where $b t$ is beginning-of-period bank balances, $y t$ is current labor income, and $R = (1 + r)$ is the constant interest factor. Actual labor income $y$ is permanent labor income $p$ modiﬁed by a transitory shock factor $𝜃$ :

where $𝔼 [𝜃 ] = 1 ∀ n > 0 t t+n$ . Permanent labor income grows by a predictable factor $G$ from period to period:

so that the expected present discounted value of permanent labor income (‘human wealth’) for an inﬁnite-horizon consumer is

where $κ$ is the ‘marginal propensity to consume’ out of total wealth $o$ .¹

Under these circumstances, RandomWalk shows that consumption will follow a random walk,

Now assume that the economy is populated by a set of measure one of consumers indexed by a superscript $i$ distributed uniformly along the unit interval. Per capita values of all variables, designated by the upper case, are the integral over all individuals in the economy, as in Aggregation, so that

In principle, we could allow each individual in this economy to experience a diﬀerent transitory and permanent shock from every other individual in each period. However, for our purposes it is useful to assume that everyone experiences the same shocks in a given period; that is $𝜃i = Θ ∀ t t t$ and $i ψ t = Ψt ∀ t$ .

Assuming (here and henceforth) that the growth factor for permanent income is $G = 1$ , ﬁgure 1 shows the path of consumption (the solid dots) for an economy populated by omniscient consumers who in periods $t − n$ for $n > 0$ had experienced $Θt − n = Ψt− n = 1$ ; that is, this economy has had no shocks to income in the past. (For convenience, the consumer is assumed to have arrived in period $t$ with $Bt = 0$ ). In period $t$ the consumer draws $Θ = 2 t$ and $Ψ = 1 t$ ; thereafter $Ψ = Θ = 1 t+n t+n$ . The ﬁgure shows $Ct − 2, Ct− 1,Ct, 𝔼t[Ct+1 ],𝔼t [Ct+2 ],...$ . Figure 2 similarly shows the path of $B$ , again as black dots.

Figure 1:Path of $C$ after a shock $Θ = 2 t$ , Omniscient Consumers

Figure 2:Path of $B$ after a shock $Θt = 2$ , Omniscient Consumers

Now suppose that not every consumer updates expectations in every period. Instead, expectations are ‘sticky’: each consumer updates with probability $Π$ in each period. Whether the consumer at location $i$ updates in period $t$ is determined by the realization of the dichotomous random variable

and each period’s updaters are chosen randomly such that a constant proportion $Π$ update in each period:

It will also be convenient to deﬁne the date of consumer $i$ ’s most recent update; we call this object $τ i t$ .

We need a notation to represent sets of consumers deﬁned by the period of their most recent update. We denote such a set by the condition on $i τt$ ; for example, the set of consumers whose most recent update, as of date $t$ , was prior to period $t − 1$ would be $i 𝒯 = {τt < t − 1 }$ . We denote the per-capita value of a variable $∙$ , among consumers in a set $𝒯$ as of date $t$ , by $∙t|𝒯$ . Dropping the $i$ superscripts to reduce clutter, per-capita consumption among households who have updated in period $t$ is therefore

In periods when expectations are not updated, the consumer continues to spend the same amount as in the most recent period when his expectations were updated.³ If the economy is large the proportion of consumers who update their expectations every period will be $Π$ .⁴ Average consumption among those who are not updating in the current period (for whom $1 − πt = 1$ ) is then

because consumption per capita among those who are not updating in the current period is (by assumption) identical to their consumption per capita in the prior period, which must match aggregate consumption per capita in the prior period because the set who do not update today is randomly selected from the $t − 1$ population.

where (13) follows its predecessor since, among consumers who have updated in period $t$ , the random walk proposition says that $𝔼t [Ct+1 |τt=t] = Ct,τt=t$ . Subtracting $C t|τt=t$ from both sides of (13) and substituting the result into (11) yields

We are ﬁnally in position to show how aggregate consumption and wealth would respond in this economy to a transitory positive shock to aggregate labor income like the one considered above for the omniscient model.

Consider the case of a positive shock of size $ˆΘt = 1$ , as before. In the ﬁrst period consumption rises only by $Π κ$ , rather than the full amount corresponding to the permanent income associated with the new level of wealth. Therefore aggregate wealth in period $t + 1$ will be greater than it would have been in the omniscient model. Similarly for all subsequent periods. Thus, in contrast with the omniscient model, the sluggish adjustment of consumption to the shock means that the shock has a permanent eﬀect on the level of aggregate wealth, and therefore on the level of aggregate consumption. (Figures 3 and 4 depict the results).

Figure 3:Path of $C$ after a shock $Θ = 2 t$ ; Sticky Expectations in Red/Gray

Figure 4:Path of $B$ after a shock $Θt = 2$ ; Sticky Expectations in Red/Gray

The sticky expectations model says that consumption growth today can be statistically related to any variable that is related to lagged consumption growth. In particular, if lagged consumption growth is related to lagged income growth (as it certainly will be), then there should be a statistically signiﬁcant eﬀect of lagged income growth on current consumption growth if expectations are sticky.

If the model derived here could be taken literally, it would suggest estimating an equation of the form

However, if there is potential measurement error in $C t$ the coeﬃcient obtained from estimating (14) would be biased toward zero for standard errors-in-variables reasons (just as regressing consumption on actual income yields a downward-biased estimate of the response of consumption to permanent income), which means that the estimate of $Π$ would be biased toward 1 (i.e. the omniscient model in which everyone adjusts all the time). Under these circumstances, direct estimation of (14) would not be a reliable way to estimate $Π$ .

For estimation methods that get around this problem see Sommer (2007), Carroll, Sommer, and Slacalek (2011), Carroll, Otsuka, and Slacalek (2011). Those papers consistently ﬁnd that the proportion of updaters is about $Π = 0.25$ per quarter, so that the serial correlation of ‘true’ consumption growth is about $(1 − Π ) = 0.75$ per quarter.

References

Carroll, Christopher D, Misuzu Otsuka, and Jiri Slacalek (2011): “How large are housing and ﬁnancial wealth eﬀects? A new approach,” Journal of Money, Credit and Banking, 43(1), 55–79.

Carroll, Christopher D., Martin Sommer, and Jiri Slacalek (2011): “International Evidence on Sticky Consumption Growth,” Review of Economics and Statistics, 93(4), 1135–1145, http://econ.jhu.edu/people/ccarroll/papers/cssIntlStickyC/.

Deaton, Angus S. (1992): Understanding Consumption. Oxford University Press, New York.

Friedman, Milton A. (1957): A Theory of the Consumption Function. Princeton University Press.

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.

Sommer, Martin (2007): “Habit Formation and Aggregate Consumption Dynamics,” Advances in Macroeconomics, 7(1), Article 21.

This appendix provides additional derivations and notation useful for simulating the model. One way of interpreting consumers’ behavior in this model is to attribute to them the beliefs that would rationalize their actions. Deﬁne $O$ as the level of wealth (human and nonhuman) that the consumer perceives. Then the Deaton deﬁnition of the permanent income hypothesis is that

and the reason consumption follows a random walk is that $κ = (r∕R )$ is precisely the amount that ensures that $𝔼 [O ] = O t t+1 t$ .

Writing the “believed” level of wealth as $O¯$ , we could then interpret the failure of the sticky expectations consumer to change his consumption during the period of nonupdating as reﬂecting his optimal forecast that, in the absence of further information, $¯O = O¯ + ...= O t+1 t+2 t$ .

To pursue this interpretation, it is useful to write the budget constraint more explicitly, as before; start with the constraint in levels, then decompose variables into ratios to permanent income (nonbold variables) and the level of permanent income:

where we permit a time subscript on $R$ and $G$ because we want to allow for the possibility that beliefs about the interest rate or growth rate might change over time.

Consider an economy that comes into existence in period $0$ with a population of consumers who are identical in every respect, including their beliefs about current and future values of the economy’s variables.

First we examine the case where neither $R$ nor $G$ can change after date $0$ . In that case, we can track the dynamics of believed and actual variables as follows.

where capturing the dynamics of the ratio of true permanent income to believed permanent income $Qt+1$ requires us to compute

with the crucially useful fact that since by assumption neither $R$ nor $G$ is changing, normalized human wealth does not change from

Matters are more complex if expectations about $R$ and $G$ are allowed to change over time.

Suppose again that we begin our economy in period $0$ with population with homogeneous views: Everyone believes $R = R0$ and $G = G0$ ; so long as these views are universally held in the population, aggregate dynamics are captured by the foregoing analysis.

Suppose, however, that in some period $n > 0$ the economy’s ‘true’ values of $R$ or $G$ change. Updating consumers see this change immediately. But nonupdaters will not discover the changed nature of the economy’s dynamics until they update again.

We capture this modiﬁcation to the model by keeping track of the aggregate values of the variables for the set of consumers who adhere to each diﬀering opinion, along with the population mass associated with the diﬀerent opinions. Speciﬁcally, suppose there are $J$ diﬀerent opinions in the population, each of whom constitutes population mass $j ℒ t$ such that $∑ ℒj = 1 ∀ t j t$ . Then for each such population, it will be necessary to keep track of their average beliefs about macroeconomic variables.

Suppose, for example, that through period $x$ there have been only $j − 1$ diﬀerent opinion groups in the population. In period $x + 1$ either $G$ or $R$ changes. We need then to deﬁne group $j$ by $Rj = Rx$ and $Gj = Gx$ and to deﬁne $A¯j ≡ A x x$ , $P¯j = P x x$ , and so on. We will henceforth need to keep track of dynamics of the consumers who remain in belief group $j$ by, e.g.,

while we need to keep track of the populations of the diﬀering groups by, e.g.,

and so on. These population dynamics continue forever, but the population of households continuing to hold any speciﬁc belief conﬁguration dwindles toward zero as time progresses.

Aggregate variables for the population as a whole can be constructed as the population-weighted sums across all the diﬀering belief groups, weighted by their masses:

and note that if beliefs change back to a conﬁguration that has been seen before it is possible to add the population mass and aggregate values of the variables associated with the new population with that belief conﬁguration to the corresponding ﬁgures for the old population that holds the same beliefs. This reduces the number of groups that the simulations must track in the case where beliefs switch between a limited number of distinct values.