February 7, 2021, Christopher D. Carroll Habits
Consider a consumer whose goal at date is to solve the problem1
where is the habit stock, and all other variables are as usually deﬁned. The DBC is
However, when habits aﬀect utility we must also specify a process that describes how habits evolve over time. Our assumption will be:
Bellman’s equation for this problem is therefore
To clarify the workings of the Envelope theorem in the case with two state variables, let’s deﬁne a function
and deﬁne the function as the choice of that solves the maximization (4), so that we have
The ﬁrst order condition for (4) with respect to is (dropping arguments for brevity and denoting the derivative of with respect to at time as ):
The intuition is as follows. Note ﬁrst that if utility is not aﬀected by habits, then and equation (8) reduces to the usual ﬁrst order condition for consumption, which tells us that increasing consumption by today and reducing it by in the next period must not change expected discounted utility. With habits, an increase in consumption today has a consequence beyond its eﬀect on tomorrow’s resources : tomorrow’s habit stock will be changed as well. An increase in consumption today of size increases the size of the habit stock which tomorrow’s consumption is compared to, and therefore reduces tomorrow’s utility by an amount corresponding to the marginal utility of higher habits tomorrow . Since is negative (higher habits make utility lower), this tells us that the RHS of equation (8) will be a larger positive number than it would be without habits. This means that the level of that satisﬁes the ﬁrst order condition will be a lower number (higher marginal utility) than before. Hence, habits increase the willingness to delay spending, and increase the saving rate.
Note that the ﬁrst order condition also implies that
when evaluated at .
Now consider the total derivative of with respect to . (To reduce clutter, I will write as ). The chain rule tells us that
so we have that
The Envelope theorem is the shortcut way to obtain this conclusion. The clearest way to use the theorem is by taking the partial derivatives of the function with respect to each of its three arguments, using the Chain Rule to take into account the possible dependency of and on :
where the Envelope theorem is what tells you that the term is equal to zero because you are evaluating the function at (and is zero by the assumed structure of the problem in which is predetermined).
Now writing out , (12) becomes
which the envelope theorem says is equivalent to
There is a potentially confusing thing about doing it this way, however: when you reach an expression like (13) it is tempting to think to yourself as follows: “ is a function of , and is also indirectly a function of , so the chain rule tells me that when I take the derivative in (13) I need to keep track of these.” In fact, you must treat and as zero here. The reason is that this is a partial derivative with respect to . The dependence of (and indirectly ) on has already been taken care of in the two terms in (12) that were equal to zero. The confusion here is caused largely by the fact that partial diﬀerentiation is an area where standard mathematical notation is basically confusing and poorly chosen.2
The shortest way to obtain the end result is, as in the single variable problem, to start with Bellman’s equation and take the partial derivative with respect to directly (treating the problem as though were a constant):
Whichever way you do it, substituting (14) into the FOC equation (8) gives
The intuition for this is as follows. The marginal value of wealth must be equal to the marginal value associated with a tiny bit more consumption. In the presence of habits, the extra consumption yields extra utility today but aﬀects value next period by (which is a negative number), the discounted consequence of which from today’s perspective is the term.
In a problem with two state variables, the Envelope theorem can be applied to each state (and indeed in general must be applied in order to solve the model).
Again let’s start the brute force way by working through the total derivative of . For this problem, the total derivative (again denoting as ) is:
so we have
Turning now to more direct use of the envelope theorem, the Chain Rule tells us
while the Envelope theorem once again says at so we obtain
since appears directly only in the part of . And once again, the shortest way to the answer is to treat as though it were a constant in the value function, which yields
From (16) this implies that
Roll this equation forward one period and substitute into equation (14) to obtain:
Note that if so that habits have no eﬀect on utility, (22) again is solved by the standard time-separable Euler equation.
Now assume that the utility function takes the speciﬁc form
which implies derivatives of
Substituting these into equation (22) we obtain,
Now assume that there is a solution in which marginal utility of consumption grows at a constant rate over time, and substitute into (25)
so marginal utility grows at rate . Note that if we assume so that habits do not matter, we again obtain the standard result that .
Now make the ﬁnal assumption that , implying of course that . Equation (26) can be rewritten
where (29) follows from (28) because and , and (32) follows from (31) because for small and .
Substituting (32) into (27) gives
Thus, this formulation of habit formation implies that the growth rate of consumption is serially correlated.
Carroll, Christopher D. (2000): “Solving Consumption Models with Multiplicative Habits,” Economics Letters, 68(1), 67–77, http://econ.jhu.edu/people/ccarroll/HabitsEconLett.pdf.