In exercise 17.9, we considered the case of me trading assets that allow you to transfer consumption from good times to bad times. Suppose again that your income during economic expansions is e_E and your income during recessions is e_R , and that the probability of a recession is δ < 0.5.

A. Also, suppose again that my tastes are risk neutral while yours are risk averse and that e_E > e_R. My consumption opportunity endowment, however, is the reverse of yours — with e_R equal to my income during economic expansions and e_E equal to my income during recessions.

(a) Draw an Edgeworth box representing the economy of you and me.

(b) Illustrate the equilibrium in this economy. Will you do in equilibrium what we concluded you would do in exercise 17.9?

(c) Next, suppose that there was a third person in our economy—your identical twin who shares your tastes and endowments. Suppose the terms of trade for transferring consumption in one state to the other remain unchanged, and suppose equilibrium exists in which everyone ends up at an interior solution. Illustrate what this would look like — given that there are now 2 of you and only one of me. (Hint: It should no longer be the case that our indifference curves within the box are tangent to one another — because equilibrium now implies that two of your trades have to be exactly offset by one of mine.)

(d) Is anyone fully insured against consumption swings in the business cycle? Is everyone?

Now continue with the example but suppose that my tastes, instead of being risk neutral, were also risk averse. Would the same terms of trade still produce an equilibrium?

(e) How do the terms of trade now have to change to support an equilibrium when all of us are risk averse?

(g) Will anyone be fully insured— i.e. will anyone enjoy the same level of consumption during recessions as during expansions?

(h) Relate your conclusion to the existence of aggregate risk in economies that experience expansions and recessions. Who would you rather be—me or you

B. Suppose that the function u(x) = lnx allows us to express your tastes over gambles as expected utilities. Also, suppose again that your income during expansions is e_E and your income during recessions is e_R, with e_E > e_R.

(a) Let p_R be defined as the price of $1 of consumption in the event that a recession occurs and let p_E be the price of $1 of consumption in the event that an economic expansion occurs. Explain why we can simply normalize p_R = 1 and then denote the price of $1 of consumption in the event of expansions as p_E = p.

(b) Using these normalized prices, write down your budget constraint and your expected utility optimization problem.

c) Solve for your demand for x_R and x_E.

(d) Repeat parts (b) and (c) for me — assuming I share your tastes but my income during recessions is e_E and my income during expansions is e_R — exactly the mirror image of your incomes over the business cycle.

(e) Assuming we are the only ones in this economy, derive the equilibrium price, or terms of trade across the two states.

(f) How much do each of us consume during expansions and recessions at this equilibrium price?

(g) Now suppose that there are two of you and only one of me in this economy. What happens to the equilibrium price?

(h) Do you now consume less during recessions than during expansions? Do I?