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