Interpreting Utility Maximization Under a Wealth Constraint

Summary

The document raises a question about a constrained expected-utility problem: maximize expected utility of terminal wealth while requiring discounted wealth to have a specified risk-neutral expectation. It focuses on a textbook exercise involving the function formed by subtracting a linear term from utility, and asks how to interpret the instruction to fix a multiplier and maximize over wealth. A logarithmic utility example is used to question whether the stated condition is correct.

The only response points to an answer based on another edition and a solutions manual; it does not reproduce the derivation or resolve the apparent mismatch. The note therefore identifies a useful issue in reading constrained optimization problems, but it does not provide a complete solution. In particular, it leaves unexplained how the multiplier relates to the inverse marginal utility function or how the exercise’s variables should be interpreted.

Key ideas

• The question concerns expected utility maximization with a constraint on discounted wealth.
• The exercise relates a utility function and a linear multiplier term.
• The author tests the statement using logarithmic utility and finds an apparent inconsistency.
• The response refers readers to a different edition and solutions manual without showing the reasoning.

Tags

# Maximizing utility subject to a wealth constraint


# Maximizing utility subject to a wealth constraint












Let $\tilde{E}$ be the risk neutral expectation, and $X_t$ the wealth that time t and $R$ the return of a risk-free investment. Consider maximizing the function $EU(X_N)$ subject to $\tilde{E}\frac{X_n}{R^N}=X_0$.

The solution is discussed in Chapter 3 of Shreve volume 1, and question 3.8.i asks to show:

> Fix $y$, and show that the function of $x$ given by $f(x)=U(x)-yx$ is maximized by $y=I(x)$. ($I(x)=\left[U'(x)\right]^{-1}$)

I might be misunderstanding what it means by "fix y", but as it stands this seems false. E.g. say $U(x)=\ln x$; then $f'(x)=x^{-1}-x/x=x^{-1}-1\not=0$.

What am I not understanding?

## Answer by BCLC (score 1, accepted)

https://quant.stackexchange.com/a/15885

From a different edition plus solutions manual.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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