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The von Neumann-Morgenstern axioms were an attempt to characterize rational decision-making in the presence of risk. The von Neumann-Morgenstern utility theorem says that if someone is vNM-rational, i.e. their preferences obey the axioms, then there exists a function $u$ from the set of outcomes to the real numbers, such that they will have to maximize the expected value of $u$ if they want to abide by their preferences. However, this function is only defined up to a linear transformation, i.e. maximizing the expected value of $u$ and maximizing the expected value of $a + bu$ yield identical results.

Now I have heard from various sources that a social welfare function based on Kaldor-Hicks efficiency can be written as a linear combination of vNM-utility functions, i.e. something of the form $a_1 u_1 + a_2 u_2 + ... a_N u_N$, where $a_i$ is the reciprocal of the $i^{th}$ individual's marginal utility of money. But I don't see how this can make any sense; can't you just rescale each of the $u_i$'s by any amounts you please, since they're only defined up to a linear transformation? Surely $a_1 u_1 + a_2 u_2 + ... a_N u_N$ and $a_1 b_1 u_1 + a_2 b_1 u_2 + ... a_N b_N u_N$ aren't equivalent social welfare functions, since e.g. we can set $b_i = 1/a_i$, thereby taking marginal utility completely out of the function.

So what's going on? What does it mean to say that the coefficients of the linear combination are reciprocals of marginal utilities, when the coefficients can be made into anything you like?

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: Does anyone know where I can find a proof that this social welfare function yields KH-efficiency? Also, I also want to know how this social welfare function should be modified if we don't assume all the von Neumann-Morgenstern axioms, so that utility functions are only defined up to a monotonic transformation, rather than being defined up to a linear transformation.

EDIT 2: I think that @QiaochuYuan's answer may clear up the scaling issue, but what about the additive constant issue? If we change $u_1$ to $u_1 + C_1$, then $a_1 u_1 + a_2 u_2 + ... + a_N u_N + C$ becomes $a_1 u_1 + a_2 u_2 + ... + a_N u_N + C + a_1 C_1$. But doesn't a_1 vary, since we usually have diminishing marginal utility of money? So wouldn't the term $a_1 C_1$ not be a constant? Or does the term $C$ also vary to accomodate that?

Keshav Srinivasan
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  • I suggest you try to write in Tex...it's also not clear what your definition of social welfare function is, does it aggregate preferences...what? – Michael Aug 27 '13 at 23:01
  • @Michael I'm not using any definition of social welfare function; I'm not even sure what the requisite properties are for a social welfare function. I'm just discussing the result that a social planner who maximizes the weighted sum of vNM-utilities, where the weight for each individual is the reciprocal their marginal utility of money, will achieve a Kaldor-Hicks-optimal result. Whether this weighted sum is a social welfare function is not relevant to my question, I think. – Keshav Srinivasan Aug 27 '13 at 23:12
  • Perhaps $u_1$ is defined to be $0$ given the current state of the world, thereby fixing its constant factor? – Christopher King Jan 06 '19 at 02:57

1 Answers1

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If you multiply $u_i$ by some scale factor $c_i$, then $a_i$ gets multiplied by $\frac{1}{c_i}$, so the product $a_i u_i$ is unchanged. The point of taking the reciprocal of marginal utility of money is to cancel out the effects of scaling (you can think of $a_i$ as having units of "inverse utilities" if you'd like).

Qiaochu Yuan
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  • Thanks, that's clear now. By the way, do you happen to know where I can find a proof that this social welfare function yield a Kaldor-Hicks efficient result? Also, if you don't assume all the vNM axioms, then the utility function is only defined up to an arbitrary monotonic transformation, i.e. f composed with u represents the same utility function as u for any increasing function f. In that case, how can we modify the social welfare function so that it's invariant under the transformation of the u_i's? – Keshav Srinivasan Aug 28 '13 at 14:19