1999).4 Thus, the present value of life-time income (2’) for an STU can be rewritten as
Vt(h, τ) = b(1 — θ)h — kt ∙ c ∙ e
(11)
+β
{τ-1-
Σ
s=1
δs [b(1 — θ)h — ks ∙ c ∙
T -1-t
e] + У δs(1 — θ)
s=τ -t
(1 — d)h + fy ki ∙ e
i=0
with β ≤ 1 and where kt is an indicator function that takes a value of kt = 0 in any subperiod
without retraining and kt = 1 in the retraining subperiod.5 If retraining happens, then kt = 1
and kt = 0 otherwise. The only but crucial difference between (11) and (2’) is that now the
discount factor from today to the next subperiod is βδ whereas δ remains the discount factor
between any two subperiods in the future.
During STU people move from subperiod t = 0 to subperiod τ — 1. We first consider the case
in which individuals have perfectly time-consistent preferences (β = 1) as a benchmark. In this
case, people rationally postpone retraining until the ultimate period of STU and then retrain
(kτ -1 = 1) if retraining is worthwhile, i.e. if
c< (1 — θ)δτ ≡ (1 — θ) ' ∖ ' .
(12)
1 — δ
Except for the determination of the timing of retraining, this is exactly the result from section
2, and with β = 1 we are back in the standard model.
Now consider agents with present-biased preferences, i.e. the case of β < 1. All future selves
of these persons prefer to retrain if condition (12) holds. The analysis below focuses on the
case where people are not aware of their time-inconsistency problem, i.e. they do not realize
that their tastes change when they get closer to the moment when they originally intended to
execute their retraining decision.6 Like the rational agents they postpone retraining. However,
in the last period of eligibility for unemployment benefits (at t = τ — 1, when the rational agents
4 The assumption that it takes one subperiod to re-educate is made for simplicity and could be generalized.
The crucial underlying assumption is that retraining has to be taken up once (or never) during unemployment.
DellaVigna and Pasermann (2005) integrate present-biased preferences into a search and matching model. They
find (theoretically and empirically) a negative effect of impatience on search intensity and exit from unemployment.
5Since the argument is independent from labor market tightness we dropped the λ,s in (11) for convenience.
6 These persons are called naive agents in O’Donoghue and Rabin (1999). Sophisticated agents would realize their
present-bias and thus execute retraining inefficiently early during the unemployment spell. Thus, sophisticated
agents will not run into the unemployment trap as discussed here.
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