1What role does education – and what is called human capital– play in the dynamics of growth and distribution in the capitalist economy? Given the advent of the so-called knowledge-based economy, this question has been receiving a fair amount of attention in the scholarly analysis and popular discussion. The role of education and human capital formation has been examined in the literature on economic growth in the last few decades. The role of education in skill formation, and the resultant division of the labor force in high- and low-skilled workers, has also been widely examined in the literatures on income distribution and international trade.
2Most of the theoretical analysis of this issue, however, has followed the neoclassical approach, which assumes that labor –and other factors of production– are fully employed. Neoclassical growth models (including the so-called new or endogenous growth models), for instance, imply that education promotes growth by making the productivity of labor increase more rapidly, and has a positive effect on income distribution by increasing wages,  although different rates of skill formation – through education – between different groups have also been argued to exacerbate income inequality.  Surprisingly, however, education has received little or no attention in classical-Marxian theories of growth and distribution, despite the obvious relevance education has for the dynamics of the capitalist economy and the fair amount of attention being given to it in broader political economy discussions.  For instance, some commentators have associated the relative neglect of the education sector with the profit squeeze and the decline in productivity growth after the period of the so-called Golden Age of Capitalism which ended in the late 1960s. 
3The purpose of this paper is to take some steps in filling this lacuna by discussing classical-Marxian ideas on education, growth and distribution, and by using these ideas selectively – and in a manner suitable for contemporary conditions - to develop a simple classical-Marxian model of growth and income distribution in which the primary role of education is to convert low-skilled workers into high-skilled workers.
4The classical-Marxian model of growth and distribution that we use for this purpose is one in which labor is not fully employed, income distribution is affected by historical, moral and political forces which are reflected in the Marxian concept of state of class struggle, and in which aggregate demand issues are absent because of the assumption that all saving is automatically invested to augment the stock of productive capital.  The classical-Marxian approach to growth has thus far focused mainly on the role of capital accumulation due to saving by capitalists, and due to technological change brought about by the response of capitalist firms to labor market conditions. It is not obvious what role education would play in the growth process within this approach. Increases in education, by increasing labor productivity, will not necessarily increase output and its growth as in neoclassical models with fully employed labor, because of the existence of unemployed workers. Regarding income distribution, the classical-Marxian framework has been concerned mainly by the question of how income is distributed between workers who receive wages and capitalists who receive profits and the consequences of this for the dynamics of capital accumulation, with some attention being given to landlords and rent (following Ricardo’s analysis of the tripartite division of output and income into rent, wages and profits),  and to financial capitalists (following Marx’s discussion of the division of the surplus or profits between financial capitalists who receive interest income and industrial capitalists who receive net profit).  How will the entry of education and human capital affect the dynamics of distribution?
5The rest of this paper proceeds as follows. Section 2 will discuss the major ideas of the classical economists and Marx on education, growth and distribution. Section 3 describes the structure of the model and relates the model to the classical-Marxian ideas. Section 4 examines the dynamics of the model by analyzing its behavior in the short and long runs. Section 5 concludes.
2 – Classical-Marxian economic thought on education, growth and distribution
6The topic of education did not play a major role in the classical analysis of growth and distribution. That analysis focused on saving, investment and in some cases technological change driven by the division of labor or mechanization in explaining growth, and in subsistence income and population dynamics or the relative bargaining power of workers and capitalists in determining distribution between wages and profits, and the monopoly over land ownership of different degrees of fertility in determining rent. Nevertheless, some of the leading classical economists did have a number of things to say about the relationship between education and growth and distributional issues.  In this section we will provide a brief discussion of some of the ideas developed by the five leading classical economists, that is, Smith, Ricardo, Malthus, J. S. Mill, and Marx, and by McCulloch because of his especially relevant contributions. In presenting these ideas of the classical economists we will examine, in turn, their discussion of the effect of education on income distribution, their analysis of the role of education in the growth process and finally, their other ideas on the interaction between the process of economic growth and education.
7Regarding the effect of education on income distribution, we may start with the ideas of the classical economists on the wages of unskilled workers and skilled workers who are educated, by which they usually meant those who received general and more technical education in institutions of learning (although they also recognized, to some extent, the importance of practical experience at work). In analyzing the causes of wage differences between different types of workers, Smith (1776, p. 118-19) stated that the “difference between the wages of skilled labour and those of common labour” was based on the principle that “it must be expected, over and above the usual wages of common labour, will replace to him the whole expence of his education, with at least the ordinary profits of an equally valuable capital.” Elsewhere he clarifies that the wage premium will cover the expense of education and not (much) more because of the response of the supply of skilled labor to deviations of the skilled labor wage from the natural price: “A man then has the natural price of his labour when it is sufficient to maintain him during the time of labour, to defray the expence of education, and to compensate the risk of not living long enough and of not succeeding in the business. When a man has this, there is sufficient encouragement to the labourer and the commodity will be cultivated in proportion to the demand” (Smith, 1976, p. 495-6). John Stuart Mill departed from Smith in focusing on the determination of the wage differential in terms of supply and demand or, what he called, monopoly. He pointed out that “a clerk from whom nothing is required but the mechanical labour of copying, gains more than an equivalent for his mere exertion if he receives the wages of a bricklayer’s labourer” and attributed such wage differentials to “monopoly, the small degree of education required being not yet even so generally diffused as to call forth the natural number of competitors” (Mill, 1848, p. 387). Mill (1848, p. 107) continued that “The number of persons fitted to direct and superintend any industrial enterprise, or even to execute any process which cannot be reduced almost to an affair of memory and routine, is always far short of demand ; as is evident from the enormous differences between salaries paid to such persons, and the wages of ordinary labour”. He also argued that “[s]ince reading and writing have been brought within the reach of the multitude, the monopoly price of the lower grade of educated employments has greatly fallen, the competition for them having been increased in an almost incredible degree.” (Mill, 1848, p. 386-7). Marx analyzed the issue of skilled labor in his discussion of how labor produces value in a commodity producing society (in which production occurs for exchange). When labor produces value, it is called abstract labor, or human labor in general. Since skilled labor can produce more than simple labor, Marx counts skilled labor as a multiple of simple labor (Capital, Vol. 1, chap. 1). Whether or not one thinks that Marx’s solution of this problem is correct, and even though his approach seems closer to Smith’s than to Mill’s, it is clear that he was interested in examining of labor values which, because prices may deviate from values, is not the same thing as analyzing the determinants of actual wage differentials.
8Turning next to the effect of education on overall wages and the wage share in total output, it may be inferred that Smith’s analysis of wage differentials implies that if education creates more skilled workers, who have higher wages than unskilled workers, the overall wage share will increase. However, this is not a certainty, because the wage of unskilled workers may also be affected by changes in the level of education, about which nothing definitive can be said a priori (if, for instance, the wage depends on the rate of growth of total output or employment, as Smith argues), but if we assume for simplicity that the natural price of unskilled labor is given, it will be the case. Malthus and Ricardo, and more explicitly, McCulloch and Mill took a different approach, focusing on the role of education (among other factors) in increasing the psychological subsistence level which, according to them, determines population growth. Malthus expected that education (at least in the form he recommended) would improve the “standard of wretchedness, a point below which they will not continue to marry and propagate their species ” of “the lower classes” (Malthus, 1798, Vol 3, p. 209). Ricardo (1951, p. 115) states, in his notes on Malthus, that “it is not difficult to conceive that with better education and improved habits, a day’s labour may become much more valuable estimated even in what are now called the necessaries of the labourer”. Mill (1848, p. 375) argued that universal education would “keep the increase of population within proper limits”. By raising the level of income at which population would grow, educational improvements would slow down population growth. He also argued that “it is impossible effectively to teach an indigent population. And it is difficult to make those feel the value of comfort who have never enjoyed it” (Mill, 1848, p. 375). Marx did not subscribe to this Malthusian population mechanism and argued, instead, that education would not have a positive effect of wages when the educated labor became unemployed due to the introduction of machinery because the skilled unemployed would bid down wages competing for the remaining jobs (Marx and Engels, 1975-2005, Vol. VI, p. 427).
9Marx and Engels noted the possibility of the division of the labor movement into factions, one consisting of “the mass of workers living in real proletarian conditions” which was revolutionary, and the other comprising “the petty bourgeois members and the labor aristocracy” which was reformist. Subsequent Marxist writers have echoed these ideas and taken the argument further. In his writings during World War I Lenin (1914-15, p. 161) followed Marx and Engels to write that “certain strata of the working class (the bureaucracy in the labour movement and the labour aristocracy ….) as well as their petit-bourgeois fellow travellers … served as the main social support of these tendencies to opportunism and reformism”. Although these ideas did not always refer specifically to the role of education in creating and maintaining differences within the labor force, they recognized the role that such divisions could play in weakening the relative power of workers, a theme that has been more fully examined by subsequent Marxist scholars (see Giddens, 1973, for instance).
10Turning next to the effect of education on growth, although Smith did not directly analyze the issue, instead emphasizing the roles of capital formation through saving, and improvements in technology due to the division of labor made possible by the expansion of the market, he indirectly considered the role of education by including investment in education as augmenting the stock of fixed capital. For instance, he wrote that fixed capital included “the acquired and useful abilities of all the inhabitants” and pointed out that “The acquisition of such talents, by the maintenance of the acquirer during his education, study, or apprenticeship, always costs a real expense, which is a capital fixed and realized, as it were, in his person. Those talents, as they make a part of his fortune, so do they likewise that of the society to which he belongs. The improved dexterity of a workman may be considered in the same light as a machine or instrument of trade which facilitates and abridges labour, and which, though it costs a certain expense, repays that expense with a profit” (Smith, 1776, p. 282). McCulloch continued in the same vein as Smith, as did the other classical economists, writing that “[m]an is as much the product of previous outlays of wealth expended upon his subsistence, education, etc., as any of the instruments constructed by his agency.” (McCulloch, 1825, p. 121). He also wrote that “[m]uch stress is … justly laid on the efficiency of machines which man has constructed to assist his undertakings ; but he is the most important of all machines, and every addition made to his skill and dexterity is an acquisition of utmost importance.” (McCulloch, 1825, p. 122). However, McCulloch went further. He emphasized, much more than his contemporaries, the role of the diffusion of knowledge in increasing growth through technological change and by preventing a few people from appropriating the result for themselves (see O’Brien, 1970, p. 280 ; 1975, p. 217).
11Beyond their examination of the effects of education on productivity and growth, the classical economists made many other more general observations about the effects education has on people, effects which are related to economic growth. First, they had a generally positive view of the effects of education on people as workers and as human beings, which presumably contributed to the better functioning of economies. Smith (1776, p. 788) stated that educated workers had fewer “delusions of enthusiasm and superstition” and “more decent and orderly than an ignorant and stupid one”. He pointed out that educated people “are more respectable, and more likely to obtain the respect of their superiors” and “less apt to be misled into any wanton or unnecessary opposition to the measures of government” (Smith, 1776, p. 788). Malthus (1798, p. 315-6) pointed that education “appears to have a considerable effect in the prevention of crimes, and the promotion of industry, morality and regular conduct”. McCulloch (1825, p. 359) opined that “nine-tenths of the misery and crime which afflict and disgrace society have their source in ignorance”. Mill (1848, p. 375) believed in the importance of universal education as a prerequisite for a good democratic government, because it would “cultivate common sense” and result in “sound practical judgment”. Marx (1867, p. 453) argued that not only would education “add…to the efficiency of production”, but was “the only method of producing fully developed human beings”.
12Second, many of them believed that education had the effect of improving the relation between workers and their employers, thereby stabilizing the social order and, presumably, promoting economic prosperity. Malthus (1798, p. 206) wrote: “The principal argument which I have heard advanced against a system of national education in England is, that the common people would be put in a capacity to read such works as those of Paine, and the consequences would probably be fatal for government. But in this subject I agree most cordially with Dr. Smith in thinking that, an instructed and well-informed people would be much less likely to be led away by inflammatory writings and much better able to detect the false declamation of interested and ambitious demagogues, than an ignorant people”. Education, McCulloch (1825, p. 134) believed, would show workers “how closely their interests are identified with those of their employers, and with the preservation of tranquility and good order”. Marx and Engels (1975-2005, Vol IV, p. 527), as one would expect, were critical of the type of education prevailing in their time, which workers found “tame, flabby, subservient to local politics and religion” and “merely a constant sermon upon quiet obedience, passivity and resignation to his fate”, and called for its replacement by a purely proletarian education.
13Third, starting with Smith, some classical economists argued that the process of economic growth would have adverse consequences on workers, which necessitated the spread of education for their mitigation. Smith (1776, p. 782) argued that although the division of labor implied increasing productivity of labor, in the progress of this division “the employment of the far greater part of those who live by labour, that is, of the great body of the people, comes to be confined to a few very simple operations ; frequently to one or two. But the understandings of the greater part of men are necessarily formed by their ordinary employments. The man whose whole life is spent in performing a few simple operations, of which the effects, too, are perhaps always the same, or very nearly the same, has no occasion to exert his understanding, or to exercise his invention, in finding out expedients for removing difficulties which never occur. He naturally loses, therefore, the habit of such exertion, and generally becomes as stupid and ignorant as it is possible for a human creature to become. The torpor of his mind renders him not only incapable of relishing or bearing a part in any rational conversation, but of conceiving any generous, noble, or tender sentiment, and consequently of forming any just judgment concerning many even of the ordinary duties of private life. Of the great and extensive interests of his country he is altogether incapable of judging…” Smith’s remedy was for the government to provide elementary education – possibly compulsory – in every parish (Smith, 1776, p. 785). Marx followed Smith and argued that the increasing division of labor and mechanization contributed to the alienation, dehumanization and deskilling of labor. This theme which has been further explored by many subsequent Marxist scholars (see, especially, Braverman, 1974), but it may be questioned whether Marx’s theory can imply a different outcome, one in which mechanization leads to increasing skills and learning by doing (see Adler, 1990).
14We will end this section with five comments about these classical-Marxian ideas. First, since the classical economists did not provide a systematic analysis of the effect of education on distribution and growth, there are several gaps in their approach. For instance, while they argued that education increases the productivity of workers, they also usually expected it to increase wages, especially that of skilled workers ; thus, the effects on profits, and hence on accumulation and growth, are unclear. Second, there are many classical ideas which can be used to construct a more systematic classical-Marxian theory of education, growth and distribution. For instance, there are concrete ideas on how education affects wage differentials between skilled and unskilled workers and how education increases productivity and the rate technological change. Third, there are many interesting but vague ideas with unclear effects which are difficult to incorporate with any degree of precision in an analysis of growth and distribution, which include the effects of education on social stability. Fourth, there are some ideas which have proved to be rather controversial, which include the classical-Marxian views on the effects of mechanization and technological change on labor skills. Finally, there are some ideas which are arguably of limited relevance for current economic conditions. The Malthusian population mechanism, for instance, may not be operative in many countries today because on the numerous checks on population growth and because, thanks to public policy, the poor do not routinely starve to death as they did at the time of the classical economists. Some of the classical arguments about the beneficial effects of education on morals and habits have arguably more to do with the spread of basic education, which has already been achieved – although arguably with different degrees of effectiveness – in many parts of the world, than with higher levels of education,
3 – Structure of the model
15In developing a classical model of education, growth and distribution we draw on the classical-Marxian perspective, which – as noted earlier – examines growth determined by saving and capital accumulation with unemployment of labor. We characterize the classical-Marxian economy as one in which: there are two basic classes, capitalists and workers ; the distribution of income is influenced by the state of class struggle between them and there are unlimited supplies of labor ; capitalists save a fraction of their income while workers consume their entire income ; all saving in the economy is automatically invested since there is no problem of aggregate demand ; and saving and investment implies capital accumulation which expands output and employment. The simplest classical-Marxian model has only two classes – capitalists who own the capital stock and workers who earn only wage income which is a fixed share of total production, and there are fixed coefficients of production. The essential characteristic of the classical-Marxian approach is that growth depends on saving and capital accumulation and since saving is done only (or at least primarily) by capitalists, a more unequal distribution of income which increases the share of profits in income, leads to higher growth. Of course, this simple characterization has to be amended to take into account the role of education and the existence of high- and low-skilled workers.
16The general approach we follow is to extend standard models in the classical-Marxian tradition discussed in the previous section to incorporate two kinds of labor – high-skilled and low-skilled, the quantities employed of which are given by H and L, and which receive real wages wH and wL. We define the ratio of skilled to unskilled wage as
18which represents the skill premium, so that typically one would expect ? > 1. With capital, we therefore have three inputs used in the economy, which correspond to three classes in society, capitalists who own physical capital and organize production, and workers who have one unit of labor, either skilled or unskilled. We examine a closed economy in which the government has no fiscal functions, and firms produce one good which can be used both for consumption and for (capital) investment. The rest of this section presents the model, relates it to some classical-Marxian ideas discussed in the previous section, compares it to some standard neoclassical ideas, and derives some preliminary implications.
19The technology is as follows. We suppose that there is only one sector and indeed productivity increases – which derive from learning-by-doing processes and innovation activity by high-skilled workers, and depend on the amount of educated workers – are non-rival and spread across firms instantaneously. Production uses fixed coefficients input-output relations with capital and a mixture of skilled and unskilled labor as inputs into production. The productivity of skilled and unskilled labor is given at a point in time by AH and AL, respectively, and the maximum output that can be produced by a unit of capital is k. High and low-skilled workers employed in the production of the consumption good are partially substitutable. To be specific, the production function of the standard firm is:
21where Y is the output of the good, K is the amount of capital, and f is an index of the two types of labor. This production function is in line with standard heterodox assumptions in rejecting the substitutability between labour and capital, but in principle it allows for some substitutability between the two types of labour. It may be noted that, by distinguishing qualitatively between high- and low-skilled workers, we are departing from much of the neoclassical growth-theoretic approach which assumes that education simply makes low-skilled workers more productive by increasing their quantity in effective units of labor, and coming closer to those classical-Marxian notions which see workers of different skills as comprising different classes. Our approach is also closer to the neoclassical trade-theoretic literature which distinguishes between low-skilled and high-skilled workers (see Wood, 1990).
22In the rest of this paper, for the sake of analytical convenience, and without significant loss of generality, we shall make some simplifying assumptions. First, we shall assume that high-skill workers are more productive at all t, and that their productivity advantage remains constant over time. Formally, there is a scalar ? ? 1, such that AH = ?AL, all t. (This encompasses the special case with ? = 1, all t.) To assume that ? is constant seems reasonable (if not necessary) in a steady state, such that if any loss of generality occurs, this only has to do with the analysis of the transition path.
23Second, we stipulate that the demands for high-skilled and low-skilled labor by firms are as follows:
25where and ?0 is a positive parameter. Similarly
27where , and ?1 and ?2 are positive parameters. Clearly, and . Equations (3) and (4) can be interpreted either as the reduced form of a general model with profit-maximizing, perfectly competitive firms (with one representative firm, with all firms being identical).  Or, from a macro perspective, and more in line with a classical-Marxian approach, this can be seen as a concise way of capturing the average behavior of capitalist firms in the economy.
28The markets for the two kinds of workers are modeled as follows. Low-skilled workers are in unlimited supply, and along standard neo-Marxian lines we assume that the real wage of these workers is determined exogenously by the relative bargaining power of low-skilled workers and firms, or what has been called the “state of class struggle”.  We parameterize this state in terms of the real wage of low-skilled workers in terms of their efficiency factor, so that given the state of class struggle, an increase in AL will result in a proportionate increase in wL. The market for high-skilled workers is flexprice, and the skill premium adjusts in response to the excess demand for high-skilled workers, given the supply of these workers at a point in time, denoted as Hs, and given wL. Our assumption reflects the approach of Mill which, as we saw in the previous section, takes the view that the wage differential between high- and low-skilled workers is determined by the supply of high-skilled workers. In our model the low-skilled worker wage serves as a reference point, and given the skill premium, high low-skilled wages increase the high-skilled wage proportionately. Formally, there exists a given scalar, ?, such that wL = ? AL, and, given Hs, at any t, ? solves .
29Given the assumptions on the labor market, in what follows we shall use the symbols H and L, to denote the quantities of high- and low-skilled workers traded. The level of ? is determined by the relative bargaining power of low-skilled workers, and can be thought of as representing the state of class struggle. This parameter will be a determinant of what is left for capitalists to pay high-skilled workers and for their profits. To be precise, the share of low-skilled workers in total income is given by ?c(?)k. Given H, K, and AL, ? is given in the short run, so that ? determines the low-wage share. The remainder is left for distribution to capitalists and high-skilled workers.
30We formalize the relationship between the use of high-skilled labor and labor productivity growth by assuming that the rate of growth of labor productivity of high-skilled workers depends positively on the amount of high skilled labor in efficiency units as a ratio of the stock of capital. With a simple linear functional form, and denoting rates of growth by overhats, we assume that there exist positive scalars ?0 and ?1 such that
32Here we measure high-skilled labor input as a ratio of capital stock as a scaling factor representing the size of the productive economy. We assume that all firms are identical, so that, for instance, AH can be thought of as representing average productivity of high-skilled workers. Further, by assumption the productivity differential between the two types of workers remains constant over time, so that , all t, and we can write
34In other words, we conceptualize innovations as non-rival products of learning-by-doing with an immediate spillover to low-skilled workers. We adopt the view that low-skilled workers are employed in routine production activities, while high-skilled workers are innovators. Unger (2007, p. 96-97), who distinguishes such roles in terms of an idea about the mind, expresses it as follows:
We know how to repeat some of our activities, and we do not know how to repeat others. As soon as we learn how to repeat an activity we can express our insight in a formula and embody the formula in a machine … The not yet repeatable part of our activities – the part for which we lack formulas and therefore also machines – is the realm of innovation, the front line of production. In this realm, production and discovery become much the same thing.
36We therefore assume that education converts workers who could only do repetitive activities into discoverers and innovators, although they continue to be engaged in some routine activities, activities which are qualitatively different from those of low-skilled workers. However, in line with Unger’s approach, we also assume that the discoveries and innovations produced by high-skilled workers ultimately become routines (perhaps they are embodied into machines) thus increasing also low-skilled workers’ productivity.
37This approach continues to stress the qualitative difference between high- and low-skilled workers, by making only high-skilled workers contribute to labor productivity growth, departing from the neoclassical growth-theoretic approach – noted earlier – that assumes that there are only quantitative differences between workers with different skills. Our approach follows McCulloch’s ideas, mentioned in the previous section, emphasizing the importance of education for technological innovation and diffusion.
38Low-skilled labor is converted into high-skilled labor through the process of education. The dynamics of the stock of high-skilled labor H is given by the following equation, where ? > 0,
40for all ? ? 1, whereas for all . According to equation (6), the change in the stock of high-skilled workers depends on three things. First, it depends on the demand for education which, in turn, depends positively on the skill premium, which increases the ’return’ to education. Thus no one will seek education if the skill premium is zero. Second, it depends on the size of the stock of high-skilled workers, both by increasing the availability of mentors and educators, and by increasing the support for, and access to, education (for instance, a higher stock implies a higher number added from high-skilled worker families). Third, it depends on a parameter, ?, which captures the openness of the education system, either through government policy or through the degree of exclusivity of the education system. Easier access to low-cost public education and greater access to student loans and grants, and a more open private education system which is less elitist on the basis of class and income would increase ?.
41Our approach follows Mill, in stressing the importance of restrictions on the availability of education, rather than Smith, who can be taken to suggest that education will expand till the wage differential between high- and low-skilled workers is determined by the (exogenously-given) real cost of education. It may also be noted that our approach differs from much of neoclassical theory which focuses on the choice individuals make regarding whether to become educated or not, or the amount of education they will obtain. This approach makes the amount of human capital accumulated depend on individual preferences (reflected, for instance, in their rate of time preference) and the returns to schooling (that is, how much a worker can increase his or her wages by obtaining more education). Since in our approach the wage differential affects the rate of education, it is not inconsistent with the choice approach. However, it stresses other factors, such as the degree of access to education, and the wage differential may reflect increases in the opportunity to obtain education because of subsidies provided by businesses who react to the relative cost of educated workers. Our approach is therefore less specific than the neoclassical one, but we consider this lack of specificity to be a virtue because it opens up space for other determinants of the spread of education, which are crowded out in the neoclassical approach.
42Concerning the consumption and saving behavior in the economy we shall assume that workers – both high-skilled and low-skilled – do not save, but consume their entire income ; capitalists save a fixed fraction, s, of their profits.
43The income of profit recipients, or capitalists, is given by
45where r is the rate of profit. Total consumption expenditure in the economy is therefore given by
47This implies that saving is given by the standard equation,
49Finally, we assume that capitalists save in order to invest, so that saving and investment are always equal. This version of Say’s law is a standard assumption of the classical-Marxian modeling approach (although aggregate demand considerations were discussed by Malthus, and especially, Marx). Together with equation (9) it implies
51This implies that there is no effective demand problem, so that, given the existence of unemployed low-skilled workers, we have
53We end this section with the comment that the approach adopted in our model takes a simple view of the effect of education on the economy, that is, it transforms low-skilled workers into high-skilled workers, and that high-skilled workers have a more complex role in the economy than just being an input into production, that is, they increase the efficiency of both low-skilled and high-skilled workers through the process of innovation, and help in the process of education, as family members, mentors or educators. This formulation of the role of education in the economy is admittedly a narrow one, being much more specific than the multidimensional role given to it in the general classical-Marxian and radical approaches to education. This literature incorporates broad issues such as the role of education in affecting social stability and the relation between workers, capitalists and the state (as mentioned in the previous section), weakening the position of workers by dividing them into groups based on their level of education, in creating and strengthening the perception of upward socio-economic mobility and thereby increasing tolerance for income inequality, indoctrination and socialization, and easing the process of the extraction of labor (and hence labor productivity and profits) from labor power (see Bowles and Gintis, 1975, 1976). We could add to these the role that education may have in creating more informed political participants and a discerning electorate. Although our analysis does give a multidimensional role to more educated labor by making them innovators, educators and directly productive workers of a special kind, it is not broad enough to incorporate these broader political economy issues. Some of these issues, however, can be addressed in our model by examining the effects of appropriate parametric changes which capture changes, for instance, in the bargaining position of workers which are the result of the system of education rather than something which depends quantitatively on the importance of educated and high-skilled workers in the economy.  Incorporating others would involve introducing different levels of education, which would complicate our model.
4 – The dynamics of the economy
54We examine the dynamics of the model by considering two runs. In the short run we assume that the levels of K, H and AL are fixed, and the equations of the model solve for Y, L, ?, r and I from equations (3), (4), (7), (10), and (11). The rate of profit is given by
56For given levels of AL, K, and H, we may solve for the equilibrium value of ? as shown in Figure 1. This is given by
Determination of the skill premium in the short run
Determination of the skill premium in the short run
58In the long run we allow K, H and AL to change. Assuming away the depreciation of capital without loss of generality, the change in capital stock is given by
60and changes in H and AL are governed by equations (6) and (5a).
61It is convenient to examine the time path of the economy in the long run by defining the state variable h = ALH/K. Because the rate of growth of h is given by
63we can substitute from equations (5b), (6), and (11) through (14), to obtain, using the definition of h,
65The economy is defined as being in long-run equilibrium when , that is, h is stationary. The determination of both short- and long-run equilibrium can be examined in terms of Figure 2. In the four-quadrant diagram, the south-west quadrant represents equation (3), essentially setting b = h from Figure 1, showing equation (13). The north-west quadrant shows the technological change curve ; equation (5a). The southeast quadrant shows the capital accumulation curve and the high-skilled labor accumulation curves, the former is obtained from equations (10)-(12) and (14), and is given by
67and the latter is derived from equation (6).
Long-run dynamics and equilibrium
Long-run dynamics and equilibrium
68Subtracting the value of from that of for each value of ? gives the HK curve shown in the same quadrant.
69In the short run, h is given, and the south-west quadrant solves for the short-run equilibrium value of ?. The north-west quadrant then determines ?. The south-east quadrant determines the values of . The north-east quadrant shows the resultant levels of and , which is a point on the short-run equilibrium curve SR. Other initial values of h trace out other possible short-run equilibria, and hence the entire SR curve. Points on it above the 450 line imply that , or that h rises over time. In the long run h increases, yielding another short-run equilibrium on the SR curve till long-run equilibrium is attained at the intersection E1 of the SR curve and the 45° line (provided the SR line is flatter than the 45° line at E1, which is required for stability). Long-run equilibrium is attained at h*.
70It is immediate to prove from equation (16) that the economically meaningful equilibrium with a positive skill premium, ?1 > 1, exists if , and it is dynamically stable. The latter inequality is more likely to hold the lower , and ?, and the higher s, k. If technical progress is too strong, or if profits and capital accumulation are too slow, then the dynamics of innovation may dominate and lead the economy to an explosive path with increases in the rates of accumulation and technological change and relative expansion of education and the high-skilled labor force. The process will come to an end when the skill premium disappears or, what is more likely, when the skill premium is just sufficient to cover the real expenses in obtaining education, as in Smith’s analysis. We will return to this issue below.
71The key question of interest, however, is the effect of the education system on the long-run equilibrium.
72Suppose and consider the long-run equilibrium with ?1 > 1. At this equilibrium, an increase in ? - reflecting a more open educational system, or an increase in public funding for education – shifts the curve in Figure 2 to the right, increasing the short-run equilibrium value of . It therefore shifts the KH line to the left, and the SR curve also up and to the left. In the long run, h begins to increase, and the new long-run equilibrium will occur at a higher level of h. At the new long-run equilibrium (compared to the initial equilibrium) the levels of capital accumulation and labor productivity growth will all increase in long-run equilibrium, as can be verified from the figure. The rate of growth of the economy would therefore increase with ?. The long-run rate of growth of education accumulation may increase or decrease depending on the relative strength of the increase in ? as compared to the decrease in ?. The share of income going to low-skilled workers, , decreases because the number of low-skilled workers employed is a positive function of ?, even though the state of class struggle is given by assumption (and so is the productivity of capital). Given equation (3), instead, the share of income going to high-skilled workers is constant. Therefore, given the decrease in the share going to low-skill workers, both the rate of profit and the share of total income going to the capitalists, r/k, will rise. Thus income is redistributed from (low-skilled) workers to capitalists. This, of course, is why the rate of capital accumulation in the economy is speeded up, as we saw earlier. 
73It is worth noting that, by using similar arguments, it is possible to draw analogous conclusions concerning changes in the other parameters of the model. Suppose again that . First, a sufficiently small improvement in the conditions of technical change, ?0 or ?1, will have the same effect as an improvement in the educational system. Unlike changes in the education system, however, changes in the processes generating technological innovations will shift the SR curve upwards by shifting the curve and the growth rate of human capital will actually fall. Second, a sufficiently small deterioration in the bargaining position of workers, ?, and a sufficiently small increase in the savings rate, s, will increase the growth rate of the main state variables while increasing the equilibrium value of the skill premium. Graphically, the latter changes imply a shift in the curve.
74If technological change responds strongly to h, or if exogenous technical progress is too strong, such that:
75, then the economy moves on an explosive path whereby at all values of h. Therefore starting with any value of h, the economy will experience increases in h and increasing rates of capital accumulation and technological change. If ? falls too low, increases in H will no longer occur, but h will keep increasing as technological change occurs faster than capital accumulation. This kind of knowledge-driven increase in knowledge, however, is unlikely to occur in practice, and the ? function is likely to flatten out, so that a stable equilibrium will be attained. Similar problems arise in economies with very low saving rates and capital productivity, or very high ?.
5 – Conclusion
76This paper has analyzed the role of education in determining the dynamics of growth and income distribution between capitalists, high-skilled and low-skilled workers in capitalist economies along classical-Marxian lines by discussing what the classical economists had to say on the issue and by developing a simple classical-Marxian dynamic model of growth and distribution, borrowing from the writings of the classical economists in ways that seem to us relevant under contemporary conditions. In so doing, we have attempted to fill a lacuna in the literature on the classical-Marxian approach, which has neglected the formal analysis of the effects of education and skill formation on distribution and growth, and issue which many observers find to be a central feature of contemporary capitalist knowledge-based economies.
77The model developed in this paper has allowed us to examine the growth and distributional consequences of greater openness in the education system, which allows more low-skilled workers to become high-skilled workers. The model shows that an expansion in education will promote growth and have beneficial distributional effects, but not along standard orthodox neoclassical lines as discussed in neoclassical growth theory. For instance, while in neoclassical full employment models education has a positive effect on output and growth directly by increasing effective labor supply, in the classical model of this paper, the growth effect is the consequence of distributional changes. In our model, greater openness in the education system increases the rate of growth of the stock of high-skilled workers by increasing access to education, reduces the skill premium by increasing the supply of high-skilled workers, reduces the employment of low-skilled workers relative to high-skilled workers by making the latter relatively cheaper, reduces the total wage share in income (this occurs due to reduction in the low-skilled labor share given our assumptions regarding the state of class struggle and the elasticity of demand for high-skilled workers), and therefore increases the rates of profit, capital accumulation, and growth of output and the real wages of both low- and high-skilled workers (by increasing the growth of the productivity coefficients of both type of labor due to the increase in the rate of technological change caused by the greater spread of education).
78These results capture some of the major ideas of the classical economists on the role of education on growth and distribution to the extent that they seem relevant for contemporary conditions, although we cannot claim to have taken into account all of their wide-ranging, complex and controversial ideas.
79The model developed here is a simple one which could be modified in various ways. Several simple extensions of the models may be particularly interesting: allowing high-skilled workers to save and hold capital, and thereby have mixed class interests ; allowing the wage premium to change slowly with the possibility that some high-skilled workers find low-skilled jobs (being chosen above low-skilled workers) ; and examining the effects of education and the consequent possible weakening of working class solidarity on the relative bargaining power of low-skilled labor workers. If the approach developed here is found to be useful, it can also be modified to deal with additional complications, by considering different kinds of education – say primary and secondary, and higher education – and effects of education beyond those on labor productivity.
We are grateful to Richard Arena, the participants in the Analytical Political Economy Workshop (London) and the History of Macroeconomics conference (Paris) and an anonymous referee for comments and suggestions on earlier versions of the paper. The usual disclaimer applies.
Department of Political Science, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: aduttCEP_idPAS_d_ISBN_pu2011-02s_sa06_art06_img0nd.edu.
(Corresponding author) Department of Economics, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom. E-mail: firstname.lastname@example.org.
See Uzawa (1965) and Lucas (1988).
Bénabou (1996), for instance, shows how minor differences in education technology, wealth and preferences can result in widening disparities between income groups that cumulate over time.
A recent text on growth theory written mainly from a classical-Marxian perspective, by Foley and Michl (1999) does not have any discussion of the role of education in economic growth. We are, in fact, unaware of any heterodox dynamic model of growth and distribution, let alone a classical-Marxian one, which analyzes the role of education, apart from a post-Keynesian model in Dutt (2010). In that model, however, aggregate demand plays a central role, while in the classical-Marxian growth theory tradition, aggregate demand issues are neglected.
Glyn et. al. (1990) argue that the slowdown in productivity growth after the late 1960s can be traced at least in part to the growing tendency, found even during the period of the Golden Age, to exclude the mass of unskilled workers from the search for new technologies, which was conducted in specialist divisions like research and development departments, while technological improvements require greater involvement of production workers. This exclusion can be explained partly by the adoption of organizational methods of command and control, and the slowdown of the expansion of education and skill formation among workers. Lipietz and Leborgne (1996) link these economic problems with the abandonment of investment in education.
For a recent treatment of growth and distribution which adopts this basic classical-Marxian approach, although with several complications, see Foley and Michl (1999). The basic model is discussed and compared to other models in Marglin (1984) and Dutt (1990), where it is called the neo-Marxian model. A classic early exposition is due to Goodwin (1967) who, however, introduces cyclical complications due to endogenous wage dynamics.
See Pasinetti (1960) for a classic early treatment.
See Dutt (1989).
For a discussion of the ideas of the classical economists on education in general, see O’Donnell (1985).
The implications of this assumption are examined in Dutt and Veneziani (2010), where a more general version of the model presented in the present paper is analyzed.
Note that we do not invoke the Malthusian population mechanism in determining distribution, for the reason mentioned in the previous section.
See Dutt and Veneziani (2010) for a formal analysis of some of these issues.
It should be noted that the property of this model that the share of income going to high-skilled workers is constant is an artefact of our simplifying assumptions regarding the high-skilled labor demand function. This allows us to obtain stark results on the effect of education on growth, which is the main focus of the analysis, without significant loss of generality. As shown in Dutt and Veneziani (2010), with more general production functions, and hence labor demand functions, the share of income going to high-skilled workers will also decrease provided the elasticity of substitution between the two types of workers is not too high.