September 2, 2017, Christopher D. Carroll Habits

Consumption Models with Habit Formation

1 The Problem

Consider a consumer whose goal at date t  is to solve the problem1

      T- t
      ∑     n
max       β  u(ct+n, ht+n )
      n=0
(1)

where ht+n  is the habit stock, and all other variables are as usually defined. The DBC is

mt+1   =   (mt  -  ct)R  +  yt+1.                       (2)
However, when habits affect utility we must also specify a process that describes how habits evolve over time. Our assumption will be:
ht+1   =   ct.                                (3)

Bellman’s equation for this problem is therefore

v (m   ,h )  =   max    u (c ,h ) +  βv    ((m   -  c)R  +  y   ,c ).     (4)
  t   t  t        {ct}       t   t       t+1     t     t       t+1  t

To clarify the workings of the Envelope theorem in the case with two state variables, let’s define a function

vt(mt, ht, ct)  =   u(ct,ht ) + βvt+1 ((mt  -  ct)R +  yt+1, ct)       (5)
and define the function ct (mt, ht)  as the choice of ct  that solves the maximization (4), so that we have
vt (mt, ht)  =   vt(mt,  ht,ct(mt,  ht)).                   (6)

1.1 Optimality Conditions

1.1.1 The First Order Condition

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 ftx  ):

          c     (  h        m   )
 0  =   u t + β  v t+1  - Rv t+1                         (7)
 c        (    m       h  )
ut  =   β   Rv t+1  - v t+1   .                           (8)

The intuition for this equation is as follows. Note first that if utility is not affected by habits, then vht+1 =  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 R ϵ  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 mt+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 vht+1   . Since vht+1   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 ct  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

dvt-  =   0                                  (9)
dct
when evaluated at ct = ct (mt, ht)  .

1.1.2 Envelope Conditions

Now consider the total derivative of vt(mt, ht, ct(mt, ht ))  with respect to mt  . (To reduce clutter, I will write dct(mt,  ht)∕dmt  as dct∕dmt  ). The chain rule tells us that

                      ◜=◞0◟ ◝        (                                   )
 dv-        dct       dht              dct
----t  =   -----uct + -----uht + β   ( ----)(vht+1 -  Rvmt+1 ) + Rvmt+1     (10)
dmt        dmt        dmt              dmt
           (      )              (               )
       =     -dct-      (uc +  β  vh    - Rvm     )    + βRvm              (11)
             dmt        ◟ -t--------t◝+1◜-------t+1--◞           t+1
                     =0  at c  =  c (m  ,h ) from  (7 )
                             t     t   t   t
so we have that
 m       -dvt-
vt   =   dm   |ct=ct(mt,ht)                         (12)
             tm
     =   βRv  t+1.                                  (13)

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  :

v (m   ,h )  =   v (m   ,h ,c (m   ,h ))                                (14)
  t   t  t       -t    t  t   t   t  t
         m       -∂vt-    ∂vt- ∂ht--   ∂vt- -∂ct-    ∂vt-∂ct- ∂ht--
        vt   =         +            +             +                     (15)
                 ∂mt      ∂ht ◟∂m◝t◜◞    ∂◟c◝t◜◞ ∂mt      ∂ct ∂ht ◟∂m◝t◜◞
                                =0      =0                     =0
where the Envelope theorem is what tells you that the ∂v--∕∂ct
   t  term is equal to zero because you are evaluating the function at ct = ct(mt,  ht)  (and ∂ht ∕∂mt  is zero by the assumed structure of the problem in which ht  is predetermined).

Now writing out ∂v--∕∂mt
   t  , (15) becomes

           ∂
vmt  =   ----- [βvt+1 ((mt  - ct)R  + yt+1, ht+1 )]             (16)
         ∂mt
     =   βRvm    .                                              (17)
              t+1

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     =  c
  t+1      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  ∕∂m
   t     t  and ∂h    ∕ ∂m
   t+1      t  as zero here. The reason is that this is a partial derivative with respect to mt  . 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 mt  directly (treating the problem as though ct  were a constant):

 vt (mt, ht)  =   u (ct,ht) +  βvt+1 ((mt  - ct)R  + yt+1, ht+1 )      (18)
vm  (m  ,h )  =   βRvm    (m     ,h    ).                              (19)
  t    t  t            t+1    t+1    t+1

Whichever way you do it, substituting (17) into the FOC equation (8) gives

 m        c       h
vt   =   ut +  βv t+1.                            (20)

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   c
u t  but affects value next period by vh
 t+1   (which is a negative number), the discounted consequence of which from today’s perspective is the    h
βv t+1   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 vt  . For this problem, the total derivative (again denoting dct(mt,  ht)∕dht  as dc  ∕dh
   t    t  ) is:

dv        dc                ( dh             dm          )
--t-  =   ---tuc +  uh +  β   ---t+1-vh   +  ----t+1-vm                  (21)
dht       dht   t    t         dht    t+1     dht    t+1
                            (                                    )
      =   dct-uc +  uh +  β   dht+1--dct-vh   +  dmt+1--dct-vm          (22)
          dht   t    t          dct  dht  t+1      dct  dht   t+1

           h    dct-     ( c        h        m    )
      =   ut +  dh       ◟ut-+-β-(v-t+◝1◜---Rv-t+1-)◞                     (23)
                   t
                     =0  at ct = ct (mt, ht) from  (7 )
so we have
  h      dvt-
v t  =   ----|ct=ct (mt,ht)                          (24)
         dht
     =   uh .                                      (25)
           t

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

              ◜=◞0◟ ◝
         ∂v-  ∂mt     ∂v-    ∂v-- ∂ct
vht  =   ----t -----+  ---t+  ---t ----
        ∂mt   ∂ht     ∂ht    ∂ct  ∂ht
while the Envelope theorem once again says ∂vt-∕∂ct =  0  at ct =  ct(mt, ht )  so we obtain
  h      ∂vt-
v t  =   ----                                (26)
         ∂ht
     =   uh                                  (27)
          t
since ht  appears directly only in the u(ct,ht )  part of vt(mt, ht, ct)  . And once again, the shortest way to the answer is to treat ct  as though it were a constant in the value function, which yields
vt (mt, ht)  =   u (ct,ht) +  βvt+1 ((mt  -  ct)R  +  yt+1,ct)        (28)
 h                 h
vt (mt, ht)  =   u t .                                               (29)

From (20) this implies that

 m        c       h
vt   =   ut +  βu t+1.                            (30)
Roll this equation forward one period and substitute into equation (17) to obtain:
                      [               ]
uct + βuht+1  =   R β  uct+1 +  βuht+2                     (31)
Note that if   h       h
u t+1 =  ut+2 =  0  so that habits have no effect on utility, (31) again is solved by the standard time-separable Euler equation.

Now assume that the utility function takes the specific form

u (c,h )  =   f(c -  αh )                          (32)
which implies derivatives of
uc   =   f′                                   (33)

uh   =   - αf ′.                               (34)

Substituting these into equation (31) we obtain,

  ′       ′             ′          ′
ft -  αβf t+1  =   R β[ft+1 -  αβf t+2]                   (35)

Now assume that there is a solution in which marginal utility of consumption grows at a constant rate over time, ft′=  kft′+1   and substitute into (35)

  ft′+1 (k -  αβ )  =   R β[f′t+2(k -  α β) ]                 (36)
   ′                       ′
kft+2 (k -  αβ )  =   R β[ft+2(k -  α β) ]                 (37)

               k  =   R β                                  (38)
so marginal utility grows at rate 1∕R β  . Note that if we assume α =  0  so that habits do not matter, we again obtain the standard result that u ′(ct) =  R βu ′(ct+1 )  .

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

                     - ρ
1  =   R β (zt+1∕zt )                             (39)
Now expand z    ∕z
  t+1   t
ct+1  - αct        ct+1∕ct -  α
------------ =   ---------------                                    (40)
ct - αct- 1      1 -  αct - 1∕ct
                 1 +  Δ  log  c    - α
             ≈   -------------t+1-----                              (41)
                  1 - α  + α Δ  log ct
                 1 -  α +  Δ  log c
             =   ------------------t+1                              (42)
                  1 - α  + α Δ  log ct

             =   1-+--(Δ--log-ct+1)∕(1----α-)                       (43)
                  1 + (α ∕ (1 - α ))Δ  log ct
                      (        )
                        ---1---
             ≈   1 +    1 -  α   (Δ  log ct+1 -  αΔ  log ct)        (44)
where (41) follows from (40) because ct+1 ∕ct =  1 + (ct+1 -  ct)∕ct ≈  1 +  Δ log ct+1   and ct- 1∕ct =  (ct - (ct - ct- 1))∕ct ≈  1 - Δ  log ct  , and (44) follows from (43) because for small η  and ϵ,(1 +  η)∕ (1 + ϵ) ≈  1 +  η -  ϵ  .

Substituting (44) into (39) gives

                          (        )
          1  ≈   R β(1 +    ---1---   (Δ log c    -  α Δ log c ])- ρ
                            1 -  α            t+1             t
                               (        )
                                 --1----
          0  ≈   log [R β ] - ρ   1 -  α   (Δ  log  ct+1 - α Δ  log ct)

Δ  log ct+1  ≈   (1 -  α )ρ- 1(r - ϑ) +  α Δ log ct.

Thus, this formulation of habit formation implies that the growth rate of consumption is serially correlated.

References

   carroll, christopher d. (2000): “Solving Consumption Models with Multiplicative Habits,” Economics Letters, 68(1), 67–77, http://econ.jhu.edu/people/ccarroll/HabitsEconLett.pdf.