![TECHNICAL CHANGE AND THE AGGREGATE
PRODUCTION FUNCTION *
Robert M. Solow
JN this day of rationallydesignedeconometric
studies and super-input-output tables, it
takes something more than the usual "willing
suspensionof disbelief" to talk seriouslyof the
aggregateproduction function. But the aggre-
gate production function is only a little less
legitimate a concept than, say, the aggregate
consumption function, and for some kinds of
long-run macro-models it is almost as indis-
pensable as the latter is for the short-run. As
long as we insist on practicingmacro-economics
we shall need aggregaterelationships.
Even so, therewould hardly be any justifica-
tion for returningto this old-fashionedtopic if
I had no novelty to suggest. The new wrinkle
I want to describe is an elementary way of
segregatingvariationsin outputper head dueto
technical change from those due to changes in
the availability of capital per head. Naturally,
every additional bit of information has its
price. In this case the price consists of one new
requiredtime series, the share of laboror prop-
erty in total income, and one new assumption,
that factors are paid their marginalproducts.
Since the former is probably more respectable
than the other data I shall use, and since the
latter is an assumption often made, the price
may not be unreasonablyhigh.
Before going on, let me be explicit that I
wouldnot try to justify what follows by calling
on fancy theorems on aggregation and index
numbers.' Either this kind of aggregate eco-
nomics appeals or it doesn't. Personally I be-
long to both schools. If it does, I think one can
draw some crude but useful conclusions from
the results.
TheoreticalBasis
I will first explain what I have in mind
mathematicallyand then give a diagrammatic
exposition. In this case the mathematicsseems
simpler. If Q represents output and K and L
representcapital and laborinputs in "physical"
units, then fhe aggregate production function
can be written as:
Q = F(K,L;t). (I)
The variable t for time appears in F to allow
for technical change. It will be seen that I am
using the phrase "technicalchange"as a short-
hand expression for any kind of shift in the
production function. Thus slowdowns, speed-
ups,improvementsin the educationof the labor
force, and all sorts of things will appear as
"technicalchange."
It is convenientto begin with the specialcase
of neutral technical change. Shifts in the pro-
duction function are definedas neutral if they
leave marginalrates of substitution untouched
but simply increase or decrease the output at-
tainable from given inputs. In that case the
productionfunctiontakes the special form
Q = A(t)f (K,L4) (ia)
and the multiplicativefactorA(t) measuresthe
cumulatedeffectof shifts over time. Differenti-
ate (ia) totally with respectto time and divide
by Q and one obtains
? A DJa.K DIL
=+ A - +A -
Q A DK Q DL Q
where dots indicate time derivatives. Now de-
fine wk -3Q K
andWL= aQ
L
the rela-
DK Q DL Q
tive shares of capital and labor, and substitute
in the above equation (note that DQ/DK=
A Df/3K, etc.) andthereresults:
-+WK-+WL ~~~~~(2)Q A K L
* I owe a debt of gratitude to Dr. Louis Lefeber for sta-
tistical and other assistance, and to Professors Fellner,
Leontief, and Schultz for stimulating suggestions.
1 Mrs. Robinson in particular has explored many of the
profound difficulties that stand in the way of giving any
precise meaning to the quantity of capital ("The Production
Function and the Theory of Capital," Review of Economic
Studies, Vol. 2I, No. 2), and I have thrown up still further
obstacles (ibid., Vol. 23, No. 2). Were the data available, it
would be better to apply the analysis to some precisely de-
fined production function with many precisely defined in-
puts. One can at least hope that the aggregate analysis
gives some notion of the way a detailed analysis would
lead.
[ 3I2 ]](https://image.slidesharecdn.com/solow1957-101108183053-phpapp02/85/Solow1957-2-320.jpg)
Solow1957
- 1. Technical Change and the Aggregate Production Function Author(s): Robert M. Solow Source: The Review of Economics and Statistics, Vol. 39, No. 3 (Aug., 1957), pp. 312-320 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1926047 Accessed: 26/10/2010 15:37 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. 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 a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=mitpress. 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 a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review of Economics and Statistics. http://www.jstor.org
- 2. TECHNICAL CHANGE AND THE AGGREGATE PRODUCTION FUNCTION * Robert M. Solow JN this day of rationallydesignedeconometric studies and super-input-output tables, it takes something more than the usual "willing suspensionof disbelief" to talk seriouslyof the aggregateproduction function. But the aggre- gate production function is only a little less legitimate a concept than, say, the aggregate consumption function, and for some kinds of long-run macro-models it is almost as indis- pensable as the latter is for the short-run. As long as we insist on practicingmacro-economics we shall need aggregaterelationships. Even so, therewould hardly be any justifica- tion for returningto this old-fashionedtopic if I had no novelty to suggest. The new wrinkle I want to describe is an elementary way of segregatingvariationsin outputper head dueto technical change from those due to changes in the availability of capital per head. Naturally, every additional bit of information has its price. In this case the price consists of one new requiredtime series, the share of laboror prop- erty in total income, and one new assumption, that factors are paid their marginalproducts. Since the former is probably more respectable than the other data I shall use, and since the latter is an assumption often made, the price may not be unreasonablyhigh. Before going on, let me be explicit that I wouldnot try to justify what follows by calling on fancy theorems on aggregation and index numbers.' Either this kind of aggregate eco- nomics appeals or it doesn't. Personally I be- long to both schools. If it does, I think one can draw some crude but useful conclusions from the results. TheoreticalBasis I will first explain what I have in mind mathematicallyand then give a diagrammatic exposition. In this case the mathematicsseems simpler. If Q represents output and K and L representcapital and laborinputs in "physical" units, then fhe aggregate production function can be written as: Q = F(K,L;t). (I) The variable t for time appears in F to allow for technical change. It will be seen that I am using the phrase "technicalchange"as a short- hand expression for any kind of shift in the production function. Thus slowdowns, speed- ups,improvementsin the educationof the labor force, and all sorts of things will appear as "technicalchange." It is convenientto begin with the specialcase of neutral technical change. Shifts in the pro- duction function are definedas neutral if they leave marginalrates of substitution untouched but simply increase or decrease the output at- tainable from given inputs. In that case the productionfunctiontakes the special form Q = A(t)f (K,L4) (ia) and the multiplicativefactorA(t) measuresthe cumulatedeffectof shifts over time. Differenti- ate (ia) totally with respectto time and divide by Q and one obtains ? A DJa.K DIL =+ A - +A - Q A DK Q DL Q where dots indicate time derivatives. Now de- fine wk -3Q K andWL= aQ L the rela- DK Q DL Q tive shares of capital and labor, and substitute in the above equation (note that DQ/DK= A Df/3K, etc.) andthereresults: -+WK-+WL ~~~~~(2)Q A K L * I owe a debt of gratitude to Dr. Louis Lefeber for sta- tistical and other assistance, and to Professors Fellner, Leontief, and Schultz for stimulating suggestions. 1 Mrs. Robinson in particular has explored many of the profound difficulties that stand in the way of giving any precise meaning to the quantity of capital ("The Production Function and the Theory of Capital," Review of Economic Studies, Vol. 2I, No. 2), and I have thrown up still further obstacles (ibid., Vol. 23, No. 2). Were the data available, it would be better to apply the analysis to some precisely de- fined production function with many precisely defined in- puts. One can at least hope that the aggregate analysis gives some notion of the way a detailed analysis would lead. [ 3I2 ]
- 3. TECHNICAL CHANGE AND PRODUCTION FUNCTION 3I3 From time series of Q/Q, Wk, K/K, w,, and L/L or their discrete year-to-year analogues, we could estimate A/A and thence A(t) itself. Actually an amusing thing happens here. Nothing has been said so far about returns to scale. But if all factor inputs are classified either as K or L, then the available figuresal- ways show WKand wi. adding up to one. Since we have assumed that factors are paid their marginal products, this amounts to assuming the hypotheses of Euler's theorem. The cal- culus being what it is, we mightjust as well as- sume the conclusion, namely that F is homo- geneotusof degreeone. This has the advantage of makingeverythingcome out neatly in terms of intensive magnitudes. Let Q/L = q, K/L = k, wI, = I - WK; note that q/q = Q/Q - L/L etc., and (2) becomes q=-+WK- (2a) q A k Now all we need to disentangle the technical change index A(t) are series for output per man hour, capital per man hour, and the share of capital. So far I have been assuming that technical changeis neutral. But if we go back to (i) and carryout the same reasoningwe arriveat some- thing very like (2a), namely q = F t + Wk k (2b) q F Zt k It can be shown,by integratinga partial dif- ferential equation, that if F/F is independent of K and L (actually underconstant returnsto scale only K/L matters) then (i) has the spe- cial form (ia) and shifts in the production functionare neutral. If in additionF/F is con- stant in time, say equal to a, then A(t) = el' or in discreteapproximationA(t) (I + a)'. The case of neutral shifts and constant re- turnsto scale is noweasily handledgraphically. The production function is completely repre- sented by a graph of q against k (analogously to the fact that if we know the unit-output isoquant, we know the whole map). The trouble is that this function is shifting in time, so that if we observepoints in the (q,k) plane, their movementsare compoundedout of move- ments along the curve and shifts of the curve. In Chart i, for instance, every ordinateon the curve for t = i has been multipliedby the same factor to give a neutral upward shift of the productionfunction for period 2. The problem is to estimate this shift from knowledge of points P1 and P. Obviously it would be quite misleadingto fit a curve through raw observed points like P1, P., and others. But if the shift factor for each point of time can be estimated, the observedpoints can be correctedfor techni- cal change, and a productionfunction can then be found.> The natural thing to do, for small changes, is to approximatethe period2 curve by its tan- gent at P., (or the period i curve by its tangent at P1). This yields an approximatelycorrected point P1,, and an estimate for A A/A, namely P12P11qj * But k1P12 = q.>- q/3k A k and hence P12PI = q2 - q1- 3q/3k A k = A q - 3q/3k Ak and A A/A = P12P1/qj = A q/q - 3q/3k (k/q) A k/k= A q/q-w, A k/k which is exactly the content of (2a). The not-neces- sarily-neutral case is a bit more complicated, but basically similar. CHIART I q P_t ___ t:2 Q2-- - - - - - - -- -- - - -r- I |I I Ik Q, I~~~~~kk 2 Professors Wassily Leontief and William Fellner inde- pendently pointed out to me that this "first-order" approxi- mation could in principle be improved. After estimating a production function corrected for technical change (see below), one could go back and use it to provide a second approximation to the shift series, and on into further itera- tions.
- 4. 3I4 THE REVIEW OF ECONOMICS AND STATISTICS An Applicationto the U.S.: 1909-1949 In orderto isolate shifts of the aggregatepro- duction function from movements along it, by use of (2a) or (2b), three time series are needed: output per unit of labor, capital per unit of labor, and the share of capital. Some roughand readyfigures,togetherwith the obvi- ous computations, are given in Table i. The conceptually cleanest measureof aggre- gate output would be real net nationalproduct. But long NNP series are hard to come by, so I have used GNP instead. The only difference this makesis that the shareof capital has to in- clude depreciation. It proved possible to re- strict the experimentto private non-farm eco- nomic activity. This is an advantage (a) be- cause it skirts the problemof measuringgovern- ment output and (b) because eliminatingagri- culture is at least a step in the direction of homogeneity. Thus my q is a time series of realprivatenon-farmGNP per manhour,Ken- drick's valuable work. The capital time series is the one that will really drive a purist mad. For present pur- poses, "capital"includesland, mineraldeposits, etc. Naturally I have used Goldsmith's esti- mates (with government, agricultural, and consumer durables eliminated). Ideally what one wouldlike to measureis the annualflow of capital services. Instead one must be content witha less utopianestimateof the stock of capi- tal goods in existence. All sorts of conceptual problemsarise on this account. As a single ex- ample, if the capital stock consisted of a mil- lion identical machines and if each one as it wore out was replacedby a more durablema- chine of the same annual capacity, the stock of capital as measured would surely increase. But the maximalflow of capital services would be constant. There is nothingto be done about this, but something must be done about the fact of idle capacity. What belongs in a pro- ductionfunctionis capital in use, not capital in place. Lackingany reliableyear-by-yearmeas- ure of the utilization of capital I have simply reduced the Goldsmithfigures by the fraction of the labor force unemployed in each year, thus assuming that labor and capital always suffer unemployment to the same percentage. This is undoubtedlywrong, but probably gets closer to the truth than making no correction at all.3 The share-of-capitalseries is anotherhodge- podge, pieced together from various sources and ad hoc assumptions (such as Gale John- son's guess that about 35 per cent of non-farm entrepreneurialincomeis a returnto property). Only after these computations were complete did I learn that Edward Budd of Yale Univer- sity has completeda careful long-termstudy of factor shares which will soon be published. It seems unlikely that minor changes in this in- gredient would grossly alter the final results, CHART 2 0.0 A -ao6 _ Q04 0.02 0.02I -004- -008 i I I i _ I I 1909 1919 1929 1939 949 CHART 3 2.0 A(t) 1.8_ 1.6- 14- 1.2_ 1.0 I0 I I 9 1 94o90 1919 1929 1939 1949 3Another factor for which I have not corrected is the changing length of the work-week. As the work-week shortens, the intensity of use of existing capital decreases, and the stock figures overestimate the input of capital serv- ices.
- 5. TECHNICAL CHANGE AND PRODUCTION FUNCTION 315 but I have no doubt that refinementof this and the capital time-serieswouldproduceneaterre- sults. In any case, in (2a) or (2b) one can replace the time-derivatives by year-to-year changes and calculate A q/q - Wk A k/k. The result is an estimate of A F/F or A A/A, depending on whether these relative shifts appear to be neutral or not. Such a calculation is made in Table i and shownin Chart2. Thence,by arbi- trarily setting A (I909) = i and using the fact that A(t + i) = A(t) (I + A A(t)/A(t)) one can successively reconstruct the A(t) time series, which is shown in Chart3. TABLE I. -DATA FOR CALCULATION OF A(t) % labor Capital Shareof Priv. nonfarm Employed force stock Col. I propertyin GNP per capital per employed ($ mill.) x Col. 2 income manhour manhour A A/A A(t) Year (I) (2) (3) (4) (5) (6) (7) (8) I909 9I.I I46,I42 I33,135 .335 $.623 $2.o6 -.OI7 I.000 I910 92.8 I50,038 I39,235 .330 .6i6 2.I0 .039 .983 I9II 90.6 I56,335 I4I,640 .335 .647 2.I7 .C02 I.02I I9I2 93.0 I59,97I I48,773 .330 .652 2.2I .040 I.023 I9I3 9I.8 I64,504 I5I,OI5 .334 .68o 2.23 .007 I.064 I9I4 83.6 I7I,5I3 I43,385 .325 .682 2.20 -.028 I.07I I9I5 84.5 I75,37I I48,188 *344 .669 2.26 .034 I.04I I9I6 93-7 I78,35I i67,II5 .358 .700 2-34 -.0I0 I.076 I9I7 94.0 i82,263 I71,327 .370 .679 2.2I .072 I.o65 I9I8 94.5 i86,679 I76,4I2 .342 .729 2.22 .OI3 I.I42 I9I9 93.I I89,977 I76,869 .354 .767 2-47 -0.76 I.I57 I920 92.8 I94,802 i80,776 .3I9 .72I 2.58 .072 I.069 I92I 76.9 20I,49I I54,947 .369 .770 2.55 .032 I.I46 I922 8I.7 204,324 I66,933 *339 .788 2.49 .OII I.I83 I923 92.I 209,964 I93,377 .337 .809 2.6I .oi6 I.I96 I924 88.o 222,I13 I95,460 .330 .836 2-74 .032 I.2I5 I925 9I.I 23I,772 2I1,I98 .336 .872 2.8I -.0I0 I.254 I926 92.5 244,6ii 226,266 .327 .869 2.87 -.005 I.24I I927 90.0 259,I42 233,228 .323 .87I 2.93 -.007 I.235 I928 90.0 27I,089 243,980 .338 .874 3.02 .020 I.226 I929 92.5 279,69i 258,714 .332 .895 3.o6 -.043 I.25I I930 88.I 289,29I 254,865 *347 .88o 3.30 .024 I.I97 I93I 78.2 289,056 226,042 .325 .904 3-33 .023 I.226 I932 67.9 282,73I I9I,974 .397 .879 3.28 .0II I.I98 I933 66.5 270,676 i8o,ooo .362 .869 3-IO .072 I.2II I934 70-9 262,370 i86,020 .355 .92I 3.00 .039 I.298 I935 73-0 257,8io i88,20I *35I *943 2.87 .059 I-349 I936 77.3 254,875 I97,0I8 .357 .982 2.72 -.0I0 I-429 I937 8i.o 257,076 208,232 .340 .97I 2.7I .02I I-4I5 I938 74-7 259,789 I94,062 .33I I.000 2.78 .048 I.445 I939 77.2 257,314 i98,646 .347 I.034 2.66 .050 I-5I4 I940 8o.6 258,o48 207,987 .357 I.082 2.63 .044 I.590 I94I 86.8 262,940 228,232 *377 I.I22 2.58 .003 i.66o I942 93.6 270,063 252,779 .356 I.I36 2.64 .oi6 I.665 I943 97-4 269,76i 262,747 .342 i.I8o 2.62 .07I I.692 I944 98.4 265,483 26i,235 .332 I.265 2.63 .02I I.8I2 I945 96.5 26i,472 252,320 .3I4 I.296 2.66 -.044 I.850 I946 94.8 258,05I 244,632 .3I2 I.2I5 2.50 -.OI7 I-769 I947 95-4 268,845 256,478 .327 I-I94 2.50 .oi6 I-739 I948 95.7 276,476 264,588 .332 I.22I 2.55 .024 I-767 I949 93.0 289,360 269,I05 .326 I.275 2.70 ... I.80g NOTES AND SOURCES: Column (i): Percentage of labor force employed. 1909-26, from Douglas, Real Wages in the United States (Boston and New York, 1930), 460. 1929-49, calculated from The Economic Almanac, 1953-54 (New York, 1953), 426-28. Column (2): Capital Stock. From Goldsmith, A Study of Saving in the United States, Vol. 3 (Princeton, 1956), 20-21, sum of columns 5, 6, 7, 9, 12, 17, 22, 23, 24. Column (3): (I) x (2). Column (4): Share of property in income. Compiled from The Economic Almanac, 504-505; and Jesse Burkhead, "Changes in the Func- tional Distribution of Income," Journal of the American Satistical Association, Vol. 48 (June 1953), 192-2I9. Depreciation estimates from Goldsmith, 427. Column (5): Private nonfarm GNP per man hour, 1939 dollars. Kendrick's data, reproducedin The Economic Almanac, 490. Column (6): Employed capital per man hour. Column (3) divided by Kendrick's man hour series, ibid. Column (7): A A/A = A (5)/(5) - (4) X A (6)/(6). Column (8): From (7).
- 6. 3I6 THE REVIEW OF ECONOMICS AND STATISTICS I was temptedto end this sectionwith the re- markthat the A(t) series, which is meant to be a rough profile of technical change, at least looks reasonable. But on second thought I de- cided that I had very little priornotion of what would be "reasonable" in this context. One notes with satisfactionthat the trendis strongly upward; had it turned out otherwise I would not now be writingthis paper. There are sharp dips after each of the World Wars; these, like the sharp rises that preceded them, can easily be rationalized. It is more suggestive that the curve shows a distinct levelling-offin the last half of the I92o's. A sustainedrise beginsagain in I930. There is an unpleasantsawtoothchar- acter to the first few years of the A A/A curve, whichI imagineto be a statistical artifact. The Outlinesof TechnicalChange The reader will note that I have already drifted into the habit of calling the curve of Chart 2 A A/A instead of the more general A F/F. In fact, a scatter of A F/F against K/L (not shown) indicates no trace of a rela- tionship. So I may state it as a formal conclu- sion that over the period I909-49, shifts in the aggregateproductionfunction netted out to be approximatelyneutral. Perhaps I should recall that I have definedneutrality to mean that the shifts were pure scale changes, leaving mar- ginal rates of substitution unchangedat given capital/labor ratios. Not only is A A/A uncorrelatedwith K/L, but one might almost conclude from the graph- that A A/A is essentially constant in time, ex- hibitingmoreor less randomfluctuationsabout a fixed mean. Almost, but not quite, for there does seem to be a break at about I930. There is some evidence that the average rate of prog- ress in the years I909-29 was smallerthan that from I930-49. The first 2I relativeshifts aver- age about 9/Ia of one per cent per year, while the last I9 average214 per cent peryear. Even if the year I929, which showed a strong down- wardshift, is moved fromthe first groupto the second, there is still a contrast between an averagerate of I.2 per cent in the firsthalf and I.9 per cent in the second. Such post hoc splitting-up of a period is always dangerous. Perhaps I should leave it that there is some evidence that technical change (broadly inter- preted) may have acceleratedafter I929. The over-all result for the whole 40 years is an average upward shift of about I.5 per cent per year. This may be comparedwith a figure of about .75 per cent per year obtained by Stefan Valavanis-Vailby a differentand rather less generalmethod,for the period I869-I948.4 Another possible comparison is with the out- put-per-unit-of-input computations of Jacob Schmookler,5which show an increase of some 36 per cent in outputper unit of input between the decades1904-13 and 1929-38. OurA(t) rises36.5 per cent between I909 and I934. But these are not really comparableestimates,since Schmookler'sfiguresinclude agriculture. As a last general conclusion, after which I will leave the interested reader to his own imn- pressions, over the 40 year period output per man hour approximatelydoubled. At the same time, accordingto Chart 2, the cumulative up- wardshift in the productionfunctionwas about 8o per cent. It is possible to argue that about one-eighthof the total increase is traceable to increasedcapitalper manhour,and the remain- ing seven-eighths to technical change. The reasoningis this: real GNP per man hour in- creased from $.623 to $I.2 75. Divide the latter figure by I.809, which is the I949 value for A(t), and thereforethe full shift factor for the 40 years. The result is a "corrected"GNP per man hour, net of technical change, of $.705. Thus about 8 cents of the 65 cent increase can be imputed to increased capital intensity, and the remainderto increasedproductivity.6 Of course this is not meant to suggest that the observed rate of technical progress would have persisted even if the rate of investment had been much smaller or had fallen to zero. Obviouslymuch,perhapsnearly all, innovation must be embodiedin new plant and equipment to be realized at all. One could imagine this process taking place without net capital for- 4 S. Valavanis-Vail, "An Econometric Model of Growth, U.S.A. i869-i953," American Economic Review, Papers and Proceedings, XLV (May I955), 2I7. 5J. Schmookler, "The Changing Efficiency of the Ameri- can Economy, I869-I938," this REViEW (August 1952), 226. ' For the first half of the period, I909-29, a similar com- putation attributes about one-third of the observed increase in GNP per man-hour to increased capital intensity.
- 7. TECHNICAL CHANGE AND PRODUCTION FUNCTION 3I7 mation as old-fashioned capital goods are re- placed by the latest models, so that the capital- labor ratio need not change systematically. But this raisesproblemsof definitionandmeas- urement even more formidable than the ones already blithely ignored. This whole area of interesthas beenstressedby Fellner. For comparison, Solomon Fabricant7 has estimated that over the period I87I-195I about go per cent of the increase in output per capita is attributable to technical progress. Presumably this figure is based on the stand- ard sort of output-per-unit-of-input calcula- tion. It mightseem at firstglance that calculations of output per unit of resource input provide a relatively assumption-freeway of measuring productivity changes. Actually I think the im- plicit load of assumptions is quite heavy, and if anything the method proposedabove is con- siderablymore general. Not only does the usual choice of weights for computingan aggregate resource-inputinvolve somethinganalogousto my assumptionof com- petitive factor markets,but in addition the cri- terion output . a weighted sum of inputs wouldseem tacitly to assume (a) that technical change is neutral and (b) that the aggregate productionfunction is strictly linear. This ex- plains why numerical results are so closely parallel for the two methods. We have already verifiedthe neutrality, and as will be seen sub- sequently, a strictly linear productionfunction gives an excellent fit, though clearly inferior to someXalternatives.8 The AggregateProductionFunction Returning now to the aggregate production function, we have earned the right to write it in the form (ia). By use of the (practically unavoidable) assumptionof constantreturnsto scale, this can be furthersimplifiedto the form q = A(t)f(k,), (3) which formedthe basis of Chart i. It was there noted that a simple plot of q against k would give a distorted picture because of the shift factorA(t). Each point wouldlie on a different member of the family of production curves. But we have now provided ourselves with an estimate of the successive values of the shift factor. (Note that this estimate is quite inde- pendent of any hypothesis about the exact shape of the production function.) It follows from (3) that by plotting q(t) /A (t) against k(t) we reduce all the observed points to a single memberof the family of curves in Chart I, andwe can then proceedto discuss the shape of f(k,i) and reconstructthe aggregateproduc- tion function. A scatter of q/A against k is shownin Chart4. Consideringthe amountof a prioridoctoring which the rawfigureshave undergone,the fit is remarkablytight. Except, that is, for the layer of points which are obviously too high. These maverick observations relate to the seven last years of the period, 1943-49. From the way they lie almost exactly parallel to the main CHART 4 74 q *~~~~~~~~~~~~~ A A72- 70~ ~ ~ ~ ~ ~~~_70 -#O0 S s 68 - , 66 - .64 _ *# 62 1 1 1 I I 1 20 22 24 2.6 28 3.0 3.2 k 3.4 S. Fabricant, "Economic Progress and Economic Change," 34th Annual Report of the National Bureau of Economic Research (New York, I954). 8 For an excellent discussion of some of the problems, see M. Abramovitz "Resources and Output Trends in the U.S. since I870," American Economic Review, Papers and Pro- ceedings, XLVI (May I956), 5-23. Some of the questions there raised could in principle be answered by the method used here. For example, the contribution of improved quality of the labor force could be handled by introducing various levels of skilled labor as separate inputs. I owe to Prof. T. W. Schultz a heightened awareness that a lot of what appears as shifts in the production function must represent improvement in the quality of the labor input, and therefore a result of real capital formation of an im- portant kind. Nor ought it be forgotten that even straight technical progress has a cost side.
- 8. 318 THE REVIEW OF ECONOMICS AND STATISTICS scatter, one is temptedto concludethat in I943 the aggregate production function simply shifted. But the whole earlierprocedurewas de- signed to purify those points from shifts in the function, so that way out would seem to be closed.I suspectthe explanationmay lie in some systematic incomparability of the capital-in- use series. In particular during the war there was almost certainly a more intensive use of capital services through two- and three-shift operation than the stock figures would show, even with the crude correction that has been applied. It is easily seen that such an underesti- mate of capital inputs leads to an overestimate of productivityincrease. Thus in effect each of the affectedpoints should really lie higher and toward the right. But further analysis shows that, for the ordersof magnitudeinvolved, the net result would be to pull the observations closer to the rest of the scatter. At best this might account for I943-I945. There remains the postwar period. Although it is possiblethat multi-shiftoperationremained fairly widespreadeven after the war, it is un- likely that this could be nearly enough to ex- plain the whole discrepancy.9 One might guess that accelerated amortization could have re- sulted in an underestimateof the capital stock after I945. Certainly other researchworkers, notably Kuznets and Terborgh,have produced capital stock estimates which rather exceed Goldsmith'sat the end of the period. But for the present, I leave this a mystery. In a first version of this paper, I resolutely let the recalcitrantobservations stand as they were in a regressionanalysis of Chart4, main- ly because such casual amputationis a practice I deplorein others. But after some experimen- tation it seemed that to leave them in only led to noticeable distortionof the results. So, with some misgivings, in the regressionsthat follow I have omitted the observations for I943- I949. It wouldbe betterif they could be other- wise explained away. Chart 4 gives an inescapable impression of curvature,of persistent but not violent dimin- ishing returns. As for the possibility of ap- proaching capital-saturation,there is no trace on this gross product level, but even setting aside all other difficulties,such a scatter con- fers no particular license to guess about what happens at higher K/L ratios than those ob- served. As for fitting a curve to the scatter, a Cobb- Douglas function comes immediately to mind, but then so do several other parametricforms, with little to chooseamongthem.'0 I can't help feeling that little or nothinghangson the choice of functional form, but I have experimented with several. In general I limited myself to two-parameterfamilies of curves, linear in the parameters (for computational convenience), and at least capable of exhibiting diminishing returns (except for the straight line, which on this account proved inferior to all others). The particularpossibilitiestriedwere the fol- lowing: q=a+,Bk (4a) q=a +,8log k (4b) q = a -,lk (4c) logq = a +,8 logk (4d) log q = a- 8/k. (4e) Of these, (4d) is the Cobb-Douglas case; (4c and e) have upper asymptotes; the semi- logarithmic (4b) and the hyperbolic (4c) must cross the horizontalaxis at a positive value of k and continueevermoresteeply but irrelevant- ly downward(whichmeansonly that someposi- tive k must be achieved before any output is forthcoming,but this is far outside the range of observation); (4e) begins at the originwith a phase of increasing returns and ends with a phase of diminishing returns- the point of inflection occurs at k = /3/2 and needless to say all our observed points come well to the right of this. The results of fitting these five curves to the scatter of Chart4 are shown in Table 2. The correlationcoefficientsare uniformly so high that one hesitates to say any more than 'It is cheering to note that Professor Fellner's new book voices a suspicion that the postwar has seen a substantial increase over prewar in the prevalence of multi-shift opera- tion. See Trendsand Cycles in EconomicActivity (New York, 1956), 92. '0A discussion of the same problem in a different con- text is to be found in Prais and Houthakker, The Analysis of Family Budgets (Cambridge, England, I955), 82-88. See also S. J. Prais, "Non-Linear Estimates of the Engel Curves," Review of Economic Studies, No. 52 (I952-53), 87-Io4.
- 9. TECHNICAL CHANGE AND PRODUCTION FUNCTION 3I9 that all five functions, even the linear one, are about equally good at representingthe general shape of the observedpoints. From the corre- lations alone, for what they are worth, it ap- pears that the Cobb-Douglas function (4d) and the semilogarithmic (4b) are a bit better than the others." Since all of the fitted curves are of the form g(y) = a + /8 h(x), one can view them all as linear regressions and an interesting test of goodnessof fit proposedby Prais andHouthak- ker (ibid., page 5I) is available. If the resid- uals fromeach regressionare arrangedin order of increasing values of the independent vari- able, then one would like this sequence to be disposed "randomly"about the regressionline. A strong"serial"correlationin the residuals,or a few long runs of positive residuals alternat- ing with long runs of negative residuals,would be evidence of just that kind of smooth de- parture from linearity that one would like to catch. A test can be constructed using pub- lished tables of critical values for runs of two kinds of elements. This has been done for the linear, semiloga- rithmic, and Cobb-Douglas functions. The re- sults strongl-yconfirmthe visual impressionof diminishingreturnsin Chart4, by showingthe linear function to be a systematically poor fit. As between (4b) and (4d) there is little to choose.'2 A Note on Saturation It has already been mentioned that the ag- gregate production function shows no signs of levelling off into a stage of capital-saturation. The two curves in Table 2 which have upper asymptotes (c and e) happen to locate that asymptote at about the same place. The limit- ing values of q are, respectively, .92 and .91. Of course these are both true asymptotes, ap- proachedbut not reachedfor any finitevalue of k. It could not be otherwise: no analytic func- tioncan suddenlylevel off and becomeconstant unless it has always been constant. But on the other hand, there is no reason to expect nature to be infinitelydifferentiable. Thus any conclu- sions extending beyond the range actually ob- served in Chart 4 are necessarily treacherous. But, tongue in cheek, if we take .95 as a guess at the saturationlevel of q, and use the linear function (4a) (which will get there first) as a lower-limitguess at the saturation level for k, it turns out to be about 5.7, morethan twice its presentvalue. But all this is in terms of gross output, whereasfor analytic purposeswe are interested in the net productivity of capital. The differ- ence between the two is depreciation,a subject about which I do not feel able to make guesses. If therewere morecertaintyabout the meaning of existing estimates of depreciation,especially over long periods of time, it would have been better to conductthe whole analysis in termsof net product. However, one can say this. Zero net mar- ginal productivity of capital sets in when gross marginalproduct falls to the "marginalrate of depreciation," i.e. when adding some capital adds only enoughproductto make good the de- preciation on the increment of capital itself. Now in recent years NNP has run a bit over TABLE 2 Curve a a3 r 4a .438 .09I .9982 b .448 .239 .9996 C .9I7 .6i8 .9964 d - .729 .353 .9996 e - .038 .9I3 .9980 'It would be foolhardy for an outsider (or maybe even an insider) to hazard a guess about the statistical properties of the basic time series. A few general statements can be made, however. (a) The natural way to introduce an error term into the aggregate production function is multiplica- tively: Q = (I + u)F(K,L;t). In the neutral case it is apparent that the error factor will be absorbed into the estimated A(t). Then approximately the error in AA/A will be Au/I + u. If u has zero mean, the variance of the estimated AA/A will be approximately 2(I - p) var u, where p is the first autocorrelation of the u series. (b) Sup- pose that marginal productivity distribution doesn't hold exactly, so that K/QaQ/aK = wk + v, where now v is a random deviation and wk is the share of property income. Then the error in the estimated AA/A will I v Ak/k, with variance (Ak/k)2 var v. Since KjL changes slowly, the mul- tiplying factor will be very small. The effect is to bias the estimate of AA/A *insuch a way as to lead to an over- estimate when property receives less than its marginal product (and k is increasing). (c) Errors in estimating A(t) enter in a relatively harmless way so far as the regression analysis is concerned. Errors of observation in k will be more serious and are likely to be large. The effect will of course be to bias the estimates of p downward. I The test statistic is R, the total number of runs, with small values significant. For (4a), R = 4; for (4b), R = I3. The I'% critical value in both cases is about 9.
- 10. 320 THE REVIEW OF ECONOMICS AND STATISTICS go per cent of GNP, so capital consumptionis a bit under io per cent of gross output. From Table i it can be read that capital per unit of output is, say, between 2 and 3. Thus annual depreciationis between 3 and 5 per cent of the capital stock. Capital-saturation would oc- cur whenever the gross marginal product of capital falls to .03-.05. Using (4b), this would happen at K/L ratios of around 5 or higher, still well above anything ever observed.'3 Summary This paper has suggested a simple way of segregating shifts of the aggregate production function from movements along it. The meth- od rests on the assumptionthat factors are paid their marginalproducts, but it could easily be extended to monopolistic factor markets. Among the conclusions which emerge from a crude application to American data, i909-49, are the following: i. Technical change duringthat period was neutral on average. 2. The upwardshift in the productionfunc- 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 half. 3. Gross output per man hour doubled over the interval, with 87'2 per cent of the increase attributable to technical change and the re- maining I 2 12 per cent to increased use of capital. 4. The aggregate production function, cor- rected for technicalchange,gives a distinct im- pression of diminishingreturns,but the curva- tureis not violent. IS And this is under relatively pessimistic assumptions as to how technical change itself affects the rate of capital consumption. A warning is in order here: I have left Ken- drick's GNP data in I939 prices and Goldsmith's capital stock figures in I929 prices. Before anyone uses the g's of Table 2 to reckon a yield on capital or any similar num- ber, it is necessary to convert Q and K to a comparable price basis, by an easy calculation.