Introduction to economic growth (2)

Transcription

1 Introduction to economic growth (2) EKN 325 Manoel Bittencourt University of Pretoria M Bittencourt (University of Pretoria) EKN / 49

2 Introduction Solow (1956), "A Contribution to the Theory of Economic Growth", is the benchmark paper/model for economic growth, and we study the ability of this model to explain the facts of growth and development we will see how this model explains the existence of rich and poor countries, or the reason for so much heterogeneity, in the world today, or alternatively, why some countries diverge from similar ones over time still on economic modelling, we have to bear in mind that theoretical models are based on simplifying assumptions that are sometimes heroic, or too simplistic to capture all features seen in the real world. Nevertheless, that is how science is done, it is based on models, or simplifications, that try to isolate the facts we are interested in M Bittencourt (University of Pretoria) EKN / 49

3 Introduction The first assumption is that the countries of the world produce and consume a single and homogeneous good, or output, or GDP Y the above implies that there is no international trade in the model (there is no reason for South Africa to trade Gala apples with Chile s Gala apples, a single and homogeneous product) furthermore, technology is exogenous (technology is not affected by what firms do, or alternatively, the R&D sector is not part of the model, that is a feature of the endogenous growth models of the 1980s that we see later on in the course) M Bittencourt (University of Pretoria) EKN / 49

4 Introduction So, what is a model? Why do we need them in economics? a model is simply a mathematical representation, or simplification, of some aspects or facts of a particular economy in a model we specify how the agents behave; economic agents try to optimise their utility; consumers (utility maximisation) and firms (profits maximisation, or costs minimisation) M Bittencourt (University of Pretoria) EKN / 49

5 Introduction Moreover, the agents, consumers and firms, face constraints, consumers have budget constraints, and firms technological constraints the best models are able to convey a lot of information in a simplified manner, they explain important facts, are generalisable to other countries than the USA, and are not too sensitive to changes in particular parameters (they are mathematically tractable) some simplifications that we will make use of are; individuals save a constant fraction of their income and spend a constant fraction of their time accumulating skills, or acquiring human capital (intertemporal optimisation) M Bittencourt (University of Pretoria) EKN / 49

6 The basic Solow model The Solow model is based on two equations; a production function and a capital accumulation equation. Recall that production functions describe how inputs are best combined to produce (ideally more) output Y the model considers only two inputs, capital K and labour L, and the production function is the Cobb-Douglas, Y = F(K, L) = K α L 1 α, (1) where α is a number between 0 and 1. This Cobb-Douglas production function, by definition, exhibits constant returns to scale (if we double all inputs, output will double too) M Bittencourt (University of Pretoria) EKN / 49

7 The basic Solow model In this economy firms pay labour a wage w, and r to rent capital. It is also assumed that there are a large number of firms in the economy, perfect competition (or the firms are price takers). The profit-maximising firms face the following unconstrained optimisation problem, max K,L F(K, L) rk wl Kα L 1 α rk wl, where we have to find and interpret the following first-order conditions w = δf δl = (1 α)y L, r = δf δk = α Y K, firms hire labour until the marginal product of labour equals the wage paid w, and rent capital up to the point in which the marginal product of capital is equal to the rental price r M Bittencourt (University of Pretoria) EKN / 49

8 The basic Solow model Recall that wl + rk = Y, payments to inputs must equal output, by definition, there are no profits in this model. Also, (1 α) is the output share paid to labour and α the share paid to capital, and these shares are constant over time (remember fact 5) we are interested in GDP, or output, per worker, and capital per worker, therefore we can write y Y L and k K L (or in per worker terms) we can then rewrite (1) simply as y = k α (2) figure 2.1 illustrates the above production function, with more capital per worker k, firms produce more output per worker y. However, there are diminishing returns to scale, more capital per worker displays less efficiency over time M Bittencourt (University of Pretoria) EKN / 49

9 The basic Solow model M Bittencourt (University of Pretoria) EKN / 49

10 The basic Solow model The second key equation of the Solow model describes capital accumulation, K = sy dk (3) the above equation is fundamental, and other models of growth make use of versions of it. Equation (3) is telling us that the change in capital stock equals the amount of investment sy, less the amount of depreciation dk that takes place during the production process K is the change in capital stock over time, or K t+1 K t = K t. About sy, remember that S = I, workers (or consumers) save a constant fraction of their income, or alternatively, in this model capital accumulation depends on savings. dk is a constant fraction of capital being depreciated every period. We assume that d =.05, or that machines and factories depreciate at 5% per year M Bittencourt (University of Pretoria) EKN / 49

11 The basic Solow model To understand the dynamics of output per person y in a particular economy we have to rewrite the capital accumulation equation (3) in terms of capital per person then the production function (2) informs us the amount of output per person y produced given the stock of capital per person k in this economy to do the above, we make use of a mathematical trick, take logs and then derivatives M Bittencourt (University of Pretoria) EKN / 49

12 The basic Solow model 1 k K k L =) log k = log K log L =) rates of capital per worker) 2 y = k α =) log y = α log k =) ẏ y = α k k. k = K K L L (or the growth 3 consider the Cobb-Douglas production function Y = K α L 1 α, taking logs, log Y = log K α + log L 1 α =) log Y = α log K + (1 α) log L. Now, taking derivatives on both sides w.r.t. time, d log Y dt = α d log K dt + (1 α) d log L dt =) Ẏ Y = α K K + (1 L α) L, this expression tells us that the growth rate of output is a weighted average of the growth rates of capital and labour M Bittencourt (University of Pretoria) EKN / 49

13 The basic Solow model Another assumption of the model is that the labour force participation rate is constant. The population growth rate is given by a parameter n. Therefore, the labour force growth rate is given by L L = n. For example, if n =.01, then the labour force and the population are growing at 1% per year now we can include n in our capital accumulation, in per worker terms, equation, k = sy (n + d)k this equation is telling us that changes in capital per worker is determined by investment per worker sy (which has a positive effect on k), dk, depreciation per worker has a negative effect on k, and the new term nk, population growth, which presents a negative effect on k M Bittencourt (University of Pretoria) EKN / 49

14 Solving the basic Solow model We now have the basic elements of the Solow model, now we can solve it in addition, we have been talking about a set of equations used to describe the model (the Cobb-Douglas production function and the capital K accumulation equation), and endogenous (determined within the model) and exogenous (determined outside the model) variables which are part of this model in our case, output Y and capital K are endogenous, obviously that their per worker counterparts are also endogenous, y and k M Bittencourt (University of Pretoria) EKN / 49

15 Solving the basic Solow model We also have parameters in our model, α, s and n, and an exogenous variable L with the above in mind we can (analytically) solve the model, say, find the values of the endogenous variables given the parameters and exogenous variables of our representation, or alternatively, we can vary the parameters and exogenous variables and then see how the endogenous variables adapt to changes M Bittencourt (University of Pretoria) EKN / 49

16 The Solow diagram Let s recall the two fundamental equations of the Solow model in per worker terms now, y = k α, k = sy (n + d)k let s assume that this model economy starts with a given stock of capital k 0, and a given population growth rate n, depreciation rate d and investment s with the above information we can diagrammatically see how this economy grows, or evolves over time, when we change the parameters and exogenous variables M Bittencourt (University of Pretoria) EKN / 49

17 The Solow diagram For that we make use of the Solow diagram, which consists of two curves. The first is the amount of investment per person, sy = sk α, and the second is the line (n + d)k representing the amount of new investment per worker needed to keep the amount of capital per worker constant. Bear in mind that both n and d reduce the amount of capital per worker in this economy M Bittencourt (University of Pretoria) EKN / 49

18 The Solow diagram M Bittencourt (University of Pretoria) EKN / 49

19 The Solow diagram In figure 2.2, at k 0 the amount of investment, or savings, exceeds the amount required to keep capital per worker constant (capital is deepening in this case). At k, sy = (n + d)k, or k = 0, or the steady state think of Japan and Germany after WWII, both countries had a considerable amount of their capital stocks destroyed by 1945, nevertheless both countries grew fast after 1945 (how the Solow model explains that, and how do we call that in economics?) in addition, we can include in the diagram the production function y, and the difference between y and sy represents the steady-state consumption per worker. Figure 2.3 illustrates that M Bittencourt (University of Pretoria) EKN / 49

20 The Solow diagram M Bittencourt (University of Pretoria) EKN / 49

21 Comparative statics Comparative statics are used to check how the model, or the endogenous variables, change when the parameters or the exogenous variables change, or when there is a shock to this economy given the parameters we have in the model, the shocks we consider are changes in the investment rate s and in the population growth rate n so, let s assume that an economy is in its steady-state and for some reason the consumers (or workers) in this economy decide to increase permanently s. Figure 2.4 illustrates the change, now this economy is richer than before, with higher capital per worker and higher output per worker now let s assume an increase in n (what happens to capital per worker?). Figure 2.5 illustrates that M Bittencourt (University of Pretoria) EKN / 49

22 Comparative statics M Bittencourt (University of Pretoria) EKN / 49

23 Comparative statics M Bittencourt (University of Pretoria) EKN / 49

24 Properties of the steady state By definition, the steady state is determined by k = 0. Therefore, using k = sy (n + d)k and substituting y = k α in there, and then using the condition k = 0 we get the steady state quantity of capital per worker, 1 s k 1 α =, (4) n + d substituting (4) in the production function y = k α we get the steady state quantity of output per worker, y = s n + d α 1 α equation (5) gives output in terms of the parameters of the model, therefore we have a solution for the model (5) M Bittencourt (University of Pretoria) EKN / 49

25 Properties of the steady state What is the intuition of this solution?, or "Why are we so rich and they so poor?" according to the model? the model tells us that countries with higher savings (investment) tend to be richer, these countries are accumulating more capital per worker, and therefore have more output per worker. On the other hand, the model tells us that countries with high n, tend to be poorer (less capital per worker?) the question now is, how well this theoretical prediction fits the empirical regularities? Figures 2.6 and 2.7 put real GDP per worker against the investment share of the GDP and the population growth rate in a cross section of countries. We see a slight positive correlation between real GDP per worker and the investment share to GDP, which suggests that those countries investing more are richer. Furthermore, we can also see that those countries with lower n have higher y M Bittencourt (University of Pretoria) EKN / 49

26 Properties of the steady state M Bittencourt (University of Pretoria) EKN / 49

27 Properties of the steady state M Bittencourt (University of Pretoria) EKN / 49

28 Economic growth in the simple model It is clear by now that in steady state there is no per capita growth! Output is growing, however at the same rate of population growth n and because a number of things in the model are constant, for instance, capital-output ratio and interest rate (or the return to capital), it is in accordance to some of the facts of growth seen before however, this simplified version of the model has a serious shortcoming, as it is it does not predict sustained economic growth over time. Economies grow for a while, however once they reach their steady state, growth ceases M Bittencourt (University of Pretoria) EKN / 49

29 Economic growth in the simple model We can say that the farther an economy is below its steady state, the faster that economy will grow. The opposite is also obviously true (diminishing returns to scale, or to capital) moreover, dividing the capital accumulation equation by k we get k k = skα 1 (n + d), and if α is < 1, then everything (including output y) is declining and figure 2.8 illustrates this principle of transition dynamics M Bittencourt (University of Pretoria) EKN / 49

30 Economic growth in the simple model M Bittencourt (University of Pretoria) EKN / 49

31 Technology and the Solow model From the above it is clear that for sustained growth to take place we need something else in the model. That something else is technology, therefore we must augment the model the Cobb-Douglas production function has to be augmented, Y = F(K, AL) = K α (AL 1 α ), in which A is technology and we call the above labour-augmenting technology. In other words, labour might become more productive if more technology is available (we are assuming, by default, that labour will be able to use this new technology once introduced) M Bittencourt (University of Pretoria) EKN / 49

32 Technology and the Solow model Technology is exogenous in this model economy, it comes from nowhere, it is G-d given, and it grows at Ȧ A = g now, to see the growth implications of this model, let s rewrite the production function (augmented with technology A) in terms of output y per worker, y = k α A 1 α furthermore, let s now take logs and differentiate the above, so that ẏ y = α k k + (1 α)ȧ A (6) M Bittencourt (University of Pretoria) EKN / 49

33 Technology and the Solow model Moreover, a situation in which capital, output, consumption and population are growing all at constant rates is known as the balanced growth path. Recall that Ȧ A = g, we can then conclude that g y = g k = g, along the balanced growth path, output per worker y and capital per worker k, are both growing at the rate of this exogenous technological change g the model which incorporates A is telling us that technology is the source of sustained economic growth M Bittencourt (University of Pretoria) EKN / 49

34 The Solow diagram with technology The Solow model with technology looks just like the one seen above without technology. Just bear in mind that now we have k = K AL, the ratio of capital per worker to technology the per worker to technology production function ỹ looks like, ỹ = k α, where ỹ = Y AL, or alternatively, the output-technology ratio. And the new capital accumulation equation now is, k = sỹ (n + g + d) k k = s k α (n + g + d) k with that in mind, we can move to the Solow diagram itself, figure 2.9, and the interpretation is fairly similar to the previous case without technology, if an economy is for some reason below its steady state (Germany and Japan right after WWII, or perhaps South Africa in 1992), then that economy is going to grow faster until it reaches its steady state M Bittencourt (University of Pretoria) EKN / 49

35 The Solow diagram with technology M Bittencourt (University of Pretoria) EKN / 49

36 Solving for the steady state Recall that the steady state is represented by k = 0. Solving 0 = s k α (n + g + d) k for k we get, k = s n + g + d 1 1 α substituting the above into the production function we get, ỹ = s n + g + d α 1 α rewriting the above in terms of output per worker gives, y (t) = A(t) s n + g + d α 1 α, (7) M Bittencourt (University of Pretoria) EKN / 49

37 Solving for the steady state which represents the output per worker along the balanced growth path, and it is determined by technology A, investment s and the population growth n Figure 2.10 illustrates a change in this model economy, a permanent increase of a subsidy to investment, and the diagram looks similar to the one seen before. Look at the transition dynamics taking place there, this economy grows faster until it reaches its new steady state M Bittencourt (University of Pretoria) EKN / 49

38 Solving for the steady state M Bittencourt (University of Pretoria) EKN / 49

39 Solving for the steady state Moreover, the next figures illustrate transition dynamics and the effect of this increase in investment on growth, the growth rates diminish once the economy reaches proximity to its new steady state over time (no growth effects, only level effects) M Bittencourt (University of Pretoria) EKN / 49

40 Solving for the steady state M Bittencourt (University of Pretoria) EKN / 49

41 Solving for the steady state M Bittencourt (University of Pretoria) EKN / 49

42 Solving for the steady state M Bittencourt (University of Pretoria) EKN / 49

43 Evaluating the Solow model First, according to the Solow model a country grows faster if it invests more and presents lower population growth rates, which allows for more capital per worker plus an increase in labour productivity second, the source of sustained growth is technology, however technology enters the model exogenously. Technology can offset the decreasing marginal returns to scale, K and L third, Japan and Germany after WWII are examples that illustrate the transition dynamics mechanism of growth of the model fourth, some of the NICs can also illustrate the story that countries that invest more will grow faster M Bittencourt (University of Pretoria) EKN / 49

44 Evaluating the Solow model M Bittencourt (University of Pretoria) EKN / 49

45 Growth accounting, the productivity slowdown and the new economy Technology is the engine of economic growth, it can offset the decreasing marginal returns to K and L Solow (1957), "Technical Change and the Aggregate Production Function", is Solow s second seminal paper on growth. The paper provides a simple exercise of growth accounting. For instance, the following production function, now, taking logs and derivatives, Y = BK α L 1 α, Ẏ Y = α K K + (1 L α) L + Ḃ B, is key for growth accounting. Output growth is equal to K and L growth plus the growth rate of B (a weighted average). But what is Ḃ B? It is the total factor productivity growth. Table 2.1 illustrates an exercise of growth accounting for the USA M Bittencourt (University of Pretoria) EKN / 49

46 Growth accounting, the productivity slowdown and the new economy M Bittencourt (University of Pretoria) EKN / 49

47 Growth accounting, the productivity slowdown and the new economy The table is telling us that between 1948 and 2010 the output per hour grew at 2.6% in the USA. The contribution from capital was 1.0% and the changing composition of the labour force (which takes into account that the labour force today is better educated than in the 1950s) contributed with 0.2%. Total factor productivity contributed with 1.4%. This 1.4% improvement in productivity is the residual of Solow other notable facts coming from the table is that in the 1970s total factor productivity decreased considerably, on the other hand it has increased from the 1990s onwards. Some economists speculate that the reason for that is the introduction of new technologies in the 1970s which took some time (20 years) to be internalised by workers. For instance, think of how difficult it was to operate a computer in the 1970s and how easy is to operate a laptop nowadays M Bittencourt (University of Pretoria) EKN / 49

48 Growth accounting, the productivity slowdown and the new economy Finally, a fast-growing economy like Singapore has not shown fast increase in TFP, however it has shown fast capital accumulation (physical and human). On the other hand, look at Brazil. It seems that one way or another, the Solow model can help in understanding how societies have been evolving over time M Bittencourt (University of Pretoria) EKN / 49

49 Growth accounting, the productivity slowdown and the new economy M Bittencourt (University of Pretoria) EKN / 49