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2 the Basic Solow Model Economic Growth Models: A Primer /Student's Guide, Miguel Lebre de Freitas 2 The basic Solow model “A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster” [Robert Lucas Jr.] Learning Goals: Acknowledge the distinctive feature of capital, as compared to labour and land in terms of availability to production. Understand the extent to which capital accumulation can overcome the law of diminishing returns The mechanics of the Solow growth model. Evaluation of the Solow model through the lenses of the Kaldor facts. Distinguish the effects of a change in the saving rate from those of an exogenous change in technology Discuss the optimality of the saving rate in the context of the Solow model 2.1 Introduction The neoclassical theory of economic growth was pioneered by two independent economists, the American Robert Solow and the Australian Trevor Swan 30 . The main innovation of the Neoclassical model in respect to the Malthusian Model is the replacement of land by “capital” in the production function. By “capital”, we mean machinery, buildings, and other equipment. This modification is more than a mere change in form: contrary to land, which is available in finite supply, capital can be produced and accumulated. This opens an avenue to overcome diminishing returns on labour: by allowing the capital stock to expand over time, 30 Solow, R., 1956. “A contribution to the theory of economic growth”, Quarterly Journal of Economics 50, 65-94. Swan, T., 1956. “Economic growth and capital accumulation”, Economic Record 32, 334-61. https://mlebredefreitas.wordpress.com/teaching-materials/economic-growth-models-a-primer/ 11/02/2021 75 Economic Growth Models: A Primer /Student's Guide, Miguel Lebre de Freitas the Solow model avoids the tendency for productivity to decrease with the size of population that plagues the Malthus model. In the Solow model, the long run is not characterized by a low income equilibrium trap. Still, because capital itself faces diminishing returns, capital accumulation alone cannot generate long-run growth: in the Solow model, a higher investment rate translates into a higher level of per capita income in the long run, but it does deliver sustained growth of per capita income. This chapter presents the Solow model in its simplest formulation. In Section 2.2, we describe the main assumptions of the model and we characterize the equilibrium. In Section 2.3, we compare the predictions of the model with the main stylized facts of Modern Economic Growth. In Section 2.4 we discuss how the main endogenous variables of the model respond to changes in the exogenous parameters. Section 2.5 addresses the question as to whether a government should try to influence an economy’ saving rate. Section 2.6 extends the model to the case in which the saving rate is optimally decided by individual agents. Section 2.7 introduces a growth accounting exercise to illustrate the fundamental limitation of this simple version of the Solow model, which is not to account for productivity growth. Section 2.8 concludes. 2.2 The Solow model 2.2.1 The neo-classical production function In the Malthus model, with all else constant, an increase in the size of the labour force leads to a decline in per capita income. This is a direct consequence of the Law of Diminishing Returns: as long as the availability of land is unchanged, its higher intensive use will translate into lower labour productivity. The Solow model retains the assumption of diminishing returns to labour, but explores another property of the neoclassical production function: Constant Returns to Scale (see box). Under constant return to scale, per capita https://mlebredefreitas.wordpress.com/teaching-materials/economic-growth-models-a-primer/ 11/02/2021 76 Economic Growth Models: A Primer /Student's Guide, Miguel Lebre de Freitas income may remain constant over time in spite of an expanding increase labour force, provided the stock of capital expands at the same rate as employment. To see this, let’s assume that the aggregate production function in the economy is as follows31: 1 Yt AK t N t , 0 < < 1 (2.1) where K denotes for the economy’ capital stock, N for the size of the labour force, Y for output and A is Total Factor Productivity. Dividing the production function (2.1) by the size of the workforce (N), one obtains an expression that relates per capita output to capital per worker: K y t A Ak t , (2.2) N In (2.2), y=Y/N denotes for per capita income and k=K/N is the capital-labour ratio. Equation (2.2) is called the production function in the intensive form and stresses the role of the capital-labour ratio as a key driver of per capita income. Thus, as long as capital and labour expand at the same rate, the capital labour ratio will remain constant and so will do per capita income. The interesting feature of the Solow model is that we do not need to postulate K to grow at the same rate as the labour force. As we will see next, the properties of the model are such that the capital stock, even if endogenous, will end up growing at the same rate as the labour force in the long run, ensuring that per capita income remains constant. Box 2.1 Constant returns to scale The key assumption of the neoclassical growth model is that of constant returns to scale (CRS). CRS means that if one increases the use of all inputs by a given proportion, 31 The Solow model is consistent with more general specifications for the production function. The Cobb-Douglas production function is assumed for mathematical convenience. https://mlebredefreitas.wordpress.com/teaching-materials/economic-growth-models-a-primer/ 11/02/2021 77 Economic Growth Models: A Primer /Student's Guide, Miguel Lebre de Freitas output will rise in the same proportion. For instance, duplicating the use of labour and capital we will obtain the double of output. The CRS property can easily be checked in equation (2.1): for any q 0, AqK qN 1 qY . Note that CRS is not inconsistent with the LDR: the Law of Diminishing Returns states that increasing the use of one input while holding the other inputs constant, output will grow less than proportionally. The law of diminishing returns applied when only onf input is increasing: the Constant Returns to Scale property applies when all inputs increase in the same proportion at the same time. Figure 2.1. Diminishing returns versus constant returns to scale yY N C y0 K A y A1 N1 y1 B K y A0 N0 N N N 0 1 The Figure displays the average product of labour as a function of the size of the labour force. The curve is negatively sloped because of diminishing returns (from A to B). When the capital stock increases, the schedle shifts to the right. If the capital stock and population increase in the same propostion, the average product of labour remains constant (from A to C). To illustrate the difference, we refer to Figure 2.1. The figure displays the average product of labour as a function of the labour force. The schedule is negatively sloped because of the Law of Diminishing Returns: all else constant (including the capital stock), an increase in the use of labour from N 0 to N1 implies a decline in the output per worker from y0 (point A) to y1 (point B). This is the main mechanism in the Malthusian model. In the Solow model, in contrast, the second input, capital, is allowed to expand: moreover, the steady state of the model is such that the capital stock grows exactly in the same proportion as labour. In that case, we see from (2.2) that per capita income remains constant. In terms of Figure 2.1, this is illustrate by a move from A to C: if, when labour expands from N 0 to N1 , the capital https://mlebredefreitas.wordpress.com/teaching-materials/economic-growth-models-a-primer/ 11/02/2021 78 Economic Growth Models: A Primer /Student's Guide, Miguel Lebre de Freitas stock also increases, and exactly in the same proportion (such that KL0 0 KL 1 1 ), then the capital labour ratio remains constant and so will be per capita income, y0 . In general, production functions may exhibit decreasing returns to scale (when output grows less than proportionally than all inputs) and increasing returns to scale (when output grows more than proportionally than all inputs). Decreasing returns to scale in its pure form is unlikely: if we managed to increase all inputs in a given proportion, there should be no reasons for output to respond less than proportionally. The case with increasing returns is trickier and will be addressed later in this book. 2.2.2 Main assumptions of the Solow model Consider a closed economy with no government with a large number of small firms producing a single homogeneous good, Y, using two inputs: labour (N), and capital (K). Population and the labour force are the same. Inputs are hired from households, who are also the owners of the firms and the consumers in this economy. Households spend a fraction (1-s) of their income on consumption, and save the remaining fraction, to acquire new capital. The capital stock depreciates at a constant rate, . Perfect competition and flexible prices are assumed, so full employment holds each moment in time. Since in this model there is no tendency for per capita income to approach a low level subsistence trap, one no longer needs to worry with Malthusian barriers limiting population growth.
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