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The Zipf distribution is used to model situations in which a few observations have a very high value or impact and for a large part of the total, while a very long tail of observations have medium, small, or very small values. Examples include:. Here we used it to model the distribution of files or data on our laptop or on the cloud for a specific companyranked by size.

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Create Subscribe. Many observers believe that pharmaceutical firms prefer to invest in drugs to treat diseases rather than vaccines. The population risk of such diseases resembles virtual date in zipf Zipf distribution, which makes the shape of the demand curve for a drug more conducive to revenue extraction than for a vaccine.

Based on revenue calibrations using US data on HIV risk, the revenue from a drug is about four times greater. 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. You know they can do it 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 virtual date in zipf if both are equally effective. Kremer and Snyder a 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. A numerical example can make the point clearly. The remaining ten have a high risk — to make things simple, say they are certain to contract the disease. 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.

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An expected 19 consumers end up contracting the disease — nine of the low-risk and all ten of the high-risk consumers. But the example could be modified to create a social distortion. Not all numerical examples involve such a large gap between drug and vaccine revenue.

There is a special feature of the first numerical example that not only le 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 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.

A Zipf distribution with two consumer types is bad enough for the vaccine monopolist, but matters can be even worse virtual date in zipf 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 a 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 in a diverse and growing set of microeconomics papers. Bergemann et al. Brooks uses them in the analysis of a new auction mechanism that is optimal when the seller has less information about value virtual date in zipf than bidders.

To see how important this effect might be in practice, Kremer and Snyder provide a calibration based on the distribution of HIV risk in the US population.

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While estimating a single like a prevalence rate may be easy, estimating the shape of an entire distribution is much more difficult. Rather than the linear mapping used for their Figure 6, here we redraw the figure assuming the mapping follows more sophisticated epidemiological model. The estimated distribution of disease risk is drawn in Figure 2 as the black curve.

The curve virtual date in zipf 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 le 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.

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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. Figure 3 provides a calibration for heart disease reported but not graphed in the published paper. This note highlights one: when the risk distribution has a Zipf shape, manufacturers may not be able to extract much revenue from virtual date in zipf vaccine or preventive more generallywhile 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. Consumer myopia and liquidity constraints may reduce the demand for vaccines relative to drugs risk aversion may work in the opposite direction. Kremer and Snyder b draw out the implications for general product markets beyond vaccines.

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Our narrower focus in this note on pharmaceuticals is still worthwhile. Topics: Health economics. Joel Z. Research-based policy analysis and commentary from leading economists Create Subscribe. Search form. Vaccines, drugs, virtual date in zipf Zipf distributions Michael Kremer, Christopher Snyder, Natalia Drozdoff 29 January Many observers believe that pharmaceutical firms prefer to invest in drugs to treat diseases rather than vaccines. City scale and productivity in China. John Gibson, Chao Li. Does price regulation affect the adoption of new pharmaceuticals? Rethinking urbanisation. Do Americans really pay too much for pharmaceuticals?

Antonio Cabrales. Numerical example A numerical example can make the point clearly.

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Zipf distribution There is a special feature of the first numerical example that not only le to a bias against vaccines, but makes it something of a worst-case scenario. 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. 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. Figure 3. Michael Kremer.

Christopher Snyder. Natalia Drozdoff.

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