American Economic Association
Transitional Dynamics and Economic Growth in the Neoclassical Model Author(s): Robert G. King and Sergio T. Rebelo Source: The American Economic Review, Vol. 83, No. 4 (Sep., 1993), pp. 908-931 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2117585 Accessed: 02/08/2010 15:14
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http://www.jstor.org TransitionalDynamics and Economic Growthin the Neoclassical Model
By ROBERT G. KING AND SERGIO T. REBELO*
Neoclassicaltransitional dynamics are a centralelement of standardmacroeco- nomic theory.Quantitative experiments with thefixed-savings-rate models of the 1960's showed lengthytransitions, thus potentiallyrationalizing sustained dif- ferencesin growthrates across countries. We investigate quantitative transitional dynamicsin variousneoclassical models with intertemporallyoptimizing house- holds. Lengthytransitions occur only with very low intertemporalsubstitution. Generally,when one triesto explainsustained economic growth with transitional dynamics,there are extremelycounterfactual implications. These result from the fact that impliedmarginal products are extraordinarilyhigh in the earlystages of development.(JEL E10, 010)
The neoclassical model of capital accu- with the model's central properties and its mulationdeveloped by Robert Solow (1956), intuitive mechanics.The model is so famil- TrevorSwan (1963), David Cass (1965), and iar that the reader may be skeptical that TjallingKoopmans (1965) is one of the most there is anythingnew to learn about it. By important theoretical paradigms for dy- contrast, we take the view that its transi- namic economic analysis. It has been the tional dynamicsare largelyunexplored from impetus for much theoretical research, in- a quantitative standpoint. Further, such an cluding elucidation of such key properties exploration is essential to understanding as the local and global turnpike theorems. whether the model can plausibly explain In the hands of Solow (1957), EdwardDeni- major differences in economic growth over son (1962), and others, the neoclassical time and across countries. model has further provided an empirical To learn about the relevanceof neoclassi- frameworkfor importantresearch into the cal transitional dynamics to economic sources and nature of economic growth. growth,we conductdynamic simulations us- Virtually every professional economist ing a range of parametervalues which are trained in the last two decades is familiar conventional in public finance and macro- economics. We find that the neoclassical model's outcomes are rich in the sense that a wide variety of dynamic paths may arise from a capital stock that is initially lower *King:University of Virginia, Charlottesville,VA 22901; Rebelo: University of Rochester, Rochester, than its ultimate level. For example, when NY 14627, the Catholic Universityof Portugal, and we vary the intertemporalpreference pa- Bank of Portugal,Lisbon, Portugal. This paper is a rameterswhich governsaving, we encounter revision of the first part of "Public Policy and Eco- sharply differing paths for output growth nomic Growth:Developing Neoclassical Perspectives," and the fraction of output that is invested. which was preparedfor Isaac Ehrlich'sconference on "The Problemof Development"at the State University Notably, the dynamicpaths are substantial- of New York at Buffalo in May 1988. This paper has ly more elaborate than those obtained in benefitedfrom the commentsand suggestionsof many the prior quantitative analyses of transi- individuals,including Robert Barro, James Dolmas, tional dynamicsby Ryuzo Sato (1963) and Stan Fischer, Arnold Harberger,Ken Judd, Robert Lucas, Paul Romer, and David Zervos. Comments Anthony Atkinson (1969), which employed fromworkshop participants at the Universityof Chicago Solow's (1956) assumptionof an exogenous and the Universityof Rochester are also appreciated. savingrate. However,by requiringan initial 908 VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 909 capital stock sufficientlysmall that capital much more rapid transitionsthan those ob- accumulationcontributes to long-term eco- tained by Sato (1963). nomic growth in an importantmanner, we Since the dynamic simulations all dis- also find that the neoclassicalmodel's pre- play staggeringlyhigh real interest rates in dictions are inconsistentwith observedvari- the early stages of development,Section IV ation in interest rates, asset prices, and fac- explores how this findingdepends on (i) the tor shares over time and across countries. parameters of the production function We trace this finding directly to a central (varying,for example, the elasticity of sub- aspect of Solow's (1956) neoclassical pro- stitutionand steady-statefactor shares);and duction function: a diminishing marginal (ii) commonlyencountered alterations of the productof capital,when labor input is held basic model (such as distinct production fixed. Thus, if the initial capital stock is very technologies for consumption and invest- low so that accumulationis important for ment goods, investment adjustment costs, growth, then its initial marginalproduct is internationaltrade, and vintage capital ef- correspondinglyvery high. fects). While some of these modifications The organizationof the paper is as fol- overcome the real-interest-rate implica- lows. In Section I, we outline the discrete- tions, they do so only at the cost of produc- time version of the basic neoclassicalmodel ing some other, related, counterfactualbe- that we focus on in the paper. We also havior. For example, with the introduction discuss why two recent related areas of re- of adjustment costs, the link between searchleave open the centralquestion posed marginal product of capital and the real in our paper: how importantcan the neo- interest rate is weakened. This generates classical transitional dynamics be for ob- more plausible values for the real interest served growth experiences? In Section II, rate but implies an extraordinarilyhigh ini- we review key quantitativeanalyses by Sato tial relative price of installed capital and (1963) and Atkinson (1969) that have led new investment goods (i.e., James Tobin's macroeconomiststo view transitional dy- [1969]"q"). namics as protractedand, hence, as poten- Overall,for realisticparameterizations of tially capable of explainingsustained cross- the productionfunction, our results suggest countrydifferences in growthrates. that neoclassical transitionaldynamics can In Section III, we study the transitional only play a minor role in explaining ob- dynamicsunder the assumptionthat saving served growth rates. That is, the physical- is rationallychosen by an immortal family capital accumulationprocess-which is the as in Cass (1965) and Koopmans(1965). We key mechanism behind the neoclassical begin by observingthat sevenfold growthin model's transitional dynamics-cannot ac- output per capita luckilycorresponds to key count for much growth without generating differencesin U.S. history and the interna- very large marginal products in the early tional cross section. First, it is roughly the stage of development. In our view, this ratio of current U.S. per capita real gross pushes one to think about models of en- domestic product to that of a century ago. dogenous economicgrowth which, following Second, it is roughlythe gap between poor Theodore Schultz (1961), Hirofimi Uzawa countries and the United States in 1950 (in (1965), Paul Romer (1986), and Robert the data set of Robert Summersand Alan Lucas (1988), assign a larger role to other Heston [1988]). To be conservativein our modes of accumulation, such as human- demands on the neoclassicalmodel, we re- capital formation or endogenous technical quire that capital accumulationexplain only progress.Indeed, we show that a version of half of sevenfold growth in output over a the neoclassicalmodel with a capital share century;the rest is attributedto technologi- close to unity yields protractedtransitional cal change.We explorehow transitionpaths dynamics and avoids the high initial mar- depend on parametersof preferences and ginal product; but this model is patently technology: generally, but not always, our unrealisticunless we broaden our notion of models with endogenous savingsrates yield capital accumulation. 910 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993
A. ProductionFunction, Y = f( K)
f(KT)
f(KT)/7 K0 K0 Capital Stock KT B. MarginalProduct of Capital 1,000
500
0)
K0 Kl Capital Stock KT
FIGURE 1. A) PRODUCTION FUNCTION, Y = f(K); B) MARGINAL PRODUCT OF CAPITAL
The strength of our negative results also will be used in our analysis. With minor gave us concern that our experiment was modifications,the model is that of Solow too extreme (i.e., that requiringan explana- (1956) translated to discrete time. At the tion of a majorportion of U.S. growthover heart of the model is a constant-returns-to- a centurywas too much of a task). For this scale aggregateproduction function, reason, we decided additionallyto consider a more restricted experiment suggested by Robert Barro's(1987) discussionof the neo- ( 1) Yt = F( Kt, NtXt) classicalmodel's content for the differential growth experiences of countries during the post-World War II interval. In particular, where Yt is output, Kt is physical capital, we use the idea that the 1950 levels of Nt is labor input (in man-hours)and Xt is a output per capita of "losing"countries may measure of labor productivity.' Holding be used to identify a war-induceddecline in fixed Nt and Xt, the production function physicalcapital stocks. For example, during (Fig. 1A) has positive and diminishingre- 1950-1980, Germany's per capita output turns to the reproducible factor Kt. The moved from 45 percent of U.S. output to marginal product of capital schedule has 88 percent, and Japan'smoved from 19 per- the familiarform displayedin Figure 1B. cent to 74 percent. In our alternativeexper- The additionalequations of this familiar iment, we take the United States as defining model are the resource constrainton con- the growth of the "technical frontier" and sumptionand investment, view the extraordinaryJapanese growth in the postwarperiod as a result of transitional dynamics.The result is striking:the neoclas- (2) Ct+it=Yt sical model implies that the Japanese real interestrate in 1950 shouldhave been nearly 500 percent. Section VI discusses our con- and the differenceequation for the accumu- clusions and directionsfor future research.
I. The Basic NeoclassicalModel 'Technicalchange in (1) is in labor-augmentingform so as to admit steady-state growth when technical In this section, we set out the basic neo- change and labor input grow at constant rates (see classical model of capital accumulationthat Swan, 1963;Edmund Phelps, 1966). VOL.83 NO. 4 KINGAND REBELO:TRANSITIONAL DYNAMICS AND GROWTH 911 lation of capital, form:
(3) Kt+1 -Kt =It -Mt. __r__ for 0
(4) Nt = oYNNt-1 As in the bulk of the growthliterature, we abstractfrom choice of labor supply,assum- ing each population member supplies n = (5) Xt Yxxt-1- hours, so that N, = nMt. Throughout, yz is Z's "gross"growth rate B. TransitionalDynamics (i.e., yz - 1 = [Zt - Zt- l]/Zt- 1) In the basic neoclassicalmodel, the com- Growth in the basic neoclassical model mon steady-stategrowth rate of manyof the can arise for two general reasons. First, system's variables is YXYN: there is steady-stategrowth associated with growthin productivityand population.Sec- = = = ond, there is transitionalgrowth associated (6) 'YY 'YC YK 'YXYN with movementfrom an initial capital stock toward the steady-state growth path. For This implies that many key ratios-such as example, under Solow's (1956) assumption consumption's share of output or labor's of a fixed savings rate with zero deprecia- income share-are constant in the steady tion, accumulationis given by state. (10) Kt,+-Kt=sF(Kt,nMtXt). A. SavingsBehavior Growth relative to the steady-statepath is We study the model under two alterna- then given by tive assumptions about savings behavior. The first case is Solow's (1956) assumption (11) yXyNkt+ 1 - kt = sF(kt, n) that saving (net investment)is a fixed frac- tion s of income: where k, = Kt/(M,Xt). From any initial value of ko, this difference equation con- (7) It =sYt + Kt. verges monotonicallyto a unique stationary value satisfying (YXYN - 1)k* = sF(k*, n), as Our second case is that saving is deter- demonstratedin Solow (1956), but this gen- mined by an immortalfamily's optimal con- eral property leaves open the issue of the sumptionchoices. We assume that this fam- rapidityof this transitionalgrowth. ily's preferences are of the form discussed Since along a steady-statepath per capita by Barro and Gary Becker (1989): output grows at rate yx, cross-countrydif- ferences in growth rates can only be "ex- plained" if they result from different rates (8) Ut= EBjMt7+1u(Ct+j/Mt+j). of technicalprogress. It is now widelyrecog- j =0 nized that this explanationis vacuous.If the neoclassicalmodel is to help us understand In (8), ,B is a discount factor, Mt is the more than why consumption, investment, numberof membersof the family, and q is and output move together along a growth a parameter reflecting valuation of future path, the model'stransitional dynamics have membership.The utility of per capita con- to play an important role in explaining sumption, u( ), has a constant-elasticity cross-countrygrowth differences. 912 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993
0.8 zO0.6-r
: 0.4 __ [) 0.2_- -
0 0 10 20 30 40 50 60 70 80 90 100 Years
1
0.8 -
0.6-
0.4 -
0 10 20 30 40 50 60 70 80 90 100 Years
0 0 10 20 30 40 50 60 70 80 90 100 Years
FIGURE 2. TRANSITIONAL DYNAMICS IN THE SOLOW MODEL
Notes: Output and capital are each expressed as a fraction of the steady-state value. For output, x denotes the quarter life; o, the half life; and * the three-quarter life.
II. Traditional Views of Transitional A. The Sato-Atkinson Experiments Dynamics If the neoclassicalmodel is to be used as In this section, we discuss the conven- a descriptionof actual growth experiences, tional perspective on the quantitative im- one natural question is: what fraction of portanceof transitionaldynamics. We begin long-run growth is plausibly attributed to by describingseveral key quantitativeexper- steady-statemechanics-associated with the iments performed in the 1960's with the exogenous evolutionof populationand pro- neoclassical model which indicated that ductivity-and what fraction to transitional these dynamics could be very protracted. dynamics? Then, we discussthe potential magnitudeof A key quantitativeexperiment by R. Sato transitional dynamics indicated by looking (1963) demonstratedthat output may adjust at cross-country and within-country eco- very slowly toward the steady-state path, nomic growth. Finally, we consider the suggesting that transitionaldynamics could "growth-accounting" perspective, which be responsible for a significantfraction of originates in the research of Solow (1957) the observed expansion in per capita out- and Denison (1962). put. Workingwith a Cobb-Douglasproduc- VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 913 tion function and a fixed savings rate, Sato as the fractionexplained by exogenoustech- showed that there was a very lengthy dy- nical progress.These definitionsare natural namic response to a savings rate shift in- ones: we know tT + Tx = 1 as well as that duced by fiscal policy. Using parameters T= 1 if there is no technical progress drawnfrom U.S. time series, Sato (1963 p. (Xt /X0 = 1) and Tx = 1 if the economy is 22) concludedthat "for a 10 percent adjust- alwayson a steady-statepath. In the experi- ment [in capital]4 years must pass; for a 50 ments describedhere and in Section III, we percent adjustment,30 years; for a 70 per- choose initial conditions so that TT ='X cent adjustment, 50 years; and for a 90 13 percent adjustment,100 years."2 The adjustmentprocess displayedin Fig- Figure 2 providesour version of R. Sato's ure 2 is indeed very lengthy, with transi- experiment:we explore how the economy tional intervalsclose to those described by evolves from a lower-than-steady-statecapi- Sato (1963) in the sentences quoted above, tal stock, ratherthan the consequencesof a even though our experimentsdiffer some- shift in the saving rate. We choose our what in their details.4 The capital and out- initial condition so that transitionaldynam- put measures displayedin these figures are ics, if fully completed, will explain one-half the transitional-dynamicscomponents (i.e., of the sevenfold per capita output growth the transformed variables kt = Kt /MtXt observed over 1870-1970 in the United and yt = Yt/MtXt). Further, we express States. That is, we know that cumulative each variableas a fractionof its steady-state output variation Y,/ Y0 can alwaysbe sub- value, to aid the reader in interpretingthe dividedinto exogenousgrowth (X, /X0) and pace and pattern of transitionaldynamics. transitionaldynamics F(k,)/F(ko) accord- The transitional dynamics are protracted: ing to Y, / Yo = (XM/XO)F(kO)/F(ko). We the half-life of output is about 30 years. choose ko so that F(k*)/F(ko)=xU and Further, the growth rate of output arising X, /XO = Vi (i.e., so that transitional dy- from the transitionaldynamics is quantita- namicsand technicalprogress together yield tively importantfor a lengthy period: it av- a level of output in 1970 that is seven times erages 3.2 percent over the first ten years higher than in 1870). More generally, we and then 1.7 percent, 1.1 percent, and 0.8 define percent over the next three ten-year inter- vals. Atkinson's(1969) experimentsinvolved a T= log(F(kt)/F(k0))/log(Yt / YO) model that admitted capital-augmenting technical change, so that the asymptotic as the fraction of cumulative growth ex- share of capitalwould be drivento zero and plained by transitionaldynamics. We corre- no steady-stategrowth path existed. Atkin- spondinglydefine son showed that the model might neverthe- less be consistent with the observed small tx = log(Xt /X0) /log(Yt / YO) movementsin the share of capital over 100- year periods. Thus, constancyof these fac-
2FollowingSolow (1956), Ryuzo Sato workedwith a modelwithout depreciation. Kazuo Sato (1966)showed that the saving specificationI, = sY, led to dramati- 3Our measure of the fraction of growth explained by cally faster transitionswhen depreciationwas intro- transitional dynamics is not the same as the fraction of duced. However,Kazuo Sato's results were much the growth accounted for by factor movements in growth same as RyuzoSato's when the savingspecification was accounting. The difference between these two concepts (7). For this reason, the two Sato experimentsled to is clear along the steady-state path. The latter is the consensusview that plausibleversions of the Solow yKy y'/ yy=y9: this is always greater than zero model could generateprotracted transitional dynamics. and is less than 1 unless there is no technical progress The fact that the speed of convergencetoward the (yx = 1). In contrast, the fraction of growth explained steady state depends on which arbitrarydefinition of by transitional dynamics is zero. savings rate is held fixed clearly indicates that the 4We use the parameter values listed at the begin- savingsrate must be made endogenous. ning of Section III and a saving rate s= 0.12. 914 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993
0.1 I 0 I I o I
0.08 + 4 + ( 0.06 0- 0 0 cc~ 0 + .o + o 0 * 0 4 0 *+ + 0.04 0 +* +++ 0+. + 00 0 +0 0 o O 00 + 0 0~++ gO 0 '0* %t++ 0++ + *+ 00 t~~~-0.0 ~~ . -* + 0.04 00 +0. 0~~ 0 0 ~ l +* 0+0 00 *L e . 0 - * 0 4O' 0.02 0 * + * + 0.0 0 + 0.0
-0.04-
-0.061 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Log of Real Per Capita GDP
FIGURE 3. ECONOMIC GROWTH AND INITIAL INCOME LEVEL
Key: O = 1950-1960; * = 1960-1970; + = 1970-1980. tor shares alone could not be used to judge Figure 3 shows no tendencyfor a low initial the adequacyof models of technical change level of income to be associated with high and accumulation. growthover the subsequentinterval. Taken together, the Sato and Atkinson This fact is often taken to be a strong experiments suggest that the steady state refutationof the neoclassicalmodel, but we need not be the full story about the growth are skeptical about relying on it in a world of nations and, as well, that transition- with potential heterogeneity in production al dynamicscould be a key component of possibilities, preferences, and public poli- observed growth experiences (see, for in- cies. The basis for our skepticism can be stance, LawrenceSummers, 1978 p. 23). illustratedby adding such heterogeneityto Solow's (1956) model and specializing to B. The ConvergenceImplication a Cobb-Douglasproduction function, with a being labor's share. Solow's difference Even if the process of convergence is equation then is: relativelyslow, the neoclassicalmodel does imply that it should ultimately occur; the (12) Kj,t+,-Kjt=sjAjKlta (NtjXt) Sato experimentssuggest that several dec- ades of economic data should be sufficient where j indexes countries.The dynamicsof to detect convergence.That is, other things transformedcapital are: equal, countries that begin with a relatively low capital and, hence, low income, should (13) yxyXkjt+l -kjt = sjAjkjl-a( njt) initiallygrow faster. This implicationof the model is sometimes tested using Figure 3, with a stationaryvalue which plots the logarithmof real per capita -1/a output in 1950, 1960, and 1970 versus the kj = [(XyN - 1)/(sjAjn7 )] mean annual growth rate over the subse- quent decade for the countries included in That is, we now can have heterogeneityin the Summers and Heston (1988) data set. both initial (kj o) and terminal(kj) capital Contrary to the convergence prediction, stocks. VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 915
Thus, a country may be growingfast ei- Hence, we believe that one cannot under- ther because it has a low k10 or because it stand the properties of the basic neoclassi- has a high k7: levels and growth rates can cal model without undertakinga detailed, be roughlyuncorrelated as in Figure 3. That quantitative evaluation of its properties is, we have an identificationproblem like when its parametersare restrictedby empir- that which arises when both the demand ical evidence. In our quantitative analysis and supplycurves shift. we pay particularattention to the behavior of prices (real wages and real rates of re- C. PerspectivesFrom Growth turn).This will complementthe information Accounting provided by growth-accountingexercises, which generallyignore price implications. Following Solow (1957) and Denison (1962), a basic accounting frameworkhas III. TransitionalDynamics of Quantities been used to describe economic growthand and Prices to highlight differences across time and countries. The net result of the early We now explore how the neoclassical growth-accountingstudies was (i) to stress model evolves through time if the initial the difficultyof raising the growth rate of capitalstock is low relativeto its steady-state final output by raising the rate of physical- level, againpositing that transitionaldynam- capital accumulation,since a 1-percentage- ics explain one-half of long-term U.S. point change in the growth rate of capital growth.5 However, we require that saving translatesto only a (1- a)-percentage-point be determined by optimal choices of con- change in the growthrate of output;and (ii) sumption over time, rather than fixing its generally to emphasize the importance of fractionof income.6 the "residual factor" in explaining growth Calibration.-The parameters of the (e.g., Solow [1957] estimated that only one- model are chosen as follows. First, we re- eighth of U.S. economic growthover 1909- 1949 was due to physical-capitalaccumula- tion). Since the neoclassicaltransitional dy- namics stem from the capital accumulation, 5We require that F(k*)/F(kO)=V7, with F(k*) computedfrom the familiarsteady-state condition (see these findings suggest their relative unim- the discussionat the beginningof Section IV). This portancein the growthprocess. calculation implicitly assumes that all transitional However, the growth-accountinginvesti- growthtakes place within the first century.This is an gations that followed Solow (1957) pro- approximation,since the neoclassicaldynamics are in- ceeded to make this line of argumentmore finitely-lived.However, since their effect after the ini- tial 100 years is negligible,this approximationis of no tenuous. These investigationsgenerally as- consequence. sign a much more importantrole to capital 6To compute solution paths arisingfrom intertem- accumulation(a surveyof these results can poral optimization,we must take into accounthow the be found in Angus Maddison [1987]). In a capital stock and its marginalvalue (shadow price) evolve throughtime, as is familiarfrom textbooktreat- recent and comprehensive volume, Dale ments of optimal accumulation(see e.g., Phelps, 1966 Jorgenson et al. (1987 pp. 20-1) conclude [essay 3]; Edwin Burmeisterand Rodney Dobell, 1970 that "growth in capital input is the most Ch. 11). In particular,since we are workingin discrete importantsource of growthin value added, time, we are led to a systemof differenceequations in growth in labor input is the next most im- the capital stock and shadow price. Because prefer- ences are concave and technology is convex in the portant source, and productivitygrowth is models we consider,there is a unique competitiveand the least important." In particular, these optimalpath for the economy.This path occurswhen authors estimated that capital input ac- we select the unique,initial value of the shadowprice counts for 46 percent of growthin aggregate for which the solution path satisfiesthe transversality conditionand, hence, capital convergesto the steady- output over 1948-1979, during which time state path. In appendixA of our workingpaper, we the average rate of growth per annum was describe the numerical solution methods we use to 3.42 percent. obtain our results. 916 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993 strict the production function to Cobb- Notable implications of these baseline Douglas form, Y, = A(K,)1-a(nXXtMt)a, trajectoriesare as follows, First, consump- Then, we normalize the level parameterA tion has an increasinglevel and diminishing to unity and choose the labor-shareparame- growth rate, as is familiar from analytical ter a to be 2, which accordswith the esti- results with constant-elasticityutility speci- mates reported in Maddison(1987 table 8). fications. Second, there are three results Second, we choose a constant value of per that are less expected. The pace of conver- capita hours devoted to work, n = 0.2, a gence is very rapid,with one-half of the gap selection which accords with the average between the initial level of output and its post-WorldWar II U.S. experience.7Third, stationary value eliminated in about five we select the depreciationrate 3 to be 0.10, years: rates of economic growth are very which is in the range reportedby Maddison rapid early on and then are sharply re- (1987 table 7). Fourth, we require that the duced. Third, gross investment displays a steady-statereal interest rate be 6.5 percent "hump-shaped"trajectory, although the ra- per annum, which corresponds to the an- tio of investmentto output is alwayshigher nual average real return on equity for the than its long-runvalue. Finally,the value of postwar United States. Fifth, we set the the real interest rate in the early stage of growthrate of the populationto 1.4 percent developmentis very high, approximately105 per year, which is its average value for the percent per year. United States during the period 1950-1980 Thus, a sharply different picture of the (see Barro, 1987 p. 296). Given other pa- neoclassical model's transitional dynamics rametersof the problem,including the pref- emerges from this baseline model. Over the erence parameter oc, we can compute an initial three ten-year periods, the average impliedvalue of the composite discountfac- growthrates of output are sharplydifferent tor By7 (only this compositefactor is neces- from those in the Sato experiment above, sary to carryout the simulations;we do not which are recordedin parenthesesfor com- need to take a stand on the values of f3 and parison:9.6 (3.2), 1.6 (1.7), and 1 (1.1) per- -q separately). cent. The same total long-run growth is loaded sharplytoward the earlyyears in our A. The Baseline Case currentexperiment, since the rate of invest- ment rises as agents postpone consumption The transitionaldynamics of the neoclas- in response to the high real rate of return. sical model are displayed in Figure 4. We Transitional dynamics account for more focus firston the baseline case (o, = 1), which growth in the short run at the expense of is the solid dark line in each panel of this long-rungrowth. and subsequentfigures. The seven panels of the figure depict the paths of output, con- B. Implications of Alternative Preferences sumption, investment, share of output de- voted to gross investment, growth rate of The pace of transitional dynamics is output, the real interest rate, and the real slowed considerablyif we reduce the inter- wage rate. As in the Sato experimentabove, temporal substitutabilityof consumption, output, consumption, and investment are which we do in two different ways in this "detrended" by removing X M,; we also section. First, we reduce the intertemporal express y, c, and i as fractions of their substitutionelasticity (i/o-) to 0.10, which steady-statevalues. is one estimate obtained by Robert Hall (1988). Looking at the dashed lines in the first panel of Figure 4, we see that output grows much more slowly (the half-life is 18 years instead of the five years found earlier o = 7The derivationof this numberftom the Household when 1). Surveypublished by the Bureau of Labor Statisticsis With less intertemporal substitution in discussedin King et al. (1988). preferences, agents choose smoother con- Output
------...... ------~~~~~~~~~~~~~~~~~~~~~~~~~~~~-- - - 0.8 ------0.6 - 0.4-
*2O 5 10 15 20 25 30 Time (years) Consumption Investment 1.5 I .
0.6 -~ 0.5 -
0.4- ______
0.2 0 0 10 20 30 0 10 20 30 Time (years) Time (years)
GrowthRate of Output(percentage) I/Y (percentage) 40 50 30-
20-
10 - -
10 20 30 0 10 20 30 Time (years) Time (years)
Real Interest Rate (percentage) Wage Rate 150, 1
100 0. 0.6 50 -~
0.4 -"
0> 10 20 30 0 10 20 30 Time (years) Time (years)
FIGURE 4. IMPLICATIONS OF INTERTEMPORAL PREFERENCES FOR TRANSITIONAL DYNAMICS
Key: Solid line, o- = 1; dashed line, or = 10; dash-dot line, Stone Geary. For output, x denotes the quarter life; 0, the half life; and *, the three-quarter life. 918 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993 sumptionprofiles (see the second panel), so the cases in Figure 4, the initial interest rate there is a higher initial level and smaller is roughly 105 percent at date t = 1. subsequentgrowth than in the baseline case. In turn,higher consumption requires smaller C. Implication of High Capital Shares investment.In fact, I/ Y rises throughtime, rather than declining as previously. Thus, With a Cobb-Douglas production func- the transitionaldynamics appear more like tion, the share of capital determines how those of the Sato (1963) model explored rapidly the marginalproduct of capital di- above. minishes with the capital stock. Figure 5 Second, we modify momentaryutility to displays how the pace and pattern of the be of a Stone-Geary form, u(Ct/Mt) = transitional dynamics depend on capital's log[(Ct /Mt)- C], where C denotes the share, which is 3 in the baseline model subsistencelevel of per capita consumption. (again the solid line in the figure). The elasticityof marginalutility is When capital's share is -, which we think is a plausible upper bound if one wants to - cr(Ct /Mt) = - (Ct /M)/[(Ct /Mt) -C]. interpret k as physicalcapital, the half-life of output is about eight years, relative to For this utility function, the intertemporal five in the baseline case. In Figure 5, since elasticityof substitutionis C/(Ct /Mt), and the various models have different levels of this value thus varies over time. steady-state capital per unit of efficiency In this model, there is an unstable steady labor, we have chosen to standardizeeach state at the level of sustainablecapital stock by its own steady-statevalue, thus empha- compatible with C. This low-level steady sizing differencesin shapes of paths, rather state resembles somewhat the "poverty than levels. The differencesin steady states, trap" familiarfrom the developmentlitera- however, are revealed in the I/ Y ratio: ture. That is, despite the good investment with a capital share of 2, we see a much opportunities,the country does not invest higher long-run value (about 38 percent, because productionis barely enough to at- instead of the value of 25 for the baseline tend to subsistenceconsumption and to the model). replacement of the depreciated capital The initial real interest rate is roughly34 stock. In the parameterizationexamined, percent. This is much smaller than in the we chose C to be 90 percent of production benchmarkmodel, since the marginalprod- in period 0. The growth rate of output dis- uct increases less sharply with lower than plays a hump-shapedpath, which resembles steady-state capital. However, it remains the evolution of Japan after World War II very high relative to the observations on (see Fig. 8 below) and development financial market returns discussed in the economists' descriptive accounts of the next section. growthprocess. Increasingthe capital share furtherto 0.9 The characterof transitionaldynamics is implies that the model's transitionaldynam- influenced by the fact the elasticity of in- ics are much more protracted(as capital's tertemporalsubstitution is variable, declin- share approachesunity, the adjustmentpe- ing from an initial value of 1/40 to its riod becomes arbitrarilylong). The initial steady-statevalue of 1. In the early stages, real interest rate is much lower than in the the Stone-Gearymodel looks like the "low- earlier two models, since there are much substitution"case discussed above. In the less sharply diminishingreturns to capital. latter stages, it more closely resembles the However, this version of the model also benchmarkcase. producesa wildlycounterfactual level of the These two preference alterationsshare a I/ Y ratio: it is roughly68 percent in both common interest-rateimplication: the more the long and short runs. That is, we here protracted period of high growth rates is modify the production technology so that matched by a more protractedearly period the marginalproduct of capital is less de- with high real interest rates. In all three of pendent on the level of the capital stock. Output
0.8- 0 8 X L L ~~~~~~~~~~~------
0.6 -V 0.4 0 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~------x-- 0.4 --
0.2, 5 10 15 20 25 30 Time (years) Consumption Investment 1 1.5X
026 -,, 1 0.5 .
0 10 20 30 0 10 20 30 Time (years) Time (years)
Growth Rate of Output (percentage) I/Y (percentage) 40 80
30 - 60- 20- 10 -~~~~~~~~4
20 10 20 30 0 10 20 30 Time (years) Time (years) Real Interest Rate (percentage) Time (years) 150 1 ------1 0.8- 100 0.6- 50 0.4
0 10 20 30 0 10 20 30 Wage Rate Time (years) FIGURE 5. IMPLICATIONSOF ALTERNATIVECAPITAL SHARES FOR TRANSITIONALDYNAMICS
Key: Lines represent various values of a (labor share) with capital share being 1- a: solid line, a = 2; dashed line, a-24; dash-dot line, a= . For output, x denotes the quarter life; o, the half life; and *, the three-quarter life. 920 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993
While this makes the rate of returnlower in capital must satisfy[DIF(Kt, nX) - 8] = r*, the earlystage of development,it also brings where D1F(K, nX) is the marginalproduct about a counterfactualvariation in I/ Y. In of capital and r * is the steady-staterate of the next section of the paper, we systemati- interest.9This defines a steady-statepath of cally ask how other modifications of the capital or, equivalently,a level of Kt /(nXt). production structure lead to variations in With given values for X0, yx, r*, and 8, model implications. then we can determinethe level of Ko that is compatiblewith output growingsevenfold IV. Real Interest, Marginal Productivity, over 100 years. When substituted into the and the Production Structure marginalproduct schedule, the capital stock Ko implies a value of the initial real interest When we first presented our results on rate ro. Throughoutthis section, we report the relevance of transitional dynamics to results based solely on technology, which long-termgrowth, there was a recurrentre- are calculatedin this manner. action from audiences. A particularmodifi- The procedure is illustratedin Figure 1. cation of our basic model would be sug- The value of Ko is consistentwith yo being gested as a means of avoidingthe very high one-seventh of its ultimate level (i.e., with marginalproduct of capitalin the,early stage all of output growth being attributableto of development.Then, other supportingevi- transitionaldynamics and none to technical dence for this modificationwould be intro- progress). The rate of return ro is then duced and debated. In thinking through about 800 percent. The value of K', which modifications suggested by a number of is consistent with one-half of growth attrib- seminar audiences and others of our own uted to transitionaldynamics, implies a rate design, we divided them into two groups. of returnof 105 percent. First, there are alternative parameter choices when we work within the basic neo- Cobb-Douglas Production Function.-Ta- classical model's production function. Sec- ble 1 summarizesthe predictedvalues for ro ond, there are modifications of other at- with a Cobb-Douglasaggregate production tributes(such as vintage capital, investment function under different assumptionsabout adjustmentcosts, separate productionfunc- (i) the growth rate of exogenous technical tions for consumptionand investmentgoods, progress;and (ii) parametersof the produc- or internationalcapital flows). We now dis- tion function.10Looking down the third col- cuss each of these types of extensions.
A. Alterations in the Production Function Throughout the paper, we use the operator nota- tion Di to indicate the partial derivative of a function Working with the production function with respect to its argument, so that D1F(K, nX) is the alone, we can extract initial values of the marginal product of capital. For functions of a single variable, such as F(k), we simply write DF(k) for the real interest rate without specifyingprefer- derivative. ences.8 Our procedure is as follows. We 10When technology is Cobb-Douglas, there is a sim- require that there is a time-invariantpro- ple relationship between the real interest rate and the duction function of the form, Yt= capital-output ratio: rt = (1 - a)Yt /Kt - S. This rela- = is the rate tion shows that the behavior of the capital-output F(Kt, nXt), where Yx Xt /Xt- 1 model predicts a signifi- In a ratio is also problematic, the of growth of technical change. steady cant increase of Kt / Yt over time which contrasts with state, we know that the marginalproduct of the small variation suggested by the data for this ratio (see e.g., Romer, 1987). Although the puzzling behav- ior of the real interest rate and the behavior of the capital-output ratio are two sides of the same coin, we choose to emphasize the real-interest-rate implications 8Lucas (1990) also uses this property to discuss the for two reasons: (i) the information available about neoclassical model's predictions for international capi- capital-labor ratios is restricted to few countries and tal flows. Since the computations discussed here are short time periods; and (ii) there are substantial mea- independent of the rate of population growth we treat surement problems associated with the capital-stock population as constant throughout this section. data. VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 921
TABLE 1-BEHAVIOR OF REAL INTEREST RATES IN THE NEOCLASSICAL MODEL
Real interest rate (percentage) 100 years ago Percentage Percentage (Cobb-Douglas production function) of growth growth rate duedue to to of ofexogenous exogenous Depreciation real~~~~~~~~~~~~~~End-of-period interest rate transitional technical rate Labor Capital share dynamics progress Baseline supply rT= rT= (T) (Y"- 1) model i= 0 5 = 0.25 No = 0.36 ak = 0.5 a/k = 0.9 4 percent 9 percent 0 2 6.5 6.5 6.5 43.5 6.5 6.5 4.0 9.0 25 1.5 33.6 17.2 58.3 131.4 16.8 7.4 27.0 40.3 50 1 105.5 45.5 195.5 364.2 33.7 8.4 88.0 123.0 75 0.5 295.6 120.4 558.4 980.1 61.0 9.4 294.3 341.9 100 0 798.5 318.5 1,518.5 2,609.2 105.5 10.5 676.0 921.0
TABLE 2-REAL INTEREST RATE (r0) AND CAPITAL SHARE (ak ) 100 YEARS AGO (CES PRODUCTION FUNCTION WITH ELASTICITY OF SUBSTITUTION P)
Percentage Percentage of growth growth rate due to of exogenous transitional technical 0.9 p = 0.6 p = 1.1 p = 1.25 dynamics progress
('P) YX rO ako rO ako rO ako rO ako 0 2 6.5 33.3 6.5 33.3 6.5 33.3 6.5 33.3 25 1.5 34.9 36.8 39.2 51.8 32.6 30.3 31.2 26.5 50 1 96.4 40.2 78.1 65.2 117.6 27.2 145.9 19.0 75 0.5 216.4 43.3 114.5 74.8 454.4 23.9 1,523.0 10.7 100 0 430.7 46.3 145.6 81.8 2,193.2 20.4 2,993,474.3 1.6 umn (labeled "baselinemodel"), we see that to a wide range of experimentswith Cobb- the real rate increases as the fraction of Douglas technologies:transitional dynamics economic growth attributed to transitional cannot account for a large fraction of the dynamics increases (as the fraction at- expansionin output without generatingim- tributed to technical progress falls). This plausiblevalues for the real interest rate in column accords with the marginalproduct the beginningof the period. In order for all schedule shown in Figure 1 (the real rate is the output expansionto be associatedwith roughly800 percentwhen all growthis tran- transitionaldynamics, the real interest rate sitional dynamics)and with the dynamicex- one century ago should have been higher periments displayed in Figure 4 (the real than 33 percent, unless we postulate an rate is roughly 105 percent when growth is implausiblyhigh share of capital in produc- equallysplit between technicalprogress and tion. For one-half the growth to be so ex- transitionaldynamics). plained, the real rate should have been 100 Columns 2, 4, and 5 consider perturba- percent or higher. tions of this baseline scenariowhich involve different rates of depreciation, capital CES Production Function.-Table 2 ex- shares, and terminal real interest rates. In amines the real-rate implications of CES column3, per capitahours worked are taken (constant elasticity of substitution)produc- to be 0.36 in the beginning of the period tion functions with elasticities of substitu- and 0.2 at the end of the period, so as to tion in production (p) different from the reflect the decrease in hours devoted to Cobb-Douglascase (unitary elasticity). All marketwork that occurred in the last cen- economies share the same terminal capital tury (see Maddison,1987 table A-9). stock, kT = 100. The remaining two CES Table 1 makes clear that the tension that parameters are chosen so that rT is 6.5 we identifiedin the last section carriesover percent and the share of capitalin output at 922 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993
TABLE 3-ANNUAL REAL RATES OF RETURN: SUMMARY STATISTICS, U.S. SECURITIES, 1926-1987
Average change Average real in real rate Standard Series rate of return of return error
Common stocks 6.65 0.0100 0.40 Small stocks 8.80 - 0.0024 1.07 Corporate bonds 1.83 0.0019 0.17 U.S. Treasury bills 0.42 0.0001 0.04 Long-term government bonds 1.18 0.0004 0.23
Notes: Units are percentage points. The first column reports the geometric average of returns. The last column reports Newey-West standard errors (Whitney Newey and Kenneth West, 1987) associated with the statistic reported in column 2. Source: Ibbotson and Sinquefield (1988).
time T is 3. Thus, all the economies have share of capital decreases by almost three- the same steady-state attributes (capital fold over a century). stock, real rate, and factor shares) but dif- ferent values of p. A Little Historical Evidence.-In dis- With nonunitaryelasticity of substitution cussingthe implausibilityof the neoclassical in productionthe capital share is no longer model's implicationsto this point, we have constantover time when the economy is not relied on the reader's sense that constancy on a steady-state path. It decreases over of the real interest rate over time is one of time when p < 1, and it increases for p > 1. the stylized facts of economic growth (see Table 2 shows that shifting p from 1 can Nicholas Kaldor [1961], Solow [1970], and moderate the value of ro. Suppose, for ex- Romer [1988]for discussion).12 Since this is ample, that transitional dynamics are the not strictly true, we present a table which full story(no exogenousproductivity growth) may aid the reader in thinking about the and we consider p = 0.6 (which coincides range of variation.Table 3 provideshistori- with Lucas's [1969] estimate). The associ- cal informationon alternativefinancial rates ated value of ro is 145.6, which is roughly of return for the United States over 1926- six times smallerthan that associatedwith a 1987, based on data from Roger Ibbotson Cobb-Douglasproduction." However,vary- and Rex Sinquefeld(1988). While there are ing the elasticity of substitution generates differences in average returns across assets implausible implications for the share of of varyingrisk shown in Table 3, there are capital in production(e.g., with p = 0.6, the relativelyminor differences across time. One cannot reject the hypothesisthat the mean rate of return for these various assets have been constant during this period. Further, "One might expect that with elasticitiesof substitu- long-periodevidence from Homer (1963) in- tion lower than 1 the value of ro would be higherthan dicates that rates of return were not much that associatedwith Cobb-Douglasproduction. This is differentmany centuriesago from their cur- not necessarilytrue as the last line of Table 1B shows: rent levels (see King and Rebelo [1990b]for without technologicalprogress, a decrease in the elas- ticity of substitutionfrom 0.9 to 0.6 actuallydecreases additional discussion). Looking backward ro. Whenwe lowerp the value of Ko (associatedwith a sevenfoldincrease in output) increases. If the marginal product schedule were independent of p, this would lead to a decrease in the real interest rate; but the 12This constancy led Robert Coen and Bert G. marginalproduct schedule is shifted by the decreasein Hickman(1987)-in their version of the neoclassical p so that the value of ro may increase, decrease, or growthmodel-to postulatethat entrepreneursseek to remain the same, depending on the combinationof earn a constant real rate of return rather than to these two effects. maximizeprofit. VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 923 over various stretches of time, it is simply implications. The theoretically elegant as- impossibleto find the magnitudeof interest pect of Solow's (1959) model is that vintage rate variation that is suggested by the re- capital can be aggregatedto a version of the sults of the last two sections. Thus, we must baseline model with a new parametersum- limit the role of neoclassical transitional marizing the vintage-capital effect. Other dynamics(choose low values of TT) in ex- empirically plausible shifts in parameters plaininglong-term economic growth. which we explored above were of minor consequence for the rate-of-returnimplica- B. Close Relatives of the Basic tion; this is true also for vintage capital. Neoclassical Model Figure 6 depicts some results from the al- ternativemodels of this section: the bench- In this section, we consider whether our mark (solid line) is a vintage-capitalmodel, basic results-which suggest a modest role which is visually indistinguishablefrom the for the neoclassicalmodel's transitionaldy- baseline model in Figure 4. namics-are robustwhen we alter the basic model in several ways that are relatively The Neoclassical Two-SectorModel.-One standard in applied research in macroeco- might think the results of Section III above nomics. We consider (i) vintage capital;(ii) are peculiarto the one-sector nature of the differentproduction technologies for capital model. Figure 6 sheds light on this conjec- and consumptiongoods; (iii) investmentad- ture. It summarizesthe adjustmentpath for justmentcosts; and (iv) internationalcapital a two-sector model in which both produc- flows.13To conserve on space, the discus- tion functions are Cobb-Douglaswith level sion below is deliberately impressionistic; parametersnormalized to 1. The labor share details are given in our workingpaper (King in the capital sector is taken to be 0.5. The and Rebelo, 1990b),which is availableupon labor share in the consumptionsector was request. chosen so that, along the steady-statepath, the aggregate share of labor is 2 (this im- Vintage-Capital Effects.-Discussions of plies a labor share for the consumptionin- the convergence hypothesis, such as those dustry of 72 percent). The remaining pa- of WilliamBaumol (1986) and BradfordDe rameterscoincide with those of the baseline Long (1988), suggest that a key element in model. The initial capital stock was chosen, the convergenceprocess may be the embod- as before, so that output increasessevenfold ied natureof technicalprogress. One partic- over the period considered.14Once again, ular idea is that countrieswho rebuilt their the dynamicsof this economy are remark- capital stock after World War II, such as ably similar to those of the baseline (or Germany and Japan, were able to grow vintage-capital)one-sector model. faster by virtue of investmentin new capital Further,separating out the capital sector vintages. However,altering the basic model and givingits productionfunction a greater of Section I to view technologicalprogress capital share still generates implausibleval- as embodied,along the lines of Solow(1959), ues for ro. As in the one-sector model, does not mitigate the model's interest-rate plausiblevalues for ro requirethat the share of capital in the productionfunction of the capital sector be close to 1.
"Another common version of the neoclassical model involves making labor supply endogenous. We did not pursue this alteration of the model since the near- steady-state dynamics studied in King et al. (1988) 14The value of Ko was found by trial and error. It indicate that, for standard preferences, when capital is cannot be computed directly as in the one-sector model, below its steady-state value, labor supply is greater since output, given by Yt = p,It + Ct (pt is the relative than in the steady state, leading to higher values of the price of investment), depends on the allocation of real interest rate than those for the exogenous-labor- factors of production between the consumption and supply models that we study. capital sectors. Output
0.8 -
0.4 -
0.O 5 10 15 20 25 30 Time (years) Consumption Investment
0.8- . - 0.6 -0.6 /
0.2 0.2 Q20 10 20 30 0 10 20 30 Time (years) Time (years)
Growth Rate of Output (percentage) I/Y (percentage) 40 50
30
20 -
10
00 () 10 20 30 0 10 20 30 Time (years) Time (years)
Real Interest Rate (percentage) Wage Rate 150 1------0.8
0.6-
0.4
0 , ^ . , 0.2 0 10 20 30 0 10 20 30 Time (years) Time (years)
FIGURE 6. IMPLICATIONS OF ALTERNATIVE TECHNOLOGIES FOR TRANSITIONAL DYNAMICS
Key: Solid line, vintage-capital model; dashed line, two-sector model; dash-dot line, adjustment-costs model. For output, x denotes the quarter life; o, the half life; and *, the three-quarter life. VOL.83 NO. 4 KINGAND REBELO:TRANSITIONAL DYNAMICS AND GROWTH 925
Investment Adjustment Costs. -Costs of with the marginal product of capital (the changing the capital stock have been partial derivative)denoted D1F( ). stressed as determinantsof investment de- The introduction of adjustment costs mand (e.g., Fumio Hayashi, 1982) and brings in two conflictingeffects on the real macroeconomicequilibrium (e.g., Andrew interest rate. First, the fact that the cost of Abel and OlivierBlanchard, 1983). To place increasing capital by an extra unit is now discipline on this investigation,we exploit higher than 1 [q-1 = 1 + h(Z,-I/K,-,)+ the link drawn by Hayashi between adjust- Dh(Zt-t IK,XZ, IK,t) > 1] lowers the ment costs and Tobin's q. real interest rate relative to the non-adjust- The introductionof adjustmentcosts al- ment-cost case. Second, the fact that addi- ters (2) and (3) as follows: tional capital lowers adjustment costs [(Zt /Kt)2Dh(Zt 7Kt) 0] contributesto a + highervalue of the real interest rate. Equa- (14) Yt =Ct +Zt [ h(Zt lKt)] tion (17) makes clear that low values of the real interest rate require adjustmentcosts (15) K1+1= zt + (1 - S)Kt. which implylarge values of q. We take estimates of the adjustment-cost where ZJ1 + h(Zt Kt)] is gross investment functionfrom Summers(1981) and then ex- expenditure (It). plore the transitionaldynamics of a general- The adjustment-costfunction h(-) is as- equilibriummodel with adjustmentcosts.16 sumed to be homogeneousof degree zero in Figure 6 displays the transitionaldynamics Z and K. As Hayashi(1982) has shown, this of a version of the baseline model in which makes the theory operationalsince it allows we introducedthe form of adjustmentcosts us to determineTobin's marginal q by mea- described above. While the introductionof suring average q. We assume that h(8) = 0 adjustmentcosts moderatesthe implications and Dh(8) = 0, so that the steady-state cap- of the model for ro, it does so by simultane- ital stock is not affected by the introduction ously generatingimplausibly high values for of adjustmentcosts.15 Finally,we make the Tobin's q. The average value of q in the conventionalassumption that both the ad- first five years of the simulationis 3.7. This justmentcosts and the total cost of investing value is well outside the range of values for are increasingin the rate of accumulation. q estimatedin the investmentliterature (the The value of Tobin's(marginal) q implied highest values of q reported by Summers by this model is
( 16) qt = 1 + h( Zt lKt)
+ (Zt /Kt)Dh(Zt /Kt) 16Summers (1981) showed that there is a linear relationship between Z, /Kt and q, when the and the real interest rate is given by adjustment-costfunction is: h(Zt /Kt) (17) rt= [D1F(kt, n) +(ZtKt) 2Dh(Z tKt) ]qt - (b 12)(Zt lKt a)2 {= b/2)(Zt /Kt when Zt /Kt > a
0O when Zt/Kt 15Without this assumption the adjustment-costs Estimatingthis linear relation, correcting qt for the economywould have a lower steady-statecapital stock effects of taxation,Summers (1981) obtained the fol- than the comparablestandard model. This would con- lowingestimates: b = 32.2, a = 0.088. The requirement tribute to an increase in ro. To make clear that our of no adjustmentcosts at the steady state implies that conclusionsdo not hinge on this effect, we chose to the steady-stateinvestment-capital ratio must be equal eliminateit. to a, so we set 8 equal to 0.088 -(yXyn -1). 926 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1993 [1981]for the United States during the pe- first 10 years. If we reduce the rate at which riod 1933-1978 and by Hayashi [1991] for marginaladjustment costs increase with in- Japan during the period 1956-1981 are vestment, then it is possible to generate barely above 2). Even if one views the em- more plausible values for Tobin's q, but pirical link between adjustment costs and these simply accelerate the importationof Tobin's q as tenuous, the fact that at t = 0 capital. With empirically plausible invest- it is necessary to employ four units of out- ment adjustmentcosts, then, the presence put to install one unit of capital suggests of (possiblyimperfect) internationalcapital that only implausiblylarge adjustmentcosts markets implies a very rapid process of can generate rea:sonablevalues for the real cross-countryconvergence. interest rate. V. A Case Study: Economic Growth After Implications for a Small Open Economy. World War II -One interpretationof the results of Sec- tion III is as yielding predictionsabout how The empiricalpower of economic theory real interest rates and economic activityare is at times best tested by looking at the related to the level of development in the consequences of major events which put absence of international capital markets. into the backgroundfactors other than those Under this interpretation ro - r* is the that one is primarilyinterested in investigat- model's predicted differentialbetween the ing. For example, hyperinflations and rate of return to capital in developed and wartime periods are frequently employed, underdeveloped countries. Assuming that since these involve majorchanges in money the same technologyis availablein all parts creation and government purchases. For of the world, the interest rate associated economic growth, the post-World War II with poor countries is given by the last line experiences of developed countries may of Table 1A, since these have one-seventh similarlyshed light on the transitionaldy- of the outputper capitaof the United States. namics of the neoclassicalmodel. For the baseline model, this interest-rate Barro (1987) points out a clear associa- differentialis 799 percent, which is so large tion between initiallevels of output (in 1950) that it is hard to believe that investment and the wartime positions of countries, flows from rich to poor countrieswould not which is describedin the first panel of Fig- take place, even taking into account such ure 7. He argues that it is plausiblethat the factorsas politicalrisk and transactioncosts. United States lost less capital than Japan In fact, in the simplest open-economy and Germany.With a neoclassical produc- neoclassical model, capital flows would in- tion function, these rankingsof initial capi- stantaneouslyequalize the rate of return in tal stocks would lead to the rankingof 1950 all countries. Investment adjustment costs levels of output depicted in Figure 7, which can eliminatethis unrealisticimplication for shows that countries with the lower initial capital flows. However, since the real in- levels of output displayhigher growth,lead- terest rate is pinned down in the world ing to a smaller dispersionof output levels economy, there is an even larger invest- at the end of the interval. ment response than in the closed-economy A plausible interpretation,then, is that adjustment-costsmodel that we just studied. Figure 7 reflects neoclassical transitional In turn, this requiresthat Tobin's q rise by dynamics in economic growth. Under this more than in Figure 6. Further, the entire interpretation, it is notable that Japan process of transitional growth takes place moved from about one-fifth of U.S. per much more rapidly.Replicating the experi- capita output in 1950 to about three-fourths ment in Figure 6 (in which transitionaldy- of the U.S. level in 1980 as a result of namicsexplain only one-half the cross-coun- capitalaccumulation. To investigatewhether try differences in initial output levels), we the broad outlines of Figure 7 can be ex- find that the initial Tobin's q is 8.8, that the plained with neoclassical transitional dy- half-life of output is 19 years, and that the namics, we study the evolution of the average growth rate is 4 percent over the Japanese economy within the baseline VOL.83 NO. 4 KINGAND REBELO:TRANSITIONAL DYNAMICS AND GROWTH 927 8 10 7 - 6 -United States U Jpa 05- Germany06 A~~~~~~~~~~~ - Japan '~~~~~~ '~~ Germany 24 United States 1 2 - 1950 1960 1970 1980 1950-1960 1960-1970 1970-1980 FIGURE 7. POSTWARGROWTH IN THREE COUNTRIES model that underlies Figure 4. To this end, if neoclassicaltransitional dynamics are be- we assumethat (i) the United States was on hind the exceptional postwar Japanese its steady-statepath in 1950; and (ii) Japan growth,then the 1950 interest rate in Japan and the United States have the same tech- should have been near 500 percent, given nology. We then choose the 1950 Japanese the modest amount of internationalcapital capital so that its output per capita was 19 flows towardJapan in that period. percent of the U.S. level. Figure 8 shows the transitionaldynamics VI. Conclusion of the baseline model from this initial con- dition. In our baseline model, nearly all of For three decades, the basic neoclassical the Japanesetransitional growth would have model of capital accumulation,in its various actually been accomplished in the first versions, has been the central framework decade of the postwarperiod. However, as for most researchthat relates to the process Lawrence Christiano (1989) stresses, the of economicgrowth. Indeed, for this reason, Japanese growth actually peaks in the sec- it is frequently referred to as the "growth ond postwar decade, and it is necessary to model." A central feature of this model is introduce a Stone-Gearyutility function to its assumptionof diminishingreturns to the capturethis feature of the data.'7 However, reproduciblefactor of production,physical capital. Under savings specificationsas dif- ferent of those of Solow (1956) and those of Cass (1965) and Koopmans(1965), diminish- ing returnsto capital assures that there is a 17Christiano (1989 p. 14) takes Japanese output in steady-stategrowth path toward which the 1946 as about 47-percent below trend, extrapolating economyconverges. from pre-WorldWar II data. His initialcondition is 12 The paths from a given percent of steady-statecapital; ours is 0.65 percent. capitalstock to the steady-stategrowth path Hence, his computationsimply an initial interest rate -transitional dynamics-are well knownto of about40 percent,while we find a muchhigher value. most economistsin qualitativeform and are Log(Output) 4 3 - 2 - 0 5 10 15 20 25 30 Time (years) Log(Consumption) Log(Investment) 4 ,- -, 3 , , 3 2--- 2 -1 0 10 20 30 0 10 20 30 Time (years) Time (years) Growth Rate of Output (percentage) I/Y (percentage) 80 50 60- 40 - 40- 30- 0 : 201 0 10 20 30 0 10 20 30 Time (years) Time (years) Real Interest Rate (percentage) Log(WageRate) 500 6 500 , ------1- - - 6 w,~------, 5 - 4- 3- 0 2 0 10 20 30 0 10 20 30 Time (years) Time (years) FIGURE 8. COMPARISON OF SIMULATED U.S. AND JAPANESE GROWTH Key:Solid line, United States; dashed line, Japan. VOL. 83 NO. 4 KING AND REBELO: TRANSITIONALDYNAMICS AND GROWTH 929 importantlyshaped by diminishingreturns ations unlike anything observed. This sug- to capital.When we seek to use the neoclas- gests that an importantcomponent of model sical model's transitional dynamics to ex- validation should be the study of price im- plain sustained cross-countrydifferences in plications, which have generally been rates of economic growth, however, dimin- understressedin the growthliterature. ishing returns to capital turn out to induce Throughoutthe course of this research, majorcounterfactual implications. we have received many comments from On the one hand, when one starts from other researchers,most of which suggested very low capital stocks, diminishingreturns that some straightforwardmodification of to capital induce intertemporal realloca- our setup would readily overcome the cen- tions. Thus, transitional dynamics are im- tral message of this paper. We have tracked portant only for very short periods, unless down most of these leads, but our conclu- agents have low intertemporalelasticity of sion remains unaltered:the transitionaldy- substitution in consumption. Hence, with namics of the familiarmodel of capital ac- conventionalassumptions about preferences cumulation cannot account for important it is difficultto use the neoclassicalmodel to parts of sustained cross-countrydifferences explainsustained differences in growthrates. in rates of economic development. In this regard,we reachedthe opposite con- We view our results as pointingto the use clusion to that suggestedby earlier research of models that do not rely on exogenous of R. Sato (1963)and Atkinson(1969), which technical change-"endogenous-growth" has become part of the popular wisdom as models such as those of Romer (1986) and indicated by Barro's (1987) textbook treat- Lucas (1988)-as the primary vehicle for ment of the economic growthprocess. researchon the processof economicgrowth. On the other hand, even if one makes More generally,our resultssuggest the value agentsvery unwillingto substituteover time, of a quantitativeapproach to evaluatingthe so as to deliver a sustained transitionalpe- adequacy of alternativegrowth paradigms. riod, interest rates or asset prices will dra- The neoclassicalmodel's qualitativeproper- maticallydisplay the implicationsof dimin- ties are well understoodby most economists, ishing returns.In general, we found that in but we found surprisingnew implications order for transitionaldynamics to be impor- about its properties as a growth model. In tant, the marginalproduct of capital has to newer theoretical frameworks,with general be very high in the early stages of economic propertiesas yet undocumented,the quanti- development. In the basic neoclassical tative approach will also help determine model, for example, this marginalproduct which model predictions are robust and translates directly into an implication for which are tightlydependent on specificeco- the real rate of return, implyingthat it is nomic structure.18In the process of quanti- implausiblyhigh relative to historicalobser- tative evaluation,we thus will gain a sharper vations. Notably, in order for the Japanese understandingof why models succeed or convergence toward the U.S. income level fail in explaining the pace and pattern of in the postwarera to be the result of transi- economic development. tional dynamics,the Japanese real annual interest rate would have been over 500 per- REFERENCES cent in 1950. 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