(a) Human capital and components of research and development are determining factors of economic growth. Explain using…

(a) Human capital, research and development in endogenous growth

In Solow's model long-run growth comes from exogenous technical progress. Endogenous growth models explain it from within: knowledge and skills are produced by deliberate investment, and they do not suffer from diminishing returns because ideas can be used by all (non-rival) and spill over.

Romer's R&D model (1990). Output is produced with human capital in production H_Y, labour L and a range of A specialised capital goods, so output per worker rises as the variety of inputs, that is, the stock of knowledge A, rises. Knowledge is produced in a research sector using human capital H_A: dA/dt = δH_A A, where δ is research productivity. Total human capital is H = H_Y + H_A. The growth rate of knowledge, and of output per worker on the balanced path, is g = δH_A.

Implications: (i) human capital is a determinant of growth, since growth rises with the human capital allocated to research, and economies with more educated people grow faster; (ii) R&D is a determinant, since higher research productivity δ, from better science institutions, and more resources to R&D raise g; (iii) because the stock of knowledge raises the productivity of future research (the A term), growth is sustained and there are increasing returns at the aggregate level; (iv) firms need monopoly profits from patents to invest in R&D, so a competitive market undersupplies research and a subsidy or patent protection is justified; (v) scale and policy matter, so subsidies, education and intellectual property can change the long-run growth rate, which they cannot do in the Solow model.

Lucas's model (1988) treats human capital per worker h as accumulated by devoting time (1 − u) to learning: dh/dt = δh(1 − u). Since h can be raised without limit, and has external effects on others' productivity, growth is sustained, and depends on the share of time devoted to education.

Both show that long-run growth is a matter of education, research and institutions.

(b) The steady state in the Solow model

The Solow model uses Y = F(K, L) with constant returns, a constant saving rate s, depreciation rate δ and labour growth n. In per-worker terms, with k = K/L and y = f(k), capital accumulates according to dk/dt = s f(k) − (n + δ)k.

The steady state is the capital-labour ratio k* at which dk/dt = 0, so s f(k*) = (n + δ)k*. The left side is actual investment per worker; the right side is breakeven investment, needed to equip new workers (n·k) and replace worn-out capital (δ·k).

Graphically, the concave curve s f(k) cuts the straight line (n + δ)k at k*. To the left of k*, investment exceeds breakeven investment, so k rises; to the right, it falls. The steady state is therefore stable.

At k*, capital per worker, output per worker and consumption per worker are constant, while total output and capital grow at rate n. With technical progress at rate g, per-worker output grows at g and total output at n + g. A higher saving rate raises k* and the level of income but not the long-run growth rate, and a higher n lowers k*.

(c) The golden rule of capital accumulation

The golden rule (Phelps, 1961) is the level of capital per worker, or the saving rate, that maximises consumption per worker in the steady state.

In the Solow model, steady-state consumption is c* = f(k*) − (n + δ)k*. Maximising with respect to k* gives f′(k*_gold) = n + δ: the marginal product of capital equals the breakeven rate (the rate of population growth plus depreciation). With technical progress at rate g, the condition is f′(k) = n + g + δ. The corresponding saving rate is found from s f(k_gold) = (n + δ)k_gold; for a Cobb-Douglas function y = k^α, the golden saving rate is s = α, the capital share.

Interpretation: if the economy has k* < k_gold (marginal product above n + δ), it accumulates too little capital, and a higher saving rate will raise consumption in the long run, after a short-run sacrifice. If k* > k_gold, it is dynamically inefficient, as it over-saves; a lower saving rate raises consumption at every date. The golden rule is a long-run benchmark and ignores the transition.