Lecture notes principles of Microeconomic Theory

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Romer-1820130 book February 15, 2011 9:24 6 Chapter 1 THE SOLOW GROWTH MODEL 1.1 Some Basic Facts about Economic Growth Over the past few centuries, standards of living in industrialized countries have reached levels almost unimaginable to our ancestors. Although com- parisons are difficult, the best available evidence suggests that average real incomes today in the United States and Western Europe are between 10 and 30 times larger than a century ago, and between 50 and 300 times larger 1 than two centuries ago. Moreover, worldwide growth is far from constant. Growth has been rising over most of modern history. Average growth rates in the industrialized countries were higher in the twentieth century than in the nineteenth, and higher in the nineteenth than in the eighteenth. Further, average incomes on the eve of the Industrial Revolution even in the wealthiest countries were not dramatically above subsistence levels; this tells us that average growth over the millennia before the Industrial Revolution must have been very, very low. One important exception to this general pattern of increasing growth is the productivity growth slowdown. Average annual growth in output per person in the United States and other industrialized countries from the early 1970s to the mid-1990s was about a percentage point below its earlier level. The data since then suggest a rebound in productivity growth, at least in the United States. How long the rebound will last and how widespread it will be are not yet clear. 1 Maddison (2006) reports and discusses basic data on average real incomes over modern history. Most of the uncertainty about the extent of long-term growth concerns the behav- ior not of nominal income, but of the price indexes needed to convert those figures into estimates of real income. Adjusting for quality changes and for the introduction of new goods is conceptually and practically difficult, and conventional price indexes do not make these adjustments well. See Nordhaus (1997) and Boskin, Dulberger, Gordon, Griliches, and Jorgenson (1998) for discussions of the issues involved and analyses of the biases in con- ventional price indexes. 61.1 Some Basic Facts about Economic Growth 7 There are also enormous differences in standards of living across parts of the world. Average real incomes in such countries as the United States, Germany, and Japan appear to exceed those in such countries as Bangladesh 2 and Kenya by a factor of about 20. As with worldwide growth, cross-country income differences are not immutable. Growth in individual countries often differs considerably from average worldwide growth; that is, there are often large changes in countries’ relative incomes. The most striking examples of large changes in relative incomes are growth miracles and growth disasters. Growth miracles are episodes where growth in a country far exceeds the world average over an extended period, with the result that the country moves rapidly up the world income distri- bution. Some prominent growth miracles are Japan from the end of World War II to around 1990, the newly industrializing countries (NICs) of East Asia (South Korea, Taiwan, Singapore, and Hong Kong) starting around 1960, and China starting around 1980. Average incomes in the NICs, for example, have grown at an average annual rate of over 5 percent since 1960. As a result, their average incomes relative to that of the United States have more than tripled. Growth disasters are episodes where a country’s growth falls far short of the world average. Two very different examples of growth disasters are Argentina and many of the countries of sub-Saharan Africa. In 1900, Argentina’s average income was only slightly behind those of the world’s leaders, and it appeared poised to become a major industrialized country. But its growth performance since then has been dismal, and it is now near the middle of the world income distribution. Sub-Saharan African countries such as Chad, Ghana, and Mozambique have been extremely poor through- out their histories and have been unable to obtain any sustained growth in average incomes. As a result, their average incomes have remained close to subsistence levels while average world income has been rising steadily. Other countries exhibit more complicated growth patterns. Cˆ ote d’Ivoire was held up as the growth model for Africa through the 1970s. From 1960 to 1978, real income per person grew at an average annual rate of 3.2 percent. But in the three decades since then, its average income has not increased at all, and it is now lower relative to that of the United States than it was in 1960. To take another example, average growth in Mexico was very high in the 1950s, 1960s, and 1970s, negative in most of the 1980s, and moderate— with a brief but severe interruption in the mid-1990s—since then. Over the whole of the modern era, cross-country income differences have widened on average. The fact that average incomes in the richest countries at the beginning of the Industrial Revolution were not far above subsistence 2 Comparisons of real incomes across countries are far from straightforward, but are much easier than comparisons over extended periods of time. The basic source for cross- country data on real income is the Penn World Tables. Documentation of these data and the most recent figures are available at http://pwt.econ.upenn.edu/.8 Chapter 1 THE SOLOW GROWTH MODEL means that the overall dispersion of average incomes across different parts of the world must have been much smaller than it is today (Pritchett, 1997). Over the past few decades, however, there has been no strong tendency either toward continued divergence or toward convergence. The implications of the vast differences in standards of living over time and across countries for human welfare are enormous. The differences are associated with large differences in nutrition, literacy, infant mortality, life expectancy, and other direct measures of well-being. And the welfare con- sequences of long-run growth swamp any possible effects of the short-run fluctuations that macroeconomics traditionally focuses on. During an av- erage recession in the United States, for example, real income per person falls by a few percent relative to its usual path. In contrast, the productivity growth slowdown reduced real income per person in the United States by about 25 percent relative to what it otherwise would have been. Other exam- ples are even more startling. If real income per person in the Philippines con- tinues to grow at its average rate for the period 1960–2001 of 1.5 percent, it will take 150 years for it to reach the current U.S. level. If it achieves 3 per- cent growth, the time will be reduced to 75 years. And if it achieves 5 percent growth, as the NICs have done, the process will take only 45 years. To quote Robert Lucas (1988), “Once one starts to think about economic growth, it is hard to think about anything else.” The first four 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 goal is to learn what insights they offer concerning worldwide growth and income differ- ences across countries. Indeed, the ultimate objective of research on eco- nomic growth is to determine whether there are possibilities for raising overall growth or bringing standards of living in poor countries closer to those in the world leaders. This chapter focuses on the model that economists have traditionally 3 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 Solow’s are often best understood through comparison with the Solow model. Thus understanding the model is essential to under- standing 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 per- son. Specifically, suppose that capital accumulation affects output through the conventional channel that capital makes a direct contribution to pro- duction, for which it is paid its marginal product. Then the Solow model 3 The Solow model (which is sometimes known as the Solow–Swan model) was developed by Robert Solow (Solow, 1956) and T. W. Swan (Swan, 1956).1.1 Some Basic Facts about Economic Growth 9 implies that the differences in real incomes that we are trying to under- stand 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 tech- nological progress, for example) or absent altogether (in the case of positive externalities from capital, for example). Thus to address the central ques- tions of growth theory, we must move beyond the Solow model. Chapters 2 through 4 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 takes the saving rate as exogenous and constant. Chapter 2 presents two models that make saving endogenous and potentially time-varying. In the first, saving and consumption decisions are made by a fixed set of infinitely lived households; in the second, the decisions are made by overlapping generations of households with finite horizons. Relaxing the Solow model’s assumption of a constant saving rate has three advantages. First, and most important for studying growth, it demon- strates that the Solow model’s 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 rela- tions among aggregate variables provides no way of judging 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 infinite-horizon and overlapping-generations models are built up from the behavior of individuals, and can therefore be used to discuss wel- fare issues. Third, infinite-horizon and overlapping-generations models are used to study many issues in economics other than economic growth; thus they are valuable tools. Chapters 3 and 4 investigate more fundamental departures from the Solow model. Their models, in contrast to Chapter 2’s, provide different answers than the Solow model to the central questions of growth theory. Chapter 3 departs from the Solow model’s treatment of technological pro- gress as exogenous; it assumes instead that it is the result of the alloca- tion of resources to the creation of new technologies. We will investigate the implications of such endogenous technological progress for economic growth and the determinants of the allocation of resources to innovative activities. The main conclusion of this analysis is that endogenous technological progress is almost surely central to worldwide growth but probably has lit- tle to do with cross-country income differences. Chapter 4 therefore focuses specifically on those differences. We will find that understanding them re- quires considering two new factors: variation in human as well as physical capital, and variation in productivity not stemming from variation in tech- nology. Chapter 4 explores both how those factors can help us understand10 Chapter 1 THE SOLOW GROWTH MODEL the enormous differences in average incomes across countries and potential sources of variation in those factors. We now turn to the Solow model. 1.2 Assumptions Inputs and Output The Solow model focuses on four variables: output (Y ), capital (K ), labor (L), 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(t)L(t)), (1.1) where t denotes time. Notice that 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 to production change. In particular, the amount of output obtained from given quantities of capital and labor rises over time—there is technological progress—only if the amount of knowledge increases. Notice also that A and L enter multiplicatively. AL is referred to as effec- tive labor, and technological progress that enters in this fashion is known as 4 labor-augmenting or Harrod-neutral. This way of specifying how A enters, together with the other assumptions of the model, will imply that the ratio of capital to output, K/Y, eventually settles down. In practice, capital-output ratios do not show any clear upward or downward trend over extended peri- ods. In addition, building the model so that the ratio is eventually constant makes the analysis much simpler. Assuming that A multiplies L 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 turn. Assumptions Concerning the Production Function The model’s 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 ex- ample, by doubling K and L with A held fixed) doubles the amount produced. 4 If knowledge enters in the form Y = F(AK,L), technological progress is capital- augmenting. If it enters in the form Y = AF(K,L), technological progress is Hicks-neutral.1.2 Assumptions 11 More generally, multiplying both arguments by any nonnegative constant c causes output to change by the same factor: F(cK,cAL) = cF(K,AL) for all c ≥ 0. (1.2) The assumption of constant returns can be thought of as a combination of two separate 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 likely to be 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 so output doubles. The second assumption is that inputs other than capital, labor, and knowl- edge 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, as Section 1.8 describes, 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 produc- tion function in intensive form. Setting c = 1/AL in equation (1.2) yields   K 1 F ,1 = F(K,AL). (1.3) AL AL Here K/AL is the amount of capital per unit of effective labor, and F(K,AL)/ AL is Y/AL, output per unit of effective labor. Define k = K/AL, y = Y/AL, and f (k) = F(k,1). Then we can rewrite (1.3) as y = f (k). (1.4) That is, we can write output per unit of effective labor as a function of capital per unit of effective labor. These new variables, k and y, are not of interest in their own right. Rather, they are tools for learning about the variables we are interested in. As we will see, the easiest way to analyze the model is to focus on the behavior of k rather than to directly consider the behavior of the two arguments of the production function, K and AL. For example, we will determine the behavior of output per worker, Y/L, by writing it as A(Y/AL), or Af (k), and determining the behavior of A and k. To see the intuition behind (1.4), think of dividing the economy into AL small economies, each with 1 unit of effective labor and K/AL units of capi- tal. Since the production function has constant returns, each of these small economies produces 1/AL as much as is produced in the large, undivided economy. Thus the amount of output per unit of effective labor depends only on the quantity of capital per unit of effective labor, and not on the over- all size of the economy. This is expressed mathematically in equation (1.4).12 Chapter 1 THE SOLOW GROWTH MODEL f(k) k FIGURE 1.1 An example of a production function The intensive-form production function, f (k), is assumed to satisfy f (0) =   5 0, f (k) 0, f (k) 0. Since F(K,AL) equals ALf (K/AL), it follows that  the marginal product of capital, ∂F(K,AL)/∂K, equals ALf (K/AL)(1/AL),    which is just f (k). Thus the assumptions that f (k) is positive and f (k) is negative imply that the marginal product of capital is positive, but that • it declines as capital (per unit of effective labor) rises. In addition, f ( )  is assumed to satisfy the Inada conditions (Inada, 1964): lim f (k)=∞, k→0  lim f (k) = 0. These conditions (which are stronger than needed for the k→∞ model’s central results) state that the marginal product of capital is very large when the capital stock is sufficiently small and that it becomes very small as the capital stock becomes large; their role is to ensure that the path  • of the economy does not diverge. A production function satisfying f ( ) 0,  • f ( ) 0, and the Inada conditions is shown in Figure 1.1. A specific example of a production function is the Cobb–Douglas function, α 1−α F(K,AL) = K (AL) ,0 α 1. (1.5) This production function is easy to analyze, and it appears to be a good first approximation to actual production functions. As a result, it is very useful. 5   • • • The notation f ( ) denotes the first derivative of f ( ), and f ( ) the second derivative.1.2 Assumptions 13 It is easy to check that the Cobb–Douglas function has constant returns. Multiplying both inputs by c gives us α 1−α F(cK,cAL) = (cK ) (cAL) α 1−α α 1−α = c c K (AL) (1.6) = cF(K,AL). To find the intensive form of the production function, divide both inputs by AL; this yields   K f (k) ≡ F ,1 AL   α K (1.7) = AL α = k . α−1  Equation (1.7) implies that f (k) =αk . It is straightforward to check that this expression is positive, that it approaches infinity as k approaches zero,  and that it approaches zero as k approaches infinity. Finally, f (k) = α−2 6 −(1 −α)αk , which is negative. The Evolution of the Inputs into Production The remaining assumptions of the model concern how the stocks of labor, knowledge, and capital change over time. The model is set in continuous 7 time; that is, the variables of the model are defined at every point in time. The initial levels of capital, labor, and knowledge are taken as given, and are assumed to be strictly positive. Labor and knowledge grow at constant rates: ˙ L(t) = nL(t), (1.8) ˙ A(t) = gA(t), (1.9) where n and g are exogenous parameters and where a dot over a variable ˙ denotes a derivative with respect to time (that is, X (t) is shorthand for dX(t)/dt). 6 Note that with Cobb–Douglas production, labor-augmenting, capital-augmenting, and Hicks-neutral technological progress (see n. 4) are all essentially the same. For example, to ˜ 1−α rewrite (1.5) so that technological progress is Hicks-neutral, simply define A = A ; then ˜ α 1−α Y = A(K L ). 7 The alternative is discrete time, where the variables are defined only at specific dates (usually t = 0,1,2,. . .). The choice between continuous and discrete time is usually based on convenience. For example, the Solow model has essentially the same implications in discrete as in continuous time, but is easier to analyze in continuous time.14 Chapter 1 THE SOLOW GROWTH MODEL The growth rate of a variable refers to its proportional rate of change. ˙ That is, the growth rate of X refers to the quantity X (t)/X(t). Thus equa- tion (1.8) implies that the growth rate of L is constant and equal to n, and (1.9) implies that A’s growth rate is constant and equal to g. A key fact about growth rates is that the growth rate of a variable equals ˙ the rate of change of its natural log. That is, X (t)/X(t) equals d ln X(t)/dt.To see this, note that since ln X is a function of X and X is a function of t,we can use the chain rule to write d ln X(t) d ln X(t) dX(t) = dt dX(t) dt (1.10) 1 ˙ = X (t). X(t) Applying the result that a variable’s growth rate equals the rate of change of its log to (1.8) and (1.9) tells us that the rates of change of the logs of L and A are constant and that they equal n and g, respectively. Thus, ln L(t) = ln L(0) + nt, (1.11) ln A(t) = ln A(0) + gt, (1.12) where L(0) and A(0) are the values of L and A at time 0. Exponentiating both sides of these equations gives us nt L(t) = L(0)e , (1.13) gt A(t) = A(0)e . (1.14) 8 Thus, our assumption is that L and A each grow exponentially. Output is divided between consumption and investment. The fraction of output devoted to investment, s, is exogenous and constant. One unit of output devoted to investment yields one unit of new capital. In addition, existing capital depreciates at rateδ. Thus ˙ K(t) = sY(t) −δK(t). (1.15) Although no restrictions are placed on n, g, andδ individually, their sum is assumed to be positive. This completes the description of the model. Since this is the first model (of many) we will encounter, this is a good place for a general comment about modeling. The Solow model is grossly simplified in a host of ways. To give just a few examples, there is only a single good; government is absent; fluctuations in employment are ignored; production is described by an aggregate production function with just three inputs; and the rates of saving, depreciation, population growth, and tech- nological progress are constant. It is natural to think of these features of the model as defects: the model omits many obvious features of the world, 8 See Problems 1.1 and 1.2 for more on basic properties of growth rates.1.3 The Dynamics of the Model 15 and surely some of those features are important to growth. But the purpose of a model is not to be realistic. After all, we already possess a model that is completely realistic—the world itself. The problem with that “model” is that it is too complicated to understand. A model’s purpose is to provide insights about particular features of the world. If a simplifying assump- tion causes a model to give incorrect answers to the questions it is being used to address, then that lack of realism may be a defect. (Even then, the simplification—by showing clearly the consequences of those features of the world in an idealized setting—may be a useful reference point.) If the simplification does not cause the model to provide incorrect answers to the questions it is being used to address, however, then the lack of realism is a virtue: by isolating the effect of interest more clearly, the simplification makes it easier to understand. 1.3 The Dynamics of the Model We want to determine the behavior of the economy we have just described. The evolution of two of the three inputs into production, labor and knowl- edge, is exogenous. Thus to characterize the behavior of the economy, we must analyze the behavior of the third input, capital. The Dynamics of k Because the economy may be growing over time, it turns out to be much easier to focus on the capital stock per unit of effective labor, k, than on the unadjusted capital stock, K. Since k = K/AL, we can use the chain rule to find ˙ K(t) K(t) ˙ ˙ ˙ k(t) = − A(t)L(t) + L(t)A(t) 2 A(t)L(t) A(t)L(t) (1.16) ˙ ˙ ˙ K(t) K(t) L(t) K(t) A(t) = − − . A(t)L(t) A(t)L(t) L(t) A(t)L(t) A(t) ˙ ˙ K/AL is simply k. From (1.8) and (1.9), L/L and A/A are n and g, respectively. ˙ K is given by (1.15). Substituting these facts into (1.16) yields sY(t) −δK(t) ˙ k(t) = − k(t)n − k(t)g A(t)L(t) (1.17) Y(t) = s −δk(t) − nk(t) − gk(t). A(t)L(t)16 Chapter 1 THE SOLOW GROWTH MODEL Break-even investment (n + g + δ)k sf(k) Actual investment ∗ k k FIGURE 1.2 Actual and break-even investment Finally, using the fact that Y/AL is given by f (k), we have ˙ k(t) = sf (k(t)) − (n + g+δ)k(t). (1.18) Equation (1.18) is the key equation of the Solow model. It states that the rate of change of the capital stock per unit of effective labor is the difference between two terms. The first, sf (k), is actual investment per unit of effective labor: output per unit of effective labor is f (k), and the fraction of that output that is invested is s. The second term, (n + g +δ)k,is break- even investment, the amount of investment that must be done just to keep k at its existing level. There are two reasons that some investment is needed to prevent k from falling. First, existing capital is depreciating; this capital must be replaced to keep the capital stock from falling. This is theδk term in (1.18). Second, the quantity of effective labor is growing. Thus doing enough investment to keep the capital stock (K ) constant is not enough to keep the capital stock per unit of effective labor (k) constant. Instead, since the quantity of effective labor is growing at rate n + g, the capital stock must 9 grow at rate n + g to hold k steady. This is the (n + g)k term in (1.18). When actual investment per unit of effective labor exceeds the invest- ment needed to break even, k is rising. When actual investment falls short of break-even investment, k is falling. And when the two are equal, k is constant. ˙ Figure 1.2 plots the two terms of the expression for k as functions of k. Break-even investment, (n + g+δ)k, is proportional to k. Actual investment, sf (k), is a constant times output per unit of effective labor. Since f (0) = 0, actual investment and break-even investment are equal at  k = 0. The Inada conditions imply that at k = 0, f (k) is large, and thus that the sf (k) line is steeper than the (n + g +δ)k line. Thus for small values of 9 The fact that the growth rate of the quantity of effective labor, AL, equals n + g is an instance of the fact that the growth rate of the product of two variables equals the sum of their growth rates. See Problem 1.1. Investment per unit of effective labor1.3 The Dynamics of the Model 17 . k 0 k ∗ k FIGURE 1.3 The phase diagram for k in the Solow model k, actual investment is larger than break-even investment. The Inada con-  ditions also imply that f (k) falls toward zero as k becomes large. At some point, the slope of the actual investment line falls below the slope of the break-even investment line. With the sf (k) line flatter than the (n + g +δ)k  line, the two must eventually cross. Finally, the fact that f (k) 0 implies ∗ that the two lines intersect only once for k 0. We let k denote the value of k where actual investment and break-even investment are equal. Figure 1.3 summarizes this information in the form of a phase diagram, ∗ ˙ which shows k as a function of k.If k is initially less than k , actual in- ˙ vestment exceeds break-even investment, and so k is positive—that is, k is ∗ ∗ ˙ ˙ rising. If k exceeds k , k is negative. Finally, if k equals k , then k is zero. ∗ 10 Thus, regardless of where k starts, it converges to k and remains there. The Balanced Growth Path ∗ Since k converges to k , it is natural to ask how the variables of the model ∗ behave when k equals k . By assumption, labor and knowledge are growing at rates n and g, respectively. The capital stock, K, equals ALk; since k is ∗ ˙ constant at k , K is growing at rate n + g (that is, K/K equals n + g). With both capital and effective labor growing at rate n + g, the assumption of constant returns implies that output, Y, is also growing at that rate. Finally, capital per worker, K/L, and output per worker, Y/L, are growing at rate g. 10 If k is initially zero, it remains there. However, this possibility is ruled out by our assumption that initial levels of K, L, and A are strictly positive.18 Chapter 1 THE SOLOW GROWTH MODEL Thus the Solow model implies that, regardless of its starting point, the economy converges to a balanced growth path—a situation where each variable of the model is growing at a constant rate. On the balanced growth path, the growth rate of output per worker is determined solely by the rate 11 of technological progress. 1.4 The Impact of a Change in the Saving Rate The parameter of the Solow model that policy is most likely to affect is the saving rate. The division of the government’s purchases between consump- tion and investment goods, the division of its revenues between taxes and borrowing, and its tax treatments of saving and investment are all likely to affect the fraction of output that is invested. Thus it is natural to investigate the effects of a change in the saving rate. For concreteness, we will consider a Solow economy that is on a balanced growth path, and suppose that there is a permanent increase in s. In addition to demonstrating the model’s implications concerning the role of saving, this experiment will illustrate the model’s properties when the economy is not on a balanced growth path. The Impact on Output ∗ The increase in s shifts the actual investment line upward, and so k rises. This is shown in Figure 1.4. But k does not immediately jump to the new ∗ ∗ value of k . Initially, k is equal to the old value of k . At this level, actual investment now exceeds break-even investment—more resources are being ˙ devoted to investment than are needed to hold k constant—and so k is positive. Thus k begins to rise. It continues to rise until it reaches the new ∗ value of k , at which point it remains constant. These results are summarized in the first three panels of Figure 1.5. t de- 0 notes the time of the increase in the saving rate. By assumption, s jumps up 11 The broad behavior of the U.S. economy and many other major industrialized economies over the last century or more is described reasonably well by the balanced growth path of the Solow model. The growth rates of labor, capital, and output have each been roughly constant. The growth rates of output and capital have been about equal (so that the capital-output ratio has been approximately constant) and have been larger than the growth rate of labor (so that output per worker and capital per worker have been rising). This is often taken as evidence that it is reasonable to think of these economies as Solow-model economies on their balanced growth paths. Jones (2002a) shows, however, that the underlying determi- nants of the level of income on the balanced growth path have in fact been far from constant in these economies, and thus that the resemblance between these economies and the bal- anced growth path of the Solow model is misleading. We return to this issue in Section The Impact of a Change in the Saving Rate 19 (n + g + δ)k s f(k) NEW s f(k) OLD ∗ ∗ k k k OLD NEW FIGURE 1.4 The effects of an increase in the saving rate on investment at time t and remains constant thereafter. Since the jump in s causes actual 0 investment to exceed break-even investment by a strictly positive amount, ˙ k jumps from zero to a strictly positive amount. k rises gradually from the ∗ 12 ˙ old value of k to the new value, and k falls gradually back to zero. We are likely to be particularly interested in the behavior of output per worker, Y/L. Y/L equals Af (k). When k is constant, Y/L grows at rate g, the growth rate of A. When k is increasing, Y/L grows both because A is increasing and because k is increasing. Thus its growth rate exceeds g. ∗ When k reaches the new value of k , however, again only the growth of A contributes to the growth of Y/L, and so the growth rate of Y/L returns to g. Thus a permanent increase in the saving rate produces a temporary increase in the growth rate of output per worker: k is rising for a time, but eventually it increases to the point where the additional saving is devoted entirely to maintaining the higher level of k. The fourth and fifth panels of Figure 1.5 show how output per worker responds to the rise in the saving rate. The growth rate of output per worker, which is initially g, jumps upward at t and then gradually returns to its 0 initial level. Thus output per worker begins to rise above the path it was on 13 and gradually settles into a higher path parallel to the first. 12 ˙ For a sufficiently large rise in the saving rate, k can rise for a while after t before 0 starting to fall back to zero. 13 Because the growth rate of a variable equals the derivative with respect to time of its log, graphs in logs are often much easier to interpret than graphs in levels. For example, if a variable’s growth rate is constant, the graph of its log as a function of time is a straight line. This is why Figure 1.5 shows the log of output per worker rather than its level. Investment per unit of effective labor20 Chapter 1 THE SOLOW GROWTH MODEL s t t 0 . k 0 t t 0 k t t 0 Growth rate of Y/L g t t 0 ln(Y/L) t t 0 c t t 0 FIGURE 1.5 The effects of an increase in the saving rate In sum, a change in the saving rate 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 at any point in time, but it does not affect the growth rate of output per worker on the balanced growth path. Indeed, in the1.4 The Impact of a Change in the Saving Rate 21 Solow model only changes in the rate of technological progress have growth effects; all other changes have only level effects. The Impact on Consumption If we were to introduce households into the model, their welfare would de- pend not on output but on consumption: investment is simply an input into production in the future. Thus for many purposes we are likely to be more interested in the behavior of consumption than in the behavior of output. Consumption per unit of effective labor equals output per unit of effec- tive labor, f (k), times the fraction of that output that is consumed, 1 − s. Thus, since s changes discontinuously at t and k does not, initially con- 0 sumption per unit of effective labor jumps downward. Consumption then rises gradually as k rises and s remains at its higher level. This is shown in the last panel of Figure 1.5. Whether consumption eventually exceeds its level before the rise in s is ∗ not immediately clear. Let c denote consumption per unit of effective labor ∗ on the balanced growth path. c equals output per unit of effective labor, ∗ ∗ f (k ), minus investment per unit of effective labor, sf (k ). On the balanced ∗ growth path, actual investment equals break-even investment, (n+ g+δ)k . Thus, ∗ ∗ ∗ c = f (k ) − (n + g+δ)k . (1.19) ∗ k is determined by s and the other parameters of the model, n, g, andδ; ∗ ∗ we can therefore write k = k (s,n,g,δ). Thus (1.19) implies ∗ ∗ ∂c ∂k (s,n,g,δ)  ∗ = f (k (s,n,g,δ)) − (n + g+δ) . (1.20) ∂s ∂s ∗ ∗ We know that the increase in s raises k ; that is, we know that ∂k /∂s is positive. Thus whether the increase raises or lowers consumption in the  ∗ long run depends on whether f (k )—the marginal product of capital—is more or less than n + g+δ. Intuitively, when k rises, investment (per unit of effective labor) must rise by n + g+δ times the change in k for the increase  ∗ to be sustained. If f (k ) is less than n + g +δ, then the additional output from the increased capital is not enough to maintain the capital stock at its higher level. In this case, consumption must fall to maintain the higher  ∗ capital stock. If f (k ) exceeds n + g +δ, on the other hand, there is more than enough additional output to maintain k at its higher level, and so con- sumption rises.  ∗ f (k ) can be either smaller or larger than n + g +δ. This is shown in Figure 1.6. The figure shows not only (n + g+δ)k and sf (k), but also f (k). Since consumption on the balanced growth path equals output less break- ∗ even investment (see 1.19), c is the distance between f (k) and (n + g+δ)k ∗ ∗ at k = k . The figure shows the determinants of c for three different values22 Chapter 1 THE SOLOW GROWTH MODEL f(k) (n + g + δ)k s f(k) H ∗ k k H f(k) (n + g + δ)k s f(k) L ∗ k k L f(k) (n + g + δ)k s f(k) M ∗ k k M FIGURE 1.6 Output, investment, and consumption on the balanced growth path Output and investment Output and investment Output and investment per unit of effective labor per unit of effective labor per unit of effective labor1.5 Quantitative Implications 23 ∗ of s (and hence three different values of k ). In the top panel, s is high, and ∗  ∗ so k is high and f (k ) is less than n + g +δ. As a result, an increase in the saving rate lowers consumption even when the economy has reached ∗  ∗ its new balanced growth path. In the middle panel, s is low, k is low, f (k ) is greater than n + g +δ, and an increase in s raises consumption in the long run.  ∗ Finally, in the bottom panel, s is at the level that causes f (k ) to just equal ∗ n + g+δ—that is, the f (k) and (n + g+δ)k loci are parallel at k = k . In this case, a marginal change in s has no effect on consumption in the long run, and consumption is at its maximum possible level among balanced growth ∗ paths. This value of k is known as the golden-rule level of the capital stock. We will discuss the golden-rule capital stock further in Chapter 2. Among the questions we will address are whether the golden-rule capital stock is in fact desirable and whether there are situations in which a decentralized economy with endogenous saving converges to that capital stock. Of course, in the Solow model, where saving is exogenous, there is no more reason to expect the capital stock on the balanced growth path to equal the golden- rule level than there is to expect it to equal any other possible value. 1.5 Quantitative Implications We are usually interested not just in a model’s qualitative implications, but in its quantitative predictions. If, for example, the impact of a moderate increase in saving on growth remains large after several centuries, the result that the impact is temporary is of limited interest. For most models, including this one, obtaining exact quantitative results requires specifying functional forms and values of the parameters; it often also requires analyzing the model numerically. But in many cases, it is possi- ble to learn a great deal by considering approximations around the long-run equilibrium. That is the approach we take here. The Effect on Output in the Long Run The long-run effect of a rise in saving on output is given by ∗ ∗ ∂y ∂k (s,n,g,δ)  ∗ = f (k ) , (1.21) ∂s ∂s ∗ ∗ where y = f (k ) is the level of output per unit of effective labor on the ∗ ∗ balanced growth path. Thus to find∂y /∂s, we need to find∂k /∂s.Todo ∗ ∗ ˙ this, note that k is defined by the condition that k = 0. Thus k satisfies ∗ ∗ sf (k (s,n,g,δ)) = (n + g+δ)k (s,n,g,δ). (1.22)24 Chapter 1 THE SOLOW GROWTH MODEL Equation (1.22) holds for all values of s (and of n, g, andδ). Thus the deriva- 14 tives of the two sides with respect to s are equal: ∗ ∗ ∂k ∂k  ∗ ∗ sf (k ) + f (k ) = (n + g+δ) , (1.23) ∂s ∂s ∗ where the arguments of k are omitted for simplicity. This can be rearranged 15 to obtain ∗ ∗ ∂k f (k ) = . (1.24)  ∗ ∂s (n + g+δ) − sf (k ) Substituting (1.24) into (1.21) yields ∗  ∗ ∗ ∂y f (k )f (k ) = . (1.25)  ∗ ∂s (n + g+δ) − sf (k ) Two changes help in interpreting this expression. The first is to convert it ∗ to an elasticity by multiplying both sides by s/y . The second is to use the ∗ ∗ fact that sf (k ) = (n + g +δ)k to substitute for s. Making these changes gives us ∗  ∗ ∗ s ∂y s f (k )f (k ) = ∗ ∗  ∗ y ∂s f (k ) (n + g+δ) − sf (k ) ∗  ∗ (n + g+δ)k f (k ) = (1.26) ∗ ∗  ∗ ∗ f (k )(n + g+δ) − (n + g+δ)k f (k )/f (k ) ∗  ∗ ∗ k f (k )/f (k ) = . ∗  ∗ ∗ 1 − k f (k )/f (k ) ∗  ∗ ∗ ∗ k f (k )/f (k ) is the elasticity of output with respect to capital at k = k . ∗ Denoting this byα (k ), we have K ∗ ∗ s ∂y α (k ) K = . (1.27) ∗ ∗ y ∂s 1 −α (k ) K Thus we have found a relatively simple expression for the elasticity of the balanced-growth-path level of output with respect to the saving rate. To think about the quantitative implications of (1.27), note that if mar- kets are competitive and there are no externalities, capital earns its marginal 14 This technique is known as implicit differentiation. Even though (1.22) does not ex- ∗ ∗ plicitly give k as a function of s, n, g, andδ, it still determines how k depends on those ∗ variables. We can therefore differentiate the equation with respect to s and solve for∂k /∂s. 15 ∗ We saw in the previous section that an increase in s raises k . To check that this is also implied by equation (1.24), note that n + g+δ is the slope of the break-even investment  ∗ ∗ line and that sf (k ) is the slope of the actual investment line at k . Since the break-even ∗ investment line is steeper than the actual investment line at k (see Figure 1.2), it follows ∗ that the denominator of (1.24) is positive, and thus that∂k /∂s 0.1.5 Quantitative Implications 25 product. Since output equals ALf (k) and k equals K/AL, the marginal prod-   uct of capital,∂Y/∂K,is ALf (k)1/(AL), or just f (k). Thus if capital earns its marginal product, the total amount earned by capital (per unit of effective ∗  ∗ labor) on the balanced growth path is k f (k ). The share of total income that ∗  ∗ ∗ ∗ goes to capital on the balanced growth path is then k f (k )/f (k ), orα (k ). K In other words, if the assumption that capital earns its marginal product is a good approximation, we can use data on the share of income going to ∗ capital to estimate the elasticity of output with respect to capital,α (k ). K In most countries, the share of income paid to capital is about one-third. ∗ If we use this as an estimate ofα (k ), it follows that the elasticity of output K with respect to the saving rate in the long run is about one-half. Thus, for example, a 10 percent increase in the saving rate (from 20 percent of output to 22 percent, for instance) raises output per worker in the long run by about 5 percent relative to the path it would have followed. Even a 50 percent ∗ increase in s raises y only by about 22 percent. Thus significant changes in saving have only moderate effects on the level of output on the balanced growth path. ∗ Intuitively, a small value ofα (k ) makes the impact of saving on output K low for two reasons. First, it implies that the actual investment curve, sf (k), bends fairly sharply. As a result, an upward shift of the curve moves its intersection with the break-even investment line relatively little. Thus the ∗ ∗ impact of a change in s on k is small. Second, a low value ofα (k ) means K ∗ ∗ that the impact of a change in k on y is small. The Speed of Convergence In practice, we are interested not only in the eventual effects of some change (such as a change in the saving rate), but also in how rapidly those effects occur. Again, we can use approximations around the long-run equilibrium to address this issue. For simplicity, we focus on the behavior of k rather than y. Our goal is thus ∗ ˙ to determine how rapidly k approaches k . We know that k is determined ˙ by k: recall that the key equation of the model is k = sf (k) − (n + g +δ)k ∗ ˙ ˙ ˙ (see 1.18). Thus we can write k = k(k). When k equals k , k is zero. A first- ∗ ˙ order Taylor-series approximation of k(k) around k = k therefore yields     ˙ ∂k(k)  ∗ ˙ k   (k − k ). (1.28)  ∂k ∗ k=k ˙ That is, k is approximately equal to the product of the difference between ∗ ∗ ˙ k and k and the derivative of k with respect to k at k = k . ˙ Letλ denote −∂k(k)/∂k ∗. With this definition, (1.28) becomes k=k ∗ ˙ k(t)−λk(t) − k . (1.29)

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