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ec142f09notes5 - Kata Bognar [email protected] Economics 142...

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Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2009 5 Decisions in the Presence of Risk Readings. Chapter 6. Concepts. Gambling problem, investment problem, insurance problem. Fair, favor- able and unfavorable gamble / insurance. First and second-order conditions. Arbitrage. Local risk neutrality. State prices. 5.1 Example Previously, we have characterized optimal choice under uncertainty with only a few available options, i.e. hire the applicant or not, go to restaurant A or B, etc. You also know a lot about optimal choice with a continuum of available options, you have learned a lot about that in Economics 11 and 101. Here we put these two together and talk about individual optimization under uncertainty if there are many available options to choose from. EXAMPLE: Exercise 5.1 Step 1: Define the elementary events and their respective likelihood. The elementary events are (i) theft happens with probability 1/4 and (ii) theft does not happen with probability 3/4. Step 2: Understand that the person faces a risky act and think about how to represent this act on a graph. The person has $10 , 000 with probability 3/4 and $5 , 000 with probability 1/4. We can use the graph introduced earlier with z 1 being the payoff if theft does not happen, z 2 being the payoff if theft happens. Then we can plot the point (10000 , 5000) Step 3: Understand what buying insurance means for the decision maker. It refers to a transaction between the decision maker and the insurance company such that (i) the decision maker pays a given fee to the company and in return (ii) she gets a money transfer in case a theft or accident, etc. happens. Since the fee is certain, it decreases the payoffs in each states, you may think about it as an amount needs to be paid in advance. We use the following notations: γ is the fee for a dollar coverage - it is 25 cents in the example, K is the amount of coverage bought and π is the probability of the accident - it is 1/4 in the example. Step 4: Calculate the payoffs in case of full coverage. The decision maker buys full coverage when she insures up to the total possible loss. In this case, her payoff will be the same in both states. The possible loss in the example is $5 , 000 so to fully insure the decision maker had to buy K = 5000 coverage. This level of insurance costs 0 . 25 × 5 , 000 = 1 , 250 therefore the payoffs will be 10 , 000 - 1 , 250 = 8 , 750 in both case. This risk-free act is represented by the point (8750 , 8750) on the graph. 1
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Step 5: Find the set of available acts. The decision maker can choose any level of coverage. 1 Any level of coverage defines a pair of payoffs, ( z 1 , z 2 ). If the decision maker buys K coverage, then z 1 = 10 , 000 - 0 . 25 K and z 2 = 5 , 000 + K - 0 . 25 K = 5 , 000 + 0 . 75 K are the implied payoffs. A line with a slope of - 0 . 75 0 . 25 will represent these pairs. Notice that by buying insurance the decision maker shifts money form the good state to the bad state.
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