Should Global Health be Tailored Toward the Rich? Altruism and Efficient R&D for Neglected Diseases

7 Appendix 1

This theoretical appendix demonstrates why standard subsidies to lower the price of drugs developed for third-world disease are economically inefficient. The Appendix formally demonstrates that such subsidies do not provide the correct incentives for R&D when drugs are valued not only by those who directly consume them but by altruists in the developed world as well.

Let y denote the output or number of people in the developing world treated by drug specifically designed for a third-world disease such as malaria, p(y) the private inverse demand curve in developing countries, e(y) the global monetary value to altruists when y individuals consume the drug, and c(y) the total cost function. The function e(y) is increasing in the number of people treated with the drug. Let the producer surplus π(y) (profits) and consumer surplus s(y) be denoted by:

and let y_π represent the assumed unique output that maximizes profits π.

Static (or ex post) social welfare W(y), i.e., society’s total welfare after a drug has been commercialized, is then defined by consumer and producer surplus s(y)+π(y) together with the altruistic surplus e(y) of those affected externally by the provision of the new drug:

Let y_W denote the (assumed unique) output or number of people consuming the drug that maximizes W.

An allocation (r, y) is defined as an R&D level r and an output level y after the drug has been discovered. The expected (or ex ante) social welfare given R&D r and output y is:

It represents ex post welfare weighted by the probability the drug is discovered x(r) minus the cost of innovation, which needs to be incurred in any event. The first-best R&D and output pair (r*, y*) maximizes this social welfare. Conditional on discovery of the drug, the optimal output choice is given by:

Clearly, the first-best and ex post optimal output coincide: y*=y_W. Given an optimal welfare W*, the optimal R&D is then one that solves the first-order condition:

More generally the optimal level of R&D that maximizes expected payoffs for any hypothetical ex post award z is denoted r(.) and is defined by:

In this notation, the corresponding first-best R&D takes into account the highest level of ex post welfare r*=r(W(y_W)).

The first-best allocation differs from the allocation that results when the externality of altruism toward third-world disease is corrected by Pigouvian subsidies. If r(π(y)) is the R&D induced by the patented profits, the Pigouvian output will generally not induce the first-best dynamic allocation because the R&D induced by the Pigouvian output coincides with the first-best R&D level only when:

which implies s(y_W)+e(y_W)=0. As consumer and producer surpluses are never zero under a positive externality, this condition fails to hold and Pigouvian corrections will lead to inefficient R&D levels.

8 Appendix 2

This empirical appendix develops an economic model that allows us to calculate the dollar value of global altruism toward third-world diseases such as malaria and HIV/AIDS on the basis of two pieces of information: 1) estimated foreign assistance for these diseases and 2) general estimates of the value of charitable giving.

Consider a poor nation for which the social surplus arising from the utilization of a new drug to treat third-world disease is defined by the WTP of those who directly consume treatment plus the WTP of those in the developed world who value utilization of this drug by the poor, all net of production costs:

Production is assumed to take on constant returns to scale so that c(y)=cy. The value the developed world places on utilization of this drug by the poor, i.e., the altruistic surplus, is given by e(y)=Na(y). There are N foreign altruists who provide treatment assistance to those in the poor country afflicted by the disease targeted by the drug. In this welfare function, treating those in the third world with a new drug is viewed as a public good in the sense that individuals in all wealthy, donor countries can benefit from the donor behavior of a single country. Each altruist therefore has a total WTP for providing the drug to the poor given by a(y), where a(y) is assumed to be increasing and either linear or concave in y. If p_a(y) is the downward sloping demand curve for altruism, then

The level of drug provision that maximizes W(y), y_w, can be attained by an appropriately chosen Pigouvian subsidy, δ, which we interpret to reflect foreign treatment assistance. Under the assumption that observed levels of foreign treatment assistance, e.g., toward malaria or HIV/AIDS, reflect a Pigouvian correction to the altruism externality toward these diseases, we can use this simple model to calibrate the relationship between treatment assistance and the dollar value of global altruism toward these diseases. We therefore assume that observed foreign assistance for a disease such as malaria reflects a correctly chosen Pigouvian subsidy that subsidizes spending by affected countries to the point where the MC of subsidization in poor countries equals the MB to the world of having additional people treated by these lower, subsidized prices. We assume that manufacturers practice cost-based pricing in poorer nations.