©August 28, 2019, Christopher D. Carroll Habits

Consumption Models with Habit Formation

Consider a consumer whose goal at date t is to solve the
problem^{1}

| (1) |

where h_{t+n} is the habit stock, and all other variables are as usually deﬁned. The
DBC is

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

The ﬁrst order condition for (4) with respect to c_{t} is (dropping arguments for
brevity and denoting the derivative of f with respect to x at time t as f_{t}^{x}):

The intuition for this equation is as follows. Note ﬁrst that if utility is not
aﬀected by habits, then v_{t+1}^{h} = 0 and equation (8) reduces to the usual ﬁrst
order condition for consumption, which tells us that increasing consumption by 𝜖
today and reducing it by R𝜖 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 m_{t+1}: tomorrow’s habit
stock will be changed as well. An increase in consumption today of size 𝜖
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 v_{t+1}^{h}. Since v_{t+1}^{h} 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 c_{t} 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

Now consider the total derivative of v_{t}(m_{t},h_{t},c_{t}(m_{t},h_{t})) with respect to m_{t}. (To
reduce clutter, I will write dc_{t}(m_{t},h_{t})∕dm_{t} as dc_{t}∕dm_{t}). The chain rule tells us
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 v_{t}
function with respect to each of its three arguments, using the Chain
Rule to take into account the possible dependency of h_{t} and c_{t} on m_{t}:

Now writing out ∂v_{t}∕∂m_{t}, (15) becomes

There is a potentially confusing thing about doing it this way, however: when
you reach an expression like (16) it is tempting to think to yourself as follows: “c_{t}
is a function of m_{t}, and h_{t+1} = c_{t} is also indirectly a function of m_{t}, so the chain
rule tells me that when I take the derivative in (16) I need to keep track of
these.” In fact, you must treat ∂c_{t}∕∂m_{t} and ∂h_{t+1}∕∂m_{t} as zero here.
The reason is that this is a partial derivative with respect to m_{t}. The
dependence of c_{t} (and indirectly h_{t+1}) on m_{t} has already been taken
care of in the two terms in (15) 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 m_{t} directly (treating the problem as though c_{t} were a constant):

Whichever way you do it, substituting (17) 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 u_{t}^{c} but
aﬀects value next period by v_{t+1}^{h} (which is a negative number), the
discounted consequence of which from today’s perspective is the βv_{t+1}^{h}
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
v_{t}. For this problem, the total derivative (again denoting dc_{t}(m_{t},h_{t})∕dh_{t} as
dc_{t}∕dh_{t}) is:

Turning now to more direct use of the envelope theorem, the Chain Rule tells us

From (20) this implies that

Now assume that the utility function takes the speciﬁc form

Substituting these into equation (31) we obtain,

Now assume that there is a solution in which marginal utility of consumption
grows at a constant rate over time, f_{t}^{′} = kf_{t+1}^{′} and substitute into (35)

Now make the ﬁnal assumption that f(z) = z^{1−ρ}∕(1 − ρ), implying of course
that f^{′}(z) = z^{−ρ}. Equation (36) can be rewritten

Substituting (44) into (39) 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.