Many are concerned about health conditions in American schools, particularly since children frequently catch infectious diseases from each other. One child with an illness might give the disease to all of the others at school, for example. As a result, some towns in the United States require school children to receive vaccinations before attending school.

a) Suppose that a new disease begins to spread around the United States, an ailment called "Economics Fever" (EF), in which the infected person's brain works so hard that it is difficult to eat or sleep. The disease strikes only children between the ages of 5 and 10 years. The government develops a vaccine that, if injected, completely prevents the child receiving the injection from contracting EF. The only problem is that it is painful and inconvenient for the child to have the injection to prevent the disease. Parents, who weigh the costs and benefits of vaccinations for their children, conclude that their child's utility function looks like:

Ui = Bi - DSi - aiVi

in which Ui is child i’s utility level, Bi is a constant for child i, D is the (utility) cost of the disease, if the child contracts the disease (D is the same for all children), Si indicates whether child i has caught the disease (so Si = 1 if child i is sick, and Si = 0 if child i is not sick), Vi indicates whether child i has been vaccinated (so Vi = 1 if child i is vaccinated, and Vi = 0 if child i is not vaccinated), and ai is a term that reflects the cost of the pain and inconvenience that child i and his/her family feels from obtaining the injection. Suppose that p* represents the probability that a child will get sick if he or she is not vaccinated. Then, assuming that parents maximize the utility of their children, including the expected utility cost of getting sick, under what circumstances will a parent have his or her child vaccinated? (Note that the expected utility cost of getting sick is zero for a vaccinated child, and p*D for an unvaccinated child. Assume that the government provides the vaccines for free.)

b) It turns out that p* depends on the number of children who have the disease, since children so often contract diseases from each other. If additional children contract EF, it becomes more likely that an unvaccinated child will catch EF from the others. Under these circumstances, will too many, or too few, children receive vaccinations, according to your answer to part (a)? Why?

c) The government proposes to require that all children be vaccinated against the disease. Under what circumstances would this requirement be excessive, in that too many children would receive vaccinations? How would you compare the desirability of this proposal to that of the alternative of no government intervention at all?

d) Suppose that there are 100 children, all with different ais ranging from 1-100, that D = 150, and that p* = (# unvaccinated children)/100. Then, in the absence of government regulation, how many children would be vaccinated? What would be the efficient number of children to be vaccinated? Can you think of a tax or subsidy scheme that might get this efficient outcome? [Hint: note that the fraction of unvaccinated children just equals (100-ai)/100, where ai is chosen for the child whose parents are just indifferent about having the vaccination or not. Note, also, that if (say) 80 children are vaccinated, then there are two benefits from having an additional child vaccinated: it reduces the chance of that child getting the disease, and it reduces the chance of the other 20 children contracting the disease.]