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Unformatted text preview: Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2009 4 Risk Preferences Readings. Chapter 5. Concepts. Random variable, expected value. Risk free act, risk averse, risk neutral, risk loving preferences. Certainty equivalent, risk premium. Arrow-Pratt measure of absolute/relative risk aversion. Increasing/decreasing risk. 4.1 A Few Things To Start With • In this section, we focus on monetary outcomes. The set of possible outcomes is the real line and therefore an act ˜ x : S → R is a random variable over real numbers. • There are several convenient properties of such random variables. First, we can define the expected value of a random variable as E [˜ x ] = X s ∈ S π ( s )˜ x ( s ) where ˜ x ( s ) is a particular outcome, hence an amount of money. We may sum random variables or multiply them with a scalar and we still get a random variable. • We will often use two types of graphical representations , so let us start to under- stand those. • First, we can graph the vNM utility function over money. Money measured on the x-axis and utility measured on the y-axis. (See Figure 5.2 in the book.) – A point on the graph represents a money amount and the respective utility index. – There is a way to represent lotteries over two possible outcomes as well as their expected value, expected utility, and the utility of the expected value (as you can see it on the Figure 5.2). • The other graph (which resembles to the one we often used in Economics 11) is to represent acts, preferences over acts and possibility sets when there are only two states of nature. The monetary outcome in state 1 is measured on the x-axis while the monetary outcome in state 2 measured on the y-axis. (See Figure 5.3 in the book.) – We assume that there are only two states of nature ( s 1 ,s 2 ) and we fix the probability of those ( π 1 ,π 2 ). 1 – Hence an act can be fully defined by the outcome x 1 in state 1 and the outcome x 2 in state 2. Therefore a point ( x 1 ,x 2 ) on this graph represents an act. 12 – The 45 ◦ line has a special role here. Points on that line represent the risk free acts . By a risk free act we mean an act that leads to the same the outcome in both states, so the outcome is basically certain. Points to the southeast of the 45 ◦ line represent risky acts that pay more money in state 1 than in state 2. Similarly, points to the northwest of the risk free line represent risky acts that pay more money in state 2 than in state 1. – On this graph, we can illustrate the acts with the same expected value . A straight line with a negative slope represents those acts. Remember that we have fixed the probabilities of the two states to be π 1 and π 2 . Hence the following equation defines all the acts with the expected value ¯ x : π 1 x 1 + π 2 x 2 = ¯ x What is the slope of this line? You can find the slope easily if you rearrange the equation on the following form: x 2 = ¯ x π 2- π 1 π 2 x 1 .....
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- Fall '09
- Microeconomics, decision maker, VNM utility function