Setting this equation equal to zero (the steady state condition) and substitut-

ing into the production function yields (11.39):

y * = [s/(n + δ )]±/(1’± ) (11.39)

Equation (11.39) is now in a form that enables a solution to be found for the

steady state output per worker (y*). As Jones (2001a) highlights, we can see

from equation (11.39) why some countries are so rich and some are so poor.

Assuming exogenous technology and a similar value for the capital exponent

624 Modern macroeconomics

(±), countries that sustain high rates of saving, and low rates of population

growth and depreciation, will be rich. According to the neoclassical growth

model the high-income economies have achieved their high living standards

because they have accumulated large per worker stocks of capital. However,

although the model correctly predicts the directions of the effects of saving

and population growth on output per worker, it does not correctly predict the

magnitudes. As Mankiw et al. (1992) and Mankiw (1995) argue, the gaps in

output per worker (living standards) between rich and poor countries are

much larger than plausible estimates of savings rates and population growth

predict using equation (11.39). The crux of the problem is that with ± = 1/3

there are sharply diminishing returns to capital. This implies that a tenfold

gap in output per worker between the USA and India would require a thou-

sandfold difference in the capital“labour ratios between these countries! (It

should be noted that this result is highly sensitive to the choice of ± = 1/3 for

the share of capital in GDP.)

A third problem with the Solow model is that given a common production

function (that is, exogenous technology) the marginal product of capital

should be much higher in poor countries than in rich countries. Given the

parameters of the Solow model, the observed tenfold differential in output

per worker between rich and poor countries implies a hundredfold difference

in the marginal product of capital if output gaps are entirely due to variations

in capital intensities. Such differentials in the rate of return to capital are

simply not observed between rich and poor countries. As David Romer

(1996) observes, such differences in rates of return ˜would swamp such

considerations as capital market imperfections, government tax policies, fear

of expropriation and so on and we would observe immense ¬‚ows of capital

from rich to poor countries. We do not see such ¬‚ows.™ But the rate of return

to capital in poor countries is less than expected and the anticipated massive

¬‚ows of capital from rich to poor countries have not been observed across

poor countries as a whole (Lucas, 1990b).

A fourth dif¬culty relates to the rate of convergence, which is only about

half that predicted by the model. The economy™s initial conditions in¬‚uence

the outcome for much longer than the model says it should (Mankiw, 1995).

In conclusion, it appears that within the Solow growth framework, physical

capital accumulation alone cannot account for either continuous growth of

per capita income over long periods of time or the enormous geographical

disparities in living standards that we observe. In terms of Figure 11.3, the

data on output per worker (or income per capita) that we actually observe

across the world reveal much greater disparities than those predicted by the

Solow model based on differences in capital per worker.

The new growth models emerging after 1986 depart from the Solow model

in three main ways. One group of models generates continuous growth by

The renaissance of economic growth research 625

abandoning the assumption of diminishing returns to capital accumulation.

To achieve this, Paul Romer (1986) introduced positive externalities from

capital accumulation so that the creation of economy-wide knowledge emerges

as a by-product of the investment activity of individual ¬rms, a case of

˜learning by investing™ (Barro and Sala-i-Martin, 2003). A second approach

models the accumulation of knowledge as the outcome of purposeful acts by

entrepreneurs seeking to maximize private pro¬ts; that is, technological

progress is endogenized (P. Romer, 1990). A third class of model claims that

the role of capital is much more important than is suggested by the ± term in

the conventional Cobb“Douglas production function shown in equations

(11.28)“(11.30). In their ˜augmented™ Solow model, Mankiw et al. (1992)

broaden the concept of capital to include ˜human capital™. The ¬rst two

classes of model constitute the core of endogenous growth theory whereas

the Mankiw, Romer and Weil (MRW) model constitutes what Klenow and

Rodriguez-Clare (1997a, 1997b) call a ˜neoclassical revival™. The central

proposition of endogenous growth theory is that broad capital accumulation

(physical and human capital) does not experience diminishing returns. The

growth process is driven by the accumulation of broad capital together with

the production of new knowledge created through research and development.

11.14 Endogenous Growth: Constant Returns to Capital

Accumulation

During the mid-1980s several economists, most notably Paul Romer (1986,

1987b) and Robert Lucas (1988), sought to construct alternative models of

growth where the long-run growth of income per capita depends on ˜invest-

ment™ decisions rather than unexplained technological progress. However, as

Crafts (1996) notes, the term investment in the context of these new models

refers to a broader concept than the physical capital accumulation reported in

the national accounts; research and development (R&D) expenditures and hu-

man capital formation may also be included. ˜The key to endogenous steady

state growth is that there should be constant returns to broad capital accumula-

tion™. Hence in order to construct a simple theory of endogenous growth, the

long-run tendency for capital to run into diminishing returns needs to be

modi¬ed to account for the extraordinary and continuous increases in observed

per capita incomes across the world™s economies. In the early versions of the

new endogenous growth theory the accumulation of capital plays a much

greater role in the growth process than in the traditional neoclassical model. In

many ways the work of Romer revives the earlier seminal contribution of

Arrow (1962) on ˜learning by doing™. Arrow had shown how the productivity

of labour increases with experience, and experience is a function of cumulative

investment expenditures that alter the work environment. That is, a ¬rm™s

626 Modern macroeconomics

accumulation of capital produces external effects on learning. However, as

Blaug (2002) argues, ˜it strains credulity to believe that this could account, not

just for a once-and-for-all increase in output, but also for a constant rate of

increase in total factor productivity year in year out™.

Building on Arrow™s insight, Romer broadened the concept of capital to

include investment in knowledge as well as the accumulation of physical

capital goods. Since the knowledge gained by workers in one ¬rm has public

good characteristics and is at best only partially excludable, then knowledge

spillovers occur such that investment in knowledge (R&D) by one ¬rm in-

creases the production potential of other ¬rms. No individual ¬rm can

completely internalize the positive impact that their investment in physical

and human capital has on the economy-wide stock of knowledge.

Paul Romer™s 1986 model can be illustrated by modifying the production

function. In equation (11.40) the production function includes technology (A)

as an endogenous input:

Y = F( K , L, A) (11.40)

At the micro level, the output of an individual ¬rm (j) depends on its own

inputs of capital (Kj), labour (Lj) and the economy-wide state of knowledge

(A), as indicated in equation (11.41):

Yj = F( Kj, Lj, A) (11.41)

In this formulation the growth of knowledge (technology) is assumed to

depend on the growth of capital because capital deepening fosters technologi-

cal spillovers that raise the marginal productivity of capital across the economy

as a whole. Therefore any increase in aggregate K will improve A and hence

the productivity of all ¬rms. In Romer™s (1986) endogenous growth model

the expansion of aggregate knowledge results from learning externalities

among ¬rms. In effect, the higher the level of the capital stock in an economy,

the more productive each ¬rm will be via a process of ˜learning by doing™. So

while a ¬rm™s production function exhibits constant returns to scale and

diminishing returns to capital accumulation, the aggregate production func-

tion will exhibit increasing, rather than constant, returns to scale.

One of the simplest models of endogenous growth is the AK* model shown

in equation (11.42) below (Rebelo, 1991):

Y = K ± H β = AK * (11.42)

Here A is a constant, K* represents a broad measure of capital (K± Hβ), and ±

+ β = 1. As Crafts (1995) points out, ˜models of this kind put investment

The renaissance of economic growth research 627

centre stage and see growth as an investment-driven process. There is no role

for the Solow residual.™ Therefore there is a close similarity between the AK

model and the Harrod“Domar model. In both models there are no diminish-

ing returns and hence no reason for growth to slow down as capital deepening

occurs. If one group of countries has higher average savings rates, lower

depreciation rates and lower capital“output ratios than some other group of