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Unformatted text preview: Notes on Uncertainty and Expected Utility Ted Bergstrom, UCSB Economics 210A November 21, 2011 1 Introduction Expected utility theory has a remarkably long history, predating Adam Smith by a generation and marginal utility theory by about a century. 1 In 1738, Daniel Bernoulli wrote: Somehow a very poor fellow obtains a lottery ticket that will yield with equal probability either nothing or twenty thousand ducats. 2 Will this man evaluate his chance of winning at ten thousand ducats? Would he not be illadvised to sell this lottery ticket for nine thousand ducats? To me it seems that the answer is in the negative. On the other hand I am inclined to believe that a rich man would be illadvised to refuse to buy the lottery ticket for nine thousand ducats. ... the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. 1 Even when Cournot (1838), Gossen (1858) developed recognizably utility theory in the mid nineteenth century, they gained little attention for at least another half century. 2 A ducat contained 3.5 grams of gold. I believe that at the time, a ducat was worth about 1/2 of an English pound sterling. English laborers earned about 20 pounds per year. So annual earnings for a poor fellow might be 40 ducats. If this is the case, 20,000 ducats would be 500 times the annual earnings of a poor fellow. TB 1 If the utility of each possible profit expectation is multiplied by the number of ways in which it can occur, and we then divide the sum of these products by the total number of possible cases, a mean utility [moral expectation] will be obtained, and the profit which corresponds to this utility will equal the value of the risk in question. Another remarkable feature of Bernoullis discussion is his focus on human capital and wealth rather than just current income. Bernoulli proposes that the utility function used to evaluate gambles should be a function of ones wealth, and not just current income flows. Bernoullis suggests a form for the utility function stated in terms of a differential equation. In particular, he proposes that marginal utility is inversely proportional to wealth. Thus we have du ( W ) dW = a W . for some constant a . We can solve this differential equation to find the function u . In particular, we must have u ( W ) = a ln W + b for some constants a and b ....
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This note was uploaded on 12/25/2011 for the course ECON 210A taught by Professor Bergstrom during the Fall '09 term at UCSB.
 Fall '09
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