$©$ February 7, 2021, 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 is quadratic:

2 u(c ) = − (1 ∕2 )(/c − c)

(2)

where $/c$ is the “bliss point” level of consumption.² Marginal utility is

u ′(c ) = (c − c) /

(3)

and suppose further that $Rβ = 1$ so that (1) becomes

(/c − ct) = 𝔼t [(/c − ct+1) ] 𝔼t [ct+1 ] = ct.

(4)

Deﬁning the innovation to consumption as

𝜖t+1 = ct+1 − ct, ≡ Δct+1,

(5)

the random walk proposition is simply that the expectation of consumption changes is zero:

𝔼 [Δc ] = 0. t t+1

(6)

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: Theory and Evidence,” Journal of Political Economy, 96, 971–87, Available at http://www.stanford.edu/~rehall/Stochastic-JPE-Dec-1978.pdf.