Economic Perspectives II

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**Risk-Averse and Risk-Loving Agents**

Note: in order to run the simulation referred to in this slide, go here, to the Java Applet version. You will be directed to download the latest version of the Java plug-in.

Consider an agent that derives utility u(x) = x^{r} from
payoff x. When r = 1.0, u(x) = x: every additional dollar is
just as useful to the agent as the last one obtained. This corresponds to the risk-neutral
investor A, who is playing a game with some extra money, not betting a
financial base she depends on for survival.

For r = 0.25, the utility function is u(x) = x^{1/4}, and the
agent derives an ever-decreasing utility from each additional dollar.
Investor B from the last slide's example probably has a utility curve much like this
one, for she needs the first $500 to pay her rent a lot more than the
next $500 that will finance her weekend getaway in Jamaica.

Investor B's utility curve is plotted in green to your left. Note that output x is normalized to be between 0 and 1, and that r may be altered in the text field in the upper left.

The graph demonstrates what happens when an agent evaluates
a behavior choice with an expected output of **x = 0.5**, as the
variance **v** goes between **0.05 **and **0.45**. Mathematically,
expected utility can be found by integrating u(x) between **x-v** and **x+v**.
Graphically, we obtain a good approximation by drawing a cord between the two points on u(x)
(shown in yellow), and then a horizontal line through the center of the cord
(show in blue)^{1}.
The blue line corresponds to the level of utility derived by
the agent, given the average outcome and its variance. You may want
to alter the variance, then press "Go" at the bottom of the screen to update the graph.

Note that for a risk-averse agent (r < 1), the expected utility of an option goes down as the variance in output increases. Similarly, you may enter an r > 1 to construct a risk-loving agent. Then check that she is risk-loving indeed, by making sure that her expected utility increases with increasing variance.

1. This approximation becomes less precise when the area between the green and the yellow lines becomes large with respect to the total area under the green curve between x-v and x+v, such as when r is very close to 0.