$©$ February 7, 2021, Christopher D. Carroll Durables

Durables

Consider a consumer who gets utility from a ﬂow of consumption of nondurable goods, $ct$ , as well as from a stock of durable goods, $dt$ .¹ The consumer’s goal is to

∑T max βs− tu (cs,ds ) s=t

(1)

where $ds$ is the stock of the durable good, and all other variables are as usually deﬁned.

We will assume that the stock of the durable good evolves over time according to

d = (1 − δ)d + x , t+1 t t+1

(2)

where $xt$ is period-t eXpenditure on the durable good and $δ$ is the durable good’s depreciation rate (a good with a lower value of $δ$ is said to be “more durable”).

The dynamic budget constraint is

mt+1 = (mt − ct − xt)R + yt+1.

(3)

Bellman’s equation is

vt(mt, dt− 1) = max [u(ct, dt) + βvt+1 (mt+1, dt)] , {ct,xt}

(4)

or, equivalently,

v (m ,d ) = max [u(c ,d ) + βv (m ,d )] , t t t− 1 {ct,dt} t t t+1 t+1 t

(5)

subject to

( --------=xt------- ) ◜ ◞◟ ◝ mt+1 = mt − ct − (dt − (1 − δ )dt− 1) R + yt+1

(6)

or (substituting this into (5)),

vt(mt, dt− 1) = max {u (ct,dt) + βvt+1 ((mt − ct − (dt − (1 − δ)dt− 1))R + yt+1, dt)} . {ct,dt}

Since this equation has two control variables, $ct$ and $dt$ , there are two ﬁrst order conditions:

wrt $ct$ :

uc − Rβvm = 0 t t+1 uct = R βvmt+1

(7)

wrt $dt$ :

d m d m d u t = β(Rv t+1 − v t+1 ) = R βv t+1 − βv t+1.

(8)

Note that when taking the derivative with respect to $ct$ you assume that $∂d ∕∂c = 0 t t$ and vice versa. Although the ﬁrst order conditions will deﬁne a relationship between the optimal values of $ct$ and $dt$ , there is no mechanical link that applies at this point.

Now we want to apply the Envelope theorem. Basically, the Envelope theorem says that at the optimal levels of the control variables the partial derivative of the entire value function with respect to each control variable is zero. This means that when taking the derivative with respect to a state variable you can simply ignore all terms that involve $∂ct∕ ∂mt, ∂dt ∕ ∂mt, ∂ct ∕∂dt − 1,$ and $∂dt ∕∂dt − 1.$ So, for example, the full expression for the derivative of the value function with respect to $mt$ is:

m ∂u-(ct,-dt)-∂ct- ∂u-(ct,dt)-∂dt-- v t = ∂c ∂m + ∂d ∂m [ t t t t ] ∂mt+1--- ∂mt+1----∂ct- ∂mt+1----∂dt- m d -∂dt- + + + βv t+1 + βv t+1 ∂mt ∂ct ∂mt ∂dt ∂mt ∂mt

but the Envelope theorem tells us to ignore all the terms that involve $∂c ∕∂m t t$ or $∂dt ∕ ∂mt$ ; then because the only term in that whole mess above that does not involve either $∂c ∕∂m t t$ or $∂d ∕ ∂m t t$ is $∂m ∕∂m = R t+1 t$ we have:

m m v t = R βv t+1

(9)

Applying the same Envelope theorem logic for $dt− 1$ yields:²

vdt = R(1 − δ)βvmt+1 m = (1 − δ )Rβv t+1 m = (1 − δ )vt

(10)

Think now about the case where depreciation is 100 percent ( $δ = 1$ ); from (10) it is clear that in this case $vdt = 0$ . This makes sense because in this case the ‘durable’ good is really a totally nondurable good. $dt−1 vt = 0$ because the amount that you consumed of a nondurable good last period has no direct eﬀect on your current utility (we have assumed that utility is time separable).

The $δ = 0$ case is more interesting. In this case the marginal utility of having an extra unit of durable good last period is equal to the marginal utility of having an extra unit of wealth this period. Why? Because if $δ = 0$ the durable good doesn’t depreciate at all. How much would it cost to buy another unit of durable good today? One unit of wealth. Because the durable does not depreciate from period to period and can be transformed into and out of wealth at a one-to-one price, it is exactly as valuable as a unit of wealth.

Now we want to try to derive a relationship between the contemporaneous marginal utilities of $d$ and $c$ . From (8) we have:

d m d u t = Rβv t+1 − βv t+1.

(11)

and

Rβv_t+1^m = u_t^c

and from (10) $vd = (1 − δ)vm t+1 t+1$ . Substituting these into (11):

d c m u t = ut − β (1 − δ)v t+1 (1 − δ ) = uct − --------R βvmt+1 [ R ] (1 − δ) c = 1 − --------- u t [ ] R r + δ = ------ uct R

(12)

Assuming $δ < 1$ , this equation tells us that the marginal utility in the current period of a unit of spending on the durable good is lower than the marginal utility of spending on the nondurable. Why? Because the durable good will yield utility in the future as well as in the present. What should be equated to the marginal utility of nondurables consumption is the total discounted lifetime utility from an extra unit of the durable good, not simply the marginal utility it yields right now.

Now assume the utility function is of the Cobb-Douglas form: $(c1−αdα)1−-ρ u (c,d) = 1− ρ .$ This implies that the instantaneous marginal utilities with respect to $c$ and $d$ are:

uc = (c1− αd α)− ρ(1 − α)c − αd α = (c1− αd α)− ρ(1 − α)(d ∕c)α ud = (c1− αd α)− ραc1 − αd α− 1 1− α α − ρ α− 1 = (c d ) α (d∕c )

(13)

Substituting these deﬁnitions into (12) gives:

( ) 1− α α − ρ α− 1 1− α α − ρ α r +-δ- (c d ) α (d∕c ) = (c d ) (1 − α )(d∕c ) R ( ) α r + δ -------= (d∕c ) ------ 1 − α ( ) R( ) α R d∕c = ------- ------ ≡ γ 1 − α r + δ

(14)

What this implies is that whenever the level of nondurables consumption changes, the level of the stock of durables should change by the same proportion. Because expenditures on durable goods are equal to the change in the stock plus depreciation, a change in $c$ implies spending on durables large enough to immediately adjust the stock to the new target level. (Recall that $dt$ was the stock of durable good owned in period $t$ , while spending on the durable good was deﬁned as $xt = dt − (1 − δ)dt− 1$ .)

Deﬁne $γ = dt∕ct$ as in (14). Now consider a consumer who had consumed the same amount of the nondurable good for periods $ct− 2 = ct− 1$ but who between period $t − 1$ and period $t$ learns some good news about permanent income; she adjusts her nondurables consumption up so that $c ∕c = (1 + 𝜖 ) t t− 1 t$ . This implies that the level of spending in period $t$ is:

xt = dt − (1 − δ)dt− 1 = γ [ct − (1 − δ)ct− 1] xt− 1 = γ [ct− 1 − (1 − δ)ct− 2] = γ δc t− 1 xt ∕xt− 1 = γ [ct− 1(1 + 𝜖t) − (1 − δ)ct− 1]∕ γδct− 1 𝜖t +-δ- = δ

(15)

Assuming $δ < 1$ , this equation implies that spending on durable goods should be more variable than spending on nondurable goods. For goods with a low depreciation rate, spending should be much more variable. This is true because the ratio of the stock of durables to income is much larger than the ratio of the average level of spending on durables to income.

A further implication of this model is that the degree of correlation between nondurables spending growth and durables spending growth depends on the frequency under consideration. For a given quarterly depreciation rate (say, 5 percent per quarter), the durable good will have almost completely depreciated over the course of 10 years $=$ 40 quarters because $40 0.95 = 0.12$ . According to the model, over an interval long enough for the durable to have completely depreciated, the rate of growth of spending on the durable should match the rate of growth of spending of the nondurable, because over such a long interval they are really both nondurable.

Some evidence on this proposition is provided in the Jupyter notebook available at here.

References

Carroll, Christopher D., and Wendy E. Dunn (1997): “Unemployment Expectations, Jumping (S,s) Triggers, and Household Balance Sheets,” in NBER Macroeconomics Annual, 1997, ed. by Benjamin S. Bernanke, and Julio Rotemberg, pp. 165–229. MIT Press, Cambridge, MA, http://econ.jhu.edu/people/ccarroll/macroann.pdf; Methodological Appendix: http://econ.jhu.edu/people/ccarroll/methods3.pdf; Empirical Results and Simulation Programs: http://econ.jhu.edu/people/ccarroll/cdfiles.html;.

Mankiw, N. Gregory (1982): “Hall’s Consumption Hypothesis and Durable Goods,” Journal of Monetary Economics, 10(3), 417–425.