Lecture 6: The neoclassical growth model


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1 Lecture 6: The neoclassical growth model February 23, The main sources of divergence in GDP per capita across countries A first source of difference in GDP per capita is the stock of physical capital. Physical capital consists in the all the nonlabor inputs. e.g machines, vehicles, buildings, and other pieces of equipment. For example, in 2000 the average US worker had $148,091 worth of capital to work with; in Mexico in the same year, the capital per worker was $42,991 and in India it was $ Question: where do the differences in physical capital stocks come from? Answer: from differences in investment rates, which themselves come from differences in savings (savings rates, initial income) Second source of difference in GDP per capita: productivity, i.e how much output is produced with each unit of capital. 1. First source of productivity difference: technology, i.e the amount of knowledge accessible to producers in different countries. Technology is acquired through innovations (R&D investments) and imitation of technologies from more advanced countries. 2. Second source of productivity difference: efficiency, which relates more to the organization of the economy, institutions, and so on. Of course, to get a good understanding of the differences in GDP per capita and of growth rates across countries, one must go further and understand what ultimately determines savings, R&D investments, technological diffusion, productive efficiency in firms and markets. 2 The neoclassical growth model his lecture presents a capitalbased theory of why countries differ in their levels of income per capita, and why less advanced countries may or may not converge (in per capita GDP) towards more advanced countries. 1
2 2.1 The nature of capital Five main characteristics of (physical) capital: 1. Capital is productive: it raises the amount of output a worker can produce 2. Capital is itself produced. The process of producing capital is called investment. Because it is produced, capital requires sacrifice of some consumption. In the US, in 2000, 1.76 trillion dollars, i.e 18%, were spent on investment. 3. Capital is rival in its use: only a limited amount of people can use a given piece of capital at a given moment in time; this distinguishes capital from ideas, which in turn are produced through another kind of investment: R&D investment. 4. Capital earns a return from renting it. However some capital goods like roads and ports are built and owned by governments. 5. Capital depreciates, both due to physical obsolescence and also because the arrival of new technologies make capital goods that embody old technologies, become obsolete. 2.2 The first equation of the Solow model The first equation you already know: Y = AK α L 1 α, (1) where the lefthand side is the current flow of output, and the right hand side is equal to a technology parameter A times the capital contribution times the labor contribution. Remember that: K MPK = αy, so that α = K MPK Y is the capital s share of output. Figure 3.3 shows that share for a sample of 53 countries. The average is about 1/ The second equation of the Solow model Growth rate in discrete time, between t and t +1: g t (z) = z t+1 z t z t. 2
3 Growth rate in continuous time: g t (z) = dz dt /z = z z = bz; Second equation of the Solow model: dk = sy δk, dt that is: net current capital accumulation is equal to total current investment (itself equal to total savings in equilibrium of the goods market) minus total current depreciation of the capital stock. 2.4 Steady state Fix labor supply at L =1. The steadystate level of capital is simply determined by: dk dt =0, or equivalently sy = sak α = δk, or equivalently again: K = K ss =( sa δ ) 1 1 α. This steady state is stable, in the sense that starting from a level of capital K<K ss, capital will accumulate until the capital stock reaches its steadystate level K = K ss ; similarly, starting from a level of capital K>K ss, capital will decumulate until the capital stock reaches K = K ss. To this steadystate level of capital corresponds a steady state level of output: Y ss = A(K ss ) α = A 1 α 1 s ( δ ) 1 α α. Remarks: 1. This latter equation provides us with a theory of the sources of (longrun) income differences across countries: in particular (per capita) GDP across two countries may differ either due to differences in productivity as captured by A, or because of differences in savings or investment rate rate as captured by s. Thus, suppose, that we abstract from differences in technologies and concentrate in differences in savings or investment rates, and compare the predicted ratio of income per worker in each country to income per worker in the US, to the ratio of actual incomes between that country and the US. We are very far from the 45 0 line. In particular, Uganda should have a 3
4 GDP per capita of around 33% of US GDP per capita, however the true ratio is only 3%. Why? Maybe Uganda still lies very far from its steadystate, whereas US lies close to its steadystate. 2. The Solow model predicts no growth of GDP per capita in the longrun, since output per capita converges to a fixed value Y ss. The reason for this nogrowth in steady state result, is the assumption of decreasing returns to capital accumulation and therefore of decreasing savings to output ratio, whereas depreciation occurs at a constant rate; so, eventually, depreciation catches up with savings. 2.5 Convergence The Solow model predicts crosscountry convergence: namely, a country very far from its steadystate will grow very fast, whereas a country very close from its steadystate will grow very slowly. For example, suppose that all countries have the same savings rate, productivity parameter, and depreciation rate, but the only difference between them is that they start from different levels of capital stocks (and therefore from different income levels). The growth rate of capital stock is equal to: bk = dk dt /K = sakα 1 δ, thus since α<1, we see that the higher the current level of capital, the lower the growth rate of the capital stock, and therefore thelowerthegrowthrateofoutput since by = αk. b Thus the country will lower current level of its capital stock will grow faster than the country with the higher level of its current capital stock. Note also that the Solow model leads to the following additional predictions: 1. if two countries have the same rate of investment but different levels of income, then the country with lower income will grow faster; 2. if two countries have the same level of income but different rates of investment, then the country with a higher rate of investment will have higher growth; 3. a country that raises its level of investment will experience an increase in its growth rate of income. While the Solow model predicts convergence to the same steady state for all countries or regions with similar savings rates s, depreciation rates δ and productivity parameter A, it does not explain why some countries manage to converge towards the most advanced countries and why others 4
5 stagnate and do not converge. This phenomenon we refer to as club convergence. The Solow model also does not explain why some countries started to grow very fast and then stopped converging. While the convergence prediction appears to be (partly) verified by crosscountry (or by crossstate data within the US), the prediction of zero longrun growth is at odds with the evidence that growth in developed countries has been sustained at 2 to 3% per year. So, how can extend the Solow model to account for positive longrun growth? 1. allow for technical progress in the form of a permanent growth in productivity A. However, the problem is that we cannot explain how this technological progress will be remunerated, especially since the Euler theorem implies that once you pay capital and labor at their marginal productivities, you simply exhaust total output Y and therefore there is nothing left to pay innovators. 2. allow for population growth; however this will affect the growth of total output, not the steadystate result on output per capita: suppose Y = AK α L 1 α, with L = e nt = y = Ak α, where y = Y/L and k = K/L. Also, one can show that equation (2) implies: dk = sy δk. dt So, we are back to the previous model, which implies that k converges to a steadystate level k ss and y converges to a steadystate level y ss. This implies that in the long run, total GDP Y grows at same rate as population, that is at rate n, but per capita GDP stops growing in the longrun. 2.6 A first glimpse at the AK model A first attempt at endogeneizing the longrun growth rate, was to say that even though individual firms may experience decreasing to capital accumulation at an individual level, yet, because of externalities in learning by doing across firms (or externalities in capital accumulation across firms), at the aggregate level the production function might exhibit constant returns to capital only, namely: Y = A 0 K. This view of the world identifies knowledge accumulation with capital accumulation by all firms simultaneously. 5
6 Now, let us replace the first equation in the Solow model by this equation, and put it together with the second equation of the Solow model; we have: and therefore: dk dt = sa 0K δk, bk = sa 0 δ. We thus get a positive longrun rate of growth, which depends positively on the savings rate and negatively on the depreciation rate. However, if we restore positive longrun growth, we also lose the convergence result of the Solow model since all countries no matter their current capital stock growth at the same rate. Therefore less advanced countries can no longer catch up with more advanced countries since they do not grow faster than more advanced countries. Note that the learningbydoing externality must be exactly right in order to obtain an aggregate AK function. More generally, externalities will lead us to an aggregate function of the form: Y = A 0 K β, where β>α. If β>1, then we get explosive growth ( K b increases with K), whereas if β<1 we obtain zero growth in the longrun as we are back to the Solow model but with a higher coefficient of capital. Early attempts at justifying the AK model on empirical grounds, relied on the observation that the Solow model with α =1/3 predicts too fast a speed of convergence compared to what we seem to observe across countries or across US states. However, next time we shall see that there is a simple way to extend the Solow model so as to deal with this problem: namely, by adding human capital on top on physical capital and labor as a third factor of production. In any case, from this subsection we have seen that growth models based on capital accumulation can not explain convergence and longrun growth simultaneously. And they do not account for several important aspects of convergence/divergence across countries. 6
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