Uncertainty & Risk Notes

Uncertainty & Risk Notes - ECO 317 Economics of...

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ECO 317 – Economics of Uncertainty – Fall Term 2009 Notes for lectures 2. The Expected Utility Theory of Choice Under Risk 1 Basic Concepts The individual (usually consumer, investor, or firm) chooses action denoted by a , from a set of feasible actions A . The set of states of the world is denoted by S , and probabilities are defined for events in S . Let π ( s ) denote the probability of the elementary event s . Here we take these probabilities to be exogenous. Actions can lead to different consequences in different states of the world. Let C denote the set of consequences, and F : A × S C denote the consequence function, so c = F ( a, s ) is the consequence of action a in state s . For example, an investor’s action might be a vector of proportions of the initial wealth that he puts into different kinds of assets. Then, depending upon the state of the world, e.g. interest rates rise and the dollar goes down, so bonds do badly but foreign stocks do well in dollar terms, there will be a consequence, namely his final wealth. First suppose C is finite, say C = { c 1 , c 2 , . . . c n } , and write p i ( a ) for the probability that action a yields consequence c i , that is, p i ( a ) = Pr { s | F ( a, s ) = c i } . Of course p i ( a ) may equal zero for some i . In many economic applications, the set of consequences C is an infinite continuum of consumption quantities, or amounts of income or wealth. But the same idea has the obvious generalization. Each action induces a cumulative distribution function or a density function on C , e.g. we can define p ( w ; a ) such that p ( w ; a ) dw is the probability that the action a yields wealth between w and w + dw . We will usually develop the theory for consequence sets that are finite, but use it in the more general contexts without going into the rigorous (pedantic?) math of this. Thus each action induces a probability distribution over C . Call such a probability distribution a lottery. The set of feasible actions will map into a set of feasible lotteries. Different consequence functions may yield equivalent lotteries. Example: The state space is the set of possible outcomes of the roll of a die: { 1 , 2 , 3 , 4 , 5 , 6 } . The consequence space is a set of prizes, { $1,$2,$3 } . The actions are bets. Bet 1, labeled a 1 , yields $1 if the die shows 1, 2, or 3, $2 if it shows 4 or 5, and $3 if it shows 6. Bet 2, labeled a 2 , yields $1 if the die shows an even number, $2 if it shows 1 or 5, and $3 if it shows 3. Then the two actions induce the same lottery, namely: the probability of getting $1 is 1/2, that of getting $2 is 1/3, and that of getting $3 is 1/6. In orthodox economic theory (we will consider some newer ideas such as framing and reference points later), a decision maker cares only about consequences and their probabili- ties. Therefore we can think of the choice problem as one of choosing among a set of feasible lotteries.
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  • Fall '08
  • Utility, p1, probability triangle diagram

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