# Public debt and aggregate risk

Audrey Desbonnet, Sumudu Kankanamge

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Audrey Desbonnet, Sumudu Kankanamge. Public debt and aggregate risk. 2008. halshs-00175877v2

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UNIVERSITÉ PARIS 1
- PANTHÉON - SORBONNE -

Documents de Travail du
Centre d'Economie de la Sorbonne

CES Working Papers

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Public debt and aggregate risk

Audrey DESBONNET, Sumudu KANKANAMGE

2007.42

Version révisée

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CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13

http://ces.univ-paris1.fr/cesdp/CES-docs.htm

ISSN : 1955-611X
Public debt and aggregate risk ⋆

Audrey Desbonnet<sup>a 1</sup> and Sumudu Kankanamge<sup>b 2</sup>

<sup>a</sup>University of Vienna, Austria.

<sup>b</sup>Paris School of Economics (Paris I), CNRS, France.

<footnote n="1"><sup>1</sup> The author was a member of Paris School of Economics (Paris I) when the first version of this paper was written.</footnote>

<footnote n="2"><sup>2</sup> Corresponding author. Address : CES - Université Paris I; 106-112 Boulevard de l'Hôpital - 75647 Paris France; Tel : +33661157683; Fax : +33144078231; Email : kankanamge@univ-paris1.fr</footnote>

<footnote><sup>⋆</sup> We thank Yann Algan, Antoine d'Autume, Fabrice Collard, Jean-Olivier Hairault, Per Krusell, Thomas Weitzenblum and Bertrand Wigniolle for their very helpful comments. We also thank seminar participants at the European Economic Association 2007 meeting, the AFSE 2007 meeting, the T2M 2007 meeting, the PET 2006 meeting and various seminar participants at University Paris I - Panthéon Sorbonne for comments and suggestions.</footnote>
**Résumé** : Cet article reconsidère l'impact de l'introduction de la dette publique dans le cadre de marchés incomplets et de fluctuations macroéconomiques. Aiyagari et Mc Grattan (1998) montrent que l'incomplétude des marchés et l'existence d'une contrainte d'endettement créent un motif d'épargne de précaution qui réaffirme le rôle de la dette publique. Parce qu'elle offre un moyen additionnel aux ménages de lisser leur consommation, l'introduction de la dette publique est bénéfique. Cependant, cette analyse ignore une source non négligeable d'exacerbation du risque idiosyncrasique de revenu non assurable qu'est le risque agrégé et risque donc d'avoir sous estimé le rôle de la dette. L'accroissement du taux et de la durée du chômage lors des phases de récession entame la capacité assurantielle de l'épargne de précaution. En outre, l'occurrence de faibles taux d'intérêt durant les périodes de récession rend plus coûteuse la constitution d'une épargne de précaution. L'introduction de la dette publique accroît le taux d'intérêt. L'épargne de précaution devient moins coûteuse. C'est la raison pour laquelle le niveau optimal de dette publique est égal à 5% du produit annuel et s'établit en moyenne à un niveau plus élevé que dans un environnement dépourvu de fluctuations macroéconomiques.

**Abstract** : This paper assesses the long-run optimal level of public debt in a framework where aggregate fluctuations are taken into account. Households are subject to both aggregate and idiosyncratic shocks and the market structure prevents them from perfectly insuring against risk. We find that the long-run optimal level of public debt is generally higher in a setting embedding aggregate fluctuations than in a setting without. Aggregate fluctuations modify both the cost and the motive for precautionary saving. Higher levels of public debt, by effectively reducing the cost of precautionary saving, help agents to smooth consumption when they face price and employment fluctuations.

**Mots-Clés** : dette publique, risque agrégé, épargne de précaution, contrainte de crédit

**Key words** : public debt, aggregate risk, precautionary saving, credit constraints

**JEL classification** : E32, E62, H31
# 1 Introduction

The introduction of uninsurable risk was a major step in the recent literature on public debt. Woodford (1990) is the first author to depart from the complete market framework to consider credit constraints or as he phrases it "imperfect financial intermediation". Woodford (1990) builds a simple economy embedding credit constraints and finds that public debt can be efficient because it keeps interest rates higher and closer to time preference rates. Aiyagari and McGrattan (1998) quantitatively address the question of the optimal level of public debt in a heterogeneous agent, incomplete market model. Calibrating on the U.S. economy they find the annual positive debt over GDP ratio of 2/3 to be optimal. Finally, Floden (2001) uses a similar framework to look at public debt/transfers optimal combinations. This author underlines the strong uncertainty and inequality implications behind public debt policies. Very much in the same way, the introduction of uninsurable risk in the literature on the cost of business cycles has generated rich implications. In an effort to reconsider the small welfare effect of business cycles found by Lucas (1987), both Krusell and Smith (2002) and Storesletten, Telmer and Yaron (2001) draw a link between the aggregate risk and the cross sectional distribution. In other words, the aggregate productivity shock is correlated to the individual specific shocks to produce strong distributional effects of aggregate fluctuations. In such a framework, these authors find a greater cost of business cycles than Lucas (1987). Even more interestingly, Imrohoroglu (1989) briefly suggests that economic policies could be used to help individuals reduce the cost of business cycles. In this paper, we explore this latter suggestion and its implications for the optimal level of public debt.

This paper's main objective is to introduce a framework exhibiting aggregate fluctuations à la Den Haan (1996) or Krusell and Smith (1998) to quantify the long-run optimal level of public debt. As aggregate fluctuations impact the saving behavior of agents, they are an important aspect to take into account for public debt policies that has been ignored so far. Unlike previous literature, this paper also takes some steps to reproduce the wealth distribution of the targeted U.S. economy as it has been documented that the need for public debt is very different across the population. Thus our framework can be used to decompose the effects of public debt along the cycle and also across the population. Following Bewley-Huggett-Aiyagari type models, we build a production economy with capital market imperfections where a large number of ex-ante identical infinitely-lived agents face idiosyncratic income shocks and aggregate productivity shocks. Households' saving behavior is influenced by precautionary saving motives and borrowing constraints. Private capital and government bonds, both yielding the same interest, can be claimed to insure against future risk. Government levies proportional taxes on households and
issues debt in order to finance its consumption.

In this setup, our benchmark calibration yields a long-run optimal level of public debt of 5% of GDP on an annual basis. This is to be compared to a model without aggregate fluctuations. We consider several methods of removing aggregate fluctuations to obtain a comparable long-run idiosyncratic risk model. The level of public debt obtained in those idiosyncratic models are generally lower although with one method the optimal level of debt is almost similar to the aggregate risk model. Thus we argue that the effect of aggregate risk on the optimal level of public debt is moderate. In the benchmark economy, we find that the gains of being at the optimal level of debt instead of the benchmark debt to GDP ratio of 66% amounts to 0.257% of consumption. Also, the consumption gains of being at the optimal level of debt are higher in recessions than in expansions and higher for the poorest percentile of the population. The intuition behind our results is the following. Credit constraints and uncertainty lead agents to engage in precautionary saving. The result of precautionary saving is a higher level of capital that in turn lowers the interest rate away from the time preference rate. Aggregate fluctuations, as it is correlated with the labor market process, exacerbate the level of risk faced by agents. An *employment fluctuation effect* increase both the risk of losing one's job and the time one could spend in unemployment in recessions. This strengthens households' precautionary saving motive. A *price fluctuation effect* changes the level of prices between recessions and booms making it more costly to accumulate precautionary saving in recessions. As a result of these effects, the capital stock rises and the interest rates move further away from the time preference rate. A higher level of public debt has two opposing effects here. A *crowding out effect* crowds out private capital and reduces welfare. A *cost of precautionary saving effect* increases the interest rate, reduces the cost of precautionary saving and enhances welfare. Because the *employment fluctuation effect* and the *price fluctuation effect* change the precautionary saving motive and cost, the *cost of precautionary saving effect* is stronger in an aggregate fluctuations setting than in an idiosyncratic risk setting. Thus the optimal level of public debt is higher in a setting with aggregate fluctuations. Finally, we emphasize that the optimal level of debt can be significantly higher if we modify the parameters governing the *employment fluctuation effect* so as to strengthen this effects.

The rest of the paper is organized as follows. Next section describes the benchmark economy. Section 3 details the results. In section 3.5 we characterize the optimal level of public debt for the European labor market. The last section concludes.
## 2 The Benchmark Model

Our benchmark economy is a Bewley-Huggett-Aiyagari type dynamic stochastic general equilibrium model augmented to allow aggregate fluctuations à la Den Haan (1996) or Krusell and Smith (1998) and public debt. Insurance markets are incomplete. Agents face idiosyncratic and aggregate risks and are borrowing constrained. These three assumptions lead agents into precautionary saving (Aiyagari (1994)). Our benchmark model can be related to the model in Aiyagari and McGrattan (1998) but present three deviations of importance apart from its aggregate fluctuations feature. First, the productivity process here is simpler as agents can only be employed or unemployed. Second, leisure is not valued. Those two deviations greatly simplify the model with aggregate fluctuations. Finally, there is no exogenous growth in our benchmark economy. Aiyagari and McGrattan (1998) specify an exogenous annual growth rate of 1.85%. This assumption exogenously reduces the cost of public debt for individuals since public debt interest repayments are diminished by this exogenous growth factor. It is noteworthy that making this assumption lead to a higher optimal level of public debt.

### 2.1 Firms

We assume that there is a continuum of firms which have a neoclassical production technology and behave competitively in product and factor markets. The output is given by :

$$Y_t = z_t F(K_t, N_t)$$

where K is aggregate capital and N aggregate labor used in production. The function F exhibits constant returns to scale with respect to K and N, has positive and strictly diminishing marginal products, and satisfies the Inada conditions. Capital depreciates at a constant rate δ. The economy is subject to an exogenous aggregate shock noted z. There are two possible aggregate states : a good state where z = z_g and a bad state where z = z_b. The aggregate shock follows a first-order Markov process with transition probability η_{z|z'} = Pr(z_{t+1} = z'/z_t = z). Thus η_{z|z'} is the probability that the aggregate state tomorrow is z' given that it is z today. We note η the matrix that describes the transition from one aggregate state to another such that :

$$\eta = \begin{pmatrix} \eta_{gg} & \eta_{gb} \\ \eta_{bg} & \eta_{bb} \end{pmatrix}$$
Finally, our setting assumes that inputs market are competitive. The wage $w$ and the interest rate $r$ verify :

$$r_t + \delta = z_t F_K(K_t, N_t) \qquad (1)$$

$$w_t = z_t F_N(K_t, N_t) \qquad (2)$$

## 2.2 The government

The government issues public debt and levies taxes to finance public expenses. Both the revenue of capital and labor are taxed proportionally at an identical rate $\tau$. The government's budget constraint verifies :

$$G_t + r_t B_t + T R_t = B_{t+1} - B_t + T_t$$

with

$$T_t = \tau(w_t N_t + r_t A_t)$$

$G_t$ is the level of public expenses, $B_t$ the level of public debt, $T_t$ tax revenues and $TR_t$ a lump sum transfer to households that amounts to zero at the equilibrium. Because of the aggregate fluctuations property of the model, we have to be cautious about how we close the model with respect to the government budget constraint. We proceed as follows. As there is no formal steady state in this economy because of the aggregate fluctuations property, we approximate a *pseudo steady state* by averaging aggregate variables over long periods of time. We first guess an interest rate in this *pseudo steady state* and derive a tax rate by assuming that public expenses and public debt are a constant fraction of the associated long run GDP. As this tax rate does not necessarily balance the government budget constraint along the cycle, we use the lump sum transfer $TR$ to make sure the budget will be balanced. Finally, we update our guess on the interest rate in the *pseudo steady state* until we reach a fix point. When this final step is completed, the model is closed with respect to the government budget constraint and $TR$ amounts to zero at the equilibrium<fnref n="1" />.

$A_t$ accounts for total average wealth in the economy. It is the sum of average physical capital $K$ and public debt $B$ such that :

$$A_t = K_t + B_t \qquad (3)$$

<footnote n="1">¹ More details can be found in Appendix A.</footnote>
## 2.3 Households

The economy is populated by a continuum of ex ante identical infinitely lived households of unit mass. Their preferences are summarized by the function $V$:

$$V = \mathbb{E}_0 \left\{ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j u(c_t) \right) \right\} \quad (4)$$

where $\beta$ is the discount factor. We assume that this discount factor is random. Thus the discount factor $\beta$ can differ across agents and can vary over time. We specify that the latter follows a three-states first-order Markov process. This assumption on the discount factor helps to reproduce the wealth distribution as shown in Krusell and Smith (1998). The discount factor verifies:

$$\begin{cases} \beta_0 = 1 \\ \beta_{j \ge 1} \in ]0; 1[ \end{cases}$$

$c_t$ is the household level consumption. The utility function we use has a standard CRRA specification and writes:

$$u(c) = \begin{cases} \frac{c^{1-\sigma}}{1-\sigma} & \text{if } \sigma \neq 1 \\ \log(c) & \text{if } \sigma = 1 \end{cases}$$

Agents are subject to idiosyncratic unemployment shocks. Let $s$ be the household's labor market status. A household can either be unemployed ($s = u$) or employed ($s = e$). Households are also subject to shocks at the aggregate level. Aggregate shocks exacerbate idiosyncratic unemployment risks. The unemployment rate and the unemployment duration are higher in recessions than in booms. Therefore, transitions on the labor market are correlated to the aggregate state. We note $\Pi_{zz'|ss'}$ the joint transition probability to a state $(s', z')$ conditional on a state $(s, z)$. The matrix that jointly describes the transition from a state $(s, z)$ to a state $(s', z')$ is the following:

$$\Pi = \begin{pmatrix} \Pi_{bbuu} & \Pi_{bbue} & \Pi_{bguu} & \Pi_{bgue} \\ \Pi_{bbeu} & \Pi_{bbee} & \Pi_{bgeu} & \Pi_{bgee} \\ \Pi_{gbuu} & \Pi_{gbue} & \Pi_{gguu} & \Pi_{ggue} \\ \Pi_{gbeu} & \Pi_{gbee} & \Pi_{ggeu} & \Pi_{ggee} \end{pmatrix}$$

where $\Pi_{ggee} = \text{Pr}(z_{t+1} = z_g, s_{t+1} = e | z_t = z_g, s_t = e)$.
When agents are in an employed state, they receive the wage $w$. However when agents are unemployed their income corresponds to their home production that we note $\theta$. Insurance markets are incomplete so that agents can only partially self-insure against idiosyncratic risk. Following Aiyagari and McGrattan (1998) no borrowing is allowed. The only way for households to self-insure against idiosyncratic risk is to accumulate physical capital and government bonds both yielding the same return $r$. Their overall holding in the later assets is noted $a$. Therefore a typical household solves the following problem:

$$ \max_{c_t, a_{t+1}} \mathbb{E}_0 \left\{ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) u(c_t) \right\}, $$

subject to:

$$ a_{t+1} + c_t = (1 + (1 - \tau)r_t)a_t + \chi(s_t)w_t + TR_t $$

$$ c_t \ge 0 $$

$$ a_{t+1} \ge 0 $$

with

$$ \chi(s_t) = \begin{cases} \theta & \text{if } s_t = u, \\ (1 - \tau) & \text{if } s_t = e \end{cases} $$

The existence of aggregate risk leads us to distinguish between individual state variables and aggregate state variables. The individual state variables are given by the vector $(a, s, \beta)$. The aggregate state variables are summarized by the vector $(z, \Gamma)$ where $\Gamma(a, s, \beta)$ is a distribution of agents over asset holdings, employment status and preferences. To determine the wage and the interest rate, households need to forecast the aggregate stock of physical capital. Therefore, they need to know the wealth distribution. That is why wage and interest rate depend on that wealth distribution. In the computational appendix, we explain how we avoid manipulating the wealth distribution by approximating it with some of its moments using the methodology developed in Den Haan (1996) and Krusell and Smith (1998). We detail the computational strategy we used to solve the model in appendix A.

## 2.4 Equilibrium

The recursive equilibrium consists of a set of decision rules for consumption and asset holding $\{c(a, s, \beta; z, \Gamma), a'(a, s, \beta; z, \Gamma)\}$, aggregate capital $K(z, \Gamma)$, factor prices $\{r(z, \Gamma), w(z, \Gamma)\}$, tax rate $\tau$ and a law of motion for the distribution $\Gamma' = H(\Gamma, z, z')$ which satisfy these conditions:
(i) Given the aggregate states, {$z, \Gamma$}, prices {$r(z, \Gamma), w(z, \Gamma)$} and the law of motion for the distribution $\Gamma' = H(\Gamma, z, z')$, the decision rules {$c(a, s, \beta; z, \Gamma), a'(a, \epsilon, \beta; z, \Gamma)$} solve the following dynamic programming problem :

$$v(a, s, \beta; z, \Gamma) = \max_{c, a'} \{u(c) + \beta E[v(a', s', \beta'; z', \Gamma')|(s, \beta; z, \Gamma)]\}$$

subject to :

$$c + a' = (1 + r(z, \Gamma)(1 - \tau))a + w(z, \Gamma)\chi(s) + TR$$

$$c \geq 0$$

$$a' \geq 0$$

and

$$\Gamma' = H(\Gamma, z, z')$$

(ii) Market price arrangements are :

$$r(z, \Gamma) = zF_K(K, N) - \delta$$

$$w(z, \Gamma) = zF_N(K, N)$$

(iii) Government budget constraint holds.

(iv) Capital market verify :

$$K + B = \int a'(a, \epsilon, \beta; \Gamma, z) d\Gamma$$

(v) Consistency : agents' optimization problem is satisfied given the law of motion $H$ and the law of motion is consistent with individual behavior.

## 2.5 Calibration

For the sake of comparison, the model economy is calibrated to match certain observations in the U.S. data. We let one period in the model be one quarter in the data. To avoid confusion we have converted all values to their annual equivalents in the results section. To remain simple and allow comparisons, we closely follow Krusell and Smith (1998) when calibrating the characteristics of the labor market and the aggregate risk.
### 2.5.1 Technology

We choose the production function to be Cobb-Douglas :

$$Y_t = z_t F(K_t, N_t) = z_t K_t^{\alpha} N_t^{1-\alpha} \quad 0 < \alpha < 1$$

Technology parameters are standard. The capital share of output $\alpha$ is set to 0.36 and the capital depreciation rate $\delta$ is 0.025. As Krusell and Smith (1998) we assume that the value of the aggregate shock $z$ is equal to 0.99 in recessions ($z_b$) and 1.01 in booms ($z_g$).

### 2.5.2 Preferences and discount factor

In the benchmark economy we assume a logarithmic utility function. We now detail the calibration steps to generate a realistic wealth distribution, the observed U.S. wealth Gini index and the capital-output ratio. Here, our calibration differs from Krusell and Smith (1998). In their economy agents can borrow whereas here, for the sake of comparison and simplicity, we follow Aiyagari and McGrattan (1998) and no borrowing is allowed. To reproduce the shape of the U.S. wealth distribution we first assume that unemployed agents receive income too and fix the home production income $\theta$ to be $0.10^2$. This assumption produces a large group of poor agents. Next we use the preference heterogeneity setting discussed in Krusell and Smith (1998) to generate a long thick right tail^3. We impose that the discount factor $\beta$ takes on three values $\{\beta_l, \beta_m, \beta_h\}$ where $\beta_l < \beta_m < \beta_h$ :

$$\begin{pmatrix} \beta_l \\ \beta_m \\ \beta_h \end{pmatrix} = \begin{pmatrix} 0.9750 \\ 0.9880 \\ 0.9985 \end{pmatrix}$$

Thus an agent with a discount factor $\beta_m$ is more patient than an agent with a discount factor $\beta_l$. To calibrate the transition matrix, we impose that the invariant distribution for discount factors has 10% of the population at the lowest discount rate $\beta_l$, 70% at the medium discount factor $\beta_m$ and 20% at the highest discount factor $\beta_h$. As Krussell and Smith (1998) we assume that there is no immediate transition between extreme values of the discount factors. Finally, we set the average duration of the lowest discount factor and the

<footnote n="2">2 This corresponds to about 12% of the average wage at the equilibrium.</footnote>

<footnote n="3">3 There are several ways to reproduce a thick right tail. One would be to give rich agents higher propensity to save or higher returns on saving for instance by introducing entrepreneurs (e.g., Quadrini (2000)). For the sake of simplicity we explore here the preference heterogeneity setting introduced by Krusell and Smith (1998).</footnote>
<table><caption>TAB. 1
Distribution of wealth : Benchmark model and data</caption><thead><tr><th></th><th colspan="4">Held by Top</th><th>Gini</th></tr></thead><tbody><tr><th scope="row">Source</th><td>1%</td><td>5%</td><td>10%</td><td>20%</td><td></td></tr><tr><th scope="row">Benchmark Model</th><td>22</td><td>52</td><td>71</td><td>89</td><td>.82</td></tr><tr><th scope="row">Data</th><td>30</td><td>51</td><td>64</td><td>79</td><td>.79</td></tr></tbody><thead><tr><th></th><th colspan="4">Held by Bottom</th><th></th></tr><tr><th scope="row">Source</th><td>20%</td><td>40%</td><td>60%</td><td>80%</td><td></td></tr></thead><tbody><tr><th scope="row">Benchmark Model</th><td>1</td><td>2</td><td>4</td><td>11</td><td></td></tr><tr><th scope="row">Data</th><td>0</td><td>1</td><td>6</td><td>18</td><td></td></tr></tbody></table>

highest discount factor to be 50 years (200 quarters). These assumptions yield the following transition matrix<fnref n="4" />:

$$ \Upsilon = \begin{pmatrix} 0.9950 & 0.005 & 0.0000 \\ 0.0007 & 0.9979 & 0.0014 \\ 0.0000 & 0.0050 & 0.9950 \end{pmatrix} $$

The conditions that we have described so far specify 10 parameters so 10 targets are needed. When we account for the Gini index and the capital-output ratio, we need 8 additional targets. Given our calibration strategy, those 8 targets would be 8 points from the U.S. wealth distribution. In practice, instead of targeting 8 specific points, we searched for a set of parameters so that the wealth distribution in the model economy is as similar as possible to its U.S. counterpart.

As shown in Table 1 this calibration does a fairly good job at approximating the shape and the skewness in the U.S. wealth distribution<fnref n="5" /> and yields a Gini index of 0.82 and a capital-output ratio of 10.6⁶.

<footnote n="4">⁴ For further details on the calibration of this matrix, see appendix B.</footnote>

<footnote n="5">⁵ The data we report on the U.S. distribution comes from Krusell and Smith (1998) and Budria-Rodriguez, Diaz-Gimenez, Quadrini and Rios-Rull (2002).</footnote>

<footnote n="6">⁶ The value of the capital-output ratio can change with the definition of capital. Here we adopt the definition in Quadrini (2000). Thus aggregate capital results from the aggregation of plant and equipment, inventories, land at market value, and residential structures. This definition is close to the findings of Prescott (1986) and is also used for instance in Floden and Linde (2001). This yields a capital-output ratio of 2.65 on an annual basis that we convert to its quarterly equivalent of 10.6.</footnote>
### 2.5.3 Labor market processes

For the sake of simplicity our calibration of the aggregate shock and the labor market process follows Krusell and Smith (1998). The process for $z$ is set so that the average duration of good and bad times is 8 quarters. Therefore, the transition matrix $\eta$ for aggregate state changes is defined by :

$$ \eta = \begin{pmatrix} 0.8750 & 0.1250 \\ 0.1250 & 0.8750 \end{pmatrix} $$

The average duration of an unemployment spell is 1.5 quarters in good times and 2.5 quarters in bad times. We also set the unemployment rate accordingly : in good periods it is 4% and in bad periods it is 10%. These assumptions enable us to define the transition matrixes for labor market status for each aggregate state change : $\Pi^{gg}$ for a transition from a good period to a good period, $\Pi^{bb}$ for a transition from a bad period to a bad period, $\Pi^{gb}$ for a transition from a good period to a bad period and $\Pi^{bg}$ for a transition from a bad period to a good period$^7$ :

$$ \Pi^{bb} = \begin{pmatrix} 0.6000 & 0.4000 \\ 0.0445 & 0.9555 \end{pmatrix} \quad \Pi^{bg} = \begin{pmatrix} 0.2500 & 0.7500 \\ 0.0167 & 0.9833 \end{pmatrix} $$

$$ \Pi^{gb} = \begin{pmatrix} 0.7500 & 0.2500 \\ 0.0729 & 0.9271 \end{pmatrix} \quad \Pi^{gg} = \begin{pmatrix} 0.3333 & 0.6667 \\ 0.0278 & 0.9722 \end{pmatrix} $$

Finally the joint transition matrix $\Pi$ for labor market statuses and aggregate states can be defined as :

$$ \Pi = \begin{pmatrix} \eta_{bb} \Pi^{bb} & \eta_{bg} \Pi^{bg} \\ \eta_{gb} \Pi^{gb} & \eta_{gg} \Pi^{gg} \end{pmatrix} = \begin{pmatrix} 0.5250 & 0.3500 & 0.0313 & 0.0938 \\ 0.0388 & 0.8361 & 0.0021 & 0.1229 \\ 0.0938 & 0.0313 & 0.2916 & 0.5833 \\ 0.0911 & 0.1158 & 0.0243 & 0.8507 \end{pmatrix} $$

### 2.5.4 Government

We fix the ratio of government purchases to GDP to 0.217. The debt over GDP ratio, noted $b$ is set to the quarterly value of $\frac{8}{3}$ which is equivalent to an

<footnote>$^7$Further details on this calibration can be found in Appendix B.</footnote>
annual value of $\frac{2}{3}$. Those values are the observed ratios in the U.S. as reported by Aiyagari and McGrattan (1998).

# 3 Results

We now present the results obtained with our benchmark economy. A first section reports the aggregate behavior of the model. A second examines the long-run welfare effects of public debt in an aggregate fluctuations setting. In a third, we move on to the business cycle and distributional effects of public debt. In the next to last section we compare the benchmark economy to simpler idiosyncratic risk only models. Finally, we explore an alternative calibration that changes the labor market process.

## 3.1 Public debt in an aggregate fluctuations setting

<figure><img src="image_5.png" /><figcaption>FIG. 1. Aggregate behavior of the benchmark model</figcaption></figure>

We start by discussing the aggregate behavior of our benchmark model. Our computations are reported in Figure 1. Increasing the level of public debt increases the supply of safe assets in the economy. Consequently, the before tax interest rates increases. Because the repayment of debt interests are higher,
the income tax rate increases. Nevertheless, the after tax interest rate unambiguously increases. In turn, public debt has a crowding out effect on private capital : higher levels of debt decrease the aggregate amount of private capital in the economy. The crowding out of capital induces the observed decline in output<fnref n="8" />. However, the decline in physical capital is smaller than the increase in public debt. The increase in the after-tax interest rate reduces the gap between the after-tax interest rate and the rate of time preference. The cost of postponing consumption to build up a buffer stock of saving is then reduced. Households choose to hold more assets at the steady state equilibrium. That is why the overall wealth level A, which is the combination of private capital and public debt, is higher.

## 3.2 Welfare analysis and optimal level of debt

The welfare analysis we conduct below apply to the long run optimal level of public debt with aggregate fluctuations. As there is no formal steady-state in a model with aggregate fluctuations, we consider for our welfare analysis the values of aggregate variables averaged over long periods of time for a given debt to GDP ratio<fnref n="9" />. We define the optimal level of public debt as the debt over GDP ratio that maximizes the traditional utilitarian welfare criterion μ. As explained in Lucas (1987) and Mukoyama and Sahin (2006), this criterion measures the amount of consumption that one would have to remove or add in order to make the agent indifferent between the benchmark debt over GDP ratio and some other level of public debt. It verifies :

$$ \mathbb{E}_0 \left[ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) \log((1 + \mu) c_t^{\text{bench.}}) \right] = E_0 \left[ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) \log(c_t) \right] $$

with {$c_t^{\text{bench.}}$}$_{t=0}^\infty$ the consumption stream in the benchmark model when the debt over GDP ratio is equal to $\frac{8}{3}$. {$c_t$}$_{t=0}^\infty$ is the consumption stream when the debt over GDP ratio is some other level than the benchmark level. For logarithmic utility we can show that :

$$ \mu = \exp \left( \left[ V - V^{\text{bench.}} \right] / S \right) - 1, $$

where $V^{\text{bench.}} = E_0 \left[ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) \log(c_t^{\text{bench.}}) \right]$, $V = E_0 \left[ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) \log(c_t) \right]$ and

<footnote n="8">⁸ the increase in wealth would have been the same as the increase in public debt. The steady state consumption would have been higher.</footnote>

<footnote n="9">⁹ More details can be found in Appendix A.</footnote>
$$S = \mathbb{E}_0 \left[ \sum_{t=0}^{\infty} \left( \prod_{j=0}^{t} \beta_j \right) \right].$$

The result of the introduction of public debt on welfare is a priori undetermined because of two opposing effects. The first effect is a *crowding out effect*. The crowding out of physical capital clearly reduces consumption and then welfare. Moreover the increase in the income tax rate tends to amplify the negative impact of public debt on welfare. The second effect is a *cost of precautionary saving effect*: the increase in the after-tax interest rate makes it less costly to accumulate precautionary saving in order to smooth consumption as the interest rate gets closer to the time preference rate. This second effect is welfare enhancing<fnref n="10" />. It is difficult to predict which effect overcomes the other analytically.

Figure 2 depicts the long-run optimal level of debt in the benchmark economy. In a setting embedding aggregate risk and calibrated on the U.S. economy, the optimal public debt level is 5% of output on an annual basis. Aggregate fluctuations, idiosyncratic risk and credit constraints lead agents to engage in precautionary saving in order to smooth consumption. Without public debt, the cost of precautionary saving is higher because when households accumulate, the interest rate lowers. Any level of public debt, raises the interest rate and reduces the cost of precautionary saving. As a result welfare is enhanced. However any level of public debt also crowds out private capital and reduces welfare. Here, a lower level of debt than the benchmark level increases welfare out of reducing the crowding out of private capital. At the same time, a lower level of debt than the benchmark level decreases welfare out of increasing the cost of precautionary saving. As long as the first effect dominates the second, a lower level of debt is optimal. At the optimal level, one effect exactly balances the other. For a debt over GDP ratio smaller than 5%, the consumption loss out of increasing the cost of precautionary saving is higher than the consumption gain out of reducing the crowding out of private capital.

As illustrated in Table 2, going from the benchmark level of debt to the optimal level of debt is welfare enhancing. The consumption gain of being at the optimal level of debt instead of the benchmark level is 0.257%. On the contrary, it is welfare decreasing to go to a higher level of public debt than the benchmark level. In an economy where the level of debt is 75%, the consumption loss not to be at the benchmark level (resp. optimal level) would be on average 0.072% (resp. 0.329%). These results suggest that the role the interest rate plays in reducing the cost of precautionary saving is central to understand

<footnote n="10">¹⁰ Woodford (1990) argues that welfare can be enhanced if the interest rates are kept high enough, that is, closer to time preference rates in a liquidity-constrained economy. We find the same effect here.</footnote>
<figure><img src="image_6.png" /><figcaption>FIG. 2. Welfare gain versus debt/GDP ratio in the benchmark model</figcaption></figure>

TAB. 2

Consumption gain (%) of going to the optimal level of debt in the benchmark economy

<table><thead><tr><th rowspan="2">Population</th><th colspan="3">Business Cycle</th></tr><tr><th>Average</th><th>Recessions</th><th>Booms</th></tr></thead><tbody><tr><td>All</td><td>0.257</td><td>0.267</td><td>0.247</td></tr><tr><td>Bottom 10%</td><td>0.885</td><td>0.901</td><td>0.868</td></tr><tr><td>Top 10%</td><td>-10.401</td><td>-10.535</td><td>-10.264</td></tr></tbody></table>

why agents settle for the optimal level of public debt. If we block any price movements by considering a price fixed small open economy, it appears that there is no consumption gains or losses out of the *cost of precautionary saving* effect. In the case where all prices are set to their average benchmark level for any level of public debt, there is no consumption loss out of increasing the cost of precautionary saving for a lower level of public debt than the benchmark level. In this case, there are only consumption gains out of reducing the crowding out of private capital.

### 3.3 Business cycles and distributional effects

In this section we move on to the effects of public debt along the cycle and across the distribution of the population. To generate statistics along the cycle, we compute separately the average value of our aggregate variables either when the economy is in boom or when it is in recession over the whole sequence of simulation. The last two columns of Table 2 illustrate the consumption gain or loss of being at the optimal level of debt instead of the benchmark level. In recession, the consumption gain of being at the optimal level of debt
is higher than on average. On the contrary, in expansion this consumption gain is lower. This is due to a *price fluctuation effect*: in recessions, the cost of postponing consumption to build up a buffer stock of savings is higher because the interest rate is lower as shown in Table 3. In expansion, we have the opposite: interest rates are higher and the cost of precautionary saving is lower. The fluctuation of prices is not the only relevant effect we need to account for to explain the optimal level of public debt. When we take aggregate risk into consideration, an *employment fluctuation effect* modifies the precautionary saving motive. Along the cycle, the unemployment rate and duration increase in recessions and reduce in booms. In recessions, the precautionary motive becomes stronger: employed agents face a higher risk of losing their job and unemployed agents find it more difficult to find a job. In recessions agents want to save more for precautionary motives because of the *employment fluctuation effect* but at the same time the *price fluctuation effect* raises the cost of saving. Thus public debt helps reducing the cost of precautionary saving in recessions. In booms, agents want to save less because of the *employment fluctuation effect*. At the same time the *price fluctuation effect* makes it less costly to save. Thus public debt is less useful in booms.

<caption>TAB. 3</caption>

Macro variables along the cycle in the benchmark economy

<table><thead><tr><th rowspan="3">Statistics</th><th colspan="6">Level of debt (% of output)</th></tr><tr><th colspan="3">66</th><th colspan="3">5</th></tr><tr><th>Booms</th><th>Average</th><th>Recessions</th><th>Booms</th><th>Average</th><th>Recessions</th></tr></thead><tbody><tr><td>Agg. Capital</td><td>12.23</td><td>12.17</td><td>12.10</td><td>12.58</td><td>12.52</td><td>12.46</td></tr><tr><td>Output</td><td>1.18</td><td>1.14</td><td>1.11</td><td>1.20</td><td>1.16</td><td>1.12</td></tr><tr><td>Before Tax Interest Rate (%)</td><td>0.99</td><td>0.90</td><td>0.80</td><td>0.93</td><td>0.83</td><td>0.74</td></tr><tr><td>Before Tax Wage</td><td>0.7902</td><td>0.7900</td><td>0.7899</td><td>0.7982</td><td>0.7981</td><td>0.7980</td></tr></tbody></table>

The overall consumption gains or losses of a change in the level of public debt is shared very differently in the population. To show that we decompose the welfare gains of going from the benchmark level of public debt to the optimal level across the population. The last two rows of Table 2 show this decomposition. For instance, the row *Bottom 10%* shows the welfare gap between the 10% least fortunate people living in an economy with the benchmark level of public debt and those living in an economy with the optimal level of public debt. This decomposition closely matches a decomposition by wealth levels as the lowest (resp. highest) expected utilities refer to people who have experienced the highest unemployment (resp. employment) spells and who end up with the lowest (resp. highest) level of assets. The poorest agents are better off with a lower level of public debt than the optimal level. Going from the benchmark level of debt to the optimal debt level leads to an increase in consumption of 0.885% for the poorest 10% of the population. In the meantime, the richest 10% of the population would lose as much as 10.401% of consumption. This is explained by the fact that rich people's income is mainly capital income
whereas poor people's income is mostly labor income. Thus when the level of public debt is higher, poor people suffers from the reduction in output caused by higher tax rates, lower wages and crowding out of capital. On the contrary, as interest rates raise with higher level of public debt, rich people are better off. The same type of effects are discussed in Ball and Mankiw (1995) and Floden (2001).

## 3.4 Optimal level of debt without aggregate fluctuations

In this section we look at setups without aggregate fluctuations and compare them with our benchmark economy. There are several ways of eliminating business cycles<fnref n="11" /> to derive a comparable long run idiosyncratic model. We look at three methods.

### 3.4.1 Imrohoroglu (1989) method

In the spirit of Lucas (1987) we replace the aggregate stochastic process with its conditional mean. We then follow Imrohoroglu (1989) to derive the labor market process. The transition probabilities of this economy are set so that the average rate of unemployment and the average duration of unemployment are the same between this economy and the benchmark economy. All other calibrated parameters are kept to their benchmark values, especially time preference rates and the risk aversion parameter <fnref n="12" />. This model is similar to the benchmark model in many ways: higher levels of debt raise interest rates and crowd out private capital, overall wealth increases with debt and taxes are higher. But as the risk faced by agents is lower, agents save less and interest rates are higher in this model.

Figure 3 (top) depicts the welfare profile we find in this idiosyncratic risk model. The optimal level of public debt is 2.5% of output on an annual basis. This level is lower than the level we found in the benchmark economy. This is because there is no *price fluctuation effect* and *employment fluctuation effect* in this economy. In the benchmark economy, as the unemployment rate and the unemployment duration increase during recessions, the precautionary motive is stronger. Moreover, in the benchmark economy the interest rate and the physical capital are smaller in recessions. Therefore, it is more costly to save for precautionary motive. That is why the need for public debt is more important in the benchmark economy. In the idiosyncratic model the risk is lower because

<footnote n="11">¹¹ For a survey, see Barlevy (2004).</footnote>

<footnote n="12">¹² For greater details on the models without aggregate risk and their calibration, see appendix C.</footnote>
<figure><img src="image_7.png" /><figcaption>FIG. 3. Welfare gain versus debt/GDP ratio in idiosyncratic models</figcaption></figure>

agents are subject to a less risky labor market and at the same time they no
longer face price fluctuations. The precautionary motive is weaker here than
in the benchmark model and agents save less. As a consequence interest rates
are at a higher level in this economy.

3.4.2 *Brute-force averaging*

This method is related to the first one above. We again set the aggregate pro-
cess to its conditional mean. The transition probabilities on the labor markets
are generated as follows. We take the exact same transition process as in the
benchmark economy and repeat it over very long periods of time. Thus we
create a history of transition on the labor market conditional on the aggre-
gate risk faced by agents. In the end of this simulation, we are left with a
sequence of aggregate shocks and corresponding labor market situations for
individuals. We then brute-force average the sequence of labor market transi-
tions out. To do so, we count the numbers of transitions from unemployed to
unemployed, from unemployed to employed, from employed to employed and
from employed to unemployed over the entire simulated sequence and then take the mean. As we simulate over very long periods of time, we respect the law of large numbers. After brute-force averaging over the simulated series, we are left with a transition matrix on the labor market. The matrix we get by brute-force averaging is very similar to the matrix obtained by the Imro-horoglu method above. Figure 3 (middle) depicts the welfare profile we find in this idiosyncratic risk model. The optimal level of public debt is 2.5% of output on an annual basis. We can characterize this optimal level in the same way as we did above.

### 3.4.3 Conditional approach

In this section we consider an alternative method of removing business cycles to obtain a long-term idiosyncratic model by conditioning on the aggregate state. First, as above, we equate the aggregate process to its conditional mean. To obtain the labor market process, we condition on the aggregate state. For instance, we compute the probability of being employed tomorrow conditional on being employed today $\pi_{ee}$ in the idiosyncratic model as follows :

$$ \pi_{ee} = \Pr(z = z_g|s = e) * (\Pi_{bgee} + \Pi_{ggee}) + (1 - \Pr(z = z_g|s = e)) * (\Pi_{gbee} + \Pi_{bbee}), $$

$$ \text{where for instance } \Pi_{ggee} = \Pr(z' = z_g, s' = e|z = z_g, s = e). $$

This means that to find who will remain employed in the next period conditional on being employed today, we consider all occurrences of being employed tomorrow conditional on being employed today in every aggregate state of the benchmark calibration and we weight them accordingly by the probability of being in either good or bad state conditional on being employed. We apply the same method to derive $\pi_{uu}$. The remaining probabilities are found by applying the raws summing to unity property of the idiosyncratic transition matrix. This method of removing aggregate risk yield a more persistent labor process than the methods above. Notably, the probability of remaining unemployed tomorrow conditional on being unemployed today is higher. Thus the unemployment rate is not equal to its average value in the benchmark model. Here the unemployment rate is higher and amounts to 7.33% instead of an average value of 7% in the benchmark model.

We find that the optimal level of debt is higher here than with the methods above. The annual optimal debt to GDP ratio is only slightly lower than 5% and is almost identical to the optimal level of debt found is the benchmark model with aggregate risk. This result is depicted in Figure 3 (bottom). It is straightforward to see why the optimal level of debt is higher with this
methods. As both the persistence of the labor market process and the unem-
ployment rate are higher, public debt plays a greater role in reducing the cost
of precautionary saving here than with the methods above. We illustrate this
in the next section with the benchmark economy.

3.5 *Higher unemployment rate and longer unemployment spells*

Our benchmark calibration reflects the behavior of the U.S. labor market. In this section we consider an alternative calibration of the labor market, that is a step towards reproducing European labor market features although we do not take into consideration or reproduce the employment benefit system found in Europe. We follow the methodology used in Algan and Allais (2004) and the data set of Blanchard and Wolfer (2000). We only modify the labor market features and leave the rest of the calibration unchanged. We fix the unemployment rate to be 13% in recessions and 7% in booms. We also set the duration of an unemployment spell to be 6 quarters in recessions and 4 quarters in booms. We now find the optimal level of debt to be 30% on an annual basis as depicted in Figure 4. Longer unemployment spells and higher unemployment rates tend to raise the optimal level of debt.

<figure><img src="image_8.png" /><figcaption>FIG. 4. Welfare gain versus debt/GDP ratio in the model with higher unemployment rate and longer unemployment spells.</figcaption></figure>

This calibration purposely strengthens the *employment fluctuation effect* :
unemployed agents have a harder time finding a job in this economy when
compared to the benchmark economy and employed agents face a higher risk of
loosing their jobs. In recessions, this is amplified. The precautionary motive is
stronger here than in the benchmark economy. The harder it is for households
to smooth consumption and the higher is the need for public debt. Here a higher level of public debt is needed to help households effectively smooth consumption.

# 4 Conclusion

This paper reconsidered the optimal level of public debt in an environment of aggregate fluctuations. Our benchmark model calibrated on the U.S. economy finds that a positive public debt level of 5% of output on a annual basis is optimal. This level is generally higher than in an economy without aggregate fluctuations. Our benchmark economy shows that in an aggregate fluctuations setting, households are subject to an *employment fluctuation effect* and to a *price fluctuation effect*. These effects make the precautionary saving motive stronger and at the same time the cost of saving higher. Our results suggest that the higher supply of safe assets induced by higher levels of public debt tends to raise the interest rate. This helps households to reduce the cost of precautionary saving and make smoothing of consumption easier. As a result there are welfare gains in the economy. However for the poorest agents, higher levels of debt are not optimal. Poor agents rely mainly on labor income and are dependent on higher wages and lower taxes. Public debt decreases wages and increases tax level thus poor agents prefer lower levels of public debt. We also emphasize that longer unemployment spells tend to raise the optimal level of public debt.
# Appendix

## A Computational strategy

We solve the model using the methodology developed by Den Haan (1996) and Krusell and Smith (1998). They show that agents need only a restrictive set of statistics about the wealth distribution to determine prices. This set includes the mean of the wealth distribution and the aggregate productivity shock. A linear prediction rule based only on the average level of capital provides an accurate prediction. This result comes from the near linearity of the decision rule $a'(a, s, \beta; z, \Gamma)$. As the aggregate capital stock is mainly held by rich people who have approximately the same propensity to save, next period's aggregate capital is accurately predicted by current period's aggregate capital. In our model, we assume the following law of motion:

$$ \log(\bar{K}') = a_0 + a_1 z + a_2 \log(\bar{K}) $$

where $\bar{K}'$ and $\bar{K}$ denote respectively the average stock of capital of the next period and of the current period. Thus the strategy is the following:

(Step 1) : Given a set of parameter values $(a_0, a_1, a_2)$ for the law of motion, we solve the individual problem. To solve the individual problem, we iterate on the Euler equation:

$$ U'(c) = E[\beta'U'(c')(1 + r'(z', \Gamma')(1 - \tau))/s, \beta; z, \Gamma], $$

on a discrete grid until a fix point is found. When the borrowing constraint bind, the solution can be deduced from the budget constraint.

(Step 2) : Given the parameter values for individual decision rules, we solve the aggregate problem i.e. the coefficients of the law of motion.

(Step 3) : If the parameters $(a_0, a_1, a_2)$ found are close to the parameter values used to solve (Step 1), the algorithm has converged. Otherwise (Step 1), (Step 2) and (Step 3) are repeated until convergence.

(Step 4) : Because of the aggregate fluctuations property and the government budget constraint, on more step is necessary. As there is no formal steady state in an economy with aggregate fluctuations, we approximate a *pseudo steady state* by averaging aggregate variables over long periods of time. We first make a guess for the interest rate in this *pseudo steady state*, and with the long run budget constraint of the government, we derive a valid tax rate
for this pseudo steady state. Also, the amount of public debt in the economy is defined according to this pseudo steady state by taking the appropriate ratio of the GDP in the pseudo steady state. Given the tax rate we just defined, we execute the above steps 1, 2 and 3. When those steps have converged, we update our guess of the interest rate in the pseudo steady state until a fix point is found.

When step 4 is complete, the pseudo steady state interest rate coincide with the average interest rate in the actual economy with aggregate fluctuations, the government budget constraint is balanced and the lump sum transfer to individuals amounts to zero at the equilibrium.

## B Calibration

We now show how we derived the transition matrices for aggregate state changes ($\eta$), for joint transition between aggregate states and labor market statuses (Π) and for discount factor changes ($\Upsilon$).

### B.1 Aggregate state change transition matrix

To deduce the aggregate state change transition matrix $\eta$, we solve the following system :

$$ \left. \begin{array}{l} \eta_{gg} = \eta_{bb} \\ \eta_{bg} = \eta_{gb} \\ \eta_{gg} + \eta_{gb} = 1 \\ \eta_{bb} + \eta_{bg} = 1 \\ \eta_{bg} = \frac{1}{8} \end{array} \right\} \Rightarrow \begin{pmatrix} 0.875 & 0.125 \\ 0.125 & 0.875 \end{pmatrix} $$

As we assumed that the duration of a boom or a recession is the same, we deduce the two first equations. Moreover, the duration of a cycle is set to 8 quarters, it follows that $\eta_{bg} = Pr(z_{t+1} = g/z_t = b) = \frac{1}{8}$ and $\eta_{gb} = Pr(z_{t+1} = b/z_t = g) = \frac{1}{8}$.
## B.2 Matrix for joint transition between aggregate states and labor market statuses

The determination of the matrix Π that describes the transition between unemployment and employment requires the identification of the aggregate shock (whether we are in a recession or in a boom). The transition matrix Π is built thanks to the matrix η and to the transition matrixes Π<sup>gg</sup>, Π<sup>bb</sup>, Π<sup>gb</sup> and Π<sup>bg</sup>. Π verifies :

$$ \Pi = \begin{pmatrix} \eta_{bb} \Pi^{bb} & \eta_{bg} \Pi^{bg} \\ \eta_{gb} \Pi^{gb} & \eta_{gg} \Pi^{gg} \end{pmatrix} $$

We assumed that in recessions the duration of the unemployment, that we note *durub*, amounts to 2.5 quarters and the unemployment rate $u_b$ is set to 10%. In booms, the duration of unemployment, *durug*, is equal to 1.5 quarters and the unemployment rate, $u_g$, is set to 4%. From this information, we can deduce the matrices Π<sup>gg</sup> and Π<sup>bb</sup>.

The transition matrix Π<sup>gg</sup> corresponds to the case $(z, z') = (g, g)$. It verifies :

$$ \Pi^{gg} = \begin{pmatrix} \Pi_{uu}^{gg} & \Pi_{ue}^{gg} \\ \Pi_{eu}^{gg} & \Pi_{ee}^{gg} \end{pmatrix} $$

Solving the system below gives the values of Π<sup>gg</sup><sub>ee</sub>, Π<sup>gg</sup><sub>eu</sub>, Π<sup>gg</sup><sub>ue</sub> and Π<sup>gg</sup><sub>uu</sub> :

$$ \left\{ \begin{array}{l} \Pi_{ee}^{gg} + \Pi_{eu}^{gg} = 1 \\ \Pi_{ue}^{gg} + \Pi_{uu}^{gg} = 1 \\ \Pi_{ue}^{gg} = \frac{1}{durug} \\ \Pi_{ee}^{gg} = 1 - \frac{u_g \Psi_{ue}^{gg}}{1 - u_g} \end{array} \right. \implies \Pi^{gg} = \begin{pmatrix} 0.3333 & 0.6667 \\ 0.0278 & 0.9722 \end{pmatrix} $$

The transition matrix Π<sup>bb</sup> corresponds to the case $(z, z') = (b, b)$. It verifies :

$$ \Pi^{bb} = \begin{pmatrix} \Pi_{uu}^{bb} & \Pi_{ue}^{bb} \\ \Pi_{eu}^{bb} & \Pi_{ee}^{bb} \end{pmatrix} $$

Solving the system below gives the values of Π<sup>bb</sup><sub>ee</sub>, Π<sup>bb</sup><sub>eu</sub>, Π<sup>bb</sup><sub>ue</sub> and Π<sup>bb</sup><sub>uu</sub> :
$$
\left\{
\begin{array}{l}
\Pi_{ee}^{bb} + \Pi_{eu}^{bb} = 1 \\
\Pi_{ue}^{bb} + \Pi_{uu}^{bb} = 1 \\
\Pi_{ue}^{bb} = \frac{1}{duryb} \\
\Pi_{ee}^{bb} = 1 - \frac{u_b \Pi_{ue}^{bb}}{1 - u_b}
\end{array}
\right.
\quad \Rightarrow \quad
\Pi^{bb} = \begin{pmatrix}
0.6 & 0.4 \\
0.0445 & 0.9555
\end{pmatrix}
$$

When the cycle changes the unemployment rate changes. The transitions bet-
ween unemployment and employment get modified. We make the same as-
sumptions as Krusell and Smith (1998) :

$$
\begin{cases}
\Pi_{uu}^{bg} = \text{Pr}(\epsilon_{t+1} = u^g / \epsilon_t = u^b) = 0.75\Pi_{uu}^{gg} \\
\Pi_{uu}^{gb} = \text{Pr}(\epsilon_{t+1} = u^g / \epsilon_t = u^b) = 1.25\Pi_{uu}^{bb}
\end{cases}
$$

The probability to remain unemployed when the next period is a recession (resp. boom), increases (resp. decreases) since by assumption the unemployment rate is higher in recession than in boom.

The transition matrix $\Pi^{bg}$ corresponds to the case $(z, z') = (b, g)$. It verifies :

$$
\Pi^{bg} = \begin{pmatrix} \Pi_{uu}^{bg} & \Pi_{ue}^{bg} \\ \Pi_{eu}^{bg} & \Pi_{ee}^{bg} \end{pmatrix}
$$

The system below gives us $\Pi_{ee}^{bg}$, $\Pi_{eu}^{bg}$, $\Pi_{ue}^{bg}$ and $\Pi_{uu}^{bg}$ :

$$
\left\{
\begin{array}{l}
\Pi_{ee}^{bg} + \Pi_{eu}^{bg} = 1 \\
\Pi_{ue}^{bg} + \Pi_{uu}^{bg} = 1 \\
\Pi_{uu}^{bg} = 0.75\Pi_{ue}^{bg} \\
\Pi_{ee}^{bg} = \frac{((1-u_g) - u_b \Pi_{ue}^{bg})}{1-u_b}
\end{array}
\right.
\quad \Rightarrow \quad
\Pi^{bg} = \begin{pmatrix} 0.25 & 0.75 \\ 0.0167 & 0.9833 \end{pmatrix}
$$

The transition matrix $\Pi^{gb}$ corresponds to the case $(z, z') = (g, b)$.

$$
\mathbf{\Pi}^{gb} = \begin{pmatrix} \Pi_{ee}^{gb} & \Pi_{eu}^{gb} \\ \Pi_{ue}^{gb} & \Pi_{uu}^{gb} \end{pmatrix}
$$

The system below gives us $\Pi_{ee}^{gb}$, $\Pi_{eu}^{gb}$, $\Pi_{ue}^{gb}$ and $\Pi_{uu}^{gb}$ :
$$
\left\{
\begin{array}{l}
\Pi_{ee}^{gb} + \Pi_{eu}^{gb} = 1 \\
\Pi_{ue}^{gb} + \Pi_{uu}^{gb} = 1 \\
\Pi_{uu}^{gb} = 1.25\Pi_{uu}^{bb} \\
\Pi_{ee}^{gb} = \frac{((1 - u_b) - u_g\Pi_{ue}^{gb})}{1 - u_g}
\end{array}
\right.
\implies \Pi^{gb} = \begin{pmatrix} 0.75 & 0.25 \\ 0.0729 & 0.9271 \end{pmatrix}
$$

B.3 Matrix for discount factor changes

We assumed that the discount factors follow a three-states first-order Markov
process. Therefore, the matrix describing the transition from the discount
factor $\beta_i$ to the discount factor $\beta_j$ is the following :

$$
\Upsilon = \begin{pmatrix} \Upsilon_{ll} & \Upsilon_{lm} & \Upsilon_{lh} \\ \Upsilon_{ml} & \Upsilon_{mm} & \Upsilon_{mh} \\ \Upsilon_{hl} & \Upsilon_{hm} & \Upsilon_{hh} \end{pmatrix}
$$

As we assumed that there is no immediate transition between $\beta_l$ and $\beta_h$ as
in Krusell and Smith (1998), it involves that $\Upsilon_{lh} = \Upsilon_{hl} = 0$. Moreover, as
we set the duration of the extreme states ($\beta_l$ and $\beta_h$) to 50 years namely 200
quarters, we have $\Upsilon_{lm} = \frac{1}{200} = \Upsilon_{hm}$. Solving the following system gives us the
transition matrix $\Upsilon$ :

$$
\left.
\begin{array}{l}
\Upsilon_{ll} + \Upsilon_{lm} + \Upsilon_{lh} = 1 \\
\Upsilon_{ml} + \Upsilon_{mm} + \Upsilon_{mh} = 1 \\
\Upsilon_{hl} + \Upsilon_{hm} + \Upsilon_{hh} = 1 \\
\Upsilon_{lh} = \Upsilon_{hl} = 0 \\
\Upsilon_{lm} = \frac{1}{200} = \Upsilon_{hm} \\
\Upsilon_{ml} = \frac{\Pr(\beta_t = \beta_l)\Upsilon_{lm}}{\Pr(\beta_t = \beta_m)} \\
\Upsilon_{mh} = \frac{\Pr(\beta_t = \beta_h)\Upsilon_{hm}}{\Pr(\beta_t = \beta_m)}
\end{array}
\right\}
\Rightarrow
\Upsilon =
\begin{pmatrix}
0.995 & 0.005 & 0 \\
0.0007 & 0.9979 & 0.0014 \\
0 & 0.005 & 0.995
\end{pmatrix}
$$
## C Model without aggregate risk

We now briefly detail the model without aggregate risk and its calibration. Most of this model is similar to the benchmark model, thus we underline only differences.

### Model

As this model serves comparison purposes most of the benchmark assumptions remain unchanged. The assumptions about the representative firm are similar with the exception of the technical progress $z$. In the absence of aggregate risk, $z$ is fixed to its average value. In the absence of aggregate risk, the budget constraint of the government is:

$$G_t + r_t B_t = B_{t+1} - B_t + T_t$$

Household preferences are unchanged. In the absence of aggregate risk, the matrix that describes the transition on the labor market becomes:

$$\pi = \begin{pmatrix} \pi_{uu} & \pi_{ue} \\ \pi_{eu} & \pi_{ee} \end{pmatrix}$$

with $\pi_{uu} = \text{Pr}(s_{t+1} = u|s_t = u)$. When there is no aggregate risk, it is no longer necessary to distinguish the nature of the cycle. In the absence of aggregate risk, the state variables are summarized by the vector $(a, s, \beta)$. The program the household solves is:

$$v(a, s, \beta) = \max_{c, a'} \{u(c) + \beta E[v(a', s', \beta')|(s, \beta)]\} \quad (C.1)$$

subject to:

$$c + a' = (1 + r(1 - \tau))a + w\chi(s) \quad (C.2)$$

$$c \ge 0 \quad (C.3)$$

$$a' \ge 0 \quad (C.4)$$

with

$$\chi(s) = \begin{cases} \theta & \text{if } s = u, \\ (1 - \tau) & \text{if } s = e \end{cases}$$
## Equilibrium

The recursive equilibrium consists of a set of decision rules for consumption and asset holding {$c(a, s, \beta), a'(a, s, \beta); z, \Gamma$}, aggregate capital K, factor prices {$r, w$}, tax rate $\tau$ satisfying these conditions :

(1) Given the prices {$r, w$}, the decision rules {$c(a, s, \beta), a'(a, \epsilon, \beta)$} solve the dynamic programming problem (C.1) subject to the constraints (C.2), (C.3) and (C.4)

(2) Market price arrangements are :

$$r = \alpha z K^{\alpha-1} N^{1-\alpha} - \delta \\ w = (1-\alpha) z K^{\alpha} N^{-\alpha}$$

(3) Government budget constraint is balanced.

(4) Capital Market clears when :

$$K + B = \int a'(a, \epsilon, \beta) d\Gamma(a, s, \beta)$$

with $\Gamma(a, s, \beta)$ the distribution of agents over asset holdings, employment status and preference.

## Calibration

We present here the calibration strategy in an economy without aggregate risk. The calibration of the preferences, the discount factors and the behavior of the government are unmodified. The calibration of z and the characteristics of the labor market differ in the absence of aggregate risk. In an economy without aggregate risk, z is constant and set to the average value of the aggregate shock, namely the unit value. As we consider three alternative calibration of the idiosyncratic labor process, we detail them sequentially.

In the Imrohoroglu method, the unemployment rate u and the unemployment spells duru are respectively set to 7% and 2 quarters (we average over good and bad periods in the benchmark model). These two assumptions define the transition matrix $\pi$ :
$$ 
\left\{ 
\begin{array}{l} 
\pi_{uu} + \pi_{ue} = 1 \\ 
\pi_{eu} + \pi_{ee} = 1 \\ 
\pi_{ue} = \frac{1}{duru} \\ 
\pi_{ee} = 1 - \frac{u\pi_{ue}}{1-u} 
\end{array} 
\right. 
\Rightarrow 
\begin{pmatrix} 
0.5 & 0.5 \\ 
0.0376 & 0.9624 
\end{pmatrix} 
$$

In the brute-force averaging method, we first simulate the transition process and find the transition probabilities. This yields the following transition matrix:

$$ \begin{pmatrix} \pi_{uu} & \pi_{ue} \\ \pi_{eu} & \pi_{ee} \end{pmatrix} = \begin{pmatrix} 0.50749 & 0.49251 \\ 0.03708 & 0.96292 \end{pmatrix} $$

With this matrix, we derive an unemployment rate of 7%.

In the conditional approach, we use our definitions of transition probabilities to derive the following transition matrix:

$$ \begin{pmatrix} \pi_{uu} & \pi_{ue} \\ \pi_{eu} & \pi_{ee} \end{pmatrix} = \begin{pmatrix} 0.53423 & 0.46577 \\ 0.03685 & 0.96315 \end{pmatrix} $$

With this matrix, we derive an unemployment rate of 7.33%.
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