For at least the past one 100 years, the growth of industrialised economies has been remarkably balanced; output per worker has increased at a roughly constant rate, while the capital-output ratio, the real return on capital, and the shares of capital and labour in national income have remained fairly constant (Kaldor 1961, Jones 2015). Understanding why economies exhibit balanced growth is important for policymakers interested in how new technologies and changing institutions will affect long-run economic performance and the division of income between capital and labour. In recent work (Grossman et al. 2016), we study the causes of balanced growth and argue that it results from schooling levels increasing over time. This suggests investment in education has a crucial, but previously overlooked role in shaping long-run growth dynamics.
The origins of balanced growth: A puzzle
In the neoclassical growth model where output is produced using capital and labour, the conditions under which a balanced growth path exists are well known (Uzawa 1961). Either the elasticity of substitution between capital and labour must equal one, or technological change must augment the productivity of labour but not of capital. For 50 years, growth economists have focused their attention on production technologies that satisfy at least one of these conditions.
But there is a problem. The size of the elasticity of substitution between capital and labour is much debated, but the balance of evidence suggests it is well below one. Reviewing firm-level estimates of this elasticity, Chirinko (2008) concludes that “the weight of the evidence suggests a value of [the elasticity of substitution] in the range of 0.4 to 0.6”. In addition, the relative price of capital has fallen over time (Gordon 1990, Greenwood et al. 1997, Cummins and Violante 2002). Figure 1 shows that between 1947 and 2013 the relative price of investment goods fell at around 2% per year, which suggests capital-augmenting technical change has occurred. Taken together these observations imply neither of the Uzawa conditions for balanced growth hold empirically.
Figure 1. US relative price of equipment, 1947-2013
Source: Federal Reserve Bank Economic Data (FRED), Series PIRIC and PERIC.
Schooling: A possible solution
Figure 2 shows average years of schooling measured at age 30 for all cohorts of native American workers born between 1876 and 1982. Clearly, schooling has been rising over time. Perhaps investment in education provides a way to square balanced growth with the failure of the Uzawa conditions. To explore this possibility, we extend the neoclassical growth model by allowing output to depend not only on capital and labour, but also on the economy’s education level. The usual formulation where labour and schooling can be combined into a single ‘human capital’ input, as assumed in the Uzawa-Lucas growth model (Uzawa 1965, Lucas 1988), does not solve the puzzle. However, under the assumption that schooling rises over time we find that balanced growth can occur even without the Uzawa conditions holding, but only if capital accumulation raises the marginal product of schooling relative to the marginal product of labour. Thus, balanced growth requires the production technology to exhibit a form of capital-skill complementarity.
Figure 2. US education by birth cohort, 1876-1982
Source: Goldin and Katz (2007) and additional data from Lawrence Katz.
Balanced growth with endogenous schooling
The fact that a balanced growth path can exist when schooling rises over time does not mean it will exist in a market economy where individuals make optimal schooling decisions. To understand the consequences of optimising behaviour we study the equilibrium dynamics of three competitive economies where the production technology depends on capital, labour, and schooling and the elasticity of substitution between capital and labour is less than one:
- A ‘time-in-school’ economy where each individual only lives for an instant and chooses what fraction of her lifetime to devote to schooling. Workers that spend longer in school are more productive when they join the labour force.
- A ‘manager-worker’ economy where again individuals only live for an instant, but education is a discrete choice and educated individuals become managers while all other individuals are workers. In this economy aggregate output depends on capital and the supplies of skilled managers and unskilled workers, as in Krusell et al. (2000).
- A more realistic ‘overlapping generations’ economy where successive cohorts of individuals live for a finite amount of time and choose how many years to spend in school before joining the workforce.
For each of these economies we characterise a class of production functions for which the economy will exhibit balanced growth. Balanced growth occurs because, in equilibrium, capital accumulation raises the returns to education and this causes schooling to increase at exactly the rate needed to offset the effect of capital-augmenting technical change on capital’s share of income.
On the balanced growth path, growth in output per capita results from both capital-augmenting and labour-augmenting technical change, and the division of income between capital and labour is constant. The overlapping generations model is also consistent with many other features of the US experience. In particular, years of schooling by birth cohort rises linearly over time as seen empirically in Figure 2, the returns to schooling satisfy a Mincer-like wage equation and the labour force participation rate declines with time.
Capital’s share of income
There is some evidence that in recent years, growth in the US and elsewhere has slowed, while capital’s share of income has risen (Elsby et al. 2013, Karabarbounis and Neiman 2014). To understand the relationship between the rate of technical change and capital’s steady state income share, we calibrate the balanced growth path of the overlapping generations model. Interestingly, we find that a decline in the rate of technical change leads to both a growth slowdown and an increase in capital’s share of income. This illustrates how changes in the economic environment may affect both long-run economic performance and the division of income between capital and labour.
The observation that the US and many other industrialised economies have experienced balanced growth over long periods is a key fact that informs our understanding of the growth process. Yet the Uzawa conditions for balanced growth are inconsistent with empirical findings. We suggest a simple and compelling way to resolve this problem. When capital and schooling are sufficiently complementary, increases in schooling offset the effect of capital deepening on the capital share and ensure growth remains balanced. Our research not only highlights the importance of education for growth, but also has novel implications for how the aggregate production function should be specified and how capital accumulation affects inequality. We hope that future research will shed further light on these questions.
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