According to conventional wisdom, because of the free rider problem, compulsory vaccination against infectious disease results in a better outcome than the Nash equilibrium in which each person freely chooses whether or not to be vaccinated. Suppose that there is an arbitrarily large population of people faced with an epidemic of an infection disease. An individual's payoff is 1 in the event she remains uninfected and 0 in the event she is infected. There is a vaccine against the disease. The vaccine is effective in reducing the chances of an infection. An individual who is not vaccinated faces a greater risk of contracting the infectious disease, ceteris paribus. The risk of infection also depends upon the proportion of the population who have been vaccinated. Suppose x is the proportion of the population that gets vaccinated. The probability that a vaccinated person gets the disease is 0 (that is, the vaccine is perfectly effective). The probability that an unvaccinated person gets the disease is 0.7 0.4x. The costs of getting vaccinated are heterogeneous in the population. Model the population as a continuum of people distributed uniformly on unit interval [0, 1]. For a person at 'location' [0, 1], her cost of getting vaccinated is equal to . You can think of people located at incurring a 'travel cost' of to get their vaccination at a vaccination center 'located' at 0. This is not a necessary interpretation, however. Let V stand for the action of getting vaccinated and N stand for the action of not getting vaccinated.

(a) Explain why the payoff to a person of choosing V is

uV (x, ) = 1 ,

given that she is located at and the proportion of the population choosing to get vaccinated is x.

(b) Explain why the payoff to a person of choosing N is

uN (x, ) = 0.3 + 0.4x,

given that she is located at and the proportion of the population choosing to get vaccinated is x. Model the vaccination decisions of people in the population as a static game, where each person must choose V or N. Note that x [0, 1], since x is a proportion.

In Nash equilibrium, each person must independently choose whichever of V or N yields a better payoff (unless there's a tie). Since there is a continuum of people and choices are made independently, when a person is making her decision, she understands that her choice does not affect x, the proportion of the population choosing V .

(c) Explain why the following two facts are true in the Nash equilibrium of the vaccination game: (1) if a person with vaccination cost = r gets vaccinated in the equilibrium, then every person with vaccination cost < r also gets vaccinated in equilibrium. (2) if a person with vaccination cost = s gets vaccinated in equilibrium, then every person with vaccination cost > s also gets vaccinated in equilibrium. (Hint: Consider what happens to the gain in payoff switching from N to V (and vice versa) as varies, holding x fixed.)

(d) Given the above fact about vaccination decisions in equilibrium, for each value of x, there must be some value of , call it (x), such that people with vaccination cost equal to (x) are indifferent between V and N. What is (x)? Note: it is a function of x (as well as the parameter b).

(e) For each x, we can understand (x) to be the marginal vaccine taker, and everyone with vaccine cost lower than that gets vaccinated. Given that is distributed uniformly on [0, 1], what is fraction of people with (x)?

(f) Recall that x is also the proportion of people who get vaccinated. So finally we can solve for the equilibrium, by recognizing that x must equal the proportion of people with (x). Use this equation to solve for the equilibrium values of x and . (g) You should find that the values of x and are equal in equilibrium. Which assumption is responsible for this coincidence?