Definition of The Exponential Utility Function: Exponential utility is a particular functional form for the utility function. Some versions of it are used often in finance.

Here is the simplest version. Define U() as the utility function and w as wealth. a is a positive scalar parameter.
U(w) = -e^-aw

is the exponential utility function.

Now consider events over time. An agent might have a utility function mapping possible streams of consumption into utility values. Here is one way this is often parameterized:

Define (b) as a constant discount rate known to the agent. It's a scalar that is between zero and one, and usually thought of as near one.

Define t as a time subscript that starts at zero and increases over the integers, either to some fixed T or to infinity.
Define c(t) as the amount the agent gets to consume at each t, and {c(t)} as the series of consumptions for all relevant t. c(t) is random here. its value is not known but its distribution is assumed known to the agent.
Let E[] be the expectations operator that takes means of distributions.

Using this notation a common dynamic version of exponential utility is:
u({c_t} = the sum over all t of (b)^tE[-e^-ac(t)]

Whether this utility function describes observed investment decisions is discussable and testable. It is not often discussed, however. If clear information on that becomes known to this author, it will be added here.
Most uses of the exponential utility function in finance are driven by these aspects: (a) its analytic tractability; e.g. that it can be differentiated with respect to choice variables that affect future wealth w or consumption c(t); (b) for some applications it aggregates usefully, meaning that if every agent has this exact utility function and they can buy securities then a representative agent can be defined which also has this analytically convenient form and for whom the securities prices would be the same. It's convenient for computing securities prices in some abstract economies to use that representative agent. There are "no wealth effects" -- that is, the amount of risky securities that the agent wants to hold is not a function of his own wealth, as long as he can borrow infinitely (which is often assumed for tractability in these models.) (Econterms)

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