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Macroeconomics
Johns Hopkins University

The Random Walk Model of Consumption

Authors
Affiliations
Johns Hopkins University
Econ-ARK
Johns Hopkins University
Econ-ARK

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

The consumption Euler equation when future consumption is uncertain takes the form[1]

u′(ct)=βREt[u′(ct+1)].\uFunc^{\prime}(\cRat_{t}) = \Discount \Rfree \Ex_{t}[\uFunc^{\prime}(\cRat_{t+1})].
(1)

Suppose the utility function is quadratic:

u(c)=−(1/2)(c−c)2\uFunc(\cRat) = -(1/2) (\cancel{\cRat}-\cRat)^{2}
(2)

where c\cancel{\cRat} is the “bliss point” level of consumption.[2] Marginal utility is

u′(c)=(c−c)\uFunc^{\prime}(\cRat) = (\cancel{\cRat}-\cRat)
(3)

and suppose further that Rβ=1\Rfree \Discount = 1 so that (1) becomes

(c−ct)=Et[(c−ct+1)]Et[ct+1]=ct.\begin{aligned} (\cancel{\cRat}-\cRat_{t}) & = \Ex_{t}[(\cancel{\cRat}-\cRat_{t+1})] \\ \Ex_{t}[\cRat_{t+1}] & = \cRat_{t}. \end{aligned}
(4)

Defining the innovation to consumption as

ϵt+1=ct+1−ct≡Δct+1,\begin{aligned} \epsilon_{t+1} & = \cRat_{t+1}-\cRat_{t} \\ & \equiv \Delta \cRat_{t+1}, \end{aligned}
(5)

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

Et[Δct+1]=0.\Ex_{t}[\Delta \cRat_{t+1}] = 0.
(6)

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

Footnotes

  1. See the The Envelope Theorem and the Euler Equation for the derivation of the Euler equation in the perfect foresight case; we will show later that the consequence of uncertainty is simply to insert the expectations operator.

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  2. Assume that the consumer is sufficiently poor that it will be impossible for them ever to achieve consumption as large as c\cancel{\cRat}.

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References
  1. Hall, R. E. (1978). Stochastic Implications of the Life-Cycle/Permanent Income Hypothesis: Theory and Evidence. Journal of Political Economy, 86(6), 971–987. 10.1086/260724
Consumption Theory
Consumption Under Perfect Foresight and CRRA Utility
Consumption Theory
Consumption Functions and the Permanent Income Hypothesis