Endogenous Growth Theory

Recall that in the Harrod-Domar, Kaldor-Robinson, Solow-Swan and the Cass-Koopmans growth models, we have maintained, either explicitly or implicitly, that technical change is "exogenous". In the Schumpeter version, this was not true: we had "swarms" of inventors arising under particular conditions. The Smithian and Ricardian models also had technical change arising from profit-squeezes or, in the particular case of Smith, arising because of previous technical conditions.

Allyn A. Young (1928) had argued for the resurrection of the Smithian concept in terms of increasing returns to scale: division of labor induces growth which enables further division of labor and thus even faster growth. The idea that technological change is induced by previous economic conditions one may term "endogenous growth theory".

The need for a theory of technical change was there: according to some rather famous calculations from Solow (1957), 87.5% of growth in output in the United States between the years 1909 and 1949 could be ascribed to technological improvements alone. Hence, what is called the "Solow Residual" - the g(A) term in the growth equation given earlier, is enormous. One of the first reactions was to argue that by reducing much of that influence to pure capital improvements, capital-intensity seem to play a larger role than imagined in these 1957 calculations - Solow does go on to argue, for instance, that increased capital-intensive investment embodies new machinery and new ideas as well as increased learning for even further economic progress (Solow, 1960).

However, Nicholas Kaldor was really the first post-war theorist to consider endogenous technical change. In a series of papers, including a famous 1962 one with J.A. Mirrlees, Kaldor posited the existence of a "technical progress" function. that per capita income was indeed an increasing function of per capita investment. Thus "learning" was regarded as a function of the rate of increase in investment. However, Kaldor held that productivity increases had a concave nature (i.e. increases in labor productivity diminish as the rate of investment increases). This proposition, of course, falls short of Solow's insistence on constant returns. asdsadasdasda

K.J. Arrow (1962) took on the view that the level of the "learning" coefficient is a function of cumulative investment (i.e. past gross investment). Unlike Kaldor, Arrow sought to associate the learning function not with the rate of growth in investment but rather with the absolute level of knowledge already accumulated. Because Arrow claimed that new machines are improved and more productive versions of those in existence, investment does not only induce productivity growth of labor on existing capital (as Kaldor would have it), but it would also improve the productivity of labor upon all subsequent machines made in the economy.

The trick is to utilize the concept that while firms face constant returns, the industry or economy as a whole takes increasing returns to account. This can be easily formalized. Taking our old Cobb-Douglas production function, Y = AKaL1-a, there is constant returns to scale for all inputs together (since a + (1-a) = 1). Therefore, as noted in the Solow model, it might seem as if output per capital and consumption per capita does not grow unless the exogenous factor, A, grows too. To endogenize A, let us first establish the Cobb-Douglas production function for each individual firm:

Yi = AiKia Li1-a

where, one can note, the output of an individual firm is related with capital, labor as well as the "augmentation" of labor by Ai. Arrow (1962) assumed that Ai, the technical augmentation factor, might thus written look specific to the firm, but it is in fact related to total "knowledge" in the economy. This knowledge and experience, Arrow argued, is common to all firms: a free and public good (i.e. non-competitive consumption).

So the first question is how knowledge is accumulated. Arrow argued that it arises from past cumulative investment of all firms. Let us call this cumulative investment G. Thus, Arrow assumed that the technical augmentation factor is related to economy-wide aggregate capital in a process of "learning-by-doing". In other words, the experience of the particular firm is related to the stock of total capital in the economy, G, by the function:

Ai = Gz

thus, as the physical capital stock G accumulates, knowledge used by a particular firm also accumulates by a proportion z such that 0 < 1. Transferring to the production function for an individual firm, then:

Yi = GzKia Li1-a

where, note, only G does not have a subscript i (i.e. is not particular to the firm), it is a productive force external to the firms (i.e. a Marshallian externality) and assumed a free public good. This force is free and any firm employing it will not implicate on another firm's consumption: it is freely-available knowledge. Thus, we still maintain CRS at the firm level, but