Bellman Equations for the DMP Search Model
Boromeus Wanengkirtyo
Simplified Worker
The worker’s Bellman equation is the net present value of being employed, or
unemployed. (This is why it is sometimes called the value function.)
Take the simplest case of a worker that gets wage w infinitely. The value of
employment W is:
1
W=
w+
1+r
1
1+r
2
w+
1
1+r
3
w+...
where r is the discount rate. The first wage is discounted because we assume that
the it is paid at the end of the month.
Let’s simplify this. You can use the fact that this is an infinite sequence, like in
the lecture notes. Or you can convert it to recursive form, by noting that:
2
3
4
1
1
1
w+
w+
w+...
1+r
1+r
1+r
1
1
W=
w+
W
1+r
1+r
1
W=
1+r
⇒
Rearranging this, then you get rW = w. This is the flow (per period) utility.
Two asset pricing interpretations:
1. The worker gets W upfront, parks it at a bank and gets interest payments
rW = w each month.
2. The NPV of a job W is the value of a perpetuity that pays w for infinite
periods, discounted at the rate r. Hence, W =
w
r.
Worker
In the DMP search model, the worker can also get fired and become unemployed.
In the simplest version of the model, this is an exogenous probability λ (the ‘job
separation rate’). So the Bellman equations become:
2
3
1
1
(1 − λ ) w +
(1 − λ )2 w + · · · +
1+r
1+r
2
3
1
1
1
λ
U+λ
( 1 − λ )U + λ
(1 − λ )2 U + . . .
1+r
1+r
1+r
1
W=
w+
1+r
1
where U is the value of being unemployed (see the next section).
Simplifying by re-defining W recursively:
2
3
1
1
(1 − λ ) w +
(1 − λ )2 w + · · · +
1+r
1+r
2
3
1
1
λ ( 1 − λ )U +
λ (1 − λ )2 U + . . .
1+r
1+r
1
1
1−λ
W=
w+
λU +
W
1+r
1+r
1+r
1−λ
W=
1+r
⇒
Simplifying again, we get the flow utility: rW = w + λ(U − W )
Asset pricing interpretation:
• The additional term in the Bellman equation is the change in value, when
you change the state, multiplied by the probability of it happening.
Unemployed
rU = b + p(W − U )
where b is the unemployment benefits and p is the ‘job finding rate’. In the next
seminar, we will endogenise this. (Note that the job separation rate λ is constant
and exogenous).
Also note that even if there are no unemployment benefits, the value of unemployment is still positive.
The firm
The assumption is that firms post vacancies, which are filled with the (endogenous) probability q.
The value of a filled vacancy to the firm is:
rJ = y − w − λ( J − V )
and the value of an open vacancy to the firm is:
rV = −k + q( J − V )
where y is the output that matched worker produces, and k is the vacancy posting
cost.
2