Definitions

# Exponential utility

In economics exponential utility refers to a specific form of the utility function, used in many contexts because of its convenience when uncertainty is present. Formally, exponential utility is given by:

$u\left(c\right)=-e^\left\{-a c\right\}$,

where $c$ is consumption and $a$ is a constant.

Exponential utility implies constant absolute risk aversion, with coefficient of absolute risk aversion equal to

$frac\left\{-u\text{'}\text{'}\left(c\right)\right\}\left\{u\text{'}\left(c\right)\right\}=a.$

Though isoelastic utility, exhibiting constant relative risk aversion, is considered more plausible (as are other utility functions exhibiting decreasing absolute risk aversion), exponential utility is particularly convenient for many calculations. Specifically, under exponential utility, expected utility is given by:

$E\left(u\left(c\right)\right)=E\left(-e^\left\{-a \left(c+epsilon\right)\right\}\right),$

where E is the expectation operator. With normally distributed noise, ie,

$epsilon sim N\left(mu, sigma^2\right),!$

E(u(c)) can be calculated easily using the fact that

$E\left(e^\left\{-a epsilon\right\}\right)=e^\left\{-a mu + frac\left\{a^2\right\}\left\{2\right\}sigma^2\right\}.$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9085785746574402, "perplexity_flag": "middle"}