The Laibson Model of Time Inconsistency

vt(mt;χt) = u(χt(mt)) + β Et[vt+1((mt −χt(mt))R+ yt+1)] (1) vt(mt;χt) = u(χt(mt)) + δβ Et[vt+1((mt −χt(mt))R+ yt+1)] (2) Notice that these functions are well defined for any consumption function χt(mt) that is feasible; they are not Bellman equations because they do not assume that the consumption function χt is optimal. For example, these functions would be well defined for χt(mt) = mt, or for χt(mt) = 1, or for many other potential consumption rules. What these functions capture is the value of behaving according to the rule χt in the current period, under two possible assumptions about discounting of the future: Either next period’s value is discounted by the factor β (for vt) or by δβ (for vt). Now consider two possible candidates for χt: