Ergodicity economics is a new branch of economic theory that notes the conceptual difference between time averages and expectation values, which coincide only for ergodic observables. It postulates that individual agents maximise the time average growth rate of wealth, known widely as growth optimality. This contrasts with the dominant behavioural model in economics, expected utility theory, in which agents maximise expectation values of changes in psychologically transformed wealth. Historically, growth optimality was explored for additive and multiplicative gambles. Here we apply it to a general class of wealth dynamics, extending the range of economic situations where it may be used. Moreover, we show a correspondence between growth optimality and expected utility theory, in which the ergodicity transformation in the former is identified as the utility function in the latter. This correspondence offers a theoretical basis for choosing utility functions and predicts that wealth dynamics are strong determinants of risk preferences.
We now compare the two expressions (Eq. 15) and (Eq. 16). Clearly the value of r in (Eq. 15) cannot depend on the way in which the diverging time period is partitioned, so the length of interval t must be arbitrary and can be set to the value of t in (Eq. 16), for consistency we then call um(t) = um(t). Expressions (Eq. 15) and (Eq. 16) are equivalent if the successive additive increments, um(t), are distributed identically to the un in (Eq. 16), which requires only that they are stationary and independent.
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Thus we have a condition on u(t) which suces to make r = (cid:104)r(cid:105), namely that it be a stochastic process whose additive increments are stationary and independent. This means that u(t) is, in general, a Levy process. Without loss of realism we shall restrict our attention to processes with continuous paths. According to a theorem stated in [4, p. 2] and proved in [2, Chapter 12] this means that u(t) must be a Brownian motion with drift, du = audt + budW, where dW is the innitesimal increment of the Wiener process.
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By arguing backwards we can address concerns regarding the existence of r. If u follows the dynamics specied by (Eq. 17), then it is straightforward to show that the limit r always exists and takes the value a. Consequently the decision criterion (Eq. 7) is equivalent to the optimisation of r, the time-average growth rate. The process x(t) may be chosen such that (Eq. 17) does not apply for any choice of u(x). In this case we cannot interpret EUT dynamically, and such processes are likely to be pathological. This gives our central result: For EUT to be equivalent to optimisation over time, utility must follow an additive stochastic process with stationary increments which, in our framework, we shall take to be a Brownian motion with drift. (14) (15) (16) (17) 4
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This is a fascinating general connection. If the physical reason why we observe non-linear utility functions is the non-linear eect of uctuations over time, then a given utility function encodes a corresponding stochastic wealth process. Provided that a utility function u(x) is invertible, i.e. provided that its inverse, x(u), exists, a simple application of Ito calculus to (Eq. 17) yields directly the SDE obeyed by the wealth, x. Thus every invertible utility function encodes a unique dynamic in wealth which arises from a Brownian motion in utility. This is explored further below.
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