23
The caseworkers base their decisions on the communicated health. The model is thus
based on asymmetric information regarding true health status. However, in order to gain
analytical tractability we simplify the decisions made by the uninformed part (i.e., the
caseworkers). We assume that there is a baseline exogenous exit rate to DB,
0
d
,
and a corresponding baseline exit rate to employment
0
e
.
20
The individual exit rate
to DB is assumed to be given by
d
, and the corresponding exit rate back to work is
assumed to be given by
/
e
. Increasing the signal of bad health thus decreases
(increases) the exit rate to employment (disability insurance). The sickness benefits
b
is
taken to be lower than the wage,
0
b w
. Without loss of generality we let the benefit
be zero when on DB, which means that the state value of DB is zero (
0
d
V
).
The state values while employed (
( , )
e
V h w
) and as sick absent (
( , , )
s
V h w
) are now
given as
( , )
e
V h w w h
(2)
and
( , , )
( , , )
( ( , )
( , , ))
s
s
T
d s
e
s
V h w b
V h w
V h w V h w
(3)
where
0
denotes the subjective discount rate.
5.2
Optimal Choice of Communicated Health
When characterizing the optimal choice of
, we treat two types of individuals
separately: i) those who weakly prefer work to sickness absence, that is,
( , )
( , , ) 0
e
s
V h w V h w
, and ii) those who prefer sickness absence to work, that is,
( , )
( , , ) 0
e
s
V h w V h w
. Those who prefer work will set
0
and thereby directly flow
back to work since
0
lim /
e
. For those who strictly prefer sickness absence over
work we will have an interior solution. It is easily confirmed that the second order
condition is satisfied at the optimum when
( , )
( , , ) 0
e
s
V h w V h w
holds. We may
therefore rely on the first order condition given by
21
2
( , , )
( , , )
( ( , )
( , , )) 0
s
s
T
d s
e
s
V h w
V h w
V h w V h w
(4)
Solving for
( , )
e
V h w
and
( , , )
s
V h w
from equations (2) and (3) and combining with
equation (4) gives an equation that determines
( , ;
)
T
h w
, that is, the optimal level of
20
The model is set up in continuous time.
21
Note that the optimal choice of
will not change over time. It is therefore irrelevant whether the
individual reevaluates the choice of
tomorrow or not. Technically, the indirect effect of
, on the
right hand side, will be zero due to the envelope property.
