Bernoulli’s Exposition

Bernoulli (Daniel) Exposition Of A New Theory On The Measurement Of Risk

The translator notes that by risk is meant risky propositions (gambles), and this paper is really about utility. Bernoulli gives this example to show that utility is not proportional to money:

 “Somehow a very poor fellow obtains a lottery ticket that will yield with equal probability either nothing or twenty thousand ducats. Will this man evaluate his chance of winning at ten thousand ducats? Would he not be ill-advised 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 ill-advised to refuse to buy the lottery ticket for nine thousand ducats. If I am not wrong then it seems clear that all men cannot use the same rule to evaluate the gamble.”

Hence Bernoulli develops his theory:

 “[T]he 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.”

 He notes that there are exceptional cases: 

“[A] rich prisoner who possesses two thousand ducats but needs two thousand ducats more to repurchase his freedom, will place a higher value on a gain of two thousand ducats than does another man who has less money than he.”

But he regards the normal case as: 

“[A]ny increase in wealth, no matter how insignificant, will always result in an increase in utility which is inversely proportionate to the quantity of goods already possessed.”

Here ‘goods’ is regarded as anything that may be valued:

“For the great majority the most valuable portion of their possessions so defined will consist in their productive capacity, this term being taken to include even the beggar’s talent :a man who is able to acquire ten ducats yearly by begging will scarcely be willing to accept a sum of fifty ducats on condition that he henceforth refrain from begging or otherwise trying to earn money.”

On gambling, Bernoulli deduces that: 

“[I]n many games, even those that are absolutely fair, both of the players may expect to suffer a loss.]

On the other hand, he shows that the poorer you are the better it is to take out insurance and the richer you are the better it is to offer it. It is also better to split goods between ships, reduce the risk as he measures it. These seem reasonable conclusions, thus justifying his notion. 

David’s Conclusion

Bernoulli is concerned with typical short-run decisions that resemble gambles, as on whether a ship will come in. He notes that utility is normally a decreasing function of wealth and hence changes with wealth. He recognizes that there are exceptions but does not consider wider uncertainties nor – explicitly – risk aversion.

Dave Marsay


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