Technical Change and the Aggregate Production Function ⑧ Robert M. Solow The Review of Economics and Statistics, Vol. 39, No. 3. (Aug., 1957), pp. 312-320. Stable URL: http: //links. jstor.org/sici?sici=0034-6535%28195708%2939%3A3%3C312%3ATCATAP%3E2.0.CO%3B2-U The Review of Economics and Statistics is currently published by The MIT Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http: //www.jstor.org/about/terms. html. jstor's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of journal or multiple copies of articles,and you may use content in the JSTOR archive only for your personal, non-commercial use. P Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http: //www. jstor.org/journals/mitpress. html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to creating and preserving digital archive of scholarly journals. For more information regarding JSTOR, please contact support @jstor.org. http://www.jstor.org/ Sat Sep1609:49:422006
TECHNICAL CHANGE AND THE AGGREGATE PRODUCTION FUNCTION Robert m. solow this day of rationally designed econometric draw some crude but useful conclusions from udies and super-input-output tables, it the results takes something more than the usual"willing suspension of disbelief"to talk seriously of the Theoretical basis aggregate production function. But the aggre- will first explain what I have in mind gate production function is only a little less mathematically and then give a diagrammatic legitimate a concept than, say, the aggregate exposition. In this case the mathematics seems consumption function, and for some kinds of simpler. If e represents output and K and L long- run macro-models it is almost as indis- represent capital and labor inputs in"physical pensable as the latter is for the she ggregate pr long as we insist on practicing macro-economics can be written as all need aggregate relationshi Q= F(K, L; t) Even so, there would hardly be any justific- The variable t for time appears in Fto allow I had no novelty to suggest. The new wrinkle using the phrase"technical change""as a short I want to describe is an elementary way of f hand expression for any kind of shift in the segregating variations in output per head due to s production function. Thus slowdowns, speed technical change from those due to changes in ups, improvements in the education of the labor the availability of capita head. Naturally, force, and all sorts of things will appear as every additional bit of information has its "technical change. price. In this case the price consists of one new It is convenient to begin with the special case hhik required time series, the share of labor or prop- of neutral technical change. Shifts in the pro- nen erty in total income, and one new assumption, duction function are defined as neutral if they that factors are paid their marginal products. leave marginal rates of substitution untouched Since the former is probably more respectable i but simply increase or decrease the output at than the other data I shall use, and since the i tainable from given inputs. In that case the latter is an assumption often made, the price production function takes the special form may not be unreasonably high Before going on, let me be explicit that I e=A(t)f(K, L) would not try to justify what follows by calling and the multiplicative factor A(t)measures the on fancy theorems on aggregation and index cumulated effect of shifts over time. Differenti numbers.Either this kind of aggregate eco- ate(ra) totally with respect to time and divide nomics appeals or it doesn't. Personally I be- by Q and one obtains long to both schools. If it does, i think one can af I owe a debt of gratitude to Dr, Louis Lefeber for sta- Q. k0 a Q Leontief, and Schultz for stimulating suggestions where dots indicate time derivatives. Now de- Mrs. Robinson in particular has explored many of the profound difficulties that stand in the way of giving any fine zek aQ K OQ L precise meaning to the quantity of capital ("The Production Function and the Theory of Capital, Review of economic tive shares of capital and labor, and substitute Studies, Vol. 21, No. 2), and I have thrown up still further bstacles(ibid, Vol 23, No. 2). Were the data available, it in the above equation (note that aQ/aK would be better to apply the analysis to some precisely de- A of/aK, etc. )and there results gives some notion of the way a detailed analysis would Q A (2) 3
TECHNICAL CHANGE AND PRODUCTION FUNCTION 3I3 From time series of e/, wm, K/K, w, and so that if we observe points in the(q, k)plane, their movements are compounded out of move- L/L or their discrete year-to-year analogues, ments along the curve and shifts of the curve we could estimate A/A and thence A(t) itself. In Chart I, for instance, every ordinate on the Actually an amusing thing happens here. curve for t I has been multiplied by the same Nothing has been said so far about returns to factor to give a neutral upward shift of the scale. But if all factor inputs are classified production function for period 2. The problem i either as K or L, then the available figures al- is to estimate this shift from knowledge of ays show wx and wz adding up to one. Since points Pi and P2. Obviously it would be quite we have assumed that factors are paid their misleading to fit a curve through raw observed marginal products, this amounts to assuming points like Pi, P2 and others. But if the shift the hypotheses of Euler's theorem. The cal- factor for each point of time can be estimated culus being what it is, we might just as well as- the observed points can be corrected for techni sume the conclusion, namely that F is homo- cal change and a production function can then geneous of degree one. This has the advantage be found. 2 of making everything come out neatly in terms of intensive magnitudes. Let Q/L=9, K/L CHART I k, wer=I-wk; note that q/q=Q/0-L/L ↑:2 etc, and(2) becomes A k (2a) Now all we need to disentangle the technical q, hange index A(t) are series for output per man hour, capital per man hour, and the share capital. So far i have been assuming that technical change is neutral. But if we go back to(I)and carry out the same reasoning we arrive at some thing very like(2a), namely q F (2b) The natural thing to do, for small changes, g F ot is to approximate the period 2 curve by its tan It can be shown, by integrating a partial dif- gent at P2 (or the period I curve by its tangent ferential equation, that if F/F is independent at Pi). This yields an approximately corrected of K and L(actually under constant returns to point Pu, and an estimate for A A/A, namely scale only k/L matters) then(1)has the spe- P12P1/q1. But k,Pi2=q2- aq/ak a k and function are neutral. If in additio0ni/ F is con-3/ak△kand△A/A=P1/y=△9、° cial form (ra)and shifts in the production hence P12P1=q2-q1-ag/ak4k=4 stant in time, say equal to a, then A(t)=eat aq/ak(k/g)Ak/k=a g1g-zex A k/k which in discrete approximation A(t)=(I+a) is exactly the content of(2a). The not-neces The case of neutral shifts and constant re- sarily-neutral case is a bit more complicated turns to scale is now easily handled graphically. but basically similar The production function is completely repre- Professors Wassily Leontief and William Fellner inde- ented by a graph of q against k(analogously mation could in principle be improved. After estimating to the fact that if we know the unit-output a production function corrected for technical change(se trouble is that this function is shifting in time, tiong e could go back and use it to provide a isoquant, we know the whole map). The below), or tion to the shift series, and on into further itera-
THE REVIEW OF ECONOMICS AND STATISTICS An Application to the U.S. 1909-1949 closer to the truth than making no correction In order to isolate shifts of the aggregate pro- duction function from movements along it, by CHART 2 use of (2a) or(2b), three time series are needed: output per unit of labor, capital per 94/ unit of labor, and the share of capital. Some rough and ready figures, together with the obvi- ous computations, are given in Table I The conceptually cleanest measure of aggre- gate output would be real net national product But long NNP series are hard to come by, so I have used GNP instead. The only difference this makes is that the share of capital has to in clude depreciation. It proved possible to re- strict the it to pr is an advantage (a) be- cause it skirts the problem of measuring govern- ment output and(b)because eliminating agri- culture is at least a step in the direction of homogeneity. Thus my g is a time series of real private non-farm GNP per man hour, Ken The share-of-capital series is another hodge- drick’ s valuable work ther from various sources The capital time series is the one that will d ad hoc assumptions(such as Gale John ally drive a purist mad. For present pur son,s guess that about 35 per cent of non-farm poses,"capital"includes land, mineral deposits, entrepreneurial income is a return to property) etc. Naturally I have used Goldsmith's esti- did I learn that edward Budd of yale univer mates (with government, agricultural, and consumer durables eliminated). Ideally what CHAH one would like to measure is the annual fow of ital services. Instead one must be content with a less utopian estimate of the stock of capi tal goods in existence. All sorts of conceptual problems arise on this account. As a single ex- ample, if the capital stock consisted of a mil lion identical machi nd if each wore out was replaced by a more durable ma chine of the same annual capacity, the stock of 1 capital as measured would surely increase. But the maximal flow of capital services would be constant. There is nothing to be done about nething must be done about the dle place. Lacking any reliable year-by-year meas- factor shares which will soon be published.It ure of the utilization of capital I have simply seems unlikely that minor changes in this in- reduced the Goldsmith figures by the fraction gredient would grossly alter the final results, of the labor force unemployed in each year, Anothe for which I have not corrected is the thus assuming that labor and capital always: changing le work-week. As the work-week suffer unemployment to the same percentage. and the st This is undoubtedly wrong, but probably gets ices overestimate the input of capital serv-
TECHNICAL CHANGE AND PRODUCTION FUNCTION but I have no doubt that refinement of this and on whether these relative shifts appear to be the capital time-series would produce neater re- neutral or not. Such a calculation is made in sults Table I and shown in Chart 2. Thence by arbi In any case, in(2a)or(2b) one can replace trarily setting A(Igog )=I and using the fact che time-derivatives by year-to-year changes that A(t+I)=A(t)(I+A A(t/A(t))one and calculate A q/- wr A k/k. The result can successively reconstruct the A(t) time is an estimate of A F/F or AA/A, depending series, which is shown in Chart 3 TABLE I.-DATA FOR CALCULATION OF A() 哪,路 △A/A 46,42 I64504 I9I4 r75371 I481I88 I7835 86,679 95460 I254 r235 271o89 I,226 25 I930 r,r97 I 2II 26237 I298 I349 357 340 4 59789 573 47 58048 357 2,940 377 r94 270,063 252779 356 r943 2 I.6g2 44 6,235 I,8I2 261,472 25232 I296 I 850 6 8,5I 244,632 3I2 327 “器黑学能我器题 United States (Boston and New York Column (a) ol. 3(Princeton, 1956), 20-21, sum of columns Column (4): soare Distrietiton of ncom., ned of mhe american satistics a'ssoidtion, 'von.d 4 g une kgsa Igh-nges id epre clain ollars,Kendrick's data, reproduced in The Economic Almanac, 490. Column(: E payed saps ser man 3t 670 divided by Kendricks man hour series Column
THE REVIEW OF ECONOMICS AND STATISTICS I was tempted to end this section with the re- evidence that technical change(broadly int mark that the A(t)series, which is meant to be preted)may have accelerated after Gzg. ter- a rough profile of technical change, at least The over-all result for the whole 4o years is looks reasonable. But on second thought I de- an average upward shift of about I 5 per cent cided that I had very little prior notion of what per year. This may be compared with a figure would be "reasonable "in this context. One of about 75 per cent per year obtained by notes with satisfaction that the trend is strongly Stefan Valavanis-Vail by a different and rather upward; had it turned out otherwise I would less general method, for the period I869-I948. not now be writing this paper. There are sharp Another possible comparison is with the out dips- after each of the World Wars; these, like- put-per-unit-of-input computations of Jacob sharp rises that preceded them, can easily Schmookler 5 which show an increase of some be rationalized. It is more suggestive that the 36 per cent in output per unit of input between curve shows a distinct levelling-off in the last the decades Ig04-I3 and I929-38. Our A(t) half of the I920's. A sustained rise begins again rises 36. 5 per cent between Igog and I934.But in I930. There is an unpleasant sawtooth char- these are not really comparable estimates, since acter to the first few years of the AA/A curve, Schmookler's figures include agriculture which I imagine to be a statistical artifact As a last general conclusion, after which I ill leave the interested reader to his own in The Outlines of Technical Change pressions, over the The reader will note that I have already time, according to Chart 2, the cumulative up Chart 2 A A/A instead of the more general ward shift in the production function was about A F/F. In fact, a scatter of A F/F against, 8o per cent. It is possible to argue that about K/L (not shown)indicates no trace of a rela- one-eighth of the total increase is traceable to tionship. So I may state it as a formal conclu- increased capital per man hour, and the remain- sion that over the period Igog-49, shifts in the ing seven-eighths to technical change. The aggregate production function netted out to be reasoning is this: real gnp per man hour in approximately neutral. Perhaps I should recall creased from $.623 to $I 275. Divide the latter that I have defined neutrality to mean that the figure by 1. 809, which is the I949 value for shifts were pure scale changes, leaving mar- A(t), and therefore the full shift factor for the ginal rates of substitution unchanged at given 40 years. The result is a"corrected"GNP per capital labor ratic man hour, net of technical change, of $ 705 fot only is A A/A uncorrelated with K/L, Thus about 8 cents of the 65 cent ind crease car but one might almost conclude from the graph! be imputed to increased capital intensity, and that AA/A is essentially constant in time, ex- the remainder to increased productivity.t that hibiting more or less random fluctuations about Of course this is not meant to suggest that a fixed mean. Almost, but not quite, for there the observed rate of technical progress would does seem to be a break at about 1930. There have persisted even if the rate of investment is some evidence that the average rate of prog- had been much smaller or had fallen to zero ress in the years I9o9-29 was smaller than that obviously much, perhaps nearly all, innovation he last io average 2 ]4 per cent per year. Even i to be realized at all. One could imagine this ard shift, is moved from the first group to the ' S. Valavanis-Vail, "An Econometric Model of growth d. there is still U.S.A. I869-1953, American Economic Review, Papers and average rate of I. 2 per cent in the first half and Efficiency of the Ameri- I 9 per cent in the second. Such post hoc can Economy, 1869-1938, plitting-up of a period is always dangerous. o For the first half of th Perhaps i should leave it that there of the observed increase some in gNP per man-hour to
TECHNICAL CHANGE AND PRODUCTION FUNCTION 3I7 mation as old-fashioned capital goods are re- The Aggregate Production Function placed by the latest models, so that the capital- Returning now to the aggregate production labor ratio need not change systematical lunction, we have earned the right to write it But this raises problems of definition and meas- in the form (ra). By use of the(practically urement even more formidable than the ones already blithely ignored. This whole area of scale. this can be further simplified to the form interest has been stressed by fellner For comparison, Solomon Fabricant has q=A(1)f(k,), estimated that over the period I87I-I95I which formed the basis of Chart I. It was there about go per cent of th e increase in output per noted that a simple plot of q against h would capita is attributable to technical progress. give a distorted picture because of the shift Presumably this figure is based on the stand- factor A(t). Each point would lie on a different ard sort of output-per-unit-of-input calcula- member of the family of production curves But we have now provided ourselves with an It might seem at first glance that calculations estimate of the successive values of the shift of output per unit of resource input provide factor. (Note that this estimate is quite inde- pendent of any hypothesis about the exact a relatively assumption-free way of measuring shape of the production function. )It follows productivity changes. Actually I think the im- from (3)that by plotting q(t)/A(t) against plicit load of assumptin ats to a if anything the method proposed above is con- single member of the family of curves in Chart I, and we can then proceed to discuss the shape Not only does the usual choice of weights for of f(k, I)and reconstruct the aggregate pro computing an aggregate resource-input involve tion function. a scatter of /A againy duc- something analogous to my assumption of com- shown in Chart 4 petitive factor markets, but in addition the cr terion output a weighted sum of inputs CHART 4 would seem tacitly to assume (a) that technical I change is neutral and (b) that the aggregate production function is strictly linear. This ex- plains why numerical results are so closely parallel for the two methods. We have already verified the neutrality and as will be seen sub gives an excellent fit, though clearly inferior to some alternatives …°“· rS. Fabricant, " Economic Progress and Economic Change, "34th Annual Report of the National Bureau of Economic Research(New York, I954) For an excellent discussion of some of the problems, see M Abramovitz"Resources and Output Trends in the U.S. a2 since I87o, American Economic Review, Papers and Pro ceedings, XLVI (May I956),5-23. Some of the qu here raised could in principle be answered by the method sed here. For example, the contribution of improved Considering the amount of a priori doctoring as levels of skilled labor as separate inputs. I owe to which the raw figures have undergone, the fit is W. Schultz a heightened awareness that a lot of remarkably tight. Except, that is, for the layer appears as shifts in the production function must of points which are obviously too high. These represent improvement in the quality of the labor input, maverick observations relate to the seven last portant kind. Nor ought it be forgotten thar on of an im- and therefore a result of real capital format even straight years of the period, I943-49. From the way technical progress has a cost side. they lie almost exactly par rallel to the main
3 THE REVIEW OF ECONOMICS AND STATISTICS scatter,one is tempted to conclude that in I943 ishing returns. As for the possibility of ap- the aggregate production function simply proaching capital-saturation, there is no trace shifted. But the whole earlier procedure was de- on this gross product level, but even setting signed to purify those points from shifts in the aside all other difficulties, such a scatter con function, so that way out would seem to be fers no particular license to guess about what closed. I suspect the explanation may lie in some happens at higher K/L ratios than those ob systematic incomparability of the capital-in- served use series. In particular during the war there As for fitting a curve to the scatter, a Cobb- was almost certainly a more intensive use of Douglas function comes immediately to mind capital services through two- and three-shift but then so do several other parametric forms, operation than the stock figures would show, with little to choose among them. 0 I cant help even with the crude correction that has been feeling that little or nothing hangs on the choice applied. It is easily seen that such an underesti- of functional form, but I have experimented mate of capital inputs leads to an overestimate with several. In general I limited myself to of productivity increase. Thus in effect each of two-parameter families of curves, linear in the the affected points should really lie higher and parameters(for computational convenience oward the right. But further analysis shows and at least capable of exhibiting diminishing that, for the orders of magnitude involved, the returns(except for the straight line, which on net result would be to pull the observations this account proved inferior to all others closer to the rest of the scatter The particular possibilities tried were the fol- At best this might account for I943-I945 There remains the postwar period. Although it is possible that multi-shift operation remained q=a+ Bk fairly widespread even after the war, it is un q= a+B log k (4b) likely that th ld be nearly B/k plain the whole discrepancy One might guess 108 q=a-E log q=a+B that accelerated amortization could have re- sulted in an underestimate of the capital stock after I945. Certainly other research workers. Of these,(4d) is the Cobb-douglas notably Kuznets and Terborgh, have produced (4c and e )have upper asymptotes the capital stock estimates which rather exceed logarithmic(4b)and the hyperbolic(4c)must cross the horizontal axis at a positive value of Goldsmith's at the end of the period. But for k and continue ever more steeply but irrelevant- the present, I leave this a mystery In a first version of this paper, I resolutely y downward which means only that some posi- let the recalcitrant observations stand as they forthcoming, but this is far outside the range were in a regression analysis of Chart 4, main- of observation);(4e)begins at the origin with ly because such casual amputation is a practice i deplore in others. But after some experimen- a phase of increasing returns and ends with a tation it seemed that to leave them in only led phase of diminishing returns-the point of to noticeable distortion of the results. So, with inflection occurs at k= B/2 and needless to some misgivings, in the regressions that follow say all our observed points come well to the The results of fitting these five curves to the 949. It would be better if they could be other- scatter of Chart 4 are shown in Table 2 wise explained away. The correlation coefficients are uniformly so Chart 4 gives an inescapable impression of high that one hesitates to say any more tha curvature, of persistent but not violent dimin- of the same problem in a different con- It is cheering to note that Professor nd in Prais and Houthakker, The Analysis icion that the postwar has s(Cambridge, England, I955),82-88. See imates of the Engel ends and Cycles in Econom of Economic Studies, No. 52(I952-53) York,I956),92
TECHNICAL CHANGE AND PRODUCTION FUNCTION This has been done for the linear, semilog urve ithmic, and Cobb-Douglas functions. The re- -34 sults strongly confirm the visual impression of diminishing returns in Chart 4, by showing the bcde linear function to be a systematically poor fit. 9I76I89964 As between(4b) and(4d)there is little to 353999 that all five functions even the linear one ar about equally good at representing the genera It has already been mentioned that the ag shape of the observed points From the corre- gregate production function shows no signs of lations alone, for what they are worth, it ap- levelling off into a stage of capital-saturation pears that the Cobb-Douglas function(4d) The two curves in Table 2 which have upper and the semilogarithmic(4b)are a bit better asymptotes(c and e) happen to locate that than the others 11 asymptote at about the same place. The limit Since all of the fitted curves are of the form ing values of q are, respectively, 92 and 97 g(y)=a+B h(=), one can view them all as Of course these are both true asymptotes, ap- linear regressions and an interesting test of proached but not reached for any finite value of goodness of fit proposed by Prais and Houthak It could not be otherwise: no analytic func ker (ibid., page 5I)is available. If the resid- tion can suddenly level off and become constant uals from each regression are arranged in order unless it has always been constant. but on the of increasing values of the independent var]- other hand, there is no reason to expect nature able, then one would like this sequence to be to be infinitely differentiable. Thus any conclu disposed"randomly"about the regression line. sions extending beyond the range actually ob- A strong"serial "correlation in the residuals, or served in Chart 4 are necessarily treacherous a few long runs of positive residuals alternat- But, tongue in cheek, if we take. 5 as a guess ing with long runs of negative residuals, would at the saturation level of g, and use the linear be evidence of just that kind of smooth de- function(4a)(which will get there first)as a parture from linearity that one would like to lower-limit guess at the saturation level for k, catch. a test can be constructed using pub- it turns out to be about 5-7, more than twice its lished tables of critical values for runs of two present value inds of elements But all this is in terms of ross output whereas for analytic purposes we are interested I It would be foolhardy fo ard a guess about outsider (or maybe even in the net productivity of capital. The differ- nade, however. (a)The natural way to introduce an error about which i do not feel able to make guesses apparent that the error factor will be absorbed into the of existing estimates of depreciation, especially will be au/i+u. If u has zero mean, the variance of the better to conduct the whole analysis in terms of where p is the pinau productivity distribution doesn't hold However, one can say this. Zero net correlation of the u series.(b)Sup- net product random deviation and wm is the share of property income. ginal productivity of capital sets in when gross Then the ariance(Ak/k)var v. Since K/L changes slowly, the mul- depreciation, i.e. when adding some capital estimate when property recei as to lead to an over- adds only enough product to make good the de- stimate of Aa/ A in such a w less than its marginal preciation on the increment of capital itself product(and k is increasing).(c)Errors in estimating A(e) Now in recent years NNP has run a bit over nter in a relatively harmless way so far as the serious and are likely to be large. The effect will of course be with small values significant. For (4a),R=4 for (4b) to bias the estimates of B downward R= I3. The I% critical value in both
32 THE REVIEW OF ECONOMICS AND STATISTICS go per cent of GNP, so capital consumption is od rests on the assumption that factors are paid a bit under Io per cent of gross output. From their marginal products, but it could easily be Table I it can be read that capital per unit of extended to monopolistic factor market output is, say, between 2 and 3. Thus annual Among the conclusions which emerge from a depreciation is between 3 and 5 per cent of the crude application to American data, Igo9-49 capital stock. Capital-saturation would oc- are the following cur whenever I. Technical change during that period was capital falls to 03-05. Using(4b), this would neutral on average happen at K/L ratios of around 5 or higher The upward shift in the production func- still well above anything ever observed. 13 tion was, apart from fluctuations, at a rate of about one per cent per year for the first half of the period and 2 per cent per year for the last This paper has suggested a simple way of half 3. Gross output per man hour doubled over segregating shifts of the aggregate production the interval, with 87/2 per cent of the increase function from movements along attributable to technical change and the re- And this is under relatively pessimistic assumptions maining 1272 per cent to increased use of as to how technical change itself affects the rate of capital mption. A warning is in order here: I have left Ken rick,s gnP data in I939 prices and Goldsmith' s capital 4. The aggregate production function, cor- cock figures in 1929 prices. Before anyone uses the B's of rected for technical change, gives a distinct im ber, it is necessary to convert o and k to a comparable price pression of diminishing returns, but the curva basis, by an easy ture is not violent
