2.1 Recently two separate strands of traditional economic theory (human capital theory and neoclassical growth theory) have been brought together to address this issue though the focus has been on externalities to education in general rather than specifically higher education. Traditional human capital theory has focused on the private rather than social gains from education and has therefore had little to say about externalities. Similarly neoclassical growth theory, in the tradition of Solow (1956), being based on assumptions of competitive markets, predicted that all factors of production would be rewarded according to their social marginal contribution to production. Thus private and social returns are always equalised and there is no scope for externalities. Traditional growth theory also included only capital and homogeneous labour as factors of production so that there was no role for education to play in the creation of 'human capital'.

2.2 Growth theory generally distinguishes between those influences on an economy's long-run growth rate from those which affect growth only in the short-run, with the main focus being on the former. It should be remembered however that short-run growth effects imply permanent changes in the economy's income level. Thus, for example, where theory does not establish a role for higher education among the determinants of long-run growth, if it can establish that more education would temporarily raise the growth rate then that economy would be wealthier into the indefinite future than a similar economy with less education. Education may therefore have both 'level effects' and 'growth effects'. In addition, since the 'short run' in the context of growth theory is often thought of in terms of decades, even short-run effects could be worthwhile policy objectives.3

The Solow model

2.3 To understand the role played both by education and externalities in the new growth theories it is necessary to begin with the simple Solow (1956) model. This depicts a firm's4 output, Y, as a function of three variables: capital, K, labour, L, and knowledge or the 'effectiveness of labour', At.5 Thus

Y = Ka(AtL)1-a 0 < a < 1 (E1)

2.4 Knowledge production is assumed to be independent of both the capital and labour inputs and to be freely available (like "manna from heaven") to all firms, but notice that it appears multiplicatively with labour in (E1) indicating that knowledge operates by 'augmenting' labour (making it more 'effective') rather than via capital.6 The exponents a and (1-a) measure the relative contribution of the two inputs - capital and 'effective labour'. By summing to unity they capture the assumption that there are constant returns to scale in production (eg. if all factor inputs are doubled simultaneously this will lead to an exact doubling of output). E1 describes the determinants of the level of output but can readily be transformed into an equation describing the growth of output, so that:

y = ak + (1-a)(a + l)   (E2)

where lower case letters represent the proportional rates of growth of their upper case equivalents. This can be rewritten as:

y - l = ak' + a   (E2.1)

where y - l is the growth of output per worker, and k' is the growth of capital per effective worker (K/AL). To see what the neoclassical growth model predicts, we can simplify matters by assuming that there is no labour force growth (annual inflows exactly match annual retirements) - a situation not too far removed from the reality of many OECD countries. This means that, in terms of equation (E2.1), y equals the growth of income per worker (ie. labour productivity).7

2.5 There are two important features of this model which recent growth theories have challenged:

  • If markets are competitive, the contributions of each factor input to output - a and (1-a) - are equal to their respective shares in total income (output). For all firms in an economy taken together this could be approximated by the National Accounts breakdown into wage and non-wage income.
  • If people save a constant proportion of their incomes8, capital per effective worker must be constant in the long run, so that k' = 0 in (E2.1) and therefore per capita income growth is entirely determined by the growth of knowledge, a. Increasing the savings (= investment) ratio can raise an economy's income level (permanently) by raising the growth rate of capital (and income) in the short-run, but since the ratio of savings to income cannot go on increasing indefinitely, investment cannot cause income to grow permanently. Countries that invest more will be wealthier but will not grow faster. And, since the only source of long-run growth is technical progress (or 'knowledge accumulation'), which by assumption is occurring at an exogenous rate, income growth rates are beyond the control of firms - and governments.

2.6 Our present interest is in the role of higher education for growth and the possibility of externalities. In this respect the key outcomes of the neoclassical model are:

  • How knowledge accumulation occurs is left unspecified and there is no human capital in the model so there is no direct role for education. However, the fact that knowledge is specified as labour-augmenting is suggestive of the possibility that it could be related to education.
  • All output is paid as income to either capital or labour, so there is no income 'left over' to act as a reward or incentive for the accumulation of knowledge.
  • There are no externalities to knowledge accumulation - each (homogeneous) worker reaps the benefits of exogenous technical progress in proportion to their contribution to output.

New growth theory

2.7 In the neoclassical model there is no explicit role for education (at any level) and no externalities - capital owners and workers are independent inputs and each is fully rewarded for their contributions to output. If the Solow model is supported empirically it would imply a negligible role for HE externalities.9 Empirical issues will be discussed in more detail in Section 3 but it is worth noting at this stage that an important motivation for the new growth theories is the apparent inconsistency between estimates of the marginal productivity of capital and capital's share in income. For example, applying the Solow model to international data, Mankiw, Romer D and Weil (1992), predict the capital share (from estimates of its marginal product) at about 60%, yet observed capital shares are around 25-35%. Capital (labour) appears to be much more (less) important for growth than the Solow model would suggest. Perhaps then the data will be better explained by a model which gives a greater role to labour-related capital - human capital?

2.8 Recent growth theories have attempted to model these processes both by introducing human capital explicitly into production functions and by allowing for the possibility of externalities. Although higher education is not typically the specific focus of attention, there is a prima facie case for a role for HE because of its twin outputs of research which generates new knowledge, and graduates embodying potentially labour-augmenting training. In particular, for major industrialised countries we might reasonably think of 'labour' as workers embodying minimum education (acquired during years 5-16) and 'human capital' as the skill acquired in post-16 education to which HE obviously makes the major contribution. This is less true for developing countries where 'minimum education' (common to all workers) can be at the primary level. Here, human capital accumulation may be mainly through additional primary or secondary education.

2.9 Human capital is introduced in new growth theories both with and without externalities. The two main approaches are:

  • the incorporation of human capital as a factor input, for example by adapting the Solow model (see Mankiw, Romer D and Weil (1992); Romer D (1996);
  • explaining the process of knowledge accumulation by relating it directly to human capital accumulation, or indirectly via research and development (R&D) activity (see Lucas, 1988; Romer P M, 1986, 1990a).

Human capital as a factor input

2.10 Three types of model of this sort can be distinguished:

  • 'sources of growth' equation models;
  • an augmented Solow model;
  • endogenous growth models in which an education sector produces human capital for use in the production sector.

We consider each of these in turn.

A. 'Sources of growth' equation models
2.11 Sources-of-growth equations are typically based on an aggregate Cobb-Douglas production function such as equation (E1) above which, when differentiated gives a relationship between the growth of output and the growth of factor inputs. For example, adapting (E1) to include human capital, H, gives:

Y = AKaLbHg a + b + g = 1 (E3)

(E3) can be written as:

y = a + ak + bl + gh   (E3.1)

where, again, lower case letters represent growth rates for their upper case equivalents and a is the growth of total factor productivity (TFP). This approach is more of an empirical taxonomy than a theory and has been used in two ways. Firstly, as the basis for so-called 'Barro regressions' (see Barro, 1991) in which the parameters a, b and g are estimated to identify the relative contribution of each input (and can be extended to other inputs and/or determinants of the 'a' term). Secondly, (E3.1) forms the basis for 'growth accounting' exercises in which values for the parameters a, b and g are imposed (eg. from data on factor shares) and applied to factor input and TFP growth rates (Denison, 1967, 1985; Maddison, 1982, 1991). In both these cases therefore aggregate output (or output per capita) growth is a function, inter alia, of the rate of growth of human capital.

B. Augmented Solow Model
2.12 Mankiw, Romer D and Weil (1992) have recently demonstrated that if a production function such as (E1) is augmented to include human capital so that

Y = KaHb(AtL)1-a-b   (E4)

and solved for the equilibrium growth rate in the manner of the Solow model, this yields a (per capita) income growth equation with physical capital and human capital investment rates (ie. as ratios of GDP) entering separately among the arguments. Alternatively the initial level of human capital can replace the human capital investment rate. For our purposes there are two interesting features of this approach. Firstly, by proposing a role for the human capital investment rate it provides a link between educational expenditures and growth. Secondly there are still constant returns to all three factors (K, AL and H), and diminishing returns to the two reproducible factors (K and H), but if empirically a and b were each approximately equal to one-third, the model would be compatible with the evidence on factor shares discussed above.

2.13 Both approaches (A) and (B) suggest a role for education in general, and post-16 education in particular in developed countries, but there are no externalities to education. Educated workers have no effects on their uneducated colleagues and each type of labour is rewarded for its efforts fully in wages/salaries.

C. Endogenous growth models
2.14 Lucas (1988) models human capital in a firm's production function in a manner analogous to the augmented Solow model (an 'internal' effect) and also allows for an 'external effect' whereby the average level of human capital in the economy affects individual firms' outputs but is not taken account of in their profit-maximising decisions. Individual workers decide on their time allocation between acquiring education and working in the 'production' sector on the basis of standard (intertemporal) utility maximisation. The Lucas production function for firm j can be represented as:

Yj = AKjb(Hj)1-bHag   (E5)

where Ha is the average level of human capital across all firms and g captures the externality effect on output. Notice that unlike the Mankiw, Romer D and Weil (1992) approach there are constant returns to the firm's two reproducible factors (Kj and Hj) but increasing returns to all factors so long as g > 0. An important feature of the Lucas representation is that, unlike the Solow model and even if there is no external effect (g = 0), long run growth is now a function of investment in both physical and human capital. There is therefore an important role for education in the long-run, as well as the short-run. This arises from the assumption of constant returns to the aggregate of the two types of capital.10 It follows that in principle this model could be tested by testing for constant returns to Kj and Hj in a firm or industry level production function and/or for increasing returns at the aggregate or economy-wide level, which would allow a value for the externality effect, g, to be inferred.

2.15 Sala-i-Martin (1996a) has proposed a more disaggregated form of (E5) which decomposes the externality effect. Thus

Yj = AKjb(Hj)1-b(Hj/Nj)ej(H/N)e (E5.1)

where N is the level of employment. Here ej captures an 'intra-firm externality' (the effect of educated workers on their colleagues within the firm) and e captures an 'inter-firm externality', with (H/N) representing the average human capital to employment ratio in the economy as a whole, as in Lucas (1988). Note that the intra-firm externality is not an externality in the usual sense of the term since it is not external to the firm's profit-maximising calculation. It is however an effect external to the worker and there may still be a case for subsidising graduates' education to persuade them to undertake the optimal level - ie. including spillovers to other workers within the same firm. The inter-firm externality on the other hand could be internalised by subsidising firms who employ more graduates, rather than subsidising the graduates themselves.

2.16 Models such as (A) and (B) above treat human capital as a 'private good'; that is, since education is embodied in the individual worker the skills which education creates are best thought of as 'rival' and 'excludable'. The Lucas approach in (C) however, while maintaining rivalness (use of a worker's education skills in one activity preclude their use in others) allows for some non-excludability - some of the gains from education spill over to others. A possible criticism of the Lucas model is that, while it is relatively easy to conceive of abstract 'knowledge' being at least partially non-excludable (see below), it is not clear why educated individuals are unable to maintain property rights over the productivity gains from their education.

Rest of Chapter