，总数量与人均数量相等。每个人生活两个时期，在第一个时期拥有两单位劳动服务，第二个时期没有，其效用函数与消费呈线性关系： U=c_(1)+betac_(2)U=c_{1}+\beta c_{2} ，其中 0 < beta < 10<\beta<1 。在每个国家内，增长路径按以下方式确定。
2.1 The general sector 2.1 总体部门
There is one multi-purpose “general” good, produced by labor and a continuum of specialized intermediate goods according to the production function: 存在一种多用途的“通用”商品，由劳动与一系列专门化的中间商品按照生产函数生产而成：
Z_(t)=P^(1-alpha)int_(0)^(1)A_(t)(i)^(1-alpha)x_(t)(i)^(alpha)di,quad0 < alpha < 1Z_{t}=P^{1-\alpha} \int_{0}^{1} A_{t}(i)^{1-\alpha} x_{t}(i)^{\alpha} d i, \quad 0<\alpha<1
where x_(t)(i)x_{t}(i) is the input of the latest version of intermediate good ii and A_(t)(i)A_{t}(i) is the productivity parameter associated with it. The general good is used for consumption, as an input to R&D and also as an input to the production of intermediate goods. 其中， x_(t)(i)x_{t}(i) 是最新版本的中间产品 ii 的投入， A_(t)(i)A_{t}(i) 是与其相关的生产率参数。一般商品用于消费，作为研发的投入，同时也作为中间产品生产的投入。
The general good is produced under perfect competition, so the price of each intermediate good equals its marginal product: 在完全竞争下，社会总产品得以生产，因此每种中间产品的价格等于其边际产品：
(We use the general good as numéraire, and P=1P=1 ). （我们以一般商品作为计价单位，并使用 P=1P=1 表示。）
2.2 Intermediate sectors 2.2 中间部门
For each intermediate good ii there is one person born each period t-1t-1 who is capable of producing an innovation for the next period. This person is called the i^(th)i^{t h} innovator in t-1t-1, and if she succeeds (innovates) then she will be the i^(th)i^{t h} incumbent in tt. Let mu_(t)(i)\mu_{t}(i) be the probability that she succeeds. Then: 对于每种中间产品 ii ，每个时期都会出生一个人 t-1t-1 ，他有能力在下一时期生产出一种创新。此人被称为 i^(th)i^{t h} 创新者，如果她成功（创新），则将成为 i^(th)i^{t h} 现任者。设 mu_(t)(i)\mu_{t}(i) 为她成功的概率。则：
A_(t)(i)={[ bar(A)_(t)," with probability "mu_(t)(i)],[A_(t-1)(i)," with probability "1-mu_(t)(i)]}A_{t}(i)=\left\{\begin{array}{cc}
\bar{A}_{t} & \text { with probability } \mu_{t}(i) \\
A_{t-1}(i) & \text { with probability } 1-\mu_{t}(i)
\end{array}\right\}
where bar(A)_(t)\bar{A}_{t} is the world technology frontier, which grows at the constant rate g > 0g>0, taken as given for now. The fact that a successful innovator gets to implement bar(A)_(t)\bar{A}_{t} is a manifestation of technology transfer, of the kind that Keller (2002) calls “active”; that is, domestic R&D makes use of ideas developed elsewhere in the world. ^(12){ }^{12} 其中， bar(A)_(t)\bar{A}_{t} 表示世界技术前沿，其增长率为常数 g > 0g>0 ，目前暂且视为已知。成功创新者能够实施 bar(A)_(t)\bar{A}_{t} 的事实，体现了技术转移的一种形式，即凯勒（2002）所称的“主动型”技术转移；也就是说，国内研发利用了世界其他地区开发出的技术理念。 ^(12){ }^{12}
In each intermediate sector where an innovation has just occurred, the incumbent is able to produce any amount of the intermediate good using as the sole input one unit of the general good per unit of intermediate good. In addition, in every intermediate sector there is an unlimited number of people capable of producing copies of the latest generation of that intermediate good at a unit cost of chi > 1.^(13)\chi>1 .^{13} 在每个刚刚发生创新的中间部门中，现有企业能够使用每单位中间产品消耗 1 单位通用产品作为唯一投入，生产任意数量的中间产品。此外，在每个中间部门中，存在无限数量的人员能够以单位成本 chi > 1.^(13)\chi>1 .^{13} 生产最新一代中间产品的复制品。
So in sectors where an innovation has just occurred, the incumbent will be the sole producer, at a price equal to the unit cost of the competitive fringe, ^(14){ }^{14} whereas in noninnovating sectors where the most recent incumbent is dead, production will take place under perfect competition with a price equal to the unit cost of each producer. In either event the price will be chi\chi, and according to the demand function (2) the quantity demanded will be: 因此，在创新刚刚发生的新兴行业中，现有企业将成为唯一的生产者，其价格等于竞争边缘企业的单位成本， ^(14){ }^{14} 而对于那些没有进行创新的行业，其中最近的现有企业已经退出市场，生产将在完全竞争下进行，价格等于每个生产者的单位成本。无论哪种情况，价格均为 chi\chi ，根据需求函数(2)，需求量为：
It follows that an unsuccessful innovator will earn zero profits next period, whereas the profit of an incumbent will be pi_(t)(i)=pi bar(A)_(t)\pi_{t}(i)=\pi \bar{A}_{t}, where pi=(chi-1)(alpha//chi)^((1)/(1-alpha))\pi=(\chi-1)(\alpha / \chi)^{\frac{1}{1-\alpha}}. 由此可知，失败的创新者在下一个时期将获得零利润，而现有企业的利润为 pi_(t)(i)=pi bar(A)_(t)\pi_{t}(i)=\pi \bar{A}_{t} ，其中 pi=(chi-1)(alpha//chi)^((1)/(1-alpha))\pi=(\chi-1)(\alpha / \chi)^{\frac{1}{1-\alpha}} 。
2.3 Aggregate behavior 2.3 集合行为
Define the country’s “average productivity” A_(t)A_{t} as: 定义该国的“平均生产率” A_(t)A_{t} 为：
A_(t)=int_(0)^(1)A_(t)(i)diA_{t}=\int_{0}^{1} A_{t}(i) d i
Substituting (3) into (1) we see that gross output of the general good will be: 将（3）代入（1），我们得到一般商品的总产出为：
Z_(t)=zetaA_(t)Z_{t}=\zeta A_{t}
where zeta=(alpha//chi)^((alpha)/(1-alpha))\zeta=(\alpha / \chi)^{\frac{\alpha}{1-\alpha}}. 在哪里 zeta=(alpha//chi)^((alpha)/(1-alpha))\zeta=(\alpha / \chi)^{\frac{\alpha}{1-\alpha}} .
In equilibrium the probability of innovation will be the same in each sector: mu_(t)(i)=mu_(t)\mu_{t}(i)=\mu_{t} for all ii; therefore average productivity evolves according to: 在均衡状态下，每个部门的创新概率将相同： mu_(t)(i)=mu_(t)\mu_{t}(i)=\mu_{t} ，对于所有 ii ；因此，平均生产率随时间演变的规律为：
That is, the productivity parameter will equal bar(A)_(t)\bar{A}_{t} in the fraction mu_(t)\mu_{t} of sectors that innovated at t-1t-1, but will remain equal to A_(t-1)(i)A_{t-1}(i) in the 1-mu_(t)1-\mu_{t} sectors that did not innovate at t-1t-1, and since innovations are distributed randomly across sectors the average value of A_(t-1)(i)A_{t-1}(i) among non-innovating sectors will equal the economy-wide average A_(t-1)A_{t-1}. 也就是说，生产率参数在创新水平为 t-1t-1 的部门中占 mu_(t)\mu_{t} 的分数部分将等于 bar(A)_(t)\bar{A}_{t} ，但在未在 t-1t-1 进行创新的 1-mu_(t)1-\mu_{t} 个部门中，生产率参数将保持为 A_(t-1)(i)A_{t-1}(i) 。由于创新在各部门中随机分布，因此未进行创新的部门中 A_(t-1)(i)A_{t-1}(i) 的平均值将等于全经济平均值 A_(t-1)A_{t-1} 。
Define the country’s normalized productivity as: 定义该国的标准化生产率为：
Normalized productivity is an inverse measure of the country’s distance to the technological frontier, or its “technology gap”. It follows that the gap evolves according to: 标准化生产率是衡量一个国家与技术前沿距离的反向指标，即其“技术差距”。由此可推，该差距随以下因素演变：
Since the general sector is perfectly competitive, the wage rate w_(t)w_{t} will be the marginal product of labor in producing the general good: 由于一般部门处于完全竞争状态，工资率 w_(t)w_{t} 将等于生产一般商品的劳动边际产品：
The fact that w_(t)w_{t} is proportional to domestic productivity A_(t)A_{t} plays an important role in what follows. For as we shall see it implies that technology investment in a country that is credit-constrained will be strictly proportional to A_(t)A_{t}. w_(t)w_{t} 与国内生产率 A_(t)A_{t} 成正比这一事实对后续分析具有重要意义。因为正如我们将要看到的，这意味着在信贷受限的国家，技术投资将严格与 A_(t)A_{t} 成正比。
Value added in the general sector is wage income, whereas value added in the intermediate sectors is profit income. Per-capita GDP is the sum of value added in all sectors: 一般部门的增加值是工资收入，而中间部门的增加值是利润收入。人均国内生产总值是所有部门增加值的总和：
In each sector the R&D\mathrm{R} \& \mathrm{D} investment needed to innovate at any given rate mu_(t)\mu_{t} is governed by the cost function: 在每个行业中，以任何给定速度进行创新所需的投资 R&D\mathrm{R} \& \mathrm{D} 由成本函数决定：
where N_(t-1)N_{t-1} is the quantity of general good that must be invested. We multiply widetilde(n)\widetilde{n} by bar(A)_(t)\bar{A}_{t} to recognize the “fishing-out” effect; the further ahead the frontier moves the more difficult it is to innovate. This effect is crucial in what follows. 其中， N_(t-1)N_{t-1} 表示必须投资的一般商品数量。我们将 widetilde(n)\widetilde{n} 与 bar(A)_(t)\bar{A}_{t} 相乘以识别“技术边界推进”效应；技术边界越向前推进，创新难度越大。这一效应在后续分析中至关重要。
In our analysis below, we shall make extensive use of the inverse of the R&D cost function widetilde(n)\widetilde{n}. Namely, an intermediate producer who invests the amount n bar(A)_(t)n \bar{A}_{t} in R&D\mathrm{R} \& \mathrm{D} will innovate next period with probability: ^(15){ }^{15} 在以下分析中，我们将广泛使用研发成本函数的倒数 widetilde(n)\widetilde{n} 。具体而言，一个中间生产者在当前时期投资 n bar(A)_(t)n \bar{A}_{t} 于 R&D\mathrm{R} \& \mathrm{D} ，则其在下一个时期进行创新的概率为： ^(15){ }^{15}
eta < beta pi < eta+delta.\eta<\beta \pi<\eta+\delta .
This condition guarantees that the equilibrium probability mu_(t)\mu_{t} will always lie strictly between 0 and 1. 该条件保证平衡概率 mu_(t)\mu_{t} 始终严格位于 0 和 1 之间。
In equilibrium mu_(t)\mu_{t} will be chosen so as to maximize the expected net payoff: 在平衡状态下， mu_(t)\mu_{t} 将被选择为使预期净收益最大化的值：
in each sector, subject to credit constraints. 在每个行业，受信用约束的。
2.5 Equilibrium innovation under perfect credit markets 2.5 完美信用市场下的均衡创新
In this section we show that if innovators had unlimited access to outside finance all economies would converge to the same growth rate. The level of each country’s growth path might be different because of country-specific differences in parameters such as beta\beta and chi\chi, but their long-run growth rates would all be the same. 在本节中，我们证明了如果创新者能够无限获取外部资金，所有经济体最终将收敛到相同的增长率。各国的增长路径水平可能因国家特有的参数差异（如 beta\beta 和 chi\chi ）而不同，但其长期增长率将完全一致。
Suppose accordingly that each innovator can borrow (from other young people) unlimited quantities at the going rate r=beta^(-1)-1r=\beta^{-1}-1 subject to a binding commitment to repay. Then mu_(t)\mu_{t} will be chosen so as to maximize (6) with no constraint. This implies that mu_(t)=mu^(**)\mu_{t}=\mu^{*}, where: 假设每个创新者可以从其他年轻人那里以当前利率 r=beta^(-1)-1r=\beta^{-1}-1 借用无限数量的资源，并承诺偿还。然后， mu_(t)\mu_{t} 将被选择以使（6）在无约束条件下最大化。这意味着 mu_(t)=mu^(**)\mu_{t}=\mu^{*} ，其中：
which grows at the same rate gg as the technology frontier bar(A)_(t)\bar{A}_{t}, as claimed. 其增长速度与技术前沿相同，如所声称的。
2.6 Credit constraints 2.6 信用约束
Now suppose that credit markets are imperfect. Each entrepreneur at the end of period tt is a young person with access to the wage income w_(t)w_{t}. Thus to invest N_(t)N_{t} in an R&D project she must borrow N_(t)-w_(t)N_{t}-w_{t}. Assume that if she pays a cost cN_(t)c N_{t} she can defraud her creditors by hiding the proceeds in the event that the project is successful. This implies that in equilibrium the entrepreneur cannot borrow more than a finite multiple of her accumulated wealth ^(17)w_(t){ }^{17} w_{t}, as in Bernanke and Gertler (1989), and therefore she cannot invest more than: 现在假设信贷市场是不完美的。每个企业在时期 tt 结束时都是一个年轻的个体，拥有工资收入 w_(t)w_{t} 。因此，若要投资 N_(t)N_{t} 于研发项目，她必须借款 N_(t)-w_(t)N_{t}-w_{t} 。假设若她支付成本 cN_(t)c N_{t} ，可在项目成功时通过隐瞒收益欺诈债权人。这意味着在均衡状态下，企业家无法借入超过其积累财富 ^(17)w_(t){ }^{17} w_{t} 的有限倍数，与伯南克和格特勒（1989）的假设一致，因此她无法投资超过：
nuw_(t)\nu w_{t}
in innovation, where nu in[1,oo)\nu \in[1, \infty) depends positively on the hiding cost cc. 在创新领域， nu in[1,oo)\nu \in[1, \infty) 与隐藏成本 cc 呈正相关。
This credit constraint will be binding if the unconstrained optimal investment n^(**) bar(A)_(t+1)n^{*} \bar{A}_{t+1} is strictly greater than the innovator’s investment capacity nuw_(t)\nu w_{t}, or equivalently, after dividing through by bar(A)_(t+1)\bar{A}_{t+1}, if: 当无约束最优投资 n^(**) bar(A)_(t+1)n^{*} \bar{A}_{t+1} 严格大于创新者的投资能力 nuw_(t)\nu w_{t} 时，该信用约束将成立，或者等价地，在除以 bar(A)_(t+1)\bar{A}_{t+1} 后，如果：
We represent financial development by the cost parameter cc, or equivalently by the credit multiplier nu\nu (or by omega\omega ), on the grounds that a highly developed financial system protects creditors by making it hard to defraud them. 我们用成本参数 cc 来表示金融发展，或者等价地用信用乘数 nu\nu （或 omega\omega ）来表示，因为高度发达的金融体系通过使欺诈行为难以实施来保护债权人。
We see from (9) that: (i) for a given level of technological development a_(t)a_{t} of the country, domestic firms are more likely to be credit constrained if financial development omega\omega is lower; (ii) for a given level of financial development omega\omega firms are more likely to be credit constrained the further the country is behind the technological frontier (i.e., the smaller is a_(t)a_{t} ). This is the “disadvantage of backwardness” induced by the existence of credit constraints. ^(18){ }^{18} 我们从（9）中可以看出：（i）对于给定的技术发展水平 a_(t)a_{t} ，如果金融发展水平 omega\omega 较低，国内企业更容易受到信贷约束；(ii) 在给定的金融发展水平 omega\omega 下，企业越落后于技术前沿（即 a_(t)a_{t} 越小），越容易受到信贷约束。这是信贷约束存在所引发的“落后劣势”。 ^(18){ }^{18}
Thus firms in more advanced countries with 因此，在更发达国家中，那些拥有
will invest the unconstrained amount n^(**) bar(A)_(t+1)n^{*} \bar{A}_{t+1} in innovation and therefore will innovate with probability mu^(**)\mu^{*}, whereas firms in less advanced countries with 将投资未受限制的金额 n^(**) bar(A)_(t+1)n^{*} \bar{A}_{t+1} 用于创新，因此将以概率 mu^(**)\mu^{*} 进行创新，而处于较不发达国家的公司则
cannot invest more than nuw_(t)=a_(t)omega bar(A)_(t+1)\nu w_{t}=a_{t} \omega \bar{A}_{t+1} and therefore will innovate with probability 无法投资超过 nuw_(t)=a_(t)omega bar(A)_(t+1)\nu w_{t}=a_{t} \omega \bar{A}_{t+1} ，因此将以一定概率进行创新。
where the innovation technology widetilde(mu)\widetilde{\mu} is given by (5). ^(19){ }^{19} 其中，创新技术 widetilde(mu)\widetilde{\mu} 由式(5)给出。 ^(19){ }^{19}
In that case a_(t+1)a_{t+1} will be determined according to: 在这种情况下， a_(t+1)a_{t+1} 将根据以下规则确定：
As in other Schumpeterian models, we suppose that the growth rate gg of the global technology frontier is determined by the pace of innovations in the leading countries, none of which are assumed to be credit constrained. For simplicity assume there is just one leader, labeled country 1 . Then: 与其他熊彼特模型类似，我们假设全球技术前沿的增长率 gg 由领先国家创新的速度决定，且假设这些国家均不存在信贷约束。为简化起见，假设仅存在一个领先国家，标记为国家 1。则：
where sigma > 0\sigma>0 is a spillover coefficient and the subscript 1 indicates a parameter value in country 1 . 其中 sigma > 0\sigma>0 表示溢出系数，下标 1 表示参数在国家 1 中的取值。
3 Theoretical implications 3 理论意义
3.1 Three dynamic patterns 3.1 三种动态模式
In general, the country’s technology gap a_(t)a_{t} will evolve according to the unconstrained dynamical system (7) when a_(t) >= a_(omega)a_{t} \geq \underline{a}(\omega) and according to the constrained system (10) when a_(t) < a_(omega)a_{t}<\underline{a}(\omega). Thus: 总体而言，该国的技术差距 a_(t)a_{t} 将根据无约束动态系统（7）演变，当 a_(t) >= a_(omega)a_{t} \geq \underline{a}(\omega) ，并根据约束系统（10）演变，当 a_(t) < a_(omega)a_{t}<\underline{a}(\omega) 。因此：
Note that H_(1)H_{1} is a linear function with positive vertical intercept and a slope between 0 and 1. Also, ^(20)H_(2){ }^{20} H_{2} is an increasing concave function when a_(t) <= min{a_(omega),1}a_{t} \leq \min \{\underline{a}(\omega), 1\}, with H_(2)(0)=0H_{2}(0)=0 and: 请注意， H_(1)H_{1} 是一个具有正垂直截距和斜率在 0 到 1 之间的线性函数。此外，当 a_(t) <= min{a_(omega),1}a_{t} \leq \min \{\underline{a}(\omega), 1\} 时， ^(20)H_(2){ }^{20} H_{2} 是一个单调递增的凹函数，且满足以下条件：
Countries will fall into three groups, defined by the level of financial development omega\omega. The evolution of the technology gap is illustrated for each case in Figures 1∼31 \sim 3 below. 各国将根据金融发展水平分为三类 omega\omega 。图 1∼31 \sim 3 以下分别展示了每类国家技术差距演变的情况。
Convergence in growth rate, no marginal effect of financial development. 增长率趋同，金融发展无边际效应。
When financial development is sufficiently high that: 当金融发展达到足够高的水平时：
so that a^(**) >= a_(omega)a^{*} \geq \underline{a}(\omega), then as shown in Figure 1,a_(t)1, a_{t} will converge asymptotically to the unconstrained steady state a^(**) > 0a^{*}>0. Per-capita GDP will be given by equation (8) in the long run, which implies that the country will grow at the same rate gg as the global technology frontier in the long run. Increases in financial development will have no marginal effect on either the steady-state growth rate or the steady-state technology gap; these converge respectively to the values gg and a^(**)a^{*} which are independent of omega.^(21)\omega .^{21} 因此，当 a^(**) >= a_(omega)a^{*} \geq \underline{a}(\omega) 时，如图 1,a_(t)1, a_{t} 所示，系统将渐近收敛于无约束稳态 a^(**) > 0a^{*}>0 。人均 GDP 在长期内将由方程 (8) 给出，这意味着该国在长期内将以与全球技术前沿相同的速率 gg 增长。金融发展水平的提高对稳态增长率或稳态技术差距均无边际影响；这两者分别收敛至值 gg 和 a^(**)a^{*} ，且与 omega.^(21)\omega .^{21} 无关。
Figure 1: A country with the highest level of financial development 图 1：金融发展水平最高的国家
2. Convergence in growth rate with a level-effect of financial development. 2. 增长率的趋同与金融发展水平效应。
When the level of financial development is neither too high nor too low, so that: ^(22){ }^{22} 当金融发展水平既不高也不低时，即： ^(22){ }^{22}
(eta g)/(1+g) <= omega < (n^(**))/(a^(**))\frac{\eta g}{1+g} \leq \omega<\frac{n^{*}}{a^{*}}
then H(a^(**)) < H_(1)(a^(**))H\left(a^{*}\right)<H_{1}\left(a^{*}\right), so a_(t)a_{t} cannot converge to the unconstrained steady state a^(**)a^{*}. From (11) we have: 然后 H(a^(**)) < H_(1)(a^(**))H\left(a^{*}\right)<H_{1}\left(a^{*}\right) ，因此 a_(t)a_{t} 无法收敛到无约束稳态 a^(**)a^{*} 。由 (11) 可得：
Therefore, as shown in Figure 2, a_(t)a_{t} will converge to a limit widehat(a)\widehat{a} that is strictly positive (except in the borderline case where (eta g)/(1+g)=omega\frac{\eta g}{1+g}=\omega and widehat(a)=0\widehat{a}=0 ) but less than a^(**)a^{*}. In the long run, per-capita GDP will be: 因此，如图 2 所示， a_(t)a_{t} 将收敛到一个严格正的极限值 widehat(a)\widehat{a} （除边界情况外，即 (eta g)/(1+g)=omega\frac{\eta g}{1+g}=\omega 和 widehat(a)=0\widehat{a}=0 ），但小于 a^(**)a^{*} 。从长远来看，人均 GDP 将为：
This country will also grow at the rate gg in the long run, because widehat(Y)_(t)\widehat{Y}_{t} is strictly proportional to bar(A)_(t)\bar{A}_{t}, as is Y_(t)^(**)Y_{t}^{*}. Increases in financial development will have no marginal effect on the steady-state growth rate but they will have a positive marginal effect on the 该国长期增长率也将保持在 gg 的水平，因为 widehat(Y)_(t)\widehat{Y}_{t} 与 bar(A)_(t)\bar{A}_{t} 严格成正比，而 Y_(t)^(**)Y_{t}^{*} 亦然。金融发展水平的提升不会对稳态增长率产生边际影响，但会对
steady-state technology gap widehat(a)\widehat{a}, because they shift the curve H_(2)(a_(t))H_{2}\left(a_{t}\right) up in Figure 2.^(23)2 .{ }^{23} According to (12) increases in financial development will also have a positive effect on the country’s steady-state per-capita GDP because of both the direct effect on widetilde(mu)\widetilde{\mu} and the indirect effect working through widehat(a)\widehat{a}. 稳态技术差距 widehat(a)\widehat{a} ，因为它们使曲线 H_(2)(a_(t))H_{2}\left(a_{t}\right) 在图 2.^(23)2 .{ }^{23} 中向上移动。根据（12），金融发展水平的提高也将对该国的稳态人均 GDP 产生积极影响，这是由于其对 widetilde(mu)\widetilde{\mu} 的直接影响以及通过 widehat(a)\widehat{a} 发挥的间接影响。
Figure 2: A medium level of financial development 图 2：中等水平的金融发展
3. Divergence in growth rate, with a growth-effect of financial development. 3. 增长率的分化，伴随金融发展带来的增长效应。
When the level of financial development is sufficiently low that: 当金融发展水平足够低时，以下情况将出现：
omega < (eta g)/(1+g)\omega<\frac{\eta g}{1+g}
then H(a^(**)) < H_(1)(a^(**))H\left(a^{*}\right)<H_{1}\left(a^{*}\right) and H^(')(0) < 1H^{\prime}(0)<1, so a_(t)a_{t} will converge to zero, as shown in Figure 3. The following argument shows that in this case the rate of productivity growth, defined as G_(t)=A_(t+1)//A_(t)-1G_{t}=A_{t+1} / A_{t}-1, will approach a limiting value that is strictly between 0 and gg. By l’Hôpital’s rule: 然后 H(a^(**)) < H_(1)(a^(**))H\left(a^{*}\right)<H_{1}\left(a^{*}\right) 和 H^(')(0) < 1H^{\prime}(0)<1 ，因此 a_(t)a_{t} 将收敛到零，如图 3 所示。以下论证表明，在这种情况下，生产率增长率，定义为 G_(t)=A_(t+1)//A_(t)-1G_{t}=A_{t+1} / A_{t}-1 ，将趋于一个严格介于 0 和 gg 之间的极限值。根据洛必达法则：
Thus the steady-state growth rate will be strictly less than the frontier growth rate gg and will be strictly increasing in the country’s level of financial development. ^(24){ }^{24} 因此，稳态增长率将严格小于边际增长率 gg ，并且将随着该国金融发展水平的提高而严格增加。 ^(24){ }^{24}
Figure 3: The lowest level of financial development 图 3：金融发展的最低水平
3.2 Summary 3.2 摘要
In summary, the two main implications of our theory are that: 综上所述，我们理论的两个主要启示是：
the likelihood that a country will converge to the frontier growth rate increases with its level of financial development, and 一个国家收敛到前沿增长率的可能性随着其金融发展水平的提高而增加，并且
in a country that converges to the frontier growth rate, financial development has a positive but eventually vanishing effect, ceteris paribus, on the steady-state level of per-capita GDP relative to the frontier. 在一个趋近于前沿增长率的国家，金融发展对人均 GDP 相对于前沿的稳态水平具有正向但最终消逝的影响，其他条件不变的情况下。
4 Credit and convergence: Evidence 4 信用与融合：证据
In this section we confront our theoretical predictions with evidence. After describing our data, we test implications 1 and 2 above with a cross-country growth regression involving an interaction term between the log of initial GDP per capita and financial development. ^(25){ }^{25} This test provides strong evidence for our model and for the general proposition that whether or not a country converges to the frontier growth rate depends on its level of financial development. 在本节中，我们将理论预测与实证证据进行对比。在描述数据后，我们利用包含人均初始 GDP 与金融发展交互项的跨国增长回归分析，检验上述推论 1 和推论 2。 ^(25){ }^{25} 该检验为我们的模型以及“一国是否收敛于前沿增长率取决于其金融发展水平”这一一般命题提供了强有力的证据。
4.1 Data 4.1 数据
We do not have a direct empirical measure of the parameter nu\nu or omega\omega which our theory takes as an indicator of financial development. Instead we follow the usual practice of using a measure of financial intermediation to proxy for financial development. We analyze crosssectional data ^(26){ }^{26} on 71 countries over the period 1960-1995, taken from Levine, Loayza and Beck (2000) (LLB) who found a strongly positive and robust effect of financial intermediation on short-run growth in a regression with initial GDP on the right hand side. We follow LLB in using private credit as our preferred measure of financial development. This is the value of credits by financial intermediaries to the private sector, divided by GDP. It is LLB’s preferred measure because it excludes credit granted to the public sector and credit granted by the central bank and development banks. We also report results below using alternative measures. 我们没有直接的经验测量指标来衡量理论中作为金融发展指标的参数 nu\nu 或 omega\omega 。因此，我们遵循惯常做法，使用金融中介指标来代理金融发展水平。我们分析了 1960 年至 1995 年期间 71 个国家的截面数据，数据来源于 Levine、Loayza 和 Beck（2000 年）（LLB）。他们在回归分析中发现，金融中介在短期内对经济增长具有显著正向且稳健的影响，其中初始 GDP 作为因变量。我们遵循 LLB 的做法，将私人信贷作为金融发展的首选指标。该指标为金融中介机构向私人部门提供的信贷总额与 GDP 之比。LLB 选择该指标是因为其排除了公共部门获得的信贷以及中央银行和开发银行提供的信贷。我们还在下文报告了使用替代指标的分析结果。
Figures 4 and 5 show that the raw data are roughly consistent with implications 1 and 2. Figure 4 plots the average growth rate of per-capita GDP over the sample period against the average level of financial development. Except for the countries with the three highest growth rates, which are clearly above their steady-state values, the scatter diagram appears consistent with a positive effect of financial development on growth that vanishes at approximately Greece’s level of financial development ( 39%39 \% ), as predicted by the implication 1. Figure 5 plots the average log of per-capita GDP on the vertical axis. It appears consistent with a positive effect of financial development on the level of GDP which vanishes once financial development has reached approximately Canada’s level (61%), as predicted by implication 2. 图 4 和图 5 表明，原始数据与推论 1 和推论 2 大致一致。图 4 绘制了样本期间人均 GDP 平均增长率与金融发展平均水平的关系。除增长率最高的三个国家明显高于其稳态值外，散点图与金融发展对增长的正向影响在希腊的金融发展水平（ 39%39 \% ）左右消失的预测一致，这与推论 1 的预测相符。图 5 以人均 GDP 的对数值为纵轴。该图与金融发展对 GDP 水平的正向影响一致，且该影响在金融发展达到加拿大水平（61%）时消失，这与推论 2 的预测一致。
Figure 4: Financial development and long-run growth of per-capita GDP 图 4：金融发展与人均 GDP 的长期增长
Figure 5: Financial development and long-run average per-capita GDP 图 5：金融发展与长期人均国内生产总值平均值
These figures do not control for the effects of initial GDP or any other possible influences on a country’s growth path. Nor do they deal with the problem of possible endogeneity of financial development. For these we turn to the following regression results. 这些数据并未控制初始 GDP 或其他可能影响一国增长路径的因素。此外，它们也未解决金融发展可能存在的内生性问题。对于这些问题，我们将参考以下回归分析结果。
4.2 Growth regression with an interaction term 4.2 包含交互项的增长回归分析
Our theoretical model can be approximated by the following growth regression: ^(27){ }^{27} 我们的理论模型可以用以下增长回归方程近似表示： ^(27){ }^{27}
where gg denotes the average growth rate of per-capita GDP, FF the average level of financial development, yy the initial (1960) log of per-capita GDP, X_(i)X_{i} a set of other regressors and epsi_(i)\varepsilon_{i} a disturbance term with mean zero. Country 1 is the technology leader, which we take to be the United States. This is a standard growth regression except for the interaction term F_(i)*(y_(i)-y_(1))F_{i} \cdot\left(y_{i}-y_{1}\right). 其中， gg 表示人均 GDP 的平均增长率， FF 表示金融发展的平均水平， yy 表示人均 GDP 的初始（1960 年）对数值， X_(i)X_{i} 表示其他自变量， epsi_(i)\varepsilon_{i} 表示均值为零的扰动项。国家 1 是技术领先者，我们将其设为美国。这是一个标准的增长回归模型，唯一不同之处在于包含了交互项 F_(i)*(y_(i)-y_(1))F_{i} \cdot\left(y_{i}-y_{1}\right) 。
Define widehat(y)_(i)-=y_(i)-y_(1)\widehat{y}_{i} \equiv y_{i}-y_{1}, country ii 's initial relative per-capita GDP. Under the assumption that beta_(y)+beta_(fy)F_(i)!=0\beta_{y}+\beta_{f y} F_{i} \neq 0 we can rewrite (13) as: 定义 widehat(y)_(i)-=y_(i)-y_(1)\widehat{y}_{i} \equiv y_{i}-y_{1} 为国家 ii 的初始相对人均 GDP。在假设 beta_(y)+beta_(fy)F_(i)!=0\beta_{y}+\beta_{f y} F_{i} \neq 0 的条件下，我们可以将 (13) 重写为：
where the steady-state value widehat(y)_(i)^(**)\widehat{y}_{i}^{*} is defined by setting the RHS of (13) to zero: 其中稳态值 widehat(y)_(i)^(**)\widehat{y}_{i}^{*} 通过将式 (13) 的右边设为零来定义：
that depends on financial development. 这取决于金融发展。
A country can converge to the frontier growth rate if and only if the growth rate of its relative per-capita GDP depends negatively on the initial value widehat(y)_(i)\widehat{y}_{i}; that is if and only if the convergence parameter lambda_(i)\lambda_{i} is negative. Thus the likelihood of convergence will increase with financial development (implication 1 above) if and only if: 一个国家能够收敛到前沿增长率的充要条件是：其相对人均 GDP 的增长率与初始值 widehat(y)_(i)\widehat{y}_{i} 呈负相关；即收敛参数 lambda_(i)\lambda_{i} 为负值。因此，金融发展程度越高（如上文推论 1 所示），收敛的可能性就越大，但这一关系成立的充要条件是： beta_(fy) < 0\beta_{f y}<0
Since this implication constitutes the central proposition of our theoretical model, our main objective in estimating (13) will be to see whether or not the estimated interaction coefficient is indeed significantly negative. 由于这一推论构成了我们理论模型的核心命题，我们在估计（13）时的主要目标是判断估计得到的交互作用系数是否确实显著为负。
It follows from (14) that the long-run effect of financial development on relative output is: 由（14）式可得，金融发展对相对产出的长期影响为：
Assume that all countries lag the United States in steady state: widehat(y)_(i)^(**) <= 0\widehat{y}_{i}^{*} \leq 0. Then if (16) holds, financial development will have a positive long-run effect on per-capita GDP of each (non-leader) country that converges if and only if beta_(f) >= 0\beta_{f} \geq 0. For then the numerator of (17) will be positive. Moreover, this effect will eventually vanish (when FF reaches the leader’s level) if and only if the direct effect is equal to zero: 假设所有国家在稳态时都落后于美国： widehat(y)_(i)^(**) <= 0\widehat{y}_{i}^{*} \leq 0 。那么，如果（16）成立，金融发展将对每个（非领先）国家的每人平均 GDP 产生长期正向影响，且这种影响仅在 beta_(f) >= 0\beta_{f} \geq 0 成立时才收敛。因为此时（17）的分子将为正值。此外，这种效应最终将消失（当 FF 达到领先国家水平时），当且仅当直接效应等于零：
beta_(f)=0\beta_{f}=0
So if we were to find that (18) held in addition to our main prediction (16), this would corroborate implication 2. If instead we were to find that beta_(f) > 0\beta_{f}>0 then the estimated effect of financial development on widehat(y)_(i)^(**)\widehat{y}_{i}^{*} would never vanish, even for the leader, whereas beta_(f) < 0\beta_{f}<0 would imply a negative effect for countries close to the leader. 因此，如果我们发现（18）在我们的主要预测（16）成立的情况下也成立，这将支持推论 2。如果相反，我们发现 beta_(f) > 0\beta_{f}>0 ，那么金融发展对 widehat(y)_(i)^(**)\widehat{y}_{i}^{*} 的估计效应将永远不会消失，即使对于领先国家也是如此，而 beta_(f) < 0\beta_{f}<0 则意味着对于接近领先国家的国家，金融发展将产生负面效应。
4.2.1 Regression results 4.2.1 回归分析结果
The financial development variable FF in (13) may be endogenous because of feedback from growth to finance, or because of the common effects of omitted variables on both growth and finance. Moreover, endogeneity of FF is likely to entail endogeneity of the interaction variable F*(y-y_(1))F \cdot\left(y-y_{1}\right). To deal with this problem we estimated (13) using instrumental variables, instrumenting for FF and F*(y-y_(1))F \cdot\left(y-y_{1}\right) using legal origins ( LL ) and legal origins interacted with initial relative output (L*(y-y_(1)))\left(L \cdot\left(y-y_{1}\right)\right). 金融发展变量 FF 在（13）中可能具有内生性，这是由于增长对金融的反馈效应，或是由于被省略变量对增长和金融的共同影响。此外， FF 的内生性很可能导致交互变量 F*(y-y_(1))F \cdot\left(y-y_{1}\right) 的内生性。为解决这一问题，我们采用工具变量法估计了(13)，以法律起源（ LL ）作为工具变量，同时将法律起源与初始相对产出 (L*(y-y_(1)))\left(L \cdot\left(y-y_{1}\right)\right) 的交互项作为工具变量。
Legal origins is a set of three zero-one variables, used first in the economics literature by La Porta et al. (1997,1998)(1997,1998) and further extended to all 71 countries by LLB, indicating whether the country’s legal system is based on French, English or German traditions (the omitted case is Scandinavian). La Porta et al. argue that the main effect of LL is on the rights of investors and creditors. LLB conclude that LL constitutes a good set of instruments for financial development because they were established too long ago to suffer from reverse causation, they have a strong effect on financial development and their main effects on growth should be through financial channels. We used the interacted variables L*(y-y_(1))L \cdot\left(y-y_{1}\right) as additional instruments to model the interaction term F*(y-y_(1))F \cdot\left(y-y_{1}\right), because using LL without L*(y-y_(1))L \cdot\left(y-y_{1}\right) resulted in too much collinearity between the fitted values of FF and F*(y-y_(1))F \cdot\left(y-y_{1}\right) 法律起源是一组三个二进制变量，首次由 La Porta 等人 (1997,1998)(1997,1998) 在经济学文献中提出，随后由 LLB 扩展至所有 71 个国家，用于表明该国法律体系是否基于法国、英国或德国传统（未提及的案例为斯堪的纳维亚国家）。拉波塔等人认为， LL 的主要影响在于投资者和债权人的权利。LLB 得出结论， LL 构成了一组良好的金融发展工具，因为它们建立时间过久，不会受到反向因果关系的影响，对金融发展有显著影响，且其对增长的主要影响应通过金融渠道实现。我们使用交互变量 L*(y-y_(1))L \cdot\left(y-y_{1}\right) 作为额外工具变量来建模交互项 F*(y-y_(1))F \cdot\left(y-y_{1}\right) ，因为仅使用 LL 而不使用 L*(y-y_(1))L \cdot\left(y-y_{1}\right) 会导致 FF 和 F*(y-y_(1))F \cdot\left(y-y_{1}\right) 的拟合值之间存在过多共线性。
to identify the crucial coefficients beta_(f)\beta_{f} and beta_(fy)\beta_{f y}. We defer further discussion of the instruments until the next section. 确定关键系数 beta_(f)\beta_{f} 和 beta_(fy)\beta_{f y} 。我们将在下一节中进一步讨论这些工具。
Our main results are presented in the first column of Table 1, which reports the slopecoefficient estimates for the case where there are no other regressors XX. These results show that financial development interacted with initial relative output has a significantly negative effect ( beta_(fy)=-0.061 < 0\beta_{f y}=-0.061<0 ), bearing out the main implication of the theory to the effect that convergence depends positively on financial development. They also fail to reject the hypothesis that the direct effect of financial development beta_(f)\beta_{f} is zero, thus bearing out our theoretical implication of a positive but vanishing steady-state effect. ^(28){ }^{28} 我们的主要结果如表 1 第一列所示，该列报告了在不考虑其他自变量的情况下对斜率系数的估计值 XX 。这些结果表明，金融发展与初始相对产出之间的相互作用具有显著的负面影响（ beta_(fy)=-0.061 < 0\beta_{f y}=-0.061<0 ），这与理论的主要推论一致，即收敛程度正向依赖于金融发展。此外，这些结果也未能拒绝金融发展直接效应为零的假设，从而验证了我们理论中关于正向但稳态效应趋于消失的推论。
TABLE 1 HERE 表 1 见此处
These findings are significant quantitatively as well as statistically, because they imply that countries will indeed belong to different convergence clubs. Specifically, a country can converge to the frontier growth rate if and only if its convergence parameter (15) is negative; that is, if and only if its level of private credit exceeds the critical value: 这些发现具有重要的定量和统计意义，因为它们表明各国确实会属于不同的收敛俱乐部。具体来说，一个国家只有在它的收敛参数（15）为负值时，才能收敛到前沿增长率；也就是说，只有当它的私人信贷水平超过临界值时，才能收敛到前沿增长率。
which according to our estimates equals 25 percent. Just over half the countries in our sample ( 37 of 71 ) exceed this critical value. Figure 6 shows the estimated convergence parameter as a function of private credit, over the observed range of FF, with 2 -standarddeviation bands. As indicated in Table 2, the estimated parameter is at least two standard deviations below zero for 30 countries, the group most likely to converge in growth rate, and two standard deviations above zero for 7 countries, those most likely to diverge. The average estimated convergence parameter in the sample is -0.82 , which implies an annual convergence rate of almost 5%5 \%. 根据我们的估算，这一比例相当于 25%。在我们的样本中，超过一半的国家（71 个中的 37 个）超过了这一临界值。图 6 显示了估计的收敛参数与私人信贷的关系，在观察到的范围 FF 内，并以 2 个标准差的带宽表示。如表 2 所示，估计参数在 30 个国家中至少低于零两个标准差，这些国家最有可能实现增长率收敛；在 7 个国家中高于零两个标准差，这些国家最有可能出现增长率分化。样本中估计的平均收敛参数为-0.82，这意味着年收敛率接近 5%5 \% 。
Figure 6: Estimated convergence parameter over the observed rate of private credit. Positive values imply nonconvergence. 图 6：私营部门信贷增长率的估计收敛参数。正值表示未收敛。
TABLE 2 HERE 表 2 见此处
Another measure of the economic significance of our parameter estimates is the size of the implied effect of financial development on a converging country’s steady-state relative output. As predicted by implication 2 of our theoretical analysis, this effect is a diminishing function of financial development. Specifically, a one-standard-deviation increase in private credit ( 28 percentage points) would raise steady-state GDP by 21 percent in Belgium, the (estimated) converging country with the smallest level of private credit. But the effect would be less than 8 percent in every other converging country, and less than 1 percent for each of the 30 “most likely to converge” countries. 衡量我们参数估计的经济意义的另一项指标是金融发展对一个收敛国家稳态相对产出的隐含影响大小。正如我们理论分析的推论 2 所预测的，这一影响是金融发展的递减函数。具体而言，私人信贷增加一个标准差（28 个百分点）将在比利时（估计的私人信贷水平最低的收敛国家）使稳态 GDP 增长 21%。但在其他所有收敛国家中，该效应将低于 8%，而在 30 个“最有可能收敛”的国家中，每个国家的效应均低于 1%。
The next two columns of Table 1 show that our results are robust to the inclusion of other regressors. Specifically, column 2 uses LLB’s policy conditioning set, which includes average years of schooling in 1960, government size, inflation, the black market exchangerate premium and openness to trade. Column 3 uses their full conditioning set, which includes the policy conditioning set plus measures of political stability and ethnic diversity. As these two columns indicate, the sign, size and significance of the crucial coefficients beta_(f)\beta_{f} and beta_(fy)\beta_{f y} remain virtually unchanged across alternative conditioning sets. 表 1 的下一两列表明，我们的结果对其他自变量的纳入具有稳健性。具体而言，第 2 列使用了 LLB 的政策控制变量集，其中包括 1960 年的平均受教育年限、政府规模、通货膨胀率、黑市汇率溢价以及贸易开放度。第三列使用了他们的完整条件集，该集在政策条件集的基础上增加了政治稳定性和民族多样性的衡量指标。如这两列所示，关键系数 beta_(f)\beta_{f} 和 beta_(fy)\beta_{f y} 的符号、大小和显著性在不同条件集下几乎保持不变。
The remaining columns report the results when three alternative measures of financial development are used. The first is liquid liabilities, which is currency plus demand and interest-bearing liabilities of banks and non-bank financial intermediaries, divided by GDP. This is a commonly used measure of financial development, although it includes liabilities backed by credits to the public sector and may involve double counting. The second alternative measure is bank assets, the ratio of all credits by banks (but not other financial intermediaries) to GDP. The third is commercial-central bank, the ratio of commercial bank assets to the sum of commercial plus central bank assets, which has been used by others although it is not so much a measure of financial development as a measure of what fraction of 剩余的列报告了使用三种替代性金融发展指标时所得的结果。第一种是流动性负债，即货币加上银行和非银行金融机构的活期存款和利息负债，除以国内生产总值（GDP）。这是金融发展的一种常用指标，尽管它包括由公共部门信贷支持的负债，可能涉及重复计算。第二种替代指标是银行资产，即银行（不包括其他金融中介机构）发放的所有贷款与 GDP 之比。第三种是商业银行与中央银行之比，即商业银行资产与商业银行资产与中央银行资产之和之比，尽管该指标并非主要用于衡量金融发展，但仍被其他研究者采用。
credit is issued by private intermediaries. Our main results ( beta_(fy) < 0\beta_{f y}<0 and beta_(f)=0\beta_{f}=0 ) are robust to all three alternative measures, although in the case of commercial-central bank (our least preferred measure ex ante) the coefficient estimates all lose their statistical significance. As in the case of private credit, in all three cases the sign, size and significance of the crucial coefficients beta_(f)\beta_{f} and beta_(fy)\beta_{f y} remain virtually unchanged across alternative conditioning sets. ^(29){ }^{29} 信贷由私人中介机构发放。我们的主要结果（ beta_(fy) < 0\beta_{f y}<0 和 beta_(f)=0\beta_{f}=0 ）在所有三种替代测量方法下均具有稳健性，尽管在商业银行与中央银行（我们事前最不倾向的测量方法）的情况下，系数估计值均失去了统计显著性。与私人信贷的情况类似，在所有三种情况下，关键系数 beta_(f)\beta_{f} 和 beta_(fy)\beta_{f y} 的符号、大小和显著性在不同条件集下几乎保持不变。 ^(29){ }^{29}
We checked the robustness of our results against outliers by removing all countries with a residual more than three standard deviations from zero and then re-estimating. We also did this using two standard deviations instead of three. We did this for each of the first 9 cases in Table 1. The coefficient beta_(fy)\beta_{f y} never changed sign and its statistical significance was always even larger than reported in Table 1, while beta_(f)\beta_{f} was never significantly different from zero. Thus it seems that the results reported in Table 1 are not driven by outliers. 我们通过移除所有残差超过零三个标准差的国家，然后重新估计，来检验结果对异常值的稳健性。我们还使用两个标准差代替三个标准差进行了同样的操作。我们对表 1 中的前 9 个案例分别进行了此操作。系数 beta_(fy)\beta_{f y} 的符号从未改变，且其统计显著性始终大于表 1 中报告的值，而 beta_(f)\beta_{f} 从未显著不同于零。因此，表 1 中报告的结果似乎不受异常值驱动。
4.2.2 Instruments 4.2.2 仪器
We tested the strength of our instruments with the usual F-tests of joint significance in the first-stage regressions of FF and F*(y-y_(1))F \cdot\left(y-y_{1}\right). The p-values reported in the first two rows of the lower panel of Table 1 indicate that the instruments passed this test at the 1%1 \% level in all three equations involving private credit, our preferred measure of financial development, in all equations involving bank assets and in all but one involving liquid liabilities. The instruments passed at the 10%10 \% significance level in all equations not involving commercialcentral bank. Because of our a priori doubts as to the suitability of the commercial-central bank measure, we believe that the other three measures are telling us the right message. 我们通过第一阶段回归中 FF 和 F*(y-y_(1))F \cdot\left(y-y_{1}\right) 的联合显著性 F 检验，测试了我们工具变量的强度。表 1 下表第一行和第二行的 p 值表明，在所有涉及私人信贷（我们首选的金融发展指标）的方程中，在所有涉及银行资产的方程中，以及在所有涉及流动负债的方程中（除一个外），工具变量均通过了该检验，显著性水平为 1%1 \% 。在所有不涉及商业银行-中央银行变量的方程中，工具变量均在 10%10 \% 显著性水平上通过检验。由于我们事先对商业银行-中央银行指标的适用性存有疑虑，我们认为其他三项指标传递了正确的信息。
These results confirm and extend similar findings by LLB. However, we have added to their analysis the three interacted instruments L*(y-y_(1))L \cdot\left(y-y_{1}\right), and it is important that they have additional explanatory power. Accordingly the third row of the lower panel of Table 1 reports the p-value of an F-test of the hypothesis that all three interacted instruments are insignificant in both first-stage regressions. The hypothesis is rejected at the 1%1 \% level in all equations except those using the suspect commercial-central bank measure. Thus our addition of the interacted instruments does not appear to have created a “many-instruments” problem. 这些结果证实并扩展了 LLB 的类似发现。然而，我们在他们的分析中添加了三个交互变量 L*(y-y_(1))L \cdot\left(y-y_{1}\right) ，并且这些交互变量具有额外的解释力这一点至关重要。因此，表 1 下表的第三行报告了 F 检验的 p 值，该检验假设三个交互变量在第一阶段回归中均不显著。除使用可疑商业银行-中央银行指标的方程外，该假设在所有方程中均在 1%1 \% 水平上被拒绝。因此，我们添加交互变量似乎并未引发“变量过多”问题。
From here on we omit the commercial-central bank measure from our analysis, on the grounds that for our purposes it is a priori inferior to the other measures and behaves empirically very differently than the others. 从这里开始，我们将在分析中省略商业银行与中央银行的措施，因为就我们的研究目的而言，该措施在理论上优于其他措施，且在实证上表现出与其他措施截然不同的特征。
To be valid our legal-origins instruments must not affect growth through any channel other than finance, since otherwise the effects we are attributing to finance might actually be effects of these non-financial channels. This restriction might appear questionable because for example different legal systems could result in different regulatory environments that affect barriers to entry as argued by Djankov et al. (2000). Therefore we tested the restriction using the standard Sargan test, whose null hypothesis is that the instruments are uncorrelated with the IV residuals. If our instruments were affecting growth through an omitted non-financial variable, then the Sargan test should reject the null. However, 要使我们的法律起源工具有效，它们必须不会通过金融渠道以外的任何其他渠道影响增长，否则我们归因于金融的影响实际上可能是这些非金融渠道的影响。这一限制可能看似有问题，因为例如不同的法律体系可能导致不同的监管环境，从而影响进入壁垒，正如 Djankov 等人（2000）所论证的。因此，我们使用标准的 Sargan 检验来测试这一限制，其原假设是工具变量与因变量的残差不相关。如果我们的工具变量通过一个被遗漏的非金融变量影响经济增长，那么 Sargan 检验应拒绝原假设。然而，
the large p-values reported in the fourth row of the lower panel of Table 1 show that the instruments pass the test in all cases. 表 1 下表第四行的较大 p 值表明，所有情况下工具变量均通过检验。
Again, these results confirm and extend the findings of LLB with respect to the 3 main instruments LL. We tested the specific validity of our interacted instruments L*(y-y_(1))L \cdot\left(y-y_{1}\right) with a C-test. The large p-values in fifth row of the lower panel of Table 1 indicate that the instruments pass this test in all cases. The large p-values in the sixth row indicated that we also cannot reject the exogeneity of initial relative income. 再次，这些结果证实并扩展了 LLB 关于 3 个主要工具的发现 LL 。我们通过 C 检验测试了交互工具 L*(y-y_(1))L \cdot\left(y-y_{1}\right) 的特定有效性。表 1 下表第五行的较大 p 值表明，所有情况下这些工具均通过了该检验。第六行中的大 p 值表明，我们也无法拒绝初始相对收入的外生性。
Another way to test for instrument validity is to include in the equation those variables that represent the alternative non-financial channels through which the instruments might affect growth. If these non-financial channels are at work then the new regressors should rob our financial variables of explanatory power. To some extent the results of Table 1 already constitute such a test, but the conditioning sets there do not include any interaction terms between the extra regressors and initial relative output. So they leave open the possibility that our main result, the strong negative effect on growth of the interaction between financial development and initial relative output, is coming from the explanatory power of the interacted instruments L*(y-y_(1))L \cdot\left(y-y_{1}\right) and that this explanatory power derives from correlation between the interacted instruments and some omitted interacted variable. 另一种验证工具有效性的方法是在方程中纳入那些代表工具可能影响增长的替代非金融渠道的变量。如果这些非金融渠道确实发挥作用，那么新的自变量应会削弱金融变量的解释力。表 1 的结果在一定程度上已构成此类检验，但其条件集未包含额外自变量与初始相对产出之间的交互项。因此，这留下了这样一种可能性：我们的主要结果——金融发展与初始相对产出之间相互作用对增长的强烈负面影响——可能源于相互作用变量 L*(y-y_(1))L \cdot\left(y-y_{1}\right) 的解释力，而这种解释力又源于相互作用变量与某些被遗漏的相互作用变量之间的相关性。
Table 5 below provides strong evidence that this theoretical possibility is not what is driving our results. As we explain in more detail below, Table 5 reports the estimates that result from including each of a long list of alternative regressors, including one that measures regulatory entry barriers, both directly and interacted with initial relative output. But in no case does the inclusion affect our main results, and in no case does the alternative regressor or its interaction have significant explanatory power, except for one marginally significant effect that appears to have the wrong sign. If our legal-origins instruments are working through some non-financial channel then it must be one that cannot be measured or has not been brought to our attention. 表 5 下文提供了强有力的证据，表明这种理论上的可能性并非驱动我们研究结果的因素。如我们在下文详细解释的，表 5 报告了在模型中纳入一系列替代解释变量（包括一个衡量监管准入壁垒的变量）后所得的估计结果，这些变量既被单独纳入，也与初始相对产出进行了交互分析。但在任何情况下，纳入这些变量均未影响我们的主要结果，且替代解释变量或其交互项均未显示出显著的解释力，唯一例外是一个边际显著的效果，且其符号似乎与预期相反。如果我们的法律起源工具变量是通过某种非金融渠道发挥作用，那么该渠道必定是无法测量或尚未被我们注意到的。
Our final check on instrument validity was to re-estimate Table 1 using alternative instruments. Specifically, we used the log of settler mortality, which Acemoglu, Johnson and Robinson (2001) have argued is a good instrument for modern institutions in formerly colonized countries. To model the interacted financial development variable we also used the log of settler mortality interacted with initial relative output as a second instrument. The results are displayed in Table 3 below. 我们对工具变量有效性的最终检验是使用替代工具变量重新估计表 1。具体而言，我们采用了 Acemoglu、Johnson 和 Robinson（2001）认为适用于前殖民国家现代制度的工具变量——定居者死亡率的对数。为了建模金融发展与其他变量的交互作用，我们还使用了定居者死亡率对数与初始相对产出交互作用的组合作为第二个工具变量。结果如表 3 所示。
TABLE 3 HERE 表 3 见此处
This re-estimation produces support for our main hypotheses ( beta_(fy) < 0\beta_{f y}<0 and beta_(f)=0\beta_{f}=0 ), because the estimated beta_(fy)\beta_{f y} is always negative and the estimate of beta_(f)\beta_{f} is always statistically indistinguishable from zero. The statistical significance of beta_(fy)\beta_{f y} is generally much lower than in Table 1, but we attribute this largely to the smaller sample size. Data on settler mortality are available only for 41 ex-colonies in our 71-country data set. 重新估算的结果支持了我们的主要假设（ beta_(fy) < 0\beta_{f y}<0 和 beta_(f)=0\beta_{f}=0 ），因为估算的 beta_(fy)\beta_{f y} 始终为负值，而 beta_(f)\beta_{f} 的估算值在统计上与零无显著差异。 beta_(fy)\beta_{f y} 的统计显著性通常远低于表 1，但我们认为这主要归因于样本规模较小。定居者死亡率数据仅在我们 71 个国家数据集中的 41 个前殖民地中有记录。
We prefer to work mainly with our legal-origins instruments rather than settler mortality because we do not want to throw 30 countries out of our data set and because in this data set the settler mortality instruments have relatively little explanatory power for the two financial development variables, as indicated by the large p-values of the first-stage F-tests 我们更倾向于主要使用法律起源变量而非定居者死亡率变量，因为我们不希望将 30 个国家排除在数据集之外，而且在该数据集中，定居者死亡率变量对两个金融发展变量的解释力相对较弱，这一点从第一阶段 F 检验的大 p 值中可以看出。
*With the usual caveat we thank Daron Acemoglu, Alberto Alesina, Jess Benhabib, Sean Campbell, Sebnem Kalemli-Ozcan, Ross Levine, Andrei Shleifer, David Weil, three anonymous referees and participants at the 2003 NBER Summer Institute, McMaster University, the 2003 Canadian Macroeconomics Study Group, the Federal Reserve Board, Brigham Young University, New York University and Harvard University for helpful comments. Cristina Santos and Stylianos Michalopoulos provided excellent research assistance. *如往常一样，我们感谢达龙·阿西莫格鲁（Daron Acemoglu）、阿尔贝托·阿莱西纳（Alberto Alesina）、杰斯·本哈比布（Jess Benhabib）、肖恩·坎贝尔（Sean Campbell）、塞尔宾·卡莱姆利-奥兹坎（Sebnem Kalemli-Ozcan），罗斯·莱文（Ross Levine）、安德烈·谢勒菲尔德（Andrei Shleifer）、大卫·韦尔（David Weil）、三位匿名审稿人以及 2003 年国家经济研究局夏季研讨会、麦克马斯特大学、2003 年加拿大宏观经济研究小组、联邦储备委员会、杨百翰大学、纽约大学和哈佛大学的与会者提出的宝贵意见。克里斯蒂娜·桑托斯（Cristina Santos）和斯蒂利亚诺斯·米哈洛普洛斯（Stylianos Michalopoulos）提供了卓越的研究协助。 ^(†){ }^{\dagger} Department of Economics, Harvard University, Cambridge, MA 02138. E-mail: p_aghion@harvard.edu ^(†){ }^{\dagger} 哈佛大学经济学系，马萨诸塞州剑桥市，邮编 02138。电子邮件：p_aghion@harvard.edu ^(‡){ }^{\ddagger} Department of Economics, Brown University, Providence, RI 02912. E-mail: ^(‡){ }^{\ddagger} 布朗大学经济学系，罗德岛普罗维登斯市，邮编 02912。电子邮件： peter_howitt@brown.edu ^("§ "){ }^{\text {§ }} División de Economía, CIDE, Carretera México Toluca 3655, 01210 México, D.F., México. E-mail: david.mayer@cide.edu ^("§ "){ }^{\text {§ }} 经济部，CIDE，墨西哥城-托卢卡公路 3655 号，01210 墨西哥城，墨西哥。电子邮件：david.mayer@cide.edu
^(1){ }^{1} The richest group was Western Europe in 1820 and the “European Offshoots” (Australia, Canada, New Zealand and the United States) in 1998. The poorest group was Africa in both years. ^(1){ }^{1} 1820 年最富裕的地区是西欧，而 1998 年最富裕的地区是“欧洲的分支”（澳大利亚、加拿大、新西兰和美国）。在两个年份中，最贫困的地区都是非洲。 ^(2){ }^{2} For example, Barro and Sala-i-Martin (1992), Mankiw, Romer and Weil (1992) and Evans (1996). ^(2){ }^{2} 例如，巴罗和萨拉-伊-马丁（1992）、曼昆、罗默和韦尔（1992）以及埃文斯（1996）。
^(3){ }^{3} See Aghion and Howitt (1998), Howitt (2000), Acemoglu, Aghion and Zilibotti (2002), and Howitt and Mayer-Foulkes (2002). The last of these papers implies three convergence groups, analogous to the three groups of the present paper, but the disadvantage of backwardness that prevents some countries from converging in that paper arises from low levels of human capital rather than from credit-market imperfections. ^(3){ }^{3} 参见阿吉翁和霍维特（1998）、霍维特（2000）、阿西莫格鲁、阿吉翁和齐利博蒂（2002），以及霍维特和梅耶-福尔克斯（2002）。这些论文中的最后一篇暗示了三个收敛群体，与本文中的三个群体类似，但该论文中阻碍某些国家收敛的落后性缺陷源于人力资本水平低下，而非信贷市场不完善。 ^(4){ }^{4} See Arrow (1969) and Evenson and Westphal (1995). ^(4){ }^{4} 参见 Arrow（1969）和 Evenson 与 Westphal（1995）。 ^(5){ }^{5} Cohen and Levinthal (1989) and Griffith, Redding and Van Reenen (2001) have also argued that R&D by the receiving country is a necessary input to technology transfer. ^(5){ }^{5} 科恩和莱文塔尔（1989）以及格里菲斯、雷丁和范·里嫩（2001）也认为，接受国的研究与开发是技术转移的必要投入。 ^(6){ }^{6} Grossman and Helpman (1991) and Barro and Sala-i-Martin (1997) also model technology transfer as taking place through a costly investment process, which they portray as imitation; but in these models technology transfer always leads to convergence in growth rates except in special cases studied by Grossman and Helpman where technology transfer is inactive in the long run. ^(6){ }^{6} 格罗斯曼和赫尔普曼（1991）以及巴罗和萨拉-伊-马丁（1997）也将技术转移建模为通过一个昂贵的投资过程实现，他们将这一过程描绘为模仿；但在这些模型中，技术转移总是导致增长率趋同，除非在格罗斯曼和赫尔普曼研究的特殊情况下，技术转移在长期内处于不活跃状态。 ^(7){ }^{7} A similar assumption has been shown elsewhere to be helpful in accounting for the fact that productivity growth rates have remained stable in OECD countries over the second half of the 20th Century despite a steady increase in R&D expenditures. See Jones (1995) and Howitt (1999). ^(7){ }^{7} 类似的假设在其他研究中也被证明有助于解释这样一个事实：尽管研发支出在 20 世纪后半叶持续增加，经合组织国家生产率增长率仍保持稳定。参见琼斯（1995）和霍维特（1999）。
^(8){ }^{8} See Banerjee (2003) for a comprehensive survey of this literature. ^(8){ }^{8} 有关此文献的全面综述，请参见班纳吉（Banerjee，2003）。 ^(9){ }^{9} In contrast, in our model countries face a productivity-adjusted cost of innovation which increases with its distance to the technological frontier. It is the interplay between credit constraints and this technological heterogeneity which generates the possibility of long-term divergence. ^(9){ }^{9} 相比之下，在我们的模型中，各国面临的创新成本（经生产率调整后）会随着其与技术前沿的距离增加而上升。正是信贷约束与技术异质性之间的相互作用，为长期分化提供了可能性。 ^(10){ }^{10} See the surveys by Levine (1997, 2003), and the book by Demirgüç-Kunt and Levine (2001). ^(10){ }^{10} 参见莱文（Levine，1997，2003）的调查研究，以及德米尔古奇-昆特（Demirgüç-Kunt）和莱文（Levine，2001）合著的专著。
^(11){ }^{11} Linear utility implies that people are indifferent between investing in any country, whether technologically or financially developed or not. We assume that all investment is locally financed, but if beta\beta were the same across all countries we could allow perfect capital mobility with no change in the analysis. Extending our analysis to the case of strictly concave utility would allow us to analyze the possibility and implications of capital flowing from less to more financially developed economies in accordance with Lucas’s (1990) oft-cited observation that capital flows from poor to rich countries rather than the reverse. ^(11){ }^{11} 线性效用假设人们对投资于任何国家（无论其技术或金融发展水平如何）都持无差异态度。我们假设所有投资均由本地资金融资，但若 beta\beta 在所有国家中相同，则可允许资本完全自由流动而分析结果不变。将分析扩展到严格凹效用函数的情况下，我们可以分析资本从金融发展程度较低的经济体流向金融发展程度较高的经济体的可能性及其影响，这与卢卡斯（1990）常被引用的观察结果一致，即资本流向富裕国家而非贫困国家。
^(12){ }^{12} In Aghion, Howitt and Mayer-Foulkes (2004) we extend our analysis and results to the more general case in which innovations do not result in an immediate jump to the frontier, so that: ^(12){ }^{12} 在阿吉翁、霍伊特和梅耶-福尔克斯（2004）的研究中，我们将分析和结果扩展到更一般的案例，即创新并不立即导致跃升至前沿，因此：
A_(t)(i)={[b bar(A)_(t)+(1-b)A_(t-1)," with probability "mu_(t)(i)],[A_(t-1)(i)," with probability "1-mu_(t)(i)]}A_{t}(i)=\left\{\begin{array}{cc}
b \bar{A}_{t}+(1-b) A_{t-1} & \text { with probability } \mu_{t}(i) \\
A_{t-1}(i) & \text { with probability } 1-\mu_{t}(i)
\end{array}\right\}
where 在哪里
A_(t)=int_(0)^(1)A_(t)(i)diA_{t}=\int_{0}^{1} A_{t}(i) d i
is the average domestic productivity at date tt and bb is a real number between 0 and 1 . 是日期 tt 时的国内平均生产率， bb 是介于 0 和 1 之间的一个实数。 ^(13){ }^{13} Thus imitation of a successful innovation is costless within a country, whereas we shall assume below that, because of the well documented fact that technologies work differently in different countries, moving a domestic sector up to the world technology frontier is costly and requires a positive R&D investment. ^(13){ }^{13} 因此，在一个国家内部模仿成功的创新是无成本的。然而，我们将在下面假设，由于技术在不同国家运作方式不同的这一已被充分证实的事实，将国内产业提升至世界技术前沿是需要成本的，并且需要正向的研发投资。 ^(14){ }^{14} This requires the further assumption that chi < alpha^(-alpha)\chi<\alpha^{-\alpha}, which we now make. ^(14){ }^{14} 这需要进一步假设 chi < alpha^(-alpha)\chi<\alpha^{-\alpha} ，我们现在接受这一假设。
^(16){ }^{16} The result that a^(**)a^{*} is strictly less than one reflects the fact that no country, even the most technologically advanced in terms of its average productivity, will ever be the world leader in all intermediate sectors simultaneously, because of the randomness of the innovation process. Thus the model is consistent with the evidence of Baily and Solow (2001) to the effect that different countries are technology leaders in different industries. ^(16){ }^{16} 结果表明， a^(**)a^{*} 严格小于 1，这反映了这样一个事实：即没有任何国家，即使是平均生产率最高的国家，也永远不可能在所有中间部门同时成为世界领导者，这是由于创新过程的随机性所致。因此，该模型与 Baily 和 Solow（2001）的证据一致，即不同国家在不同行业中是技术领导者。 ^(17){ }^{17} See Appendix A. ^(17){ }^{17} 参见附录 A。
^(18){ }^{18} Our model implies that, holding the credit multiplier nu\nu (or omega\omega ) constant, among those countries that are financially constrained external financing (equal to (nu-1)w_(t)(\nu-1) w_{t} ) is bigger in those that are closer to the technological frontier. However, the opposite is true among those countries that are not constrained, as the amount of external financing is then entirely determined by the gap between the R&D\mathrm{R} \& \mathrm{D} cost n^(**) bar(A)_(t)n^{*} \bar{A}_{t}, which is proportional to the frontier productivity bar(A)_(t)\bar{A}_{t}, and the amount of internal finance which is proportional to current domestic productivity. ^(18){ }^{18} 我们的模型表明，在信用乘数 nu\nu （或 omega\omega ）保持不变的情况下，对于那些面临金融约束的国家而言，外部融资（等于 (nu-1)w_(t)(\nu-1) w_{t} ）在那些更接近技术前沿的国家中更大。然而，对于不受约束的国家，情况则相反，因为外部融资的规模完全由 R&D\mathrm{R} \& \mathrm{D} 成本 n^(**) bar(A)_(t)n^{*} \bar{A}_{t} （与技术前沿生产率 bar(A)_(t)\bar{A}_{t} 成正比）与内部融资规模（与当前国内生产率成正比）之间的差距决定。 ^(19){ }^{19} This raises the question of why a constrained entrepreneur at t-1t-1 would not instead target a lower technology level B_(t) < bar(A)_(t)B_{t}<\bar{A}_{t}, which would be less expensive given the assumption that the cost of innovating at a given rate is proportional to the targeted technology level. In Aghion, Howitt and Mayer-Foulkes (2004) we answer the question by showing that this alternative would be dominated, from the entrepreneur’s point of view, by the strategy of always targeting the frontier. This relies on the fact that the innovation function widetilde(mu)(n)\widetilde{\mu}(n) has an elasticity less than one, which in turn follows from the fact that the innovation cost widetilde(n)(mu)\widetilde{n}(\mu) is strictly convex with widetilde(n)(0)=0\widetilde{n}(0)=0. ^(19){ }^{19} 这引发了一个问题：为什么在 t-1t-1 的约束下，企业家不会选择瞄准较低的技术水平 B_(t) < bar(A)_(t)B_{t}<\bar{A}_{t} ，毕竟在假设创新成本与目标技术水平成正比的情况下，较低技术水平的创新成本会更低。在阿吉翁、霍伊特和梅耶-福尔克斯（2004）的研究中，我们通过证明从企业家的角度来看，这种替代策略会被始终瞄准前沿的策略所支配，从而回答了这一问题。这依赖于创新函数 widetilde(mu)(n)\widetilde{\mu}(n) 的弹性小于 1，而这一性质又源于创新成本 widetilde(n)(mu)\widetilde{n}(\mu) 对 widetilde(n)(0)=0\widetilde{n}(0)=0 的严格凸性。
^(20){ }^{20} See footnote 15 above. ^(20){ }^{20} 参见上文第 15 注。 ^(21){ }^{21} That differences in the credit multiplier omega\omega within this high financial-development range do not affect the long-run technological gap results from the fact that the incentive constraint underlying omega\omega (see Appendix A) only places an upper bound on the amount borrowed by the entrepreneur. As soon as this constraint ceases to bind, then omega\omega becomes irrelevant in determining the dynamics of productivity. A different model of credit constraints, e.g. one that would rely on ex ante moral-hazard considerations and a continuous effort choice, might generate the possibility that differences in financial development always affect long-run productivity. ^(21){ }^{21} 信用乘数在这一高金融发展水平区间内的差异不会影响长期技术差距，这是因为支撑 omega\omega （见附录 A）的激励约束仅对企业家借贷金额设定了上限。一旦该约束不再约束，则 omega\omega 在决定生产率动态中便不再相关。一种不同的信贷约束模型，例如基于事前道德风险考虑和连续努力选择的模型，可能导致金融发展差异始终影响长期生产率。
^(22){ }^{22} Note that ^(22){ }^{22} 注意：
(eta g)/(1+g) < (n^(**))/(a^(**))\frac{\eta g}{1+g}<\frac{n^{*}}{a^{*}}
^(24){ }^{24} Aghion, Howitt and Mayer-Foulkes (2004) show that per-capita GDP grows at the same asymptotic rate as productivity in this case. ^(24){ }^{24} 阿吉翁、霍伊特和梅耶-福尔克斯（2004）指出，在这种情况下，人均国内生产总值的增长率与生产率的增长率相同，且两者都趋于一个渐近值。
^(25){ }^{25} We do not pursue a panel-data approach because we believe that financial development is imperfectly measured and persistent, which means that its growth effects are likely to be underestimated by a paneldata approach relative to a cross-section approach. (See Hauk and Wacziarg, 2004.) This may explain why Benhabib and Spiegel (1997, 2000) found no significant interaction between initial GDP and financial development using panel data on 92 countries from 1960-85. ^(25){ }^{25} 我们不采用面板数据方法，因为我们认为金融发展存在测量不完善和不持续的特性，这意味着其增长效应相对于截面数据方法而言，可能被面板数据方法低估。(见 Hauk 和 Wacziarg，2004。)这可能解释了为什么 Benhabib 和 Spiegel（1997，2000）在对 1960 年至 1985 年间 92 个国家的面板数据进行分析时，未发现初始 GDP 与金融发展之间存在显著的交互作用。
^(26){ }^{26} See Appendix B for detailed description and sources of data. ^(26){ }^{26} 详见附录 B 中的详细说明及数据来源。
^(27){ }^{27} See Appendix C of our (2004) working paper for the details of the approximation. ^(27){ }^{27} 有关近似方法的详细说明，请参见我们 2004 年工作论文的附录 C。
^(28){ }^{28} The wide confidence intervals for beta_(f)\beta_{f} are also consistent with a quantitatively large direct effect of financial development, although as pointed out below the point estimate of beta_(f)\beta_{f} indicates that for most converging countries the effect will be quantitatively quite small. ^(28){ }^{28}beta_(f)\beta_{f} 的宽泛置信区间也与金融发展具有显著的直接效应一致，尽管如下面所指出的， beta_(f)\beta_{f} 的点估计表明，对于大多数收敛国家而言，该效应在量上将相当微小。
^(29){ }^{29} Although our theory does not rule out non-financial determinants of steady-state output and growth, the fact that our estimated effects of financial development are independent of other conditioning variables suggests that we can safely treat the influence of those other determinants as part of the error term in the equations with empty conditioning sets. ^(29){ }^{29} 尽管我们的理论并未排除非金融因素对稳态产出和增长的影响，但金融发展对稳态产出和增长的估计效应与其他控制变量无关这一事实表明，我们可以安全地将这些其他决定因素的影响视为方程中控制变量为空时的误差项。
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