Suppose you own a business that does well during economic expansions but not so well during recessions which happen with probability δ. Let xE denote your consumption level during expansions and let xR denote your consumption level during recessions. Unless you do something to diversify risk, these consumption levels are E = (eE ,eR ) where eE is your income during expansions and eR your income during recessions (with eE > eR). Your tastes over consumption are the same during recessions as during expansions and you are risk averse. For any asset purchases described below, assume that you pay for these assets from whatever income you have depending on whether the economy is in recessions or expansion.
A. Suppose I own a financial firm that manages asset portfolios. All I care about as I manage my business is expected returns, and any asset I sell is characterized by (p, bR , bE ) where p is how much I charge for 1 unit of the asset, bR is how much the asset will pay you (as, say, dividends) during recessions and bE is how much the asset will pay you during expansions.
(a) Is someone like me—who only cares about expected returns—risk averse, risk loving or risk neutral?
(b) Suppose that all the assets I offer have the feature that those who buy these assets experience no change in their expected consumption levels as a result of buying my assets. Derive an equation that expresses the price p of my assets in terms of δ, bR and bE.
(c) What happens to my expected returns when I sell more or fewer of such assets?
(d) Suppose you buy 1 asset (p,bR ,bE ) that satisfies our equation from (b). How does your consumption during expansions and recessions change as a result?
(e) At what rate do assets of the kind I am offering allow you to transfer consumption opportunities from expansions to recessions? On a graph with xE on the horizontal and xR on the vertical axis, illustrate the “budget line” that the availability of such assets creates for you.
(f) Illustrate in your graph your optimal choice of assets.
(g) Overall output during recessions is smaller than during expansions. Suppose everyone is risk averse. Is it possible for us to all ends up doing what you concluded you would do in (f)? (We will explore this further in exercise 17.10.)
B. Suppose that the function u(x) = xα is such that we can express your tastes over gambles using expected utility functions.
(a) If you have not already done so in part A, derive the expression p(δ,bR ,bE ) that relates the price of an asset to the probability of a recession δ, the dividend payment bR during recessions and the dividend payment bE during economic expansions assuming that purchase of such assets keeps expected consumption levels unchanged.
(b) Suppose you purchase k units of the same asset (bE , bR )which is priced as you derived in part (a) and for which (bR −bE ) = y > 0. Derive an expression for xR defined as your consumption level during recessions (given your recession income level of eR ) assuming you purchase these assets. Derive similarly an expression for your consumption level xE during economic expansions.
(c) Set up an expected utility maximization problem where you choose k — the number of such assets that you purchase. Then solve for k.
(d) How much will you consume during recessions and expansions?
(e) For what values of α is your answer correct?
(f ) True or False: So long as assets that pay more dividends during recessions than expansions are available at “actuarially fair” prices, you will be able to fully insure against consumption shocks from business cycles.
(g) Could you accomplish the same outcome by instead creating and selling assets with (bE > bR)?