Natural disasters are local phenomena — impacting a city or a part of a state but rarely impacting the whole country, at least if the country is geographically large. To simplify the analysis, suppose there are two distinct regions that might experience local disasters.
A. Define “state 1” as region 1 experiencing a natural disaster, and define “state 2” as region 2 having a natural disaster. I live in region 2 while you live in region 1. Both of us have the same risk averse and state-independent tastes, and our consumption level falls from y to z when a natural disaster strikes. The probability of state 1 is δ and the probability of state 2 is (1−δ).
(a) Putting consumption x1 in state 1 on the horizontal axis and consumption x2 in state 2 on the vertical, illustrate an Edgeworth box assuming you and I are the only ones living in our respective regions. Illustrate our “endowment” bundle in this box.
(b) Suppose an insurance company wanted to insure us against the risks of natural disasters. Under actuarially fair insurance, what is the opportunity cost of state 2 consumption in terms of state 1 consumption? What is the opportunity cost of state 1 consumption in terms of state 2 consumption? Which of these is the slope of the actuarially fair budget in your Edgeworth Box?
(c) Illustrate the budget line that arises from the set of all actuarially fair insurance contracts within the Edgeworth Box. Where would you and I choose to consume assuming we are risk averse?
(d) How does this outcome compare to the equilibrium outcome if you and I were simply to trade state-contingent consumption across the two states?
(e) Suppose there were two of me and two of you in this world. Would anything change?
(f) Now suppose that the two of me living in region 2 go to a local insurance company that operates only in region 2. Why might this company not offer us actuarially fair insurance policies?
(g) Instead of insurance against the consequences of natural disasters, suppose we instead considered insurance against non-communicable illness. Would a local insurance company face the same kind of problem offering actuarially fair insurance in this case?
(h) How is the case of local insurance companies insuring against local natural disasters similar to the case of national insurance companies insuring against business cycle impacts on consumption? How might international credit markets that allow insurance companies to borrow and lend help resolve this?
B. Suppose that, as in exercise 17.10 the function u(x) = lnx allows us to represent our tastes over gambles as expected utilities. Assume the same set-up as the one described in A.
(a) Let p1 be defined as the price of $1 of consumption if state 1 occurs and let p2 be the price of $1 of consumption in the event that state 2 occurs. Set p2 = 1 and then denote the price of $1 of consumption in the event of state 1 occurring as p1 = p and write down your budget constraint.
(b) Solve the expected utility maximization problem given this budget constraint to get your demand x1 for state 1 consumption as well as your demand x2 for state 2 consumption.
(c) Repeat (a) and (b) for me
(d) Derive the equilibrium price. Is this actuarially fair?
(e) How much do we consume in each state?
(f) Does the equilibrium price change if there are 2 of you and 2 of me?
(g) Finally, suppose that the two of me attempt to trade state-contingent consumption just between us. What will be the equilibrium price?
(h) Will we manage to trade at all?
(i) Can you illustrate this in an Edgeworth Box? Is the equilibrium efficient?