February 7, 2021, Christopher D. Carroll RandomWalk
This handout derives the Hall (1978) random walk proposition for consumption.
The consumption Euler equation when future consumption is uncertain takes the form1
Suppose the utility function is quadratic:
where is the “bliss point” level of consumption.2 Marginal utility is
and suppose further that so that (1) becomes
Deﬁning the innovation to consumption as
the random walk proposition is simply that the expectation of consumption changes is zero:
This means that no information known to the consumer when the consumption choice was made can have any predictive power for how consumption will change between period and (or for any date beyond ).
Hall, Robert E. (1978): “Stochastic Implications of the Life-Cycle/Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, 96, 971–87, Available at http://www.stanford.edu/~rehall/Stochastic-JPE-Dec-1978.pdf.