Basic References: Hall and Taylor, Chapter 4; Study Guide, Chapter 4. A. OVERVIEW In this chapter, HT first review the determinants of the aggregate supply of goods and services in an economy in any given period T: (a) the labor force [quantity and quality]; (b) the stock of productive capital [quantity and quality]; (c) and technology. They then provide an extended discussion of the notion of "potential real GDP," previously introduced in brief terms in Chapter 3. Specifically, HT interpret "potential real GDP" as potential (i.e., full employment) aggregate supply. Thus, potential GDP in any period T measures an economy's #capabilities# for producing goods and services during period T, conditional on the given labor force, capital stock, and state of technology. For concreteness, HT generally take T = one year. The remaining portions of chapter 4 discuss various factors affecting the growth of potential GDP over time. B. SOME PRELIMINARY SUPPLY-SIDE CONCEPTS The #real wage# is the nominal wage divided by the cost of living as measured by some appropriate price index--e.g., the CPI or the GDP implicit price deflator P. In this chapter HT take the latter index P as the price index and define: (4.1) W(T) = nominal wage during period T Total labor earnings during T = ---------------------------------- Total number of hours worked during T = Average hourly wage during T. W(T) (4.2) ------ = real wage during period T. P(T) The #potential employment for period T# is defined to be the volume of employment N*(T) that would obtain in period T, given existing incentives such as fringe benefits, unemployment insurance, etc., if real wages were fully flexible and responsive to demand and suply pressures. (4.3) N*(T) = the potential (or "natural") level of employment for period T = amount of employment which would obtain in period T, given existing incentives such as fringe benefits, unemployment insurance, etc., if real wages were fully flexible and responsive to demand and supply pressures. Suppose the maximum amount Y of real GDP which can be produced from any given amount (N,K) of labor and capital used in production, given the current technology A, is described by an #aggregate production function# (4.4) Y = F(N,K,A) . It is assumed that the production function F(N,K,A) is an increasing function of N, K, and A---that is, the partial derivatives of F(N,K,A) with respect to N, K, and A, denoted by F_N, F_K, and F_A, are assumed to be strictly positive. Thus, for any given K and A, F(N',K,A) is strictly greater than F(N'',K,A) if and only if N' > N'', and similarly for changes in K (holding N and A fixed) and A (holding N and K fixed). The level of real GDP, Y*(T), which would result in period T if employment N were equal to potential employment N*(T), and the existing capital stock K(T) were fully in use, is called the #potential (or "natural") level of real GDP#. (4.5) Y*(T) = the potential (or "natural") level of real GDP for period T = F(N*(T),K(T),A), where N*(T) = potential employment for period T K(T) = existing stock of capital for period T The unemployment rate U*(T) which prevails when real GDP coincides with potential real GDP is called #the natural rate of unemployment#. Currently, U*(T) seems to be about 6 percent in the U.S. The reason the natural rate of unemployment is not zero is that there is always #frictional unemployment#---e.g., people in the process of changing jobs, or engaging in job search, acquiring new training, etc. Some economists thus refer to U*(T) as the period T rate of #voluntary# unemployment. C. MARKET EQUILIBRIUM INTERPRETATION FOR POTENTIAL EMPLOYMENT AND OUTPUT We will now undertake a more detailed understanding of the market equilibrium interpretation for N*(T) and Y*(T) which HT and other macroeconomists ascribe to. #THE LABOR DEMAND CURVE#: The #labor demand curve# shows the maximum amount of labor which profit-maximizing firms would be willing to hire at each possible different real wage, taking as given the existing capital stock. Profit maximization requires that this demand for labor be such that the marginal product of labor is just equal to the real wage. Recall that the marginal product of labor is the incremental change in produced output due to an incremental change in labor input. Thus, letting N^D denote the labor demand of firms, the requirement that the marginal product of labor be equal to the real wage can be expressed in the form (4.6) F_N(N^D,K(T),A) = W/P, where the left-hand-term in (4.6) denotes the derivative of the aggregate production function F(N,K,A) with respect to N, evaluated at (N^D, K(T), A). Under normal production circumstances, one would expect to see a downward sloping labor demand curve in the N-W/P plane. W/P | | | | | | ----------------------------- N 0 Fig. 4.1: Downward Sloping Labor Demand Curve #THE LABOR SUPPLY CURVE#: The #labor supply curve# shows the maximum amount of labor which households would be willing to supply at each different real wage. Letting N^S denote household labor supply, it will be supposed that the labor supply curve takes the form N^S = h(W/P). Note that W/P measures the "price" (opportunity cost) of an hour of leisure. Consequently, in analogy to other goods, one might expect to see the demand for leisure #fall# (and hence the supply of labor #rise#) as the real wage W/P increases, implying an upward sloping labor supply curve. Call this the "price effect" of an increase in W/P. Nevertheless, an increase in W/P also represents a direct increase in #income# (or wealth = accumulated income); and this increase in income is more substantial the longer the increase in W/P is expected to last. If leisure is a "normal good" in the sense that demand for leisure increases with increases in income, then an #increase# in W/P, viewed as an increase in income, would tend to #increase# the demand for leisure and hence to #decrease# the supply of labor. Call this the "wealth effect." The wealth effect and price effect thus tend to move the supply of labor in opposite directions, implying that the #net# change in labor supply following upon a wage increase is theoretically ambiguous. Labor supply might increase in response to an increase in the real wage, resulting in an upward sloping labor supply curve (price effect outweighs wealth effect), or it might decrease, resulting in a backward bending labor supply curve (wealth effect outweighs price effect). In Fig. 4.2, for example, the labor supply curve becomes backward bending (wealth effect outweighs price effect) for real wages higher than (W/P)'. W/P | | | (W/P)'|............... | . | . ----------------------------- N 0 N' Fig. 4.2: Backward Bending Labor Supply Curve This theoretical ambiguity about the slope of the labor supply curve is reflected in empirical estimates showing that estimated labor supply curves tend to be fairly steeply sloped. Actual labor supply thus seems to be fairly insensitive overall to changes in the real wage, suggesting that the wealth and price effects are offsetting one another. #DETERMINATION OF N*(T), (W/P)*(T), and Y*(T) AS A LABOR MARKET EQUILIBRIUM# N*(T) and (W/P)*(T) are then jointly and simultaneously determined as the point of intersection of the demand and supply curves for labor. Given this derivation, N*(T) and (W/P)*(T) are also referred to as the period T #full employment# levels for employment N and the real wage rate W/P. W/P | | | | (W/P)*(T) |...................... LE | . | . | . ------------------------------------------------- N 0 N*(T) Fig. 4.3: Labor Market Equilibrium at Point LE Once N*(T) is determined, Y*(T)---the potential real GDP for period T---is determined as the maximum amount of output which can be produced using N*, taking as given the existing capital stock K(T). Given this derivation, Y*(T) is also referred to as the #full employment# level of real GDP. Y | | | | Y*(T) | | | | | --------------------------------------------------- N 0 N*(T) Fig. 4.4: Determination of potential (full employment) GDP HT define the "potential output line" for period T to be the vertical line at Y = Y*(T). Increasingly, macroeconomists now use the more accurate term "full employment line" for the vertical line through the full employment level of output Y*(T). We will follow this latter usage. P FE Line | | | | | | | | | | | | | | --------------------------------------- Y 0 Y*(T) Fig. 4.5: The Full Employment (Potential Output) Line SUMMARY NOTE ON TERMINOLOGY: N*(T) = "potential" or "natural" or "full employment" level of employment for period T; hereafter we will use the latter term. Y*(T) = "potential" or "natural" or "full employment" level of real GDP for period T; hereafter we will use the latter term. (W/P)*(T) = "potential" or "natural" or "full employment" level of real wage in period T; hereafter we will use the latter term. #Key Points to Note# (a) The period T full employment levels N*(T), (W/P)*(T), and Y*(T) depend #only# on the period T supply and demand curves for labor, and on the period T capital-conditioned production function F(N,K(T),A). Unless there is some change either in the labor supply schedule h(W/P), the production function F(N,K,A), or the capital stock K(T), these full employment levels will be #unchanged# over time. Note in particular that Y*(T) does not vary in response to changes in the period T market price P(T). (b) #Unanticipated# disturbances to the economy in period T which affect the period T production function, and hence change the period T full employment levels Y*(T), N*(T), and (W/P)*(T), are referred to as period T #supply shocks#. For example---as will be discussed below---the 1973 and 1979 sharp increases in the price of oil are viewed as prime examples of supply shocks. (c) But note that a variety other types of factors, both anticipated and unanticipated in period T, can affect these full employment levels over time, such as: (a) changes in the capital stock K(T) as the result of ongoing investment; (b) changes in household tastes for labor vs. leisure, which affect the labor supply function h(z); and (c) gradual changes in technology which affect the production function F(N,K,A), e.g., upward shifts in A reflecting technological improvements that permit a higher level of output for each given level of N and K. QUESTION: How would a change in the period T price of oil be reflected in our diagrams for the determination of Y*(T), N*(T), and (W/P)*(T)? First note that oil is an intermediate good and not a final good, hence its price does #not# enter into the implicit GDP deflator, P(T). In our simplified model of production, oil enters as a capital input to production. An increase in the price of oil increases the #costs# of production to each oil-using firm, and hence would result in firms cutting back on their oil usage. This cutback in the use of oil in turn implies that less output can be produced for any given quantity of other inputs to production. In particular, then, the period T production function F(N,K(T),A) shifts down for each level of N as K(T) shifts down in response to the decrease in oil usage. The downward shift in the production function F(N,K(T),A) due to the downward shift in K(T) implies less capital input is available per worker at each level of N. This in turn will generally imply a decrease in the #productivity# of labor for each given level of N, and hence a downward shift in the period T labor demand curve. Assuming the period T labor supply curve is unaffected, this implies a drop in the full employment level of employment N*(T), and hence also in potential output Y*(T). For example, it has been estimated that the increase in the relative price of energy resulting from the 1979-1989 oil price shock resulted in a 5.7% drop in output in the U.S. Indeed, the U.S. went into a recession following both the 1973 and 1979-1980 oil price shocks, with negative real GDP growth in each case, combined with a drop in the real wage rate. #Keynesian vs. Real Business Cycle Interpretation of Employment Fluctuations# In Keynesian-type macro models, fluctuations in observed employment N(T) over time are generally interpreted as fluctuations #around# the trend line defined by the graph of the full employment level of employment, N*(T). In particular, Keynesians believe that "involuntary unemployment", as measured by the difference between supply and demand at any given real wage, is a major recurring problem in the downward (recessionary) phase of the business cycle. In contrast, during the past ten years an alternative school of thought has arisen, referred to as the "real business cycle school." This school argues that employment N(T) in each period T is #always# at the intersection of the labor supply and demand curves, and hence #always# equal to its full employment level N*(T). According to this view, changes in employment over time #represent fluctuations in N*(T) itself.# It is argued that these fluctuations in N*(T) are due entirely to various types of supply shocks---in particular, to direct shocks to the production function ("productivity shocks"). If the latter school of thought is right, it would have tremendous implications for government policy. Traditionally, Keynesians have viewed the proper role of government as minimizing "involuntary unemployment" in each period T. In contrast, the real business cycle interpretation implies that fluctuations in employment represent the natural and desirable responses of private optimizing agents to various productivity shocks, hence government should not attempt to offset these fluctuations. Although the real business cycle theory has generated quite a research industry, many economists argue that the empirical support for this theory to date is not compelling. They argue that adverse "productivity shocks" in practice are not large enough to explain, in and of themselves, the timing and magnitudes of the recessions experienced by the U.S.; and that shocks to aggregate demand (particularly money shocks) are equally, if not more, important in understanding the business cycle. To test the real business cycle theory and other theories of the business cycle, it is essential to have a way of measuring the magnitude and sign of productivity shocks. One approach to this measurement is through the "Solow growth formula." This formula is developed in the next section. D. THE ECONOMIC GROWTH FORMULA #OVERVIEW# In most years, potential real GDP grows relative to its previous magnitude. Focusing on the supply side of the economy, three factors contribute to the growth of potential real GDP: (1) growth in the quantity and/or quality of the labor force; (2) growth in the quantity and/or quality of the capital stock; (3) changes in "total factor productivity" stemming from technological improvements in the efficiency with which labor and capital inputs are transformed into output. Growth in the #quantity# of labor and capital inputs, and to some extent the #quality# of labor and capital inputs, is in principle measurable; but changes in total factor productivity are more problematic. Many researchers attempt to measure changes in total factor productivity---or "productivity change" for short---by defining productivity change as a residual change in actual #measured# real GDP after changes in the quantity and quality of labor and capital inputs have been accounted for. Schematically: [Change in Productivity] = [Total change in real GDP] - [Change in real GDP due to change in N usage] - [Change in real GDP due to change in K usage], where labor and capital input measures have been adjusted to account for changes in quality. For example, N might be measured in hours of "unskilled labor." An hour of skilled labor would then be counted as more than one hour of "unskilled labor." This was the approach taken by Robert Solow, a 1987 Nobel prize winner in economics. The "Solow growth formula" is now so well known, and so extensively used in empirical and policy research, that it is important we take a bit of time to understand its derivation, and its potential uses and limitations. #THE SOLOW GROWTH FORMULA# Suppose the production function F(N,K,A) for period T takes the particular form (4.7) Y = Af(N,K), so that the technology factor A multiplies some function f(N,K) of N and K. The technology factor A is then an index of #total (or overall)# factor productivity in period T. If A grows by one percent during period T, it means that, by the end of period T, the economy can produce one percent more output (real GDP) from each given amount of labor N and capital K than it was capable of producing at the beginning of period T. More generally, the representation (4.7) for the aggregate production function implies that any growth in Y from one period to the next must be attributable either to a change in total factor productivity, A, or to a change in labor usage, N, or to a change in capital usage, K. Robert Solow used representation (4.7) to derive a formula connecting the percentage rate of change in Y with the percentage rates of change in A, N, and K. The derivation of this formula is discussed in the appendix to Chapter 4 of Hall and Taylor. Hall and Taylor use discrete time changes [Y(T+1)-Y(T)] in place of derivatives dY(T)/dT, so that their derivation is heuristic rather than rigorous. Derivatives will be used in the following discussion so that the relations obtained are exact rather than approximate. The basic math fact used to derive this formula is as follows: If V is a time-dependent variable which is expressable as a #product# of two other time-dependent variables X and Z, i.e., if (4.8) V(T) = X(T) x Z(T), then the percentage time-rate of change of V at T is given by the #sum# of the percentage time-rates of change of X and Z, i.e., DV(T) DX(T) DZ(T) (4.9) ------- = ----- + ------- , V(T) X(T) Z(T) where DV(T) = dV(T)/dT, the derivative of V respect to time T, and similarly for X and Z. For the case at hand, Y plays the role of V, A plays the role of X, and f(N,K) plays the role of Z. Thus, recalling that period T real GDP is given by Y(T) = A(T) x f(N(T),K(T)), DY(T) DA(T) Df(N(T),K(T)) (4.10) ------- = ------ + ------------- . Y(T) A(T) f(N(T),K(T)) Recalling the standard formula for total differentiation of a function of two variables---i.e., Dh(u,v) = h_uDu + h_vDv for any differentiable function h(u,v)---the far right term in (4.10) can be further broken down as follows. Df(N(T),K(T)) DN(T) DK(T) (4.11) ------------- = M_N(T) ----- + M_K(T) ----- , f(N(T),K(T)) Y(T) Y(T) where M_N(T) = A(T)f_N(N(T),K(T)) denotes the period T marginal product of labor and M_N(T) = A(T)f_K(N(T),K(T)) denotes the period T marginal product of capital. Suppose firms face "competitive" labor and capital input markets, and hence take prices as given. Profit-maximization by firms then requires that labor and capital inputs N and K are used up to the point where their marginal products M_N and M_K are equal to their real rates of return. The real rate of return on capital K is defined to be (4.12a) R^K ---- = the real rate of return on capital K (rent, P interest, dividend payments, economic profits) whereas the real rate of return on labor N is defined to be (4.12b) W --- = the real wage rate . P Consequently, profit-maximization requires (4.13a) Df(N(T),K(T)) W(T) M_N(T) = A(T) ------------- = ------ ; DN(T) P(T) (4.13b) Df(N(T),K(T)) R^K(T) M_K(T) = A(T) --------------- = -------- . DK(T) P(T) Dividing through by A(T), one obtains (4.14a) W(T) Df(N(T),K(T)) --------- = ----------------- ; P(T)A(T) DN(T) (4.14b) R^K(T) Df(N(T),K(T)) ---------- = ---------------- . P(T)A(T) DK(T) Combining (4.10), (4.11), and (4.14), recalling Y = Af(N,K), and dropping time subscripts for ease of presentation, DY DA W DN R^K DK (4.15) ---- = ---- + --- ---- + ---- ---- Y A P Y P Y or DY DA WN DN R^KK DK ---- = ---- + ---- ---- + ------ ---- . Y A PY N PY K "labor "capital share" share" The ratio WN/PY is the fraction of nominal GDP (i.e., the fraction of PY) that is paid out to labor, and the ratio R^KK/PY is the fraction of nominal GDP paid out to capital. These ratios have been remarkably stable over time, equalling approximately .7 and .3, respectively. Substituting these empirical share estimates into (4.15), we finally arrive at Solow's famous formula expressing growth in national income as the sum of time-rates of change in productivity, labor, and capital inputs: SOLOW GROWTH FORMULA: DY DA DN DK (4.16) ---- = ---- + .7 ---- + .3 ---- . Y A N K [Compare Hall and Taylor, equation (4-7).] Re-arranging terms in (4.16), one then obtains DA DY DN DK (4.17) ---- = ---- - .7 ---- - .3 ---- . A Y N K Since all terms on the right-side of (4.17) are measurable in principle, (4.17) gives an empirically operational way to measure "productivity growth" DA/A. #RELATIVE IMPACTS OF CHANGES IN THE THREE GROWTH FACTORS# According to the Solow growth formula (4.16), a 1% increase in the growth rate of real GDP requires nearly a 3.3% increase in the growth rate of physical capital. That is: (4.18) 1% increase in DK/K ---> .3% increase in DY/Y ; hence (4.19) (1/.3)% = 3.3% increase in DK/K --> 1% increase in DY/Y. However, also according to (4.16), employment growth has over twice the leverage of capital growth. Specifically: (4.20) 1% increase in DN/N ----> .7% increase in DY/Y or (4.21) (1/.7)% = 1.43% increase in DN/N --> 1% increase in DY/Y. Finally, the most leverage of all is obtained from a change in productivity growth. Specifically, (4.22) 1% increase in DA/A -------> 1% increase in DY/Y . ACTUAL BREAKDOWN FOR THE U.S.: In Figure 4-7, Hall and Taylor give estimates for the factor contributions DN/N, DK/K and DA/A to real GDP growth, DY/Y, for the U.S. from 1961 to 1990. In order to smooth out business cycle fluctations, the estimated factor contributions are averaged over ten-year periods. The data reveal a relatively increasing contribution to GDP growth from both capital investment, DK/K, and labor growth, DN/N, over this time period. However, there appears to have been a marked decrease in the relative contribution from total factor productivity growth, DA/A. A more dramatic presentation of similar empirical findings can be found in the work of E. Denison, an economist at the Brookings Institution. According to Denison, the annual real GDP growth rate DY/Y over 1929-1982 averaged about 2.92%. The breakdown by source, averaged over this period, is as follows: 1929-1982: DY DA DN DK (4.23) ---- = ---- + .7 ---- + .3 ---- . Y A N K 2.92% 1.02% 1.34% 0.56% On the other hand, the breakdown by source of average real GDP over the shorter more recent period 1973-1982 is strikingly different: 1973-1982: DY DA DN DK (4.24) ---- = ---- + .7 ---- + .3 ---- . Y A N K 1.55% -0.27% 1.13% 0.69% In other words, Denison estimated that any given combination of labor and capital inputs would have produced #less# output in 1982 than in 1973. This is yet another manifestation of the "productivity puzzle" mentioned in the first class lecture. The main message of Denison's findings is that the slowdown in output growth is #not# due to a slowdown in the growth of labor or capital inputs. Rather, the slowdown is due largely to a slowdown in productivity growth. #POLICY IMPLICATIONS OF THE SOLOW GROWTH FORMULA ESTIMATES# Various explanations have been offered for the productivity slowdown, but no compelling single explanation has been found. Some economists have argued that it simply reflects a measurement problem---the underestimation of face-paced #quality# changes in both labor and capital inputs, resulting in an underestimation of productivity growth. Other economists have pointed to increased regulations in the workplace, the lingering impact of the 1973 and 1979-1980 oil price shocks, and a slower rate of technological innovation. But recently a consensus seems to be forming across all sectors of the U.S. economy that a large part of the slowdown must be attributed to a decine in educational quality, leading to a sharp decline in the human capital embodied in the labor force. One very important form of human capital in decline seems to be entrepreneurial skill. See: John H. Bishop, "Is the Test Score Decline Responsible for the Productivity Slowdown?," American Economic Review, (March 1989), pp. 178-197. W. J. Baumol, "Entrepreneurship: Productive, Unproductive, and Destructive," Journal of Political Economy (October 1990), Part 1, pp. 893-921. Moreover, there now seems to be widespread agreement that the deteriorating nature of the U.S. economy's infrastructure (i.e., its highways, bridges, dams, airports, and other publicly owned capital) is also a significant factor in the productivity slowdown. Over the past twenty years, U.S. government investment in infrastructure has taken place at a low rate, relative both to American experience and to rates of infrastructure investment in other developed countries. The primary way to stimulate output growth through increased labor #quality# is increased expenditures on human capital, e.g., education, training, health and safety measures, etc. An increase in labor #usage# might be encouraged by increasing labor supply at each real wage by lowering marginal tax rates on wage income, and/or by increasing labor demand at each real wage by reducing private firm sector responsibility for provision of key worker benefits, e.g., health insurance. The primary way to stimulate output growth through increased capital #quality# is through increased expenditure on research and development (R&D). An increase in capital #usage# (as opposed to capital quality) might be stimulated by various types of tax credits or deductions (e.g., an investment tax credit). Finally, output growth might be stimulated through increased productivity growth (increases in DA/A) by encouraging more efficient exploitation of economies of scale, better organization of the workplace, better product distribution systems, better allocation of resources (e.g., job availability services), an improved work ethic, etc. Government itself might increase its expenditures on these items, or it might encourage the private sector to increase expenditures on these items by some type of tax and/or transfer policy. The success of past and current government attempts to implement such policies will explored at a later point in the course after the complete HT model has been assembled. One aggressive approach which has been proposed for increasing productivity growth is an industrial policy---a growth strategy in which the government attempts to influence the nation's pattern of industrial development through taxes, subsidies, or regulation. Critics argue that there is no good reason to expect that the government will be better than the private sector at picking the most promising technologies and oversee their development. See: Gene Grossman, "Promoting New Industrial Activities: A Survey of Recent Arguments and Evidence," OECD Economic Studies, Spring 1990, pp. 87-125. #REAL BUSINESS CYCLE THEORY RE-VISITED# The real business school has one important fact on its side. When changes in productivity growth are measured by changes in DA/A, in accordance with the Solow growth formula, the changes are large. Moreover, empirical estimates for DA/A increase at the same time that growth rates for Y and employment N rise---namely, during boom times---and decline sharply in a recession. The real business cycle view is that these changes in the empirical estimates for DA/A represent exogenous changes in productivity growth which #cause# the ups and downs in real GDP growth. Critics of the real business cycle theory give another interpretation for these measured changes in DA/A. They argue that these measured changes in DA/A are #caused by# changes in DY/Y and DN/N rather than vice versa. Specifically, they argue that three factors make measured changes in productivity growth move with changes in output and employment growth: (a) #Labor hoarding#: Firms find it costly to lay off skilled workers when production declines and rehire them when production increases. Hence DN/N remains fairly steady even during downturns and upturns. Thus, productivity #per worker# tends to drop in times of low production (excess workers are still kept on the payroll) and rise during times of high production (excess workers now become more productive). (b) #Market power#: Firms with market power are able to pay a real wage #lower# than the marginal product of labor. Productivity measurement methods which presume workers are paid real wages equal to their marginal product are biased because they give workers too little credit. (c) #Increasing returns#: With increasing returns to scale, an across-the-board 1% increase in inputs results #more# than a 1% change in output. Standard productivity growth measurements attribute this difference to productivity growth rather than to increasing returns. Consequently, every general expansion in inputs is bound to result in a concurrently measured increase in "productivity growth" and vice versa.