Human capital formation and business cycle persistence

Journal of Monetary Economics 42 (1998) 67—92

Human capital formation and business cycle persistence Roberto Perli!,*, Plutarchos Sakellaris" ! Department of Economics, University of Pennsylvania, Philadelphia, PA 19104, USA " Department of Economics, University of Maryland, College Park, MD 20742, USA Received 5 August 1996; received in revised form 28 July 1997; accepted 27 August 1997

Abstract In this paper we examine the role of the formation of human capital in propagating shocks over the business cycle. We show that a two-sector equilibrium business cycle model with human capital is able to generate persistence in the growth of output and other aggregate variables comparable to that observed in the post-war US data. A key feature is the relatively low elasticity of substitution between skilled and unskilled labor in the production of human capital. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: E32; E10; J24 Keywords: Human capital; Equilibrium business cycles; Persistence; Low elasticity of substitution

1. Introduction In this paper we consider the role of human capital acquisition in generating persistence in cyclical movements. We show that in a two-sector equilibrium business cycle model cyclical movements of labor into and out of a human capital formation sector give rise to a substantial propagation mechanism. Our emphasis on the role of such labor movements is motivated by recent empirical evidence that there is significant substitution between education and competing labor activities during the business cycle (see Dellas and Sakellaris, 1995). Several authors have pointed out that standard real business cycle (RBC) models do not have a satisfying internal, or endogenous, propagation

* Corresponding author. E-mail: [email protected] 0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 3 2 ( 9 8 ) 0 0 0 1 1 - 7

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mechanism. This is evident from the fact that the time series properties of simulated output growth (or first differences in the log of output) strongly resemble those of the innovations to technology, or of other relevant shocks.1 These models fail to account for features of the data such as the positive serial correlation of output growth and the shape of its power spectrum. Artificial output growth data simulated by standard RBC models have autocorrelation function very close to zero (positive or negative), and power spectrum that is flat across all frequencies including business cycle ones. Standard RBC models fail to display persistence in output growth because of the pattern of response of labor and capital to technology shocks that they embody. In these models labor (and, thus, output) increases immediately in response to a positive technology shock. For the following discussion, we assume that this shock is persistent but not permanent. In subsequent periods labor goes monotonically back to its steady state level thereby reducing the level of output.2 In contrast, in the model economy that we describe in this paper labor input, broadly defined to include human capital, continues to increase after the shock causing output growth to be persistent. In our model both the physical good and human capital are produced according to a Cobb—Douglas technology with capital and labor as factors of production; labor, in turn, is an aggregation of ‘unskilled’ labor (or work-hours) and ‘skilled labor’ whose efficiency depends on the level of human capital. The key feature of the model is that the elasticity of substitution between the two types of labor in the production of human capital (education sector) is relatively low. When a shock hits the goods sector the agents want to use more skilled and unskilled labor in the physical sector. If the elasticity of substitution between the two is low enough in the education sector, it may not be optimal to divert these factors from the education to the physical sector immediately because in that way too little human capital would be produced for next period. In other words, it may be optimal to transfer labor to the goods sector gradually in order to facilitate the accumulation of more human capital for the future. For empirically plausible parameters, the slow release of labor to the goods sector results in strong output growth for a few periods following the impact period of the shock. Our model is able to replicate quite well the properties of key aggregate variables of the US economy as captured by statistics describing volatility, comovement with output and persistence in levels. Along the lines of such standard statistics the two-sector model performs like standard RBC models. 1 A partial list of papers addressing this issue includes Cogley and Nason (1995), Rotemberg and Woodford (1996), and Watson (1993). 2 This is so despite the fact that capital increases mildly at first and then returns to its steady-state level. This may be seen, for instance, in the impulse response functions in Fig. 2 of King et al. (1988a) p. 220. At the standard parameterizations, movements in the capital stock are not strong enough to impart persistence in output growth.

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A notable fact is that the correlation between hours and average labor productivity in the model is not unrealistically high. Most importantly, however, we find that our model can replicate various features of post-war US business cycles that standard RBC models cannot. We employ the generalized Q statistic (suggested by Cogley and Nason, 1995) to show that the autocorrelation properties of the growth in output as simulated by the model are similar to those observed in the data. We reach similar conclusions for consumption and investment (though not for employment). The power spectrum of simulated output growth is remarkably similar to the empirical one. After applying the Watson procedure Watson, 1993) we find that the model does not need to be augmented with substantial error in order to match the second moment properties of post-war US output growth. The lower bound on the variance of the error at business cycle frequencies is 10% of the variance in the actual data. This is almost one fourth the corresponding lower bound for a standard RBC model. The fit for consumption, employment and investment is not as good, though (in the case of the first two variables) still better than in the standard RBC model. Several papers have addressed the persistence problem in RBC models recently. Among them we mention: Andolfatto (1996), Beaudry and Devereux (1995), Burnside and Eichenbaum (1996), Perli (1998), and Wen (1995). These papers can be broadly classified into two groups. Some papers, such as Perli (1998) and Wen (1995) rely on increasing returns. Perli (1998a, b) argues that persistence in growth rates is easily achieved in a one sector RBC model with home production driven by a technology shock when returns to scale are increasing enough to give rise to multiple equilibria. This multiplicity is not exploited at all, since no sunspot shocks are assumed. He also shows that the distance between the spectra of the model and of the data growth rates is minimized for a degree of increasing returns that allows for multiple equilibria. Wen (1995), instead, presents a one-sector model with habit formation where present and past leisure are complements rather than substitutes, and with increasing returns. Using the Watson (1993) procedure he shows that his model fits the US data quite well. This is, however, accomplished by relying on strong increasing returns to labor, so that the aggregate labor demand is upward sloping. Papers in a second group, such as Andolfatto (1996), Beaudry and Devereux (1995), and Burnside and Eichenbaum (1996) rely on ‘rigidities’ in the labor market. Beaudry and Devereux (1995) present a model with variable factor utilization rates, organizational capital, and efficiency wages, and show that it is able to generate persistent fluctuations in levels with shocks that are only mildly autocorrelated. Burnside and Eichenbaum (1996) present a one-sector model that displays variation in unobserved labor effort and in capital utilization rates together with one-period lags in adjusting employment in response to shocks. Their model generates an autocorrelation function of output growth with

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a positive first lag, and subsequent ones close to zero. Finally, Andolfatto (1996) uses a search-theoretic approach to describe the labor market. Since search is costly, firms have an incentive to hoard labor. In this sense, the model is similar to Burnside and Eichenbaum (1996), where labor hoarding was an assumption. We believe that our contribution to the literature that addresses the absence of an internal propagation mechanism in standard RBC models is interesting for at least three reasons. First, in contrast to the papers described above we do not rely on nonstandard features. Our model, which is a two-sector RBC model, emphasizes the role that different elasticities of substitution across sectors can have for the propagation over time of technology shocks, and it does this by staying close to the original RBC paradigm of a stochastic optimal growth model without distortions.3 A second interesting feature of our contribution is that in our model the propagation mechanism arises endogenously without the presence of labor market rigidities. The low magnitude of the elasticity of substitution in the human capital sector relative to the physical sector induces a persistent response to a technology shock in the physical sector. The agents would want to reallocate labor and human capital toward that sector but an immediate and complete adjustment would reduce the amount of human capital available next period. The optimal response is to spread the adjustment over time. This is similar to the effect of adjustment costs in helping to propagate shocks over time. We achieve this, however, without imposing arbitrary convex functions as in Cogley and Nason (1995) or lags in adjusting employment, as in Burnside and Eichenbaum (1996). Finally, in emphasizing the role of human capital formation for business cycles we are addressing recent empirical evidence that there is significant substitution between education and competing labor activities during the business cycle. Dellas and Sakellaris (1995) provide evidence that the propensity of 18- to 22-year-old high school graduates to enroll in college is significantly countercyclical. They examine CPS micro data for the period of 1968—88 and find that a 1% increase in the unemployment rate is associated with about a 2% increase in college enrollments. Their simulations suggest that cyclical fluctuations of the aggregate economy may have been associated with significant swings in enrollments. For instance, in 1982 when the unemployment rate stood about 1.9 points higher than the previous year, the recession was associated with a net increase in college enrollment of about 232,000. This number refers to 3 Other authors have explored the effects of the introduction of human capital into an RBC model. King et al. (1988b) provide an example of an economy with a separate human capital sector and endogenous growth in which there is a closed-form solution. Einarsson and Marquis (1994) also formulate a RBC model with a human capital sector. They show that their model can resolve the counterfactually high correlation between productivity and hours of employment displayed by standard RBC models. None of the above addresses the issue of business cycle persistence.

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movements into the human capital sector of 18- to 22-year-old high school graduates only but is, nonetheless, substantial when contrasted with the net reduction in employment observed between October 1981 and October 1982 of about 1.163 million. The paper is organized as follows. In Section 2 we present our model and in Section 3 we discuss its propagation mechanism in detail. In Section 4 we simulate the model and assess its performance with respect to both the conventional RBC statistics and the persistence of growth rates. We offer some concluding remarks in Section 5.

2. The model We assume that the economy consists of identical infinitely-lived agents who divide their time between leisure (l), skilled work (s) and unskilled work (n). There are two productive sectors in this economy. The first one produces a homogeneous physical good, y, which can be either consumed or invested. We call this the ‘goods sector’. The second sector involves activities of formal training resulting in increases of individuals’ human capital, h, which enhances their productivity when doing skilled work. We call this the ‘human capital sector’. The agents’ utility function is separable in consumption (c) and leisure (l) and displays constant intertemporal elasticity of substitution (CIES): l1~c!1 u(c ,l )"ln c #A t . t t t 1!c Production in the goods sector is done according to a constant returns to scale and Cobb—Douglas technology with capital, k , and labor, ¸ , as factors of 1 1 production. Labor is a CES aggregate of unskilled labor, n , and skilled labor, 1 s . The production function is therefore 1 F (k , s , h , n )"z ka1[an( #(1!a)(s h )(](1~a1)@(, y 1t 1t 1t 1t t 1t 1t 1t 1t where z is a technology shock and the elasticity of substitution between skilled t and unskilled labor is 1/(1!/). The production function in the human capital sector is also Cobb—Douglas with constant returns to scale in physical capital, k , and total labor, ¸ , which is 2 2 a CES aggregate of unskilled labor, n , and skilled labor, s : 2 2 F (k , s , h , n )"x ka2[bnt #(1!b)(s h )t](1~a2)@t, t 2t 2t 2t 2t h 2t 2t 2t 2t where x is a technology shock that affects the production of human capital. t Note, that the elasticity of substitution between skilled and unskilled labor, 1/(1!t), may be different in this sector. The total physical and human capital,

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and skilled and unskilled labor in the economy are simply the respective sums across the two sectors. The two types of capital are subject to geometric depreciation at rates given by the parameters d and d for physical and human k h capital respectively. The above specification of the production functions contains several realistic features. The division between skilled and unskilled labor captures the fact that the efficiency of labor in some production tasks increases over time as a result of formal training. Increases in efficiency (that is, in human capital) are embodied, as in Lucas (1988).4 However, the human capital production function that we employ here is a generalization in allowing capital and unskilled labor to play a role. We provide a more detailed discussion in Section 4.1 of factor inputs measurement in the human capital sector. A key part of the persistence mechanism in this economy is that skilled and unskilled labor be substitutable with different degrees of difficulty in each of the two sectors. We argue in Section 4.1 that, indeed, it is considerably harder to substitute the two forms of labor in the education sector. The complete problem of the representative agent is

C

D

= (1!n !s )1~c!1 t t max E + bt ln c #A 0 t 1!c ct,kit,nit,hit,sit t/0 subject to

"(1!d )k #z ka1[an( #(1!a)(s h )(](1~a1)@(!c ; 1t 1t t t`1 k t t 1t 1t h "(1!d )h #x ka2[bnt #(1!b)(s h )t](1~a2)@t; t`1 h t t 2t 2t 2t 2t z "zf exp(u ); t t`1 t`1 x "xm exp(v ); t t`1 t`1 k "k #k ; t 1t 2t h "h #h ; t 1t 2t n "n #n ; t 1t 2t s "s #s . t 1t 2t The innovations to the goods production technology shock, u , and to the t human capital production technology shock, v , are assumed to be normal and t iid with mean 0 and variances p2 and p2 respectively, and covariance p . v uv u k

4 The concavity of the representative agent’s problem is not guaranteed when human capital is embodied and individuals choose leisure in addition to consumption. On this point see Benhabib and Perli (1994), and Ladron-de-Guevara et al. (1995). To address this possibility we checked for the concavity of our problem at the particular calibrations we performed and found it to hold. In an earlier version of this paper we formulated a model that was guaranteed to be concave because human capital was not embodied. The results on persistence were qualitatively similar and are available upon request. Since the output of education and formal training activities is best thought to be embodied we present here only results under this assumption.

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This problem can be solved using standard dynamic programming techniques.5 The first order conditions with respect to c, h , k , n , n , s and s are 1 1 1 2 1 2 MºC "c~1"bj , (1) t t t`1 (2) j MPh "h ) MPh , t`1 2t t`1 1t (3) j MPk "h ) MPk , t`1 2t t`1 1t (4) Mºn "A(1!n !s )~c"bj ) MPn , t t t`1 1t 1t (5) Mºn "A(1!n !s )~c"bh ) MPn , t t t`1 2t 2t (6) Mºs "A(1!n !s )~c"bj ) MPs , t t t`1 1t 1t Mºs "A(1!n !s )~c"bh ) MPs , (7) 2t t t t`1 2t together with the usual two transversality conditions. In the above expressions, denote the undiscounted expected value of the derivative of the j , and h t`1 t`1 value function with respect to k and h respectively both evaluated at time t#1, and MPw is the marginal product of variable w in the respective sector where t t w is employed. Using the envelope conditions and after some manipulation we t have j

"E MºC (1!d #MPk ), t`1 t t`1 k 1t`1 1!d #MPh h 2t`1. h "E Mºs ) t`1 t 2t`1 MPs t`1 2 This model is too complicated to solve analytically, so we resort to numerical techniques. Note that here we concentrate on business cycles and we do not add a deterministic trend to z or x . We use the steady-state values of variables in the t t model to match averages of the data for calibration purposes (see Section 4.1). We use Eqs. (1)—(7) together with the laws of motion for k, h, x and z to find the steady state of our economy. Next we use the first order conditions to express five of the control variables in terms of the remaining two controls, the endogenous state variables (k,h), and the exogenous state variables (z,x). For example, one can get approximate expressions for k , h , s , s and c in terms of 1 1 1 2 k, h, z, x, n and n . 1 2 In this way, the system is completely described by six expectational difference equations in the six variables k, h, z, x, n , and n . These equations can then be 1 2 linearized around the steady state, which yields the system: p

"Jp #He , t`1 t t`1

5 Only seven of the nine choice variables are free; k and h can be obtained from the states k and 2 2 h and from the controls k and h . 1 1

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where J and H are matrices of first derivatives, p is defined as (k, h, z, x, n , n ), 1 2 and e is a vector of innovations and forecast errors. The model can then be simulated taking into account the two transversality conditions, which force the motion of the system to take place on the stable manifold of the saddle point.

3. The propagation mechanism Many of the existing RBC models do not have a satisfying propagation mechanism: the simulated output growth is almost always slightly negatively autocorrelated, whereas in the US data it is positively autocorrelated at least for the first two or three lags. The reason lies in the behavior of capital (the endogenous state variable) and of labor (the other input to goods production). Specifically, a technology shock leads to persistent, but small period-to-period capital growth and to non-persistent increase in employment. Our focus in this paper is on labor movements in particular so we present now a discussion of lack of persistence in employment movements in the standard RBC model. Consider a positive and permanent technological shock. In any one-sector RBC model based on the stochastic optimal growth model the following condition must hold at any point in time: Mºl "MPn, Mºc

(8)

i.e. the marginal rate of substitution between leisure and consumption must equal the marginal product of labor. The technological shock increases MPn and, therefore, labor. Consumption also increases, because output is now higher than before. Thus, at impact consumption and labor move in the same direction. In subsequent periods consumption continues to increase toward its higher steady state, while labor decreases and returns to its (unchanged) steady state.6 To understand why this is so let us examine what happens to each component of (Eq. (8)). As consumption increases, its marginal utility decreases. At the same time the increases in the capital stock provide too small a boost to the MPn to maintain the equality in Eq. (8) and, thus, labor has to decrease, at least under standard assumptions on the utility and production functions.7 The combination of dynamic responses described above leads to output rising above its new (higher) steady state in the impact period. Subsequently, output declines smoothly at a slow rate. The generated autocorrelation of output growth is close

6 The new and old steady states for labor are identical if the utility function is CIES; they may be different in other cases. Under most conventional utility functions, however, n always ‘overshoots’ the new steady state following a technology shock. 7 On this point see also Barro and King (1984) and Rotemberg and Woodford (1996).

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to zero or slightly negative.8 One way to obtain a satisfying propagation mechanism is through persistent movements in labor. The model presented in the previous section is able to do that in a setup where the notion of labor is extended to include not just hours but also human capital.9 A key requirement is that the elasticity of substitution between skilled and unskilled labor be relatively high in the production of physical capital and relatively low in the production of human capital. Suppose, for expositional purposes, that the aggregator between the two types of labor in the production of human capital is Leontief, so that they can be used only in a fixed proportion. Then, a positive, persistent shock to the production of the physical good, makes agents want to increase the aggregate labor in that sector. A positive, persistent shock to the goodsproducing sector makes agents desire increased long-term levels of physical and human capital in this sector and an identical level of leisure. In the short term they increase the supply of time available for employment and schooling due to the high real interest rate. As regards physical and human capital, however, these are fixed in the period when the shock occurs. Moving them away from the education sector would cause a substantial decline in the production of new human capital due to the fixed proportions technology (that is, if s h is moved, 2 2 also n has to be moved). Optimality requires that the agents shift input factors 2 from the education to the goods sector slowly as new human capital is generated. This slow transferance of human capital to the goods-producing sector contributes to the shock propagation and leads to persistent output growth. Another way to view this persistence mechanism is through (Eq. (8)). Note that this equation still holds, for n instead of n, and it may be obtained by 1 dividing (Eq. (4)) by (Eq. (1)). In our model, consumption and labor in the goods sector move together not only at impact, but also for a few periods thereafter. This is optimal because the marginal product of unskilled labor, MPn , in2 creases substantially for a while even as labor, n , increases itself. The marginal 2 product of unskilled labor shifts upward, due to the continuing inflow of more and better-skilled workers to the goods sector.10 As a consequence, output growth is persistent. Note that this mechanism does not indeed require the labor aggregator in the human capital sector to be Leontief. It only requires that the elasticity of 8 This holds if the elasticity of aggregate labor supply is high. For an illustration of this point see Burnside and Eichenbaum (1996) who provide the impulse response function of labor and output for the indivisible labor model of Rogerson (1988) and Hansen (1985) with a share of labor in output equal to 0.66. 9 We thank an anonymous referee for comments leading us to substantially improve the exposition here. 10 To make this mechanism work, the share of human capital in physical production must not be too low. The actual speed at which the transfer occurs depends also on the share of physical capital in the production of human capital: a higher share of capital increases the speed.

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substitution between the two types of labor be lower than the corresponding elasticity in the goods sector. Of course, this difference must be substantial to generate persistence comparable to that observed in the data and we argue below that such elasticity differences are empirically plausible. 3.1. Human capital accumulation and adjustment costs Cogley and Nason (1995) find that models with some form of adjustment friction in the labor market such as convex costs of adjusting labor or lags in employment adjustment can generate persistent output growth. Additionally, Burnside and Eichenbaum (1996) discuss that their model’s feature of oneperiod lag in the adjustment of employment to any shock is fundamental in generating persistence in output growth. We provide here a discussion of the relation between our two-sector model and such one sector models with costs of adjustment in the labor market. Mussa (1976, 1977) has shown that a two-sector model has a cost-of-adjustment interpretation as long as the second sector produces an accumulated factor of production (such as human capital) subject to diminishing returns. In this case the production possibility frontier between the (output of the) consumption good and additions to human capital is concave and the total marginal cost of human capital accumulation in terms of forgone output is upward-sloping.11 Alternatively, a one-sector model with convex costs to adjusting the employment of human capital also generates a trade-off between consumption goods production and human capital accumulation that is described by a concave transformation curve and upward-sloping marginal cost of human capital accumulation. Under both situations, firms producing the physical good will find it optimal to spread over time the adjustment of human capital employment (and thus of output) in response to a shock. In our model, such persistent response arises from endogenous sectoral reallocations rather than through imposed frictions. We should note that whereas adjustment friction in employment can generate output growth, convex costs of adjusting physical capital cannot, a fact that was demonstrated by Cogley and Nason (1995). The latter form of adjustment costs does generate strong persistent movements in investment but since this is a flow variable and constitutes only a small fraction of the (highly durable) capital stock the overall effect on the endogenous state variable and output fluctuations is small.

11 Mulligan and Sala-i-Martin (1993) also show that with constant point-in-time returns (which amounts to the conditions: k "k t#k t and h "h t#h t) the production possibility frontier t 1 2 t 1 2 is concave as long as the two sectors have different (diminishing marginal return) technologies.

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4. Simulations In this section we calibrate the model and present the results of various simulations. First, we present the standard RBC set of statistics, namely the standard deviations of the simulated series relative to output, their correlation with output and their AR(1) coefficient. Then, we turn our attention to the dynamic properties of output and other variables. We elect the model of King et al. (1988a) (henceforth KPR) as the standard RBC model against which we compare our results. 4.1. Calibration In order to map our model to the data of the US economy we use the following general guidelines. We consider the human capital sector (HCS) to consist of formal education and formal job training whereas the goods sector (GS) consists of all other output measured in the NIPA. Since some inputs in the HCS are not measured in the NIPA, such as the value of students’ time, they have to be imputed. The key task is to calibrate the HCS production function. We first determine the share of inputs in formal education. Here we make the assumption that skilled labor is the teachers and professional staff, while unskilled labor are the students. Then, we determine the share of inputs in formal training. Finally, we aggregate the two human capital augmenting activities using relative weights calculated from relative expenditures in 1989. We proceed now to the details of the calibration of the model. The share of capital in physical production, a , is set at 0.42, based on NIPA statistics and 1 following King et al. (1988a). This implies a share of total labor of 0.58. We assume that the elasticity of substitution between the two types of labor in the production of the physical good is one, i.e. that /"0 and the production function is Cobb—Douglas in the three variables k , s h and n . Following 1 1 1 1 Mankiw et al. (1992), we set the share of human capital in total labor to 50% (a"0.5).12 The calibration of the production function for human capital is considerably more difficult. The key difficulty seems to be lack of reliable data for certain components of human capital accumulation and for the human capital stock. A related problem is that, to the best of our knowledge, there are no empirical 12 Mankiw et al. (1992) assume that the minimum wage is the wage paid to a person with zero human capital, which is consistent with the definition of h in our model. They report that in the United States the ratio of the minimum wage to the average wage in manufacturing was 0.3:0.5. Under profit maximization this wage premium is equal to [1!a !(1!a)(1!a )]/[1!a ], 1 1 1 implying that 1!a ranges from 0.5 to 0.7. We use a value of 1!a"0.5 as the empirical results of Mankiw et al. (1992) suggest. We also tried the mid-point, 1!a"0.6, and our persistence results were qualitatively similar.

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estimates of human capital production functions with physical and human capital, and raw labor as separate inputs. As our interest in this paper is on business cycles we ignore investment in health and focus on human capital accumulation that takes place at school and on the job. The US Department of Education compiles extensive data on expenditures for schooling and enrollments but there are no comparable official data on investment in job training. Whenever possible we use data from both human capital investment activities in our calibration. It seems natural to consider that formal education and job training are described by different production functions. In formal education we identify with skilled labor the teachers, administrators and other professional staff and with unskilled labor the students. We assume that the share of the skilled in total labor in the steady state is 50%, as in the physical sector. Using unpublished data from Jorgenson and Fraumeni (1992), we calculate the average share of capital in educational output at 0.08 implying a share of skilled and unskilled labor of 0.46 each.13 It seems reasonable to think that the training process is more similar to physical production than is the schooling process. This implies that training is more capital-intensive than schooling. Since we do not have data on the share of different inputs in job training we assume that the production function in job training is the same as that in the physical sector.14 To arrive at the share of inputs in total human capital production we need measures of the relative importance of expenditures on education and expenditures on job training. Clotfelter (1991) reports total educational expenditures of $331 billion in 1989, about 6.8% of GNP. Including a measure of the opportunity cost of time for higher education students raises expenditures to $462 billion. We estimate the opportunity cost of time of high school students at $44 billion, thus bringing total educational expenditures to $506 billion.15 Mincer (1993) estimated expenditures on formal job training in 1976, which extrapolated to 1987 amounted to $165 billion (1989 dollars).16 In contrast, Bartel (1989) 13 The data capture production inputs in the US education sector for the period from 1948 to 1986. The capital input is comprised of educational buildings and equipment. We treat as skilled labor the part of labor input that corresponds to educational staff. Since we do not have estimates of the value of the unskilled labor input (the value of students’ time) we make the assumption that this is equal to the value of the skilled labor input. 14 Mankiw et al. (1992) maintain this assumption for the whole human capital accumulation sector. 15 There were approximately 12.6 million students enrolled in grades 9—12 in the Fall of 1989 (see Table 56, US Department of Education (1994)). We assume that these students gave up 25 hours of work per week for 9 months at $4 per hour. 16 Mincer (1993), Chapters 9 and 13, used information in the 1976 PSID Time Use Survey of time allocation on the job during a week’s period. He calculated total annual costs of job training to be about $56 billion in 1976 and, employing some projections, $150 billion in 1987 (current dollars) or $165 billion (1989 dollars).

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estimated that $60 billion (1989 dollars) was spent on firm-provided training in 1987. Thus, according to these data, educational expenditures account for between 75% and 90% of formal investment in human capital, which implies that the average share of capital in the human capital sector ranges between 0.11 and 0.17. We set a "0.11.17 2 As discussed above, we assume that it is more difficult to substitute raw labor and human capital in this sector relative to the physical sector. This implies, of course, that the production function is not Cobb—Douglas and that the shares of s h and n are not constant. We treat the substitutability between human 2 2 2 capital and raw labor in the human capital sector as relatively low.18 We set t"!1, which implies a value of 0.5 for the elasticity of substitution. We set b"0.5 so that in the steady state the share of skilled labor (s h ) in total labor in 2 2 the human capital sector is 50%. We now turn to specifying the rest of the parameters. We have already assumed that the utility function is logarithmic in consumption. We also assume it is linear in leisure as in many RBC models, i.e. c"0. The value of h is chosen so that in steady state total labor amounts to 33% of free time. The discount factor b is set at 0.9898, which is equivalent to a 4% discount rate at annual frequency. The constant A is arbitrarily set to one, and B is fixed at 1.41, so that, in steady state, the output of the human capital sector is 12.6% of the output of the goods sector.19 The depreciation rates for physical and human capital, d and d , are both set at 0.025.20 We consider shocks to the physical sector only, k h 17 We chose the low end of the range to balance the extreme assumption we had made previously that the share of capital in job training is 0.42, as high as in physical production. Our results on persistence do not change when we use a value for a of 0.15. 2 18 There is no direct evidence on t. However, an indication of low substitutability is provided by the high correlation between the numbers of teachers and of students in the US elementary and secondary education, which was 0.78 for the period of 1960 to 1994. These annual data are contained in Table 64 of US Department of Education (1994). We detrended the data using up to cubic trend terms. The correlations for elementary and secondary education separately were 0.51 and 0.86, respectively. Similarly, there is a high correlation between teachers and non-professional support staff. 19 We calculated this proportion as follows. Our conservative estimate of total expenditures in the human capital sector in 1989, as outlined in a previous paragraph, is $566 billion. Of this amount, $331 billion of educational expenditures and $60 billion of formal job training expenditures are accounted in NIPA and have to be subtracted from GNP (about $4868 billion) in order to obtain an estimate of total expenditures in the goods sector. The ratio of output in the two sectors is, then, 566/4477"0.126. 20 This value for d is probably on the high end of the reasonable range. Estimates of this h depreciation rate at the individual level vary widely. Mincer and Ofek (1982) estimated it at 3.3% to 7.6% per year, whereas Heckman (1976) estimated it at 3.7 to 8.9%. To arrive at a figure appropriate for the aggregate economy one needs to add the proportion of total human capital stock embodied in new retirees. The rough calculations in Stokey and Rebelo (1995) suggest that retirees embody about 2.5% to 4.0% of the total stock of human capital on a yearly basis.

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i.e. we set x "1 for all t. The technology shocks are assumed to be highly t persistent with a persistence parameter f"0.98. The variance of the innovations is fixed at 0.0007, which is the variance of the innovation in the Solow Residual in the US data estimated from a Cobb—Douglas production function with capital and labor only as inputs. 4.2. The data Most of the macroeconomic data that we use in this paper are the same as those used by King et al. (1991), and by Watson (1993), updated to the fourth quarter of 1994. In particular, output is defined as GNP minus gross government expenditure (GNPQ!GGEQ); consumption is defined as durables plus nondurables plus services (GCQ); investment is total private fixed investment (GIFQ); labor is nonagricultural employment minus government employment all multiplied by average weekly hours ((LHEM!LPGOV) * LCHC). The first three variables are expressed in constant 1987 dollars, and all the variables are divided by population of age greater than or equal to 17 (PAN17). All the series come from Citibase where they are denoted with the acronyms listed in parentheses here. The series for total human capital that we use is an index for quality of labor input constructed by Jorgenson et al. (1987). This is an index that reflects changes in the quality of labor due to education only. The original series for h is annual. t 4.3. Standard deviations and correlations with output We first compute the usual RBC statistics, namely the standard deviations of the simulated time series and their correlations with output. A problem that we face is that all our macroeconomic series are quarterly whereas the human capital series is annual. We decided not to convert all data to the annual frequency and calibrate the model accordingly because we are particularly interested in the information on quarterly persistence. For comparison to the US series on human capital we transform the artificial quarterly human capital series generated from the model to annual frequency. Thus, the statistics in Table 1 in column h refer to annual data, whereas in the columns y, k,i,c,H refer to quarterly data. Output in the goods sector is denoted by y and total hours (n #s #s ) by H. All series, empirical and model-generated, are 1 1 2 Hodrick—Prescott filtered. To make comparisons easier we report standard deviations (in the first two rows of Table 1) as fractions of the corresponding statistic for output.21 21 The volatility in our model’s output series is 2.44, higher than that of US output, which is 1.86 (if GNP is divided by working-age population) or 2.20 (if GNP is not divided by population).

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Table 1 RBC statistics y

k

h

i

c

H

Standard deviation US data

1.00 1.00

0.22 0.38

0.25 0.20

3.01 2.82

0.39 0.49

0.98 0.86

Correlation with output US data

1.00 1.00

0.15 0.28

0.64 0.17

0.99 0.96

0.97 0.76

0.80 0.86

AR(1) coefficient US data

0.89 0.86

0.98 0.91

0.66 0.37

0.89 0.76

0.83 0.84

0.73 0.90

Notes: Within each of the three groups of rows, the first row displays the statistic for model generated output (y), capital (k), human capital (h), investment in physical capital (i), consumption (c) and hours (H), whereas the second row displays the corresponding statistic for the US data.

As one can see from Table 1 the statistics obtained from the model are roughly comparable to the ones obtained from the data. There are however a few problems, namely: the total human capital is too correlated with output and too strongly autocorrelated; the first-order autoregressive coefficient of labor is slightly low; physical capital is about half as correlated with output as it is in the US data. Standard one-sector RBC models generate too high a correlation between hours and average labor productivity as pointed out, for example, by Christiano and Eichenbaum (1992). Our model performs well with respect to this correlation (0.41 versus 0.36 in the US data), and this seems to confirm the intuition in Einarsson and Marquis (1994), that models with human capital have the potential to solve that puzzle because they introduce intersectoral flows in raw labor that enhance the substitution into and out of employment in the goods sector. 4.4. Impulse response functions We turn now our attention to the dynamic propagation properties of the model. A first interesting source of information in this respect are the impulse responses of the model to a technology shock; they are reported in Fig. 1. As one can see, they all display a ‘hump shape’, a clear sign of persistence, since it implies that the shock is propagated for three or four more periods after impact. Note also that consumption and raw labor in the physical sector increase together not only in period 1, but also until period 5. This confirms the intuition given above that if these two variables move together for a few periods after the occurrence of the shock, the model will have a fairly strong propagation

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Fig. 1. Impulse responses with t"!1.

mechanism. In contrast, the impulse response functions in Fig. 2 correspond to a model where the elasticity of substitution between skilled and unskilled labor is the same in both sectors (t"0). This model does not display hump-shaped impulse response functions and does not embody a significant propagation mechanism much like the standard one-sector RBC model (see Cogley and Nason, 1995). 4.5. Autocorrelation functions A second indicator of persistence is the autocorrelation function (ACF) of the growth rates of output and other variables of interest. We report these ACFs in Fig. 3, where the solid lines correspond to the model and the dashed lines to the data. The model ACF’s were obtained by running 1000 simulations, computing the ACF for each simulation, and then taking the sample average. If the propagation mechanism of the model is strong enough, the autocorrelations should not be close to zero or negative at all lags. Note that the ACFs for the US data are always positive for the first two or three lags.22 The results are in line 22 Only the first two lags are significantly different from zero for y, i and n, and only the second for c.

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Fig. 2. Impulse responses with t"0.

with what one would expect by looking at the impulse responses: the model’s ACFs are positive for more lags than the data ACFs for output, consumption and investment growth. A problem arises here with total hours, which do not display much persistence. For purposes of comparison we report in Fig. 3 also the ACFs for model-generated variables under a parameterization that does not generate persistence (t"0). We also report in Fig. 4 the ACFs for the KPR model. These two models display autocorrelations that are negative at all lags, with the exception of consumption, which is always strongly positive. We perform, next, a test proposed by Cogley and Nason (1995). We compute generalized Q statistics to evaluate the match between the ACF of the data and that of the model. We calculate this statistic as Q"(c !c )@»K ~1(c !c ), D M D M where c is the ACF estimated from the data and c was estimated by averaging D M the ACF’s of 1000 samples simulated by the model. The covariance matrix of these simulated ACF’s forms an estimate of the metric of comparison, »K , between sample and theoretical ACF’s. Following Cogley and Nason (1995) we choose to evaluate the first 8 lags of the ACF in order to preserve power for the test. The quantity Q, then, is approximately distributed as a s2 variable with 8 degrees of freedom. To get a meaningful estimate of » we introduce a small

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Fig. 3. Autocorrelation functions with different values of t.

Fig. 4. Autocorrelation functions for the KPR model.

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Table 2 Generalized Q statistics

Q p-value

y

c

i

H

9.11 0.333

14.81 0.063

7.19 0.516

16.47 0.036

Notes: The table reports generalized Q statistics for the autocorrelation functions of model generated output (y), consumption (c), investment in physical capital (i) and hours (H). These functions are truncated at lag 8.

variance for the shock x to the human capital sector. We use a variance of t 0.0001 which leaves the previous results unaltered and at the same time produces a non-singular covariance matrix. The results we obtain are reported in Table 2. At the 5% significance level, we can not reject the null hypothesis that the autocorrelation functions of US output, investment, and hours could have been generated by the model. The null hypothesis is only marginally rejected for total hours. By comparison, Cogley and Nason (1995) reject the null in almost all the RBC models that they analyze, except when convex adjustment costs to capital and labor are included. Note that Cogley and Nason (1995) only deal with the ACF of output growth and the same is true of Burnside and Eichenbaum (1996). Our failure to reject the null (for three out of the above four variables) should not be interpreted as statistical evidence that the US data were generated by our model. We are not claiming that ours is the true model. The above Q statistic evaluates how likely it is that the model generates a realization of the variable of interest with unconditional second moment properties equal to those found in the US data. Thus, the Q test is useful in evaluating how well the model does in generating persistence in the growth output (and other variables) of the form observed in the US data. Moreover, it allows comparisons with other models in the literature and, in this respect, our model seems to perform better than the ones analyzed by Cogley and Nason (1995). 4.6. The power spectrum We now turn our attention to the spectra of the growth rates of the four series of interest generated from our model and provide another measure of the effectiveness of the model’s propagation mechanism. In particular, we use here the procedure proposed by Watson (1993) to measure how well the model fits the data. The procedure augments the variables generated by the RBC model with the ‘minimum amount’ of approximation error in order to match the unconditional second moment properties of the actual data. The economic

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Table 3 Relative mean square approximation error y

c

i

H

First differences Our model King—Plosser—Rebelo

0.17 0.52

0.52 0.69

0.55 0.21

0.46 0.74

HP-filtered Our model King—Plosser—Rebelo

0.11 0.40

0.49 0.58

0.52 0.29

0.48 0.64

Levels, periods 6—32 Our model King—Plosser—Rebelo

0.09 0.38

0.50 0.56

0.53 0.30

0.47 0.63

Notes: Relative mean square approximation error (RMSAE) is the lower bound of the variance of the approximation error divided by the variance of the model generated series. This is constructed from the representation that minimizes the unweighted trace of the error spectrum. The table provides a comparison of RMSAE for the series of output (y), consumption (c), investment in physical capital (i) and hours (H) generated by our model and the King et al. (1988a) model.

model may be considered to fit the data well if the size of this error is relatively small. A lower bound on the ratio of the mean square approximation error to the variance of the US data may be calculated and used to judge the goodness of fit. We call this lower bound the relative mean square approximation error (RMSAE). The Appendix contains a brief description of the procedure to calculate the RMSAE. To establish an analogy with a familiar concept the RMSAE is, roughly speaking, similar to a lower bound on 1!R2 from a hypothetical regression. The lower is 1!R2 in the regression the better the fit of the statistical model underlying it. A comparison of the results for our model with those for the KPR model may be made from Table 3. As one can see the improvement over the KPR model is substantial with respect to output and hours and smaller with respect to consumption. Our model’s performance, on the other hand, is considerably worse with respect to investment. The improvement in labor is interesting, given that this variable’s autocorrelation function was the only one that failed the Q test (see Table 2). Still, the spectrum of H does not have a peak at the business cycle frequencies. What Table 3 says is that, whereas one needs an error with a variance of at least 52% of the variance of the actual data to reconcile the KPR first differences of output with US first differences of output, the lower bound of the variance of

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the error needed to do the same with our model is just 17%. The improvement is even more evident when one looks at the business cycle frequencies, i.e. at the results for the HP-filtered data and for the levels restricted to the periods of 6 to 32 quarters. Of course, the Watson procedure is not a formal statistical test. Moreover it provides only a lower bound for the variance of the approximation error. This notwithstanding, and considering also all the evidence summarized above, we can say that our model seems to have a better fit than the KPR model, except for investment. 4.7. Sensitivity analysis We have shown that with the particular parameters that we chose to calibrate the model we obtain persistence properties comparable to those observed in the data. However, in light of the fact that the calibration of the production function for human capital is novel in this paper it is essential to analyze how sensitive are our results on persistence to variations in the parameter values. The parameters in question (and their baseline values) are: 1) the share of physical capital (a "0.11), 2) the share of unskilled in total labor (b"0.5), 2 and 3) the elasticity of substitution between skilled and unskilled labor (1/(1!t)"0.5). Keeping the other parameters constant, persistence in output is displayed as long as a is not more than 0.15. This seems reassuring considering that the high 2 end of likely values for a is 0.17, as discussed in Section 4.1. Limited substituta2 bility between skilled and unskilled labor in the production of human capital is a key element of the propagation mechanism of our model, as explained in Section 3. For plausible values of a , persistence is still displayed for elasticities 2 up to about 0.6 (correspondingly, values of t up to about !0.7). At higher elasticities of substitution the propagation mechanism breaks down. In response to a positive shock to the goods sector agents shift skilled labor from the human capital to the goods sector fast. The production of new human capital itself is not harmed as additional unskilled labor switches out of enjoying leisure and substitutes well for skilled labor. Finally, the share of unskilled in total labor employed in the human capital sector is not important for obtaining persistence. The results are qualitatively similar for a wide range of values of b. Incidentally, the same holds true for the share of unskilled in total labor employed in goods production, a. It is of interest to note that if c is set to a high value, implying a low intertemporal elasticity of substitution of leisure, any persistence disappears. To understand this consider a positive shock to goods production. In response, the agents release little extra time from leisure to the other two activities. That means that the initial increase in output is low and any subsequent releases of hours and extra human capital from the human capital sector are small and do not lead to pronounced, persistent movements.

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So far we have assumed that there is no variability in the productivity of the human capital sector. We explore now the implications of activating the shock to human capital production. In doing this we face the problem of how to calibrate this shock’s variance and its correlation with the technology shock in the goods sector. In Section 4.1 we discussed at length the difficulties of measuring the output of the human capital sector. It does not seem possible to use NIPA data to estimate the relative magnitude of the two technology shocks. We resort to the assumption that the two shocks have the same variance, and then scale them both so that the variance of simulated output is unchanged. In light of the fact that education is a much smaller and, it seems, less volatile sector than the goods sector, the assumption of equal variances seems to be the extreme opposite of the assumption of zero variance maintained so far for the human captial sector.23 We experiment with various values for the correlation between the two shocks, ranging from #1 to !1. Our simulations indicate that the correlation between the shock to human capital production and the goods technology shock is important for the persistence results. Correlations below 0.65 compromise the persistence results and, in particular, persistence disappears with values below 0.3. A shock to the education sector that goes in the opposite direction of the technology shock works against the intuitive argument for propagation given in Section 3. Due to the positive shock, the goods production sector draws labor from leisure and output increases dramatically. In contrast, new human capital production is low, due to the negative shock. Subsequent transfers of new human capital to the goods sector are too small to generate significant persistence in output growth. We believe that the hypothesis that shocks to the two sectors are strongly positively correlated is empirically plausible.

5. Conclusions In this paper we have presented a two-sector model of business cycles with human capital, with the key feature that the elasticity of substitution between skilled and unskilled labor in the production of human capital itself is relatively low with respect to the same elasticity in the production of the physical good. We demonstrated that the model has a strong propagation mechanism and that it is able to improve substantially the performance of other standard RBC models with respect to the persistence of output and consumption growth. The results are not conclusive, on the other hand, for the persistence of hours and investment growth as some statistics indicate that the model improves also with 23 An alternative extreme assumption is to set the variances of the two shocks equal and scale them so that the variance of simulated output matches the variance of US output. This does not affect our sensitivity results on persistence below.

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respect to these variables, while some other statistics indicate the opposite. This problem should be addressed in future research. Another possibly interesting line of future research is the exploration of the effects of time to build human capital on the results. In this paper we assume that it takes just one period, i.e. one quarter, to produce new human capital. We expect that time to build human capital in a two-sector model should enlarge persistence provided the elasticities of substitution among factors are different across sectors.24 Overall, it seems to us that the line of research according to which shocks propagate throughout the economy because of differences in elasticities of substitution across sectors is a promising one. We note that this mechanism of generating persistence is more general than our application of it in this paper. In particular, it is not essential that the alternative capital input produced in the economy be human capital. For example, suppose that the second stock is that of organizational capital, which is not embodied in workers. If organizational capital is not very substitutable with labor in producing more organizational capital the model could display persistent business cycles much like the model described in this paper. Our emphasis on human capital in this paper is motivated by the empirical evidence of Dellas and Sakellaris (1995) that there is significant substitution during the business cycle between education and competing labor activities.

Acknowledgements We would like to thank Jess Benhabib, Michael Binder, Michele Boldrin, Satyajit Chatterjee, Tim Cogley, John Haltiwanger, Boyan Jovanovic, Bob King, Richard Rogerson, Marcelo Veracierto, Randall Wright, an anonymous referee, and seminar participants at the Federal Reserve Bank of Philadelphia, University of Maryland, Universite´ de Montreal, the University of Pennsylvania, 1996 Econometric Society Summer Meetings and the 1996 NBER Summer Institute for valuable comments; Barbara Fraumeni for providing some of the data used here, and Mark Watson for providing the program used in Section 4.6. The first version of this paper was circulated in December 1995.

Appendix A. Measure of fit for the calibrated model The procedure used in subsection 4.6 can be briefly described as follows (see Watson, 1993 for more detail). Under the assumption that the US and the artificial data are jointly covariance stationary, one can define the error as 24 Cogley and Nason (1995) show that one-sector models with time to build physical capital (see also Rouwenhorst, 1991), do not display persistence.

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simply the difference between the two data sets, u "D !M , where D indicates t t t t the US data and M the model generated data. Then the autocovariance t generating function (ACGF)25 of u can be written as A (q)"A (q)#A (q)!A (q)!A (q), u D M DM MD where A is the ACGF for k, A is the cross ACGF for k and l, q,e~iu, and k kl u takes values in [!n,n]. Whereas A (q) and A (q) can be easily computed, D M further assumptions are needed to compute the cross ACGF. In particular, Watson (1993) chooses A (q) so that the variance of the error is minimized DM subject to the constraint that the joint ACGF for the data and the model is positive semidefinite. A lower bound on the variance of the error relative to the variance of the data at each frequency is A (q) r (u)" u jj , j A (q) D jj where j indicates the variable of interest (y, c, i or H in our case) and Q denotes, jj in general, the jth diagonal element of matrix Q.26 Integrating the numerator and denominator of r (u) separately and taking their ratio defines the relative j mean square approximation error (RMSAE) for the model and provides an overall measure of fit. To compute r it is convenient to use the spectrum instead of the ACGF. The j spectrum of the data is estimated from a cointegrated sixth-order VAR with variables (*y, y!c, y!i, H). The spectrum of the model can be easily computed analytically from the state-space representation of the model itself, i.e. from p

"Jp #He , t`1 t t M "Up , t t where p,(k, h, z, x, n , n ), M,(y, c, i, H), J and U are matrices obtained from 1 2 the linearization of the model around the steady state, and H is a matrix that controls the variance of the errors included in e&N(0, I). In this way the spectrum of the levels of the model for each frequency u is S (q)"[(I!J ) q)~1H] ) [(I!J ) q)~1H]@ M 25 All the results below can be interpreted in terms of spectra, which are proportional to the ACGF. 26 The minimum error representation we choose minimizes the unweighted trace of the spectral density of u, A (q), for each frequency. u

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and the spectrum of the first differences is *S (q)"[(1!q)I] ) S (q) ) [(1!q)I]@, M M where again q,e~iu. A (q) is proportional to S (q) or to *S (q), depending on M M M whether one is interested in the levels or in the first differences.27

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