Uzawa–Lucas Human Capital Growth Model

The Uzawa–Lucas model is a two-sector model of economic growth in which output of a final good is produced with physical and human capital, and human capital is itself produced by devoting a fraction of non-leisure time to education and training. The structure was first set out by Hirofumi Uzawa in his 1965 article "Optimum Technical Change in an Aggregative Model of Economic Growth" in the International Economic Review, and was developed into a theory of cross-country growth differences by Robert E. Lucas Jr. in "On the Mechanics of Economic Development" in the Journal of Monetary Economics in 1988. The model belongs to the family of endogenous growth theory, in which long-run growth is a result of choices made inside the model, and it contrasts with the Solow–Swan model, where growth in output per worker depends on an exogenous rate of technological progress.

The economy contains identical infinitely lived households with a population normalized to one. Each household owns physical capital K and human capital h, and allocates a share u of its working time to producing the final good and the remaining share 1 − u to accumulating human capital.

Goods production. Output is produced with a Cobb–Douglas technology:

Y = A K^β (u h L)^1−β h_a^γ

where A is a constant, L is the number of workers, and uhL is effective labor devoted to goods. The term h_a^γ is Lucas's external effect: the average level of human capital in the economy, h_a, raises the productivity of every worker, but each individual ignores that his own accumulation contributes to the average. Setting γ = 0 recovers the Uzawa case without externalities.

Human capital production. Human capital accumulates according to

dh/dt = δ h (1 − u)

with δ a positive constant. The law is linear in h, so there are constant returns to the reproducible factor in the education sector. This linearity is what prevents diminishing returns from choking off growth: the more human capital a person has, the more efficiently the next unit is produced.

Preferences and resources. Households maximize the discounted sum of utility from consumption, usually with constant relative risk aversion, subject to the physical capital accumulation constraint dK/dt = Y − C − δ_KK. The two control variables are consumption and the time allocation u; the state variables are K and h.

A balanced growth path is a solution in which consumption, physical capital, and output per capita grow at a common constant rate, and u is constant. Along such a path the growth rate of human capital is g_h = δ(1 − u), which depends directly on how much time is devoted to schooling. Physical capital, output, and consumption per worker grow at a rate that combines g_h with the externality parameter.

In the version without externalities, the growth rate of consumption is determined by the interplay of the intertemporal elasticity of substitution, the discount rate ρ, and the productivity of education:

g = (δ − ρ) / σ

where σ is the coefficient of relative risk aversion, under a common parameterization. The result is intuitive: higher productivity of the schooling sector, more patient households, and a greater willingness to substitute consumption over time all raise the growth rate. Economies that are more patient or that have more efficient educational technology therefore grow faster in the long run, and the model makes permanent growth differences a function of preferences and technology rather than a transitory adjustment to a steady state.

With externalities, the decentralized equilibrium and the social optimum diverge. Individuals devote too little time to learning because they do not internalize the productivity gains accruing to others, so the competitive growth rate is lower than the planner's. The wedge justifies policies such as education subsidies, in the spirit of correcting a Pareto-inefficient allocation arising from an unpriced spillover.

The balanced path is a saddle point in the space of the ratio of physical to human capital and the consumption ratio. Starting from an initial condition with low human capital relative to physical capital, the economy adjusts by raising the time devoted to schooling above its long-run level and gradually reducing it, so growth is initially high in the education sector. An important feature is that the model produces conditional convergence in the ratio of the two capital stocks but not convergence in levels of income across countries with different initial human capital, since a permanent difference in h is persistent.

Analytical work after Lucas by Caballé and Santos, Mulligan and Sala-i-Martin, and others characterized the dynamics and showed that local indeterminacy is possible under certain externality values, so the model can generate multiple equilibria in which beliefs affect growth.

Solow–Swan. In the Solow model, capital has diminishing returns, so growth in output per worker ceases unless exogenous technology improves. The Uzawa–Lucas model removes the dependence on exogenous technology by making the accumulable factor, human capital, subject to constant returns. Mankiw, Romer, and Weil in 1992 retained the Solow structure but added human capital as a factor, finding that the augmented model explained about 80% of cross-country income variation in a sample of countries; their approach treats human capital as a stock with diminishing returns, so it differs conceptually from Lucas's engine of growth.

Romer. In Paul Romer's 1990 model of endogenous technological change, growth comes from the production of new ideas in a research sector, and ideas are non-rival, producing increasing returns at the economy level. In the Uzawa–Lucas model human capital is rival and embodied in people, so it cannot be copied at zero cost and dies with its owner. This distinction affects policy: the Romer framework emphasizes patents and research subsidies, while the Lucas framework points to schooling, on-the-job training, and the city as a place of external learning. Lucas's 1988 paper also discusses the role of cities as generators of human capital externalities, tying growth to agglomeration.

The empirical record is mixed. Cross-country regressions of growth on schooling attainment show a weaker relation in first-difference specifications than in levels, which is hard to reconcile with a model where changes in education time map directly into growth rates. Measurement problems are severe, because years of schooling do not capture differences in school quality, and the time cost of schooling in practice includes forgone earnings, class size, and public provision, which the model abstracts from. Critics also point out that, because the growth rate depends on a knife-edge linearity assumption in the human capital technology, small departures toward diminishing returns restore a stationary long-run level of income per capita.

The model has nevertheless influenced growth accounting and the design of development policy. It sharpens the idea that raising the rate of return to schooling and reducing barriers to human capital formation, such as credit constraints for education, have growth effects beyond a one-time level shift. Extensions add finite lifetimes, fertility choice, the quality-quantity tradeoff for children, and public education, and variants have been used to study the demographic transition, the effect of inequality on growth, and the sources of persistent international income differences, themes also visible in the measurement of inequality through tools such as the Gini coefficient.

1. Uzawa, H. (1965). Optimum technical change in an aggregative model of economic growth. International Economic Review, 6(1), 18–31. https://doi.org/10.2307/2525621
2. Lucas, R. E., Jr. (1988). On the mechanics of economic development. Journal of Monetary Economics, 22(1), 3–42. https://doi.org/10.1016/0304-3932(88)90168-790168-7)
3. Romer, P. M. (1990). Endogenous technological change. Journal of Political Economy, 98(5, Part 2), S71–S102. https://doi.org/10.1086/261725
4. Mankiw, N. G., Romer, D., & Weil, D. N. (1992). A contribution to the empirical economics of growth. Quarterly Journal of Economics, 107(2), 407–437. https://doi.org/10.2307/2118477