3.1.1 Assumptions

These equilibria are based on two assumptions. First, assets are in zero net supply and production only necessitates labor ($\lambda =0$ in the production function). The first consequence is that the total amount of saving must be equal to the total amount of borrowing among households. The second consequence is that the real wage is only the technology level in each period $w_t=A_t$. Second, it is assumed that the borrowing limit is 0, $\overline{a}=0$. As agents can't borrow, there are no assets in which agents can save: $a_t^i=0$ for all agents i in any period t. These equilibria are not interesting for generating a realistic cross-section of wealth, but they can be interesting to investigate the behavior of the economy facing time-varying uninsurable risk. Indeed, the price of any asset is determined by the highest price than any agent is willing to pay.

Denoting {1} the employed agents and {0} the unemployed agents, one can now state the problem recursively. The value functions for employed and unemployed agents are (I write these functions with the time index to facilitate the reading, although it is not necessary in this recursive formulation).

$\begin{array}{cc}\hfill V(a_t,h_t,\{1\})=& \max {}_{c_t,l_t,a_{t+1}}U(c_t,l_t)+\beta \mathbb{E}({\alpha }_{t+1}V(a_{t+1},h_{t+1},\{1\})\hfill \\ \hfill & +(1-{\alpha }_{t+1})V(a_{t+1},h_{t+1},\{0\}))\hfill \\ \hfill & a_{t+1}+c_t=A_t{l}_t+a_t(1+r_t)\hfill \\ \hfill & a_{t+1}⩾0\hfill \end{array}$

and

$\begin{array}{cc}\hfill V(a_t,h_t,\{0\})=& \max {}_{c_t,a_{t+1}}U(c_t,\delta )+\beta \mathbb{E}({\rho }_{t+1}V(a_{t+1},h_{t+1},\{0\})\hfill \\ \hfill & +(1-{\rho }_{t+1})V(a_{t+1},h_{t+1},\{1\}))\hfill \\ \hfill & a_{t+1}+c_t=\delta +a_t(1+r_t)\hfill \\ \hfill & a_{t+1}⩾0\hfill \end{array}$

where the expectation operator $\mathbb{E}$ is taken for the aggregate shock h only.

As no agent can save, we have $a_{t+1}=0$ for all agents, and one can thus see that all employed agents consume $c_{1,t}=A_t{l}_{1,t}$ and supply the same quantity of labor $l_{1,t}$, whereas unemployed agents simply consume $c_{0,t}=\delta$ and the labor supply is obviously given by $l_{0,t}=\delta$.

The equilibrium can be derived using a guess-and-verify structure. Indeed, for general values of the parameters derived below, unemployed agents are credit-constrained: they would like to borrow, and employed agents would like to save. As a consequence, they are the marginal buyer of the asset (although in zero-net supply) and make the price.

Deriving the first-order conditions of the previous program and then using these values, one finds

$\begin{array}{cc}\hfill A_t{U}_c(c_{1,t},l_{1,t})& =-U_l(c_{1,t},l_{1,t})\hfill \\ \hfill U_c(c_{1,t},l_{1,t})& =\beta \mathbb{E}(1+r_{t+1})(\alpha {}_{t+1}{U}_c(c{}_{1,t+1},l{}_{1,t+1})+(1-{\alpha }_{t+1})U_c(c{}_{0,t+1},\delta ))\hfill \end{array}$

and the conditions for unemployed agents to be credit-constrained at the current interest rate is

$U_c(c_{0,t},\delta )>\beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{U}_c(c{}_{0,t+1},\delta )+(1-{\rho }_{t+1})U_c(c{}_{1,t+1},l{}_{1,t+1}))$

Specification of the functional forms. Assume that

$\{\begin{array}{cc}U(c,l)=\frac{c^{1-\sigma }-1}{1-\sigma }-\chi \frac{l^{1+\frac{1}{\varphi }}}{1+\frac{1}{\varphi }}\hfill & \text{if }\sigma \ne 1\hfill \\ U(c,l)=\log (c)-\chi \frac{l^{1+\frac{1}{\varphi }}}{1+\frac{1}{\varphi }}\hfill & \text{if }\sigma =1\hfill \end{array}$

σ is the curvature of the utility function (not directly equal to risk aversion to the endogenous labor supply), and ϕ is the Frisch elasticity of labor supply, ranging from 0.3 to 2 in applied work (see Chetty et al., 2011 for a discussion). χ is a parameter scaling the supply of labor in steady state. With this specification one finds

(7) $\begin{array}{ccc}\hfill A_t{c}_{1,t}^{-\sigma }\hfill & \hfill =\hfill & \hfill \chi l_{1,t}^{\frac{1}{\varphi }}\hfill \\ \hfill c_{1,t}^{-\sigma }\hfill & \hfill =\hfill & \hfill \beta \mathbb{E}(1+r_{t+1})({\alpha }_{t+1}{c}_{1,t+1}^{-\sigma }+(1-{\alpha }_{t+1}){\delta }^{-\sigma })\hfill \end{array}$

and the conditions for unemployed agents to be credit-constrained at the current interest rate is

${\delta }^{-\sigma }>\beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{\delta }^{-\sigma }+(1-{\rho }_{t+1})c_{1,t+1}^{-\sigma })$

From the budget constraint of employed agents $c_{1,t}=A_t{l}_{1,t}$ and the labor choice in (7), we obtain

$c_{1,t}^{1+\varphi \sigma }=A_t^{1+\varphi }/{\chi }^{\varphi }$

The technology process is the following

$A_t=e^{a_t}$

where $a_t$ is an AR(1) process specified above.

Assume that three shocks hit the economy: A shock to the technology level, $a_t$, a shock to the probability to stay employed ${\alpha }_t$ and a shock to the probability to stay unemployed ${\rho }_t$, which are AR(1) processes. More formally,

$\left[\begin{array}{c}\hfill a_t\hfill \\ \hfill {\alpha }_t-\overline{\alpha }\hfill \\ \hfill {\rho }_t-\overline{\rho }\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill {\rho }^a\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \text{0}\hfill & \hfill {\rho }^{\alpha }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }^{\rho }\hfill \end{array}\right]\left[\begin{array}{c}\hfill a_{t-1}\hfill \\ \hfill {\alpha }_{t-1}-\overline{\alpha }\hfill \\ \hfill {\rho }_{t-1}-\overline{\rho }\hfill \end{array}\right]+\left[\begin{array}{c}\hfill {ϵ}_t^a\hfill \\ \hfill {ϵ}_t^{\alpha }\hfill \\ \hfill {ϵ}_t^{\rho }\hfill \end{array}\right]$

where the innovations ${ϵ}_t^a$, ${ϵ}_t^{\alpha }$, and ${ϵ}_t^{\rho }$ are white noise with standard deviation equal to ${\sigma }^a$, ${\sigma }^{\alpha }$, and ${\sigma }^{\rho }$ respectively, $\mathcal{N}(0,{\left({\sigma }^a\right)}^2),\mathcal{N}(0,{\left({\sigma }^{\alpha }\right)}^2),\mathcal{N}(0,{\left({\sigma }^{\rho }\right)}^2)$. In the previous processes, the covariation between the exogenous shocks are 0, but alternative specifications are easy to introduce. The steady-state value of ${\alpha }_t$ is $\overline{\alpha }$ and the steady-sate level of ${\rho }_t$ is $\overline{\rho }$.

Steady state. To use perturbation methods, we first solve for the steady sate and then consider first-order deviation from the steady state. In steady state $A=1$, and we get from the two equations in (7),

(8) $c_1={\chi }^{-\frac{\varphi }{1+\sigma \varphi }}$

and

$1+r=\frac{1}{\beta }{(\overline{\alpha }+(1-\overline{\alpha }){\left(\frac{\delta }{c_1}\right)}^{-\sigma })}^{-1}$

Putting in some numbers allows estimating the order of magnitude. Consider the period to be a quarter. The previous equality shows that the effect of uninsurable risk on the interest rate is the consumption inequality between employed and unemployed agents (irrespective of labor-supply elasticity for instance). Chodorow-Reich and Karabarbounis (2014) estimate a decrease in consumption of non-durable goods of households falling into unemployment between 10% and 20%. As a consequence, one can take the conservative value $\frac{\delta }{c_1}=0.9$. The quarterly job loss probability is roughly 5% and $\alpha =0.95$ (see Challe and Ragot, 2016 for a discussion), and the discount factor is $\beta =0.99$. One finds a real interest rate $r=0.45\text{\%}$ for $\sigma =1$, and $r=-0.002$ when $\sigma =2$. In the complete market case, we have $\alpha =1$ and $1+r=1/\beta$. We find $r=1\text{\%}$. As is well known, market incompleteness contributes to a smaller steady-state interest rate compared to the complete market case (see Aiyagari, 1994 for a discussion).

4.1.1 Assumptions

The first class of equilibria is based on two assumptions.

First, it is assumed that agents choose their labor supply when employed and that the disutility of labor supply is linear. If c is consumption and l is labor supply, the period utility function is

$U(c,l)=u(c)-l$

The implication of this assumption is that the first-order condition for labor supply pins down the marginal utility of consumption of employed agents. This assumption is used in Scheinkman and Weiss (1986) and in Lagos and Wright (2005) to simplify heterogeneity in various environments.

The second assumption is that the credit constraint is tighter than the natural borrowing limit

(10) $\overline{a}>-\delta /r$

where r is the steady-state interest rate. This concept is introduced by Aiyagari (1994), and it is the loosest credit constraint, which ensures that consumption is always positive. The implication of this assumption is that unemployed agents will hit the credit constraint after a finite number of periods of unemployment. This property is key to reduce the state space, and we discuss it below.

4.1.2 Equilibrium Structure

To simplify the exposition, the equilibrium is presented using a guess-and-verify strategy. For the sake of clarity, the time index is kept to variables (although not necessary in the recursive exposition). Assume that all employed agents consume and save the same amount in each period t, $c_{0,t}$ and $a_{0,t+1}$ respectively. In addition, assume that all agents unemployed for k periods consume and save the same amount, denoted $c_{k,t}$ and $a_{k,t+1}$, for $k⩾1$ respectively. In addition, assume that agents unemployed for L periods are credit-constrained, and that this number is not time-varying (L is an equilibrium object). This last assumption is important and will be justified below.

Denote as $V_k(a_t,X_t)$ the value function of agents in state $k=0,1,...$ (0 is employed agents, here), where $X_t$ is the set of variables specified below that are necessary to form rational expectations.³

We have for employed people

$\begin{array}{c}\hfill V_0(a_t,X_t)={\max }_{c_{0,t},a_{0,t+1}{l}_t}u(c_{0,t})-l_t+\beta \mathbb{E}({\alpha }_{t+1}{V}_0(a_{0,t+1},X_{t+1})\hfill \\ \hfill +(1-{\alpha }_{t+1})V_1(a_{0,t+1},X_{t+1}))\hfill \\ \hfill a_{0,t+1}+c_{0,t}=w_t{l}_t+a_t(1+r_t)\hfill \\ \hfill a_{0,t+1}⩾-\overline{a}\hfill \end{array}$

and for all unemployed people, $k⩾1$

$\begin{array}{c}\hfill V_k(a_t,X_t)=\max u(c_t)-\delta +\beta \mathbb{E}({\rho }_{t+1}{V}_0(a_{k,t+1},X_{t+1})\hfill \\ \hfill +(1-{\rho }_{t+1})V_{k+1}(a_{k,t+1},X_{t+1}))\hfill \\ \hfill a_{k,t+1}+c_{k,t}=\delta +a_t(1+r_t)\hfill \\ \hfill a_{k,t+1}⩾-\overline{a}\hfill \end{array}$

As credit constraints bind for agents unemployed for $k⩾L$ periods, we have for these agents $a_{k,t+1}=-\overline{a}$.

We can derive the set of first-order conditions. For employed agents

$\begin{array}{ccc}\hfill u^{\prime }(c_{0,t})\hfill & \hfill =\hfill & \hfill 1/w_t\hfill \\ \hfill u^{\prime }(c_{0,t})\hfill & \hfill =\hfill & \hfill \beta \mathbb{E}(1+r_{t+1})({\alpha }_{t+1}{u}^{\prime }(c_{0,t+1})+(1-{\alpha }_{t+1})u^{\prime }(c_{1,t+1}))\hfill \end{array}$

For unemployed agents, for $k=1...L-1$.

$u^{\prime }(c_{k,t})=\beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{u}^{\prime }(c_{k+1,t+1})+(1-{\rho }_{t+1})u^{\prime }(c_{0,t+1}))$

Note that when $L=1$, such that the credit constraints bind after one period of unemployment, then the previous equations don't exist. This case is studied more precisely below.

The conditions define a system of $2(L+1)$ equations.

$\begin{array}{ccc}\hfill 1/w_t& \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\alpha }_{t+1}/w_{t+1}+(1-{\alpha }_{t+1})u^{\prime }(c_{1,t+1}))\hfill \\ \hfill u^{\prime }(c_{k,t})& \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{u}^{\prime }(c_{k+1,t+1})+(1-{\rho }_{t+1})/w_{t+1}),\hfill \\ \hfill & \hfill \hfill & \text{for }k=1...L-1\hfill \\ \hfill a_{k,t+1}+c_{k,t}& \hfill =\hfill & \delta +a_{k-1,t}(1+r_t),\text{for }k=1...L\hfill \\ \hfill u^{\prime }(c_{0,t})& \hfill =\hfill & 1/w_t\hfill \\ \hfill a_{L,t}& \hfill =\hfill & -\overline{a}\hfill \end{array}$

These equations form a system in the $2(L+1)$ variables ${(c_{k,t},a_{k,t})}_{k=0...L}$. These equations confirm the intuition that all employed agents consume and save the same amount.

How is this possible? This comes from the labor choice, which provides some insurance. Indeed, as soon as an unemployed agent for k period in period $t-1$ becomes employed in period t, then they work the necessary amount, denoted as $l_{k0,t}$ to consume $c_{0,t}$. This amount is given by the budget constraint of employed households

(11) $\begin{array}{cccc}\hfill l_{k0,t}\hfill & \hfill =\hfill & \hfill (a_{0,t+1}+c_{0,t}-a_{k,t}(1+r_t))/w_t,\hfill & \hfill k=0,...,L-1\hfill \\ \hfill l_{k0,t}\hfill & \hfill =\hfill & \hfill (a_{0,t+1}+c_{0,t}+\overline{a}(1+r_t))/w_t,\hfill & \hfill k=L,...,\infty \hfill \end{array}$

(The previous equation is indeed valid for $k=0$.) Finally, note that for agents $k⩾L+1$, we simply have, from the budget constraint

$c_{k,t}=\delta -r_t\overline{a}$

Thanks to the assumption about the credit constraint given by (3) and the assumption of small aggregate shocks (to use perturbation methods), this amount will be positive.

This almost concludes the description of the agent's decision. The last step is to follow the number of employed and of each type of unemployed agent. Denote as $n_{k,t}$ the number of agents in state $k=0,...,L$ in each period t. We have the law of motion of each type of agent

$\begin{array}{ccc}\hfill n_{0,t}\hfill & \hfill =\hfill & \hfill {\alpha }_t{n}_{0,t-1}+(1-{\rho }_t)(1-n_{0,t-1})\hfill \\ \hfill n_{k,t}\hfill & \hfill =\hfill & \hfill {\rho }_t{n}_{k-1,t-1},\hfill & \hfill \text{for }k=1,...,\infty \hfill \end{array}$

The first equation states that the number of employed agents is equal to the number of employed agents who keep their job (first) term, plus the number of unemployed agents, which is $1-n_{0,t-1}$, who find a job. The second equation states that the number of agents unemployed for k periods at date t, are the number of agents unemployed for $k-1$ periods at the previous date who stay unemployed.

The number of k0 agents (i.e. employed agents at date t, who were unemployed for k periods at date $t-1$) is

$n_{k0,t}=(1-{\rho }_t)n_{k,t-1}$

The number of credit-constrained agents is denoted as $n_t^c$ and is simply

$n_t^c=1-\sum _{k=0}^{L-1}{n}_{k,t}$

In this equilibrium, the capital market equilibrium is simply

$K_t=\sum _{k=0}^{L-1}{n}_{k,t}{a}_{k,t}-n_t^c\overline{a}$

and

$L_t=\sum _{k=0}^{L-1}{n}_{k0,t}{l}_{k0,t}+(1-{\rho }_t)n_{t-1}^c{l}_{L0,t}$

Here we used the fact that all constrained agents work the same amount when they find a job, due to condition (11).

Due to this assumption, the marginal utility of all employed agents is $u^{\prime }(c_t)=1/w_t$ in all periods. As a consequence, this marginal utility does not depend on the history of agents on the labor market, what considerably simplifies the equilibrium structure.

4.1.3 The System

We can now present the whole system of equations:

$\begin{array}{ccc}\hfill 1/w_t& \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\alpha }_{t+1}/w_{t+1}+(1-{\alpha }_{t+1})u^{\prime }(c_{1,t+1}))\hfill \\ \hfill u^{\prime }(c_{k,t})& \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{u}^{\prime }(c_{k+1,t+1})+(1-{\rho }_{t+1})/w_{t+1}),\hfill \\ \hfill & \hfill \hfill & \text{for }k=1,...,L-1\hfill \\ \hfill a_{k,t+1}+c_{k,t}& \hfill =\hfill & \delta +a_{k-1,t}(1+r_t),\text{for }k=1,...,L\hfill \\ \hfill u^{\prime }(c_{0,t})& \hfill =\hfill & 1/w_t\hfill \\ \hfill a_{L,t}& \hfill =\hfill & -\overline{a}\hfill \\ \hfill l_{k0,t}& \hfill =\hfill & (a_{0,t+1}+c_{0,t}-a_{k,t}(1+r_t))/w_t,\text{for }k=0,...,L-1\hfill \\ \hfill l_{L0,t}& \hfill =\hfill & (a_{0,t+1}+c_{0,t}+\overline{a}(1+r_t))/w_t\hfill \\ \hfill n_{0,t}& \hfill =\hfill & {\alpha }_t{n}_{0,t-1}+(1-{\rho }_t)(1-n_{0,t-1})\hfill \\ \hfill n_{1,t}& \hfill =\hfill & (1-{\alpha }_t)n_{0,t}\hfill \\ \hfill n_{k,t}& \hfill =\hfill & {\rho }_t{n}_{k-1,t-1},\text{for }k=2,...,L-1\hfill \\ \hfill n_{00,t}& \hfill =\hfill & {\alpha }_t{n}_{0,t}\hfill \\ \hfill n_{k0,t}& \hfill =\hfill & (1-{\rho }_t)n_{k,t-1},\text{for }k=1,...,L-1\hfill \\ \hfill n_t^c& \hfill =\hfill & 1-\sum _{k=0}^{L-1}{n}_{k,t}\hfill \\ \hfill K_t& \hfill =\hfill & \sum _{k=0}^{L-1}{n}_{k,t}{a}_{k,t}-n_t^c\overline{a}\hfill \\ \hfill L_t& \hfill =\hfill & \sum _{k=0}^{L-1}{n}_{k0,t}{l}_{k0,t}+(1-{\rho }_t)n_{t-1}^c{l}_{L0,t}\hfill \\ \hfill r_t& \hfill =\hfill & \lambda A_t{K}_t^{\lambda -1}{L}_t^{1-\lambda }-\mu \hfill \\ \hfill w_t& \hfill =\hfill & (1-\lambda )A_t{K}_t^{\lambda }{L}_t^{-\lambda }\hfill \end{array}$

This system is large but finite. There are $5L+8$ equations for $5L+8$ variables ${({(c_{k,t},a_{k,t+1},l_{k0,t})}_{k=0,...,L},{(n_{k,t},n_{k0,t})}_{k=0,...,L-1},n_t^c,K_t,L_t,r_t,w_t)}_{t=0...\infty }$. For any process for the exogenous shocks $({\alpha }_t,{\rho }_t,A_t)$ one may think that it is possible to simulate this model. This is not the case because one key variable is not determined: L.

4.1.4 Algorithm: Finding the Value of L

The value of L can be found using steady-state solutions of the previous system. The idea is to find the steady-state for any L and iterate over L to find the value for which L agents are credit constrained, whereas $L-1$ agents are not. The algorithm to find L and the steady state is the following (finding the steady state is not difficult because the problem is block-separable). I simply drop the time subscript to denote steady-state values.

4.1.5 Simulations

Once the steady-state value of L and the steady-state value of the variables are determined, one can simulate the model using standard perturbation methods. One can use DYNARE to simulate first and second approximations of the model, compute second moments, and so on. The distribution of wealth is summarized by the vector ${(n_{k,t},a_{k,t})}_{k=0,...,L-1}$ and belongs to the state space of agents $X_t$ to form rational expectations. In these simulations, one has to check that the aggregate shock is small enough such that L is indeed constant over time. One must thus check that the condition (12) is satisfied not only in the steady state but also during the simulations.

4.1.6 References and Limits

In the Bewley–Huggett–Aiyagari environment, Algan et al. (2011) use this framework to investigate the impact of money injections in a model where agents use money to self-insure against idiosyncratic shocks. Challe et al. (2013) use this assumption to study the effect of an increase in public debt on the yields curve in an environment where agents use safe assets of various maturities to self-insure against idiosyncratic risk. They show that an increase in idiosyncratic risk decreases both the level and the slope of the yield curve. In addition, an increase in public debt increases both the level and the slope of the yield curve. LeGrand and Ragot (2016a) use this assumption to consider insurance for aggregate risk in these environments. Introducing derivative assets, such as options, in an environment where agents use a risky asset to self-insure against idiosyncratic risk, they show that the time-variations in the volume of traded derivative assets are consistent with empirical findings.

This framework is interesting to investigate the properties of time-varying precautionary saving in finance (for instance to study asset prices) but it is not well suited for the macroeconomy. Indeed, the elasticity of labor supply is much too high compared to empirical findings (the Frisch elasticity is here infinite, whereas it is between 0.3 and 1 in the data, see Chetty et al., 2011 for a discussion). In addition, all employed agents consume the same amount, which is pinned down by the real wage, which is obviously a counterfactual. For this reason, other frameworks with positive trade have been developed.

4.2.1 Assumptions

The model relies on three assumptions.

First, instead of introducing linearity in the labor supply, the linearity is in the utility of consumption. More precisely, it is assumed that there exists a threshold $c^{⁎}$ such that the period utility function is strictly concave for $c<c^{⁎}$ and linear for $c⩾c^{⁎}$. The linear-after-a-threshold utility function was introduced by Fishburn (1977) in decision theory to model behavior in front of gains and losses differently.

The period utility function is thus

(13) ${\tilde{u}}^{\prime }(c)=\{\begin{array}{cc}{u}^{\prime }(c)\hfill & \text{if }c<c^{⁎},\hfill \\ \eta \hfill & \text{if }{c}^{⁎}\le c,\hfill \end{array}$

where the function $u(.)$ is increasing and concave. The slope of the utility function must be low enough to obtain global concavity, that is $u^{\prime }(c^{⁎})>\eta$.

Second, the borrowing limit is assumed to be strictly higher than the natural borrowing limit, as before.

Third, it is assumed that the discount factor of agents is such that all employed agents consume an amount $c_t>c^{⁎}$ and all unemployed agents consume an amount $c_t<c^{⁎}$.

4.2.2 Equilibrium Structure

To save some space, we now focus on the households' program. Assume that labor is inelastic as a useful benchmark (introducing elastic labor is very simple in this environment). All employed agents supply one unit of labor.

We have for employed people

$\begin{array}{c}\hfill V_0(a_t,X_t)={\max }_{c_{0,t},a_{0,t+1}{l}_t}\tilde{u}(c_{0,t})-1+\beta \mathbb{E}({\alpha }_{t+1}{V}_0(a_{0,t+1},X_{t+1})\hfill \\ \hfill +(1-{\alpha }_{t+1})V_1(a_{0,t+1},X_{t+1}))\hfill \\ \hfill a_{0,t+1}+c_{0,t}=w_t+a_t(1+r_t)\hfill \\ \hfill a_{0,t+1}⩾-\overline{a}\hfill \end{array}$

and for all unemployed people, $k⩾1$

$\begin{array}{c}\hfill V_k(a_t,X_t)=\max \tilde{u}(c_t)-\delta +\beta \mathbb{E}({\rho }_{t+1}{V}_0(a_{k,t+1},X_{t+1})\hfill \\ \hfill +(1-{\rho }_{t+1})V_{k+1}(a_{k,t+1},X_{t+1}))\hfill \\ \hfill a_{k,t+1}+c_{k,t}=\delta +a_t(1+r_t)\hfill \\ \hfill a_{k,t+1}⩾-\overline{a}\hfill \end{array}$

One can solve for the order conditions following the same steps as before. Using the same notations as in the previous section, denote as k agents the agents unemployed for k periods. Assuming that credit constraints are binding after L periods of unemployment, one can find consumption and saving choices.

The key difference between this environment and the one presented in the previous section is that employed agents will not differ according to their labor supply (which is inelastic), but by their consumption level. Denote as $c_{k0,t}$ the consumption at date t of employed agents who were unemployed for k periods at date $t-1$, and denote (as before) as $c_{k,t}$ (for $k=1,...,L$) the consumption of unemployed agents who are unemployed for k periods at date t. The households are now described by the vector ${(a_{k,t},c_{k0,t})}_{k=0,...,L}$ and ${(c_{k,t})}_{k=1,...,L}$ solving

$\begin{array}{ccc}\hfill \eta & \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\alpha }_{t+1}\eta +(1-{\alpha }_{t+1})u^{\prime }(c_{1,t+1}))\hfill \\ \hfill u^{\prime }(c_{k,t})& \hfill =\hfill & \beta \mathbb{E}(1+r_{t+1})({\rho }_{t+1}{u}^{\prime }(c_{k+1,t+1})+(1-{\rho }_{t+1})\eta ),\hfill \\ \hfill & \hfill \hfill & \text{for }k=1,...,L-1\hfill \\ \hfill a_{k,t+1}+c_{k,t}& \hfill =\hfill & \delta +a_{k-1,t}(1+r_t),\text{for }k=1,...,L\hfill \\ \hfill a_{0,t+1}+c_{k0,t}& \hfill =\hfill & w_t+a_{k,t}(1+r_t),\text{for }k=0,...,L\hfill \\ \hfill a_{L,t}& \hfill =\hfill & -\overline{a}\hfill \end{array}$