What kind of math problem is this?
- To: mathgroup at smc.vnet.net
- Subject: [mg20896] What kind of math problem is this?
- From: "Seth Chandler" <SChandler at uh.edu>
- Date: Sat, 20 Nov 1999 01:07:09 -0500 (EST)
- Organization: University of Houston
- Sender: owner-wri-mathgroup at wolfram.com
This may be a little off topic, but I was hoping someone here could help. Let r and f be real numbers between 0 and 1 inclusive. How does one determine if there is a function u that satisfies this relation and, if so, what the function is. InverseFunction[u][f u[x] + (1 - f)u[y]] == (1 - r)(f x + (1 - f)y) + r x For those interested in the origins of this math problem, it is as follows. The "certainty equivalent wealth" associated with some lottery in which the payoffs are x (the bad payoff) and y (the good payoff) is traditionally thought to be equal to the inverse utility of the expected utility of the lottery. That is, it is thought to be the amount which, if held with certainty, would provide the same "utility" as the lottery. Sometimes, this formulation of certainty equivalent wealth is a bit messy. Thus, others have proposed a simpler formulation that does not rely on utility functions. Under this method, which game theorist Martin Shubik and others have used, the certainty equivalent wealth of a lottery is set equal to something between the worst case of the lottery and the expected value of the lottery. The "where" in between is determined by a coefficient of risk aversion (r). If r is zero, the lottery is valued at the expected value. If r is one, the lottery is valued at the worst case. My question is whether there is any utility function that could emulate the behavior of this simpler formulation. Unfortunately, I can't figure out what branch of mathematics is relevant to the question, or, more importantly, I supposed, how to solve it. If Mathematica could be used in finding the answer, that would be great. Seth J. Chandler Associate Professor of Law University of Houston Law Center