# Economic Growth Given Machine IntelligenceRobin Hanson*hanson@econ.berkeley.eduSchool of Public Heal

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Economic Growth Given Machine IntelligenceRobin Hanson∗hanson@econ.berkeley.eduSchool of Public Health, 140 Warren Hall,University of California, Berkeley, CA 94720-7360 USAâ€ AbstractA simple exogenous growth model gives conservative estimates of the economic implications of machine intelligence. Machines complement human labor when they become moreproductive at the jobs they perform, but machines also substitute for human labor by takingover human jobs. At ﬁrst, expensive hardware and software does only the few jobs wherecomputers have the strongest advantage over humans. Eventually, computers do most jobs.At ﬁrst, complementary eﬀects dominate, and human wages rise with computer productivity. But eventually substitution can dominate, making wages fall as fast as computer pricesnow do. An intelligence population explosion makes per-intelligence consumption fall thisfast, while economic growth rates rise by an order of magnitude or more. These results arerobust to automating incrementally, and to distinguishing hardware, software, and humancapital from other forms of capital.1. IntroductionSince at least 1821 economists have recognized that dramatic consequences for wages andpopulation can result from machines which can directly substitute for human labor (Ricardo,1821; Samuelson, 1988). Yet since then machines seems to have mostly complementedhuman labor; new machine technology has seemed to raise, not lower, the demand forskilled labor. If anything, computers seem to have intensiﬁed this trend. The standardeconomic view seems to be that this trend will continue into the indeﬁnite future (Simon,1977).Some economists, however, including a few famous ones (Keynes, 1933; Leontief, 1982),have forecast that machines will eventually substitute for most human labor. Popularfears of automation do not seem to have abated much, and many computer and roboticsresearchers have made strong predictions that just as technology ﬁrst complemented horsesin transportation (e.g., carriages), but later substituted for them (e.g., cars), computerswhich now complement human labor will eventually be intelligent enough to substitute formost such labor (Nilsson, 1985).Today, Artiﬁcial Intelligence and Robotics are long-standing ﬁelds of engineering whoseexplicit goals are to develop principles of design to eventually enable machines to accomplishall human cognitive and physical tasks. These ﬁelds have made impressive progress overrecent decades, though they also clearly have a long way to go. Another possible route toadvanced machine intelligence is â€œuploading.â€ An upload is a computer simulation of theparticular neuronal connections observed in a particular human brain. If the observations∗. I thank Doug Bailey and Michael Wellman for helpful discussions, and the Robert Wood Johnson Foundation for ﬁnancial support.â€ . http://hanson.berkeley.edu 510-643-1884 FAX: 510-643-86141and simulation are accurate enough, the simulated brain should be able to accomplish anycognitive task that the original brain could (Hanson, 1994). With a well-engineered androidbody, an upload could also accomplish all human physical tasks.What are likely to be the long-term economic consequences of continued progress indeveloping machine intelligence? Beyond disputes over when machines have complementedvs. substituted for human labor, very little research seems to have been done on thisquestion.This paper examines the economic consequences of machine intelligence by using standard economic theory to construct a series of simple but increasingly more realistic formalmodels of economies where machines can both complement and substitute for human labor.Before describing these models in detail, we ﬁrst summarize qualitatively the nature of ourmodels and the conclusions they suggest.2. Overview of Model and ResultsAs with models in any ﬁeld, a good economic model includes the important relevant featuresof a system, while abstracting away from minor features whose inclusion would mainlyobscure relationships between important features.We are here interested in the wage, population, and economic growth consequences ofmachine intelligences. We thus focus here on how the amounts and prices of the majorinputs to modern economic production change with time, paying special attention to theinputs provided by machine intelligences. Speciﬁcally, we consider human labor, humancapital (e.g., education and training), computer hardware, computer software, and otherforms of capital (e.g., factories, roads, and improvements to land).To allow machines to both complement and substitute for human labor, we consider acontinuum of jobs or roles in our economy, roles which either humans or machines can ﬁll.Each job strongly complements all the others, in the sense that having one job done betterincreases the value (i.e, marginal product) of having the other jobs done well. Thus whencomputers do some jobs and humans do others, computers strongly complement humans.In ﬁlling a particular role, however, a computer can also directly substitute for a human.If the relative advantage of humans over computers varies from job to job, and if computerhardware and software gets cheaper faster than comparable human hardware and software,then computers slowly take over more and more jobs.We consider both an overall technology level, which can allow more total production fromthe same inputs, and computer-speciﬁc technology levels, which allows computer inputs tobe produced more cheaply from other inputs. These technology levels vary slowly, andso there is no special date when machine intelligence is â€œinventedâ€ or achieved. Rather,computers become more widely used as they slowly get â€œbetter,â€ i.e., can accomplish thesame tasks more cheaply.While we will usually assume that these technology levels change â€œexogenously,â€, i.e.,independently of the rest of our model parameters, we also give an example of an â€œendogenousâ€ growth where technology improves with production experience. Growth implicationsare much more dramatic with endogenous growth, as is typical for such models. To avoidsuspicions that our dramatic implications are due to peculiar model features, this paperfocuses on producing conservative estimates via exogenous technological growth.2We assume that the product of the economy can be either consumed or used to producemore of any kind of capital (i.e., human, hardware, software, or other). We assume this newcapital is immediately available for further production, and we ignore capital depreciationor decay. We assume that investor arbitrage ensures that the price of capital reﬂects itsmarginal rate of return, and we consider competitive wages, set by the marginal product ofhuman labor.As steady exponential growth seems to describe the world economy over the last halfcentury, economists typically seek steady growth solutions to such models. To allow this, weassume diminishing returns to all inputs (to avoid super-exponential growth), and assumeinvestor preferences are such that interest rates remain constant under steady growth. (Sinceour models have no uncertainty, there is no need to distinguish risk-less from market ratesof return.)We assume a ﬁxed rate of human population growth. We typically also assume thathumans devote a ﬁxed fraction of their hours to work, though at one point we show that aﬁxed work fraction is a plausible consequence of other assumptions under steady growth.Given all of these assumptions and other modeling choices, these models can reconcileour historical experience with machines which have mostly complemented human labor withprojections of dramatic consequences from mature machine intelligence. This is because thebehavior of our models depend on whether machines do a small or large fraction of jobs.When computers are very expensive, they do few jobs. Human labor is very importantthen, and so human labor wages rise as the economy grows. During this phase, computersare mostly irrelevant to growth, which is mainly determined by improving general technologymagniﬁed by increases in ordinary forms of capital.When computers do most jobs, human labor is relatively unimportant, and whetherhuman wages rise or fall depends on whether owners of capital place a strong special valueon services that only humans can provide. If they do, human wages can rise with theeconomy again, but if not, then human wages fall faster than computer prices now do.During this phase, economic growth is much faster. Some conservative parameter choicessuggest it could be an order of magnitude faster or more. Faster growth is due in part to thefact that computer technology is now more important, and computer technology improvesfaster than general technology.Faster growth is also due to the fact that, in contrast with a slowly growing humanpopulation, we assume that the population of machine intelligences can increase as fastas needed to keep up with the demand for labor. The population of machine intelligencesgrows very fast in all periods, growing faster than the rate at which computer prices fall plusthe rate at which the economy grows. A Malthusian analysis of population is appropriatefor this era, and per-intelligence consumption levels fall rapidly. Per-human consumptionlevels can rise rapidly, however, if humans own a ﬁxed fraction of capital.We now derive these results explicitly in a series of three simple but increasingly realisticmodels of an economy containing machine intelligences. The ﬁrst model is extremely simple,containing just enough structure to say something about how machine intelligence mighteﬀect growth rates and wages. We show how this simple model is eﬀected by choices ofleisure time, by a reside of exclusively human jobs, and by learning by doing. The next twomodels show that the results of this ﬁrst model are robust to two extentions: to smoothingthe transition via a continuum of job types, and to including â€œhumanâ€ capital in the analysis.33. The Simplest ModelLet us now take a standard neo-classical (Solow-Swan) growth model with diminishingreturns, and modify it minimally to include ordinary computers and machine intelligence.First we have a Cobb-Douglas production equation,Y = Y (A, L, K, M ) = ALα K β M γ ,(1)where Y is the rate at which stuﬀ can be produced, A is a general technology level, L islabor, M is ordinary computer capital, and K includes all other forms of capital (includingâ€œhumanâ€ capital, e.g., education and training). Note that the marginal products for theseinputs, YL , YK , YM (the partial derivative of Y with respect to these inputs), satisfy YL =αY /L, YK = βY /K, and YM = αY /M .In a competitive economy with α + β + γ = 1, the parameters α, β, γ describe the shareof total product Y paid to each input to production, L, K, M . Competition would ensurethat each unit of an input received its marginal product, and with α + β + γ = 1, totalmarginal product equals total average product. Thus, for example, α would describe thefraction of total product paid to labor L as wages, and β would describe the fraction paidas interest on capital K invested.Let us assume instead that α + β + γ < 1, giving diminishing returns to the set ofall inputs. In this case, total marginal product is less than total product, and all we cansay is that competitively priced units should be paid their marginal product. Thus thecompetitive wage for a unit of labor is YL and α, β, γ give lower bounds on the share oftotal product paid to each input when all units of an input are paid at least their marginalproduct.To allow for the possibility of machine intelligence, let labor L = H + U , where H ishuman labor, and U is labor by machines intelligent enough to substitute for human labor.(Ordinary computers complement human labor.) Let us also assume that the spendingequation isY = C + K + P (M + U ),where C is the rate of consumption, K , M , U are time rates of change of these forms ofcapital, and P is the â€œpriceâ€ (really cost) of computers.Note that the main reason for distinguishing computer capital from other capital inour model is because computer hardware prices have been falling much faster than otherprices for a long time. Thus whether some computer-related capital, such as software, isconsidered part of M or part of K depends on whether its price has been falling more likecomputer hardware prices, or more like other forms of capital.Let us assume that exogenous functions determine the growth in population H(t) andimprovements in technology A(t), P (t). Note that we do not model any variation in leisuretime or fertility. We have also assumed zero capital depreciation, and that producing moreof each form of capital requires the same relative shares α, β, γ of other forms of capital.We have further assumed that all forms of capital, including machine intelligence, can beinstantly produced and are then instantly available to aid production.Let us also assume that the interest rate I that investors demand remains constant whentotal product Y grows exponentially, i.e., when ln Y is constant, where ln F ≡ (ln F ) =4F /F and F is the time derivative of F for any F . The interest rate should be constant, forexample, when each investor consumes some constant fraction of total product, and whenall investors have the same inter-temporal elasticity of substitution (e.g., the same discountrate and logarithmic utility on their consumption rate; see (Barro, 1995) p.65). Finally,let us assume that investment arbitrage eliminates opportunities to obtain returns greaterthan I by varying K, M, U .Arbitrage sets the marginal product of capital. Consider a small increase in generalcapital production, ﬁnanced by a small reduction in consumption, followed a time ∆t laterby an equal general capital decrease. If the extra product from this extra capital is consumedinstead of invested, then capital amounts for the economy before and after this ∆t periodremain unchanged. If interest I is constant over this period, investors are indiﬀerent to thisplan if∆t0e−It YK (t)dt = 1 − e−I∆t .This implies YK = I both for I constant and for I varying, since in the latter case we canconsider the limit as ∆t goes to zero.Arbitrage also sets the marginal product of computer capital. A small increase incomputer capital followed by a matching decrease ∆t later makes investors indiﬀerent if∆t0e−It YM (t)dt = P (0) − P (∆t)e−I∆t = P (0)(1 − e∆ ln P −I∆t )This is solved byYM = (I − ln P )P.(2)Note how declining computer prices make computer investments less attractive; investorswould rather wait till prices fall further.If computer prices P are initially very high, then very few will be bought. If computerprices then fall, more ordinary computers will appear. At ﬁrst machine intelligence wouldbe prohibitively expensive, and so none would be produced, with U = 0. As the computerprice P falls, investments in machine intelligence U would remain unattractive as long asYL < YM . Since YL = αY /H in this period, competitive wages are always proportional toper-capita production. Thus when production rises, wages rise, as is our current experience.In contrast, when investments in U become attractive, so that machine intelligences areproduced, with U > 0, arbitrage should ensure that spending on M vs. U gives equalproﬁts. This requires that YL = YM . Such arbitrage requires that those who help pay tocreate new machine intelligences be repaid from the wages such machines will earn. Thisis possible via either autonomous machines who repay their debts, or via directly ownedmachines.By our computer marginal product equation (2), YL = YM implies that wages willbe falling with computer price P unless interest rates rise very rapidly. And for steadyconsumption growth, interest rates must be constant. Since competitive wages are proportional to per-intelligence production, rapidly declining wages imply rapidly declining5per-intelligence production, due to an intelligence population growing much faster than total production. Wages might well fall below human subsistence levels, if machine subsistencelevels were lower.This model seems to conﬁrm the intuition that machine intelligence has Malthusianimplications for population and wages. Note, however, that these results may be consistentwith a rapidly rising per-capita income for humans, if humans retain a constant fraction ofcapital, perhaps including the wages of machine intelligences, either directly via ownershipor indirectly via debt.Since our results have been expressed in terms of marginal products, let us express theproduction equation (1) in these terms, as in−α −β −γY 1−α−β−γ = αα β β γ γ AYL YK YM .Expressed in terms of growth rates, this is(1 − α − β − γ) ln Y = ln A − α ln YL − β ln YK − γ ln YM .(3)To see what eﬀect a transition to machine intelligence may have on steady state growthrates, let us assume a steady decline in computer prices, with ln P a constant. Since weearlier assumed constant interest rates under constant product growth, we have ln YK = 0and ln YM = ln P . We also know that when U = 0, ln YL = ln Y − ln H, and when U > 0,ln YL = ln YM .Putting this all into the growth rate equation (3), we getln Y =ln A + α ln H − γ ln PËœËœ,1−β−γËœwhich for α = α and γ = γ is valid when U = 0, and for α = 0 and γ = α + γ is validËœËœËœËœfor U > 0. Thus steady state growth rates are diﬀerent with machine intelligence, and thediﬀerence can be thought of as a change in the product shares.To see how diﬀerent the growth rates are, consider some conservative parameter values.Let the product shares be α = .25, β = .5, and γ = .02, let annual growth rates be 1.5% forthe human population H and 1% for general technology A, and let computer prices P halveevery two years. Without machine intelligence, world product grows at a familiar rate of4.3% per year, doubling every 16 years, with about 40% of technological progress comingfrom ordinary computers. With machine intelligence, the (instantaneous) annual growthrate would be 45%, ten times higher, making world product double every 18 months! If theproduct shares are raised by 20%, and general technology growth is lowered to preserve the4.4% ﬁgure, the new doubling time falls to less than 6 months.Such rapid growth may seem so far from historical experience that it should be rejectedout of hand. However, an empirical projection of historical trends in world economic growthmakes a median prediction that the economy will transition to a doubling time of one totwo years around 2025 (Hanson, 1998). While this prediction has large uncertainties, itsuggests that we not reject out of hand the predictions of this model of machine intelligence.(Coincidentally, 2025 is also roughly when long-steady computer hardware price trends givecheap computers with hardware as powerful as the human brain, and so is a favorite datefor predicting when machine intelligence will arrive for those who believe hardware is thelimiting factor (Moravec, 1998).)64. Some VariationsBefore considering more complex substitutes for the above model, let us consider somesimple variations on it.We have so far assumed that human labor is proportional to the human population.What if people instead choose the fraction of time they devote to work based on theirpreferences for consumption, leisure time, and the fulﬁllment of working?Imagine that a human owns some constant fraction of total output Y , due to owningsome constant fraction of labor and each form of capital. And imagine that her utility is asimple product of terms for consumption, leisure, and work,e−δt ( Y (t) + YL (t)l(t))ξ l(t)κ (1 − l(t))ζ dt,where l(t) is the fraction of time spent working. Work fraction l(t) will stay constant if eitherthe ratio of product Y to wages YL stays constant, as it does before machine intelligence, orif wages become negligibly small, as they eventually do after machine intelligence. Naturally,people work fewer hours when wages are negligibly small.If people consume some constant fraction of total product, then people may demandcertain services which can only be performed by humans. There may, for example, be astatus value in being served by a real human rather than a machine imitation. In this case,human labor share α doesnâ€™t drop all the way to zero, and if α holds constant, wages riseËœËœËœwith total world product even after machine intelligence, according to YL = αY /H. (Notethat the work fraction l(t) can still remain constant.)Having some exclusively human jobs slows growth somewhat. In our previous numericalexample, if α falls from .25 to only .05, instead of to zero, then the economic doubling timeËœfalls from 16 years to 27 months, instead of to 18 months.We have assumed constant exogenous rates of improvement for computer and othertechnology. Let us now consider the simplest endogenous growth model, learning by doing(Solow, 1997), where technological ability goes as some power of total experience so far, asinA∝φt−∞P ∝Y (s)ds−ψt−∞M (s)dsfor φ, ψ < 1. For steady growth, i.e., ln Y a constant, this implies ln A = φ ln Y and− ln P = ψ ln Y .Substituting these into our growth equation (3), we getln Y =α ln HËœ.1 − β − φ − γ /(1 − ψ)ËœThis equation is valid when this expression is positive. Otherwise, we no longer havediminishing returns and steady growth solutions are no longer possible; our model predictsa continually accelerating growth rate.Given our previous conservative parameter values, the values φ = .23, and ψ = .89reproduce our familiar economic doubling time of 16 years arises via a 1% annual growth ingeneral technology A, and a halving of computer prices P every two years. Using these φ, ψ7values, lowering α just a little, from .25 to .241, reduces the economic doubling time from 16Ëœyears to 13 months. (In our exogenous growth model, this just reduced the doubling timeto 14 years.) Reducing α further to .24 eliminates diminishing returns and steady growthËœsolutions entirely.Clearly endogenous growth models are capable of producing much more dramatic implications of substituting machine intelligence for human labor. To remain conservative,however, we stick with exogenous growth models.5. A Continuum of Job TypesIn our simplest model, computers stay conﬁned to a small sector of the economy untilcomputers are suddenly equally likely to take on any human job, at which point machineintelligence begins substituting for human labor on a massive scale. In reality, automationseems to increase more incrementally, with computers slowly taking over more types of jobsonce done by humans.Let us now consider a continuum of job types θ, distributed uniformly on the interval[0, 1]. In the limit of many small job types each contributing to the total product, we have1ln Y = ln A + β ln K + ρ0ln f (θ)dθ(4)where f (θ) is the contribution of job type θ to the total product.Each job can be done by either a human or a computer, and the human has a job-typespeciﬁc productivity advantage a(θ). Thusf (θ) = a(θ)h(θ) + m(θ)where h(θ) and m(θ) are the human and computer labor densities across the job types.Total human and computer labor is given by1H=01h(θ)dθM=0m(θ)dθ.Note that in this model we do not explicitly distinguish between ordinary computers andmachine intelligence. This distinction is implicit instead, in the type of job the computerdoes. Some job types θ may require more intelligence than others, and the human vs.computer advantage a(θ) may then reﬂect this diﬀerence.Note also that while a computer may substitute for a particular human in a particularjob, each job complements all the others, and so each computer is complementing all humanworkers in all other jobs. Thus this model is not in spirit contrary to our observationof ubiquitous cases where individual computers strongly complement individual humanworkers.We again assume a falling computer â€œpriceâ€ P , so that the expenditure equation becomes1Y =C +K +P0This again implies YK = I and YM = P (I − ln P ).8m (θ)dθ.Our production equation (4) suggests we deﬁne a density marginal Yf (θ) = ρY /f (θ).The chain rule then gives marginal productsYl(θ) = a(θ)Yf (θ) = a(θ)Ym(θ) .If arbitrage ensures that marginal products are equal across job types with positive densities,then h(θ1 ), h(θ2 ), m(θ1 ), m(θ2 ) > 0 implies Yh(θ1 ) = Yh(θ2 ) and Ym(θ1 ) = Ym(θ2 ) . These implya(θ1 ) = a(θ2 ), however.Thus if we assume a (θ) > 0, there can be at most one type θ where we could haveboth h(θ), m(θ) > 0. We call this border type b. For all θ < b, h(θ) = 0 and for all θ > b,