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The probability

of finding a job is $p(a)$ where $p$ is an increasing, strictly concave,

and twice differentiable function of $a$ that satisfies

and twice differentiable function of $a$ that satisfies

$p(a) \in [0,1]$ for $a \geq 0$, $p(0)=0$.

The consumption good is nonstorable.

Once a worker has found a job, he is beyond the planner's grasp.

* This is Shavell and Weiss's assumption, but not Hopenhayn and Nicolini's.

* Hopenhayn and Nicolini allow the unemployment insurance agency to

* This is Shavell and Weiss's assumption, but not Hopenhayn and Nicolini's.

* Hopenhayn and Nicolini allow the unemployment insurance agency to

impose history-dependent taxes on previously unemployed workers.

* Since there is no incentive problem after the worker has found

* Since there is no incentive problem after the worker has found

a job, it is optimal for the agency to provide an employed worker with

a constant level of consumption.

* Hence, Hopenhayn and Nicolini's insurance agency imposes

* Hence, Hopenhayn and Nicolini's insurance agency imposes

a permanent per-period history-dependent tax on a previously

unemployed but presently employed worker.

Therefore,

$$ V^e = {u(w) \over (1-\beta)} .

$$

V^e = {u(w) \over (1-\beta)} .

$$ (eq:hugo2)

% \EQN hugo2

Now let $V^u$ be the expected discounted present value of utility for an

unemployed worker who chooses consumption, effort pair $(c,a)$

V^u = \max_{a \geq 0} \biggl\{ u(0) - a + \beta \left[

p(a) V^e + (1-p(a)) V^u \right] \biggr\} .

$$ (eq:hugo3)

%EQN hugo3

The first-order condition for a maximum is

$$

\beta p'(a) \left[V^e - V^u \right] \leq 1 ,

$$ (eq:hugo4)

%\EQN hugo4

with equality if $a>0$.

form the basis for

an iterative algorithm for computing $V^u = V_{\rm aut}$.

* Let $V^u_j$ be

* Let $V^u_j$ be

the estimate of $V_{\rm aut}$ at the $j$th iteration.

* Use this value

* Use this value

in equation {eq}`eq:hugo4` and solve

for an estimate of effort $a_j$.

* Use this value in a version of equation

* Use this value in a version of equation

{eq}`eq:hugo3` with $V^u_j$ on the right side

to compute $V^u_{j+1}$.

* Iterate to convergence.

* Iterate to convergence.

### Unemployment Insurance with Full Information

where minimization is subject to the promise-keeping constraint

$$

V \leq u(c) - a + \beta

V \leq u(c) - a + \beta

\left\{ p(a) V^e + [1-p(a)] V^u \right\}.

$$ (eq:hugo6)

respect to $c, a$, and $V^u$, respectively, are

$$

\begin{align} \theta & = {1 \over u'(c)}\,, \cr

\begin{aligned} \theta & = {1 \over u'(c)}\,, \cr

C(V^u) & = \theta \left[ {1 \over \beta p'(a)} -

(V^e - V^u) \right]\,, \cr

C'(V^u) & = \theta\,.

\end{align}

\end{aligned}

$$ (eq:hugo7)

The envelope condition $C'(V) = \theta$ and the third equation

spell.

Thus, the unemployed worker's consumption $c$ and search effort $a$ are both fully smoothed

during the unemployment

spell.

during the unemployment spell.

But

the worker's consumption is not smoothed across states of

But the worker's consumption is not smoothed across states of

employment and unemployment unless $V=V^e$.

### The incentive problem

The worker is free to choose $a$, which puts expression {eq}`eq:hugo4`, the worker's first-order condition under autarky,

back in the picture.

* We are assuming that the worker's

* We are assuming that the worker's

best response to the unemployment insurance arrangement is

completely characterized by the first-order condition {eq}`eq:hugo4`,

an instance of the so-called first-order approach to incentive problems.

respect to $c, a$, and $V^u$, respectively, are

$$

\begin{align} \theta & = {1 \over u'(c)}\,, \cr

\begin{aligned} \theta & = {1 \over u'(c)}\,, \cr

C(V^u) & = \theta \left[ {1 \over \beta p'(a)} - (V^e - V^u) \right]

\,-\, \eta {p''(a) \over p'(a)} (V^e - V^u) \cr

& = \,- \eta {p''(a) \over p'(a)} (V^e - V^u) \,, \cr

C'(V^u) & = \theta \,-\, \eta {p'(a) \over 1-p(a)}\, ,

\end{align}

\end{aligned}

$$ (eq:hugo8)

where the second equality in the second equation in {eq}`eq:hugo8` follows from strict equality

represent condition {eq}`eq:hugo4` as

$$

V^u \geq V^e - [\beta p'(a)]^{-1},

V^u \geq V^e - [\beta p'(a)]^{-1},

$$

with equality if $ a > 0$.

Our strategy involves finding a function $C(V)$ -- the expected cost of giving the worker value $V$ -- that satisfies the Bellman equation:

$$

C(V) = \min_{c,a,V^u} \{c + \beta\left[1-p(a)\right]C(V^u)\}

C(V) = \min_{c,a,V^u} \{c + \beta\left[1-p(a)\right]C(V^u)\}

$$ (eq:yad3)

To solve this model, notice that in equations {eq}`eq:hugo21` and {eq}`eq:hugo22`, we have analytical solutions of $c$ and $a$ in terms of (at most) promised value $V$ and $V^u$ (and other parameters).

There is a **carrot-and-stick**

aspect to the replacement rate and search effort schedules:

* the **carrot** occurs in the forms of high compensation and low search

* the **carrot** occurs in the forms of high compensation and low search

effort early in an unemployment spell.

* the **stick** occurs in the low compensation and high effort later in

* the **stick** occurs in the low compensation and high effort later in

the spell.

We shall encounter a related carrot-and-stick feature in our other lectures about dynamic programming squared.