Traditional growth models Pasquale Tridico

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1 1. EYNESIN THEORIES OF ECONOMIC GROWTH The eynesian growth models are models in which a long run growth path for an economy is traced out by the relations between saving, investements and the level of output. The Harrod- Domar model was the most used in the literature before the neoclassical model of the middle 1950s. Both authors (Harrod in 1939, and Domar in 1946) developed it independently and it is a model that uses two key concepts from eynesian economics: the multiplier, and the accelerator of investments. The importance of savings (S) and investment (I) is the central line in the work of Harrod and Domar. In particular, the Domar model (1946) starts from the eynesian multiplier (1/1-c), where c is the propensity to consume) as follows: From the eynesian multiplier theory, we know that the variation of income Y= (1/1-c) I or Y/ I = (1/s) Y = ( 1 ) I (2.1) s Investments increase capital stock: (No depreciation) I =. Given that v = / X, where X is productive capacity, and is capital. The Hypothesis here is that the ration / X is constant through time. Remembering that I = we will have I = v 1 = v X X I X = I v (2.2) In order to have equilibrium the effective demand Y should be equal to productive capacity X. Changing in Y= X (2.1.) and (2.2) or (1/s) I = I / v we will have: I/ I = s/ v 1 (2.3) Hence, according to this theory, there are two determinants of the rate of growth of a country. The first looks at the relationship between changes in the capital stock of a country (capital-output ratio, v). This shows how much new capital is needed to create a given amount of new national income. The second element of the model considers the relationship between savings and national income (savings ratio, s), and this shows how much is saved from a given amount of national income. The model indicates how these two ratios affect the rate of growth. Therefore, the growth rate of basic Domar relationship for an economy is the following: g=s/v (2.4) 1 We know that I/ I = s/v is equal to Y/ Y (=g). In fact from (2.1.): Y= (1/s) I divide by Y. Y/Y= (1/s) I/Y s=s/y and S=I in equilibrium. Hence, Y/Y= (Y/I ) I/Y Y/Y= (Y/I ) I/Y Y/Y= I/I = g = s/v

2 Essentially, the higher the savings ratio, the more an economy will grow; and the higher the capital-output ratio, the lower the rate of growth, because it shows low efficiency of capital in the economy. The Harrod model (1939) reaches the same results introducing however the fundamental elements of expectations (e) concerning investments. In his model, the decisions about investments are taken on the basis of the accelerator principle, i.e. they depend on expected variation of Y: I e t v( Yt 1 t Y ) In order, for the expectations of investors to be verified, it is needed that: Yt+1=Yt+1 expected. Hence: I t v( Yt 1 t Y ) Where, Yt+1-Yt= Y Hence: sy= v Y Y /Y= s/v = Gg Gg = guaranteed rate of growth = ratio between Propensity to save s and capital/outcome v (like in Domar). Gg is called guaranteed (not of equilibrium tout court) because although in here investors are satisfied and have the same rate of development as the one expected, this is an instable equilibrium. Gg is the rate of growth at which I (and Y) have to grow in order to have equilibrium between Demand and Supply. Given s e v one can get the Gg, the rate that if verified will guarantee to the investor the production in the same amount which will lead him to make orders and purchases that will maintain the same rate of development (Harrod, 1939). Gg guarantees that the firms expectations concerning the demand in the following period are actually happening, hence, the aggregate demand will be equal to the productive capacity (the supply). In order to have a guaranteed path of economic growth one needs that I increases through time in line with the productive capacity, so that the increased productive capacity finds its correspondence in the market and a stable growth of the aggregate demand. t the same time to higher saving higher investments must correspond otherwise a fall in the aggregate demand will occur. Nevertheless, growth is instable. If for any reason the actual rate of growth get outside from the Gg path, the market will not generate self-regulating equilibrium forces; on the contrary, the actual path of growth will diverge more and more 2. 2 If S>D, the excess of S (the lack of D) will increase more because investors will invest less and less with decreasing marginal productivity of capital; in this situation, actual growth g will be lower and lower: g<gg. Instead, if D>S inflation will be higher and higher, because increasing marginal productivity of capital will stimulate growing investments, ( and actual growth will be higher than Gg: g>gg)

3 In Harrod-Domar approach governments and economic policies are called into play its role as guarantor of full employment of productive factors, i.e., capital and labor. The basic strength of the Harrod-Domar model is its simplicity. The data requirements are small, and the equation is easy to use and to estimate. Generally speaking, in the absence of severe economic shocks (such as financial crises or large changes in export or import prices), the model can do a reasonable job of estimating expected growth rates in most countries over very short period of time (a few years). nother strength is that Harrod-Domar model makes it clear that saving is crucial for income to grow over time. The model, however, has some major weaknesses. lthough saving is regarded as highly significant, modern growth theory takes into account a broad set of growth factors. The theory of Harrod and Domar considers savings to be a sufficient condition for growth and development. In other words, if an economy saves, it will grow, and if it grows, it must develop. ggregate savings are largely determined by national income, so if income is low, savings will not be accumulated. Modern theory tends to see savings as a necessary but not sufficient condition for growth. Perhaps the most important limitation in the model derives from the rigid assumption of fixed capital-to-labor, capital-to-output, labor-to-output ration, which imply very little flexibility in the economy over time. In order to keep these ratios constant, capital, labor and output must all grow at exactly the same rate, which is highly unlikely to happen in real economies. In this model, the economy remains in equilibrium with full employment of the labor force and the capital stock only under the very special circumstances that labor, capital and output all grow at the same rate. final weakness of the Harrod-Domar model is the absence of any role of technological change. dvances in technology generally are thought to play a critical role in long-term growth and development by contributing to increased productivity of all factors of production. Finally, in the Harrod-Domar model, the major role is plaied by the State who manage aggregate demand and investements: in other words in order to guarantee full employment and development, the state must play a crucial role in the economy. Hence, in this model, little room is left to non-economic factors, as far as market economy problems are solved by the State. However, in a situation where the State does not intervene in the economy (and this is the case today in most of advanced economies where laissez-faire and market economy rules dominate more and more), the role of non-economic factors (such as institutions, social capital, trust, cooperation, etc) emerge as important factors which cannot be neglected.

4 2. THE NEOCSSIC GROWTH MODES The most important neoclassical growth model, also known as the Solow-Swan model or exogenous growth model, aims to explain long-run economic growth. n exogenous change is one that comes from outside the model and is unexplained by the model. Neoclassical growth models try to find an explanation of long run economic growth by looking at productivity, capital accumulation, technological progress and population growth. The most important contributions to existing growth models came from the work done by Solow (1956) and Swan (1956) who independently developed relatively simple models. In 1987, Solow received the Nobel Prize in Economics for his work. The most important hypothesis of the Solow model are the following: substitubility between and ; variation of different combinations of and ; I depends on i (interest rates). The hedge knife (the Harrod instability) is solved through the variation of v= /Y allowed for by prices flexibility (prices of factors). Moreover, Solow assumes that the number of workers growth at the same rate of population growth: t = 0e nt, hence, population growth and work force growth exhibit exponential growth: t = n. t Capital instead accumulates in the following way: = I δ = sy δ Where is the change in capital stock over time Individuals save always the same fraction (with s<1, > 0 as a fraction of Y) ggregate saving = to gross I sy=s=i depreciates over time (with δ > 0, < 1 as a fraction of ) Divide = I δ = sy δ by = s Y δ = s Y δ Very importantly, accumulation of capital per workers occurs in the following way:

5 k if if if k k k k k k Briefly, capital per workers grows when investments (which increase capital stock ) grow faster than labour force (, or population growth n as stated above). On the contrary when capital is not sufficient and grow less than (because investments are not sufficient), capital per worker decreases. The Solow s model develops then as follows (the function of production used by Solow is a Cobb-Douglas function, with Constant Elasticity Substitution (CES), Homogeneous of first degree, which respect the Euler Theorem: y Y 1 1 k, with : y=f(k α ) (*) where y is output per worker (that is, y=y/) and k is capital per worker (k=/). While, α is the capital coefficient which indicates the share of capital (and 1- α is the share of labour) over GDP (Y). α is necessarily lower than 1 because Solow assumes the fundamental hypothesis of decreasing return of capital. It means that at higher levels of capital per worker, the same amount of new investment leads to a smaller increase in output. Each addition of a unit of capital per worker yields smaller and smaller growth in output per worker. Moreover, the analysis of the Solow s fundamental capital accumulation equation, with the crucial hypothesis of decreasing return of capital (and labor), shows that each country reaches a steady-state point of accumulation when the capital-output ratio is constant. From the capital k k. Y s Y s ; and n y n s n k. accumulation per worker equation developed above, we know that: Hence, we reach the second fundamental equation of Solow

6 Δk=sy ( n+d)k (**) where investment per worker, sy, increases growth of Δk, while depreciation per worker, dk, limits growth of Δk. n, the population growth rate (which is equal to labor growth because of the full employment hypothesis) reduces k too in the same direction as capital depreciation does. In the basic Solow diagram (Figure 1.1), we combine the two fundamental equations of Solow (*) and (**). Point is the only place where the amount of new savings sy is exactly equal to the amount of new capital needed to compensate the growth of population (or workforce n) and depreciation d of capital (n+d). For each value of k, n indicates how much of Investment is needed in order to keep / constant. Point is called the steady-state level of capital per worker and the output per worker. lthough output per worker is constant, total output continues to grow at the rate n, the same rate the population and workforce grow. In other words, at the steady-state GDP (Y) grows at the rate n, but GDP per capita (y) is constant (average income remains unchanged). Figure 1. The Solow growth model diagram The Solow model predicts that / will stabilize at ko where Investments (sy) are equal to n+d ( they offset in ko population growth and depreciation). ko is then the Steady state of the Solow model. nd: if if : k k : k k o o sy ( n ) k k 0 sy ( n ) k k 0..

7 The powerful conclusion of Solow model concerns the convergence towards a steady state, contrary to the Harrod model, without the public intervention. In fact before the steady state, capital deepening is occurring because capital is cheaper (interest rate is low) with respect to wages. The economy expands and GDP per capita increases (between k1 and ko). In k2, on the contrary, n+d is higher than sy, hence investment required to keep full employment are not sufficient, and unemployment increases. solution is found in the flexibility of prices (of wages and interest rates), and the substitution between the factors of production (labour and capital). Hence, wages decrease and interest rates increase; it follows that capital accumulation decreases because interest rates is high (capital became relatively more expensive). In ko capital widening occurs (i.e. capital increases at the same rate of n, so capital per worker does not increase and GDP per worker does not increases too). The Solow model, as described to this point, is a powerful tool for analyzing the interrelationship between saving, investment, population growth and output growth. If ko (the initial level of per worker), α, s, n and depreciation of capital, are known, one can establishe whether capital per worker will grow or not (so the economy will grow or not). This has also very important policy implications as far as policy can influence. However, the unsettling conclusion of the basic model is that, once the economy reaches its long-run potential level of income, economic growth simply matches population growth, with no chances for sustained increases in average income. Solow argued that high-income countries have been able to reach sustained growth in per capital income over long period of time because of technological progress that has allowed output per worker to continue to grow. Technology increases the rate of growth of GDP, and allows for countries to reach sooner the steady state. Therefore, with introduction of technology, the model shows the possibility of an economy experiencing sustained growth in per capital income. This mechanism provides a plausible explanation for why the industrialized countries never seem to reach a steady state with constant output per worker but instead we can observe the growth in output per worker. mong the strengths of the Solow model there is that the model focuses on the important role of factor accumulation and productivity (including technology) as the proximate determinants of the steady state. However, it does not give the answer to the question what are the determinants of factor accumulation and productivity that affects the rate of economic growth. The model does not provide a theory of sustained long-run economic growth. It implies that policy has no long-run effect on growth. Policy measures such as tax cuts or investment subsidies affect the steady state but not long-run growth rate.

8 The Solow model takes as given the saving rate, growth of the labor supply, the skill level of the workforce, and the rate of technological change (moreover, technology is considered to be a free good ). These assumptions help simplify the model, but as a result we learn little about the underlying determinants of these parameters, their origins, and how they might change during the process of development. Further, the fact that capital runs into diminishing returns means that the model does not lead to sustained economic growth. Eventually, the new investment is only just sufficient to offset depreciation, the capital stock and output stop to grow and the economy settles down to a steady state (Weil, 2007). 3. ENDOGENOUS GROWTH THEORIES The Solow model s results leave policy makers with little or no room for policy deliberations or initiatives since economic growth is determined by exogenous variables and in the end all the countries would converge towards the same level of steady-state, because the technological progress is a free good available to all the countries. However, the results of the Solow s model did not materialize empirically, and in general, countries/regions are not converging towards the same level of income nor towards the same rate of growth. In order to explain the lack of convergence, a new growth theory (the endogenous growth theory) at the end of 1980s emerged. new generation of models moving beyond the assumption of an exogenously fixed saving rate, workforce skill level and pace of technological change. Economists have begun to develop more sophisticated models in which one or more of these variables are determined within the model (that is, these variables become endogenous to the model). ccording to the new endogenous theory, each country has its own path of development that depends on the technological progress the country is able to reach, given the level of organization, social capability, institutional framework and imitation processes at play in the given territory (bramowitz, 1986). For instance, D Uva and De Siano (2011) investigated the evolution of the gap between Italian regions and Italy as a whole during the period of 1980 to They used different time series approaches to test for the presence of the stochastic and β-convergence. Their results showed that most of the Italian regions do not converge in an actual way (i.e. there is no the simultaneous presence of both types of convergence). These models depart from Solow framework by assuming that the national economy is subject to increasing return to scale, rather than constant return to scale. doubling of capital, labor, and other factors of production leads to more than a doubling of output. To the extent this occurs, the impact of investment on both physical capital and human capital would be larger than

9 suggested by Solow. This is possible because of beneficial effects from investment in research and education which influence not only on firms and individuals but also have a positive spillover effect on other actors in the economy. Theories of endogenous growth are based on the assumption that knowledge also has some of the features of a public good, because it is imperfectly excludable and therefore can produce positive externalities (Romer, 1990; Westlund, 2006). The endogenous model literature started to make important progress after the work of Romer (1986). However, even before, aldor (1962) showed that there is no evidence of significant convergence in growth rates of per capita income between countries in the world. The main determinant of the divergence is the way countries accumulate technological progress which is endogenous, in a sense that new ideas are embedded in new products and new investments, which in turn bring about new ideas in each country; hence the process is circular, and it brings about improvement in labour productivity continuously, as indicated by the chain below: capital investments ideas new investments new process, new products ideas new capital new investments. new process, new products ideas new capital This process brings about increasing return to scale, which is the fundamental point of the endogenous growth model in contrast with the neoclassical ideas where higher level of capital and of investments brings about lower marginal returns of each factor (and constant return to scale). In advanced and modern economies innovation, technological progress, ideas, knowledge are the drivers of economic growth. These drivers are pushed up by investments, which increase the production frontier of technology. However, these drivers do not come out of the blue, they are not free good, easy to access by each country. On the contrary, each country can (or cannot) access to them in different ways, at different speed and paces, determining in this way divergences in income levels among countries. The institutional framework is essential to accumulate knowledge and innovation (Dosi, 1988). The national system of innovation is determined by government strategies (Mazucato, 2014). The basic model which describes the essential results of the endogenous growth model is the so called model where represents both physical and human capital of an economy, and is a certain level of technology. stands for the set of capital factors reproducible in the broad sense, which can be also knowledge (education) and capital goods created by public spending. The nonaccumulating factors (for example unskilled abour) are of no importance. The fact that is reproducible in the production process is a great innovation of this model. In fact, the neoclassical theory has always assumed that cannot be used alone in the production process, but it needs to be

10 associated with factors which are not reproducible such as land and labour (unskilled labour). Hence, if labour is non-reproducible, the marginal productivity of capital will be decreasing. On the contrary if labour (skilled labour) can be considered reproducible, the marginal productivity of capital will no longer be decreasing. In other words, the marginal productivity of capital is decreasing if the associated factor (i.e. labour) which reproduces capital is not able to reproduce itself. But if skilled labour become able to reproduce it-self through continuous improvement of human capital then this limit is overcame and capital will not have decreasing returns. In most of advanced economies, today one can assume that skilled labour - the human capital - is reproducible. In this way,, from the model above can be considered in a broad sense embedding both human and physical capital. g y g kt g t It n t s n t t n s n In this model the returns to scale are still constant (or increasing, if has an exponent >1), but the productivity of (embedding both human and physical capital) is constant, and the growth rate is non convergent. In its simple version the model can be represented by the following figure: gk, gy s n kt Figure 2. The growth rate of income per capita is (s) - n. Hence, the saving rate and the level of technology affect the rate of growth of long run as opposed to what happens in the Solow model. Convergences among countries will not occur, while countries can have their own path of development (on the basis of parameters s and ). Countries with similar parameters can have

11 similar paths of development. Increasing s (hence capital investment) will also have long term effects and will increase both economic growth and economic levels, contrary to the neoclassical results where higher levels of capital will not lead to higher levels of economic growth in the long run. In the model productivity is endogenous in the sense that it depends on the size of, that with various technologies in the economies, can be affected by a country effect. nd, an increase of s increases the growth rate not only temporarily but also permanently. t the same time, if capital per worker decreases, this does not cause a temporary increase in the rate of growth, but with less (human capital, skills, ideas, physical cap, technology) countries will grow less and convergence will not occur. n evolution of the model is represented by the following specification: Where b b Y b 1b d, with : d 1 1 is the specification of the production function at micro level, for the single firm with constant return; while d is external to firms (which accounts for the well know externality phenomenon), and is the composite capital good/public good (knowledge, public education, university, etc) with increasing returns able to compensate the traditional decreasing marginal productivity of physical capital. Romer, in his model (1990) of economic growth included TP, which is driven by R&D. He considered these technological changes to be endogenous by introducing the search for new ideas by researchers interested in profiting from their inventions. He emphasized ideas that drive progress as specific types of goods considering them as non-rival in contrast to other goods. This focus on non-rivalry nature of ideas and knowledge as the basic form of capital suggests natural changes in the formulation of the standard aggregate growth model. In contrast to models in which capital exhibits diminishing marginal productivity, knowledge will grow without bound. Hence, investments in human capital and R&D are necessary in order to increase the productivity of labour and capital. The Romer s model is the following: Y a ( y) In this context, we have constant return to scale in and y, whereas all three variables, i.e., y and (respectively physical capital, labour and stock of knowledge) exhibit increasing return to scale together in the production function. Here, the main difference with neoclassical growth model, where grows exogenously at constant rate, is that is endogenous in the model. The growth of the stock depends on number of new ideas in one period, such as: 1a

12 , where 0<λ<1 Where, is the number of researchers; the average productivity of research depends on the number of researchers. Hence, y+ = total number of workers: = workers in R&D; S R&D =/ (the share of workers in the R&D sector). δ is the productivity of the single researcher. Instead, the parameter φ indicates the productivity of the whole research activity. In Romer δ and φ are constant. Moreover, because 0<λ<1 it follows that. increases but with decreasing return, i.e., it has positive externality but less than one because of possible duplication among researchers. If the invention in the past raises productivity of the single researcher today we will have that δ increases (with δ >1). However, with time it becomes more and more difficult to produce new ideas, hence δ reduces (with δ <1). If φ > 0, then the productivity of research increases with the stock of ideas that already has been discovered; if φ < 0 then the productivity declines because once so many inventions have been already made, new inventions became difficult to discover; whereas φ = 0 the productivity of research is independent of the stock of knowledge. In Romer s model economic growth is equal to the rate of growth of technological progress G y =G k =G, which is the following: Using log and derivative: 1 0 g n n g 1 (1 ) long a balanced growth path, the growth rate of number of researchers must be equal to population growth. long a balanced growth path the relationship is constant. However, this will be constant only if numerator and denominator grow at the same rate, so that the difference is equal

13 to zero. Hence, the long-term growth is determinate by the parameters λ and φ for ideas and rate of growth of researchers. The policy suggestion coming from the Romer model is clear and powerful: economic growth is driven by the size of research and development which in turn expands over time if the number of researchers grow. This occurs when the governments invest more in R&D. This result is completely opposed to the Solow results, where a higher n reduces the level of per capita income along a balanced growth path. On the other hand, more capital is required in order to keep / constant, but runs with diminishing returns. In endogenous growth models, instead, there is no such a diminishing return, and more people (higher n, possibly with a higher share of R&D) generate more ideas and increases growth.