Are We Approaching an Economic Singularity? Information Technology and the Future of Economic Growth REVISED. William D. Nordhaus 1 January 2, 2017

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1 Are We Approaching an Economic Singulariy? Informaion Technology and he Fuure of Economic Growh REVISED William D. Nordhaus 1 January 2, 2017 Absrac Wha are he prospecs for long-run economic growh? The presen sudy looks a a more recenly launched hypohesis, which I label Singulariy. The idea here is ha rapid growh in informaion echnology and arificial inelligence will cross some boundary, afer which economic growh will increase sharply as an ever-acceleraing pace of improvemens cascade hrough he economy. The paper develops a growh model ha feaures Singulariy and presens several ess of wheher we are rapidly approaching Singulariy. The key quesion for Singulariy is he subsiuabiliy beween informaion and convenional inpus. The ess sugges ha he Singulariy is no near. I. Inroducion Wha are he prospecs for long-run economic growh? One prominen line of economic hinking is he rend oward sagnaion. Sagnaionism has a long hisory in economics, beginning prominenly wih Malhus and surfacing occasionally in differen guises. Prominen hemes here are he following: Will economic growh slow and perhaps even reverse under he weigh of resource depleion? Will overpopulaion and diminishing reurns lower living sandards? Will unchecked CO2 emissions lead o caasrophic changes in climae and ecosysems? Have we depleed he sore of poenial grea invenions? Will he aging sociey lead o diminished innovaiveness? There is a vas lieraure on he poenial sources of sagnaion. In he modern era, he Limis o Growh school was an early compuerized modeling effor ha produced scenarios for overshoo and decline in living sandards (see Meadows e al 1972 and 1 The auhor is Serling Professor of Economics; Cowles Foundaion, Yale Universiy, New Haven, CT 06511, USA. Mailing address: P. O. Box , Yale Saion, New Haven, CT , USA. william.nordhaus@yale.edu. The auhor declares no financial conflic. Version is Singulariysub docx. This version replaces he earlier draf of Sepember

2 Ranjens e al 1990). Gordon (2016) argued ha a decline in fundamenal invenions may slow growh of leading counries. Some foresee a long period of demand-side sagnaion in he wake of he long recession ha began in 2008 (see Summers 2014), alhough his looks less plausible in 2017 for he US given he economic plaform of he new governmen. However, he presen sudy looks a he opposie idea, a recenly launched hypohesis which I label Singulariy. The idea here is ha rapid growh in informaion echnology and arificial inelligence will cross some boundary, afer which economic growh will rise rapidly as an ever-increasing pace of improvemens cascade hrough he economy. The mos prominen exponens are compuer scieniss (see he nex secion for a discussion and references), bu a sof version of his heory has recenly been advanced by some economiss as well (Brynjolfsson and McAfee, 2014, Varian 2016). The purpose of his sudy is wo-fold. Firs, I lay ou some of he hisory, curren views, and analyical basis for rapidly rising economic growh. Nex, I propose several diagnosic ess ha migh deermine wheher Singulariy is occurring and apply hese ess o recen economic behavior in he Unied Saes. In he end, I hope ha he analysis and ess will allow us o keep a running scoreboard as o wheher he economic universe is on a sagnaionis or acceleraing pah or possibly in ha middle ground of seady growh. II. Arificial Inelligence and he Singulariy For hose wih a background primarily in economics, he presen secion is likely o resemble science ficion. I will explain he hisory and a modern view abou how he rapid improvemens in compuaion and arificial inelligence (AI) have he poenial o increase heir produciviy and breadh o he exen ha human labor and inelligence will become increasingly superfluous. The sandard discussion in compuer science has no explici economic analysis and leaves open imporan economic issues ha will be addressed in laer secions. I will be useful o summarize he argumen before giving furher background. The produciviy of compuers and sofware has grown a phenomenal raes for more han a half-cenury, and rapid growh has coninued up o he presen. Developmens in machine learning and arificial inelligence are aking on an increasing number of human asks, moving from calculaions o search o speech recogniion, psychoherapy, and roboic aciviies on he road and balefield. A he presen growh of compuaional capabiliies, some have argued, informaion echnologies will have he skills and inelligence of he human brain iself. For discussions of he background and rends, see Moravec (1988), Kurzweil (2000, 2005), Schmid and Cohen (2013). 2

3 Cos of compuer power (2003$/MUCP) The foundaion of he acceleraionis view is he coninuing rapid growh in he produciviy of compuing. One measure of he produciviy is he cos of a sandardized operaion in consan prices, shown in Figure 1. The coss of a sandard compuaion have declined a an average annual rae of 53% per year over he period There may have been a slowing in he speed of chip compuaions over he las decade, bu he growh in parallel, cloud, and high-performance clusers as well as improvemens in sofware appears o have offse he slowing of hardware speed for many applicaions. Compuer scieniss projec he rend shown in Figure 1 ino he indefinie fuure. A some poin, hese projecions move from compuer science o compuer science ficion. They involve improved convenional devices and evenually quanum compuing. If highqubi quanum compuing becomes feasible, hen compuing will be essenially free and he consrains on arificial inelligence will largely be ones of sofware and engineering (see paricularly Moravec 1988, Kurzweil 2005). 1.E+04 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 1.E-08 1.E-10 1.E Figure 1. The progress of compuing measured in cos per compuaion per second deflaed by he price index for GDP in 2006 prices 3

4 Source: Nordhaus (2008) updaed by he auhor using PassMark from hp:// resuls as of April One imporan milesone will come when inexpensive compuers aain he compuing capaciy of he human brain. Curren esimaes are ha he compuaional capaciy of he human brain is in he range of compuaions per second (someimes measured as flops or floaing poin operaions per second). The fases supercompuer as of 2016 was clocked a 93 x flops, and he speed of supercompuers has been growing a a rae of 82% per year over he period (Top500, 2016). A his rae of increase, supercompuers will reach he upper level of flops by Compuaional speed does no easily ranslae ino human inelligence, bu i would provide he raw maerial for scieniss o work wih. Ohers have pu he dae a which human inelligence would be aained by compuers from a decade o a cenury ino he fuure. As compuer scieniss look furher ino heir crysal ball, hey foresee arificial inelligence moving oward superinelligence, which denoes inellec ha is much smarer han he bes human brains in pracically every field, including scienific creaiviy, general wisdom and social skills. (Bosrum, 2006) A he poin where compuers have achieved superinelligence, we have reached he Singulariy where humans become economically superfluous in he sense ha hey make no difference o economic performance. Superinelligen compuers are he las invenion humans would make, as described by he mahemaician Irving Good (1965) as follows: Le an ulrainelligen machine be defined as a machine ha can far surpass all he inellecual aciviies of any man however clever. Since he design of machines is one of hese inellecual aciviies, an ulrainelligen machine could design even beer machines; here would hen unquesionably be an inelligence explosion, and he inelligence of man would be lef far behind. Thus he firs ulrainelligen machine is he las invenion ha man need ever make. This poin a which he rae and breadh of echnological change will be so grea is someimes call he Singulariy in a sense analogous o passing over he even horizon ino a black hole here he even horizon is where he forces of compuer inelligence leave no room for human inervenions. 2 2 The noion of Singulariy of superinelligence is ofen aribued o John von Neumann. The only reference o Singulariy comes in an appreciaion by Sanislaw Ulam o von Neumann in 1958: One conversaion cenered on he ever acceleraing progress of echnology and changes in he 4

5 Before ourselves falling ino he even horizon of acceping he Singulariy hypohesis, we need o clarify some of he implici economic assumpions ha lie behind i. This will be he purpose of he nex secion. III. Hisorical perspecives on Singulariy in economics Socieal Singulariy is a recen heory, bu concerns abou displacemen of humans by machines have been persisen for more han wo cenuries. The concerns ended o focus on disappearance of paricular jobs or occupaional caegories. Wih he rise of compuers, he major concern has been he replacemen of unskilled labor by compuers and robos. Macroeconomic concerns abou rapid produciviy growh and auomaion, as i was called in he early days, focused firs on he poenial for saiaion of human wans and a crisis eiher of unemploymen or superabundan leisure. This was he heme of J.M. Keynes s essay, The Economic Prospecs for Our Grandchildren (1930). He hough ha wih rapid echnological growh, he problem of humaniy will be how o use his freedom from pressing economic cares, how o occupy he leisure, which science and compound ineres will have won for him, o live wisely and agreeably and well. Alhough we are close o he hundred-year mark, here is no sign ha humans have found hemselves saiaed wih goods or leisure. One of he earlies aemps o deal wih he macroeconomic implicaions of compuerizaion was Herber Simon in The Shape of Auomaion (1965). Simon was unique in he inellecual hisory of he acceleraionis debae in being a pioneering compuer scienis as well as a leading economis. Wriing a half-cenury ago, he was a selfdescribed echnological radical. He wroe, I believe ha, in our ime, compuers will be able o do anyhing a man can do. (pp. xii-xiii) A he same ime, he was no wha I will call an acceleraionis, holding ha compuers and auomaion will conribue o a coninuing, bu no grealy acceleraed, rise in produciviy. (p. xiii) As we show below, i seems likely ha if, as Simon believed, compuers can do anyhing humans can do, hen produciviy would grealy accelerae. Simon s pioneering analysis is summarized in Appendix A. He shows an imporan resul abou facor prices ha i is impossible in he neoclassical framework o have boh a falling rae of profi and immiseraion of he working classes (a formal analysis is in Samuelson 1957). Bu his analysis was unable o deal wih he poenial of rapidly growing mode of human life, which gives he appearance of approaching some essenial Singulariy in he hisory of he race beyond which human affairs, as we know hem, could no coninue. Ulam (1958). 5

6 capial produciviy in he case where he share of capial in naional oupu is rising raher han sable. There is much abou robos bu remarkably lile wriing on Singulariy in he modern macroeconomic lieraure. While rend produciviy growh has clearly risen from he period before he Indusrial Revoluion, he workhorse models oday assume seady produciviy and real income growh ino he fuure. Joseph Zeira (1998) examines he implicaions of biased echnological change wih bounded growh. David Hemous and Moren Olsen (2014) examine a model of endogenous growh wih auomaion. They show an ineresing paern of wage growh for low-skilled workers. Daron Acemoglu and Pascual Resrepo (2016) examine a growh model wih auomaion ha has a balanced growh pah (more on his below). Similarly, Jeffrey D. Sachs, Seh G. Benzell, and Guillermo LaGarda (2015) analyze a roboic economy wih immiseraion. They find ha he rise of he robos is more likely o lower he welfare of young workers and fuure generaions when he saving rae is low, high- and low-auomaion goods are more subsiuable in consumpion, and when radiional capial is complemenary o labor. These papers do no produce a Singulariy, even wih he rise of he robos. The poenial for acceleraing economic growh has arisen occasionally as a curiosiy in he lieraure on endogenous echnological change. The key feaure of he endogenous echnology models is ha knowledge is a produced inpu. One formulaion would be ha knowledge growh is proporional o he inpus ino he producion process. For his analysis, A is echnological knowledge, Y is oupu, a fracion λ of oupu is devoed o invenive inpus. A da / d is knowledge growh, and knowledge growh is a funcion of invenive inpus, as in A ( Y). To simplify his grealy, assume ha oupu is produced wih labor, and ha labor grows a a consan growh rae n. This implies ha A A A L A L A If β > 1, which corresponds o increasing reurns o 1 / ( ) / ( ). invenive inpus, hen he growh rae of oupu ends o infiniy (see paricularly Romer 1986, 1990). The prospec of unbounded growh raes has no been aken seriously in he empirical growh lieraure for boh echnical and empirical reasons. The empirical reasons are ha produciviy growh has no acceleraed in recen years. The echnical reason is ha i has unaracive assumpions abou he knowledge-generaion funcion, paricularly he lack of diminishing reurns o invenive inpus. For useful discussions of he shorcomings of he model, see Jones (1995, 1995a). Noe also ha his mechanism for rapid growh in endogenous growh models involves he echnological producion funcion raher han he subsiuion of informaion for labor ha we emphasize here. 6

7 A final poenial source of rising produciviy growh comes from he benign version of Baumol s cos disease. Baumol and his co-auhors emphasized he poenial for lowproduciviy-growh indusries o have rising coss, and poenially o slow aggregae economic growh (see Baumol and Bowen 1965, Baumol 1967, Baumol, Blackman, and Wolff 1985). However, depending upon he subsiuion parameers, he impac could be o raise raher han lower aggregae produciviy growh. This migh be called Baumol s cos euphoria and will be examined below. IV. Key insigh on he source of Singulariy While he impressive behavior of compuers is suggesive, i does no address he cenral quesion of he impac of IT on he economy as a whole. The rapid growh in he produciviy of compuers or informaion echnology such as shown in Figure 1 has no necessary implicaion for aggregae economic growh. The reason is ha he economy does no run on bis alone, eiher on he demand side or he supply side. Consumers may love heir iphones, bu hey canno ea he elecronic oupu. Producion requires scarce maerial inpus ( suff ) in he form of labor, energy, and naural resources as well as informaion for all goods and services. As will become clear, he quesion abou Singulariy for he long run involves he subsiuion properies beween informaion and oher suff in producion. Here is he general resul: Cenral mechanism for Singulariy: If informaion and convenional suff (noninformaion inpus or oupus) are elasic subsiues eiher in consumpion or in producion, hen growh will rise, perhaps exremely rapidly. Singulariy, in he sense of unbounded growh, can only arise wih elasic subsiuion on he supply side. If informaion and convenional suff are inelasic in producion and consumpion, hen rapid improvemens in informaion echnology will evenually be irrelevan o he economy. Pu more precisely, and as will be developed below, we migh hink ha Singulariy can arise from eiher he demand or he supply side. Rapid growh in boh cases resuls from subsiuion oward high-growh inpus or oupus and away from sagnan inpus or oupus. On he demand side, rapid growh would occur if preferences are such ha consumer spending moves increasingly oward high-produciviy-growh indusries as relaive prices change. This is Baumol s cos euphoria. On he supply side, Singulariy would occur if producion has sufficien subsiuabiliy ha he inpu bundle moves oward rapidly improving informaion capial as growh proceeds. I begin wih he demand side and hen move o he supply side. 7

8 V. The Baumol effec and demand-side growh euphoria I begin by describing he forces from he demand side ha migh lead o rapid growh. These are he mirror image of Baumol s cos disease, and will be called Baumol s cos or growh euphoria. The idea a he simples level is ha secors wih relaively rapid produciviy growh have relaively rapid price declines and will herefore generally experience a rise in relaive consumpion levels. The key quesion for he growh in aggregae consumpion is wheher hose secors wih relaively rapid produciviy growh have rising or falling shares in nominal expendiures. If low-produciviy-growh secors dominae, ha will produce sagnaion; if high-produciviy growh secors dominae, ha will produce rapid aggregae growh. Baumol and his co-auhors appeared o hold ha he rend poined oward sagnaionism because of rising expendiure shares of sagnan secors. We can use a wosecor example o undersand he forces a work. Assume ha he economy has wo secors call hem informaion and handicrafs produced by a single composie inpu. Oupu in each secor is linear in he composie inpu wih divergen produciviy rends. The raes of produciviy growh are very high and very low, respecively. According o he Baumol mechanism, relaive prices will be changing rapidly in favor of informaion. If demand subsiuion is inelasic (echnically, if he elasiciy of subsiuion in demand beween o wo goods is less han one), hen handicrafs evenually dominae expendiures, and he rae of growh of consumpion will approach he rae of growh of produciviy in he handicrafs secor. By conras, if subsiuion is elasic (he elasiciy of subsiuion in demand beween o wo goods is greaer han one), hen informaion dominaes consumpion, and he growh in consumpion ends o he growh rae in he informaion secor. So here he criical parameer is he elasiciy of subsiuion in he demand beween he wo kinds of goods. However, unlike supply-side effecs discussed below, high demand-side subsiuabiliy does no lead o a Singulariy. For a more deailed analysis, see Appendix B. If we move o a muli-secor example, he analysis is analogous bu more complicaed. The analysis is laid ou in Nordhaus (2008) and will be summarized here. Assume he growh rae of he ideal index of consumpion is given by an almos ideal demand sysem in which consumpion growh in each secor is a funcion of he growh in relaive prices of he good and an income effec. If we assume ha he income elasiciies are uncorrelaed wih he changes in relaive prices, hen he average change in shares for each good will be deermined by he average change in he relaive price of ha good imes he price-elasiciy of demand minus 1 imes he relaive price movemen. So his is he analog of he wo-secor example where he price-elasiciy replaces he elasiciy of 8

9 subsiuion. The aggregae effec is hen he weighed average of his erm plus errors due o exogenous growh raes plus income effecs. VI. Empirical ess of Baumol s growh euphoria We can es for he Baumol or demand-side growh euphoria by looking a he relaionship beween he shares of differen goods in oal consumpion and he rends in relaive prices. In a prior sudy of rends of major indusries for he U.S., I deermined ha here was a endency for indusries wih falling relaive produciviy and rising relaive prices o have rising shares of nominal oupu and of employmen. This was consisen wih he rend idenified by Baumol and his colleagues cied above of he cos disease. I concluded, There is a negaive associaion of produciviy growh wih he growh in nominal oupu. In oher words, sagnan indusries end o ake a rising share of nominal oupu; however, he relaionship is only marginally saisically significan. An alernaive approach for he presen sudy focuses on consumpion as a more naural place o examine subsiuion paerns. We can es he impac of he composiion by examining wheher hose secors ha have he mos rapid decline in prices end o have rising or declining shares in expendiures. The idenifying assumpion ha is ha prices move inversely wih echnological change, ha echnological change is exogenous, and ha he errors in he demand relaions are independen of echnological shocks. Sricly speaking, i would be sufficien for a single secor o have rapidly falling prices and ake over he enire consumpion bundle, bu we examine he more limied ask of examining he rends of all componens. The BEA has developed long-erm daa on consumpion expendiures and prices saring in These daa include 89 disinc secors ranging in size from owneroccupied housing o food provided on he farm. In our analysis, we ake a simple regression of he log of expendiure change on he log of price change for differen periods. The coefficiens differ by secor and period (see Appendix Table A-3). The general paern is for posiive coefficiens, indicaing inelasiciy of subsiuion for demand. If we examine he enire period from 1929 o 2012 or pooled sub-periods of he oal period, here is a clear indicaion of inelasiciy of subsiuion. These resuls are consisen wih he analysis in Nordhaus (2008), which focuses on producion paerns. An alernaive looks a major informaion-echnology secors (see Appendix Table A-4). These are defined as elecommunicaions, video services, informaion equipmen, inerne services, elephone, and phoographic services. This new economy group shows a 9

10 differen paern from he oaliy of indusries. The prices of he new economy services in oal have been declining seadily, and he shares have risen during all subperiods. However, a saisical analysis of he 6 new-economy secors along he lines of Table A-3 does no show a consisen paern of elasic demand. These resuls indicae ha he Baumol effec of changing shares in consumpion is a force for sagnaion raher han acceleraion. In plain English, he secors ha are experiencing he mos rapid price declines are also experiencing sligh declines in expendiure shares. This endency means ha growh in aggregae consumpion would slow over ime if he underlying echnological rends were sable. However, he impac of changing shares on he aggregae growh in consumpion has hisorically been exremely small in he order of minus 0.1% per year. The reason is ha he shares of high- and lowproduciviy-growh indusries have no changed appreciably over he las wo decades. So his firs es indicaes no sign of rapid demand-side growh, or Baumol s euphoria. I is worh repeaing, however, ha even in he exreme case, his would no lead o a Singulariy of he kind seen on he supply side. VII. Supply-side Singulariy A second acceleraionis mechanism involves subsiuion in producion. We can again sar wih a wo-secor model, similar o ha of Simon above, o moivae he analysis. In his model, here are wo facors of producion and a single composie oupu ha can be used for eiher consumpion or invesmen. One inpu is eiher fixed or slowly growing, and i is usefully hough of as labor. The oher inpu is assumed o be informaion capial, which is produced by a rapidly improving echnology. In he simple wo-inpu model, analogous o he Baumol effec, he key parameer is he elasiciy of subsiuion in producion. If he elasiciy of subsiuion is greaer han one, hen informaion capial akes an increasing share of inpus, and he growh of produciviy rises. If he elasiciy of subsiuion is less han one, hen informaion capial s share in producion declines over ime, and he growh in aggregae produciviy ends oward he growh of he relaively fixed facor, labor. In he uni-elasic Cobb-Douglas case, produciviy growh ends o a consan rae. There are clearly oher cases as well, such as muliple goods and muliple inpus, which are discussed below. However, he analysis is exremely simple in he onegood/wo-inpu case. So i seems bes o sar here and see wha we find. To develop he model furher, we use a sandard closed-economy neoclassical growh model wih a consan savings rae and wih a paricular modificaion. Assume ha labor is growing a a consan rae n and ha all echnological change is capial-augmening 10

11 a a consan and rapid rae. In effec, we consider only informaion capial as an endogenous variable and sweep all oher inpus ino labor. The model is sraighforward. Oupu and capial growh are given by (1) Y F( A K, L ) (2) K / sy K So he growh of oupu is: (3) g( Y ) [ g( A ) g( K )] (1 ) g( L ) Here, is he elasiciy of oupu wih respec o capial, which would equal capial s share of naional income in a perfecly compeiive economy. Combining he equaions, we ge: (4) g( Y ) [ g( A ) sy / K ] (1 ) n For a Cobb-Douglas economy, consan (5) g( Y ) n [ / (1 )] g( A ), which implies ha This is a sraighforward balanced growh pah ha ends oward consan growh. For our purposes, he more ineresing cases are where he elasiciy of subsiuion beween capial and labor ( ) is bounded away from one. In he case of inelasic producion ( 1), he compeiive share of capial ends o zero, and he growh rae ends o he sagnaionis case of zero growh in per capia oupu: (6) g( Y ) (1 ) n n The acceleraionis case is where he elasiciy of subsiuion is bounded above one ( 1). The algebra in he general case is complicaed, so simplify by assuming ha he rae of growh in informaion produciviy is a consan h. Because producion shows elasic subsiuion and effecive informaion grows more rapidly han labor, he elasiciy of oupu wih respec o capial (or he compeiive share of capial) ends o one. As h 1, Y ca K, so Y / K ce. This leads o producion becomes linear in capial, or he Singulariy resul: h( T ) (7) g( Y ) h sy / K h s( YT / KT ) e. 11

12 Growh rae of oupu and wages, share capial (%) The surprise here is ha he growh of oupu is unbounded. In effec, he economy is jus informaion produced by informaion capial, which is produced by informaion, which in urn is producing informaion ever faser every year. We don need o push his resul o he absurd limi. Raher he hree key poins are (1) he value share of informaion capial in he inpu bundle is ending oward uniy, (2) as a resul he conribuion of informaion capial is rising, and finally (3) because informaion capial is a produced inpu, he growh rae of oupu is acceleraing. A numerical example This resul is so surprising ha we can perform numerical analyses o make sure i is no a misake or a possibiliy for disan millennia. To ge a flavor for he dynamics, perform a simple simulaion. Assume ha labor is consan, ha all echnological change is capial-augmening a 10% per year, and ha he elasiciy of subsiuion beween labor and informaion capial is Figure 2 shows a ypical simulaion of he share of capial and he growh raes of oupu and wages. Growh goes off he chars afer abou 70 years. (The resul is he same as long as effecive capial grows faser han labor, bu wih slower K growh, he ime o he Singulariy is lenghened grealy.) 200% 180% 160% 140% 120% 100% 80% 60% 40% 20% 0% Share of capial Growh rae of oupu Wage growh Year Figure 2. Simulaion of a growh model wih rapid echnological change in capial and elasic subsiuion beween labor and capial. i 12

13 A second surprising resul concerns he impac of rapidly growing growh on wages. Wages grow increasingly rapidly in his specificaion: wage growh reaches 200% per year in year 80. Capial evenually ges virually all he cake, bu he crumbs lef for labor which are really small pieces of he increasingly huge mounains of cake are sill growing a a phenomenal rae. The exac iming depends upon he parameers, bu wih elasic producion and rapid capial produciviy, he paern always looks like Figure 2. VIII. Tess for Supply-Side Singulariy Are we heading for he Singulariy? If so, how far off is our rendezvous? Opimiss believe ha superinelligence could be achieved in a few decades based on he progress in compuing power. We can apply he economic models developed above o examine observable economic variables ha can disinguish supply-side acceleraionism from sagnaion or seady growh. We examine he following five ess: 1. The elasiciy of subsiuion beween labor and capial above one 2. A rising growh rae of produciviy 3. A rising income share of capial 4. An acceleraing rise in he capial-oupu raio 5. A rising share of informaion capial in oal capial Tes 1: Elasiciy of subsiuion beween capial and labor greaer han one We begin wih evidence on he elasiciy of subsiuion beween capial and labor (EOSKL). Appendix C reviews he evidence from earlier sudies and recen daa. A brief summary is ha he aggregae EOSKL is in he neighborhood of, and canno be reliably disinguished from, 1. There is no reason why i should be consan over ime, and i appears o be higher in recen years han in earlier years if he income share of capial is assumed o equal he elasiciy of oupu wih respec o capial. So on his parameer ha is so criical o undersanding he likelihood of he Singulariy, he daa speak sofly if a all. Appendix C also examines he elasiciy of subsiuion beween oher inpus and informaion capial (EOSIK). The evidence here suggess an elasiciy of subsiuion of informaion capial for oher inpus ha is greaer han 1. However, he mos recen evidence indicaes ha he elasiciy for he equipmen componen may be less han uniy since Tes 2: Rising produciviy growh The mos imporan implicaion of he acceleraionis growh model is ha oupu 13

14 growh is rising. This will show up as eiher rising labor produciviy (LP) growh or rising oal facor produciviy (TFP) growh. While his is clearly a cenral predicion, i does no provide a srong differenial diagnosis because he rising produciviy growh could come from oher sources. Figure 3 shows an esimae of oal facor produciviy for he U.S. for he period by decade using wo sources. Produciviy growh rae looks like an invered U. I rose from he lae 19h cenury and peaked in he 1950s, bu has slowed o a crawl since For he laes period, , mulifacor produciviy growh rose a an average annual rae of 1.0% per year. Figure 3 shows dramaically ha here is no sign of any acceleraion of mulifacor produciviy as of he mos recen daa for he U.S. Even wih he poenial biases discussed below, his es is decidedly negaive. 4.0% 3.5% 3.0% Produciviy growh by decade (% per year) Gordon Alernaive 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% Decade ending in year Figure 3. Toal facor produciviy growh, Alernaive measure of produciviy growh from Rober Gordon and as compiled by he auhor. For he period , he source is BLS esimae of mulifacor produciviy. Source: Nordhaus (2016). ii 14

15 Tes 3: Rising share of capial? A cenral diagnosic forecas of Singulariy is a rising share of capial in naional income. (Noe ha he share in he growh model is he elasiciy of oupu wih respec o capial. Tha parameer is no observable, so we use he income share, which would equal he elasiciy under compeiive condiions.) Figure 4 shows he rend in he income share of capial (sricly speaking, all income oher han labor compensaion) over he period One secoral concep is he enire economy, while he oher is he non-farm business secor. The laer is beer measured and provides a cleaner definiion of capial income han he former, which includes a large componen in owner-occupied housing as well as governmen capial. Noe ha capial income in he daa include many elemens oher han he ne reurn o capial, such as depreciaion, royalies on minerals, ineres income, income of proprieors, and some labor income. Some analyss suspec ha a subsanial par of he increase in capial s share is eiher mismeasuremen or is due o housing, so he esimaes here are probably an upper bound on he share change (Elsby, Hobijn, and Şahin 2103 and Rognlie 2015) Capial share: nonfarm business Capial share: all economy Figure 4. Trend in share of capial for US in overall economy and non-farm business secor, Source: U.S. Bureau of Economic Analysis and U.S. Bureau of Labor Saisics iii 15

16 Appendix Table A-5 shows regressions of he shares in he wo secors wih and wihou breaks in rend in 1990 and Boh show a small upward rend of abou 0.2 percenage poins per year since 1990 and close o 0.5 percenage poins per year since This rend is supporive of he acceleraionis hypohesis a he raw daa level. However, since we do no have a good undersanding of he reasons for he rise in capial s share, furher research would be necessary o deermine wheher here is a link beween his rise and a rapid rise in capial produciviy (paricularly in informaion capial). Projecing fuure rends such as hose of capial s share in Figure 4 is clearly a primiive exercise. However, projecions are useful o give some perspecive on when he Singulariy migh become more apparen. Our simulaion model shown in Figure 2 indicaes ha he acceleraion in oupu becomes quie apparen (wih he growh rae crossing he 20% per year hreshold) when capial s share crosses he 80% level. A he rae of increase from a forecas using he regression model underlying he las se of esimaes in Table A-5 (+0.47% per year), he 80% rae will no be reached unil 2100 (plus or minus 20 years depending on he sample period of he regression). So while he es is posiive, Singulariy is apparenly many decades in he fuure using his diagnosic es. Tes 4: Acceleraing rise in he capial-oupu raio? Anoher imporan diagnosic concerns he real capial-oupu raio. As is seen in he growh model skeched above, he capial sock (in efficiency unis) will rise increasingly rapidly relaive o oupu. The rise will come because informaional capial grows rapidly, and also because informaional capial akes a larger share of he capial sock. 3 For his es, we can look a he rends in he real capial-oupu raio since 1960 (see Appendix Table A-6). The capial socks shown are differen componens of privae capial correced for qualiy by he Bureau of Economic Analysis and oher governmen agencies. The oupu measure is gross business produc. The firs line shows ha he overall capial-oupu raio has been falling slowly over his period, alhough i has been close o consan in he las subperiod. Looking a he informaion capial componens, hese have been rising relaive o oupu, bu only a modes raes. In any case, he overall conribuion of informaional capial has been oo small o lead o a rising capial-oupu raio. 3 The relaionship beween ess 3 and 4 is he following. Le p K = he price of capial goods, V = curren-cos value of capial, K = quaniy of capial, p Y = he price of oupu, Y = quaniy of oupu, and Q = nominal value of oupu. Quaniies in boh cases are measured as nominal values divided by price indexes. The real capial oupu raio in es 4 is K/Y = (p KK/p YY)/(p K/p Y), while he price relaives in es 3 are p K/p Y. So he difference is he share of capial in naional oupu. 16

17 Tes 5: Share of informaion capial rising? A furher es is ha informaional capial should be a rising share of he capial sock. Indeed, as he economy approaches he Singulariy, he share of informaional capial should approach 100%. Appendix Table A-7 shows he shares of informaional capial in oal privae asses. (These are he curren-cos ne sock of privae fixed asses.) I is clear ha informaional capial is becoming a more imporan par of he capial sock. The growh is paricularly srong in inellecual propery producs. Surprisingly, compuers and informaion processing equipmen have seen a declining share over he laes period. So his es would appear o conform o he Singulariy view, alhough here is sill a long way o go before hese secors dominae invesmen. To deermine wheher an inflecion poin is in he near fuure, we projec he share of informaional capial ino he fuure a he growh rae for he period. Our numerical example suggess ha he growh rae begins o accelerae when he capial share exceeds 80% of income. Our exrapolaion of Table A-7 indicaes ha his would no occur wihin he nex cenury, so he Singulariy appears a bes disan by his es. Tes 6: Mismeasuremen One concern wih he ess above is ha ha produciviy growh is underesimaed. Are measuremen errors hiding rapid produciviy growh? Hal Varian, he chief economis a Google, argues ha here is an explosion of produciviy underway because of he devices, apps, and oher digial innovaions coming ou of Silicon Valley. There is a lack of appreciaion for wha s happening in Silicon Valley because we don have a good way o measure i. (WSJ 2015). Appendix D reviews evidence on mismeasuremen. The conclusions here are hree. Firs, here is evidence ha he growh in real oupu is underesimaed in several secors. This is clear in he example of cell phones. Second, mismeasuremen of produciviy is imporan only for final goods. To he exen ha we have underesimaed produciviy growh of inermediae inpus, his would show up as mulifacor produciviy (MFP) growh in final oupus. Finally, we have performed a es of mismeasuremen by looking ouside IT. If inpus of informaion echnologies were improving rapidly bu no capured in he measured inpus of informaion capial, his would show up as rapid growh and perhaps acceleraion of MFP in non-it indusries. I simply is no happening in a broadbased fashion. So his es is also negaive on Singulariy hrough

18 IX. Summary of Tess for Singulariy Table 1 shows a summary of he six supply-side ess of Singulariy. Four of he six ess are negaive or ambiguous for Singulariy while wo are weakly posiive. We can also calculae for he wo posiive ess how far we are from he poin of Singulariy. I define Singulariy as a ime when he economic growh rae crosses 20% per year. Using simple exrapolaion for he wo posiive ess, he ime a which he economy migh plausibly cross he Singulariy is 100 years or more. Source Table 1. Resuls of he Singulariy ess and ime o Singulariy Source: Earlier figures and ables. iv Resul of es Time unil singulariy Tes 1: Elasiciy of subsiuion beween capial and labor greaer han one Amibiguous x Tes 2: Acceleraing produciviy growh Negaive x Tes 3: Rising share of capial Posiive 100 years + 20 years Tes 4: Acceleraing rise in capialoupu raio Negaive x Tes 5: Share of informaion capial rising Posiive > 100 years Tes 6. Rising MFP in indusries ouside of informaion echnologies because of mismeasuremen Negaive x 18

19 X. Inerpreaions and Elaboraions The heory and ess proposed above raise several issues of inerpreaion. I consider some imporan ones in his secion. a. Old wine in new boles? One reacion o he economics of he Singulariy is ha i is jus a coninuaion of pas rends. An early reader of his paper commened ha machines can fly, compuers can carry ou he mos complex calculaions, bu hese feas have no made humans obsolee. Moreover, i akes years or decades for businesses o learn how o apply new echnologies like seam and elecriciy o he economy. Big daa can make big calculaions, bu hey have no replaced docors or nurses, and have made virually no inroads ino carpenry. While demurring o hese observaions, i is imporan o disinguish he heory of he Singulariy from earlier innovaions. Seam, elecriciy, Inerne shopping hese clearly are mulipliers of human work. The naure of he Singulariy is differen: i is o replace he unique feaure of human labor, which is is inelligence. Labor produciviy doubles when he number of pilos in an airplane goes from wo o one, bu i goes up infiniely for when he number of pilos goes from one o zero for drones. Anoher way o hink of he prospec of Singulariy is as a race beween ask auomaion and ask innovaion. The scope of human asks has wo dimensions, laeral and emporal. The laeral dimension involves all he hings humans do oday, such as piloing or carpenry, while he emporal dimension reflecs he fac ha innovaion expands he lis of asks, including new ones such as code-wriing in Julia or designing auonomous cars. Perhaps AI can evenually ake on mos of oday s laeral lis of asks wih decades of work and rillions of lines of code. Bu asks are evolving, so by he ime AI has masered he asks of 2016, a whole new lis of asks have sprung up. This is he analysis developed in Daron Acemoglu and Pascual Resrepo (2016). They have a framework in which asks previously performed by labor are auomaed, while innovaion creaes more complex asks in which labor has a comparaive advanage. While his picure of a race beween ask auomaion and ask innovaion helps undersand he naure of evoluion of human work, i misses he cenral feaure of Singulariy. Wih he developmen of superinelligen machines, he ask of auomaing asks is aken over by ask-auomaing machines. If hese machines become ruly superinelligen, he pace of ask-auomaion will surpass he pace of ask-innovaion. All of his is of course highly speculaive, bu i helps show why he Singulariy differs from earlier paerns of ask evoluion. 19

20 b. Violaions of basic physical laws? An objecion ha migh arise is wheher he Singulariy violaes basic laws of naure. All processes need minimal energy, and energy is limied if superabundan. Oher poenial limiing resources are fresh and clean waer, oxygen, and exoic minerals o build machines. Some would invoke he second law of hermodynamics, which holds ha increasing order mus be offse by increasing disorder elsewhere. The issues here are oo deep o be adequaely reaed in he presen sudy. While some resources are indeed needed for all producion processes, he inpus can in heory be reduced sharply, and poenially as rapidly as producion increases. This is vividly illusraed for compuaion. An early compuer was he ENIAC (shown a he upper lef in Figure 1). I required abou 150 kw o operae, or approximaely 55 was per floaing poin operaion (flop). A deskop compuer oday requires abou 75 was o produce flops. While his is only an approximaion, his calculaion indicaes ha he energy requiremen for compuaion has declined by a facor of 10,000,000,000,000. In recen years, he energy use has declined a approximaely he rae of improvemen of compuers. So he boom line on resources is ha improvemens in maerial use and miniaurizaion can overcome he physical limiaions on acceleraing growh. As Richard Feynman said, There is pleny of room a he boom. c. Heerogeneous labor in he growh model The Simon-ype growh model of informaion and produciviy analyzed above has he shorcoming ha i assumes heerogeneous capial and labor. Heerogeneous oupu is considered in he Baumol example. We consider in his secion he ineresing implicaions of adding heerogeneous labor o he analysis. Economiss have generally found ha skilled workers are more adapable o rapid changes in informaion echnology han middle-skilled, manual, or unskilled workers. The process is summarized nicely by Auor (2014): Rouine asks [are ones] ha follow an exhausive se of rules and hence are readily amenable o compuerizaion. Rouine asks characerisic of many middle-skilled cogniive and manual aciviies, such as bookkeeping, clerical work and repeiive producion asks. Because he core asks of hese occupaions follow precise, wellundersood procedures, hey are increasingly codified in compuer sofware and performed by machines. This force has led o a subsanial decline in employmen in clerical, adminisraive suppor and, o a lesser degree, producion and operaive employmen [135] 20

21 We can exend he Simon model o include heerogeneous labor by considering some polar cases. Assume as one example ha unskilled labor is a perfec subsiue for informaional capial, while he oher inpu is skilled labor. As above, skilled labor has high bu imperfec subsiuabiliy wih capial. We hen direcly apply he analysis above. The marginal produc and wage of unskilled labor fall proporionally wih capial prices. More realisically, if here is a reservaion wage for unskilled labor, say because of income suppor, skilled labor is essenially worhless and disappears from he labor marke. Is his an absurd resul? As a hisorical analog, consider he fae of human adding machines of he nineeenh cenury. As explained in Nordhaus (2007), here was a revoluion in he employmen of human calculaors around In his 19 h cenury book on calculaion, Oron wries, To be able o add wo, hree or four columns of figures a once, is deemed by many o be a Herculean ask, and only o be accomplished by he gifed few, or in oher words, by mahemaical prodigies. (Oron 1866, p. v) The prodigies who could add up columns of numbers rapidly, called lighning calculaors, were a a premium. Indeed, John D. Rockefeller was a champion lighning calculaor before he urned o being a champion monopolis. The adven of calculaors changed all ha. Aside from quiz shows, here is zero demand oday for lighning calculaors. Such would be he fae of unskilled labor in his simple wo-labor model as we approached he Singulariy. Wha of skilled labor? In he simple wo-labor-inpu model described here, skilled labor would have he same fuure as labor in he one-labor Simon model. Is share in naional income would end o zero as capial ook over he economy. Bu skilled labor would be fully employed, and is wages would begin o rise rapidly as shown in Figure 2. We would see social and economic polarizaion wih a vengeance. Perhaps he paern of impacs would be reversed, as is suggesed by Auor (2014). Perhaps he work of skilled labor would be subsiued by informaion echnology while unskilled labor would be he only group no suscepible o subsiuion by informaion echnology. Perhaps, paiens would be diagnosed and reaed by compuers raher han docors. Classes would be augh online by compuerized insrucors and virual eaching assisans raher han academic scribblers. Cenral banks would finally, in Milon Friedman s vision, be run by a compuerized rule raher han imperfec discreion. Workers jus hook up he moniors, plug in he machines, and make sure ha he Fed has he laes operaing sysem. Since he skill ladder is a wo-way sree, skilled workers would abandon heir professional degrees as he skilled jobs disappear and all humaniy becomes unskilled apprenices o compuers. We are hen back o he Simon model, bu in his case wih he one facor being unskilled labor. Surprisingly, here is much greaer labor-marke equaliy han in he firs example. So wih heerogeneous labor, he fae of differen skill groups depends criically on heir subsiuabiliy wih informaion. 21

22 d. The euhanasia of he laboring classes As growh acceleraes wih superinelligen capial, he rae of reurn on capial and real ineres raes fall o zero. This was an oucome envisioned by J.M. Keynes in a chaper from The General Theory (Keynes 1935). [There would be an] increase he sock of capial up o a poin where is [marginal produc] had fallen o a very low figure. Now, [his] would mean he euhanasia of he renier, and, consequenly, he euhanasia of he cumulaive oppressive power of he capialis o exploi he scarciy-value of capial. Keynes s analysis predaed he pioneering work on producion funcions ha clarified he key role of he elasiciy of subsiuion on facor shares, and as a resul he saw only one of several possible oucomes. Keynes s scenario described a growh pah in which he elasiciy of subsiuion beween labor and capial is less han one; accumulaion in he inelasic case herefore drives no only he rae of reurn o zero bu also he share of capial o zero. However, he acceleraionis case leads o he opposie oucome, where he share of capial goes o uniy. In his oucome, we hus would see he euhanasia of he laboring classes in he sense ha all of income evenually goes o he owners of capial. Workers would be well-paid bu would conrol a vanishing par of naional oupu. However, as long as corporaions own mos of he capial, and people or human insiuions (including governmens hrough axaion) own corporaions, capial income will indirecly flow hrough o humans. Since naional income equals naional oupu, average income will be growing increasingly rapidly. How his will play ou in erms of individual equaliy or inequaliy goes beyond economics o poliics, ax and benefi sysems, and he naure of dynasic savings. Will he incomes be capured by he Schumpeerian classes he innovaors who design machines and wrie sofware for hem? Or by he wealhy who subver insiuions o increase heir wealh? By hose who are he las humans who are complemens raher han subsiues for informaion, perhaps as gardeners or bulers? Perhaps by hose who conrol he inelligen machines before hey ake over? Forunaely, he euhanasia of he laboring classes is far off and will flash warning signals so ha, if i does occur, humans will have ime o conemplae he social srucures of such an era. 22

23 e. Auonomous agens in warfare Mos people oday are focusing on he growing hrea of compuer hacking ino elecions, ax reurns, and probably our homes as well. Less prominen bu perhaps more ominous is he use of informaion echnology in warfare. There are very powerful incenives o develop auonomous and roboic aciviies because of he winner-ake-all naure of miliary echnologies and because he dangers of comba make naions averse o risking lives. The key word in he las paragraph is auonomous. The US Deparmen of Defense defined hese as weapon sysems ha, once acivaed, can selec and engage arges wihou furher inervenion by a human operaor. This definiion suggess he abiliy of such sysems o assess he siuaional conex on a balefield and o decide on he required aack according o pre-programmed rules and balefield informaion. Some of he key developmens in IT warfare are he following. Drone aircraf such as he Predaor have he capabiliy o idenify arges and fire missiles. Daksh is a baeryoperaed remoe-conrolled robo on wheels ha can recover and defuse bombs. Guardium is a small Israeli ank-like surveillance vehicle ha operaes compleely auonomously o guard he Gaza border. PackBos are a series of small robos used o idenify bombs, collec air samples in hazardous sies, and sniff for explosives. SWORDS is a small American anklike vehicle ha is remoe conrolled a his ime. The Samsung SGR-A1 is a Souh Korean miliary robo senry, armed wih sensors and a machine gun ha can operae auonomously and is designed o replace human counerpars in he demiliarized zone a he Souh and Norh Korea border. More advanced versions of hese are under developmen. I is possible o envision ha a rogue naion will develop geneically engineered super-humans o figh alongside robos. While he auomaion of warfare is only in is infancy, we can examine he impac o dae. The share of compensaion in oal oupu for US defense spending has risen slighly over he las wo decades, so on ha es he acceleraionis hypohesis is no suppored. Bale deahs in recen wars (in Iraq and Afghanisan) are down sharply from earlier wars (Vienam and Korea), and his is undoubedly in par due o beer informaion and smar weapons. The success of cyberweapons in warfare (as far as we can ell from public sources) is limied o dae, perhaps seing back Iran s nuclear program by a year or so, bu here are many frighening scenarios. The gruesome wars of he presen (in Syria o ake one example) rely largely on las-generaion echnology of bombs and orure. So he boom line on he role of IT in miliary echnologies is ha i has no moved subsanially oward replacing human labor up o now. 23