Innovation appears to be getting harder. At least, that’s the conclusion of Bloom, Jones, Van Reenen, and Webb (2020). Across a host of measures, getting one “unit” of innovation seems to take more and more R&D resources.

To take a concrete example, although Moore’s law has held for a remarkable 50 years, maintaining the doubling schedule (twice the transistors every two years) takes twice as many researchers every 14 years. You see similar trends for medical research - over time, more scientists are needed to save the same number of years of life. You see similar trends for agriculture - over time, more scientists are needed to increase crop yields by the same proportion. And you see similar trends for the economy writ large - over time, more researchers are needed to increase total factor productivity by the same proportion. Measured in terms of the number of researchers that can be hired, the resources needed to get the same proportional increase in productivity doubles every 17 years.

There are lots of issues with any one of these numbers. I’ve written about some of them (on the recent total factor productivity slowdown here, and on agricultural crop yields here). But taken together, the effects are so large that it does look like something is happening: it takes more people to innovate over time.

Why?

The Burden of Knowledge

A 2009 paper by Benjamin Jones, titled The Burden of Knowledge and the Death of the Renaissance Man, provides a possible answer (explainer here). Assume invention is the application of knowledge to solve problems (whether in science or technology). As more problems are solved, we require additional knowledge to solve the ones that remain, or to improve on our existing solutions.

This wouldn’t be a problem, except for the fact that people die and take their knowledge with them. Meanwhile, babies are (inconveniently) born without any knowledge. So each generation needs to acquire knowledge anew, slowly and arduously, over decades of schooling. But since the knowledge necessary to push the frontier keeps growing, the amount of knowledge each generation must learn gets larger. The lengthening retraining cycle slows down innovation.

Age of Achievement

A variety of suggestive evidence is consistent with this story. One line of evidence is the age when people begin to innovate. If people need to learn more in order to innovate, they have to spend more time getting educated and will be older when they start adding their own discoveries to the stock of knowledge.

Brendel and Schweitzer (2019) and Schweitzer and Brendel (2020) look at the age of academic mathematicians and economists when they publish their first solo-authored article in a top journal: it rose from 30 to 35 over 1950-2013 (for math) and 1970-2014 (for economics). For economists, they also look at first solo-authored publication in any journal: the trend is the same. Jones (2010) (explainer here) looks at the age when Nobel prize winners and great inventors did their notable work. Over the twentieth century, it rose by 5 more years than would be predicted by demographic changes. Notably, the time Nobel laureates spent in education also increased - by 4 years.

Brendel and Schweitzer (2019) and Schweitzer and Brendel (2020) also point to another suggestive fact that the knowledge required to push the frontier has been rising. The number of references in mathematicians and economists’ first solo-authored papers is rising sharply. Economists in 1970 cited about 15 papers in their first solo-authored article, but 40 in 2014. Mathematicians cited just 5 papers in the 1950s in their debuts, but over 25 in 2013.

Outside academia, the evidence is a bit more mixed. In Jones’ paper on the burden of knowledge, he looked at the age when US inventors get their first patents and found it rose by about one year, from 30.5 to 31.5, between 1985 and 1998. But this trend subsequently reversed. Jung and Ejermo (2014), studying the population of Sweden, found the age of first invention dropped from a peak of 44.6 in 1997 to 40.4 in 2007. And a recent conference paper by Kaltenberg, Jaffe, and Lachman (2020) found the age of first patent between 1996 and 2016 dropped in the USA as well.

That said, there is some other suggestive evidence that patents these days draw on more knowledge - or at least, scientific knowledge - than in the past. Marx and Fuegi (forthcoming) use text processing algorithms to match scientific references in US and EU patents to data on scientific journal articles in the Microsoft Academic Graph. The average number of citations to scientific journal articles has grown rapidly from basically 0 to 4 between 1980 and today. And as noted in a previous newsletter, there’s a variety of evidence that this reflects actual “use” of the ideas science generates.

Splitting Knowledge Across Heads

But that’s only part of the story. In Jones’ model, scientists don’t just respond to the rising burden of knowledge by spending more time in school. They also team up, so that the burden of knowledge is split up among several heads.

The evidence for this trend is pretty unambiguous. The rise of teams has been documented across a host of disciplines. Between 1980 and 2018, the number of inventors per US patent doubled. Brendel and Schweitzer also show the number of coauthors on mathematics and economics articles has also risen sharply through 2013/2014. Wuchty, Jones, and Uzzi (2007) has also documented the rise of teams in scientific production through 2000.

We can also take inspiration from Jones (2010) and look at Nobel prizes. The Nobel prize in physics, chemistry, and medicine has been given to 1-3 people for most of the years from 1901-2019. When more than one person gets the award, it may be because multiple people contributed to the discovery, or because the award is for multiple separate (but thematically linked) contributions. For example, the 2009 physics Nobel was one half awarded to Charles Kuen Kao "for groundbreaking achievements concerning the transmission of light in fibers for optical communication", with the other half jointly to Willard S. Boyle and George E. Smith "for the invention of an imaging semiconductor circuit - the CCD sensor."

The figure below gives the average number of laureates per contribution, over the preceding 10 years. For the physics and chemistry awards, there’s been a steady shift: in the first part of the 20th century, each contribution was usually assigned to a single scientist. In the 21st centruy, there are, on average, two scientists awarded per contribution. In medicine, there was a sharp increase from 1 scientist per contribution to a peak of 2.6 in 1976, but has slightly declined since then, though it remains above 2.

According to Jones’ the reason for teams is that teams can bring more knowledge to a problem than an individual. If that’s the case, then innovations that come from teams should tend to perform better than those created by individuals, all else equal. For both patents and papers, that’s precisely what Ahmadpoor and Jones (2019) find. For teams of 2-5 people, the bigger the team the higher the citations the paper/patent receives (though the extent varies by field). Wu, Wang, and Evans (2019) also find the bigger the team, the more cited are patents, papers, and software code.

The Death of the Renaissance Man

By using teams to innovate, scientists and innovators reduce the amount of time they need to spend learning. They do this by specializing in obtaining frontier knowledge on an ever narrower slice of the problem. So Jones’ model also predicts an increase in specialization.

In Jones’ paper, specialization was measured as the probability solo-inventors patented in different technological fields within 3 years on consecutive patents. The idea is the less likely they are to “jump” fields, the more specialized their knowledge must be. For example, if I apply for a patent in battery technology in 1990 and another in software in 1993, that would indicate I’m more of a generalist than someone who is unable to make the jump. Jones used data on 1977 through 1993, but in the figure below I replicate his methodology and bring the data up through 2010. Between 1975 and 2005, the probability a solo-inventor patents in different technology classes, on two consecutive patents with applications within 3 years of each other, drops from 56% to 47%.

(While the probability does head back up after 2005, it remains well below prior levels and it's possible this is an artifact of the data - see the technical notes at the bottom of this newsletter if curious)

Schweitzer and Brendel exploit the JEL classification system in economics. These classifications can be aggregated up to the level of one of 9 fields, and Brendel and Schweitzer look at the probability an economist hops from one field to another between two solo-authored publications that are published within 3 years. Among all articles listed on EconLit, it's fallen in half, from 33% to 14% between 1973 and 2014. Restricting attention to top ten publications, it fell even more sharply, from 28% to 0%(!) in 2014.

Lastly, let’s consider the Nobel prizes again. Since Nobel prizes are awarded for substantially distinct discoveries, winning more than one Nobel prize in physics, chemistry, or medicine, may be another signifier of multiple specialties. There have been just three Nobel laureates to win more than one physics, chemistry, or medicine Nobel prize: Marie Curie (1903, 1906), John Bardeen (1956, 1972), Frederick Sanger (1958, 1980). If it takes as long as 25 years to receive a second Nobel prize, then we can be sure there was no multiple-winner between 1958 and 1994. There were 218 Nobel laureates between 1959 and 1994, compared to 207 between 1901 and 1958. That means there were 3 multiple Nobel laureates in the first 207, and 0 in the second 218.

Why are ideas getting harder to find?

Bloom, Jones, Van Reenen and Webb (2020) document the productivity of research is falling: it takes more inputs to get the same output. Jones (2009) provides an explanation for why that might happen. New problems require new knowledge to solve, but using new knowledge requires understanding (at least some) of the earlier, more basic knowledge. Over time, the total amount of knowledge needed to solve problems keeps rising. Since knowledge can only be used when it’s inside someone’s head, we end up needing more researchers. And that’s precisely the dimension that Bloom et al. (2020) use to measure the declining productivity of research - it does take more researchers to get the same innovation.

A few closing thoughts.

First, while the evidence discussed above is certainly consistent with Jones’ story, stronger evidence would be nice. Most of the above evidence is about how things have changed over time. But we should also be able to see differences across fields. The story predicts fields with “deeper” knowledge requirements should have bigger teams and more specialization. Jones (2009) provides evidence this is indeed the case for patents, but as far as I know, no one else has updated his work or extended this line of evidence into academia and other domains.

Second, Jones’ model isn’t the only possible explanation for the falling productivity of research. Arora, Belenzon, Patacconi, and Suh (2020) suggest the growing division of labor between universities and the private sector in innovation may be at fault. As universities increasingly focus on basic science and the private sector on applied research, there may be greater difficulty in translating science into applications. Bhattacharya and Packalen (2020) suggest the incentives created by citation in academia have increasingly led scientists to focus on incremental science, rather than potential (risky) breakthroughs. Lastly, it may also be that breakthroughs just come along at random, sometimes after long intervals. Maybe we are simply awaiting a new paradigm to accelerate innovation once again.

Third, where do we go from here? Is innovation doomed to get harder and harder? There are a few possible forces that may work in the opposite direction.

If breakthroughs in science and technology wipe the slate clean, rendering old knowledge obsolete, then it’s possible the burden of knowledge could drop. In fact, Jung and Ejermo (2014) suggest this may be a reason why the age of first patent declined in the mid-1990s: digital innovation became relatively easy and did not depend on deep knowledge. It would be interesting to see if the three measures discussed above tend to reverse in fields undergoing paradigm shifts.

On the other hand, the burden of knowledge may, itself, make breakthroughs more difficult! As discussed in more detail in a previous newsletter, there is some evidence that teams are less likely to produce breakthrough innovations. This might be because it’s harder to spot unexpected connections between ideas when they are split across multiple people’s heads. In that case, the burden of knowledge can become self-perpetuating.

Alternatively, if knowledge leads to greater efficiency in teaching, so that students more quickly vault to the knowledge frontier, that could also reduce the burden of knowledge. Lastly, it may be possible for artificial intelligence to shoulder much of the burden of knowledge. Indeed, artificial general intelligence could hypothetically upend this whole model, if it is disrupts the cycle of retraining and teamwork that is required of human innovators. I suppose we’ll know more in 20 years.

Technical Notes

For patent data, I use US patentsview data and their disambiguated inventor data. To calculate the probability of jumping fields, I use the primary US patent classification 3-digit class (as in Jones 2009). This patent classification system was discontinued in mid-2015, and it’s possible this is a contributing factor to the uptick observed after 2005. A patent applied for in 2006 only “counts” as a possible field jump if there was a second patent applied for before 2010 and granted before the classification system was discontinued in 2015. This selection effect might be result in an increasingly unrepresentative sample of patents.

Note: I’m experimenting with Substack’s audio features. This week you can read the newsletter below, or listen to me read it by clicking the link above. Thanks!

“No amount of real resources devoted to medical research would have helped European society in 1348 to solve the riddle of the Black Death.” - Joel Mokyr (1998)

There is no currently existing human vaccine for covid-19. Can we force one into existence by promising to spend a lot on it? Is there some price at which we can “buy” a covid-19 vaccine in the next year?

That’s the premise of a proposal by economists Susan Athey, Michael Kremer, Christopher Snyder, and Alex Tabarrok. They propose the US government commit in advance to paying a substantial price for a specified number of vaccine doses: something like $100 each for the first 300 million. The idea is that the potential of winning $30bn will induce pharma companies to pour resources into vaccine development.

Covid-19 and the Profit Motive

We have lots of reasons to believe that a promise to pay more for a covid-19 vaccine would induce more covid-19 vaccine work. Academic research is moving so fast these days that we already have good evidence that pharma companies are extremely responsive to profit signals around covid-19. Bryan, Lemus, and Marshall (2020) track the number of covid-19 therapies at any stage of development, as well as the number of academic publications related to covid-19, to produce this stunning figure:

The black line that is shooting off to the top of the chart is the total number of therapies or publications related to covid-19, as measured against the number of days since the beginning of the pandemic/epidemic. The various dashed lines correspond to the number of therapies and publications for other diseases and/or pandemics (Ebola, Zika, H1N1, and breast cancer). Two things are immediately apparent.

First, covid-19 research is much higher than research related to other pandemic diseases. Second, the gap between covid-19 and other diseases has widened as the magnitude of the covid-19 pandemic becomes clearer. It seems obvious these differences are entirely driven by the difference in demand for a covid-19 therapy, both relative to other drugs and over time, rather than some scientific breakthrough that made it suddenly easier to do covid-19 research. So the above figure is strong evidence that pharma companies respond to profit opportunities and would probably respond further if the government promised to buy a working vaccine at a higher price than the market would normally support.

But dig into the data a bit deeper and there is something troubling. While a vaccine would be the most useful therapy, an unusually large share of the therapies under development are drugs, rather than vaccines. And while it would be nice if an existing drug turned out to be a useful therapy for covid-19, it seems more likely a new disease will require a new kind of drug. But repurposed drugs, rather than novel therapies account for an unusually large share of trials.

This difference has grown over time, as the scope of the pandemic widened. And the divergence between vaccines vs. drugs, and novel drugs vs. repurposed ones, is significantly larger for covid-19 than for Ebola, Zika, and H1N1.

This suggests the rising profitability of a covid-19 treatment is pushing ever more firms to focus on therapies that are not necessarily the best treatment for the disease, but which are most likely to get to the market soon. Vaccines tend to be harder than drugs, and novel drugs tend to be harder than repurposing existing drugs.

It turns out the above evidence is quite consistent with existing research on how medical research responds to market demand. We have good evidence that government promises to pay more for vaccines would likely induce more vaccine research. But the evidence we have also suggests such a policy is most effective at bringing to market a vaccine that does not require much more R&D (but read the ending of this newsletter for caveats).

Markets for Vaccines

The kind of program Athey, Kremer, Snyder, and Tabarrok are proposing is called an Advance Market Commitment, and it’s been successfully tried before. In 2007, a coalition of governments and the Gates Foundation pledged $1.5bn towards the production of 200 million annual doses of a pneumococcal conjugate vaccine for developing countries. If a manufacturer would supply the vaccine at a price of no more than $3.50 per dose, the advance market commitment would top up the rest with a share of the $1.5bn pledged. The program launched in 2009 and in 2010 GSK and Pfizer each committed to supply 30 million doses annually. This amount was increased over time, and a third supplier entered in 2019. Annual distribution exceeded 160 million doses annually by 2016.

Uptake of the pneumococcus vaccine was much faster than uptake for vaccines for a different virus without an advance market commitment (rotavirus). So the advance market commitment seems to have worked.

But there's an important caveat to all this: very little R&D was required to develop the pneumococcal conjugate vaccine. When it was selected, vaccines for similar diseases in developed countries already existed, and vaccines covering the strains in developing countries were already in late-stage clinical trials. So in this case, the advance market commitment pushed firms to quickly build up manufacturing and distribution capacity, but it didn’t push them to do extensive R&D since none was needed.

This is the only time a large-scale advance market commitment has been tried. But that’s not the only place we can look for evidence.

Finkelstein (2004) identifies three US policy changes that increased the profitability of vaccines for some diseases but not others. She then looks to see if firms respond by creating more new vaccines for the affected diseases, relative to the unaffected diseases. Indeed, they do. Let's dig in a bit more.

The three policies Finkelstein uses are (1) the 1991 CDC recommendation that all infants be vaccinated against Hepatitis B; (2) the 1993 decision for Medicare to fully cover the cost of influenza vaccination for Medicare recipients and; (3) the 1986 creation of the Vaccine Injury Compensation Fund which indemnified vaccine manufacturers from lawsuits relating to adverse effects for some specified vaccines. In each of these three cases, policy choices made vaccines for some diseases more profitable, but had no effect on other diseases.

As a control group, Finkelstein considers various sets of alternative diseases that were not affected by these policies, but which otherwise share some of the same characteristics as the affected diseases. All told, she has data on preclinical trials, clinical trials, and vaccine approvals for 6 affected diseases and control groups consisting of 7-26 other diseases, over 1983-1999.

Diseases where policy increased profitability saw an additional 1.2 clinical trials per year and an additional 0.3 new approved vaccines per year (but only 7 years after the policy took effect), as compared to controls. So the promise of more profit did pull in more vaccine development.

But the effect only travels so far up the research stream. When Finkelstein looks farther up the development pipeline, the effect disappears. Affected diseases had no more preclinical trials than the control group. This suggests firms responded to the increased profit by pulling vaccines already far along off the shelf and putting them into clinical trials. But if it stimulated more basic research, the effect was too small to be detected.

Markets for Drugs

There is also a rich vein of research on the extent to which general pharma R&D (not vaccines) respond to changes in the size of the market for different health products. Dubois, Mouson, Scott-Morton, and Seabright (2015) look at the link between potential profits and innovation in the context of global pharmaceutical innovation. They've got a data on drug sales in 14 major countries, which they use to make estimates of the size of the market for different categories of therapeutic medicine. Their goal is to see how changes in the size of the market for a drug change the propensity to develop new drugs for the market. In this case, they're holding the measure of innovation to a relatively high bar: a newly approved drug, marketed in one of their 14 countries, that is also a new chemical entity (i.e., not a modification of an existing drug).

One challenge is that better drugs can, themselves, change the size of the market. Suppose for example, that new drugs just come along randomly as a result of serendipity. In that case, potential profit doesn't actually induce firms to develop new drugs. But if these new drugs find a market, and we're measuring the size of the market by looking at spending on drugs, then we'll create a misleading correlation between the "size" of the market and the number of new drugs. In this case, the number of drugs is "causing" the size of the market, rather than vice-versa. To avoid this, they use a statistical technique (instrumental variables) to pull out the parts of demand that vary due to demographics and overall GDP growth (neither of which should be affected by drug innovation over the 11-year period they work with).

When they do this, they find that bigger markets do indeed lead to more drugs. On average, when the market for a therapeutic category grows by 10%, there are 2.6% more new chemical entities approved over a given time period.

But how scientifically novel are these new drugs? Suggestive evidence comes from Acemoglu and Linn (2004), who perform a similar exercise as Dubois, Mouson, Scott-Morton and Seabright (2015), but on US rather than global sales data. When the market for different diseases in the US changes due to shifting demographics, how does this change the flow of new drug approvals for those diseases? Acemoglu and Linn find the effect of a bigger market is much, much stronger for generic drugs than for new molecular entities.

More direct evidence comes from Dranove, Garthwaite, and Hermosilla (2020) who also investigates this question in the context of global drug development over 1997-2018. They use the US Medicare Part D extension to see if the promise of higher profits leads led firms to pursue more scientifically novel drugs.

The basic idea is that Medicare Part D extended medicare to pay for enrollee's pharmaceutical drugs beginning in 2006. This created a big new market for drugs used by Medicare enrollees (US residents aged 65 and up). Dranove, Garthwaite, and Hermosilla have data on worldwide pharmaceutical company drug trials, and they want to see if companies run more trials on scientifically novel drugs in response to the new opportunities created by Medicare part D.

To measure the scientific novelty of a drug, Dranove, Garthwaite, and Hermosilla count the number of times the specific "target-based action" of the drug has been explored in previous drug trials (of similar or stronger intensity). A target based action comprises the specific (targeted) biological entity and the mechanism used to modify its function: for example, a p38 MAP kinase inhibitor is a target-based action that targets the p38 mitogen-activated protein kinases and inhibits its function. If this target-based action has never before been used in a clinical trial, then a drug using it is considered maximally novel. The more often it has been previously used, the less novel.

With this measure in hand and data on 76,161 clinical trials on 36,002 molecules, Dranove, Garthwaite, and Hermosilla look to see if therapeutic areas with greater profit potential in the wake of Medicare Part D see more clinical trials for scientifically novel drugs. While they do find that more exposed therapeutic areas do see a small increase in trials for the most novel kinds of drugs, once again the effect is much stronger for the least novel drugs. Over 2012-2018 the number of trials for the least novel group of drugs increased 106%, while the number of trials for the most novel group increased just 14% (with most of the gains coming in the second half of that period).

Can We Buy a Covid-19 Vaccine?

So back to covid-19. Can an advance market commitment “buy” a vaccine that doesn’t yet exist? Or are we in the same position as Joel Mokyr’s medieval kings, whose wealth can’t buy any treatment for the bubonic plague until someone thinks of the germ theory of diseases?

First, the studies above suggest these policies do work, but are most effective if the vaccine does not require too much more R&D. Does a covid-19 vaccine require a lot more research? I don’t know. On the one hand, there hasn’t been a human vaccine for this class of virus before. On the other hand, there have been vaccines for veterinary applications (innovation in human and animal health has a lot of similarities), and there seems to be no shortage of options.

Second, the size of the proposed policy is enormous relative to what’s been tried before. So even if these policies normally only work weakly on vaccines that are far from approval, it may be that we still observe a large effect simply because we’re pouring so much money into it.

Third, one of the goals of the Athey, Kremer, Snyder, Tabarrok proposal is explicitly to build manufacturing capacity for vaccines before they are proven, so that we can mass produce them as soon as we find one that works. To the extent building capacity is not a problem that requires R&D, than advance market commitments should work very well. In general, an advance market commitment is only one (albeit big) part of a set of complementary incentives the authors recommend to push and pull a vaccine to market. Give the whole thing a read!

Suppose there was a website called Optimal Kickstarter. Like the actual Kickstarter, it lets people propose projects to be funded, and it lets people crowdfund projects. But there are two differences.

First, Optimal Kickstarter only funds public goods: successful projects are made freely available to all. Second, Optimal Kickstarter has a benevolent patron who supplements the contributions of the crowd in a pre-specified formula. When you fund a project through Optimal Kickstarter, it essentially lets you buy support for a project with someone else’s money.

The way the website works is that every project has a calculator to explain how much patron money you can purchase for the project. For example: maybe there’s an open source software program that you think would be really useful. It is currently funded at $88,209. The site tells you that you can increase the funding to this project by $595 for the price of $1, $1,333 for $5, $1888 for $10, and so on (you need a calculator because the price of patron money is not constant).

Why is it called Optimal Kickstarter? Because the formula determining how much of the patron’s money you can “buy” for a project is from a paper by Buterin, Hitzig, and Weyl (2019) (preprint), which optimally funds public goods.

The trouble with Public Goods

Why so complicated? Public goods are projects that are, by their nature, enjoyable by many users at once and costly to exclude people from using them. Think open source software, art, research, national defense, herd immunity, etc. These goods cannot be efficiently provided by the private market. Since the good can be simultaneously enjoyed by many at once, efficient provision would make it free. But if the good is given away for free, it is impossible to cover the fixed costs necessary to create it. And if you try to charge a price greater than zero (to cover fixed costs), it’s doubly inefficient since it’s costly to prevent people from accessing the good. You have to waste more effort wrapping the project in IP, DRM, etc.

Such goods can be crowd-funded, but under standard economic models, they’ll be drastically underfunded. This is because individuals only consider the private benefits they derive from contributing, not the benefits that accrue to other users. Suppose, for example, for $1,800 programmers can optimize the program mentioned above and make it run 50% faster. Let’s say the value of that is $10 per person. If there are 180 users, the total value of that optimization justifies the cost.

But each user only derives $10 of value themselves so no one has an incentive to provide the full $1800 worth of funding. All the users could band together and provide $10 each, but if contribution is voluntary, this doesn’t solve the problem either. For any individual, if they withhold their $10, the project only raises $1790. That’s nearly $1800, enough to optimize the program by nearly 50% and generate, say, $9.95 worth of value to every consumer. Given this, a selfish individual should just free-ride on the contributions of others, enjoying $9.95 in value for free, instead of spending $10 to get $10 in value. But it doesn’t stop there. The next user can save $10 and enjoy a bit less than $9.95 in value by not donating either. Indeed, this logic continues all along the chain and the project ends up getting a tiny fraction of the funding it should.

Solving the Public Goods Problem

But notice that Optimal Kickstarter doesn’t have this problem. If I chip in $10, the project’s funding rises by $1888, because the patron supplements my contribution. This is enough to fully cover the cost of optimizing the software, meaning I enjoy the full $10 of benefit. If I don’t chip in $10, the software isn’t optimized and I don’t get to free ride on the improvement. So I might as well spend the $10.

The actual formula linking the contributions of the crowd to the funding level of the project is derived from Buterin, Hitzig, and Weyl (2019), who propose a decentralized way of optimally funding public goods. The actual formula isn’t that complicated, but neither is it very intuitive. Let c1 be the contribution of person 1, c2 be the contribution of person 2, and so on. The formula for N contributors is:

Essentially, you take the square root of everyone’s contribution, add all those up, then square that.

In the example I’ve been using, I assumed there were 99 funders, each of whom contributed $9. The square root of $9 is $3, so the total funding of the project is (99 x $3)^2 = $88,209. When you throw in another $10, the square root of $10 is $3.16, so the total funding of the project rises to (99x$3 + $3.16)^2 = $90,097, which is $1,888 more than $88,209. Hence, for $10, you buy $1888 in support for the project.

This formula lets the crowd provide information to the patron about how valuable the public good is, via their contributions. The more contributors, the more funding the patron provides. Notice that if there were just one person who benefited from the public good, and therefore one contributor, the patron doesn’t provide any support:

No more support is needed in this case: the whole problem is when others benefit from your contribution. If it’s just you that benefits, then it’s easy for you to decide how much to fund.

But as the number of contributors grow, there is a stronger and stronger signal that more people benefit from this public good. If there are two contributors each contributing $9, the patron supplies an additional $18. If there are three $9 contributors, the patron supplies an additional $54. If there are 99 contributors ($9 each), the patron supplies an additional $87,318.

Meanwhile, this system ensures people give credible signals about the benefits they receive from the public good since, at the end of the day, they do actually have to spend money to purchase patron support. And the fact that your contributions are supercharged sidelines the free rider problem.

It’s Complicated

In the real world, such a system would have some complications.

For one, each person’s decision about how much to contribute should actually depend on how much others are submitting. But, done properly, this should all be done simultaneously. If I’m the first person to contribute to a project, it is hard to know how much patron support my contribution actually buys, because it will depend on what other people ultimately give. Buterin, Hitzig, and Weyl suggest the program would have to be iterative, with some initial guesses about how what the other contributions would look like, followed by some rounds of fine-tuning in response to better information about how much everyone else actually gives.

Second, the patron’s funds are not actually unlimited. Buterin, Hitzig, and Weyl discuss ways the formula can be modified to take into account the reality that project support may be limited. That shifts you away from the optimal solution, but you still end up with something much better than crowd-funding, and where the patron learns a lot about the value of public goods through contributions.

Third, there are ways to game this system. Splitting contributions among several sock puppet contributors, for example, is a way to buy more support than is warranted. Indeed, a fraudster able to funnel his money through multiple “contributors” could use Optimal Kickstarter as a money pump, if they are able to create projects that simply pay the fraudster from patron money. Fraud detection mechanisms would need to be implemented in a real world implementation.

There are many other issues to be considered. But one of my favorite things about Glen Weyl’s work is that in this, and his many other radical proposals, he always tempers utopian visions with advocacy for an incremental implementation. Start small. See how it works. Correct problems and move forward.

Unfortunately, Optimal Kickstarter doesn’t exist. But I think it’s a cool idea. It would be a great public good. Just the kind of website that I bet would get a lot of funding from Optimal Kickstarter.

Appendix

If there are any curious and mathematically inclined readers out there, here’s a short proof for why Optimal Kickstarter is indeed optimal (warning: calculus).

Denote total funding for a project by F. The value of the public good to person i is Vi(F), and the total value of the project is the sum of Vi(F) across all individuals i. Assume Vi(F) is concave, continuous, etc. The optimal level of funding F for the public good satisfies:

This is just the level where a dollar of funding buys a dollar of value (when we add up the value of the public good across all people).

An individual using optimal kickstarter only considers the value they receive, but also only cares about the cost of their own money. They are maximizing Vi(F) - ci, which at the optimum satisfies:

This is where a dollar of individual i’s money buys a dollar of value to individual i. The key idea is that a dollar of individual i’s money raises total funding F by more than $1.

The partial derivative of F with respect to ci, comes from the optimal funding formula, and is:

Using this, we rearrange the individual’s optimality condition to yield:

Last, we add up these conditions across all individuals i. That gives us:

Which is the precisely the condition for the optimal level of funding F.

I recently posted a working paper called The Case for Remote Work that some of you might find interesting. It’s kind of like this newsletter, in that it surveys a lot of different academic papers to make an argument. Also like this newsletter, it’s written in a way that I hope is accessible to non-specialists. But unlike this newsletter, it’s 31 pages (plus 6 pages of references).

From the abstract:

The case for remote work goes well beyond its use during the covid-19 global pandemic. Over the last ten years, research from a variety of subdisciplines in economics and other social sciences collectively makes a strong case for the viability of remote work for the long-run. This paper brings this research together to argue remote work (also called telework) is likely to become far more common in the future for four reasons.

1. The productivity of individual workers who switch to remote work is comparable or higher than their colocated peers, at least in some industries.

2. Matching firms to geographically distant workers is becoming easier thanks to technological and social developments.

3. Remote workers tend to be cheaper because workers value geographic flexibility and the ability to work remotely.

4. The benefits of knowledge spillovers from being physically close to other knowledge workers has been falling and may no longer exist in many domains of knowledge.

While the prevalence of remote work (pre-covid-19) is small, I show it was already rising rapidly with plenty of room to continue growing. Finally, I argue remote work has positive externalities and should be promoted by policy-makers.

Much of the material under item 4 above is drawn from papers that have been discussed in this newsletter. But the rest is probably new material to regular readers of this newsletter.

Back to your regular coverage of new work on the economics of innovation in a few weeks!

Hey everyone; thanks for reading this newsletter. My original goal for this project was for it to be a weekly delivered on Thursdays. But it turns out time is scarce when you have two kids under five at home due to the global pandemic. So delivery will be ad-hoc for awhile.

Knowledge spillovers, in the economics of innovation literature, are when a firm makes use of knowledge discovered by other firms. The existence of knowledge spillovers is a classic reason why there may be underinvestment in R&D. When a firm decides how much R&D to do, it weighs the costs it bears against the benefits it expects to receive - not the benefits all firms expect to receive.

Of course, just because something could happen in theory doesn’t mean it happens in practice. So how big a deal are spillovers anyway? A couple studies using data on patents suggests spillovers are really important. More knowledge spills over than stays put.

Clancy (me), Heisey, Moschini, and Ji (2019) have a forthcoming paper that looks at this question for the specific case of agriculture. We wanted to see how much agricultural innovation drew on ideas developed outside of agriculture. So we identified all US patents for a variety of agricultural subsectors over 1976-2018 and tried to measure the share of “knowledge flows” from outside agriculture to inside it. We used a few different approaches, each imperfect in different ways, but hopefully adding up to something convincing.

First, we looked at the citations patents make to other patents. In most cases, more than half of the citations go to patents we don’t classify as belonging to agriculture, or to patents belonging to firms whose patent portfolio is mostly non-agricultural.

Second, we looked at the citations patents make to academic journal articles. Again, most of those citations don’t go to journals we classify as agricultural science journals.

Third, we look for “novel concepts” in the text of the abstracts and titles of agricultural patents. For our purposes, a “novel concept” is a string of one-to-three word text that is common after 1996, but absent from agricultural patents before then. (We also manually go through all these concepts to verify they correspond to technological ideas). We then look to see how many of these novel concepts are already out there, in non-agricultural patents. It turns out, most of them are.

As the figure above makes clear, there is some significant variation, even within agriculture: newly patented plants really do mostly rely mostly on agricultural R&D. Veterinary drugs don’t. But if you pick one of the six fields below at random, one of our five indicators at random, and a specific knowledge flow proxy (citation or concept) at random, the chance it corresponds to a flow from out of agriculture to inside it is about 65%. That suggests spillovers are the main source of ideas in agriculture!

What about outside of agriculture? Azoulay, Graff Zivin, Li, and Sampat (2015) also find a big role for knowledge spillovers in medicine. They identify 315,982 life sciences patents granted between 1980 and 2012, plus the citations those patents make to scientific journal articles in PubMed, plus the grant acknowledgements those journal articles make to NIH funded grants. In this way, they can trace out the link between an NIH grant for basic scientific research and a patent.

One nice thing about NIH grants is that they’re assigned to different disease areas. So one thing the authors do in this paper is look to see if grants contribute to patents that belong to different disease areas than what was funded: how often does a grant for cancer research contribute to a patent for diabetes medication (for example)?

To do that, they have to assign patents to different disease areas. Their approach is to use the disease area associated with the plurality of cited publications; i.e., if 35% of cited papers are associated with cancer grants, and no other disease has more than 35% of citations, then the patent is assigned to cancer. By this method, grants are slightly more likely to be cited by a patent from a different disease than they are from a patent associated with the grant’s disease. In short - spillovers in the basic science underlying different diseases are substantial.

But again; this is another specific area. Can we say anything systematic?

Bloom, Schankerman, and Van Reenen (2013) tries. This paper uses the set of all publicly traded US firms over 1980-2001 in an effort to assess how R&D by one firm affects others. Unlike the previous two papers, it comes in with beliefs about what kinds of firms are likely to have strong spillovers and then checks those beliefs by seeing if they are correlated with predicted outcomes. Essentially, they create measures for spillovers based on (1) the amount of R&D a firm does (more R&D leads to more potential spillovers to all firms) (2) adjusted for each firm by the degree of overlap in the kinds of patents they own (more R&D spills over to firms with similar technologies in their patent portfolios).

Then, with these measures of potential spillovers across different firms, they look to see if those measures are correlated with things in the expected ways: do more spillovers (by this measure) lead to more patents, productivity or profits, for example? It’s actually a lot more complicated than that (I took a shot at explaining how this paper works to non-specialists here). But suffice it to say they have nice estimates of how R&D by one firm affects every other firm in their sample. This lets them estimate the private return on R&D and the social return on R&D.

To see the difference, suppose Apple is deciding whether to spend another dollar on R&D. The increase in Apple’s profits due to that dollar are the private returns to R&D. The increase in Apple’s profits, and Google’s profits, and all other publicly traded firms, is the social return on R&D, as measured in this paper. If there’s a strong link between spillovers and profits, than the social return might be large. If there’s no link between their hypothesized measure of spillovers, than the social return might simply be the private return.

They find the private return on R&D is 21%; but the social return is 55%. Again - more than half the value of R&D comes from it’s impact on other firms!