Vaccines, drugs, and Zipf distributions

Michael Kremer, Christopher Snyder, Natalia Drozdoff 29 January 2016



“Private companies find vaccines less financially rewarding than drugs. In 2001, the global marketplace for therapeutic drugs exceeded $300 billion, whereas worldwide vaccine sales were only about $5 billion... It is not hard to understand why major pharmaceutical companies, capable of developing drugs and preventive vaccines, generally invest in drugs that patients must take every day rather than shots given only occasionally. Drug company executives have investors to answer to, after all.”
— Patricia Thomas, author of Big shot: Passion, politics, and the struggle for an AIDS vaccine, quoted in Thomas (2002)

“Curing AlDS? … [T]hat's like Cadillac making a car that lasts 50 years. You know they can do it... but they ain't gonna do nothing that f------ dumb."
— Chris Rock “Ain’t No Money in the Cure” bit from 1999 television special, Bigger and Blacker

As the quote from commentator Patricia Thomas—and the even more colourful one from comedian Chris Rock— illustrate, many observers worry that pharmaceutical manufacturers may lack adequate incentives to invest in vaccines because they are much less lucrative than drugs. A drug is more lucrative, according to the quotes, because it can generate a stream of revenue from the consumer rather than just a single payment. From a neoclassical perspective, this explanation is problematic because a consumer should be willing to pay a lump sum for the vaccine equal to the present discounted value of the stream of benefits provided, the same present discounted value as for a drug if both are equally effective.

Kremer and Snyder (2015a) provide a different explanation for manufacturers to be biased against vaccines that is fully consistent with a neoclassical perspective. Vaccines are bought by consumers who have not yet contracted the disease and thus have considerable private information about their disease risk. The act of purchasing a drug reveals the consumer’s disease status, thus reducing that source of private information. Holding constant consumers’ mean value for the two products—after all, both relieve the burden of the same disease—the shape of the demand curve for a drug can be much more conducive to extracting revenue than for a vaccine.

Numerical example

A numerical example can make the point clearly. Consider a population of 100 consumers, 90 of whom have a low disease risk, say 10%. The remaining ten have a high risk – to make things simple, say they are certain to contract the disease. Suppose the disease generates harm equal to the loss of $100 and pharmaceuticals of either form are costless to produce and administer and are perfectly effective. Suppose consumers are risk neutral and fully rational.

What price would a profit-maximising monopolist charge for a vaccine sold to consumers on the private market? The firm has the choice of a broad or narrow strategy. It can try the broad strategy of serving the whole market. The most it can charge is the low-risk consumers’ willingness to pay, which is equal to the expected avoided harm of $10 (the 10% chance times the $100 harm). Revenue equals $10 per consumer times 100 consumers for a total profit of $1,000. Alternatively, the firm can try the narrow strategy of just targeting high-risk consumers, charging each of the 10 of them $100. The producer surplus from this strategy is also $1,000, so in fact the firm is indifferent between the two pricing strategies.

Now consider a drug monopolist’s pricing problem. An expected 19 consumers end up contracting the disease – nine of the low-risk and all ten of the high-risk consumers. Each can be charged the full $100 of avoided harm for a total expected producer surplus of $1,900, nearly twice that from the vaccine.

The monopolist’s bias toward drugs is not cause for concern in this example because both products can lead to the first-best outcome, so a social planner would be indifferent between them. But the example could be modified to create a social distortion. One can show that the R&D cost can be much higher for the drug ($900 more in fact) or the efficacy much lower (as low as 53% effective) before the monopolist would switch to developing the vaccine instead, generating a deadweight loss amounting to nearly half of the total disease burden.

Not all numerical examples involve such a large gap between drug and vaccine revenue. Consider another example in which all 100 consumers now have the same 19% chance of contracting the disease (a number picked to maintain the disease prevalence in the population from the first example). A vaccine manufacturer can now earn $1,900 by charging the $19 expected avoided harm to all 100 consumers, matching drug revenue.

Zipf distribution

There is a special feature of the first numerical example that not only leads to a bias against vaccines, but makes it something of a worst-case scenario. The distribution of disease risk takes the form of a power law. Power-law distributions have the property that an increase in the value is accompanied by a proportionate decrease in the probability of observing a value at least that high. Indeed, the numerical example is a special case of a power law, called a Zipf distribution, in which the values and probabilities scale not by some arbitrary constant but in exact inverse proportion. In particular, moving from low- to high-risk consumers increases disease risk by a factor of ten but reduces the number of consumers having at least that disease risk by the same factor of ten.

Zipf distributions of consumer values present something of a worst case to the vaccine monopolist because it earns the same revenue regardless of the price it charges, in effect leading all pricing strategies to be equally unattractive. The drug is sold after consumers learn their disease status – when consumer values are all the same, and thus no longer have a Zipf distribution.

Figure 1. Zipf distributions of disease risks for various prevalence rates

A Zipf distribution with two consumer types is bad enough for the vaccine monopolist, but matters can be even worse if the Zipf distribution involves a continuum of types, illustrated in Figure 1. Drug revenue is proportional to the whole area under the curve, because this can be shown to be equal to disease prevalence. Vaccine revenue is proportional to the area of a rectangle inscribed underneath. This shape minimises the ratio of the rectangle to the area underneath the curve and thus minimises the ratio of vaccine to drug revenue. Kremer and Snyder (2015a) show that the revenue ratio for any distribution of disease risk can be decomposed into two factors: how much the distribution resembles one of these Zipf curves, and how prevalent the disease is (with less prevalent diseases being relatively worse for the vaccine monopolist, as the figure suggests).

Zipf distributions of consumer values (alternatively called equal-revenue distributions) provide the key to unlocking results in a diverse and growing set of microeconomics papers. Bergemann et al. (2015) use them to prove a general result on the ambiguous welfare effects of price discrimination. Brooks (2013) uses them in the analysis of a new auction mechanism that is optimal when the seller has less information about value distributions than bidders. A growing literature in computer science uses them to bound worst cases for approximately optimal mechanisms in different settings (see Hartline and Roughgarden 2009 for an early such reference) Gabaix’s (2009) survey points to a diverse set of economic phenomena that follow a Zipf distribution, including firm size, income, CEO compensation, and stock-price changes, as well as social and natural phenomena ranging from city size, to word frequency, to earthquakes.

Zipf-similarity of HIV risk

To see how important this effect might be in practice, Kremer and Snyder (2015) provide a calibration based on the distribution of HIV risk in the US population. While estimating a single number like a prevalence rate may be easy, estimating the shape of an entire distribution is much more difficult. The authors take a first pass at this problem by assuming that the number of sexual partners, taken from the 2010 National Health and Nutrition Examination Survey (NHANES), determines HIV risk. Rather than the linear mapping used for their Figure 6, here we redraw the figure assuming the mapping follows more sophisticated epidemiological model.

Figure 2. Zipf-similarity of the distribution of HIV risk 


The estimated distribution of disease risk is drawn in Figure 2 as the black curve. The curve is quite similar to the associated Zipf distribution for the same prevalence level, drawn as the grey curve. Coupled with the low prevalence of HIV in the US population, the Zipf similarity leads to a low ratio of vaccine revenue (proportional to the area of the blue rectangle, the largest one that can be inscribed under the black curve) to the area under the whole black curve, which is proportional to drug revenue. This calibration suggests that revenue from an HIV vaccine would only amount to about a quarter that from a drug.

Other diseases

Human papillomavirus (HPV), by contrast, is much more prevalent than HIV. A calibration using the same distribution of sexual partners and the Kaplan model, but now substituting HPV prevalence, more than doubles the vaccine to drug revenue ratio. The calibrations suggest that the bias against vaccines should be much worse for HIV than HPV, providing a provocative explanation of why there is an HPV vaccine, but none for HIV.

Kremer and Snyder’s (2015a) theory is not specific to sexually transmitted diseases or even infectious diseases. Figure 3 provides a calibration for heart disease (reported but not graphed in the published paper). The calibration estimates the distribution of heart-disease risk by entering NHANES subjects’ information into the online risk calculator based on the Framingham Heart Study of ten-year heart attack risk, allowing for factors such as age, gender, diabetes, smoking, blood chemistry, etc. While not as Zipf-similar as HIV, the figure suggests that a vaccine (or preventive more generally) for heart disease would provide only around 40% of the revenue of that from a drug (or treatment more generally).

Figure 3. Zipf-similarity of the distribution of heart-attack risk


Incentives for pharmaceutical R&D depend on a complex array of factors. This note highlights one: when the risk distribution has a Zipf shape, manufacturers may not be able to extract much revenue from a vaccine (or preventive more generally), while a drug (or more generally any treatment sold after disease status is realised) may still be quite lucrative. Calibrations for actual disease distributions suggest that this factor may be more than a second-order concern for HIV, heart attacks, and other diseases.

Of course, a range of other factors might also bias firms against developing vaccines. For infectious diseases, consumers may rely on the ‘herd immunity’ provided by others’ vaccination, reducing their demand for a vaccine relative to a drug that treats symptoms but does not prevent transmission. Consumer myopia and liquidity constraints may reduce the demand for vaccines relative to drugs (risk aversion may work in the opposite direction). Vaccines may prove to be technologically more complex than drugs for some diseases (although this factor may not, strictly speaking, represent a bias in that a social planner would be similarly deterred by high R&D costs).

More generally, R&D incentives will be limited for any product with a demand curve that resembles those in Figure 1, whether vaccines or widgets. Kremer and Snyder (2015b) draw out the implications for general product markets beyond vaccines. They show that in a globalised economy in which demand increasingly depends on the distribution of world income, the Zipf shape of this distribution has disturbing implications for future R&D incentives. Our narrower focus in this note on pharmaceuticals is still worthwhile. The comparison of vaccines to drugs provides a laboratory of sorts in which the level of demand can be held relatively fixed—both products target the same disease with the same overall burden—while changing the demand curve’s shape depending on the distribution of consumer values for products sold before or after disease status is resolved.


Bergemann, D, B Brooks and S Morris (2015) ‘‘The limits of price discrimination,” American Economic Review, 105: 921–957.

Brooks, B A (2013) ‘‘Surveying and selling: Belief and surplus extraction in auctions,’’ University of Chicago working paper.

Gabaix, X (2009) ‘‘Power laws in economics and finance,’’ Annual Review of Economics, 1: 255–293.

Hartline, J D and T Roughgarden (2009) ‘‘Simple versus optimal mechanisms,’’ Proceedings of the 10th ACM Conference on Electronic Commerce, Northwestern University.

Kremer, M and C M Snyder (2015a) “Preventives versus treatments,” Quarterly Journal of Economics, 130: 1167–1239.

Kremer, M and C M Snyder (2015b) “Worst-case bounds on R&D and pricing distortions: Theory and disturbing conclusions if consumer values follow the world income distribution,” Harvard University working paper.

Thomas, P (2002) “The economics of vaccines,” Harvard Medical International World, September/October.



Topics:  Health economics

Tags:  vaccine, drug, treatment, preventive, pharmaceuticals, R&D, disease, health, population risk, HIV, AIDS, Zipf distribution, US, immunity

Gates Professor of Developing Societies in the Department of Economics, Harvard University

Joel Z. and Susan Hyatt Professor in the Department of Economics, Dartmouth College

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