Models of Exogenous Growth

The marginalist or “neoclassical” school of economic thought seeks to explain income distribution in a symmetrical way via the relative scarcities of the factors of production, labour, “capital”, and land.

The idea of an economic system growing exclusively because some exogenous factors make it grow has variously been put forward in the history of economic thought as a standard of comparison. For example, in chapter 5 of book V of his Principles, first published in 1890, Alfred Marshall (1890 [1977]: 305) introduced the “famous fiction of the ‘Stationary state’... to contrast the results which would be found there with those in the modern world”. By relaxing one after another of the rigid assumptions defining the stationary state, Marshall sought to get gradually closer to the “actual conditions of life”. The first relaxation concerned the premise of a constant (working) population:

The Stationary state has just been taken to be one in which population is stationary. But nearly all its distinctive features may be exhibited in a place where population and wealth are both growing, provided they are growing at about the same rate, and there is no scarcity of land: and provided also the methods of production and the conditions of trade change but little; and above all, where the character of man himself is a constant quantity. For in such a state by far the most important conditions of production and consumption, of exchange and distribution will remain of the same quality, and in the same general relations to one another, though they are all increasing in volume. (Marshall (1890 [1977]: 306)

The resulting economic system grows at a constant rate which equals the exogenous rate of growth of population.

We encounter essentially the same idea in Gustav Cassel’s (1918 [1932]) Theory of Social Economy. The model of exogenous growth delineated by Cassel can be consid­ered the proximate starting point of the development of neoclassical growth theory. In chapter 4 of book I of the treatise, Cassel presented two models, one of a stationary economy, the other of an economy growing along a steady-state path.

In his first model Cassel assumed that there are z (primary) factors of production. The quantities of these resources and thus the amounts of services provided by them are taken to be in given supply. General equilibrium is characterized by the equality of supply and demand for each factor service and for each good produced and the equality of the price of a good and its cost of production. The resulting sets of equations consti­tute what is known as the “Walras-Cassel model” (Dorfman et al. 1958: 346). It satis­fies the then going criterion of completeness: there are as many equations as there are unknowns to be ascertained.

Cassel (1932: 152-3) then turned to the model of a uniformly progressing economy. Although described only verbally, he introduced the model in the following way:

We must now take into consideration the society which is progressing at a uniform rate. In it, the quantities of the factors of production which are available in each period... are subject to a uniform increase. We shall represent by [g] the fixed rate of this increase, and of the uniform progress of the society generally.

In Cassel’s view this generalization to the case of an economy growing at an exogenously given and constant rate does not cause substantial problems. The previously developed set of equations can easily be adapted appropriately, “so that the whole pricing problem is solved”. Cassel thus arrived at basically the same result as Marshall.

The neoclassical growth models of the 1950s and early 1960s differ from the growth version of the Walras-Cassel model in the first five of the following six important respects:

1. They are macro-models with a single produced good only which could be used both as a consumption good and as a capital good.

2. The number of primary factors of production is reduced to one, homogeneous labour (as in Solow 1956, 1970; Swan 1956), or two, homogeneous labour and homogeneous land (as in Swan 1956; Meade 1961).

3. The all-purpose good is produced by means of labour, capital, that is, the good itself, and possibly land.

4. There is a choice of technique, where technical alternatives are given by a macro­economic production function, which is homogenous of degree one with positive and decreasing marginal productivities with respect to each factor of production.

5. Savings are proportional to net income, that is, a “Keynesian” saving function is assumed.

6. Say’s law holds, that is planned savings are taken to be equal to planned investment at all times. There is no separate investment function.

Focusing attention on the models with a single primary factor (labour), in steady-state equilibrium:

where 5 is the (marginal and average) propensity to save, f(k) is the per unit of labour or per capita production function, k is the capital-labour ratio (where labour is measured in terms of efficiency units), and g is the steady-state growth rate of capital (and labour, and income, and so on). In steady-state equilibrium output expands exactly as the exogenous factors make it grow. Note that assuming 5 > 0 presupposes that the exogenous factors are growing at some positive rate. In these models the steady-state rate of growth is exog­enous. Outside steady-state equilibrium the rate of growth can be shown to depend also on the behavioural parameter of the system, that is, the propensity to save (and invest), but that parameter plays no role in determining the long-term rate of growth.

While these models are aptly described as models of exogenous growth, they can also be described as models of endogenous profitability. Since in the one-good framework adopted by the authors under consideration the rate of profits r equals the marginal productivity of capital,

the two equations are able to determine a relationship between the rate of profits and the steady-state rate of growth.