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Module 1: Decision Making Under Uncertainty
Information Economics (Ec 515) · George Georgiadis
Today, we will study settings in which decision makers face uncertain outcomes.
- Natural when dealing with asymmetric information.
- Need to have a model of how agents make choices / behave when they face uncer- tainty.
- Prevalent theory: Expected utility theory.
States of the world (or states of nature):
- Relevant pieces of information that are mutually exclusive.
- The state of the world a↵ects your payo↵ (or utility, or welfare).
An Example: Will Greece default on its debt or not?
Two possible “states of nature”: default (D), or no default (N).
- An investor’s return may be a↵ected by the state of nature.
- This may a↵ect whether you invest in stocks or cash.
Agent has two options: invest in cash or in stocks. If the agent invests in stocks:
- Return equal to 5% under state N.
- Return equal to -10% under state D.
If he invests in cash:
- Return equal to 0% under either state.
In which assets should the agent invest?
- Need a model of decision making under uncertainty.
Preferences over lotteries
Let X be a set of “prizes”.
- For instance, X could be monetary payo↵s (returns). In this case X = R.
A “lottery” is a function p : X! [0, 1] such that P x 2 X p(x) = 1.
- p(x) is the probability with which lottery p pays x 2 X.
Example: X = { 1 , 10 , 100 }, p(1) = 0.4, p(10) = 0.2, p(100) = 0.4.
Let L(X) denote the set of all lotteries on X.
A lottery p 2 L(X) is non-trivial if it has at least two distinct prizes with positive probability.
Expected utility
Let ⌫ be a “preference relation” over lotteries in L(X).
- The preference relation ⌫ tells us how the agent whose decisions we are studying ranks the lotteries in L(X).
- For two lotteries p, q, p ⌫ q means that p is preferred to q.
- p q means p is strictly preferred to q (p ⌫ q and q ✏ p).
We would like there to be a utility function function u : X! R such that p ⌫ q if and only if (^) X
x 2 X
p(x)u(x)
X
x 2 X
q(x)u(x)
In this case, we can evaluate lotteries by computing their expected payo↵s.
Under certain conditions such a utility function exists.
- ⌫ is complete and transitive:
- For any pair of lotteries p, q, either p ⌫ q, or q ⌫ p (or both).
- If p ⌫ q and q ⌫ r, then p ⌫ r.
- ⌫ satisfies the independence axiom:
Definition 1. The certainty equivalent of a lottery p is the value x cp such that u
x cp
P
x 2 X p(x)u(x). The agent is indi↵erent between facing a lottery p or obtaining x cp for sure.
For any lottery p, let x (^) p =
P
x 2 X p(x)x^ be the expected payment of lottery^ p. An agent is risk-averse if, for all non-trivial lotteries p 2 L(X), x cp < x (^) p. An agent is risk-neutral if, for all non-trivial lotteries p 2 L(X), x (^) p = x cp.
An agent is risk-loving if, for all non-trivial lotteries p 2 L(X), x cp > x (^) p.
Jensen’s inequality: if u is strictly concave and p is a non-trivial lottery, then X x 2 X
p(x)u(x) < u(x (^) p )
If u(·) is strictly convex, the opposite inequality holds. If u is linear, then
P
x 2 X p(x)u(x) =^ u(x^ p^ ). An agent with utility function u(·) is:
- risk-averse i↵ u(·) is strictly concave (u 00 (x) < 0 for all x).
- risk-neutral i↵ u(·) is linear (u 00 (x) = 0 for all x).
- risk-loving i↵ u(·) is strictly convex (u 00 (x) > 0 for all x).
The risk premium of lottery p is x (^) p x cp.
The following statements are equivalent:
P
x 2 X p(x)u(x)^ < u(x^ p^ ) for all non-trivial lotteries^ p^2 L(X).
- x cp < x (^) p for all non-trivial lotteries p 2 L(X).
- The risk premium of lottery p is positive for all non-trivial lotteries p 2 L(X).
- u is strictly concave (u 00 ).
Measuring risk aversion
Absolute risk aversion
Suppose an individual has wealth w.
This individual faces the following choice: a sure gain of z or a lottery p.
- In first case, he gets u(w + z) for sure.
- In second case, he gets an expected payo↵ of
P
x 2 X p(x)u(w^ +^ x). How does this agent’s choice depends on his wealth w?
If the agent’s willingness to take the lottery increases with wealth, we say that he has decreasing absolute risk aversion (ARA).
- If agent has decreasing ARA, then if he is willing to take lottery when his wealth is w 1 , he will also be willing to take the lottery when his wealth is w 2 > w 1.
Analogous definitions for increasing ARA and constant ARA.
Coe cient of absolute risk aversion: A(x) = u u^00 0 (^ (xx)) :
- If A(x) is decreasing (or constant, or increasing), then agent with utility u has decreasing (or constant, or increasing) absolute risk aversion.
Examples:
- u(x) = e ↵x^ ) A(x) = ↵ (CARA).
- u(x) = p x ) A(x) = (^21) x (decreasing ARA).
Let u 1 (x) be a utility function, and let u 2 (x) = g (u 1 (x)) (with g 0 > 0 , g 00 < 0). Then, A 1 (x) < A 2 (x) for all x.
- If A 1 (x) < A 2 (x), then agent 1 is less risk-averse than agent 2.
