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
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 :
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
while the ﬁrst order condition from the maximization problem implies that
Now note that for there is a simple identity linking and :
(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)
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.”)