February 7, 2021, Christopher D. Carroll Laibson
February 7, 2021, Christopher D. Carroll Laibson
This handout provides a simple example of a discrete-time solution to the problem of a consumer with a self-control problem a la Laibson (1997).1
Suppose a value function 
exists for period 
. Then for any
period-
consumption function 
we can define
![vt(mt; χχχt) = u(χχχt (mt )) + β 𝔼t[vt+1 ((mt − χχχt (mt ))R + yt+1 )]
𝔳t(mt; χχχt) = u(χχχt (mt )) + δ β 𝔼t[vt+1 ((mt − χχχt (mt ))R + yt+1 )]](Laibson5x.svg) | (1) |
Notice that these functions are well defined for any consumption function
that is feasible; they are not Bellman equations because they do not
assume that the consumption function 
is optimal. For example, these
functions would be well defined for 
, or for 
, or for
many other potential consumption rules.
What these functions capture is the value of behaving according to the rule
in the current period, under two possible assumptions about discounting of
the future: Either next period’s value is discounted by the factor 
(for 
) or
by 
(for 
).
Now consider two possible candidates for 
:
![ct(mt ) = argmax u(c ) + β 𝔼t [vt+1((mt − c)R + yt+1 )]
c
𝔠 (m ) = argmax u(c ) + δβ 𝔼 [v ((m − c)R + y )]
t t c t t+1 t t+1](Laibson16x.svg) | (2) |
If we solve the problem recursively using 
in every period, we obtain the
standard time consistent solution. (Think about why).
The Laibson alternative is to suppose that there is something special about
“now”: Next period’s value is discounted not only by the standard geometric
discount factor 
, but also by an extra factor 
(Laibson argues that at an
annual frequency the appropriate value of 
is about 0.7). This may
reflect the fact that certain areas of the brain associated with emotional
rewards are activated only by instant gratification, and are not activated by
thoughts of future gratification (see, e.g., Cohen, Laibson, Loewenstein, and
McClure (2004)).
It is clear from comparing the equations in (2) that the consumer with Laibson preferences will consume more in the current period, because he values future rewards less.
More insight about the solution can be obtained from the modified Euler equation that can be derived for the Laibson problem. This is derived as follows.
Note first that if 
the Envelope theorem implies that
 | (3) |
while the first order condition from the maximization problem implies that
![′ m
u (ct) = δR β 𝔼t [v t+1 (mt+1 )]
m
= δvt (mt; 𝔠t)](Laibson23x.svg) | (4) |
Now note that for 
there is a simple identity linking 
and
:
 | (5) |
(to see this, multiply the first equation in (1) by 
and note that the difference
between the result and the second equation is 
). Now differentiate
(5)
 | (6) |
Thus,
 | (7) |
If 
, this collapses to the usual consumption Euler equation. However, if
(the Laibson case), the equation says several interesting things. First,
note that since 
and 
and 
are all positive, the contribution of
the “Laibson” term in (7) is to reduce the RHS of the equation. In order to
match a lower RHS, the LHS must be smaller. But a smaller marginal utility of
consumption implies a higher level of consumption - so the Laibson consumer
spends more.
Second, notice that the magnitude of the “present-bias” effect depends on the
size of next period’s marginal propensity to consume 
. If the MPC is small,
the size of the Laibson bias will be small.
Finally, notice that this model nicely captures the commonplace psychological tension in which the cost of deviating from the optimal plan in a single period may be trivially small (“eating dessert this one time will not make me fat”), but the consequences of perpetual deviation could be quite large (“but if I give in to temptation this time, maybe that means I will always give in.”)