where $\beta \in (0,1)$ and $u(c)$ is strictly increasing, twice differentiable,

and strictly concave.

We assume that $u(0)$ is well defined.

We require that $c_t \geq 0$ and $ a_t \geq 0$.

employed.

The probability

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

of finding a job is $p(a)$.

$p$ is an increasing, strictly concave,

and twice differentiable function of $a$ that satisfies

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

```{note}

When we compute examples below, we'll use assume the same $p(a)$ function as

{cite}`Hopenhayn_Nicolini_97`, namely, $p(a) = 1 - \exp(- r a)$, where $r$ is a parameter

that we'll calibrate to hit the same target that {cite}`Hopenhayn_Nicolini_97` did, namely,

an empirical hazard rate of leaving unemployment.

```

The consumption good is nonstorable.

An unemployed worker has no savings and cannot borrow or lend.

An **insurance agency** or **planner** is the unemployed worker's only source of consumption

smoothing over time and across states.

The unemployed worker's only source of consumption

smoothing over time and across states is an **insurance agency** or **planner**.

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

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

optimally.

It satisfies the Bellman equation

Value $V^u$ satisfies the Bellman equation

$$

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

Another benchmark model helps set the stage for the model with private information that we ultimately want to study.

In this model, the unemployment agency has full information about the unemployed work.

We study optimal provision of insurance with

full information.

We temporarily assume that an unemployment insurance agency has full information about the unemployed worker.

An insurance agency can set both

the consumption and search effort of an unemployed person.

We assume that the insurance agency can control both the consumption and the search effort of an unemployed worker.

The

agency wants to design an unemployment insurance contract to give

The agency wants to design an unemployment insurance contract to give

the unemployed worker expected discounted utility $V > V_{\rm aut}$.

The planner wants to deliver value $V$ efficiently,

meaning in a way that minimizes expected

discounted cost, using $\beta$ as the discount factor.

The agency, i.e., the planner, wants to deliver value $V$ efficiently,

meaning in a way that minimizes an expected present value discounted costs, using $\beta$ as the discount factor.

We formulate the optimal insurance problem

recursively.

Given $V$, the planner assigns first-period pair $(c,a)$ and promised

continuation value $V^u$, should the worker be unlucky

and not find a job.

continuation value $V^u$ next period if the worker is unlucky

and does not find a job this period.

$(c, a, V^u)$ are chosen to be functions of $V$ and to

satisfy the Bellman equation

The planner sets $(c, a, V^u)$ as functions of $V$ and to

satisfy the following Bellman equation for associated cost function $C(V)$:

$$

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

Let $\theta$ be a Lagrange multiplier

on constraint {eq}`eq:hugo6`.

At an interior solution, the first-order

At an interior solution, first-order

conditions with

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

of {eq}`eq:hugo7` imply that $C'(V^u) =C'(V)$.

Strict convexity of $C$ then

implies that $V^u =V$

implies that $V^u =V$.

Applied repeatedly over time,

$V^u=V$ makes

### Incentive Problem

The preceding efficient insurance scheme requires that the insurance agency

control both $c$ and $a$.

The preceding efficient insurance scheme assumes that the insurance agency

controls both $c$ and $a$.

It will not do for the insurance agency

simply to announce $c$ and then allow the worker to choose $a$.

The insurance agency cannot simply provide $c$ and then allow the worker to choose $a$.

Here is why.

It **increases** the unemployed worker's consumption $c$ and **decreases** his search

effort $a$.

But the prescribed

The prescribed

search effort is **higher** than what the worker would choose

if he were to be guaranteed consumption level $c$ while he

remains unemployed.

fact that the insurance scheme is costly, $C(V^u)>0$, which imply

$[ \beta p'(a) ]^{-1} > (V^e - V^u)$.

But look at the worker's

Now look at the worker's

first-order condition {eq}`eq:hugo4` under autarky.

It implies that if search effort $a>0$, then

$[\beta p'(a)]^{-1} = [V^e - V^u]$, which is inconsistent

with the preceding inequality

$[ \beta p'(a) ]^{-1} > (V^e - V^u)$ that prevails when $a >0$ under

the social

insurance arrangement.

with the inequality

$[ \beta p'(a) ]^{-1} > (V^e - V^u)$ that prevails when $a >0$ when the agency controls

both $a$ and $c$.

If he were free to choose $a$, the worker would therefore want to

fulfill {eq}`eq:hugo4`, either at equality so long as $a >0$, or by setting

$a=0$ otherwise.

Starting from the $a$ associated with

the social insurance scheme,

he would establish the desired equality

the full-information social insurance scheme in which the agency controls both $c$ and $a$,

the worker would establish the desired equality

in {eq}`eq:hugo4` by *lowering* $a$, thereby decreasing

the term $[ \beta p'(a) ]^{-1}$ (which also lowers $(V^e - V^u)$

when the value of being

Thus, since the worker does not take the

cost of the insurance scheme into account, he would choose a search

effort below the socially optimal one.

effort below the socially optimal, full-information level.

The efficient contract

The full-information contract thus

relies on the agency's ability to control *both* the unemployed

worker's consumption *and* his search effort.

## Private Information

Following Shavell and Weiss (1979) {cite}`Shavell_Weiss_79` and

Hopenhayn and Nicolini (1997) {cite}`Hopenhayn_Nicolini_97`, now assume that the unemployment insurance agency cannot

observe or enforce $a$, though it can observe and control $c$.

Following {cite}`Shavell_Weiss_79` and

{cite}`Hopenhayn_Nicolini_97`, now assume that the unemployment insurance agency cannot

observe or control $a$, though it can observe and control $c$.

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.

Given a contract, the individual will choose search effort according to

first-order condition {eq}`eq:hugo4`.

This fact leads the insurance agency

to design the unemployment insurance contract to respect this restriction.

This fact motivates the insurance agency

to design an unemployment insurance contract that respects this restriction.

Thus, the recursive contract design problem is now to minimize the right side of equation

Thus, the contract design problem is now to minimize the right side of equation

{eq}`eq:hugo5` subject to expression {eq}`eq:hugo6` and the incentive constraint {eq}`eq:hugo4`.

Since the restrictions {eq}`eq:hugo4` and {eq}`eq:hugo6` are not linear

and generally do not define a convex set, it becomes difficult

and generally do not define a convex set, it becomes challenging

to provide conditions under which the solution to the dynamic

programming problem results in a convex function $C(V)$.

* Sometimes this complication can be handled by convexifying

the constraint set through the introduction of lotteries.

the constraint set by introducing lotteries.

* A common finding is that optimal plans do not involve

lotteries, because convexity of the constraint set is a sufficient

but not necessary condition for convexity of the cost function.

* Following Hopenhayn and Nicolini (1997) {cite}`Hopenhayn_Nicolini_97`, we therefore proceed under the assumption that $C(V)$ is strictly convex in order to characterize the optimal solution.

* In order to characterize the optimal solution, we follow Hopenhayn and Nicolini (1997) {cite}`Hopenhayn_Nicolini_97` by hopefully proceeding under the assumption that $C(V)$ is strictly convex.

Let $\eta$ be the multiplier on constraint {eq}`eq:hugo4`, while

$\theta$ continues to denote the multiplier on constraint {eq}`eq:hugo6`.

But now we replace the weak inequality in {eq}`eq:hugo6` by an equality.

The unemployment insurance agency cannot award a higher utility than

* We do this because the unemployment insurance agency cannot award a higher utility than

$V$ because that might violate an incentive-compatibility constraint

for exerting the proper search effort in earlier periods.

of the incentive constraint {eq}`eq:hugo4` when $a>0$.

As long as the

insurance scheme is associated with costs, so that $C(V^u)>0$, first-order

insurance scheme is associated with costs, so that $C(V^u)>0$, the first-order

condition in the second equation of {eq}`eq:hugo8` implies that the multiplier $\eta$ is strictly

positive.

search effort $a$ rises as $V^u$ falls, i.e., it rises with the duration

of unemployment.

The duration dependence of benefits is designed to provide

incentives to search.

The of benefits on the duration of unemployment is designed to provide the worker an

incentive to search.

To see this, from the third equation of {eq}`eq:hugo8`, notice how

To understand this, from the third equation of {eq}`eq:hugo8`, notice how

the conclusion that consumption falls with the duration of

unemployment depends on the assumption that more search effort

raises the prospect of finding a job, i.e., that $p'(a) > 0$.

the expected lifetime utility in autarky,

$V_{\rm aut}$.

To compute the upper bound,

To compute an upper bound,

represent condition {eq}`eq:hugo4` as

$$

### Parameter Values

For the other parameters we have just loaded in the above Python code, we'll set brate the net interest rate $r$ to match the hazard rate -- the probability of finding a job in one period -- in US data.

For the other parameters appearing in the above Python code, we'll calibrate parameter $r$

that pins down the function $p(a) = 1 - \exp(- r a)$ to match an observerd hazard rate -- the probability that an unemployed worker finds a job each -- in US data.

In particular, we seek an $r$ so that in autarky `p(a(r)) = 0.1`, where `a` is the optimal search effort.

First, we create some helper functions.

```{code-cell} ipython3

# The probability of finding a job given search effort, a and interest rate r.

# The probability of finding a job given search effort, a and parameter r.

def p(a,r):

return 1-np.exp(-r*a)

V^u = \max_{a} \{u(0) - a + \beta\left[p_{r}(a)V^e + (1-p_{r}(a))V^u\right]\}

$$ (eq:yad1)

At the optimal choice of $a$, we have the first order condition for this problem as:

At the optimal choice of $a$, we have first-order necessary condition:

$$

\beta p_{r}'(a)[V^e - V^u] \leq 1

$$ (eq:yad2)

with equality when a >0.

Given an interest rate $\bar{r}$, we can solve the autarky problem as follows:

Given a value of parameter $\bar{r}$, we can solve the autarky problem as follows:

1. Guess $V^u \in \mathbb{R}^{+}$

2. Given $V^u$, use the FOC {eq}`eq:yad2` to calculate the implied optimal search effort $a$

return error

```

Since the calibration exercise is to match the hazard rate under autarky to the data, we must find an interest rate $r$ to match `p(a,r) = 0.1`.

Since the calibration exercise is to match the hazard rate under autarky to the data, we must find a parameter $r$ to match `p(a,r) = 0.1`.

The function below `r_error` calculates, for a given guess of $r$ the difference between the model implied equilibrium hazard rate and 0.1.

This will be used to solve for the a calibrated $r^*$.

We'll use this to compute a calibrated $r^*$.

```{code-cell} ipython3

# The error of our p(a^*) relative to our calibration target

```{code-cell} ipython3

r_calibrated = sp.optimize.brentq(r_error_Λ,1e-10,1-1e-10)

print(f"Interest rate to match 0.1 hazard rate: r = {r_calibrated}")

print(f"Parameter to match 0.1 hazard rate: r = {r_calibrated}")

Vu_aut = sp.optimize.fsolve(Vu_error_Λ,15000,args = (r_calibrated))[0]

a_aut = invp_prime(1/(params.β*(params.Ve-Vu_aut)),r_calibrated)

print(f"Check p at r: {p(a_aut,r_calibrated)}")

```

Now that we have calibrated our interest rate $r$, we can continue with solving the model with private information.

Now that we have calibrated our the parameter $r$, we can continue with solving the model with private information.

+++

### Computation under Private Information

+++

Our approach to solving the full model is a variant on Judd (1998) {cite}`Judd1998`, who uses a polynomial to approximate the value function and a numerical optimizer to perform the optimization at each iteration.

Our approach to solving the full model follows ideas of Judd (1998) {cite}`Judd1998`, who uses a polynomial to approximate the value function and a numerical optimizer to perform the optimization at each iteration.

```{note}

For further details of the Judd (1998) {cite}`Judd1998` method, see {cite}`Ljungqvist2012`, Section 5.7.

```

In contrast, we will use cubic splines to interpolate across a pre-set grid of points to approximate the value function. For further details of the Judd (1998) {cite}`Judd1998` method, see {cite}`Ljungqvist2012`, Section 5.7.

We will use cubic splines to interpolate across a pre-set grid of points to approximate the value function.

+++

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).

Notice that in equations {eq}`eq:hugo21` and {eq}`eq:hugo22`, we have analytical solutions of $c$ and $a$ in terms of promised value $V$ and $V^u$ (and other parameters).

We can substitute these equations for $c$ and $a$ and obtain the functional equation {eq}`eq:hugo23` that we want to solve.

We can substitute these equations for $c$ and $a$ and obtain the functional equation {eq}`eq:hugo23`.

```{code-cell} ipython3

return C_new,V_star,cons_star,a_star

```

The below code executes steps 4 and 5 in the Algorithm until convergence to a function $C^*(V)$.

The following code executes steps 4 and 5 in the Algorithm until convergence to a function $C^*(V)$.

```{code-cell} ipython3

def solve_incomplete_info_model(self,Vu_grid,Vu_aut,tol = 1e-6,max_iter = 10000):

+++

Using the above functions, we create another instance of the parameters with the correctly calibrated interest rate, $r$.

Using the above functions, we create another instance of the parameters with our calibrated parameter $r$.

```{code-cell} ipython3

##? Create another instance with the correct r now

+++

We want to graph the replacement ratio ($c/w$) and search effort $a$ as functions of the duration of unemployment.

Let's graph the replacement ratio ($c/w$) and search effort $a$ as functions of the duration of unemployment.

We'll do this for three levels of $V_0$, the lowest being the autarky value $V_{\rm aut}$.

### Interpretations

The downward slope of the replacement ratio when $V^u > V_{\rm aut}$ is a consequence of the

the planner's limited information about the worker's search effort.

planner's limited information about the worker's search effort.

By providing the worker with a duration-dependent schedule of replacement ratios, the planner induces the worker in effect to reveal

his/her search effort to the planner.

who has been unemployed for a long time.

The planner believes that a worker who has been unemployed a long time is unlucky, not that he has

done anything wrong (i.e., has not lived up to the contract).

done anything wrong (e.g.,that he has not lived up to the contract).

Indeed, the

contract is designed to induce the unemployed workers to search in