# ( ) = ( ) = ( + ) which means that both capital and output grow permanently at a constant rate

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1 1 Endogenous Growth We present two models that are very popular in the, so-called, new growth theory literature. They represent economies where, notwithstanding the absence of exogenous technical progress, output per capita grows permanently. 1.1 AK Model Consider the production function = which is linear in the aggregate capital stock. Assume population grows at rate. Denoting per-capita variables with small letters, the growth rate of output per capita is therefore = equal to the growth rate of capital per capita. The Solow growth equation, in per capita terms, is = ( + ) = ( + ) which means that both capital and output grow permanently at a constant rate = ( + ) As long as ( + ) this economy displays positive long-run growth, notwithstanding the absence of exogenous productivity growth. This class of models where output per capita grows without the need of exogenous technical progress are called endogenous growth models. Notice that this (linear) economy is a limiting case of the Solow model as the capital share 1 When =1 the decreasing returns in production which are the force that impede permanent growth in the standard Solow model, disappear and output is produced with constant returns to capital. 1

2 An alternative, complementary, way to explain why this economy displays endogenous growth is that the reproducible factor (capital in this case) is produced without decreasing marginal returns, i.e. investment (new capital) can be generated with a production structure that is linear in physical capital = = The lesson we learn from the model is about the mathematical structure of models that is needed to generate long-run growth. We need some way to stop the decreasing marginal returns to the factors of production. Finally, note that in the model there is no conditional convergence. Take two countries with the same set of parameters ( ) but two different initial conditions for capital 0 (and hence income 0 = 0 ) The initial gap will perpetuate forever, since the two economies will grow at the same rate. 1.2 Human Capital Consider an economy without physical capital, but with human capital, where output is produced with = [ ] 1 The term denotes the fraction of time endowment (normalized to 1) devoted to production by each worker. So, in per capita terms = 1 (1) where = Human capital per worker is accumulated over time with = (1 ) (2) i.e., the fraction of time 1 is devoted to human capital accumulation. For example, one could think of 1 as the fraction of years spent in school, or fraction of time devoted to on-the-job learning at work. What is the growth rate of output per capita in this economy? Differentiating (1) with respect to time, we arrive at = +(1 ) (3) We are interested in a balanced growth path where { } grow at the same, constant rate forever. This is the equilibriumwealways focusoningrowthmodels 2

3 becausethedatasuggestthat,inthepast,thegrowthexperienceonmanycountrieshas followed this pattern [remember the Kaldor facts?]. Now, can grow at a constant rate indefinitely? No, it s a fraction of time, so along the balanced growth path it must be constant at some value independent of time. 1 Hence, =0and, using (3) and (2) we conclude that the growth rate of output per capita in the long-run is = = (1 ) (4) Output growth is faster the larger is the share of human capital in production the higher is the productivity of human capital accumulation and the more time is devoted to learning (1 ) Why do we have endogenous growth in this model? Because there are no decreasing returns in the production of the reproducible factor, human capital in this case, i.e., the accumulation function of human capital (2) is linear. How do we take stock of the human capital model? If we stare at equation (4) we learn three lessons: 1. The productivity of the education sector (the parameter ) translatesintofaster growth. Countries with better education system, which make the best use of resources to train students, which have the best teachers, etc., will produce more human capital. 2. The share of human capital in production translates into faster growth. Think of Nigeria and the US. In Nigeria the share of agriculture in production is much larger than in the US and agriculture is not a very human capital-intensive sector. Thus human capital accumulation in Nigeria cannot generate too much growth anyway because it impacts only a small fraction of sectors in the economy. 3. The time devoted to education and training (1 ) translates into faster growth. This is something we saw in the stylized facts about income per capita differences: years of schooling of the population and income per capita are very positively correlated. 1 The value of along the balanced growth path can be determined precisely as a function of the parameters of the model, but we omit this derivation here. 3

4 1.3 Research and Development (R&D) We now develop a growth model based on R&D. By R&D we denote all the innovative activities done by the set of people who are, broadly speaking, innovators (scientists, researchers, managers,...). R&D activities advance the frontier of knowledge of society, hence its productivity. However, note that only few countries in the world (the most advanced ones, e.g. the U.S., Europe, Japan) devote large resources to research activities, so this is a model that at most can explain the different growth experiences of those countries. To think about North-South differences (e.g., US and Nigeria) one should explain why underdeveloped countries do not adopt innovations developed by the North, not so much why they do not innovate themselves. Lack of human capital may be one of the reasons. Let s present the model. Consider an economy where labor grows at rate. At every point in time, workers can allocate their time endowment between two sectors: production and research and development ( & ). The production sector produces output with = [ ] 1 where is the level of knowledge generated through & and is the time devoted to production. In per capita terms (i.e., dividing by ), the above equation becomes = [ ] 1 The research sector produces new ideas with = [(1 ) ] (5) hence new ideas necessitates past ideas and time from workers (e.g., scientists) to be developed. When 1 there are decreasing returns to scientists in the production of ideas and when 1 there are decreasing returns to existing knowledge. Along the balanced growth path output and capital grow at the same rate and the fraction of time devoted to labor in each sector has to remain constant (similarly to the human capital model), thus = +(1 ) = (6) What is the growth rate of knowledge? Dividing equation (5) by we obtain = 1 [(1 ) ] (7) 4

5 Differentiating once more this equation with respect to time, and imposing the balanced growth conditions that 1) the growth rate of knowledge is constant (since the growth rate of output is constant by assumption), and 2) the allocation of labor in production is constant at,wehave 0=( 1) + which implies that output grows at the long-run rate Some remarks are in order: = = = Growth here is in part exogenous, because it depends on exogenous population growth. Indeed these models are called semi-endogenous growth models. The reason we have long-run growth in income per capita thanks to population growth is that as population grows, more workers can be allocated into the & sector which offsets the decreasing returns to ideas when 1 2. Clearly, when =1and ideas are produced without decreasing returns, growth becomes fully endogenous. From equations (6) (7) and =1we obtain = = [(1 ) ] (8) This equation also clarifies that we don t need population growth in this case to have permanent growth. A problem arising in this case though is that there are scale effects in growth, i.e. the larger is the economy, in terms of population, the faster should be growth. This conjecture which is not supported by the empirical evidence... although that was before China and India took off. 1.4 The Role of Government Policy: Exogenous vs Endogenous Growth There is a key difference between the effect of policies in the Solow model with exogenous technical change and in endogenous growth models. In endogenous growth models, government policies have effects on the long-run permanent growth rate of the economy, 5

6 while in the Solow model with exogenous growth they only affect the steady-state income per capita, and therefore, only transitional growth towards the new steady state. For example, a subsidy to time spent in education in the human capital accumulation economy will increase (1 ) and the long-run growth rate of output, whereas a subsidy to saving in the Solow model only increases steady-state output and affects growth temporarily, until the economy converges to the new steady-state. Many of the structural reforms that would induce faster long-run growth, such as a subsidy to education would induce, in the short run, a decrease in the growth rate, as the economy diverts resources away from production into accumulation of human capital. Therefore, in the transition towards the new growth regime, output may be growing at a slower rate, or may be even lower. The nature of structural reforms is that they trade lower output in the short run for higher output in the long run. For this reason, structural reforms are so difficult to accept. 6

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