Richard Melson

September 2006

Growth Theory


I - Classical Growth Theory
After Marx

II - Keynesian Growth

III - Neoclassical Growth
(A) Neoclassical Growth Theory
(1) The Solow-Swan Growth Model
(2) Solving the System
(3) Adjustment Processes: Solow vs. Harrod
(B) Neoclassical Growth: Empirical Implications
(1) Paradoxes of Growth
(2) The Convergence Hypotheses
(3) Poverty Traps
(C) Technical Progress

IV - Multisector Growth
(1) Introduction
(2) The Uzawa Two-Sector Growth Model
(3) Optimal Two-Sector Growth
(4) Heterogeneous Capital and Growth
(5) Turnpikes

V - Optimal Growth
(1) Introduction
(2) The Ramsey Exercise
(3) Golden Rule Growth
(4) Intertemporal Social Welfare
(5) The Cass-Koopmans Optimal Growth Model
(6) Conclusion

VI - Monetary Growth Theory

VII - Endogenous Growth

Adam Smith

When Adam Smith wrote his famous 1776 treatise, he called it An Inquiry into Nature and Causes of the Wealth of Nations. Some have taken this as indicating that he was concerned primarily with economic growth. In this way, Smith moved away from the Cantillon-Physiocratic system which concentrated on "natural equilibrium" of circular flows, and brought back into economics what had been the Mercantilists' pet concern.

Smith posited a supply-side driven model of growth. Succinctly we can lay out the story via the simplest of production functions:

Y = (L, K, T)

where Y is output, L is labor, K is capital and T is land, so output is related to labor and capital and land inputs. Consequently output growth (gY) was driven by population growth (gL), investment (gK) and land growth (gT) and increases in overall productivity (g). Succinctly:

gY = f(g, gK, gL, gT)

Population growth, Smith proposed in the traditional manner of the time, was endogenous: it depends on the sustenance available to accommodate the increasing workforce. Investment was also endogenous: determined by the rate of savings (mostly by capitalists); land growth was dependent on conquest of new lands (e.g. colonization) or technological improvements of fertility of old lands. Technological progress could also increase growth overall: Smith's famous thesis that the division of labor (specialization) improves growth was a fundamental argument. Smith also saw improvements in machinery and international trade as engines of growth as they facilitated further specialization.

Smith also believed that "division of labor is limited by the extent of the market" - thus positing an economies of scale argument. As division of labor increases output (increases "the extent of the market") it then induces the possibility of further division and labor and thus further growth. Thus, Smith argued, growth was self-reinforcing as it exhibited increasing returns to scale.

Finally, because savings of capitalists is what creates investment and hence growth, he saw income distribution as being one of the most important determinants of how fast (or slow) a nation would grow. However, savings is in part determined by the profits of stock: as the capital stock of a country increases, Smith posited, profit declines - not because of decreasing marginal productivity, but rather because the competition of capitalists for workers will bid wages up. So lowering the living standards of workers was another way to maintain or improve growth (although the counter-effect would be to reduce labor supply growth).

Despite increasing returns, Smith did not see growth as eternally rising: he posited a ceiling (and floor) in the form of the "stationary state" where population growth and capital accumulation were zero.

David Ricardo

Smith's model of growth remained the predominant model of Classical Growth. David Ricardo (1817) modified it by including diminishing returns to land. Output growth requires growth of factor inputs, but, unlike labor, land is "variable in quality and fixed in supply". This means that as growth proceeds, more land must be taken into cultivation, but land cannot be "created". This has two effects for growth: firstly, increasing landowner's rents over time (due to the limited supply of land) cut into the profits of capitalists from above; secondly, wage goods (from agriculture) will be rising in price over time and this then cuts into profits from below as workers require higher wages. This, then, introduces a quicker limit to growth than Smith allowed, but Ricardo also claimed (at first) that this decline can be happily checked by technological improvements in machinery (albeit, also with diminishing productivity) and the specialization brought by trade, although he also had stationary states.

However, in the third edition of his Principles, Ricardo modified his position on machinery. He claimed that, in fact, machinery displaces labor and that the labor "set free" might not be reabsorbed elsewhere (because capital is not simultaneously "set free") and thus merely create downward pressure on wages and thus lower labor income. In order to reabsorb this extra labor without this effect, then the rate of capital accumulation must be increased. But there is no obvious mechanism for this to happen -- particularly given the tendency described above for profits and thus savings to decline over time.

Ricardo's portrait is somewhat more pessimistic than Smith's. The ultimately dismal portrait, however, was painted by T.R. Malthus (1796) with his famous claim that population growth was not so easily checked and would quickly outstrip growth and cause increasing misery all around. John Stuart Mill improved little upon Ricardo, perhaps only to emphasize the need for control of population growth to put a brake on declining growth and his view of stationary states as wonderful things to achieve.

Karl Marx

Karl Marx (1867-1894) modified the Classical picture once again. For "modern" growth theory, Marx's achievement was critical: he not only provided, through his famous "reproduction" schema, perhaps the most rigorous formulation to date of a growth model, but he did so in a multi-sectoral context and provided, in the process, such critical ingredients as the concept of a "steady-state" growth equilibrium.

We analyze Marx's theory in more detail elsewhere. Here we are interested in merely sketching the "story" he sought to tell and how it differed from the earlier Classicals. Firstly, unlike Smith or Ricardo, Marx did not believe that labor supply was endogenous to the wage. As a result, Marx had wages determined not by necessity or "natural/cultural" factors but rather by bargaining between capitalists and workers and this process would be influenced by the amount of unemployed laborers in the economy (the "reserve army of labor", as he put it). Marx also saw profits and "raw instinct" as the determinants of savings and capital accumulation.

Thus, contrary to Smith, he saw a declining rate of profit doing nothing to stem capital accumulation and bring the stationary state about, but only as an inducement for capitalists to further reduce wages and thus increase the misery of labor.

Like the Classicals, Marx believed there was a declining rate of profit over the long-term. The long-run tendency for the rate of profit to decline is brought about not by competition increasing wages (as in Smith), nor by the diminishing marginal productivity of land (as in Ricardo), but rather by the "rising organic composition of capital".

Marx defined the "organic composition of capital" as the ratio of what he called constant capital to variable capital. It is important to realize that constant capital is not what we today call fixed capital, but rather circulating capital such as raw materials. Marx's "variable capital" is defines as advances to labor, i.e. total wage payments, or heuristically, v = wL (where w is wages and L is labor employed).

The rate of profits, Marx claimed, are defined as:

r = s/(v+c)

where r is the rate of profit, s is the surplus, and (v+c) are total advances (constant and variable). Surplus, s, is the amount of total output produced above total advances, or s = y - (v+c), where y is total output. It is important to note that for Marx only labor produces surplus value. This was to become a sore point of debate between the Neo-Ricardians and the Neo-Marxians in later years. Marx called the ratio of surplus to variable capital, s/v, the "exploitation rate" (surplus produced for every dollar spent on labor).

Marx referred to the ratio of constant to variable capital, c/v, as the organic composition of capital (which can be viewed as a sort of capital-labor ratio). Notice that dividing numerator and denominator of r by v we obtain:

r = (s/v)(v/(v+c)))

so the rate of profit can be expressed as a positive function of the exploitation rate (s/v) and a negative function of the organic composition of capital (c/v)).

Marx then argued that the exploitation rate (s/v) tended to be fixed, while the organic composition of capital (c/v) tended to rise over time, thus the rate of profit has a tendency to decline. Why? The basic logic can be thought of as follows. For simplicity, assume a static economy (no labor supply growth). As the surplus accrues to capitalists and, necessarily, capitalists invest that surplus into expanding production, then output will rise over time while the labor supply remains constant. Thus, the labor market gets gradually "tighter" and so wages will rise. Thus, v (= wL) rises and r falls.

But this decline in r is temporary. There are forces at work which will restore profit rate What are these forces? In Cantillon, Smith, et al., a rise in wages would induce population growth which would then loosen the labor markets and bring wages back down again. Marx does not accept this story. For Marx, wages are set by "bargaining" in the labor market. Thus, there is no "extra supply of labor" being encouraged by the higher wages. However, Marx argued, capitalists can boost their profit rate back up by introducing labor-saving machinery into production -- thereby releasing labor into unemployyment.

There are two effects of this. Firstly, notice that v declines because labor (L) is released. But, concurrently, the employment of machinery implies that constant capital, c, rises. Thus, the introduction in labor-saving machinery does not seem to change anything: the fall in v from less labor is counteracted by the rise in c, so it seems that c/v stays constant. This is where the second effect kicks in: the concurrent expansion in the unemployed -- the "reserve army of labor" -- will, by itself, influence the labor bargaining process and reduce wages down to subsistence. Thus v declines further. So, on the whole, the net effect of a labor-saving technology is to raise c/v, i.e. to reduce the rate of profit.

But notice that v declines further because labor is released. So, both the w and the L part of v = wL declines. But, concurrently, the employment of machinery implies that constant capital rises, thus c rises. Thus, the fall in L is counteracted by the rise in c, so that, on the whole, v

So, in sum, the organic composition of capital, c/v, falls. Profits, consequently, are increased.

Thus, the L part of v = wL declines and so r = s/(v+c) comes back up. There is a double effect in that, of course, the release of labor is not automatically absorbed by higher investment so that a "reserve army of labor" is created. In this manner, at the bargaining table, firms will be at an advantage relative to their employees, so that wages decline (or at least are prevented from rising further).

But this is merely a temporary respite. Profits will be reinvested, output will grow again, labor markets will tighten once more and the whole process will repeat itself. The problem is that the second time around, there is less labor to lay off. Recall, L was already reduced in the first round. Introducing more machinery reduces L further -- and, via several rounds, further and further -- until there is hardly any more L that can be released. When the system gets to the point that there are no more laborers to be fired, then there is nothing to bring s/v back up. The profit rate declines and firms will begin going bankrupt.

The bankruptcy of firms means a sudden release of even more labor and capital into the market, depressing prices tremendously. Firms which remain active will thus be able to buy the bankrupt smaller firms and thus acquire more labor and capital at very cheap rates -- indeed, cheaper than their proper "value". This increases the

The unemployed, thus, act as a "reserve army of labor" and bring wages back down to a manageable level.

However, the introduction of labor-saving capital and laying off of workers means that c rises while v falls, i.e. the organic composition of capital rises. It is easy to notice that a constant s/v and a rising c/v will necessarily reduce the profit rate (to see this, just notice that r can be rewritten as: r = (s/v)(v/(v+c))). Thus, there is a natural tendency for the rate of profit to fall.

One way to prevent this decline in r would be to increase the exploitation rate in proportion to which variable capital declines relative to constant capital. The manner of increasing the exploitation rate, Marx claimed, was up to the devilish imagination of the capitalist. Technological progress in the form of machinery or division of labor were not wholly beneficial ways of improving growth either. Marx took on Ricardo's idea that machinery is labor-saving and leads to a disproportional adjustment: the rate of release of labor does not accompany the rate of re-absorption of that labor, so that there tends to be permanent "technological" unemployment which can be used to bring down the wage. One does not even need to undertake it: technological improvement is also a way capitalists can increase their leverage over labor merely by threatening it with mechanization. Whereas Marx contended that division of labor was a way of generating the "alienation" of the working classes and thus tie them more dependently to the production process - thereby, again, reducing the bargaining position of labor.

The issue of trade, another possible check to the decline in profit rate, was seen by Marx as an inducement to produce on an even greater scale - thereby increasing the organic composition of capital further (and reducing profit quicker). The connection between trade with non-capitalist economies to prevent of the decline in profit rate was for later Marxians like Rosa Luxemburg (1913) to propose in their theories of imperialism.

However, despite all their efforts, Marx claimed that there were social limits to the extent to which capitalists could increase the exploitation rate, while no such thing limited the growing organic composition of capital. Consequently, Marx envisioned that greater and greater cut-throat competition among capitalists for that declining profit. Then a crisis occurs: large firms buy up the small firms at cheaper rates (i.e. below , and thus the total number of firms declines. This will boost the surplus value as firms can now purchase capital.

As capital becomes more concentrated,

increasing the tendency for capital to be concentrated in fewer and fewer hands,

combined with the greater misery of labor would culminate in ever greater "crises" which would destroy capitalism as a whole.

Marx had only temporary "stationary states",

punctuating the secular tendency to breakdown.

Growth Theory

September 15, 2006