Preventives Versus Treatments Redux: Tighter Bounds on Distortions in Innovation Incentives with an Application to the Global Demand for HIV Pharmaceuticals

Kremer and Snyder (2015) show that this bias may be reversed when income (or more generally willingness to pay) covaries sufficiently negatively with disease risk (see their Proposition 18). The bias against vaccines described in the footnoted paragraph above arises in a setting with little or no income variation (covered by their Proposition 3) or in a setting with independent income and disease-risk distributions (covered by their Proposition 16).

For theoretical analyses of vaccine markets in the presence of epidemiological externalities, see Brito et al. (1991), Francis (1997), Geoffard and Philipson (1997), Gersovitz (2003), Gersovitz and Hammer (2004; 2005), Chen and Toxvaerd (2014); as well as our own work (Kremer et al. 2012).

Kremer and Snyder (2015) was initially circulated as a series of National Bureau of Economic Research working papers (Kremer and Snyder 2003, 2013). The international calibration that was provided in Sect. 6 of the 2013 working paper but cut from the 2015 published version became the germ of the present paper. Besides Fig. 4, the other calibrations as well as all of the theoretical results are new developments.

Kremer and Snyder (2016) also include calibrations of international demand. Since they analyze general product markets, their calibrations include only income—not disease risk. Unlike the present calibrations, that paper accounts for within-country heterogeneity by allowing each country to have a different lognormal distribution of income.

We do not deny the importance of epidemiological externalities—indeed some of our other work (Kremer et al. 2012) focuses exclusively on such externalities—but want to focus on other distortions in this paper. Epidemiological externalities can be shut down as a source of distortion by assuming that the pharmaceuticals, while preventing individuals from experiencing disease symptoms, do not slow transmission of an infectious disease. An alternative way to shut down epidemiological externalities would be to consider non-infectious conditions such as heart attacks. The alternative interpretation of demand reflecting purchases by a national agent discussed next can incorporate some forms of epidemiological externality.

For example, if we allow for positive values of \(c_v\) and \(c_d\), the normalization \(c_v = c_d\) equalizes the cost of producing a dose but introduces a bias in the aggregate cost of a universal pharmaceutical program. In particular, universal vaccination would be more costly than universal drug treatment by a factor equal to the reciprocal of the prevalence rate. In addition, the benchmark parameters are associated with the most extreme worst-case bounds under some conditions; see Proposition 12 from Kremer and Snyder (2015).

Tirole (1988) proposes slightly different expressions for relative deadweight loss, dividing by first-best social surplus rather than disease burden. By Eq. (2), our relative concepts coincide with his for benchmark parameters.

To trace out the precise connection between the series of propositions provided here and our past results, Proposition 1 here superficially resembles Proposition  2 of Kremer and Snyder (2015) but they are subtly different. The previous result applied to the case in which both products could be produced but there is heterogeneity in X alone. Proposition 1 here allows for heterogeneity in both X and Y but assumes only product j can be produced. In fact, Proposition 1 here is a corollary of Theorem 1 of Kremer and Snyder (2016) for the special case of benchmark parameters. The translation of that result into the present context is somewhat involved, so instead we provide a direct proof here. Propositions 2 and 3 are new results. The assumptions behind Proposition 3 are identical to those behind Proposition 15 of Kremer and Snyder (2015), so the results are directly comparable. Proposition 3 tightens the previous bound.

Adapting the formula from Lemma 1 of Kremer and Snyder (2015) to the present context, for the vaccine market we havewhere \(N = \sum _{i=1}^I N_i\) is total population size and \((XY)_{(I)}\) denotes the maximum order statistic for the product of \(X_i\) and \(Y_i\): the maximum value for \(X_i Y_i\) across countries. For the drug market,where \(Y_{(I)}\) denotes the maximum value of \(Y_i\) across countries.

The branches of \(\text { LW}\) are built-in functions in standard mathematical software packages including Mathematica, Matlab, and R. Other ways to compute \(\underline{\rho }(\mu ^0_j)\) besides Eq. (10) include reading the value from the graph in Kremer and Snyder (2015, Fig. 4) or taking the value from the tabulation in Kremer and Snyder (2016, Table 2).

We call this a “calibration” rather than an “estimation” exercise because we assume convenient forms for demand and cost and we fix certain important parameters (including \(c_j\), \(s_j\), \(r_j\), and the income elasticity) rather than estimating them from price and quantity data.

The most general form that preserves the property of constant income elasticity is \(A_i Y_i^\varepsilon \), which allows the leading coefficient to vary across consumers by taking it to be a random variable. We do not allow for that source of heterogeneity because doing so would introduce a third random variable characterizing consumers in a country; this would contradict the maintained assumption that consumers are fully characterized by just \(X_i\) and \(Y_i\). A form that is more general but does not introduce a third source of heterogeneity is \(A Y_i^\varepsilon \), with a leading coefficient A that is constant across consumers. Our specification of willingness to pay normalizes \(A=1\). For most of our analysis, this normalization is without loss of generality since all of our surplus calculations will be expressed as a proportion of disease burden; A is a scale factor which divides out of the proportion. In our analysis of prices, which is done in levels, the combined normalizations \(A = 1\) and \(H_i = 1\) comport with World Health Organization procurement thresholds, as will be discussed below.

With benchmark values of the parameters \(s_j = r_j = 0\), one can showWith general values of \(s_j\) and \(r_j\), the expressions become considerably more complicated. Among other things, with imperfect efficacy, a consumer who purchases a vaccine that turns out to be ineffective may later purchase the drug as well.

The UNAIDS website that we used as the source for the aggregate trends displayed in Fig. 2 would be a natural source for country-level HIV data. However, the website seems to have expunged current and historical data for a substantial number of countries, including the United States. We thus relied on a historical publication (UNAIDS 2004a) to recover the country-level HIV data.

Our data include all countries with substantial populations except for Iraq, North Korea, Saudi Arabia, and Turkey, which are excluded because of missing HIV data. Other sources (UNAIDS 2004b) report very low prevalence rates for these countries, so their omission likely has little effect on our results.

To see the improvement that Proposition 2 entails over previous results, compare the tight bound of 62% reported here to the bound from Proposition 15 of Kremer and Snyder (2015), equal to \(\rho _\mathrm {max}^0 - \rho _\mathrm {min}^0 = 44 - 38 = 6\)%. The new bound point-identifies worst-case deadweight loss at 62% of B. The old bound tells us that the supremum on deadweight loss lies somewhere in the interval between 6 and 100% of B, a fairly uninformative statement in this calibration.

Eliminating the Harberger triangle entirely would require almost double the subsidy, $1867.

Several caveats apply to the model of national-government purchases. First, governments must be assumed to purchase at a uniform posted international price. If the firm were instead allowed to post country-specific prices, the outcome would be equivalent to the scenario with perfect price discrimination. If governments were instead allowed to bargain with the firm, this raises a new scenario not yet analyzed. It is easy to see that if parties engage in Nash bargaining, they will arrive at the social optimum. Let \(\alpha \) be the firm’s bargaining share. Then the producer-surplus ratios in the bargaining model would be \(\rho ^0_v = \rho ^0_d = \alpha \). Whether the deadweight-loss supremum is higher or lower in this bargaining scenario than in the uniform-posted-price baseline depends on \(\alpha \): if \(\alpha = 0.5\), then the deadweight-loss supremum is lower in the bargaining scenario, but the reverse is true for sufficiently low \(\alpha \). A second caveat is that the national government’s purchase must be tied to a commitment to universal access for all citizens—as, for example, Brazil committed to for ARTs in 1996 (Reich and Bery 2005)—rather than targeting the rich or otherwise higher demand consumers. A third caveat regards the interpretation of harm relieved by the pharmaceuticals. As noted at the beginning of Sect. 6, if we interpret the \(H_i = 1\) normalization as a year’s course of the drug, which extends life by a year, the parallel interpretation for the vaccine would involve a booster each year to maintain protection. If the vaccine is assumed to provide permanent protection, relieved harm would have to be scaled up by the discounted stream of expected DALYs saved.