MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT

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1 MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT ECONOMICS OF GROWTH AND INNOVATION Lecture 1, January 23, 2004 Theories of Economic Growth 1. Introduction 2. Exogenous Growth The Solow Model Mandatory Readings: Romer, D., Advanced Macroeconomics, McGraw Hill Chapter 1.

2 THEORIES OF ECONOMIC GROWTH

3 Introduction Standards of living differ among parts of the world by amounts that defy comprehension. It is common knowledge that the real output per capita in rich countries such as the US, Japan and some Western European countries is greater than that of countries like Mozambique and Bangladesh by a factor of twenty or more. It is also notorious that almost every country is considerably richer today than it was a century ago. Since the implications of these differences in standards of living in space and time affect almost all aspects of life, it is essential to understand what determines economic growth.

4 Given its unquestionable importance, economic growth has been the subject of due attention of economists since the early ages. The theoretical developments in the later half of the last century identified technological progress as the main driving force of long term growth. It is therefore not surprising that the most recent theoretical contributions have focused on the role of R&D and innovation in fostering growth. In particular, it became apparent that understanding growth entails understanding what leads profitmaximising firms to engage in research and innovative activities. The prominent role of technology and technical progress on economic growth makes the topics addressed in this course of particular relevance for a Master programme on engineering and technology management. For if the agents that deal with the creation and development of technology at the private and public levels get it right, the impact on economic growth and so on the society s well-being can be very significant.

5 Exogenous Growth The Solow Model This lecture will focus on the Solow Model, which was developed in 1956 by Robert Solow, a Nobel laureate, and that has been the main theoretical tool for economic growth until the 1980s. In spite of being very limited and largely inappropriate to account for the growth dynamics of modern economies, in particular the disparities of economic growth across time and space, the importance of the Solow model transcends by far its historical relevance. The Solow model is the starting point for almost all analysis of economic growth. Even models that depart significantly from Solow s are best understood through comparison with the Solow model. In this sense, the Solow model is a crucial tool for the economics of growth since and above all, it constitutes a benchmark against which all other growth theories compare to.

6 Assumptions The Production Function In the Solow model, output (Y) is produced through the combination of three inputs: capital (K), Labour (L) and Knowledge (A), according to the following production function: Y (t) = F (K(t), A(t)L(t)) (1) Two features of the production function should be noted: i) Time, which is denoted by t, does not enter the production function directly. That is, there is not a deterministic force that makes output to grow overtime. Output will only increase if inputs increase. ii) A and L enter multiplicatively. AL is referred as effective labour. This implies that technical progress affects the marginal product of labour only and it is called labour-augmenting or Harrod-neutral technical progress. This assumption is quite convenient as it not only simplifies the analysis a great deal but also possesses some desirable theoretical and empirical properties.

7 Assumptions About the Production Function Constant returns to scale (CRS): The production function is assumed to exhibit CRS in its two arguments, K and AL. This means that: F (ck, cal) = cf (K, AL), c > 0 The CRS assumption can be thought as combining two different assumptions: i) The economy is big enough for the specialisation gains to have been exhausted. For very low levels of output, a country can gain from further specialisation (just like an individual firm), by increasing the amount of the inputs used and have an increase in output proportionately greater than that of inputs. ii) The inputs not contemplated in the production function (especially those that are available in fixed proportions) are relatively unimportant. That is, we are assuming that all natural resources (eg. land) are not crucial for growth (as it seems to be the case), otherwise the country s production function could not exhibit CRS.

8 The assumption of CRS allows to work with the production function in intensive form. Setting c = 1/AL in the previous equation, we have: ( ) K F AL, 1 = 1 F (K, AL) AL Notice that K/AL is the amount of capital per unit of effective labour, F (K, AL)/AL = Y/AL is the amount of output per unit of effective labour. y = Y/AL, f(k) = F (k, 1) we can re-write the production function as: Defining k = K/AL, y = f(k) (2) This way of defining the production function amounts to looking at the production per unit of effective labour. Since the economy exhibits CRS, we know that the country s output is given by: Y = AL y.

9 The intensive-form production function, f(k), is assumed to satisfy the following assumptions: f(0) = 0, f (k) > 0, f (k) < 0, lim f (k) =, lim f (k) = 0 k 0 k } {{ } Inada Conditions Each of the five assumptions has its own implications: i) The first implies no inputs, no output. ii) The second and third imply that the marginal product of capital is positive but decreasing in the input level (law of diminishing returns). iii) The Inada conditions are crucial to ensure that the economy does not diverge and imply that the marginal product of capital is very large for small quantities of capital and becomes very small when the amount of capital per unit of effective labour is very high.

10 Any function that satisfies this assumptions has a graph with a particular shape(see fig 1): It is easy to show that the Cobb-Douglas function satisfy those assumptions, which makes it a quite attractive function to use, since it is simple and also a good first approximation to actual production functions (see p of Romer, 1996).

11 The Evolution of Inputs into Production The model is set in continuous time and so we need to specify the laws of motion of each of the inputs in order to solve the model. The initial levels of capital, labour and knowledge are taken as given. Labour and knowledge grow at constant rates: L(t) = nl(t) A(t) = ga(t) (3) where n and g are exogenous parameters and a dot over a variable denotes a derivative with respect to time. Those laws of motion imply that the variables grow exponentially. That is, if L(0) and A(0) denote their level at time 0, the laws of motion above imply that: L(t) = L(0)e nt A(t) = A(0)e gt (4)

12 To determine the law of motion of capital, consider the following. Output is assumed to be divided into consumption and investment. The fraction devoted to investment (savings rate) is assumed exogenous and constant and denoted by s. We further assume that capital depreciates at a rate δ. Thus, investment (change in the capital stock) is given by: K(t) = sy (t) δk(t) (5) Before proceeding a note is worth mentioning. The assumptions of these model entail a degree of abstraction and simplification that in some instances seem quite gross: the government is not considered at all, the output of a country is assumed to be produced out of three inputs only, just to mention a few. As we will see later on in the course some of these assumptions are responsible for the empirical failure of the Solow model and will accordingly be relaxed to produce a more realistic setting.

13 Dynamics of the Model We now turn our attention to characterising the behaviour of the Solow model. Since the evolution of labour and knowledge is exogenous, we start by working out how capital behaves. Dynamics of k Since k is a function of K, L and A, we can use the chain rule to determine the time profile of k as follows: k = k K K + k L L + k A A k(t) = = K(t) A(t)L(t) K(t) A(t)L(t) K(t) [A(t)L(t)] 2 [A(t) L(t) + L(t) A(t)] K(t) L(t) K(t) A(t) A(t)L(t) L(t) A(t)L(t) A(t) } {{ }}{{} } {{ }} {{ } k n k g

14 Substituting K(t) for equation (5): k(t) = sy (t) δk(t) A(t)L(t) k(t)n k(t)g Y (t) = s δk(t) nk(t) gk(t) A(t)L(t) } {{ } y=f(k(t)) k(t) = sf(k(t)) (n + g + δ)k(t) (6) Equation (6) is the key equation in Solow s model. It states that the rate of change of capital per unit of effective labour its equal to the difference between two terms: i) The first is actual investment per unit of effective labour, which is equal to the amount of output saved, i.e. sf(k(t)). ii) The second is the break-even investment, or the amount of investment that has to be made as to maintain k constant. Notice that since capital depreciates at a rate δ and effective labour grows at rate n + g, capital per unit of effective labour must grow at a rate equal to n + g + δ just to keep k unchanged.

15 Figure 2 depicts the two terms of equation (6) as a function of k. In that figure it is apparent that break-even investment is proportional to k whereas actual investment its equal to a constant (s) times output. Since f(0) = 0, actual investment is equal to break-even investment when k = 0. The Inada conditions imply that for small values of k the slope of the production function (given by the marginal product of capital) is large and so the slope of sf(k) is steeper than that of break-even investment. In that case, actual investment is greater than break-even s so that k is growing over time. The Inada conditions also imply that for high levels of k the slope of the production function is small, meaning that the slope of sf(k) is smaller than that of break-even investment, so that actual investment is below break-even and so k is decreasing over time. Together, the two Inada conditions imply that the two curves must cross at some point k, where actual and break-even investment are equal and so k remains constant. Finally, the assumption that f (k) < 0 implies that the two curves intersect just once for positive values of k.

16 Figure 3, which draws k against k, is called phase diagram and summarises the dynamics of capital per unit of effective labour. As we can see in that figure, regardless of the starting level of k, it always converges to k. Balanced Growth Path Definition: Balanced growth path corresponds to a situation in which all variables in the model are growing at a constant rate. Since k always converges to k we need to know the dynamic behaviour of the remaining variables when k = k. By assumption, L and A are growing at exogenous constant rates of, respectively, n and g. The capital stock, K = ALk, when k = k (i.e. k is constant) grows at a rate n + g. With capital and effective labour both growing at the same rate, CRS implies that output (Y ) is also growing at a rate of n + g.

17 It turns out that the output per worker is growing at the rate of technological progress. The fact that the rate of technological progress, g, is determined outside the model means that growth is exogenous in this model, which is one of the major criticisms of the Solow model. After all, we are trying to explaining growth with a model that actually does not explain it! Nevertheless, the balanced growth path of the Solow model fits several of the major stylised facts about growth. For most of the industrialised countries, it is a good first approximation to assume that output, effective labour and capital all grow at constant rates. Also, it seems to be the case that empirically, output and capital grow at a similar rate, meaning that capital and output per worker have been increasing. All these properties are present in the Solow model.

18 The Impact of a Change in the Savings Rate For a host of different reasons, the parameter of the model that is most likely to be affected by policy actions (although those are considered in the Solow model explicitly) is the savings rate, s. For that reason, we will look at the impact of an increase in s on the main variables of the model. The Impact on Output The increase in s shifts the actual investment curve upwards so that k rises. Since k is equal to the old level of k, it is below the new level of k. At this level actual investment is greater than break-even, meaning that k > 0, until the new k is reached (see figure 4).

19 We now turn into the impact of the rise in s on output per worker, Y/L = Af(k). When k is constant, output per worker increases at a rate of g. However, because k rises after an increase in s, output per worker will increase at a rate faster than g. As k approaches the new k the growth rate of output per worker starts declining towards g again. The upshot is that, in the Solow model, a permanent increase in the savings rate produces only a temporary increase in the growth rate of output per worker. The impact of a permanent increase in the savings rate is summarised in figure 5. From that figure, it is apparent that a permanent change in s has a level effect but not a growth effect: it changes the economy s balanced growth path, and thus the level of output per worker but it does not affect the growth rate of output per worker on the balanced growth path. In sum, in the Solow model, only changes in the rate of technological progress have growth effects, all other have only level effects.

20 The Impact on Consumption Consumers welfare depends not on output but on the level of consumption. Therefore it is important to know what happens to consumption when the savings rate changes permanently. Consumption per unit of effective labour is equal to (1 s)f(k). At the moment s increases, consumption necessarily jumps downwards. However, as k rises towards its new k level, y increases and so does consumption per unit of effective labour. Whether consumption in the new balanced growth path rises above its original level it is not immediately clear. Let c denote consumption per unit of effective labour on the balanced growth path. Then: c = f(k ) sf(k ) (7) Since on the balanced growth path actual investment equals break-even investment, equation (7) can be written as: c = f(k ) (n + g + δ)k (8)

21 Thus, when the marginal product of capital is greater than (n + g + δ), consumption per unit of effective labour will respond positively to an increase in s. Intuitively, when k rises, investment per unit of effective labour must rise by (n + g + δ)dk for the increase to be sustained. If the marginal product of capital is lower than (n + g + δ), then the additional output brought about by the increase in k is not enough to maintain the capital stock at its higher level, and so consumption per unit of effective labour must decrease permanently. Since k is determined by s and the other parameters of the model, we can write k = k (s, n, g, δ), and so: c s = [f (k (s, n, g, δ)) (n + g + δ)] k (s, n, g, δ) } s {{ } >0 (9)

22 f(k) An Example of a Production Function k Figure 1:

23 K Actual and break-even investment AL ( n + g + δ ) k Break-even investment sf(k) Actual investment k * k Figure 2:

24 & k The phase diagram for k in the Solow model 0 k * k Figure 3:

25 &K AL Effects of an Increase in s on Investment ( n + g + δ ) k s NEW f(k) s OLD f(k) k OLD * k NEW * k Figure 4:

26 Figure 5:

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