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notes6 - Chapter 6 Choice Under Uncertainty Up until now we...

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Chapter 6 Choice Under Uncertainty Up until now, we have been concerned with choice under certainty. A consumer chooses which commodity bundle to consume. A producer chooses how much output to produce using which mix of inputs. In either case, there is no uncertainty about the outcome of the choice. We now turn to considering choice under uncertainty, where the objects of choice are not certainties, but distributions over outcomes. For example, suppose that you have a choice between two alternatives. Under alternative A, you roll a six-sided die. If the die comes up 1, 2, or 3, you get \$1000. If it comes up 4, 5, or 6, you lose \$300. Under alternative B, you choose a card from a standard 52 card deck. If the card you choose is black, you pay me \$200. If it is a heart, you get a free trip to Bermuda. If it is a diamond, you have to shovel the snow o ff of my driveway all winter. If I were to ask you whether you preferred alternative A or alternative B, you could probably tell me. Indeed, if I were to write down any two random situations, call them L 1 and L 2 , you could probably tell me which one you prefer. And, there is even the possibility that your preferences would be complete, transitive (i.e., rational), and continuous. If this is true then I can come up with a utility function representing your preferences over random situations, call it U ( L ) , such that L 1 is strictly preferred to L 2 if and only if U ( L 1 ) > U ( L 2 ) . Thus, without too much e ff ort, we can extend our standard utility theory to utility under uncertainty. All we need is for the consumer to have well de fi ned preferences over uncertain alternatives. Now, recall that I said that much of what we do from a modeling perspective is add structure to people’s preferences in order to be able to say more about how they behave. In this situation, what we would like to be able to do is say that a person’s preferences over uncertain alternatives 158

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Nolan Miller Notes on Microeconomic Theory: Chapter 6 ver: Aug. 2006 should be able to be expressed in terms of the utility the person would assign to the outcome if it were certain to occur, and the probability of that outcome occurring. For example, suppose we are considering two di ff erent uncertain alternatives, each of which o ff ers a di ff erent distribution over three outcomes: I buy you a trip to Bermuda, you pay me \$500, or you paint my house. The probability of each outcome under alternatives A and B are given in the following table: Bermuda -\$500 Paint my house A .3 .4 .3 B .2 .7 .1 What we would like to be able to do is express your utility for these two alternatives in terms of the utility you assign to each individual outcome and the probability that they occur. For example, suppose you assign value u B to the trip to Bermuda, u m to paying me the money, and u p to painting my house. It would be very nice if we could express your utility for each alternative by multiplying each of these numbers by the probability of the outcome occurring, and summing.
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notes6 - Chapter 6 Choice Under Uncertainty Up until now we...

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