OWJC: Why are Drugs More Profitable than Vaccines? (Health Economics Working Paper)
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Kremer, M. and Snyder C.M. (2003) Why are Drugs More Profitable than Vaccines? NBER Working Paper Series Reference № 9833. Available at: http://www.nber.org/papers/w9833
While I really enjoy writing about science and biotech, my D.Phil is in health economics, so I will branch a bit into this field here too. I chose this paper as it was recommended to me from a Prof I was discussing my research with.
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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 2002)
The standard use of vaccines versus drugs in healthcare management differs significantly. Vaccines are generally given one or twice as a prophylactic to a healthy population. In comparison, drugs are usually used in patients who already have the disease (so therefore a much smaller subset of the population), the patient usually takes the drug regularly for a set period of time, and there is an immediate health benefit conferred by the therapy.
Vaccines promote herd immunity. Due to the development of vaccines, we have largely eradicated many diseases that were devastating for earlier generations. The smallpox vaccine was the first vaccine to be developed, in 1796 —after a regimented global vaccination campaign beginning in 1967, the last reported case of smallpox was in 1977!
In this working paper, the authors argue that, while simple models suggest that vaccines and drugs yield the same revenue (and therefore pharmaceutical companies should be equally incentivized to develop both), in reality, pharmaceutical companies are more incentivized to develop drugs than vaccines for two reasons:
- Consumers of vaccines are heterogenous who have different likelihoods of actually contracting the disease of interest, but none at the moment of vaccine purchase will actually have the disease. If vaccines demand high prices, only those at high-risk of developing the disease will consider purchasing the vaccine. This is a small subset of the population. In comparison, drugs customers will have already contracted the disease, therefore, the company may demand the high price from everyone with the disease, whether or not the patient was previously considered high risk or low risk — this makes for a larger segment of the population and therefore larger value of the drug than the vaccine. The authors go on to develop an equation to prove this point — specifically, that in any heterogenous population, a drug yields more revenue than a vaccine.
- Vaccines interfere with the spread of disease, and therefore the more successful and widely adopted they are, the smaller the eligible population and demand for their product becomes (as they effectively “cure” their target population)
The example given in the paper to explain reason one is really nice, so I will rephrase it here:
- There are 100 total customers
- 10 have a 100% chance of developing the disease (high risk), while 90 have a 10% chance (low risk)
- The vaccine developer can charge $100k for their vaccine. Only the 10 high risk patients will be willing to pay this. Revenue = $1 million.
- Alternatively, the vaccine developer can charge $10k for their vaccine. All of the 100 are willing to pay this. Revenue = $1 million
- The drug developer charges $100k for the treatment to any patient who develops the disease. On average, 19 of the 100 will. All 19 are willing to pay since they are now sick. Revenue = $1.9 million
This is obviously a gross simplification, but it recapitulates the concept well.
CONCLUSIONS OF THE PAPER
- In a homogenous population (everyone has a known, same risk of developing the disease), a firm strictly prefers to develop either a vaccine or drug only dependent on its production price, development price, safety, and efficacy. The firm is otherwise indifferent if the values are the same.
- In a heterogenous population (the risk of developing the disease varies person to person, and the firm is unaware of an individual’s risk), the profit possible from a drug is always greater than with a vaccine
- A drug can charge higher prices, up to the value of the harm of the disease to the patient, to all who have the disease
- A vaccine can only charge an equilibrium price, at which those with a certain risk of developing the disease it prevents will be willing to pay the price. This price will be below maximum price that the drug can charge
- Vaccines cure their patient population as they are adopted and therefore shrink their consumer population and the demand for the vaccine as the risk of contracting the disease decreases
- All of the social welfare gained from a drug is translated into revenue for the firm
- Only a portion of the social welfare from a vaccine is translated into revenue for the firm, as the unvaccinated benefit from herd immunity
- In a static model, the drug can earn about double the revenue from the vaccine. This is higher in the dynamic model.
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I will go pretty in depth explaining the math in this and most of the equations developed by the author. I’ve summarized what this all means above and you do not need to read the below to understand the points of the paper. If you want to be walked through the models and equations developed in the paper to reach the above conclusions, proceed on!
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METHODS
Homogeneous Static Model
This is from the perspective of a pharmaceutical provider (“the firm”) which has the choice of developing a vaccine or a drug. This is mutually exclusive.
Variables
(the _ denotes subscript, which medium doesn’t seem to support)
j ← the medicine
j = v ← if the medicine developed is a vaccine
j = t ← if the medicine developed is a drug
i → the consumer
x_i ∈ [0, 1] ← the probability of the consumer contracting the disease
F(x_i) ← function describing the distribution of probabilities of x_i
d_i = 0 → the consumer does not develop the disease
d_i = 1 → the consumer does develop the disease
k_j ∈ [0, ∞) ← the present discounted value of the fixed cost of developing the medicine
c_j ∈ [0, ∞) ← the present discounted value of the cost of administering medicine j
e_j ∈ [0, 1] ← the efficacy of medicine j (this is a probability — the probability the medicine prevents the disease from doing harm)
σ_j ∈ [0, 1] ← the probability the patient experiences side effects from the drug
s_j ∈ [0, ∞) ← present discounted value of the harm of the side effects — this is conditional on σj
h ∈ [0, ∞) → present discounted value of the harm of the disease to the consumer if it is contracted
P_j ∈ [0, ∞) ← the present discounted value of the price the firm receives for the medicine
Π_j ← Revenue of the medicine from the perspective of the firm
x^∈[0,1] ← cut-off value where consumer is willing to buy the vaccine
Conditions
- In the homogeneous model, x_i will be a single value and not a distribution of probabilities
- x_i is known before the consumer pursues any medicine
- P_j will be discounted by the net harm of the therapy, as the consumers willingness-to-pay will decrease as σ_j increases
- A vaccine is administered before d_i is realized, while a drug is administered after d_i is realized
- A consumer will only purchase the drug if they contract the disease
Equations as Developed in the Paper
Firm’s maximum profit from the vaccine — EQN 1
In english…
expected price of the vaccine − cost of administering the medicine −cost of developing the vaccine = (probability of the consumer contracting the disease*harm expected from the disease should it be contracted*probability that the vaccine is effective) − (probability of side effects*value of harm of the side effects) − cost of administering the vaccine −cost of developing the vaccine
The profit maximizing price of the vaccine would extract all of the consumer surplus benefit of the vaccine.
The consumer’s expected net surplus is: (x_i)(h)(e_v) − (σ_v)(s_v)−(P_v) (so, the harm avoided minus the initial cost of the drug).
**A consumer will only purchase the vaccine if the cost of possibly developing the disease is higher than the cost of the vaccine. This is dependent on their personal probability of developing the disease.**
Firm’s maximum profit from the drug—EQN 2
Again, in english…
the probability of the consumer contracting the disease*(the expected price of the drug − cost of administering the medicine) − the cost of developing the drug = the probability of the consumer contracting the disease*(harm expected from the disease should it be contracted*probability that the drug is effective − probability of side effects*value of harm of the side effects − cost of administering the drug)−cost of developing the drug
The authors then go on to evaluate a set of propositions.
Proposition One
- The firm strictly prefers to develop the vaccine over the drug treatment if and only if (IFF) the following conditions are met if EQN 1 strictly exceeds EQN 2
- The firm strictly prefers to develop the drug treatment over the vaccine IFF EQN 2 > EQN 1
- The firm is indifferent if EQN 1 = EQN 2
So, if one modality is cheaper to produce or develop over the other, safer, or more effective, the firm will prefer it over the other. This is not exhaustive of all the reasons a firm may prefer one modality over the others, but it is a good start. It is also a rather obvious conclusion. So, for future analyses, these new conditions were set:
Conditions, cont.
- k_j = c_j = σ_j = 0 (so, the cost of developing and administering the medicine is $0, and the probability of side effects is 0)
- e_j = 1 for j = v or j = t (so, both the vaccine and drug are 100% effective)
This leads to EQN 1 and EQN 2 simplifying down to:
Proposition Two
- The firm is indifferent between developing the vaccine and the drug in a homogeneous consumer model
Heterogenous Static Model
The above model is used in the heterogeneous static model, but with one modifying condition:
Conditions, cont.
- In the heterogenous model, x_i will be a distribution of probabilities following the function F(x_i)
- x_i is known to the consumer, but not to the firm
Equations as Developed in the Paper, cont.
Firm’s monopoly profit from the vaccine — EQN 3
In english…
The revenue of a vaccine is equivalent to the maximum of: the harm if the disease occurs * the cutoff value * [ 1 − the distribution of the values of x^_v]
The consumer i will buy the vaccine if the price P_v < h(x_i) where h(x_i) is the expected harm from the disease (the harm had if the disease occurs * probability of having the disease).
There is a cutoff value x^_v = (P_v)/h. (Price of the drug divided by the harm expected if they contract the disease) This is just found from rearranging P_v < h(x_i). When x_i is greater or equal to x^_v, the consumer prefers to buy the vaccine than not.
Therefore, the firm’s profit is dependent on x^_v. (See left)
This means that the profit possible is the total of the price of the vaccine times the differential of the function describing the values possible for x_i (probability of contracting the disease, again which only the consumer knows), integrated from x^_v to 1, again with x^_v being the cutoff value beyond which a consumer is then unwilling to buy the vaccine as a prophylactic.
Since x_i is unknown, the firm must charge a uniform price to all consumers. The firm will want to maximize this value. So, if we define Π_v as the monopoly profit of the vaccine, we get EQN 3.
Firm’s revenue from the drug — EQN 4
Any consumer who has the disease is willing to buy the drug, as long as P_t (price of the drug) is less than h (the harm from the disease). Here, d_i = 1 , since the consumer has the disease. Therefore, the firm’s optimal price is P*_t = h. Here, the consumer will only pay if they actually contract the disease, which is defined by x_i.
So, Π_t, the maximum profit from the drug, is the integral of the harm of contracting the disease * the probability of a consumer contracting the disease * the differential of the possible values of x_i (from 0 to 1). The third portion of the equation, hE(x_i), uses the expectations operator. Basically this just means the value on average that x_i takes.
Visual Representation of the Increased Value of the Drug over the Vaccine
Below is Figure 2 from the paper. On the x axis is the probability of the consumer i contracting the disease. On the y axis is the price of the medicine. At a price of h(x^), only those whose risk to develop the disease (x_i) is greater than or equal to x^ (the barrier value we defined) will purchase the vaccine. So, only those who fall within box B will buy the vaccine.
Box B = Π_v
On the other hand, patients have a certain probability of contracting the disease, but if they do, they will buy the drug where P*_t = h. So the profit of the drug for the disease is (x_i)*h. So, Boxes A + B + C will buy the drug, = Π_t.
The increased value of the drug over the vaccine approach is Π_t − Π_v = Boxes A and C.
This leads to the next proposition by the authors:
Proposition Three
- If the population of consumers who have a probability of contracting the disease is heterogenous, then Π_t > Π_v. So, the firm will profit more from developing a drug than a vaccine.
There are three more propositions under the heterogenous static model, however I won’t go over them right now in such detail as they are less crucial to the main points of the paper, and the explanation given in the paper is straightforward and does not introduce new equations.
The result of the static model evaluation have addressed reason one to why vaccines are less profitable to pharmaceutical companies than drugs — because of the heterogeneity of their population and the masked risk of the consumers.
Next, the dynamic model will address reason two — that vaccines cure their consumers and decrease the amount of eligible consumers.
Dynamic Model
This model shows that drug treatments are more profitable than vaccines, even in a homogenous population, if vaccines cause greater reductions in disease transmission that drug treatments.
Vaccines reduce disease transmission more than drugs because (1) people spread the disease in the latency period between developing the disease and starting the drug treatment, and (2) drugs often treat the symptoms instead of curing the disease, so even if the patient is now functionally healthy, they may still be carrying the disease.
New Variables
δ ← rate at which people are born and die into the population
S ← susceptible
I ← infected
V ← vaccinated
β ← rate at which S contact I and the proportion of contacts which lead to a new infection
ξ ← fraction of newborns who are vaccinated
W_j ← social welfare
* on a variable denotes the equilibrium value for the respective variable
The population is defined by S + I + V = 1. The rate of new infections is βIS.
Rate of change of the susceptible population — EQN 5
In English…
= birth rate * (1 − vaccinated newborn fraction) − rate of new infections − susceptibles who die
Rate of change of the infected population — EQN 6
In English…
= rate of new infections− infected who die
Rate of change of the vaccinated population — EQN 7
In English…
= birthrate of vaccinated newborns − vaccinated who die
(Instead of the dot letter, I will use ` to denote the rate of change variable)
Here, the paper starts talking about trivial and non-trial steady states. Generally, trivial states/solutions are those involving the number 0. Non-trivial solutions do not include 0.
Setting S` = I` = V` = 0 from EQN 5 to 7 — EQN 8
(the * denotes the equilibrium value of this variable)
In English…
Susceptible = birth rate/rate of contacts to new infection
Vaccinated = fraction of newborns who are vaccinated
Infected = 1−fraction of vaccinated newborns −(birthrate/rate of contacts to new infection)
This holds true when ξ < 1−δ/β. (because if it was not, then I would become less than 0)
δ/β is the latent proportion of healthy individuals in the steady state before a vaccine is introduced into the population.
λ = δ/β for simplicity
Rate of new infections when a vaccine is developed — EQN 9
In English…
Rate of contact to new infection * infected population after vaccine intro’d * susceptible population after vaccine intro’d =birthrate(1−fraction of newborns vaccinated −birthrate/rate of contact to new infection)
OR
Rate of contact to new infection * infected population after vaccine intro’d * susceptible population after vaccine intro’d = birthrate * I*
Rate of new infections when a drug is developed — EQN 10
In English…
Rate of contact to new infection * infected population after drug intro’d * susceptible population after drug intro’d =birthrate(1−birthrate/rate of contact to new infection)
EQN 9 and 10 differ just by the vaccinated newborns (ξ) term. The drug therapy can be given at any point, while presumably the vaccine will be a prophylactic.
So basically what these equations mean are that the rate of new infections (so new conversions from susceptible to infected, βIS) is dependent on the rate of contact of infected to susceptible, β. This equals the birthrate times 1 minus those who are susceptible but not infected (λ = δ/β). Those who are susceptible but not infected are defined by λ.
The authors then define some more variables and minor equations:
Π_j = P_j * Q_j
W_j = social welfare
P*_j ← equilibrium medicine price
Q*_j ← equilibrium quantity sold
Π*_j ← equilibrium revenue for firm
W*_j ← equilibrium social welfare from medicine
P*_t ← profit maximizing price for drug
P*_v ← profit maximizing price for vaccine
Conditions, cont.
- Drug is taken only once, offers a full relief of symptoms
Proposition 7
- In the dynamic model with drug treatment, the equilibrium price P* = h, quantity Q*_t = δ(1 − λ), flow profit is Π*_t = hδ(1 − λ) and flow welfare is W*_t = hδ(1 − λ).
In English…
P*_t = h as the price consumers are willing to pay for the drug is equal to the financial harm of the disease. Since all newly infected buy the drug, the quantity sold Q*_t is the birthrate δ * the newly infected rate 1 − λ.
Resulting flow profit is Π*_t = (Q*_t) * (P*_t) = hδ(1 − λ).
The social benefit from the medicine is also hδ(1 − λ) as it is the harm alleviated from the drug curing the symptoms of the disease in all which acquire it.
Visual Representation of the Dynamic Vaccine Model
As the quantity of vaccines sold (Q_v) increases, the infection prevalence I_v decreases. BB’ shows the relationship between these two variables.
I^ is the infection prevalence at which when I_v is greater than or equal to, the consumers will buy the vaccine. This is represented by the blue line AA’.
AA’ gives the quantity of the vaccine sold, dependent on the infection prevalence and the price set of the vaccine. A higher price P_v shifts AA’ to the right — there needs to be a higher infection rate for consumers to be willing to pay a higher price for the vaccine. The equilibrium quantity Q*_v is denoted by the intersection of AA’ and BB’.
Proposition 8
- In the dynamic model with vaccine treatment, the equilibrium price P∗_v = h(1 −√λ), quantity is Q∗_v = δ(1 −√λ), flow profit is Π∗_v = hδ(1 −√λ)² , and flow welfare is W∗_ v = hδ(1 − √λ)
I may come back in here and explain the proof for Proposition 8, but the math is a bit hard at this point to translate into english, and is well described in the actual paper’s appendix, so I will leave it for now and just take these equations at face value. Also this article is getting really, really long…
Proposition 9
- In the dynamic model at the steady state, the firm’s profit is higher with a drug treatment than a vaccine. All of the social surplus of the drug is appropriated by the firm, but only a fraction (1 −√λ) of the vaccines’ is.
- Firm only will develop a drug, not a drug and vaccine, as the firm appropriates all of the social benefit from the drug.
All of the social welfare (W*_j) is appropriated because the firm can charge the consumer’s maximum willingness to pay to all consumers (all infected). With the vaccine, only a fraction of the eligible consumers will buy the vaccine, as to not cure the whole consumer population and therefore negate demand for the vaccine.
The unvaccinated susceptibles, while they do not purchase the vaccine, still benefit from the vaccine due to the lowered infection rate when the vaccine is present. The firm does not appropriate this social benefit.
Proposition 10
- In the limit, as the initial prevalence of the disease approaches zero, the ratio of profit from the drug treatment to the vaccine ( Π∗_t : Π∗_v ) grows without bound i.e., limλ→1(Π∗_t /Π∗_v) = ∞, and the ratio of profit to social welfare from a vaccine goes to zero, i.e., limλ→1(Π∗_v/W∗_v) = 0
Remember λ, the initial proportion of healthy individuals. As λ approaches 1 (AKA, when there is a very small prevalence of the disease within the population) the profit of the drug over the vaccine becomes infinitely large. As this number goes to 1, the ratio of profit versus social welfare of the vaccine goes to 0, AKA, the social welfare grows much larger than the firm’s profit from the vaccine. This social welfare is not captured by the firm.
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THIS IS NOT COMPLETE but I am publishing my intermittent progress in order to get feedback from others! — May 24th
