February 7, 2021, Christopher D. Carroll CARAModelWithYRisk
Consider the optimization problem of a consumer with a constant absolute risk aversion instantaneous utility function implying facing an interest rate that is constant at .^{1} The consumer’s optimization problem is
 (1) 
subject to the constraints
 (2) 
where is the consumer’s idiosyncratic income, which exhibits a randomwalk deviation from an exogenouslygrowing trend:

Bellman’s equation for this problem is
 (3) 
The ﬁrst order condition (FOC) for the CARA utility problem is
 (4) 
and the Envelope theorem tells us that
 (5) 
In the perfect foresight version of the model in which , the Euler equation will be
 (6) 
The term reﬂects the intertemporal substitution factor in consumption. Notice that intertemporal substitution takes the form of additive changes in the level of consumption in the CARA utility model, rather than multiplicative changes that aﬀect the growth rate of consumption, as in the CRRA model.
Now suppose we are interested in the case where permanent income shocks are distributed normally, . Then it turns out that the process
 (7) 
satisﬁes the FOC under uncertainty:
 (8) 
Deﬁne , so that (7) becomes:
 (9) 
The expected present discounted value of consumption is

Now we need the following fact:
Thus, the expectation of the inﬁnite horizon PDV of consumption is:
 (10) 
Given the process for income described above, we have
 (11) 
The IBC says
 (12) 
Because the intertemporal budget constraint must hold in every state of the world, the expectation of the PDV of consumption must equal current wealth plus the expectation of the PDV of income. Thus,

The term reﬂects the consumer’s idiosyncratic level of permanent income, which has no systematic growth (or decline). The next term reﬂects the MPC out of total ‘certain’ wealth, human and nonhuman. The ﬁnal term reﬂects the combination of the intertemporal substitution motive (in the term) and the precautionary motive in the term, as is evident from the fact that it equals zero if either there is no precautionary motive () or there is no uncertainty .
Note some peculiar aspects of this solution. First, observe that, marginally, the consumer spends exactly the interest income on capital, . The reason this is peculiar is that the MPC out of capital does not depend on how impatient the consumer is. Impatience is reﬂected in the change in consumption over time, but not in the level of consumption except as that is aﬀected by the budget constraint.
Second, notice that the eﬀect of income uncertainty on saving is the same in absolute dollars regardless of the level of resources or permanent income.
Caballero, Ricardo J. (1990): “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Economics, 25, 113–136, http://ideas.repec.org/p/clu/wpaper/1988_05.html.