1. The model’s inner force (the diminishing returns to scale of capital, in particular) takes the economy smoothly to its Steady State. This is the state where output and capital grow at the same rate as population (labour). This means that per-capita income and capital-labour ratio (K/L = k) are constant at the steady state.

Implications of the Neoclassical (Solow’s) Growth Model

3. An increase in saving rate (or any policy change to stimulate income) increases per-capita income but do not leave lasting effect on growth rate. The economy moves to a new steady state (with higher per-capita income, higher k and lower marginal productivity of capital). Once the economy reaches the new steady state, per-capita growth would be nil again (i.e total income will increase at the rate of population growth.)

4. An increase in population growth leads to lower per-capita income and lower k. But the economy settles back again at a new steady state whereby K and Y will be growing at the same new rate of population.

2. Once the steady state is reached, the growth of per-capita income can come only from technical progress which is exogenous and un-explainable by the

model.

Implications of the Neoclassical (Solow’s) Growth Model

7. If full capital mobility between countries exists (thus all countries would have identical k and the same capital productivity) then the growth rate as well as level of per-capita income would be the each and every country.

5. The level of per-capita income depends on k, and the growth of per-capita income (before reaching the steady state) depends on the marginal productivity of capital (thus also on k). This means that countries with relatively low k (low volume of capital stock relative to labour, as the LDCs) experience lower per-capita income but faster per capita growth (because lower K means higher capital productivity) than countries with high k.

6. In absence of international capital mobility, various countries will reach their steady state at different levels of k, thus with different levels of per-capita income. However, since per-capita growth in the steady state depends exclusively on exogenous technical progress which is available to all countries, then all countries will grow at the same rate.

Growth AccountingStart with a Cobb-Douglas production function:

Y =A Kα . L1-α

Where Y is output, K is capital , L is labour, A is a constant (to secures homogeneity of measuring units) and α is a parameter.

Cobb-Douglas function is a standard neoclassical function which exhibits:

• A considerable substitution possibility between the two inputs K and L. The elasticity of substitution is constant and equal one.

• Constant returns to scale with respect to all production inputs: α + (1 - α ) = 1.

• Diminishing returns to scale with respect to each input: α < 1 and 1- α < 1. This means that successive increases in K (or in L) lead to smaller and smaller increases in Y. Technically, the same can be expressed by saying that the first partial derivative of Y (with respect to L or to K) is positive, while the second is negative.

Growth Accounting

The partial derivative of Y with respect to labour gives the marginal productivity of labour (MPL):

ΔY/ΔL = (1- α) L1- α -1 . A Kα = (1- α ) . A Kα . L1- α / L = (1 - α ) . (Y/L). On the other hand, MPL matches the real wage rate in equilibrium. Thus:

(1- α ) Y/L = w/p (2)

The first partial derivative of Y, with respects to K, measures the increase in output due from deploying an extra unit of capital. This is the marginal productivity of capital (MPC):ΔY/Δ K = (α Kα-1 ) . A L 1- α = α . A Kα . L1- α / K = α . (Y/K)On the other hand, MPC = real rental cost of capital (the real interest rate) in equilibrium: α (Y/K) = r / p (1)

Y =A Kα . L 1-α

Growth Accounting

• Competitive equilibrium ensures that production factors receive their ‘fair’ shares of income, i.e. rewards to labour and capital equal their contributions in the production. I.e. the factors of production are paid according to their marginal productivities.

• The share of labour in total income is (MPL) . L/Y or (w/p) . (L /Y)

• Substituting the value of w/p from (2), the share of labour in Y (in the GDP) would equal (1- α) .

This is an interesting result. It affirms that the parameters of the Cobb-Douglas production function, 1- α and α, are non other than the shares of labour and capital in total GDP. Empirically, the actual share of labour in the GDP was found to be about 2/3 in the US. This means that (1- α ) = 2/3, and therefore the share of capital in the GDP is α = 1/3.

• The production function becomes then Y = AK1/3 . L2/3 , and this can be approximated by the following expression:

gy = 1/3 gk + 2/3 gl where gy, gk and gl are growth rates of income, capital and labour.

• Thus, income growth during a certain period should match the growth in capital and labour weighted by their shares of the GDP.

• If the left-hand side turned to be larger than the right-hand side, the difference (the residual) is thought to be due to improved efficiency, a result of productivity growth.

Growth AccountingTherefore, once we know the extra amounts of K and L deployed, the production function Y = A K1/3 . L2/3 gives the increase in Y (or GDP) due from these extra inputs. However, when this is compared with the actual growth rate in Y, the exercise can help to test the accuracy of the equation, and to determine the effects of other variables which are not taken into account in the production function.

When actual historic growth rates of capital, labour and GDP in the US, for the period 1890-1995, were inserted in the production function, it appeared that some 55% of the actual growth in the GDP was not explained by the function, i.e. was neither due from deployment of more capital or more labour.

This unexplained growth is called Solow’s Residual: the portion of actual growth in GDP which is not explained by the use of more inputs. Solow referred to it as “a measure of our ignorance”. Economists argue that this unexplained growth results from improvements in total factor productivity (TFP). The increases in TFP refer to improvements in efficiency, in inputs’ productivities. The efficiency gains allow for producing more output from the use of the same amounts of inputs.

Growth Accounting• Several empirical studies have been conducted (on basis

of the above explained methodology) to figure out the sources of actual growth in various parts of the world.

• For example, average annual growth rate in Latin America during 1944-85 was 5.3%. The sources of this growth were as follow:

Needless to say that the TFP is the most critical factor in determining long term growth, since all societies will confront ultimate limits on the amounts of labour and capital they can deploy. One study (Klenow et al, 1997) shows that 60% of income per capita differences among countries in 1985 cannot be explained by differences in physical or human capital. For an overview of growth accounting in the LDCs see Agénor et. al. (1996) chapter 15.

1.3% (or 25% of the total) came from more labour

2.5% (or 47% of total) came from more capital use

1.5% (or only 28% of the total growth) resulted from better TFP

Convergence

• Absolute convergence: poorer countries (i.e. countries or regions with lower k, thus with lower per-capita income) will grow faster than richer countries. This means that per-capita income in all countries will smoothly converge (and will be the same everywhere if full capital mobility prevails.)

Numerous empirical studies are devoted to test these hypothesis. The results are mixed: convergence is well documented among regions in a single country. Also conditional convergence is more or less accepted among the rich countries, although it happens at a much slower rate than predicted by the model. As to the LDCs, the difficulties of identifying the steady state (and the corresponding per-capita income) in each country lead to widely contradicting results. (see, for example, Mankiew, Romer & Weil, 1992)

One of the most important implications of the neoclassical growth model is the convergence hypothesis:

• Conditional convergence: when a pair of countries are reasonably close to their steady states, the country which is further away from its steady state will grow faster than the one which is closer. This is a weaker proposition than the above, since it permits rich countries to grow faster than the poor.

Endogenous growth models

The strong implications of the neoclassical growth model (policy irrelevance, convergence, steady states, etc.) are results of one single assumption: the diminishing marginal productivity of capital. Endogenizing technical progress (knowledge) in the model requires neutralizing this assumption. This opens the possibility for continuous capital accumulation and for unlimited growth in per-capita income.

But neutralizing the diminishing marginal productivity of capital means that the whole production function would exhibit then increasing returns to scale (rather than constant returns.) The fact that production with increasing retunes (which leads eventually to monopolization of markets) is overly incompatible with competitive general equilibrium framework explains the reluctant of economists to explicitly endogenize technical progress.

In other words, to secure boundless growth (thus to escape the inevitable convergence), a non-diminishing (increasing or constant) capital productivity should be incorporated in the model. This can be done by introducing technical progress and assuming that this progress depends on capital accumulation. Once this is done, the production function (with capital, labour and technical progress as inputs) would exhibit increasing returns to scale. With increasing returns, perfect competitive solution (where products are priced at their marginal costs, and production factors are rewarded in accordance with their marginal productivities) cannot be secured.

Endogenous growth: two approaches

The AK models. Allowing for increasing returns while maintaining the norms of competitive equilibrium is done in a somewhat ”tricky” manner. The approach assumes that capital has two different effects on output: a direct (private) effect which is rewarded by the usual marginal productivity, and an indirect social effect. The social effect is a by-product, accidental and unintended by those who make investment decisions. This by-product effect leads to higher capital productivity for the economy as a whole and eliminate the diminishing returns. The indirect effect works either via improving the efficiency of capital (Phelps’s “embodied technical progress”), or via improving the effectiveness of labour (Arrow’s ”learning by doing”) or via cultivating the ”human capital” (Lucas model)..

In their attempts to endogenize growth (and thus neutralize the diminishing marginal productivity of capital), economists have followed two different approaches: The first approach (the so-called AK models) neutralize the diminishing marginal productivity of capital while maintaining the overall environment of perfect competitive equilibrium. The second, endogenized growth and abandoned perfect competition altogether.

The AK models then attempt to resolve the dilemma between allowing for non-diminishing returns and securing competitive equilibrium by suggesting that private investment brings about certain externalities (spill-over like technical progress, education, etc) which benefits the society as a whole. The private returns on investment are still the (diminishing) marginal product but the total benefits for the economy as a whole are much higher: capital exhibits constant returns for the society as a whole. This approach is known as AK, because it suggests an aggregate (social) production functions of the type Y = AK , with no diminishing marginal product (and thus no necessary slow down or convergence.)

A true endogenous growth approach. The second approach abandoned the competitive framework explicitly (Romer, Grossman & Helpman). Technical progress (or knowledge) is taken explicitly here as a separate and independent production factor, rather than as an accidental by-product. Knowledge (ideas) has its own stock and own production (generation) process. Investment in creating new ideas is a conscious effort carried out by firms in order to acquire monopoly rent. The deployment of new ideas in producing consumers goods has increasing returns, and therefore growth can be limitless and is internally regenerated. The production and consumption of knowledge are substantially different than the case for other ‘normal’ goods. Creating new ideas requires high initial cost, but once they are discovered, ‘consuming’ them and reproducing them are almost costless. Consumption of ideas, is non-rivalrous, although potentially excludable (by patents). These unique characteristics of producing and consuming knowledge violate the principles of marginal cost pricing in the perfect competitive model.

Endogenous growth: two approaches

The AK approach was successful in escaping the strong, and probably unrealistic, implications of Solow’s model. It also allowed public policies to play a role in affecting growth (for example in encouraging public investments in project with high ‘social’ returns). Yet, the approach remained problematic and not totally convincing.

The Significance of Endogenous Growth Theory

”By placing technological change at the heart of the theory of growth, by postulating that the generation of new knowledge results from conscious decisions by economic agents, and by highlighting the special characteristics of this new knowledge, Romer is able to show that economic growth is no longer captive to the saving ratio or to exogenous technology but can indeed be directly influenced by a conscious policy of investment in new designs and, more generally, in knowledge-generating activities. In addition, because of the special nature of knowledge itself, achieving growth may require specific policy intervention to promote innovation. … Additional works is ... just beginning to enrich our understanding of the complex process of growth and to explain the facts of recent growth experiences better: … That differences in growth rates can persist for long periods of time, that globalization does promote growth, that large population size may not be sufficient for economies of scale, that intra-industry trade among the developed countries is more important than trade between developed and developing nations, that quality of human capital migrates from areas where it is scarce to areas where it is abundant, and that the spread of technology corresponds to the observed product cycle in international trade”. G. Abed, IMF