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DISCUSSION PAPERS IN ECONOMICS Working Paper No. 99-03 W(h)ither the Stock of Public Capital? Kenneth R. Beauchemin Department of Economics, University of Colorado at Boulder Boulder, Colorado January 1999 Center for Economic Analysis Department of Economics University of Colorado at Boulder Boulder, Colorado 80309 © 1999 Kenneth R. Beauchemin Kenneth R. Beaucheminl Revised: January 1999 1Department of Econonucs, University of Colorado at Boulder, Boulder, CO 80309. Phone: (303) 492-2651. Fa-x: (303) 492-8960. E-~Iail: [email protected]. I am grateful to Martin Boileau, Todd Clark, Dean Corbae, JoAnne Feeney,Beth Ingram, Narayana Kocherlakota. GeneSavin, Kei-~Iu Yi and Chuck Whiteman for helpful comments and suggestions. An earlier ,-ersionof this paper was presentedat the 1994annual meeting of the Society for Economic Dynamics and Control and the 1996 Monetary Policy Roundtable meeting hosted by the Federal ReserveBank of Kansas City. I thank the participants at these gatherings for useful feedback. Abstract This paper maintains that the commonly used measure of the aggregate stock of public capital is conceptually divergent from the "true"' public capital input in private production technologies. Consequently, the published measure has the potential to misrepresent the historic growth profile of the public capital input and the pro duct ivity of government purchases in general. As a test of this hypothesis: the identifying assumption of a standard stochastic growth model is combined with observations on output, consumption, labor hOUl-S and government purchases to deduce public capital paths that are mutually consistent with observed flows and economic theory. It is found that the inferred series grows more rapidl~' than the published series during the post-1973 period thereby pro\iding evidence that government under-investment is not an important source of declining U.S. producti,.ity Key words: public capital; productivity; gro,\-th JEL classification: E62, 040. growth. 1 Introduction In recent years, the stock of publicly-owned capital -as measured by the national income accounts -has declined relative to the private capital stock. Since 1970. the ratio of nonmilitary public capital to private capital (including consumer durables), declined from a peak value of .284 to approximately .237 in 1993. Including military capital, the ratio attains a peak of .365 in 1963 and declines to .257 in 1993. (see figure 1). Similar observations formed the basis for an extensive research program statisti<;:ally relating the diminishing rate of public capital formation to the slowdown in productivity growth that began in the early- to mid-1970s.1,2 This paper, while accepting, a priori, the notion that some portion of government purchases are ultimately useful in private production activities, questions the validity of using the measured stock of public capital to draw inferences regarding the productivity growth slowdown and as a guide to public policy. Although measurementdifficulties exist for both privately and publicly-owned capital, important differencesmake assessingthe public capital stock inherently more problematic. The primary difference is conceptual. How does one decide on the proper fraction of current government purchases to be capitalized as an input to private production technologies? Although it is hard to deny the complementary nature of private capital and "core infrastructure" items such as highways and streets, \vater systems and mass transit facilities, the link between private production and pub- lic capital structures such as hospitals, courthouses.civic centers, and conser,-ation 1Aschauer (1989a, 1989b) are early references to what became the 'public literature. capital hypothesis' This work is surveyed by Gramlich (1994). This article also contains a good description of the measured stock of public capital and its components. 2United States capital stock data are published by the Bureau of Economic Analysis in Fixed Reproducible Tangible Wealth in the United States. The series are updated annually in the August Survey of Current Business. 1 facilities is less clear. These ambiguities imply that the measured stock of public capital is potentially "too large." Alternatively, some components of government ..consumption" expenditures conceivably expand future production possibilities and, as such, are part of a broader and unmeasured concept of public capital. For exampIe, the public education expense of a teacher's salary is not capitalized even though it augments the stock of human capital and productive capacity suggesting that the measured stock of public capital is potentially "too small." These arguments indicate a potential rift between an economic theorist's conception of the public capital input and the vie" reflected by the official measures.:! This paper adopts the extreme vie\\" that the productivity-enhancing stock of pub- lic capital is not observed. To study the interaction between public investment and productivity. an alternative measurement method is proposed that combines observ- able time series with the identifying assumptions implied by a standard stochastic growth model to deducea path for the public capital input. In this sense, the study follows in the tradition of growth accounting due to Solow (1957) In its simplest form, growth accounting combines time-series data on inputs to production with a neoclassical. constant returns to scale production function to infer the history of to- tal factor productivity. This deducedseries is then interpreted as the contribution of technologic~l progress to economic growth. For the purposes of this study, this simple gro\\"th accounting is inadequate because public capital is not only taken to be productive, but also unobserved. Clearly, a richer apparatus is required to simul- taneouslyidentify the growth paths of both total factor productivity and productive 3Gramlich (1994) points out two additional problems associated with the measure of public capital in relation to private capital. First, the servicesof public capital are not sold in markets making it difficult to value public assetsaccording to the future stream of benefits that they produce. Second, once purchased,public capital items are rarely re-sold. As a consequence,economic rates of depreciation are almost never directly observedso it is difficult to ascertain service lives. 2 The plan of the paper is as follows. Section 2 sets out a modified version of a standard stochastic grov.rth model that includes a go'\-ernment that purchases consumption and investment. goods The model \\ith indivisible labor is also discussed Section 3 describes the implememation of the procedure and Section 4 presents the results and their implications for aggregate labor productivity. 2 The Model Section 5 concludes. Environment This section introduces a standard stochastic groVv"thmodel modified to include a government that purchases goods from the private sector that can be used for either investment or consumption pUl'pO~es.The first-order conditions of this model provide the restrictions necessary to identif\- the gro\\'th path of public capital. The measurement procedure \vill be executed under both the standard labor market institution with a finely divisible labor suppl~' and the indivisible labor setup of Hansen (1985) and Rogerson (1988).4 2.1 The Standard Model Consider an economy inhabited by an infinitely-lived representative agent and a government. The agent owns a constant returns to scale production technology Yt = At that transforms the labor and capital inputs into a single output good, Yt, where nt is the fraction of period-t spent \vorking, /t'ptrepresents the beginning-of-period-t stock of private capital and kgt ~imilarly denotes the stock of public capital. The 4Braun and McGratten (1993) ana Baxter and King (1993) also introduce public inputs to standard dynamic general equilibrium modelsto assessthe quantitative impacts of assorted policy experiments. 4 :omplements exogenous shock to total factor productivity is given by .-it. The parameters a and b are each contained in (0,1); the former gives the share of output earned by capital (broadly defined), and the latter attaches relative weights to the two types of capital. The parameter p E [-1,00) governs the substitutability bet\\-een private and public capital. This specification allows the model to nest a continuum of economies which are distinguished by the technological relationship between the two capital types. If p -1, public and private capital are perfect substitutes: they become perfect in the limit as p becomes infinite. Output in this environment is divided between private consumption, private investment, and government purchases: Yt = Ct+ it + gt The one-period ahead value of private capital is the sum of the current undepreciated quantity and the current flow of private investment kp,t+l kpt(1 fJp) + it (2) where 8p is the rate of depreciation on private capital.5 During period t. the government spends 9t which is financed contemporaneously with lump-sum ta.--::es levied on households. In a fashion similar to the private capital accUllrulation technology, government purchases are apportioned between consurnption and investment uses so that k9,t+l = kgt(1 -8g) + O"t9t (3) 5In creating the correspondencebetweenthe measurementsthat are available for the U.S. economy and the model variables, investment (i) is defined as the sum of businessfixed investment and householdinvestment. Household investmentcombinesthe purchase of consumer durables and new homes. Since the national income accountsdo not value the flow of senices arising from household durables, it is imputed by assuming that the flow is proportional to the deterministic steady state interest rate plus the rate of depreciation of privately-owned capital (tp). The flow is then added to the published consumption-of-servicesseries. Seethe appendLxfor details. 5 Lhe where O"tis the fraction of period-t government expenditures devoted to building the productive public capital stock and 6g is the depreciation rate of public capital. As seen belo,v. neither type of government expenditure affects the utility of private agents.6 representative agent maximizes expected lifetime utility given by (4) where /3 is the subjective discount factor of the agent and ()t is the period-t realization of an exogenous shock to preferences. Utility in anyone period depends positively on consumption and leisure; the time endowment in each period is normalized to one. Changes in {}t represent changes in the agent's willingness to substitute market activities (measured by consumption) for non-market activities (measured by leisure). An increase in (}t, for example, is strictly interpreted as a willingness of the representative agent to work longer hours which subsequently drives down the marginal product of labor. A sustained increase in the gro\vth rate of (}t, therefore, results in a period of slower labor productivity gro\vth. In eachperiod t, it is assumedthat the agentknows the history of the state given by Ot {As, (}s, 0" s, kps, kgB, 9s S .$: t} In this environment, the agent faces two external risks to the rate of return on private capital. First. the rate of return is affected directly by the total factor productivity (technology) shock At. Positive movements in At raise the marginal products of all inputs uniformly. Second,the rate of return is affectedby movementsin the amount of productive public capital available (kgt). A public investment boom raises the marginal productivity of private capital and labor while driving down the marginal UThis restriction could be rela.xed without altering the analysis provided that government pur-:hases enter the momentary utilit}, function in an additively separable fashion, 6 1 product of public capital. Both shocks influence the optimal behavior of the agent in the same way: positi,-e shocks encourage work and discourage present consumption in favor of private capital accumulation- Negative shocks naturally have the opposite influence. Since there are no distortions or externalities produced by this economy, the allocation that solves the centralized planning problem goes through as the competitive equilibrium. 7 This problem is written as follows: 00 max Eo L {3t[()t log Ct+ log (1 -nt)] t=O Sot Ct -O"t) gt + kp,t+l + kg,t-r-l = (1 -lip) kpt 1 -8g) kgt Yt given (1), (2), kpO> O. kgo > 0 and {gt}~o. The solution to this problem must satisfy equation (1) and the following first-order conditions: Ct/ (}t 1- ~ = /3Et~ Ct Ct+l { 1 -Dp = (1 -a) ~ nt (5) nt (~ ) [1 + (~ ) </>-;:1 ] -1 + Q kp,t+l Ct+ kp,t+l = Yt + (1 1 -b {jp) kpt -gt (6) (7) where ~ c/>t+l -k . p,t+l Equation (5) is the intratemporal efficiency condition equating the marginal rate of substitution between consumption and leisure to the marginal product of labor. 7The assumption of constant returns to scale coupled \vith the agent's role as a competitive profit-maximizing firm implies a portion of private output, equal in value to the stock of public capital, multiplied by its marginal product, is not distributed as income to the private factors of production. In the specifiedenvironment, the integrity of the competitive equilibrium is maintained by implicitly assuming that the governmentextracts a lump sum payment from the firm equal to this amount to maintain the zero-profit condition. 7 ences. Equation (6) is the intertemporal efficiency condition equating expected discounted marginal utility of consumption acrosstime periods. and equation (7) is the resource constraint. 2.2 The Model with Indivisible Labor It is now apparent from the intertemporal efficiency condition (6) that the time path of the unobserved preference shock, ()t, identified by the intratemporal efficiency condition (5), plays a role in determining the intertemporal allocation of resources. Consequently, the measurement of public capital is not independent of the institutional nature of the labor market" The popular alternative to the labor market structure embodied above, is the indi,isible labor mod~l of Hansen (1985) and Rogerson (1988) in which volatility in hour5 \\orked occurs through employment variation rather than the continuous adjustment of work effort. Since it is likely that \"ariation in ac- tual aggregatelabor hours is generatedby a combination of hours and employment adjustment, the measurement procedures are performed under both institutions. The model with indivisible labor is obtained with a: simple substitution of preferIf n. is the constant fraction of the time endowment that is ~'orked with some probability 7rt > 0, then per-capita hours worked is nt = 1itn. The pre,.ious preference specification (4) is then replaced b~' Eof y lOtlog Ct '+ t=O resulting in the intratemporal efficiency condition Ct/Ot (1 -0:) ~ ii/ log (1 -ii) nt All other specifications and necessary conditions are unchanged from the standard economy. 8 where'Y is a p x 1 vector of parameters. Unlike preference and technology parameters. economic theory and independent evidence provides no guidance in selecting these values. They must, however, be chosen to satisfy the first-order conditions (5) (7). Equivalently, the elements of 1 are assigned so that a sample analog of the orthogonality condition implied by the stochastic Euler equation (6) is satisfied: T-l L (Xt+l -1) Zt = 0, Zt E Slt (9) t=l where Xt+l - 1 -8p b ) 1+ ( ~ °t-'-l Yt+l + Q; -, kp,t+l -1 -p J and Zt is a vector of instruments contained in the agent's information set Slt and T is the numberof data observations. It is assumedthat there are as many instruments as there are parameters in the government's policy function so that (9) represents apxp system of equations with, as the unique solution. In other words, the procedure for assigning ,-alues to the parameters of f is the exactly-identified version of Hansen's (1982) generalized method-of-moments estimator. The parameter vector is computed using annual data from the period 1956-93 (T = 38).8 The government policy rule (8) is further restricted so that f is a linear combination of elements in the agent's information set describing the current state: f (Ot;,) = Wt 'Y (10) where Wt is a (1 x p) vector of variables observed in period t and f is a (p x 1) vector of parameters that satisfies the sample orthogonality condition (9). To facilitate computation, only the variables in Ot that do not need to be inferred from the procedure itself appear in the vector Wt With this restriction in mind~ it is assumed that Wt 1~~(} , , Yt , t Yt 1!A description of the data is contained in the appendix. 10 so that p = 4 with the corresponding parameter vector")' = (")'°"')'1"')'2"')'3)' Given that the instruments need only be observed by the agent in period t, the vector of instruments is assumed to be identical to the vector of explanatory state variables: Zt Wt It should be understood that there is no model of government behavior intended the function (10) is merely descriptive of the government:s past behavior. Viewed from this perspective, the parameter values comprising I do not have a strict economic interpretation Calibration To complete the task of identifying the shocks, it is necessaryto calibrate the model economy. The most straightforward settings are those for a and /3. In calibrating capital's share of income, it must be recognized that the flow of income deriving from the stock of public capital is not included in the official measure of U.S. national income. Furthermore. the value of the stock must be known prior to computing its income flow Given that this is not possible. a value of a 36 is chosen following the work of Kydland and Prescott (1982), Hansen (1985) and Prescott (1986) in which income from public capital is not imputed. Labor's share is the remaining 1 -Q = .64.9 The subjective discount factor is chosen to be /3 1.03-1 implying a deterministic steady-state real interest rate of 3.0 percent.10 In setting the remaining technology parameters, band p, it is useful to view the measurement procedure as a composition of level effects and growth effects. Once the initial stock of public capital is determined, a complete sequence of period-to-period growth rates are sufficient to characterize the imputed time path of public capital. 9The results are not significantly impacted by the Q = .40 setting suggested by Cooley and Prescott (1995) in accounting for income generated by (published measures of) public capital. !UAgain, reasonable adjustments of this parallleter leave the general results and conclusions intact. 11 5 Conclusion This paper began ,vith the assumption that publicly provided capital augments private production possibilities and also plays a quantitati\"ely significant role in economic growth. It also maintains that the existing measures of public capital do not accurately reflect the economic theorist's view of the true public capital input, and therefore, may not be represemati\e of the true input's historic time profile. It is shown that a time-series for public capital that is mutually consistent with a standard stochastic gro"--th model and the data on flows of aggregate output, consump- tion. governmentpurchasesand labor hours, exhibits a markedly different historical pattern than its measured counterpart. In particular, the \iew that public underinvestment is partially responsible for the slowdown in labor productivity growth is not supported It is important to bear in mind that the results depend on the identifying as- sumption of a neoclassical gro"lh model. Given the widespread application and understanding of this framework. the results serve as a benchmark from which other investigations can be launched. The intent of the research is not to replace traditional accounting methods. Different models will generate different numbers comprising the inferred history of the public capital input; the issue is whether these competing modelswill changethe generalshapeof the time profile or the basic conclusionscontained herein. It is hoped that the results of this study are sufficiently compelling to generate future research along these lines and to inform accounting-based economic measurementtechniques. 17 18 Appendix This appendix contains the sources of the raw data and definitions of the variables used in the study. Data used are annual, real aggregate data of the United States for the period 1956 to 1993 unless otherwise indicated. In cases where the raw flow data are published at a monthly or quarterly frequency: the values are averagedto an annual frequency. The data on output, consumption,and governmentpurchasesare constructed from the following Citibase variables (Source: National Income and Product Accounts, Table 1.2) la. GC.!~:-Q(personal consumption expenditures, nondurable goods) 2a. GC SQ (personal consumption expenditures, services) 3a. GC DQ (personal consumption expenditures, durable goods) 4a. G P I Q (gross private domestic investment.) 5a. GGEQ (government purchases of goods and services) Series la -5a are converted into per capita quantities using the total labor force (Citibase variable LHG). 6a. c The following aggregations are used: GCNQ + GCSQ + csd 7a. i = GPIQ + GCDQ 8a. 9 = GGEQ 9a. y = c + i + 9 where c3d is the flow of services generated by the per capita stock of consumer durables. The service flow (csd) is computed using the expression csdt (r + 8p) kht wherer = ~i3 is the deterministicsteadystate (real) interest rate and is the kht beginning of period-t real net stock of consumer durables as reported by the Bureau of Economic Analysis (Katz and Herman 1997) converted to a per capita quantity. The labor hours series used in the study is constructed from the following Citibase variables: lOa. LHOURS: lla. LHC: average weekly hours (constructed from household survey data) civilian labor force. Per capita hours are expressed as a proportion of the time endowment, which is set to 80 hours per week (16 hours per day times 5 days per week): 12a. 'n = (LHOU RBI LHC) 180. 19 REFERENCES 1] Aschauer. D. A Public Expenditure Productive' Journal of Monetary Economics, 23, 177-200, 21 D. A. (1989b) Public Capital Crowd Out Private Capital? :" Journal of Monetary Economics, 24. 171-188. [3] Baxter, ~I. and King R. G. (1993), "Fiscal Policy in General Equilibrium," American Economic Review, 83. 315-334. Braun, R. A. and McGratten, E. R. (1993), "The ~Iacroeconomics of War and Peace, NBER ~Iacroeconomics Annual, l\IIT Press: Cambridge, MA. Burnside, C., Eichenbaunl. ~1 and Rebelo, S. (1993), "Labor Hoarding and the Business Cycle," Journal of Political Economy, 100. 245-273. Cooley, T. F. and Prescott. E. C. 1995), "Economic Growth and Business Cycles T. F. Cooley (ed in Frontiers of Business Cycle Research, Princeton, NJ: Prince- ton University Press [7] Gramlich, E. M. (1994) Investment A Review Essay,1 Journal of Economic Literature. 32, 1176-1196. [8] Hansen, G. D. (1985), Labor and the Business Cycle, Journal of Mon- etary Economics, 16. 309-327. [9] Hansen, L. P. (1982), ;'Large Sample Properties of Generalized Method of Moments Estimators," Econometrica. 50, 1029-1054. [10] Ingram. B. F., Kocherlakota, ~.~ and Savin~ N. E. 'Is (1989a) Aschauer. 'Does [4] [5] [6] 'Infrastructure 'Indivisible 'Explaining 1994) Business Cycles: A Multiple Shock_-1..pproach.': Journal of Monetary Economics,34, 415-428. 20 llId 'Using 199. ,urement: 'Time Ingram. B. F, Kocherlakota, N Savin. :\" E Theory for Mea- An .:-\nalysis of the Cyclical Beha\"ior of Home Production."" Journal of Monetary Economics, 40. -4035-456. [121 Prescott, E. C. (1986), "Theory .-\head of Business-Cycle ~Ieasurement," CarnegieRochester Conference on Public Policy, 24, 11-44. [13] Katz, A. J. and Herman, S. W. (1997), "Improved Estimates of Fixed Reproducible Tangible Wealth, 1929-95,' SurI.'eyof Cu~nt K v"dland. F. E. and Prescott. , E. C (1982) c Business, 77 (May), 69-92. to Build and Aggregate Fluctua- tions," Econometrica, 50, 13-,1.5-1370. 15] Rogerson, R. (1988), "Indivisible Labor, Lotteries and Equilibrium, " Journal of Mon- etary Economics, 21, 3-16. [16] Solow, R. (1957), "Technical Change and the Aggregate Production Function," Review of Economic Studies, 39. 312-330. 21 Table I Parameter Settings a (0.20,0.25) Q {j .B 0.36 1.03-1 p 0.035 (.5,1,2..5) n 0.466 Note: Bold face indicates the benchmarkparameters (divisible labor economy used for presentationpurposesonly The ..shift length" n setting is only required in the indivisible labor economy 22 Table IIa (Divisible Labor) *The measurementexperiment with rho equal to unity is a useful benchmark for comparisonin that the simulated public to private capital ratio of 0.311 is identical to the one reported by the Bureau of Economic Analysis. 23 -0.609 -0.23 -0.48 Table lIb (Indivisible Labor) Policy rule parameters: 'Yo (constant) 7.494 3.747 'Yl (kpt/Yt) 2.915 1.458 0.729 0.292 12 (gt/Yt) 13.365 6.682 3.341 1.336 13 (()t) -6.093 -:3.047 -1.874 0.749 -1.523 Average annual gro\vth of kg/kp (0/(,) 1956 -93 -1.75 -0.88 -0.44 -0.17 1970 -93 4.97 2.45 1.22 0.49 1974 -93 7.55 3.71 1.84 0.73 0.97 0.93 Average annual growth of A (Cl( 1956 -93 1.18 1970 -93 -0.97 1974 -93 -1.46 Mean of kg/ kp: 1.04 -0.09 -0.70 0.096 0.299 24 -0.32 0.542 -0.09 0.781 FIG. 1. Public-pri\'ate capital ratios, 1956-93. Source: Fixed Reproducible Tangible Wealth in the United States, U.S. Department of Commerce. 25 0 ~ I!J ~ 0 0 ~ 0 I!J 00; 0 0 00; 0 ro (l) .c I!J I'0 CJ r., Ln t'! -, 0 '--' ' ,---~--, ,---,- 0 CD '- ... c:i Ln Ln d 0 1I1 01955 1960 1965 ,975 380 1985 Year FIG. 5. RealizaTionsof the preference shock (}t 29 !990 1995