〔QJE5601〕A contribution to the theory of economic growth


All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive.’ A “crucial” assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic. When the results of a theory seem to flow specifically from a special crucial assumption, then if the assumption is dubious, the results are suspect.
所有的理论都依赖于不那么真实的假设。正是这一点使之成为理论。理论成功的艺术,在于做一些不可避免的简化假设,使得最终结果不易受其影响。一项“关键”假设是结论所依赖的,合理且实际的关键假设是非常重要的。一个理论的结果似乎是从特殊的关键假设当中明确得出的,如果假设是含糊的,那么结果是可疑的。

I wish to argue that something like this is true of the Harrod-Domar model of economic growth. The characteristic and powerful conclusion of the Harrod-Domar line of thought is that even for the long run the economic system is at best balanced on a knife-edge of equilibrium growth. Were the magnitudes of the key parameters the savings ratio, the capital-output ratio, the rate of increase of the labor force – to slip ever so slightly from dead center, the consequence would be either growing unemployment or prolonged inflation.In Harrod’s terms the critical question of balance boils down to a comparison between the natural rate of growth which depends, in the absence of technological change, on the increase of the labor force, and the warranted rate of growth which depends on the saving and investing habits of households and firms.
我想说明的是,哈罗德-多马的经济增长模型就是如此。哈罗德-多马思想脉络的特点和强有力的结论是,即使在长期,经济体系至多是在均衡增长的刀刃上保持平衡。如果储蓄率、资本产出比率和劳动力增长率的数量从静止的中心轻微滑落,结果可能是失业率的上升或通货膨胀的延续。在哈罗德的概念中,平衡的关键问题可以归结为在没有技术变革的情况下依赖于劳动力增加的自然增长率,同依赖于家庭和企业储蓄和投资习惯的有保证的增长率之间的比较。

But this fundamental opposition of warranted and natural rates turns out in the end to flow from the crucial assumption that production takes place under conditions of fixed proportions. There is no possibility of substituting labor for capital in production. If this assumption is abandoned, the knife-edge notion of unstable balance seems to go with it. Indeed it is hardly surprising that such a gross rigidity in one part of the system should entail lack of flexibility in another.
不过,有保证的增长率和自然增长率之间的根本对立,最终源于生产发生在固定比例条件下这一关键假设。在生产中不可能用劳动代替资本。如果放弃这个假设,随之而来的就是平衡不稳定的刀刃状态。事实上,其中一部分的总量刚性会导致另一部分缺乏灵活性,在这个系统中是不足为奇的。

A remarkable characteristic of the Harrod-Domar model is that it consistently studies long-run problems with the usual short-run tools. One usually thinks of the long run as the domain of the neoclassical analysis, the land of the margin. Instead Harrod and Domar talk of the long run in terms of the multiplier, the accelerator, “the” capital coefficient. The bulk of this paper is devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions. Instead I suppose that the single composite commodity is produced by labor and capital under the standard neoclassical conditions. The adaptation of the system to an exogenously given rate of increase of the labor force is worked out in some detail, to see if the Harrod instability appears. The price-wage interest reactions play an important role in this neoclassical adjustment process, so they are analyzed too. Then some of the other rigid assumptions are relaxed slightly to see what qualitative changes result: neutral technological change is allowed, and an interest-elastic savings schedule. Finally, the consequences of certain more “Keynesian” relations and rigidities are briefly considered.
哈罗德-多马模型的一个显著特征是:它始终如一地用常用的短期工具研究长期问题。人们通常认为对长期的思考是新古典分析领域的边缘地带。相反,哈罗德和多马谈到了长期中的乘数、加速器和资本系数。本文的大部分内容都是关于长期增长的模型,它接受除了固定比例之外哈罗德-多马模型的所有假设。我转而假设,在标准的新古典主义条件下,单一的复合商品是由劳动和资本生产的。如果出现哈罗德意义上的不稳定,系统对外生给定劳动力增长率的适应性将在细微处给予解决。价格-工资-利息的反应在新古典的调整过程中起着重要的作用,所以也对它们进行了分析。而后稍微放松其他的严格假设,以观察会出现什么样的质变:允许的中性技术变化,以及具有利率弹性的储蓄计划。最后简要考察某些更加“凯恩斯主义”的关系和刚性产生后果。

假设经济中只生产一种产品,产出数量为$Y(t)$,则它同时也是经济中的实际收入。每一个时刻的产出被用于消费或是储蓄(以转化为投资),其中储蓄率为$s$,则储蓄水平为$sY(t)$,经济中的资本存量为$K(t)$,则净投资可以记作$dk/dt$或者$\dot{k}$,则有$\dot{k}=sY$,产出由资本和劳动等两种生产要素生产而成,生产可能性由生产函数$Y=F(K,L)$表示,此处假设生产函数是规模报酬不变的,排除了土地等稀缺资源(一旦引入土地,则生产函数就是规模报酬递减的了。

联立投资方程和生产函数,可得$\dot{k}=sF(K,L)$,此时,一个方程有两个未知数,此时对$L$既不诉诸劳动需求函数也不诉诸劳动供给函数,而是借鉴哈罗德的做法,引入$L(t)=L_0 e^{nt}$,其中$n$是外生人口增长率。在假定充分就业的前提下,可得$\dot{k}=sF(K,L_0 e^{nt})$,由此可得资本积累的时间路径。

一旦知道了资本和劳动的时间路径,就可以获得产出的时间路径。