The AK model and learning by doing

By the 1980s the Solow model faced three complaints. First, growth is assumed, not explained: g, the rate of technical progress, is a number the modeller supplies, and it accounts for all long-run growth per head. Second, it predicts convergence, but income per head across the world has not narrowed in the way the model suggests (Lesson 8). Third, it cannot explain very large differences between countries without implausible differences in the return to capital, which is Lucas’s paradox.

Paul Romer’s 1986 paper began from these facts: growth rates in the leading economies had not fallen as diminishing returns would suggest, and they did not converge. The way out is to find a source of growth that does not run into diminishing returns. Easterly and Levine (2001) argued the same from the other direction: most of the differences in income across countries are not explained by the accumulation of factors, which is the Solow residual of Lesson 3 again.

Drop diminishing returns. Let output be proportional to capital, $Y=AK$, where A is a constant. Per head, y = Ak. Keep everything else from Solow: a saving rate s, depreciation δ and population growth n. The accumulation equation is

Compare this with Solow’s growth rate of capital, s f(k)/k − (n + δ), which fell as k rose because f(k)/k fell. Here f(k)/k = A is constant, so the growth rate of capital is the same at every level of capital. If sA > n + δ, the economy grows at the rate γ from the start and for ever: no transition, no steady-state level. Output per head, y = Ak, grows at the same rate.

Two economies start on the same path, growing at 2% a year. In year 0 the saving rate rises in both. Move the sliders and watch what happens. In the Solow economy (blue) growth jumps and fades back to 2%, and income settles on a higher parallel path. In the AK economy (red) growth jumps to a new rate and stays there, so income pulls away without limit. This is the difference between a level effect and a growth effect from Lesson 6, on one chart.

The lesson of the lab is the policy contrast. In a Solow world, anything that raises saving or investment raises income once. In an AK world it raises growth, so even small differences in saving, taxes or the efficiency of investment compound into large differences in income. That is why endogenous growth theory became the language of growth policy.

Constant returns to capital looks like an assumption of convenience. Why would an extra machine be as productive when there are a thousand of them as when there are ten? The answer of Arrow (1962) and Romer (1986) is that capital does two things: it produces goods, and it teaches people how to make them better. Firms learn from the capital they install. If what one firm learns is available to others, because workers move and engineers copy designs, then there is a spillover.

Take firms with the production function $Y_i=K_i^{\alpha }(A{L}_i{)}^{1-\alpha }$, with diminishing returns to each firm’s own capital. Let knowledge be the economy’s whole stock of capital, A = K, which each firm takes as given. Add up over firms with total labour L:

Each firm faces diminishing returns, but the economy as a whole is an AK model with A = L^1−α. Two things follow. First, growth rises with the size of the economy (L), a scale effect that returns in Lesson 11. Second, there is an externality: a competitive firm is paid its private marginal product of capital, which is only a fraction α of the social one, so firms and households invest too little. The market growth rate is below the best one, and a subsidy to investment can raise welfare.

Lucas (1988) gave the broad-capital argument a concrete form. People can spend their time producing goods or acquiring skills. In its simplest form the growth rate of an individual’s human capital h is proportional to the share of time spent learning, 1 − u:

where φ measures how productive learning is. There are no diminishing returns to h in its own accumulation: a person who has learned more learns more easily. Output per head grows at a rate proportional to the growth of h, so growth is determined by how much time a society chooses to spend learning. Lucas added that the average level of human capital raises everyone’s productivity (a spillover again), which makes the market allocate too little time to learning.

Two pieces of evidence cut against the sharpest claims. Charles Jones (1995) looked at the OECD since 1950: investment rates in these economies rose substantially, yet growth rates did not rise with them, which is the opposite of what an AK model with a permanent growth effect predicts. And Lesson 8 showed that conditional convergence at about 2% a year fits the data, which an AK model with no transition cannot match.

A second problem is conceptual. The AK model needs the exponent on broad capital to be exactly one. At 0.99 the economy would converge to a stable steady state with no growth, and at 1.01 it would explode. This knife edge is the same kind of fragility as Harrod and Domar’s (Lesson 4), which is why many economists moved on to models in which growth comes from ideas, not from capital (Lesson 11).

Questions on the AK model, on endogenous growth against Solow, and on what raises long-run growth, from IIT JAM, UGC NET and Hyderabad. The two Hyderabad questions on saving as the source of long-run growth are the everyday version of this lesson: the key treats saving as the answer, which in a Solow world would be a level effect, so read them with the AK model in mind.

Next: ideas and R&D, where growth comes from the production of knowledge, not from the accumulation of capital.