eco200_uncertainty_fall2009

eco200_uncertainty_fall2009 - ECO 200: Microeconomic Theory...

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ECO 200: Microeconomic Theory Lecture notes on uncertainty and demand for insurance 1 Fall 2008 Carlos J. Serrano 1 Lecture notes updated on Nov. 1, 2009 at 8.28pm 1
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Summary Uncertainty Probability, expected value and variance Von Neumann utility function and contingent consumption bundles Variance and preference towards risk Risk premium and certainty equivalent Demand for insurance Fair insurance 2
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1 Uncertainty and demand for insurance 1.1 Uncertainty 1.1.1 Probability and Expected Value Let be the set of outcomes of a random variable . Fo simplicity, just consider that a random variable is a variable that can take random values in the set and only in . Assume that the number of elements in is f nite and equal to . De f nition 1 A probability is a function that maps outcomes in the set X to real numbers satisfying that X =1 ( )=1 Example 2 Suppose that there are two outcomes such { =100  =20 } ;andtha t ( )=Pr( )=0 3 and ( )=  ( )=0 7 Note that Pr( )+  ( )=1 An very useful application of the theory of probabilities is the calculation of an expected value. De f nition 3 The expected value is the probability weighted average of the payo f sassoc i - ated will all possible outcomes Example 4 Let { = 100  =20 } and  ( )=0 3 and  ( )=0 7 The expected value of the random variable is ( )=  ( ) +  ( ) =30+14=44 Sometimes it is useful to obtain measures of the dispersion of the random variable  This is particularly interesting when two random variables have the same expectation. The dispersion of a random variable is associated to risk. For instance, two mutual funds can o f er you the same expected return, but the risk can be di f erent. How do we measure
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De f nition 5 Thevar ianceo farandomevar iab le is  ( )= [( ( )) 2 ]= X =1 (Pr( )( ( )) 2 ) As you can see the variance is a measure of the dispersion of the outcomes from the mean weighted by how likely the outcomes are. Example 6 Whenthenumbero fe lemen tsinSistwo ,then  ( )=  ( )( ( )) 2 +  ( )( ( )) 2 Example 7 Let { =100  =20 } and  ( )=0 3 and  ( )=0 7  ( )=0 3 (100 44) 2 +0 7 (20
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eco200_uncertainty_fall2009 - ECO 200: Microeconomic Theory...

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