Utility Function of Wealth

It is often assumed that wealth has linear utility. For example, it is taken for granted that $2 is worth exactly double of $1. In most real-world scenarios, where the amount of wealth available is finite, this is not exactly true.

Imagine, for example, three travellers in identical situations, except

• Traveller A has $2000
• Traveller B has $1000
• Traveller C has nothing

Now, while traveller A is certainly better off than traveller B, traveller B is much better off than traveller C. Traveller B’s condition is clearly better than the midpoint between traveller A’s and traveller C’s. The first $1000 is a necessity. The additional $1000 is a convenience.

Characteristic Shape

Of course, the exact shape of the utility function of wealth depends heavily on the context. (Is the owner an individual/government/store/bank/something else? What goals does the owner have? Etc.). However, considering the usefulness of a collection of wealth in a wide variety of contexts, we know that the utility function is in general

• monotonically non-decreasing, because additional wealth is never a bad thing.
• concave down at the large scale, because of the typically diminishing returns that additional wealth brings.

So the utility function’s vague typical shape is rather universal.

This non-linearity of wealth utility functions can serve as a basis for understanding a variety of principles ranging from politics to risk management – principles that are usually otherwise explained by invoking intuitive judgements.

Application: Understanding why gambling is not rational

Most people have an understanding that gambling (even if the odds are fair) is not rational. This intuitive judgement has an underlying mathematical reason: While a fair gamble may yield zero expected wealth change, it will yield negative expected utility change.

Therefore taking a fair gamble is worse than not taking it. If the utility function of wealth was linear, a fair gamble would have been a neutral action, rather than a foolish action.

(Of course, under exceptional conditions that violate the concavity of the utility function, gambling can be rational. For example, suppose traveller B receives information that he will unavoidably be killed tomorrow if he remains in the country. A ticket out of the country costs $2000, but he has only $1000. The utility function in this scenario is no longer concave down, and gambling becomes rational. A 50% chance of doubling his money and surviving is better than nothing.)

Application: Optimal Government Size

Most people feel that extreme wealth disparity is a bad thing. Different people draw the line in different places. That is why we have revolutions. This intuitive judgement of “fairness”, again, has the same underlying mathematical principle that discourages gambling: the optimization of overall utility, rather than overall wealth.

The problem (from the government’s perspective) with 100% tax (i.e. communism) is that it kills the individual’s incentive to generate wealth. The problem with 0% tax (i.e. pure capitalism/anarchy) is that it would lead to unbounded growth in wealth disparity (and in the real world, revolution, crime, war etc.). If a wealth utility function (or stochastic process) can be determined for individuals in a particular country, it can be used, in conjunction with a “wealth generation incentive function” (or stochastic process), to derive an optimal tax schedule for the population of that country. This would automatically and optimally balance the concepts of “fairness” and “incentive” in such a way that total utility growth is optimized. This is ultimately what the tax schedules of modern democracies try to achieve, even if it is not recognized explicitly as the objective.

Application in Trading

See follow-up posts: Logarithmic Utility of Wealth

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