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Strict Concavity Connects Equal Marginal Utility to Equal Wealth

Article Quant Q&A · Author: honso
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Summary

The note addresses a basic inference in expected utility analysis: if the marginal utilities at two wealth levels are equal, when can one conclude that the wealth levels themselves are equal? The answer relies on strict concavity of the utility function. Strict concavity makes its derivative strictly decreasing, so the marginal utility function is one-to-one.

Because an injective function takes the same value only at the same input, equality of marginal utility implies equality of the corresponding wealth levels. This is a short mathematical clarification rather than a trading strategy or empirical result. The conclusion depends on strict concavity; without that assumption, marginal utility need not be strictly decreasing and the inference may not hold.

Key ideas

  • Strict concavity of utility implies that marginal utility is strictly decreasing.
  • A strictly decreasing marginal utility function is injective.
  • Equal marginal utility values therefore imply equal wealth levels under this assumption.
  • The inference may fail if utility is not strictly concave.

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# How did we get $W_g=W_b$ from $\dfrac{U'(W_g)}{U'(W_b)}=1$?


# How did we get $W_g=W_b$ from $\dfrac{U'(W_g)}{U'(W_b)}=1$?












My question is from Nicholson-Snyder's text , E-book here.

My question is here, from page 217 of the book. (I can't post image as my reputation is not enough.)

How did we get $W_g=W_b$ from $\dfrac{U'(W_g)}{U'(W_b)}=1$ ?

## Answer by Hans (score 1)

https://quant.stackexchange.com/a/9778

$U$ is usually strictly concave. So $U'$ is strictly decreasing and is therefore injective.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.