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5 The Solow modelsometimes known as the SolowSwan modelwas developed by Robert Solow (Solow, 1956) and T. W. Swan (Swan, 1956). The first three chapters of this book are therefore devoted to economic growth. We will investigate several models of growth. Although we will examine the models mechanics in considerable detail, our ultimate goal is to leam what insights they offer concerning worldwide growth and income differences across countries. This chapter focuses on the model that economists have traditionally used to study these issues, the Solow growth model. The Solow model is the starting point for almost all analyses of growth. Even models that depart fundamentally from Solows are often best understood through comparison with the Solow model. Thus understanding the model is essential to understanding theories of growth. The principal conclusion of the Solow model is that the accumulation of physical capital cannot account for either the vast growth over time in output per person or the vast geographic differences in output per person. Specifically, suppose that the mechanism through which capital accumulation affects output is through the conventional channel that capital makes a direct contribution to production, for which it is paid its marginal product. Then the Solow model implies that the differences in real incomes that we are trying to understand are far too large to be accounted for by differences in capital inputs. The model treats other potential sources of differences in real incomes as either exogenous and thus not explained by the model (in the case of technological progress, for example), or absent altogether (in the case of positive externalities from capital, for example). Thus to address the central questions of growth theory we must move beyond the Solow model. Chapters 2 and 3 therefore extend and modify the Solow model. Chapter 2 investigates the determinants of saving and investment. The Solow model has no optimization in it; it simply takes the saving rate as exogenous and constant. Chapter 2 presents two models that make saving endogenous and potentially timevarying. In the first, saving and consumption decisions are made by infinitelylived households; in the second, they are made by households with finite horizons. Relaxing the Solow models assumption of a constant saving rate has three advantages. First, and most important for studying growth, it demonstrates that the Solow models conclusions about the central questions of growth theory do not hinge on its assumption of a fixed saving rate. Second, it allows us to consider welfare issues. A model that directly specifies relations among aggregate variables does not provide a way to judge whether some outcomes are better or worse than others: without individuals in the model, we cannot say whether different outcomes make individuals better or worse off. The infinitehorizon and finitehorizon models are built up from the behavior of individuals, and can therefore be used to discuss welfare issues. Third, infinite and finitehorizon models are used to study
1.2 Assumptions 7 many issues in economics other than economic growth; thus they are valuable tools. Chapter 3 investigates more fundamental departures from the Solow model. Its models, in contrast to Chapter 2s, provide different answers than the Solow model does to the central questions of growth theory. The models depart from the Solow model in two basic ways. First, they make technological progress endogenous. We will investigate various models where growth occurs as the result of conscious decisions on the part of economic actors to invest in the accumulation of knowledge. We will also consider the determinants of the decisions to invest in knowledge accumulation. Second, the models examine the possibility that the role of capital is considerably larger than is suggested by considering physical capitals share in income. This can occur if the capital relevant for growth is not just physical capital but also human capital. It can also occur if there are positive externalities from capital accumulation, so that what capital earns in the market understates its contribution to production. We will see that models based on endogenous technological progress and on a larger role of capital provide candidate explanations of both worldwide growth and crosscountry mcome differences. We now turn to the Solow model. 1.2 Assumptions Inputs and Output The Solow model focuses on four variables: output ( ), capital (K), labor li), and "knowledge" or the "effectiveness of labor" (A). At any time, the economy has some amounts of capital, labor, and knowledge, and these are combined to produce output. The production function takes the form Y{t) = F{K{t),A{mt)), (1.1) where t denotes time. Two features of the production function should be noted. First, time does not enter the production function directly, but only through K, L, and A. That is, output changes over time only if the inputs into production change. In particular, the amount of output obtained from given quantities of capital and labor rises over timethere is technological progressonly if the amount of knowledge increases. Second, A and I enter multiplicatively. AL is referred to as effective labor, and technological progress that enters in this fashion is known as laboraugmenting or Harrodneutral? This way of specifying how A enters, If knowledge enters m the form = F(AK,L), technological progress Is capitalaugmenting. If It enters In the form = .\F(K, L), technological progress Is Hicksneutral.
Assumptions Concerning the Production Function The models critical assumption concerning the production function is that it has constant returns to scale in its two arguments, capital and effective labor. That is, doubling the quantities of capital and effective labor (for example, by doubling and I with A held fixed) doubles the amount produced. More generally, multiplying both arguments by any nonnegative constant causes output to change by the same factor: F(cK,cAL) = cF(K,AL) for all > 0. " (1.2) The assumption of constant returns can be thought of as combining two assumptions. The first is that the economy is big enough that the gains from specialization have been exhausted. In a very small economy, there are probably enough possibilities for further specialization that doubling the amounts of capital and labor more than doubles output. The Solow model assumes, however, that the economy is sufficiently large that, if capital and labor double, the new inputs are used in essentially the same way as the existing inputs, and thus that output doubles. The second assumption is that inputs other than capital, labor, and knowledge are relatively unimportant. In particular, the model neglects land and other natural resources. If natural resources are important, doubling capital and labor could less than double output. In practice, however, the availability of natural resources does not appear to be a major constraint on growth. Assuming constant returns to capital and labor alone therefore appears to be a reasonable approximation. The assumption of constant returns allows us to work with the production function in intensive form. Setting = 1/ 1 in equation (1.2) yields F(K,AL). (1.3) Growth accounting, which is described in Section 1.7, can be used to formalize the argument that natural resources are not very important to growth. Problem 1.10 investigates a simple model where natural resources cause there to be diminishing retums to capital and labor. Finally, Chapter 3 examines the implications of Increasing returns. together with the other assumptions of the model, will imply that the ratio of capital to output, KjY, eventually settles down. In practice, capitaloutput ratios do not show any clear upward or downward trend over extended periods. In addition, building the model so that the ratio is eventually constant makes the analysis much simpler. Assuming that A multiplies I is therefore very convenient. The central assumptions of the Solow model concern the properties of the production function and the evolution of the three inputs into production (capital, labor, and knowledge) over time. We discuss each in tum.
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