© February 14, 2012, Christopher D. Carroll RandomWalk

The Random Walk Model of Consumption

This handout derives the Hall (1978) random walk proposition for consumption.

The consumption Euler equation when future consumption is uncertain takes the form1

′ ′ u (ct) = βR Et [u (ct+1 )]. (1)

Suppose the utility function takes the quadratic form

u(c ) = - (1∕2 )(/c - c)2 (2)
where /c is the “bliss point” level of consumption.2 Marginal utility is
′ u (c) = (/c - c) (3)
and suppose further that R β = 1 so that (1) becomes
(/c - ct) = Et [(/c - ct+1 )] (4) Et [ct+1 ] = ct. (5)

Defining the innovation to consumption as

ϵt+1 = ct+1 - ct, (6) ≡ Δct+1, (7)
the random walk proposition is simply that the expectation of consumption changes is zero:
E [Δc ] = 0. (8) t t+1

This means that no information known to the consumer when the consumption choice ct was made can have any predictive power for how consumption will change between period t and t + 1 (or for any date beyond t + 1 ).

References

   HALL, ROBERT E. (1978): “Stochastic Implications of the Life Cycle–Permanent Income Hypothesis,” Journal of Political Economy, 86(6), 971–987.