The size of the productivity residual depends on who measured the capital

Suppose you want to know whether an economy has got better at turning work and machines into output. Nobody can see that quantity. Hours can be counted and machines inventoried, but the efficiency with which they produce output has no meter. So economists reach it by subtraction: measure how fast output grew, measure how fast the inputs grew, weight each input by its share of income, and call the difference total factor productivity. The subtraction is the method rather than a detail of it. Every other number in the exercise estimates something somebody tried to observe; the residual alone is defined rather than measured.

Robert Solow's 1957 paper did the arithmetic for the American non-farm economy from 1909 to 1949, a span over which output per hour of work roughly doubled, and attributed about one eighth of the rise to more capital per hour and the remaining seven eighths to technical change.¹ Solow meant technical change in the broadest sense, which in his usage took in the rising education of the workforce and better organisation of production as well as new machinery.¹ With that qualification the finding is modest. Without it, it says that nearly all of a half-century's improvement came from knowing how to do things better, which is how it has been read.

Ten years later the number collapsed. Dale Jorgenson and Zvi Griliches argued that the residual was large because the inputs had been measured badly, and measured them again: capital by the services it delivers rather than the stock it represents, adjusted for how hard it is worked, and labour adjusted for schooling. Their revision left the residual almost nothing to explain: properly measured, they argued, virtually all recent American economic growth was attributable to the growth of conventional capital and labour inputs.² Nothing in the American economy had changed between the two calculations. The accounting for the inputs changed, and the residual, being the leftover, absorbed all of it.

The step that did most work was a convention, disputed at once. To adjust capital for utilisation the authors multiplied their capital series by an index built from the use of electric motors in American manufacturing. Edward Denison objected that motor use in factories was no way to adjust equipment, structures, land and inventories across an entire economy; later versions narrowed the adjustment, and a reply in 1972 moved some way back towards him.³ The quarrel was about whether factory motors are a good proxy for the use of farmland, and the answer set the size of the number textbooks report as the contribution of technical progress.

A sharper case arises where two sound methods disagree about one country at one time. The primal approach adds up quantities of inputs; the dual adds up their prices instead, meaning how fast wages and the return to capital are rising. Under the assumptions growth accounting already makes the two must agree, because prices and quantities are tied by the identity that output equals what is paid out to produce it. If they disagree, one body of data is wrong. They disagree spectacularly about Singapore. Alwyn Young found that productivity growth accounted for more than two thirds of the rise in Hong Kong's output per head during the 1970s and 1980s while Singapore's was next to nil, a pattern he extended to Taiwan and South Korea.⁴⁵ Paul Krugman made this famous in 1994, calling Asian growth perspiration rather than inspiration and comparing Singapore's record to Stalin's Soviet Union.⁶

Chang-Tai Hsieh then did the same sum from the price side. For South Korea and Hong Kong his dual estimates roughly matched the primal ones; for Taiwan they came out about a percentage point a year higher, and for Singapore more than two points higher.⁷ The reason follows from the primal story. If capital accumulated as fast as Singapore's national accounts report while productivity stood still, the return to capital should have fallen steeply, because there is a limit to what more machinery earns when nothing else improves. Reasonable measures of the cost of capital in Singapore are flat.⁷ Either the return data are wrong or the investment data are, and which you believe decides whether Singapore's growth was bought or earned.

The dual is not simply the correction. Young replied in 1998 that Hsieh's implementation contained a list of errors and that, corrected on Hsieh's own data, the dual estimates sit close to the primal ones, and a later study judged Young in part too optimistic.⁸⁹ Singapore's own trade ministry has estimated the residual by the dual method, calling the result the country's productivity puzzle.¹⁰ A government computing its own residual twice indicates how much the choice matters.

Decisions ride on these numbers. Krugman's essay was read, and meant to be read, as evidence that East Asian growth was ordinary and unsustainable and that industrial policy had achieved nothing a high savings rate would not have achieved anyway, which is a strong claim about policy drawn from a subtraction whose size depends on how a statistical office measured capital. Development accounting runs the same decomposition across countries rather than years, and the consensus Francesco Caselli reports is that efficiency matters at least as much as capital in explaining why some countries are rich.¹¹ That points advice away from investment and towards institutions, and it rests on exercises that assume capital's share of income is the same everywhere.¹¹

The strongest objection is that this proves too much. Every measured quantity rests on conventions, and mismeasurement is the standard refuge of anyone who dislikes a number. The refuge is often empty: Chad Syverson found that America's productivity slowdown appeared in dozens of countries and that its size bears no relation to how intensively they produce or consume information technology, which is hard to square with blaming uncounted digital goods.¹² But the residual is unlike the other entries. Hours can be recounted and prices resurveyed; the residual has nothing to check it against, because it is whatever makes the identity balance. It cannot be shown to be wrong on its own terms, and so it is the sink into which every error in the inputs drains, including errors pointing in opposite directions that would otherwise have cancelled in the open. Robert Hall read its habit of rising in booms as evidence that price exceeds marginal cost, which requires dropping the assumption that each factor's income share equals its elasticity in production, the assumption growth accounting needs to weight its inputs.¹³

None of this argues for discarding the residual, the only summary of productivity anybody has. It argues for reporting it with its conventions attached, and for treating a large residual as a question rather than a finding. Before concluding that one country is more technologically advanced because its residual is bigger, check whether the two statistical offices measure capital the same way. Moses Abramovitz got there first, calling the unexplained part of American growth in 1956 a measure of our ignorance about the causes of economic growth and an indication of where we need to concentrate our attention.¹⁴ The phrase is quoted now as modesty about technology. It was a remark about the accounts.

Notes

1Robert M. Solow, "Technical Change and the Aggregate Production Function," Review of Economics and Statistics 39, no. 3 (August 1957), pp. 312-320, for the computation on the American non-farm economy from 1909 to 1949, for the attribution of about one eighth of the rise in output per man-hour to increased capital per man-hour and the remaining seven eighths to technical change, and for Solow's own definition of technical change in the broadest sense. The outbound network this paper was drafted on refused every request, so the article could not be opened; the sentence quoted above ("about one-eighth of the total increase is traceable to increased capital per man hour, and the remaining seven-eighths to technical change") was returned verbatim in search results quoting a scanned copy, and the figure is separately reported by the National Academies. Solow's own paper gives a different split for the sub-period 1909 to 1929, where about one third of the gain is attributed to capital intensity. Scanned copy of the article, a second hosted copy and Solow's Nobel lecture, for the breadth of the term

2Dale W. Jorgenson and Zvi Griliches, "The Explanation of Productivity Change," Review of Economic Studies 34, no. 3 (1967), pp. 249-283. The characterisation above is Nicholas Crafts's, who writes that Jorgenson and Griliches argued that, properly measured, virtually all recent American economic growth was attributable to the growth of conventional capital and labour inputs. The draft of this paper carried instead a precise pair of figures, a fall from 1.6 to 0.1 per cent a year over 1945 to 1965, attributed to another Crafts working paper; that paper has gone from the web and the figures could not be found in any reachable source, so the claim has been reduced to the one that can be checked. The Jorgenson and Griliches article itself was not opened. Crafts, Quantifying the Contribution of Technological Change, LSE working paper 79/03 and the article's bibliographic record

3Zvi Griliches, "Hedonic Price Indexes and the Measurement of Capital and Productivity: Some Historical Reflections," NBER working paper 2634 (1988), for the utilisation adjustment built from electric motor use in United States manufacturing, for Edward Denison's objection that such an index was inappropriate for adjusting equipment, structures, land and inventories across the whole economy, for the later restriction of the adjustment to non-residential structures and producers' durable equipment, and for Griliches's own retrospective judgement that he remained in sympathy with the original paper's major thrust. The 1972 reply to Denison, and the authors' partial movement towards his position, is reported in the Bureau of Economic Analysis's centennial reprint on the measurement of productivity, cited here for that point alone. NBER working paper, a bibliographic record of it and the Bureau of Economic Analysis reprint

4Alwyn Young, "A Tale of Two Cities: Factor Accumulation and Technical Change in Hong Kong and Singapore," NBER Macroeconomics Annual 1992, vol. 7, pp. 13-54, for the finding that productivity growth accounts for over two thirds of the rise in Hong Kong's output per head during the 1970s and 1980s while Singapore's is next to nil. NBER chapter record and the author's own copy

5Alwyn Young, "The Tyranny of Numbers: Confronting the Statistical Realities of the East Asian Growth Experience," Quarterly Journal of Economics 110, no. 3 (1995), pp. 641-680, for the attribution of most of the postwar growth of Hong Kong, Singapore, South Korea and Taiwan to factor accumulation, including rising participation, education, investment and the movement of labour into manufacturing. Working paper version, the London School of Economics research record and a bibliographic record

6Paul Krugman, "The Myth of Asia's Miracle," Foreign Affairs (November/December 1994), for the argument that growth in output per person in the region came almost entirely from increased inputs per head, for the characterisation of that growth as perspiration rather than inspiration, and for the comparison of Singapore to the Soviet Union under Stalin. The essay could not be opened; these points were returned in search results quoting a hosted copy of it and secondary accounts of it. The essay at Foreign Affairs and a hosted copy, whose server was returning errors at the time of review

7Chang-Tai Hsieh, "What Explains the Industrial Revolution in East Asia? Evidence from the Factor Markets," American Economic Review 92, no. 3 (2002), pp. 502-526, for the dual estimates of total factor productivity growth, for their closeness to the primal estimates in South Korea and Hong Kong, for their exceeding the primal estimates by about one percentage point a year in Taiwan and by more than two in Singapore, and for the explanation that the return to capital remained constant despite the high rate of accumulation reported in Singapore's national accounts. The flatness of reasonable user-cost measures is also reported in the working paper at note 9. Bibliographic record and a second record of the article

8Alwyn Young, "Alternative estimates of productivity growth in the NICs: a comment on the findings of Chang-Tai Hsieh," for the reply that Hsieh's implementation of the dual approach contained errors and that, corrected on Hsieh's own data, the dual estimates are not far removed from those implied by primal sources. London School of Economics research record and a catalogue record

9Federal Reserve Bank of Chicago working paper 2003-15, for the treatment of primal and dual growth accounting as complementary rather than competing, for the judgement that Young was in part too optimistic, and for the estimate that output in Singapore's favoured sector grew at nearly ten per cent a year for two decades with negative productivity growth. Working paper

10Singapore Ministry of Trade and Industry, "Singapore's Productivity Puzzle: Estimating Singapore's Total Factor Productivity Growth Using the Dual Approach," Economic Survey of Singapore, third quarter 2010, for the ministry's own estimation of the residual by the dual method and for its statement that productivity growth in Singapore has been low across decades despite an improving education profile, rising research spending and capital inflows. Ministry paper

11Francesco Caselli, "Accounting for Cross-Country Income Differences," Handbook of Economic Growth (2005), for the survey's report that the consensus holds efficiency to be at least as important as capital in explaining cross-country income differences, and for the observation that most development accounting exercises assume capital and labour shares are constant across countries. Caselli's later revisiting of the survey notes that significant revisions to the data underlying the 2005 paper have since been published. Working paper version, the author's handbook chapter and the later revisiting

12Chad Syverson, "Challenges to Mismeasurement Explanations for the US Productivity Slowdown," Journal of Economic Perspectives 31, no. 2 (Spring 2017), pp. 165-186, for the finding that the slowdown occurred in dozens of countries and that its size is unrelated to measures of those countries' production or consumption intensities of information and communication technologies. Working paper version and a hosted copy

13Robert E. Hall, "The Relation between Price and Marginal Cost in U.S. Industry," Journal of Political Economy 96, no. 5 (1988), pp. 921-947, for the use of the Solow residual's cyclical behaviour to infer that marginal cost lies well below price in some industries, which rests on relaxing the assumption that factor income shares equal output elasticities. The counter-argument, that estimates of this kind are biased by the use of value-added rather than gross-output data, is made by Norrbin and is reported in Mazumder's working paper, which is the source consulted here. Bibliographic record of Hall's article and Mazumder's paper, for the method and the criticism

14Moses Abramovitz, "Resource and Output Trends in the United States Since 1870," NBER Occasional Paper 52 (1956), reprinted from the American Economic Review, for the description of the unexplained residual as some sort of measure of our ignorance about the causes of economic growth and as an indication of where we need to concentrate our attention, and for the underlying comparison of an output index of 1,325 against an input index of 381 for 1944-53 relative to 1869-78. The paper could not be opened; the phrases above were returned in search results quoting it, and are also reported in the encyclopaedia entry on Abramovitz. NBER record of the publication and encyclopaedia entry