Note on human capital externalities

TOR EINARSSON University of Iceland

Reykjavik, Iceland

MILTON H. MARQUIS Florida State University

Tallahassee, Florida

Note on Human Capital Externalities*

This paper provides an estimate for the magnitude of the externality generated by human capital investment that is required to fit the endogenous growth model of Lucas (1988). The model is calibrated to fit selected stylized facts of the postwar U.S. economy, including the behavior of a human capital series that was constructed by Jorgenson, Gollop, and Franmeni (1987) and adjusted by Bishop (1989) for changes in the quality of the U.S. educational system. This calibration requires an externality that is somewhat larger than previously estimated. It also indicates that some "learning-by-doing" is required for the model to produce a reasonable 'allocation of nonleisure time between work and formal education.

1. Introduction King and Rebelo (1993) have argued that the transitional dynamics

implied by the standard neoclassical growth model are inconsistent with certain stylized facts about the long-run properties of U.S. data. In particular, they argue that the declining growth rates associated with the asymptotic convergence to the steady-state that are implied by the model require coun- terfactual or incredulous assumptions in order to fit the data. They interpret their findings as support for modeling growth and development as an endogenous process, as in Lucas (1988) and Romer (1986). They further stress the importance of performing quantitative assessments of "alternative theoretical paradigms," which they pursue in a linear growth model in King and Rebelo (1990). In Lucas (1988), the "engine of growth" is investment in human capital, which accumulates according to a linear technology, similar to that of Uzawa (1965). In addition, Lucas allows for the possibility that human capital generates a positive externality in the production of final goods,

*We wish to thank Jim Clouse, Christian Gilles, and Barry Hirsch, as well as seminar participants at the University of Iceland for helpful comments and discussions. This paper was written while Marquis was on leave at the Federal Reserve Board. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the Board of Governors of the Federal Reserve System, or its staff'.

Journal of Macroeconomics, Spring 1996, Vol. 18, No. 2, pp. 341-351 Copyright © 1996 by Louisiana State University Press 0164-0704/96/$1.50

341

Tor Einarsson and Milton H. Marquis

and provides some evidence from U.S. data for the 1900-57 period in support of the existence of this externality, t

The purpose of this paper is to confront the Lucas model with the postwar U.S. data, and to identify the existence and/or magnitude of the externality that is required for the model to be consistent with a set of stylized facts concerning the long-run properties in the data. To this end, we employ a human capital series constructed by Jorgenson, Gollop, and Fraumeni (1987) to quality-adjust labor inputs into the production technology. This construction is consistent with the theoretical specification in the Lucas model. Adjustments to this series are made to account for changes to the quality of the educational system in the United States using the data and procedure described by Bishop (1989). We then fit the linear model (similar to Uzawa) without the externality and find it to be inconsistent with the historical evolution of both capital stocks, output and real interest rates. The model is then calibrated to these long-run properties by introducing an externality, which when measured as an increase to the factor share on human capital, is found to lie between 0.4 and 0.6, compared with Lucas's estimate of 0.417. This provides some evidence in support of the Lucas hypothesis, in that a positive externality is required in order for his model to be compatible with the postwar U.S. data. Moreover, given that alternative methods of constructing the human capital series would have resulted in even larger estimates of this externality, we interpret our estimates as conservative. We also find that some human capital accumulation must occur via "learning- by-doing" in order for the model to generate sufficient human capital growth without simultaneously requiring that an excessive amount of nonleisure time be devoted to formal education.

2. The Model

The Maximization Problem Consider an economy composed of identical infinitely lived house-

holds, each of whom seeks to maximize lifetime utility from a sequence of consumption. For simplicity, we take leisure as fixed, such that nonleisure time is supplied inelastically to the market. The representative household has access to two types of production technologies: one for producing current output, and the other for producing human capital. Hence, at any given period, the household faces a choice in the allocation of nonleisure time

1He goes on to argue that such an externality is required if the model is to fit cross-country empirical evidence, which he interprets as suggesting a divergence in the levels of output/capita across economies. This divergence would occur in his model when the economies differ only by initial endowments.

342

Note on H u m a n Capital Externali t ies

between current production and investment in human capital, where the latter fosters future growth. This is the setup employed in Lucas (1988) and Uzawa (1965).

The maximization problem for the representative household can thus be stated as follows:

o o

max '~. u(ct) [5 t Ic,} ~ , 13 ~ (0,1), (1)

where u : 9~+ ---) 9~ is a one period utility function, c denotes consumption and [~ a subjective discount factor. We assume u to be twice differentiable, increasing, and strictly concave, that is, u' > 0, u" < 0, and u'(0) = oo.

The maximization in (1) is subject to three basic constraints. First, there is a resource constraint:

ct =f(kt , stht) + (1 - 8)kt - kt+l, st E (0,1], ~5 ~ [0,1], (2)

f : 9~2+ ---) 9~+ is a production function, (counting the multiple sh as a single ' f ac to r ' ) , k t is the beginning of period stock of physical capital, s t is effort devoted to current production (with nonleisure time normalized to one each period), h t denotes beginning of period stock of human capital, and ~ the rate of depreciation of physical capital. We assume f to be "well-behaved," that is, f > 0 , f , < 0, Vi andf j > 0, Vi ~ j , i , j = 1, 2. It should be emphasized, however, that in the aggregate, these assumptions are modified such that f2 2 need not be negative, in order to accommodate externality effects. Further specification is given in Section 3.

Second, each household has access to a production technology for human capital, given by the function, g: 9~÷ x (0,1] ---) 9~+.

ht+l = g(ht, st) - $h ht , gh > O, g~ < O, ghh = 0, (3)

where 8 h E (0, 1) is the depreciation rate on human capital. This function is the same in spirit to those used by Lucas (1988) and Uzawa (1965). The function is increasing in last period's h, but decreasing in s, since raising the fraction of nonleisure time devoted to producing current output amounts to withdrawing nonleisure time from accumulating human capital. Note that if s = 1, no formal training is taking place in the economy. Whether the stock of human capital increases or declines over time depends on ~h given the quality of the learning-by-doing technology.

Third, we have some initial conditions on the stock variables, together with standard nonnegativity constraints

h0, k0 > 0 given, and ct, ht+l, kt+t > 0, t = 0,1 . . . . . (4)

343


Carrying out the maximization problem as described by (1)-(4) yields the following pair of Euler equations:

- u [ [ f ( k , , st ht) + (1 - 6)k,-k,+~] + i?mt+~[f(kt+t, st+t ht+l) + (1-~)k t+ ] ]

x [fkt+l + (1-8)] = O. (5a)

u[[f (k, , stht) + ( 1 - 6 ) k t - kt . l ] fs , + I~ t+ l [ f (k t+ l , St+l ht+]) + ( ] - ~ ) k t + l - kt+2]

x [ fh t+i - - f s t+ l ( g h , + l - - ~h)/gst+l]gs, ---- 0 . (5b)

Equation (5a) involves a standard rule for allocation of consumption across time; equivalently, it states that investment is undertaken in physical capital to the point where the marginal utility of today's foregone consumption equals the discounted gain from increased consumption tomorrow. Equation (5b) can be interpreted as follows: The first term measures the gain from increased output today by disinvesting in human capital (i.e., by raising st). The loss tomorrow is reflected in two terms. First, the t e r m ~3Ut+lfht+lgst measures the discounted utility loss from lowered ht+l, and hence, reduced output as a result of today's disinvestment. Second, the term --~Ut+lgs.t+lfst+l(ght+l -- ~h)/gst+l shows how a unit reduction in tomorrow's work effort affects the marginal utility of consumption.

The Working of the Model The Euler equations (5a) and (5b) can be used to generate optimal

paths for kt+ 1 and s t. The optimal path for consumption, c t, follows from the budget constraint in (2), and the stock of human capital, ht+l, follows from (3). Gross output can then either be obtained from

q, = ~, + _1,, (6)

where I t = kt+ 1 - (1 - 5)kt, that is, gross investment in physical capital, or simply from the production function,f. In sum, from Equations (2), (3), (5a), (5b), and (6), we can generate optimal paths for ct, st, ht+l, kt+l, and qt : {et 8, ~* h * 1, k*

' t, t+l}t=O" The long-run properties of the model can be outlined on the assump-

tions given. It is evident from Equation (3) that a sustained growth in ht+ 1 will occur as long as st is sufficiently below unity. Further, iff]z > 0, as we have assumed, the marginal product of kt+ 1 will increase steadily, thereby constituting a permanent growth of physical capital, as well as of consumption and output.

344

Note on Human Capital Externalities

3. Modal Calibration The model is calibrated for the special case of logarithmic utility 2 and

Cobb-Douglas production technology.

u(ct) = in ct ; (7)

f(kt, stht) = Ak~(stht)l-ah~,, A > 0, (x E (0,1), r I > 0 . (8)

This production function is identical to the one employed in Lucas (1988), with h e representing externalities in production, that is, effects observed at the aggregate level, but not by the individual producer.

The production function for human capital is assumed linear in h t, as previously noted.

ht+ 1 = h t [ P l ( 1 - s t) + [ 2 s t + (1 - 5h)], (9)

where Pl, Pe > 0, and Pl > Pz in order to make human capital investment economically feasible. That is, the household must be indifferent at the margin between allocating time to formal training versus production. In the case of formal training, the returns are fully captured by the valuation placed on the increase in the household's stock of human capital; however, in the case of production, the household receives compensation in the form of labor income in addition to the increase in its human capital stock associated with learning-by-doing. As a consequence, human capital must accumulate more rapidly via formal training. Otherwise, no formal training would take place.

In choosing the parameter values, we employ the following strategy: Four of the parameters, or, [~, 8, and ~5 h, are pinned down a priori. The first, 0c is set to 0.3, which broadly corresponds to the share of physical capital in gross output observed in U.S. data over a long period of time. The discount factor, I~ = 0.96, which is equivalent to approximately a 4% annual discount rate. The depreciation rate, 8 = 0.065, is set to imply a certain share of gross investment in output, from a given capital to output ratio. For example, a capital to output ratio of 2.5, an annual growth rate for k of 1.8%, and a depreciation rate of 0.065, implies an investment to output ratio of 0.2, which is close to that observed in the postwar U.S. data. 3 We chose a rough average of estimated values from labor market studies for the rate of depreciation of

2Assuming a more general tbrm of preferences, such as u(c) = c a -~/(1 - 7) would call for a recalibration of the human capital technology parameters pl and Pz, but otherwise has no effect on the results. Raising 7 to 2, for example, would nfise s and slow growth, ceteris paribus, due to increased myopia on the part of households. Therefore, in order to maintain values ofs = 0.91 and the growth rate of human capital at 0.83% would require a recalibration of Pl and Pz from 0.100 and 0.54 to 0.109 and 0.053, respectively. This would leave all values in Table i unaffected.

3The actual series used to construct the stylized facts of interest include gross private domestic investment and real GDP, and were obtained from the CITIBASE data bank.

345


human capital, with ~h = 0.05.4 The remaining parameters, A, 11, Pl, and P2, are chosen by calibrating the model for given growth rates and certain levels for selected variables as described below.

With no externalities in production (11 = 0), the model as set out in Equations (5a), (5b), and (7)-(9), generates a transition path which converges to a balanced growth path. Hence, by deflating all quantities by, for example, ht, the model can be made stationary, such that the steady-state version of the Euler equations (5a) and (5b) become

[pl(1 -- s) + pes + 1 -- ~5 h] = ~[aAk a-1 s 1-a + (1 - 6)] , (10a)

s = (1 - 13)(1 + Pl - 5h)/(Pl -- P2), (10b)

where k - k/h. The left-hand side of (10a) is one plus the growth rate of human capital; hence, h is growing as long as [p1(1 - s) + p2s + 1 - 8 hI > 1. The term in brackets on the right side is in effect one plus the real interest rate: 1 + r. Thus, with growth, we have the familiar result that 1 + r > 13-1, that is, the real interest rate exceeds the subjective discount rate.

From Equations (10a) and (10b), we use U.S. data from the period 1950--89 to calibrate Pl and P2 in order to obtain an annual growth rate of h and an average value of s compatible with those data. During the period, h increased by 39% (or approximately 0.8% annually). To obtain an estimate for the average value of s, we calculated an estimate of the fraction of an adult's expected worklife that is not spent acquiring formal education. ~ This corresponds to an approximate value ofs = 0.91. With 8 h -- 0.05, the implied values for Pl and P2 are 0.100 and 0.054, respectively. We note here that setting P2 = ~h = 0, as in Lucas (1988), has no effect on the estimates reported in Table 1; however, this leads to an unrealistically low value for s of 0.83, suggesting that 17% (versus our estimate of 9%) of an average adult's worklife is spent acquiring formal education. ° In Table 1, predictions of the two

4Estimates of the rate of depreciation of human capital have been made by Mincer and Ofek (1982) in a study of temporary drop-outs of the workforce. They found that real wages fell by approximately 3.3% to 7.6% per year after controlling for lost tenure, age, and other factors. Other estimates range from approximately 0.6% to 13.3% per year (Johnson 1970; Johnson and Hebein 1974; Haley 1976; Heckman 1976; and Rosen 1976).

Sin 1977, the average expected worldife for men, age 16, was 38.5 years, and for women, age 16, 27.7 years. (Reference: Worklife Estimates: Effects of Race and Education, Table 1, p.3) The average for both sexes is about 33 years. In 1976, the median education level beyond the age of 16 was 2.9 years. (Reference: Statistical Abstract of the United States, 1977, Table 216, p. 136.) This would produce an estimate ofs = 1 - (2.9/33) = 0.91. Since this excludes all forms of vocational training this estimate provides an upper bound for s.

~We emphasize here that for a given 8 h, lower values of s, which imply more time is spent in formal training, involves a recalibration of Pl and pe, such that the quality of formal training relative to learning-by-doing is lower (that is, Pl is reduced and P2 is increased) in order to achieve

346


versions of the model, that is, with and without the human capital externality, are compared with three "stylized facts" concerning the long-run properties of the U.S. economy: (1) a rising physical to human capital ratio; (2) a broadly constant or slightly increasing physical capital to output ratio; and (3) a constant real interest rate. 7 This is performed in the following manner. First, we assume a physical capital to output ratio of 2.5 at the end of the transition path. Second, we assume k/h = 0.43 at the end of the period, reflecting the relative factor shares (=0.3/0.7). Third, h is assumed to grow at a rate of 0.83% annually on average; s while k is growing at 1.8% per year. Fourth, output is normalized to unity in the final year (this in turn pins down the scale parameter A). From this set of assumptions, we can deduce the evolvement of k, q, k/q, and r.

Look first at the left part of the table, which contains the model version without externalities in production. We observe that the implied growth over the entire period is about 54%, k/q rises from 2.0 to 2.5 and the real rate of interest falls substantially, from 8.5% to 5.5%, or by more than one-third. According to the data, however, we should expect approximately 75% output growth, a slightly rising k/q and roughly unchanged r. So the model seems quite off the mark. The reason is that with constant returns, there is a unique relationship between k and k/q, as discussed below, and this is not reflected in the data.

The center part of Table i contains results from the model version with

Note cont. from page 346 the observed growth in human capital. Similarly, for a given s, a higher (lower) choice for 8 h implies an increase (decrease) in the quality of formal training. In neither case are the results of Table 1 affected.

?The real interest rate series that is used for establishing the stylized facts is constructed by subtracting the inflation expectations series from the Livingston survey from the nominal long-term government bond rate. (The latter was extracted from CITIBASE.)

SThe Jorgenson-Gollop-Fraumeni series on "'labor-quality" only extends through 1979. In our calculations, we have extrapolated this series out through 1989 using the growth rate for the period 1950--59. Our rationale is that by all accounts labor productivity in the United States has risen markedly during the 1980s, after a decade of very slow to stagnant growth (1970-79). See Dean, Kunze, and Rosenblaum (1990). Among the principal reasons cited for this pattern in productivity has been the entry of inexperienced "baby boomers" and women into the workforce during the 1970s causing the productivity slowdown, followed by the acceleration in the 1980s as these two groups gained work experience. We emphasize that assuming the more rapid growth for the 1980s essentially paralleling that of the 1950s, causes our estimates to err on the side of low values for the externality. Similarly, alternative human capital series have been constructed by Jorgenson and Fraumeni (1992) and Dean, Kunze, and Rosenblaum (1990); however, these series grow even more slowly than the Bishop-adjusted Jorgenson-Gollop-Franmeni series used here. To fit the model with data using either of these two series would have required a significantly larger externality than is suggested by the results in Table 1.

347


TABLE 1. Lucas-Uzawa Equilibrium Model with Human Capital: Long-Run Properties

q = Ak ~ (sh) 1 - a . he n, (in equilibrium h e = h) a = 0.3, 5 = 0.065, 5 h = 0.05, Pl = 0.100, P2 = 0.054

A. Model without externalities, r I = 0 B. Model with externalities, A = 0.106, A = 0.077,

A = 0.222 q = 0.42 I"1 = 0.6

Year 1 40 1 40 1 40

f¢ 0.31 0.43* 0.31 0.43* 0.31 0.43*

h 4.16" 5.8 4.16" 5.8 4.16" 5.8 k 1.3" 2.5* 1.3" 2.5* 1.3" 2.5* q 0.65 1.0" 0.57 1.0" 0.53 1.0" k/q 2.0 2.5 2.28 2.5 2.45 2.5 r 0.085 0.055 0.067 0.055 0.057 0.055

NOTES: * indicates an imposed value. For convenience, output is normalized to unity at the end of the period.

~---k/h

externalities in production, implying increasing returns in the aggregate. We set q = 0.42, which is essentially the value inferred by Lucas (1988). Now the output growth has risen to around 75% over the same period, the capital- output ratio has risen somewhat and the real interest rate dropped from 6.7% to 5.5%. In general, there is a substantial improvement over the constant returns version. Further increases in "q would lower the real interest rate further in year 1 and raise the k/q ratio. However, output would start to grow excessively. For example, as shown in the right-hand side of Table 1, raising r I to 0.6 yields an interest rate of 5.7% and a k/q ratio of 2.45 in year 1, but an output growth of nearly 90% over the period. Overall, it seems that the value of ~ = 0.42 provides a lower bound for the estimate of the degree of externality.

The difference between the two versions of the model can be clearly seen by looking at the relationship between k/q and/¢ implicit in the production functions.

k/q = f~l-as~-ihe~A -1 . (11)

From (11) it is evident that k/q can remain constant despite a rising f¢ over time, if the stock of human capital is increasing sufficiently. With r I = 0, however, (11) reduces to a unique positive relationship between k/q and

348


given the stationarity ofs and A. As pointed out in Lucas (1988), for any given growth rate of human capital, the rest of the economy will grow faster along an optimal path when "q > 0. This is indeed a property that the long-run postwar U.S. data appear to exhibit. 9

4. Conclusions Human capital is seen by many to be an important omission in standard

neoclassical growth theory. Without it, King and Rebelo (1993) are unable to render the transitional dynamics implied by the theory consistent with the long-run properties of the U.S. economy. One method of introducing human capital is via a linear production technology as in Lucas (1988) and Uzawa (1965). Given this model, this paper then examines evidence that bears on the conjecture of Lucas that investment in human capital generates positive externalities in the production of final goods. Using a human capital series for the U.S. economy constructed by Jorgenson, Gollop, and Fraumeni (1987) and adjusted by Bishop (1989), we find evidence from the postwar U.S. data in support of Lucas' hypothesis. Conservative estimates of the externality correspond to a value between 0.4 and 0.6, suggesting that the number obtained by Lucas of 0.417 (using U.S. data from 1900-57) corresponds more nearly to a lower bound, within the context of his model. These results underscore the importance that human capital externalities may play in traditional growth accounting exercises.

Received: September 1994 Final version: March 1995

References Bishop, John H. "Is the Test Score Decline Responsible for the Productivity

Growth Decline?" American Economic Review 79 (March 1989): 178-97. Dean, Edwin R., Kent Kunze, and Larry S. Rosenbaum. "Productivity and

the Measurement of Change in the Composition of Labor Input." Paper presented at the International Productivity Symposium IV in Kyoto, Ja- pan, October 1-4, 1990.

Denison, Edward F. The Sources of Economic Growth in the United States. New York: Committee for Economic Development, 1961.

Haley, William. "Estimation of the Earnings Profile from Optimal Human Capital Accumulation." Econometrica 44 (November 1976): 1223-38.

~Also, see Denison (1961), who focuses on the 1900~57 period, and contains the data used by Lucas (1988).

349


Heckman, James. "'A Life-Cycle Model of Earnings, Learning, and Con- sumption." Journal of Political Economy 84 (August 1976): s11-s44.

Johnson, Thomas. "Returns from Investment in Human Capital." American Economic Review 60 (September 1970): 546-60.

Johnson, Thomas, and Frederik Hebein. "Investment in Human Capital and Growth of Personal Income, 1956-1966," American Economic Review 64 (September 1974): 604-15.

Jorgenson, Dale W., Frank M. Gollop, and Barbara M. Fraumeni. Produc- tivity and Economic Growth. Cambridge: Harvard Press, 1987.

- - . "Output of the Education Sector." In Output Measurement in the Service Sector, edited by Zvi Griliches. Chicago: University of Chicago Press, 1992.

King, Robert G., and Sergio Rebelo. "Transitional Dynamics and Economic Growth in the Neoclassical Model." American Economic Review 83 (Sep- tember 1993): 908--31.

- - . "Public Policy and Economic Growth: Developing Neoclassical Implications."]ournal of Political Economy 98 (October 1990): s 126-s 150.

Lucas, Robert E., Jr. "On the Mechanics of Economic Development." Jour- nal of Monetary Economics 22 (July 1988): 3-42.

Mincer, Jacob, and Haim Ofek. "Interrupted Work Careers: Depreciation and Restoration of Human Capital." The Journal of Human Resources 17 (1982): 13-24.

Romer, Paul M. "Increasing Returns and Long-run Growth." Journal of Political Economy 94 (October 1986): 1002-37.

Rosen, Sherwin. "'A Theory of Life Earnings." Journal of Political Economy 84 (August 1976): s45-s67.

Statistical Abstract of the United States, 1977. Department of Commerce, Bureau of Census.

Uzawa, Hirofumi. "Optimal Technical Change in an Aggregate Model of Economic Growth." International Economic Review 6 (January 1965): 18-31.

Worklife Estimates: Effects of Race and Education. U.S. Department of Labor, Bureau of Labor Statistics Bulletin 2254, February 1986.

Appendix

List of Symbols 13 = discount factor. c t = consumption.

u(ct) = period utility function. k t = physical capital stock.

350

h t

S t

s~t fCk,, sh,)

g(ht, st)

5h 1, qt

11

91

92

k r


= human capital stock. = nonleisure time devoted to current production. = effective units of labor. = production function. = net investment in human capital. -- rate of depreciation of physical capital. = rate of depreciation of human capital. = gross investment in physical capital. = out-put. = factor share on physical capital in production. = parameter characterizing the magnitude of the human capital

externality. = human capital technology parameter associated with formal

training. = human capital technology parameter associated with learning-

by-doing. = ratio of physical capital to human capital. = real interest rate.

351

Documents

Note on human capital externalities