Cowles Foundationfor Research in Economicsat Yale UniversityCowles Foundation Discussion Paper No. 2021ARE WE APPROACHING AN ECONOMIC SINGULARITY?INFORMATION TECHNOLOGY AND THE FUTUREOF ECONOMIC GROWTHWilliam D. NordhausSeptember 2015An author index to the working papers in theCowles Foundation Discussion Paper Series is located at:http://cowles.yale.edu/This paper can be downloaded without charge from theSocial Science Research Network Electronic Paper Collection:http://ssrn.com/abstract=2658259Electronic copy available at: http://ssrn.com/abstract=2658259Are We Approaching an Economic Singularity?Information Technology and the Future of Economic GrowthWilliam D. Nordhaus1September 1, 2015AbstractWhat are the prospects for long-run economic growth? The presentstudy looks at a recently launched hypothesis, which I label Singularity.The idea here is that rapid growth in computation and artificialintelligence will cross some boundary or Singularity after whicheconomic growth will accelerate sharply as an ever-accelerating paceof improvements cascade through the economy. The paper develops agrowth model that features Singularity and presents several tests ofwhether we are rapidly approaching Singularity. The key question forSingularity is the substitutability between information andconventional inputs. The tests suggest that the Singularity is not near.I. IntroductionWhat are the prospects for long-run economic growth? One prominent line ofeconomic thinking is the trend toward stagnation. Stagnationism has a long historyin economics, beginning prominently with Malthus and surfacing occasionally indifferent guises. Prominent themes here are the following: Will economic growthslow and perhaps even reverse under the weight of resource depletion? Willoverpopulation and diminishing returns lower living standards? Will unchecked CO2emissions lead to catastrophic changes in climate and ecosystems? Have weThe author is Sterling Professor of Economics; Cowles Foundation, Yale University, New Haven, CT06511, USA. Mailing address: P. O. Box 208268, Yale Station, New Haven, CT 06511-8268, USA.Email: william.nordhaus@yale.edu. The author declares no financial conflict. Version isSingularity_090115.docx.11Electronic copy available at: http://ssrn.com/abstract=2658259depleted the store of potential great inventions? Will the aging society lead todiminished innovativeness?2However, the present study looks at the opposite idea, a recently launchedhypothesis which I label Singularity. The idea here is that rapid growth incomputation and artificial intelligence will cross some boundary or Singularity, afterwhich economic growth will accelerate sharply as an ever-increasing pace ofimprovements cascade through the economy. The most prominent exponents arecomputer scientists (see the next section for a discussion and references), but a softversion of this theory has recently been advanced by some economists as well(Brynjolfsson and McAfee, 2014)At the outset, I want to emphasize that this is not a tract for or againstSingularity. Rather, the purpose is two-fold. First, I lay out some of the history,current views, and show an analytical basis for rapidly rising economic growth.Next, I propose several diagnostic tests that might determine whether Singularity isoccurring and apply these tests to recent economic behavior in the United States. Inthe end, I hope that the analysis and tests will allow us to keep a running scoreboardas to whether the economic universe is on a stagnationist or accelerating path … orpossibly in that middle ground of steady growth.II. Artificial Intelligence and the SingularityFor those with a background primarily in economics, the present section islikely to read like science fiction. It will explain the history and a modern view abouthow the rapid improvements in computation and artificial intelligence (AI) have thepotential to increase its productivity and breadth to the extent that human labor andintelligence will become increasingly superfluous. The standard discussion incomputer science has no explicit economic analysis and leaves open importanteconomic issues that will be addressed in later sections.There is a vast literature on the potential sources of stagnation. In the modern era, the “Limits toGrowth” school was an early computerized modeling effort that produced scenarios for overshootand decline in living standards (see Meadows et al 1972 and Ranjens et al 1990). Gordon (2012,2015) argues that a decline in fundamental inventions may slow growth of leading countries. Someforesee a long period of demand-side stagnation in the wake of the long recession that began in2008 (see Summers 2014).22Electronic copy available at: http://ssrn.com/abstract=2658259It will be useful to summarize the argument before giving furtherbackground. The productivity of computers and software has grown at phenomenalrates for more than a half-century, and rapid growth has continued up to thepresent. Developments in machine learning and artificial intelligence are taking onan increasing number of human tasks, moving from calculations to search to speechrecognition, psychotherapy, and robotic activities on the road and battlefield. At thepresent growth of computational capabilities, some have argued, informationtechnologies will have the skills and intelligence of the human brain itself. Fordiscussions of the background and trends, see Moravec (1988), Kurzweil (2000,2005), Schmid and Cohen (2013).The foundation of the accelerationist view is the continuing rapidgrowth in the productivity of computing. One measure of the productivity is the costof a standardized operation in constant prices, shown in Figure 1. The costs of astandard computation have declined at an average annual rate of 53% per year overthe period 1940-2012. There may have been a slowing in the speed of chipcomputations over the last decade, but the growth in parallel, cloud, and highperformance clusters as well as improvements in software appear to have offsetthat for many applications.Computer scientists project the trend shown in Figure 1 into the indefinitefuture. At some point, these projections move from computer science to computerscience fiction. They involve improved conventional devices and eventuallyquantum computing. If high-qubit quantum computing becomes feasible, thencomputing will be essentially free and the constraints on artificial intelligence willlargely be ones of software and engineering (see particularly Moravec 1988,Kurzweil 2005).31.E+04Cost of computer power (2003$/MUCP)1.E+021.E+001.E-021.E-041.E-061.E-081.E-101.E-121850190019502000Figure 1. The progress of computing measured in cost per computation per seconddeflated by the price index for GDP in 2006 pricesSource: Nordhaus (2008) updated by the author using PassMark fromhttp://www.cpubenchmark.net/, results as of April 2014.One important milestone will be when inexpensive computers attain thecomputing capacity of the human brain. Current estimates are that thecomputational capacity of the human brain is in the range of 1018 computations persecond (sometimes measured as “flops” or floating point operations per second).The fastest supercomputer as of 2015 was clocked at 3.4 x 1017 flops, and the speedof supercomputers has been growing at a rate of 82% per year over the 2007-2015period (Top500, 2015). At this rate of increase, supercomputers will reach theupper level of 1018 flops by 2017. Computational speed does not easily translate intohuman intelligence, but it would provide the raw material for scientists to work4with. Others have put the date at which human intelligence would be attained bycomputers from 10 to 100 years in the future.As computer scientists look further into their crystal ball, they foreseeartificial intelligence moving toward superintelligence, which denotes “intellect thatis much smarter than the best human brains in practically every field, includingscientific creativity, general wisdom and social skills.” (Bostrum, 2006)At the point where computers have achieved superintelligence, we havereached the “Singularity” where humans become economically superfluous in thesense that they make no difference to economic performance. Superintelligentcomputers are the last invention humans would make, as described by themathematician Irving Good (1965) as follows:Let an ultraintelligent machine be defined as a machine that can far surpassall the intellectual activities of any man however clever. Since the design ofmachines is one of these intellectual activities, an ultraintelligent machinecould design even better machines; there would then unquestionably be an“intelligence explosion,” and the intelligence of man would be left far behind.Thus the first ultraintelligent machine is the last invention that man needever make.This point at which the rate and breadth of technological change will be so great issometimes call the “Singularity” in a sense analogous to passing over the eventhorizon into a black hole – here the event horizon is where the forces of computerintelligence leave no room for human interventions.3Before ourselves falling into the event horizon of accepting the Singularityhypothesis, we need to clarify some of the implicit economic assumptions that liebehind it. This will be the purpose of the next section.The notion of Singularity of superintelligence is often attributed to John von Neumann. The onlyreference to “Singularity” comes in an appreciation by Stanislaw Ulam to von Neumann in 1958:“One conversation centered on the ever accelerating progress of technology and changes in themode of human life, which gives the appearance of approaching some essential Singularity in thehistory of the race beyond which human affairs, as we know them, could not continue.” Ulam(1958).35III. Historical perspectives on Singularity in economicsSocietal Singularity is a recent theory, but concerns about replacement ofhumans by machines have been persistent for more than two centuries. Theconcerns tended to focus on displacement of particular skills or occupationalcategories. With the rise of computers, the major concern has been the replacementof unskilled labor by computers.Macroeconomic concerns about rapid productivity growth and “automation,”as it was called in the early days, focused first on the potential for satiation of humanwants and a crisis either of unemployment or superabundant leisure. This was thetheme of J.M. Keynes’s essay, “The Economic Prospects for Our Grandchildren”(1930). He wrote:Suppose that a hundred years hence we are eight times better off than today.Assuming no important wars and no important increase in population, theeconomic problem may be solved. This means that the economic problem isnot—if we look into the future—the permanent problem of the humanrace….Must we not expect a general “nervous breakdown”? Thus for the firsttime since his creation man will be faced with his real, his permanentproblem—how to use his freedom from pressing economic cares, how tooccupy the leisure, which science and compound interest will have won forhim, to live wisely and agreeably and well.Although we are close to the hundred-year mark, perhaps humans are sufferingfrom a nervous breakdown, but there is no sign that humans have found themselvessatiated with goods or overendowed with leisure.One of the most impressive attempts to deal with the macroeconomicimplications of computerization was Herbert Simon in “The Shape of Automation”(1965). Simon was unique in the intellectual history of the accelerationist debate inbeing a pioneering computer scientist as well as a leading economist. Writing a halfcentury ago, he was a self-described “technological radical.” He wrote, “I believethat, in our time, computers will be able to do anything a man can do.” (p. xii-xiii) Atthe same time, he was not what I will call an accelerationist, holding that “computersand automation will contribute to a continuing, but not greatly accelerated, rise inproductivity.” (p. xiii) As we show below, it seems likely that if, as Simon believed,computers can duplicate humans, then productivity would greatly accelerate.6Simon’s analysis was very simple, relying on what is known as the “factorprice frontier.” This is the concept that, under highly stylized conditions, factorrewards can be summarized by the equation:waL (1 r ) aK = 1In Simon’s analysis (similar to the second model used below), output is produced bylabor and capital, there are constant returns to scale, there is one good that can beused for either consumption or new capital. In this equation,w wage rate, aL = labor input coefficient, aK = capital input coefficient, and r = realinterest rate. The price of goods is normalized to one.Simon correctly argues that technological change affects unit inputs bylowering the labor and/or capital input coefficients so that at existing factor prices,the cost of production with the new technology is less than 1. Using the notation ofthe factor-price equation (where subscripts 0 are original factor prices, and**asterisks denote the new technology) with an innovation, w0 aL (1 r0 ) aL < 1.Under competition, factor prices will rise until the cost will be equal to the price at 1,so in equilibrium, an innovative technology will raise either wage rates or interestrates or both.Simon does not deploy a formal model for his critical next step. He argues thatlabor is inelastically supplied while capital is elastically supplied (so r is close toconstant). This leads him to conclude that future changes in technology fromautomation will lead to nearly constant interest rates. He further argues(mistakenly) for a near-constant share of capital in national income, which thenimplies that “almost all the increased productivity will go to labor.” (p. 15)Simon’s pathbreaking analysis pointed to an important result about factorprices – that it impossible in the neoclassical framework to have both a falling rateof profit and immiseration of the working classes (a formal analysis is in Samuelson1957). But his analysis was unable to deal with the potential of rapidly growingcapital productivity in the case where the share of capital in national output is risingrather than stable.There is remarkably little writing on Singularity in the modernmacroeconomic literature. While trend productivity growth has clearly risen from7the period before the Industrial Revolution, the workhorse models today assumesteady productivity and real income growth into the future.The potential for accelerating economic growth has arisen occasionally as acuriosity in the economic literature. Explosive growth was explored in studies onendogenous technological change. (Similar but less explosive results are found inthe “AK model,” but those are not examined here.) The key feature of theendogenous technology models is that knowledge is a produced input. Oneformulation would be that knowledge growth is proportional to the inputs into theproduction process. Here At is technological knowledge, Yt is output, a fraction λ ofoutput is devoted to inventive inputs, dAt / dt is knowledge growth, and its growth isa function of inventive inputs, as in dAt / dt (Yt ) . To simplify this greatly,assume that output is produced with labor, and that labor grows at a constantgrowth rate n. If β > 1, which corresponds to increasing returns to inventive inputs,then the growth rate of output tends to infinity (see particularly Romer 1986, 1990).The prospect of unbounded growth rates has not been taken seriously in theempirical growth literature for both technical and empirical reasons. The empiricalreasons are that productivity growth has not accelerated in recent years. Thetechnical reason is that it has unattractive assumptions about the knowledgegeneration function, particularly the lack of diminishing returns to inventive inputs.For useful discussions of the shortcomings of the model, see Jones (1995, 1995a).A final potential source of rising productivity growth comes from the benignversion of Baumol’s cost disease. Baumol and his co-authors emphasized thepotential for low-productivity-growth industries to have rising costs, and potentiallyto slow aggregate economic growth (see Baumol and Bowen 1965, Baumol 1967,Baumol, Blackman, and Wolff 1985). However, depending upon the substitutionparameters, the impact could be to raise rather than lower aggregate productivitygrowth. This might be called Baumol’s cost euphoria and will be examined below.IV. Singularity from the demand and the supply sideTo begin with, I emphasize that rapid growth in the productivity of computersor information technology such as shown in Figure 1 has no necessary implicationfor aggregate economic growth. The reason is that the economy does not run on bitsalone, either on the demand side or the supply side. Consumers may love their8iPhones, but they cannot eat the electronic output. Similarly, at least with today’stechnologies, production requires scarce inputs (“stuff”) in the form of labor, energy,and natural resources as well as information for most goods and services.The question for the long run is the substitution properties betweeninformation and other stuff such as conventional, non-informational inputs oroutputs. Here is the general result:Major insight: If information and conventional stuff are elastic substituteseither in consumption or in production, then growth will rise, perhapsextremely rapidly. However, if information and conventional stuff areinelastic in production and consumption, then rapid improvements ininformation technology will eventually be irrelevant to the economy.Put more precisely, and as will be developed below, Singularity can arise fromeither the demand or the supply side. Both are the results of substitution towardhigh-growth inputs or outputs and away from stagnant inputs or outputs. On thedemand side, Singularity would occur if preferences are such that consumerspending move increasingly toward high-productivity-growth industries as relativeprices change. This is Baumol’s cost euphoria. On the supply side, Singularity wouldoccur if production has sufficient substitutability that the input bundle movestoward rapidly improving information capital as growth proceeds. Both, as we willsee, will lead to rapid growth over time. I begin with the demand side and then moveto the supply side.V. The Baumol effect and demand-side SingularityI begin by describing the forces from the demand side that might lead to rapidgrowth. These are the mirror image of Baumol’s cost disease, and will be calledBaumol’s cost euphoria. The idea at the simplest level is that sectors with relativelyrapid productivity growth have relatively rapid price declines and will thereforegenerally experience a rise in relative consumption levels. The key question for thegrowth in aggregate consumption is whether those sectors with relatively rapidproductivity growth have rising or falling shares in nominal expenditures.9Baumol and his co-authors appeared to hold that the trend pointed towardstagnationism because of rising expenditure shares of stagnant sectors. Forexample, Baumol, Blackman, and Wolff (1985), p. 815-816 concluded as follows: 4The [real] output shares of the progressive and stagnant sectors have in factremained fairly constant in the postwar period, so that with rising relativeprices, the share of total expenditures on the (stagnant) services and theirshare of the labor force have risen dramatically.…Unfortunately, their analysis was made with old-style (Laspeyres) output indexes,so the calculations using real output shares were biased.We can use a two-sector example to understand Baumol-type Singularity.Assume that the economy has two sectors– call them information and handicrafts –produced by a single composite input. The rates of productivity growth are veryhigh and very low, respectively. According to the Baumol mechanism, the relativelyprices will be changing rapidly in favor of information.If demand substitution is inelastic (technically, if the elasticity of substitutionin demand between to two goods is less than one and the income elasticities areunity), then handicrafts eventually dominate expenditures, and the rate of growth ofconsumption will approach the rate of growth of productivity in the handicraftssector. By contrast, if substitution is elastic (the elasticity of substitution in demandbetween to two goods is greater than one with unit income elasticities), theninformation dominates consumption, and the growth in consumption tends to thegrowth rate in the information sector. So here the critical parameter is the elasticityof substitution in the demand between the two kinds of goods.Analysis of the Baumol effectA more rigorous statement is as follows for the two-sector example. Assumethat there are two consumption goods (C1 and C2) that are information andhandicrafts, respectively. Outputs are competitively produced with a singleexogenously growing composite input, L. Productivity growth is assumed constantin each industry (at rates h1 and h2). Preferences are homothetic with a constantelasticity of substitution between the two goods, σ. Given these assumptions, pricesin the two sectors are falling at rate h1 and h2 relative to wages. Total consumption4Op. cit., p. 815-816.10as measured by an ideal index (such as the Tornqvist index) will be growing at rate1 h1 2 h2 , where 1 and 2 are the relative expenditure shares of the two goods.With some work we can show that the ratio of the shares of the two industries ischanging at the logarithmic rate of (h 1h 2 )( 1).So for example, if σ = 1.25, h1 = 10% per year and h2 =0% per year, then theshare of information will be rising at approximately 10 0.25 = 2.5% per year(percent, not percentage points). Or to take a specific example of computers(formally, Information processing equipment), the relative price decrease over thelast decade has been about 10% per year relative to other consumption. The shareof computers in 2000 was approximately 2.0%. If the elasticity of substitutionbetween computers and other goods was 1.25, then the share would grow to 2.6%after a decade. This is almost exactly the actual pattern over this period.We can also easily calculate the Baumol effect for the two sector example. Thegrowth in consumption (in the superlatively measured Tornqvist index) equals theweighted growth of consumption, 1 (t )h1 (t ) 2 (t )h2 (t ) . Under the assumptions inthe last paragraph, the growth in the index of consumption over the decade wouldincrease from 1.20% to 1.26% per year, or an increase of 0.006% per year per year.This is equal to the change in shares times the difference in the growth rates(change in shares = 0.06 %-points per year growth rate difference of 10% peryear). Note that with elastic substitution the growth rate in this model tends towardthe growth in the high-productivity-growth industry. The share of computers tendsto one, so the weighted growth rate tends toward 10% per year in the simpleexample.If we move to a multi-sector example, the analysis is analogous but morecomplicated. The analysis is laid out in Nordhaus (2008) and will be summarizedhere. Assume the growth rate of the ideal index of consumption is given by analmost ideal demand system in which consumption growth in each sector is afunction of the growth in relative prices of the good and an income effect. If weassume that the income elasticities are uncorrelated with the changes in relativeprices, then the average change in shares for each good will be determined by theaverage change in the relative price of that good times the price-elasticity of demandminus 1 times the relative price movement. So this is the analog of the two-sectorexample where the price-elasticity replaces the elasticity of substitution. The11aggregate effect is then the weighted average of this term plus errors due toexogenous growth rates plus income effects.VI. Empirical tests of demand-side SingularityWe can test for the Baumol or demand-side Singularity by looking at therelationship between the shares of different goods in total consumption and thetrends in relative prices.In a prior study of trends of major industries for the U.S., I determined thatthere was a tendency for industries with falling relative productivity and risingrelative prices to have rising nominal shares and shares of employment. This wasconsistent with the trend identified by Baumol and his colleagues cited above of thecost disease.

Cowles Foundationfor Research in Economicsat Yale UniversityCowles Foundation Discussion Paper No. 2021ARE WE APPROACHING AN ECONOMIC SINGULARITY?INFORMATION TECHNOLOGY AND THE FUTUREOF ECONOMIC GROWTHWilliam D. NordhausSeptember 2015An author index to the working papers in theCowles Foundation Discussion Paper Series is located at:http://cowles.yale.edu/This paper can be downloaded without charge from theSocial Science Research Network Electronic Paper Collection:http://ssrn.com/abstract=2658259Electronic copy available at: http://ssrn.com/abstract=2658259Are We Approaching an Economic Singularity?Information Technology and the Future of Economic GrowthWilliam D. Nordhaus1September 1, 2015AbstractWhat are the prospects for long-run economic growth? The presentstudy looks at a recently launched hypothesis, which I label Singularity.The idea here is that rapid growth in computation and artificialintelligence will cross some boundary or Singularity after whicheconomic growth will accelerate sharply as an ever-accelerating paceof improvements cascade through the economy. The paper develops agrowth model that features Singularity and presents several tests ofwhether we are rapidly approaching Singularity. The key question forSingularity is the substitutability between information andconventional inputs. The tests suggest that the Singularity is not near.I. IntroductionWhat are the prospects for long-run economic growth? One prominent line ofeconomic thinking is the trend toward stagnation. Stagnationism has a long historyin economics, beginning prominently with Malthus and surfacing occasionally indifferent guises. Prominent themes here are the following: Will economic growthslow and perhaps even reverse under the weight of resource depletion? Willoverpopulation and diminishing returns lower living standards? Will unchecked CO2emissions lead to catastrophic changes in climate and ecosystems? Have weThe author is Sterling Professor of Economics; Cowles Foundation, Yale University, New Haven, CT06511, USA. Mailing address: P. O. Box 208268, Yale Station, New Haven, CT 06511-8268, USA.Email: william.nordhaus@yale.edu. The author declares no financial conflict. Version isSingularity_090115.docx.11Electronic copy available at: http://ssrn.com/abstract=2658259depleted the store of potential great inventions? Will the aging society lead todiminished innovativeness?2However, the present study looks at the opposite idea, a recently launchedhypothesis which I label Singularity. The idea here is that rapid growth incomputation and artificial intelligence will cross some boundary or Singularity, afterwhich economic growth will accelerate sharply as an ever-increasing pace ofimprovements cascade through the economy. The most prominent exponents arecomputer scientists (see the next section for a discussion and references), but a softversion of this theory has recently been advanced by some economists as well(Brynjolfsson and McAfee, 2014)At the outset, I want to emphasize that this is not a tract for or againstSingularity. Rather, the purpose is two-fold. First, I lay out some of the history,current views, and show an analytical basis for rapidly rising economic growth.Next, I propose several diagnostic tests that might determine whether Singularity isoccurring and apply these tests to recent economic behavior in the United States. Inthe end, I hope that the analysis and tests will allow us to keep a running scoreboardas to whether the economic universe is on a stagnationist or accelerating path … orpossibly in that middle ground of steady growth.II. Artificial Intelligence and the SingularityFor those with a background primarily in economics, the present section islikely to read like science fiction. It will explain the history and a modern view abouthow the rapid improvements in computation and artificial intelligence (AI) have thepotential to increase its productivity and breadth to the extent that human labor andintelligence will become increasingly superfluous. The standard discussion incomputer science has no explicit economic analysis and leaves open importanteconomic issues that will be addressed in later sections.There is a vast literature on the potential sources of stagnation. In the modern era, the “Limits toGrowth” school was an early computerized modeling effort that produced scenarios for overshootand decline in living standards (see Meadows et al 1972 and Ranjens et al 1990). Gordon (2012,2015) argues that a decline in fundamental inventions may slow growth of leading countries. Someforesee a long period of demand-side stagnation in the wake of the long recession that began in2008 (see Summers 2014).22Electronic copy available at: http://ssrn.com/abstract=2658259It will be useful to summarize the argument before giving furtherbackground. The productivity of computers and software has grown at phenomenalrates for more than a half-century, and rapid growth has continued up to thepresent. Developments in machine learning and artificial intelligence are taking onan increasing number of human tasks, moving from calculations to search to speechrecognition, psychotherapy, and robotic activities on the road and battlefield. At thepresent growth of computational capabilities, some have argued, informationtechnologies will have the skills and intelligence of the human brain itself. Fordiscussions of the background and trends, see Moravec (1988), Kurzweil (2000,2005), Schmid and Cohen (2013).The foundation of the accelerationist view is the continuing rapidgrowth in the productivity of computing. One measure of the productivity is the costof a standardized operation in constant prices, shown in Figure 1. The costs of astandard computation have declined at an average annual rate of 53% per year overthe period 1940-2012. There may have been a slowing in the speed of chipcomputations over the last decade, but the growth in parallel, cloud, and highperformance clusters as well as improvements in software appear to have offsetthat for many applications.Computer scientists project the trend shown in Figure 1 into the indefinitefuture. At some point, these projections move from computer science to computerscience fiction. They involve improved conventional devices and eventuallyquantum computing. If high-qubit quantum computing becomes feasible, thencomputing will be essentially free and the constraints on artificial intelligence willlargely be ones of software and engineering (see particularly Moravec 1988,Kurzweil 2005).31.E+04Cost of computer power (2003$/MUCP)1.E+021.E+001.E-021.E-041.E-061.E-081.E-101.E-121850190019502000Figure 1. The progress of computing measured in cost per computation per seconddeflated by the price index for GDP in 2006 pricesSource: Nordhaus (2008) updated by the author using PassMark fromhttp://www.cpubenchmark.net/, results as of April 2014.One important milestone will be when inexpensive computers attain thecomputing capacity of the human brain. Current estimates are that thecomputational capacity of the human brain is in the range of 1018 computations persecond (sometimes measured as “flops” or floating point operations per second).The fastest supercomputer as of 2015 was clocked at 3.4 x 1017 flops, and the speedof supercomputers has been growing at a rate of 82% per year over the 2007-2015period (Top500, 2015). At this rate of increase, supercomputers will reach theupper level of 1018 flops by 2017. Computational speed does not easily translate intohuman intelligence, but it would provide the raw material for scientists to work4with. Others have put the date at which human intelligence would be attained bycomputers from 10 to 100 years in the future.As computer scientists look further into their crystal ball, they foreseeartificial intelligence moving toward superintelligence, which denotes “intellect thatis much smarter than the best human brains in practically every field, includingscientific creativity, general wisdom and social skills.” (Bostrum, 2006)At the point where computers have achieved superintelligence, we havereached the “Singularity” where humans become economically superfluous in thesense that they make no difference to economic performance. Superintelligentcomputers are the last invention humans would make, as described by themathematician Irving Good (1965) as follows:Let an ultraintelligent machine be defined as a machine that can far surpassall the intellectual activities of any man however clever. Since the design ofmachines is one of these intellectual activities, an ultraintelligent machinecould design even better machines; there would then unquestionably be an“intelligence explosion,” and the intelligence of man would be left far behind.Thus the first ultraintelligent machine is the last invention that man needever make.This point at which the rate and breadth of technological change will be so great issometimes call the “Singularity” in a sense analogous to passing over the eventhorizon into a black hole – here the event horizon is where the forces of computerintelligence leave no room for human interventions.3Before ourselves falling into the event horizon of accepting the Singularityhypothesis, we need to clarify some of the implicit economic assumptions that liebehind it. This will be the purpose of the next section.The notion of Singularity of superintelligence is often attributed to John von Neumann. The onlyreference to “Singularity” comes in an appreciation by Stanislaw Ulam to von Neumann in 1958:“One conversation centered on the ever accelerating progress of technology and changes in themode of human life, which gives the appearance of approaching some essential Singularity in thehistory of the race beyond which human affairs, as we know them, could not continue.” Ulam(1958).35III. Historical perspectives on Singularity in economicsSocietal Singularity is a recent theory, but concerns about replacement ofhumans by machines have been persistent for more than two centuries. Theconcerns tended to focus on displacement of particular skills or occupationalcategories. With the rise of computers, the major concern has been the replacementof unskilled labor by computers.Macroeconomic concerns about rapid productivity growth and “automation,”as it was called in the early days, focused first on the potential for satiation of humanwants and a crisis either of unemployment or superabundant leisure. This was thetheme of J.M. Keynes’s essay, “The Economic Prospects for Our Grandchildren”(1930). He wrote:Suppose that a hundred years hence we are eight times better off than today.Assuming no important wars and no important increase in population, theeconomic problem may be solved. This means that the economic problem isnot—if we look into the future—the permanent problem of the humanrace….Must we not expect a general “nervous breakdown”? Thus for the firsttime since his creation man will be faced with his real, his permanentproblem—how to use his freedom from pressing economic cares, how tooccupy the leisure, which science and compound interest will have won forhim, to live wisely and agreeably and well.Although we are close to the hundred-year mark, perhaps humans are sufferingfrom a nervous breakdown, but there is no sign that humans have found themselvessatiated with goods or overendowed with leisure.One of the most impressive attempts to deal with the macroeconomicimplications of computerization was Herbert Simon in “The Shape of Automation”(1965). Simon was unique in the intellectual history of the accelerationist debate inbeing a pioneering computer scientist as well as a leading economist. Writing a halfcentury ago, he was a self-described “technological radical.” He wrote, “I believethat, in our time, computers will be able to do anything a man can do.” (p. xii-xiii) Atthe same time, he was not what I will call an accelerationist, holding that “computersand automation will contribute to a continuing, but not greatly accelerated, rise inproductivity.” (p. xiii) As we show below, it seems likely that if, as Simon believed,computers can duplicate humans, then productivity would greatly accelerate.6Simon’s analysis was very simple, relying on what is known as the “factorprice frontier.” This is the concept that, under highly stylized conditions, factorrewards can be summarized by the equation:waL (1 r ) aK = 1In Simon’s analysis (similar to the second model used below), output is produced bylabor and capital, there are constant returns to scale, there is one good that can beused for either consumption or new capital. In this equation,w wage rate, aL = labor input coefficient, aK = capital input coefficient, and r = realinterest rate. The price of goods is normalized to one.Simon correctly argues that technological change affects unit inputs bylowering the labor and/or capital input coefficients so that at existing factor prices,the cost of production with the new technology is less than 1. Using the notation ofthe factor-price equation (where subscripts 0 are original factor prices, and**asterisks denote the new technology) with an innovation, w0 aL (1 r0 ) aL < 1.Under competition, factor prices will rise until the cost will be equal to the price at 1,so in equilibrium, an innovative technology will raise either wage rates or interestrates or both.Simon does not deploy a formal model for his critical next step. He argues thatlabor is inelastically supplied while capital is elastically supplied (so r is close toconstant). This leads him to conclude that future changes in technology fromautomation will lead to nearly constant interest rates. He further argues(mistakenly) for a near-constant share of capital in national income, which thenimplies that “almost all the increased productivity will go to labor.” (p. 15)Simon’s pathbreaking analysis pointed to an important result about factorprices – that it impossible in the neoclassical framework to have both a falling rateof profit and immiseration of the working classes (a formal analysis is in Samuelson1957). But his analysis was unable to deal with the potential of rapidly growingcapital productivity in the case where the share of capital in national output is risingrather than stable.There is remarkably little writing on Singularity in the modernmacroeconomic literature. While trend productivity growth has clearly risen from7the period before the Industrial Revolution, the workhorse models today assumesteady productivity and real income growth into the future.The potential for accelerating economic growth has arisen occasionally as acuriosity in the economic literature. Explosive growth was explored in studies onendogenous technological change. (Similar but less explosive results are found inthe “AK model,” but those are not examined here.) The key feature of theendogenous technology models is that knowledge is a produced input. Oneformulation would be that knowledge growth is proportional to the inputs into theproduction process. Here At is technological knowledge, Yt is output, a fraction λ ofoutput is devoted to inventive inputs, dAt / dt is knowledge growth, and its growth isa function of inventive inputs, as in dAt / dt (Yt ) . To simplify this greatly,assume that output is produced with labor, and that labor grows at a constantgrowth rate n. If β > 1, which corresponds to increasing returns to inventive inputs,then the growth rate of output tends to infinity (see particularly Romer 1986, 1990).The prospect of unbounded growth rates has not been taken seriously in theempirical growth literature for both technical and empirical reasons. The empiricalreasons are that productivity growth has not accelerated in recent years. Thetechnical reason is that it has unattractive assumptions about the knowledgegeneration function, particularly the lack of diminishing returns to inventive inputs.For useful discussions of the shortcomings of the model, see Jones (1995, 1995a).A final potential source of rising productivity growth comes from the benignversion of Baumol’s cost disease. Baumol and his co-authors emphasized thepotential for low-productivity-growth industries to have rising costs, and potentiallyto slow aggregate economic growth (see Baumol and Bowen 1965, Baumol 1967,Baumol, Blackman, and Wolff 1985). However, depending upon the substitutionparameters, the impact could be to raise rather than lower aggregate productivitygrowth. This might be called Baumol’s cost euphoria and will be examined below.IV. Singularity from the demand and the supply sideTo begin with, I emphasize that rapid growth in the productivity of computersor information technology such as shown in Figure 1 has no necessary implicationfor aggregate economic growth. The reason is that the economy does not run on bitsalone, either on the demand side or the supply side. Consumers may love their8iPhones, but they cannot eat the electronic output. Similarly, at least with today’stechnologies, production requires scarce inputs (“stuff”) in the form of labor, energy,and natural resources as well as information for most goods and services.The question for the long run is the substitution properties betweeninformation and other stuff such as conventional, non-informational inputs oroutputs. Here is the general result:Major insight: If information and conventional stuff are elastic substituteseither in consumption or in production, then growth will rise, perhapsextremely rapidly. However, if information and conventional stuff areinelastic in production and consumption, then rapid improvements ininformation technology will eventually be irrelevant to the economy.Put more precisely, and as will be developed below, Singularity can arise fromeither the demand or the supply side. Both are the results of substitution towardhigh-growth inputs or outputs and away from stagnant inputs or outputs. On thedemand side, Singularity would occur if preferences are such that consumerspending move increasingly toward high-productivity-growth industries as relativeprices change. This is Baumol’s cost euphoria. On the supply side, Singularity wouldoccur if production has sufficient substitutability that the input bundle movestoward rapidly improving information capital as growth proceeds. Both, as we willsee, will lead to rapid growth over time. I begin with the demand side and then moveto the supply side.V. The Baumol effect and demand-side SingularityI begin by describing the forces from the demand side that might lead to rapidgrowth. These are the mirror image of Baumol’s cost disease, and will be calledBaumol’s cost euphoria. The idea at the simplest level is that sectors with relativelyrapid productivity growth have relatively rapid price declines and will thereforegenerally experience a rise in relative consumption levels. The key question for thegrowth in aggregate consumption is whether those sectors with relatively rapidproductivity growth have rising or falling shares in nominal expenditures.9Baumol and his co-authors appeared to hold that the trend pointed towardstagnationism because of rising expenditure shares of stagnant sectors. Forexample, Baumol, Blackman, and Wolff (1985), p. 815-816 concluded as follows: 4The [real] output shares of the progressive and stagnant sectors have in factremained fairly constant in the postwar period, so that with rising relativeprices, the share of total expenditures on the (stagnant) services and theirshare of the labor force have risen dramatically.…Unfortunately, their analysis was made with old-style (Laspeyres) output indexes,so the calculations using real output shares were biased.We can use a two-sector example to understand Baumol-type Singularity.Assume that the economy has two sectors– call them information and handicrafts –produced by a single composite input. The rates of productivity growth are veryhigh and very low, respectively. According to the Baumol mechanism, the relativelyprices will be changing rapidly in favor of information.If demand substitution is inelastic (technically, if the elasticity of substitutionin demand between to two goods is less than one and the income elasticities areunity), then handicrafts eventually dominate expenditures, and the rate of growth ofconsumption will approach the rate of growth of productivity in the handicraftssector. By contrast, if substitution is elastic (the elasticity of substitution in demandbetween to two goods is greater than one with unit income elasticities), theninformation dominates consumption, and the growth in consumption tends to thegrowth rate in the information sector. So here the critical parameter is the elasticityof substitution in the demand between the two kinds of goods.Analysis of the Baumol effectA more rigorous statement is as follows for the two-sector example. Assumethat there are two consumption goods (C1 and C2) that are information andhandicrafts, respectively. Outputs are competitively produced with a singleexogenously growing composite input, L. Productivity growth is assumed constantin each industry (at rates h1 and h2). Preferences are homothetic with a constantelasticity of substitution between the two goods, σ. Given these assumptions, pricesin the two sectors are falling at rate h1 and h2 relative to wages. Total consumption4Op. cit., p. 815-816.10as measured by an ideal index (such as the Tornqvist index) will be growing at rate1 h1 2 h2 , where 1 and 2 are the relative expenditure shares of the two goods.With some work we can show that the ratio of the shares of the two industries ischanging at the logarithmic rate of (h 1h 2 )( 1).So for example, if σ = 1.25, h1 = 10% per year and h2 =0% per year, then theshare of information will be rising at approximately 10 0.25 = 2.5% per year(percent, not percentage points). Or to take a specific example of computers(formally, Information processing equipment), the relative price decrease over thelast decade has been about 10% per year relative to other consumption. The shareof computers in 2000 was approximately 2.0%. If the elasticity of substitutionbetween computers and other goods was 1.25, then the share would grow to 2.6%after a decade. This is almost exactly the actual pattern over this period.We can also easily calculate the Baumol effect for the two sector example. Thegrowth in consumption (in the superlatively measured Tornqvist index) equals theweighted growth of consumption, 1 (t )h1 (t ) 2 (t )h2 (t ) . Under the assumptions inthe last paragraph, the growth in the index of consumption over the decade wouldincrease from 1.20% to 1.26% per year, or an increase of 0.006% per year per year.This is equal to the change in shares times the difference in the growth rates(change in shares = 0.06 %-points per year growth rate difference of 10% peryear). Note that with elastic substitution the growth rate in this model tends towardthe growth in the high-productivity-growth industry. The share of computers tendsto one, so the weighted growth rate tends toward 10% per year in the simpleexample.If we move to a multi-sector example, the analysis is analogous but morecomplicated. The analysis is laid out in Nordhaus (2008) and will be summarizedhere. Assume the growth rate of the ideal index of consumption is given by analmost ideal demand system in which consumption growth in each sector is afunction of the growth in relative prices of the good and an income effect. If weassume that the income elasticities are uncorrelated with the changes in relativeprices, then the average change in shares for each good will be determined by theaverage change in the relative price of that good times the price-elasticity of demandminus 1 times the relative price movement. So this is the analog of the two-sectorexample where the price-elasticity replaces the elasticity of substitution. The11aggregate effect is then the weighted average of this term plus errors due toexogenous growth rates plus income effects.VI. Empirical tests of demand-side SingularityWe can test for the Baumol or demand-side Singularity by looking at therelationship between the shares of different goods in total consumption and thetrends in relative prices.In a prior study of trends of major industries for the U.S., I determined thatthere was a tendency for industries with falling relative productivity and risingrelative prices to have rising nominal shares and shares of employment. This wasconsistent with the trend identified by Baumol and his colleagues cited above of thecost disease. I concluded, “There is a nega…