Returns to scale in research

When universities or university departments produce research outputs—such as published papers—they sometimes experience increasing returns to scale, sometimes constant returns to scale, and sometimes decreasing returns to scale. At the level of nations however, R&D tends to see increasing returns to scale. These results are preliminary.

Background

“Returns to scale” refers to the responsiveness of a process’ outputs when all inputs (e.g. researcher hours, equipment) are increased by a certain proportion. If all outputs (e.g. published papers, citations, patents) increase by that same proportion, the process is said to exhibit constant returns to scale. Increasing returns to scale and decreasing returns to scale refer to situations where outputs still increase, but by a higher or lower proportion, respectively.

Assessing returns to scale in research may be useful in predicting certain aspects of the development of artificial intelligence, in particular the dynamics of an intelligence explosion.

Results

The conclusions in this article are drawn from an incomplete review of academic literature assessing research efficiency, presented in Table 1. These papers assess research in terms of its direct outputs such as published papers, citations, and patents. The broader effects of the research are not considered.

Most of the papers listed below use the Data Envelopment Analysis (DEA) technique, which is a quantitative technique commonly used to assess the efficiency of universities and research activities. It is capable of isolating the scale efficiency of the individual departments, universities or countries being studied.

Paper Level of comparison Activities assessed Results pertaining to returns to scale
Wang & Huang 2007 Countries’ overall R&D activities Research Increasing returns to scale in research are exhibited by more than two-thirds of the sample
Kocher, Luptacik & Sutter 2006 Countries’ R&D in economics Research Increasing returns to scale are found in all countries in the sample except the US
Cherchye & Abeele 2005 Dutch universities’ research in Economics and Business Management Research Returns to scale vary between decreasing, constant and increasing depending on each university’s specialization
Johnes & Johnes 1993 UK universities’ research in economics Research Constant returns to scale are found in the sample as a whole
Avkiran 2001 Australian universities Research, education Constant returns to scale found in most sampled universities
Ahn 1988 US universities Research, education Decreasing returns to scale on average
Johnes 2006 English universities Research, education Close to constant returns to scale exhibited by most universities sampled
Kao & Hung 2008 Departments of a Taiwanese university Research, education Increasing returns to scale exhibited by the five most scale-inefficient departments. However, no aggregate measure of returns to scale within the sample is presented.

Table 1: Sample of studies of research efficiency that assess returns to scale
Note: This table only identifies increasing/constant/decreasing returns to scale, rather than the size of this effect. Although DEA can measure the relative size of the effect for individual departments/universities/countries within a sample, such results cannot be readily compared between samples/studies.

Discussion of results

Of the studies listed in Table 1, the first four are the most relevant to this article, since they focus solely on research inputs and outputs. While the remaining four include educational inputs and outputs, they can still yield worthwhile insights.

Table 1 implies a difference between country-level and university-level returns to scale in research.

  • The two studies assessing R&D efficiency at the country level, Wang & Huang (2007) and Kocher, Luptacik & Sutter (2006), both identify increasing returns to scale.
  • The two university-level studies that assessed the scale efficiency of research alone found mixed results. Concretely, Johnes & Johnes (1993) concluded that returns to scale are constant among UK universities, and Cherchye & Abeele (2005) concluded that returns to scale vary among Dutch universities. This ambiguity is echoed by the remainder of the studies listed above, which assess education and research simultaneously and which find evidence of constant, decreasing and increasing returns to scale in different contexts.

Such differences are consistent with the possibility that scale efficiency may be influenced by scale (size) itself. In this framework, as an organisation increases its size, it may experience increasing returns to scale initially, resulting in increased efficiency. However, the efficiency gains from growth may not continue indefinitely; after passing a certain threshold the organisation may experience decreasing returns to scale. The threshold would represent the point of scale efficiency, at which returns to scale are constant and efficiency is maximized with respect to size.

Under this framework, size will influence whether increasing, constant or decreasing returns to scale are experienced. Applying this to research activities, the observation of different returns to scale between country-level and university-level research may mean that the size of a country’s overall research effort and the typical size of its universities are not determined by similar factors. For example, if increasing returns to scale at the country level and decreasing returns to scale at the university level are observed, this may indicate that the overall number of universities is smaller than needed to achieve scale efficiency, but that most of these universities are individually too large to be scale efficient.

Other factors may also contribute to the differences between university-level and country-level observations.

  • The country level studies use relatively aggregated data, capturing some of the non-university research and development activities in the countries sampled.
  • Country level research effort is not necessarily subject to some of the constraints which may cause decreasing returns to scale in large universities, such as excessive bureaucracy.
  • Results may be arbitrarily influenced by differences in the available input and output metrics at the university versus country level.

Limitations to conclusions drawn

One limitation of this article is the small scope of the literature review. A more comprehensive review may reveal a different range of conclusions.

Another limitation is that the research outputs studied—published papers, citations, and patents, inter alia—cannot be assumed to correspond directly to incremental knowledge or productivity. This point is expanded upon under “Topics for further investigation” below.

Further limitations arise due to the DEA technique used by most of the studies in Table 1.

  • DEA is sensitive to the choice of inputs and outputs, and to measurement errors.
  • Statistical hypothesis tests are difficult within the DEA framework, making it more difficult to separate signal from noise in interpreting results.
  • DEA identifies relative efficiency (composed of scale efficiency and also “pure technical efficiency”) within the sample, meaning that at least one country, university, or department is always identified as fully efficient (including exhibiting full scale efficiency or constant returns to scale). Of course, in practice, no university, organisation or production process is perfectly efficient. Therefore, conclusions drawn from DEA analysis are likely to be more informative for countries, universities, or departments that are not identified as fully efficient.
  • It may be questionable whether such a framework—where an optimal scale of production exists, past which decreasing returns to scale are experienced—is a good reflection of the dynamics of research activities. However, the frequent use of the DEA framework in assessing research activities would suggest that it is appropriate.

Topics for further investigation

The scope of this article is limited to direct research outputs (such as published papers, citations, and patents). While this is valuable, stronger conclusions could be drawn if this analysis were combined with further work investigating the following:

  • The impact of other sources of new knowledge apart from universities or official R&D expenditure. For example, innovations in company management discovered through “learning by doing” rather than through formal research may be an important source of improvement in economic productivity.
  • The translation of research outputs (such as published papers, citations, and patents) into incremental knowledge, and the translation of incremental knowledge into extra productive capacity. Assessment of this may be achievable through consideration of the economic returns to research, or of the value of patents generated by research.

Implications for AI

The scope for an intelligence explosion is likely to be greater if the returns to scale in research are greater. In particular, an AI system capable of conducting research into the improvement of AI may be able to be scaled up faster and more cheaply than the training of human researchers, for example through deployment on additional hardware. In addition, in the period before any intelligence explosion, a scaling-up of AI research may be observed, especially if the resultant technology were seen to have commercial applications.

This review is one component of a larger project to quantitatively model an intelligence explosion. This project, in addition to drawing upon the conclusions in this article, will also consider inter alia the effect of intelligence on research productivity, and actual increases in artificial intelligence that are plausible from research efforts.

Global computing capacity

Computing capacity worldwide was probably around 2 x 1020 – 1.5 x 1021 FLOPS, at around the end of 2015.

Support

We are not aware of recent, plausible estimates for hardware capacity.

Vipul Naik estimated global hardware capacity in February 2014, based on Hilbert & Lopez’s estimates for 1986-2007. He calculated that if all computers ran at full capacity, they would perform 10-1000 zettaFLOPS, i.e. 1022 – 1024 FLOPS.1 We think these are substantial overestimates, because producing so much computing hardware would cost more than 10% of gross world product (GWP), which is implausibly high. The most cost-efficient computing hardware we are aware of today are GPUs, which still cost about $3/GFLOPS, or $1/GFLOPSyear if we assume hardware is used for around three years. This means maintaining hardware capable of 1022 – 1024 FLOPS would cost at least $1013 – $1015  per year. Yet gross world product (GWP) is only around $8 x 1013, so this would imply hardware spending constitutes more than 13% – 1300% of GWP. Even the lower end of this range seems implausible.2

One way to estimate global hardware capacity ourselves is based on annual hardware spending. This is slightly complicated because hardware lasts for several years. So to calculate how much hardware exists in 2016, we would ideally like to know how much was bought in every preceding year, and also how much of each annual hardware purchase has already been discarded. To simplify matters, we will instead assume that hardware lasts for around three years.

It appears that very roughly $300bn-$1,500bn was spent on hardware in 2015.3 We previously estimated that the cheapest available hardware (in April 2015) was around $3/GFLOPS. So if humanity spent $300bn-$1,500bn on hardware in 2015, and it was mostly the cheapest hardware, then the hardware we bought should perform around 1020 – 5 x 1020 FLOPS. If we multiply this by three to account for the previous two years’ hardware purchases still being around, we have about  3 x 1020 – 1.5 x 1021 FLOPS.

This estimate is rough, and could be improved in several ways. Most likely, more hardware is being bought each year than the previous year. So approximating last years’ hardware purchase to this years’ will yield too much hardware. In particular, the faster global hardware is growing, the closer the total is to whatever humanity bought this year (that is, counterintuitively, if you think hardware is growing faster, you should suppose that there is less of it by this particular method of estimation). Furthermore, perhaps a lot of hardware is not the cheapest for various reasons. This too suggests there is less hardware than we estimated.

On the other hand, hardware may often last for more than three years (we don’t have a strong basis for our assumption there). And our prices are from early 2015, so hardware is likely somewhat cheaper now (in early 2016). Our guess is that overall these considerations mean our estimate should be lower, but probably by less than a factor of four in total. This suggests 7.5 x 1019 – 1.5 x 1021 FLOPS of hardware.

However Hilbert & Lopez (2012) estimated that in 2007 the world’s computing capacity was around 2 x 1020 IPS (similar to FLOPS) already, after constructing a detailed inventory of technologies.4 Their estimate does not appear to conflict with data about the global economy at the time.5 Growth is unlikely to have been negative since 2007, though Hilbert & Lopez may have overestimated. So we revise our estimate to 2 x 1020 – 1.5 x 1021 FLOPS for the end of 2015.

This still suggests that in the last nine years, the world’s hardware has grown by a factor of 1-7.5, implying a growth rate of 0%-25%. Even 25% would be quite low compared to growth rates between 1986 and 2007 according to Hilbert & Lopez (2012), which were 61% for general purpose computing and 86% for the set of ASICs they studied (which in 2007 accounted for 32 times as much computing as general purpose computers).6 However if we are to distrust estimates which imply hardware is a large fraction of GWP, then we must expect hardware growth has slowed substantially in recent years. For comparison, our estimates are around 2-15% of Naik’s lower bound, and suggest that hardware constitutes around 0.3%-1.9% of GWP.

Such large changes in the long run growth rate are surprising to us, and—if they are real—we are unsure what produced them. One possibility is that hardware prices have stopped falling so fast (i.e. Moore’s Law is ending for the price of computation). Another is that spending on hardware decreased for some reason, for instance because people stopped enjoying large returns from additional hardware. We think this question deserves further research.

Implications

Global computing capacity in terms of human brains

According to different estimates, the human brain performs the equivalent of between 3 x 1013 and 1025 FLOPS. The median estimate we know of is 1018 FLOPS. According to that median estimate and our estimate of global computing hardware, if the world’s entire computing capacity could be directed at running minds around as efficient as those of humans, we would have the equivalent of 200-1500 extra human minds.7 That is, turning all of the world’s hardware into human-efficiency minds at present would increase the world’s population of minds by at most about 0.00002%. If we select the most favorable set of estimates for producing large numbers, turning all of the world’s computing hardware into minds as efficient as humans’ would produce around 50 million extra minds, increasing the world’s effective population by about 1%.8

Figure: Projected number of human brains equivalent

Figure: Projected number of human brains equivalent to global hardware under various assumptions. For brains, ‘small’ = 3 x 10^ 13, ‘median’ = 10^18, ‘large’ = 10^25. For ‘world hardware’, ‘high’ =2 x 10^20, ‘low’ = 1.5 x 10^21. ‘Growth’ is growth in computing hardware, the unlabeled default used in most projections is 25% per annum (our estimate above), ‘high’ = 86% per annum (the apparent growth rate in ASIC hardware in around 2007).

 


 

Costs of human-level hardware

Computing hardware which is equivalent to the brain –

  • in terms of FLOPS probably costs between $1 x 105 and $3 x 1016, or $2/hour-$700bn/hour.
  • in terms of TEPS probably costs $200M – $7B, or or $4,700 – $170,000/hour (including energy costs in the hourly rate).
  • in terms of secondary memory probably costs $300-3,000, or $0.007-$0.07/hour.

Details

Partial costs

Computation

Main articles: Brain performance in FLOPS, Current FLOPS prices, Trends in the costs of computing

FLoating-point Operations Per Second (FLOPS) is a measure of computer performance that emphasizes computing capacity. The human brain is estimated to perform between 1013.5 and 1025 FLOPS. Hardware currently costs around $3 x 10-9/FLOPS, or $7 x 10-14/FLOPShour. This makes the current price of hardware which has equivalent computing capacity to the human brain between $1 x 105 and $3 x 1016, or $2/hour-$700bn/hour if hardware is used for five years.

The price of FLOPS has probably decreased by a factor of ten roughly every four years in the last quarter of a century.

Communication

Main articles: Brain performance in TEPSThe cost of TEPS 

Traversed Edges Per Second (TEPS) is a measure of computer performance that emphasizes communication capacity. The human brain is estimated to perform at 0.18 – 6.4 x 105 GTEPS. Communication capacity costs around $11,000/GTEP or $0.26/GTEPShour in 2015, when amortized over five years and combined with energy costs. This makes the current price of hardware which has equivalent communication capacity to the human brain around $200M – $7B in total, or $4,700 – $170,000/hour including energy costs.

We estimate that the price of TEPS falls by a factor of ten every four years, based the relationship between TEPS and FLOPS.

Information storage

Main articles: Information storage in the brainCosts of information storageCosts of human-level information storage

Computer memory comes in primary and secondary forms. Primary memory (e.g. RAM) is intended to be accessed frequently, while secondary memory is slower to access but has higher capacity. Here we estimate the secondary memory requirements ofthe brain. The human brain is estimated to store around 10-100TB of data. Secondary storage costs around $30/TB in 2015. This means it costs $300-3,000 for enough storage to store the contents of a human brain, or $0.007-$0.07/hour if hardware is used for five years.

In the long run the price of secondary memory has declined by an order of magnitude roughly every 4.6 years. However the rate has declined so much that prices haven’t substantially dropped since 2011 (in 2015).

Interpreting partial costs

Calculating the total cost of hardware that is relevantly equivalent to the brain is not as simple as adding the partial costs as listed. FLOPS and TEPS are measures of different capabilities of the same hardware, so if you pay for TEPS at the aforementioned prices, you will also receive FLOPS.

The above list is also not exhaustive: there may be substantial hardware costs that we haven’t included.

Brain performance in FLOPS

Five credible estimates of brain performance in terms of FLOPS that we are aware of are spread across the range from 3 x 1013 to 1025. The median estimate is 1018.

Details

Notes

We have not investigated the brain’s performance in FLOPS in detail. This page summarizes others’ estimates that we are aware of. Text on this page was heavily borrowed from a blog post, Preliminary prices for human-level hardware.

Estimates

Sandberg and Bostrom 2008

Sandberg and Bostrom project the processing required to emulate a human brain at different levels of detail.1 For the three levels that their workshop participants considered most plausible, their estimates are 1018, 1022, and 1025 FLOPS. These would cost around $100K/hour, $1bn/hour and $1T/hour in 2015.

Moravec 2009

Moravec (2009) estimates that the brain performs around 100 million MIPS.2 MIPS are not directly comparable to MFLOPS (millions of FLOPS), and have deficiencies as a measure, but the empirical relationship in computers is something like MFLOPS = 2.3 x MIPS0.89, according to Sandberg and Bostrom.3 This suggests Moravec’s estimate coincides with around 3.0 x 1013 FLOPS. Given that an order of magnitude increase in computing power per dollar corresponds to about four years, knowing that MFLOPS and MIPS are roughly comparable is plenty of precision.

Kurzweil 2005

In The Singularity is Near, Kurzweil claimed that a human brain required 1016 calculations per second, which appears to be roughly equivalent to 1016 FLOPS.4


 

Index of articles about hardware

Hardware in terms of computing capacity (FLOPS and MIPS)

Brain performance in FLOPS

Current FLOPS prices

Trends in the cost of computing

Wikipedia history of GFLOPS costs

Hardware in terms of communication capacity (TEPS)

Brain performance in TEPS (includes the cost of brain-level TEPS performance on current hardware)

The cost of TEPS (includes current costs, trends and relationship to other measures of hardware price)

Information storage

Information storage in the brain

Costs of information storage

Costs of human-level information storage

Other

Costs of human-level hardware

Research topic: hardware, software and AI

Index of articles about hardware

Related blog posts

Preliminary prices for human level hardware (4 April 2015)

A new approach to predicting brain-computer parity (7 May 2015)

Time flies when robots rule the earth (28 July 2015)

Cost of human-level information storage

It costs roughly $300-$3000 to buy enough storage space to store all information contained by a human brain.

Support

The human brain probably stores around 10-100TB of data. Data storage costs around $30/TB. Thus it costs roughly $300-$3000 to buy enough storage space to store all information contained by a human brain.

If we suppose that one wants to replace the hardware every five years, this is $0.007-$0.07/hour.1

For reference, we have estimated that the computing hardware and electricity required to do the computation the brain does would cost around $4,700 – $170,000/hour at present (using an estimate based on TEPS, and assuming computers last for five years). Estimates based on computation rather than communication capabilities (like TEPS) appear to be spread between $3/hour and $1T/hour.2 On the TEPS-based estimate then, the cost of replicating the brain’s information storage using existing hardware would currently be between a twenty millionth and a seventy thousandth of the cost of replicating the brain’s computation using existing hardware.

Costs of information storage

Cheap secondary memory appears to cost around $0.03/GB in 2015. In the long run the price has declined by an order of magnitude roughly every 4.6 years. However the rate has declined so much that prices haven’t substantially dropped since 2011 (in 2015).

Support

Cheap secondary memory appears to cost around $0.03/GB in 2015.1

The price appears to have declined at an average rate of around an order of magnitude every five years in the long run, as illustrated in Figures 1 and 2. Figure 1 shows roughly six and a half orders of magnitude in the thirty years between 1985 and 2015, for around an order of magnitude every 4.6 years. Figure 2 shows thirteen orders of magnitude over the the sixty years between 1955 and 2015, for exactly the same rate. Both figures suggest the rate has been much slower in the past five years, seemingly as part of a longer term flattening. It appears that prices haven’t substantially dropped since 2011 (in 2015).

xxx

Figure 1: Historic prices of hard drive space, from Matt Komorowski

Figure 2:

Figure 2: Historical prices of information storage in various formats, from Havard Blok, mostly drawing on John C. McCallum’s data.


 

Information storage in the brain

The brain probably stores around 10-100TB of data.

Support

According to Forrest Wickman, computational neuroscientists generally believe the brain stores 10-100 terabytes of data.1 He suggests that these estimates are produced by assuming that information is largely stored in synapses, and that each synapse stores around 1 byte. The number of bytes is then simply the number of synapses.

These assumptions are simplistic (as he points out). In particular:

  • synapses may store more or less than one byte of information on average
  • some information may be stored outside of synapses
  • not all synapses appear to store information
  • synapses do not appear to be entirely independent

We estimate that there are 1.8-3.2 x 10¹⁴ synapses in the human brain, so according to the procedure Wickman outlines, this suggests that the brain stores around 180-320TB of data. It is unclear from his article whether the variation in the views of computational neuroscientists is due to different opinions on the assumptions stated above, or on the number of synapses in the brain. This makes it hard to adjust our estimate well, so our best guess for now is that the brain can store around 10-100TB of data, based on this being the common view among computational neuroscientists.


 

  1. “…Most computational neuroscientists tend to estimate human storage capacity somewhere between 10 terabytes and 100 terabytes, though the full spectrum of guesses ranges from 1 terabyte to 2.5 petabytes. (One terabyte is equal to about 1,000 gigabytes or about 1 million megabytes; a petabyte is about 1,000 terabytes.)

    The math behind these estimates is fairly simple. The human brain contains roughly 100 billion neurons. Each of these neurons seems capable of making around 1,000 connections, representing about 1,000 potential synapses, which largely do the work of data storage. Multiply each of these 100 billion neurons by the approximately 1,000 connections it can make, and you get 100 trillion data points, or about 100 terabytes of information.

    Neuroscientists are quick to admit that these calculations are very simplistic. First, this math assumes that each synapse stores about 1 byte of information, but this estimate may be too high or too low…”

    – Wickman 2012

  2. “So it seems human-level hardware presently costs between $3/hour and $1T/hour. ” – our blog post, ‘preliminary prices for human-level hardware’.
  3. See p89. It actually says FLOPS not MFLOPS, but this appears to be an error, given the graph.
  4. ‘If we use the figure of 1016 cps that I believe will be sufficient for functional emulation of human intelligence…’ – Kurzweil, The Singularity is Near, p121
  5. In 2007, GWP was probably about $66T (in 2007 dollars). According to Hilbert & Lopez, the world could then perform 2 x 1020 IPS, which is  2 x 1014 MIPS. According to Muehlhauser & Rieber, hardware cost roughly $5 x 10-3/MIPS in 2007. Thus the total value of hardware would have been around $5 x 10-3/MIPS x 2 x 1014 MIPS = $1012 (a trillion dollars), or 1.5% of GWP.
  6. “The respective compound annual growth rates between 1986–2007 were 61% for general-purpose computations and 86% for application-specific computations, which is 10 and 14 times faster than global GDP during that period, respectively.”

    Hilbert & Lopez (2012)

  7. 2 x 1020 /1018  = 2 x 102

    1.5 x 1021/1018=1.5 x 103

  8. 1.5 x 1021 FLOPS of hardware divided by 3 x 1013 FLOPS/brain gives us 5 x 107 minds.

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