Public debt and aggregate risk

Transcription

1 Public debt and aggregate risk Audrey Desbonnet, LEDa, Université Paris-Dauphine, Sumudu Kankanamge, Toulouse School of Economics (GREMAQ). First draft: March 2006, This draft: April Abstract In this paper, we investigate the importance of aggregate fluctuations for the assessment of the optimal level of public debt in an incomplete markets economy. In our setup, households are subject to aggregate risk correlated with uninsurable idiosyncratic risk. Our main finding is that the cyclical behavior of the economy is a quantitatively important determinant of the optimal level of public debt. We report that in an economy calibrated on the US business cycle and labor market, the optimal level of public debt is 5% of GDP on an annual basis. We alter business cycle characteristics and describe the impact on the optimal level of public debt. We also underline that distributional and business cycle analysis in our economy reveal large welfare variations that contrasts with average welfare gains of a public debt policy change. Keywords : public debt, aggregate risk, precautionary saving, credit constraints. JEL classification: H60, H30, E32, E60. 1 Introduction In this paper, we investigate the importance of aggregate fluctuations for the assessment of the optimal level of public debt. The setting used to conduct this analysis is an incomplete markets economy where agents are subject to both uninsurable idiosyncratic risk and aggregate risk. Incomplete markets models in which agents face uninsurable income risk have been used to assess to what extent the government should issue public debt (e.g. Aiyagari and McGrattan (1998) or Floden (2001)). However, to the best of our knowledge, there has been surprisingly no attempt to analyze the impact of aggregate risk on the optimal level of public debt in such a non-ricardian setting so far. In this paper, we quantify the optimal level of public debt and conduct detailed distributional and business cycle effects analysis in an economy targeted to reproduce several features of the US wealth distribution. Corresponding author. Address: Toulouse School of Economics - GREMAQ - Manufacture des Tabacs - 21, allée de Brienne, Toulouse - France. sumudu.kankanamge@tse-fr.eu. Tel: +33 (0)

2 We use a minimal setup to address the issues raised above. Following the Bewley (1986)-Huggett (1993)-Aiyagari (1994) class of models, we build a production economy with capital market imperfections where a large number of ex-ante identical infinitelylived agents face idiosyncratic income shocks and aggregate productivity shocks. Aggregate fluctuations are introduced following the approach of Den Haan (1996) and Krusell and Smith (1998). Households saving behavior is influenced by a precautionary saving motive that helps to smooth consumption and borrowing constraints. Public debt and private capital assets can be claimed to insure against future risk. The model generates an endogenous distribution of asset holdings and a preference heterogeneity setting is also introduced to help produce a realistic wealth distribution. Government levies proportional taxes on households and issues public debt to finance its expenses. A key feature of our model is the cyclical behavior of the economy and its correlation with the idiosyncratic labor market risk. The literature that has reexamined the welfare cost of business cycles in models that feature both aggregate and idiosyncratic risk reveals that aggregate risk can have asymmetric welfare effects across the population 1. In such a setup, the welfare cost of aggregate risk is much greater than in the seminal paper of Lucas (1987). Moreover, Storesletten et al. (2004) report that idiosyncratic risk is strongly countercyclical: the variance of idiosyncratic labor income risk is much greater in recessions than in expansions. As aggregate risk exacerbates uninsurable idiosyncratic income risk, the motive for precautionary saving is likely to increase. Moreover, because saving decisions depend on prices, the cost of precautionary saving changes as prices fluctuate along the cycle. Both the cost and the motive for precautionary saving impact the optimal level of public debt and their interplay with the business cycle might have been underestimated by the previous literature. The main finding of the paper is that the cyclical behavior of the economy is a quantitatively important determinant of the optimal level of public debt. With our benchmark economy that follows a quarterly calibration based on the US business cycle and labor market we find that the optimal level of public debt is 20% of GDP on a quarterly basis (5% in annual equivalent). A policy that would change the debt to GDP ratio from its postwar US average 2 to this optimal level would lead to a consumption gain of 0.253%. The welfare profile of our model for a wide range of debt to GDP ratios is not as flat as what is reported in previous quantifications such as Aiyagari and McGrattan (1998). Altering the features of the business cycle lets us measure the impact on the optimal level of public debt. For instance, a symmetric 40% (percent change) increase in the duration of an unemployment spell in the cycle leads to an optimal debt to GDP ratio of 42.5% on a quarterly basis. A symmetric 40% (percent change) increase in the unemployment rate in the cycle leads to an optimal debt to GDP ratio of 45%. If we combine the previous alterations to target economies with more persistent unemployment than in the US, the optimal level of public debt shifts to 80% on a quarterly basis. We also derive a comparable model without aggregate fluctuations. In this model, the optimal level of public debt decreases to 10% of GDP on a quarterly basis. The optimal level of public debt reported in the experiments 1 See for instance Storesletten et al. (2001) or Krusell et al. (2009). 2 Aiyagari and McGrattan (1998) report that the average debt to GDP ratio in the US over the postwar period is 267% on a quarterly basis. 2

3 above is the result of opposing effects on the welfare of agents. Increasing the level of public debt crowds out private capital and raises distortionary taxes leading to a lower level of welfare. But at the same time, as the after-tax interest rate increases, the cost of precautionary saving is reduced and welfare is increased. Part of our results derive precisely from the interaction between the business cycle and this precautionary behavior of agents. We also underline that distributional and business cycle analysis in our benchmark economy reveal large variations that contrasts with the average welfare gain reported above. For instance, with a policy that would change the debt to GDP ratio from its postwar US average to the optimal level of public debt in the benchmark economy, the bottom 1% of the population would gain 2.679% of consumption whereas the top 1% would lose as much as %. This policy increases the aftertax wage and reduces the after-tax interest rate and as a result is favorable to wealth poor individuals and disadvantageous to wealth rich individuals. This example also highlights the importance of a realistic reproduction of the wealth distribution when discussing such policies. Finally, we stress that welfare gains and losses due to public debt policies differ depending on the focus on recession periods or expansion periods. A vast literature explores the implications of public debt starting from papers studying optimal taxation issues. Barro (1979) builds a public debt theory in which there is aggregate risk and a deadweight cost to tax. Lucas and Stokey (1983) characterize optimal fiscal policy in a Ramsey model where government expenditures follow a stochastic process and public debt is state contingent. Chari et al. (1994) recast the optimal taxation problem in the Lucas and Stokey (1983) model extended to allow for capital accumulation and business cycles. Aiyagari et al. (2002) revisit the result of Lucas and Stokey (1983) in a setting with only risk-free one-period debt. Finally, Farhi (2007) reconsiders the results of Aiyagari et al. (2002) in a model with capital accumulation and capital taxation. In the class of models above, households are either fully insured against idiosyncratic income risk or no such risk is considered. Therefore, these models do not take precautionary saving into account. Precautionary saving is likely to be an important source of aggregate wealth accumulation as it has been documented by Gourinchas and Parker (2002), Cagetti (2003) and Fuchs-Schundeln and Schundeln (2005). Our paper is more closely related to the literature analyzing the impact of public debt in models which depart from the representative agent assumption. In a liquidity-constrained economy, Woodford (1990) shows that raising the level of public debt may be welfare enhancing. Aiyagari and McGrattan (1998) address the question of the optimal level of public debt in an incomplete markets economy where agents face idiosyncratic labor productivity risk. They find the optimal level of public debt to be positive even though welfare gains appear to be very small. Floden (2001) extends the latter model to take public transfers into account and conduct a detailed risk sharing and welfare analysis. Desbonnet and Weitzenblum (2011) emphasize the short-run effects of an increase in public debt. In this strand of the literature, there is however no aggregate risk. Finally, Heathcote (2005) studies an economy with heterogeneous agents, idiosyncratic income risk and aggregate risk coming from the tax rate. Nevertheless, the scope of his analysis differs: he investigates the effect of a temporary tax cut financed by debt and there are no business cycles in the way it is defined in our paper. The rest of the paper is organized as follows. In the next section, we describe 3

4 our benchmark economy along with our calibration strategy and numerical solution method. Section 3 details our main results. The last section concludes. 2 The Model Our benchmark economy is a Bewley-Huggett-Aiyagari type dynamic stochastic general equilibrium model augmented to allow for aggregate fluctuations à la Den Haan (1996) or Krusell and Smith (1998) and public debt. 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 apart from the introduction of aggregate fluctuations the main differences are the following. The productivity process here is simpler as agents can only be employed or unemployed and leisure is not valued. Those two deviations greatly simplify the model with aggregate fluctuations. Then, as we want to discuss distributional effects implied by different levels of public debt, we adopt a strategy to reproduce the wealth distribution. This strategy implies a preference heterogeneity setup absent from the Aiyagari and McGrattan (1998) paper. 2.1 Firms We assume that there is a continuum of firms with a neoclassical production technology that 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, N aggregate labor and F the technology. 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 that we note 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: ( ) ηgg η η = gb η bg Finally, our setting assumes that inputs markets are competitive. The wage w and the interest rate r verify: η bb r t + δ = z t F K (K t, N t ) (1) w t = z t F N (K t, N t ) (2) 4

5 2.2 The government The government issues public debt and levies taxes to finance public expenses. Both capital and labor income are taxed proportionally at an identical rate τ. The government s budget constraint verifies: with G t + r t B t = B t+1 B t + T t T t = τ t (w t N t + r t A t ) G t is the level of public expenses, B t the level of public debt and T t tax revenues. 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. Here, we introduce the simplest setup with aggregate fluctuations and public debt. Public debt is assumed to be a one period bond. One complication that does not arise in models without aggregate fluctuations is the fact that the tax base is random and that it is, in general, not possible to exclude an exploding path for the newly contracted public debt. To overcome this difficulty, we assume that the budget constraint of the government holds in the long-run and impose a constant level of public debt corresponding to a specific debt to GDP ratio and a constant tax rate. But then, the period-by-period budget constraint of the government might not be balanced. Our strategy is thus to offset the effect of the random tax base by adjusting government expenditures. The outcome of this strategy is that, on average, government expenditures are equal to a fixed expenditures over GDP ratio but they slightly fluctuate around this ratio over the business cycle. This fluctuation is quantitatively small and government expenditures play no role in this model so that this strategy has no particular consequence 3. Moreover, we still retain some realistic features of fiscal policy and public debt. Mainly the facts that governments will try to smooth taxation across periods and should not in general be able to trade state-contingent debt. With these specifications, the period-by-period government budget constraint simplifies to the following: with G t + r t B = T t T t = τ(w t N t + r t A t ) 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: } { ( ) t V = E 0 β j u(c t ) t=0 j=0 3 It is also possible to offset the effect of the random tax base with a lump-sum transfer to the households. In our simulations, both assumptions lead to the same optimal level of public debt and quantitative results of the same order for the benchmark economy. More details on this can be found in Appendix A.2. 5 (3)

6 In the benchmark economy we assume that the utility is logarithmic. c t is household consumption and β is the discount factor. We assume that this discount factor is random so that it can differ across agents and vary over time. We specify that the stochastic process for the discount factor 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: { β0 = 1 β j 1 ]0; 1[ 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 risk. The unemployment rate and the unemployment duration are higher in recessions than in expansions. Therefore, transitions on the labor market are correlated with the aggregate state. We note Π 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: Π = Π bbuu Π bbue Π bguu Π bgue Π bbeu Π bbee Π bgeu Π bgee Π gbuu Π gbue Π gguu Π ggue Π gbeu Π gbee Π ggeu Π ggee where Π ggee = 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 θ. 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. We do not distinguish at the household level holdings of each separate asset and note overall holdings a. Therefore the recursive problem of a household is: v(a, s, β; z, Γ) = max{u(c) + βe [ v(a, s, β ; z, Γ ) (s, β; z, Γ) ] } (4) c, a subject to: c + a = (1 + r(z, Γ)(1 τ))a + χ(s) { θ if s = u, χ (s) = (1 τ)w(z, Γ) if s = e Γ = H(Γ, z, z ) c 0 a 0 The existence of aggregate risk leads us to distinguish between individual state variables and aggregate state variables. The individual state variables are given by 6

7 the vector (a, s, β). The aggregate state variables are summarized by the vector (z, Γ) where Γ(a, s, β) is a distribution of agents over asset holdings, employment status and preferences. This distribution is needed because agents use this information to predict futures prices for each possible aggregate state. Predicting the prices impose that agents use the current wealth distribution to forecast next period s aggregate capital. This is cumbersome because the wealth distribution is an infinite-dimensional object. 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). Finally, we detail the computational strategy that we use 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, β; z, Γ), a (a, s, β; z, Γ)}, aggregate capital K(z, Γ), factor prices {r(z, Γ), w(z, Γ)}, tax rate τ and a law of motion for the distribution Γ = H(Γ, z, z ) which satisfy these conditions: (i) Given the aggregate states, {z, Γ}, prices {r(z, Γ), w(z, Γ)} and the law of motion for the distribution Γ = H(Γ, z, z ), the decision rules {c(a, s, β; z, Γ), a (a, s, β; z, Γ)} solve the dynamic programming problem (4). (ii) Market price arrangements are: r(z, Γ) = zf K (K, N) δ w(z, Γ) = zf N (K, N) (iii) Government budget constraint holds. (iv) Capital market verifies: K + B = a (a, s, β; Γ, z)dγ (v) Consistency: The law of motion H is consistent with individual behavior. 2.5 Calibration The model economy is calibrated to match certain observations in the US data. We let one period in the model be one quarter in the data. 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 Technology We choose the production function to be Cobb-Douglas: Y t = z t F(K t, N t ) = z t K α t N 1 α t 0 < α < 1 Technology parameters are standard. The capital share of output α is set to 0.36 and the capital depreciation rate δ is As Krusell and Smith (1998) we assume that the 7

8 value of the aggregate shock z is equal to 0.99 in recessions (z b ) and 1.01 in expansions (z g ) Discount factor We now detail the calibration steps to generate a realistic wealth distribution, the observed US wealth Gini index and the capital-output ratio. The skewness of the wealth distribution and the existence of poor agents is an important component when conducting aggregate welfare analysis. Here, our calibration differs from Krusell and Smith (1998). In their economy, agents can borrow whereas here, we follow Aiyagari and McGrattan (1998) and no borrowing is allowed. To reproduce the shape of the US wealth distribution, we first assume that unemployed agents receive income too and fix the home production income θ to be 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 5. We impose that the discount factor β takes on three values {β l, β m, β h } where β l < β m < β h : β l β m β h = Thus, an agent with a discount factor β m is more patient than an agent with a discount factor β 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 β l, 70% at the medium discount factor β m and 20% at the highest discount factor β h. As Krusell 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 highest discount factor to be 50 years (200 quarters) to roughly match the length of a generation. These assumptions yield the following transition matrix 6 : Υ = The results of this calibration are shown in Table 1. The preference heterogeneity setting helps to reproduce the shape and the skewness of the US wealth distribution 7 and yields a Gini index of The quarterly capital-output ratio we target here is This corresponds to about 13% of the quarterly wage. This is roughly in the range of the value used by Krusell et al. (2009). 5 There are several other 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)). 6 For further details on the calibration of this matrix, see appendix C. 7 The data we report on the US distribution comes from Krusell and Smith (1998) and Budria- Rodriguez et al. (2002). 8 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 8

9 Held by Top Model 1% 5% 10% 20% Gini Benchmark model Data Held by Bottom 20% 40% 60% 80% Benchmark model Data Table 1: Distribution of wealth: benchmark economy and data Labor market process 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 η for aggregate state changes is defined by: ( ) η = 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: Π gg for a transition from a good period to a good period, Π bb for a transition from a bad period to a bad period, Π gb for a transition from a good period to a bad period and Π bg for a transition from a bad period to a good period 9 : Π bb = Π gb = ( ( ) ) Π bg = Π gg = ( ( Finally the joint transition matrix Π for labor market statuses and aggregate states can be defined as: ( ηbb Π Π = bb η bg Π bg ) η gb Π gb η gg Π gg = 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 Further details on this calibration can be found in Appendix C. ) ) 9

10 2.5.4 Government We set the (long-run) ratio of government purchases to GDP to The debt over GDP ratio, noted b is set to the quarterly value of 267% which is equivalent to an annual value of 67%. Those values are the observed ratios in the US as reported by Aiyagari and McGrattan (1998). 3 Results 16 Aggregate Capital Aggregate Wealth Interest Rate (%) Before Tax After Tax Wage Before Tax 0.7 After Tax Output Tax Rate (%) Quarterly Debt to GDP Ratio Quarterly Debt to GDP Ratio Figure 1: Aggregate behavior of the benchmark economy In this section, we present the results obtained with our simulations 10. We first report the aggregate behavior of the model before examining the long-run welfare effects of public debt in an aggregate fluctuations setting. Then we move on to the analysis of business cycle and distributional effects of public debt. We also explore calibrations that alter business cycle properties through changes in the labor market process. Finally, we compare the benchmark economy to a simpler idiosyncratic risk only model. 10 Unless stated otherwise, all our results are expressed in terms of quarterly values following our calibration scheme. 10

11 3.1 Public debt in an aggregate fluctuations setting We start by discussing the aggregate behavior of our benchmark model when our policy experiment is carried out. This experiment consists in changing the long-run debt to GDP ratio in the economy. Our computations are reported on Figure 1. Increasing the level of public debt raises the supply of assets in the economy. Consequently, the before-tax interest rate increases. Because the repayment of debt interests is 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. 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 private wealth level, 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 applies to the long-run optimal level of public debt with aggregate fluctuations. Because the aggregate variables are not constant, not even in the limit, we consider for our welfare analysis the values of aggregate variables averaged over long periods of time for a given debt to GDP ratio. We define the optimal level of public debt as the debt over GDP ratio that maximizes the traditional utilitarian welfare criterion. We follow Lucas (1987) and Mukoyama and Sahin (2006) to measure 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 ratio of public debt. The details of these computations are explained in Appendix B. Figure 2 depicts the long-run optimal level of public debt in the benchmark economy. In a setting embedding aggregate risk and calibrated on the US economy, the optimal public debt level is 20% of output on a quarterly basis (5% on an annual basis). The result of the introduction of public debt on welfare is a priori unknown because of opposing effects. The first is a crowding out effect: as it would happen in a standard overlapping generations model, a higher level of public debt crowds out capital and reduces per capita consumption and then welfare. Moreover, the increase in the income tax rate tends to amplify the negative impact of public debt on welfare. Another effect is related to the cost of precautionary saving: 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 last effect is welfare enhancing 11. It is difficult to predict which effects overcome the others analytically. 11 Woodford (1990) shows in a liquidity-constrained economy that the closer the interest rate is to the time preference rate, the higher welfare is. We have the same effect here. 11

12 Consumption Gain (%) Quarterly Debt to GDP Ratio Figure 2: Welfare gains in percent consumption in the benchmark economy Reducing the level of public debt from the benchmark level where the debt to GDP ratio is 267%, down to the optimal level where the debt to GDP ratio is 20% reduces the interest rate. The quarterly before-tax interest rate (resp. after-tax interest rate) falls from 0.896% to 0.836% (resp. from 0.612% to 0.586%). The cost of precautionary saving is increased and welfare is reduced 12. At the same time the welfare increases because there is less crowding out of private capital and the level of taxes is reduced. At the benchmark level of debt the aggregate capital level (resp. the tax rate) is (resp. 31.7%) whereas at the optimal level it is (resp. 29.9%). As long as the crowding out and distortionary taxation effects dominate the precautionary saving effect, a lower level of public debt is welfare enhancing. At the optimal level, the opposing effects balance each other. Population Average utility gains in (%) consumption Average Recessions Expansions Table 2: Welfare gains in percent consumption of being at the optimal level of public debt Changing the level of public debt from its benchmark value to the optimal one 12 The role prices play is central to explain our results and especially the role of the interest rate. To illustrate this role, we later describe a variant of the model without aggregate risk which behaves as in a small open economy. 12

13 would lead to a consumption gain of 0.253%, as illustrated in Table 2. On the contrary, it is welfare decreasing to implement a policy that targets higher levels of public debt than the benchmark one. In an economy where the level of debt is for instance 300% of GDP, the consumption loss not to be at the benchmark level (resp. optimal level) would be on average 0.07% (resp %). It is noteworthy that the welfare profile we find is not as flat as what Aiyagari and McGrattan (1998) report. In their paper, the costs of going to a broad set of levels of public debt around the optimal level are very low. They report, for instance, that being in an economy without public debt instead of their optimal economy, which is attained for a debt to GDP ratio of 267% in quarterly equivalent, would only cost about 0.08% of consumption. 3.3 Distributional and business cycle effects In this section, we move on to the effects of public debt first across the distribution of the population and then along the business cycle Distributional effects The overall consumption gains or losses of a change in the level of public debt are shared very differently across the population. We decompose consumption gains of changing the level of public debt for a number of percentiles in the population. This decomposition shows that the consumption gain for many percentiles in the population is orders of magnitude higher than the average gain reported earlier. Moreover, gains of rich and poor percentiles are not symmetrical. Table 3 documents that a change in public debt from the benchmark level to the optimal level leads to an increase in consumption for poorer percentiles of the population. The bottom 1% of the population gains as much as 2.679% of consumption whereas the bottom 10% of the population gains 2.529%. Most of the agents in the lower percentiles are unemployed and impatient. There is a jump in the consumption gain when the bottom 20% are considered as they gain as much as 3.386%. This is explained by the fact that the new agents in the bottom 20% when compared to the bottom 10% are mostly employed. This effect also applies to the bottom 30%, 40% and 50% although the consumption gain slowly reduces: as agents become richer, the adverse effect of a lower interest rate has a greater impact. Contrastingly, the richest percentiles in the population suffer when the same policy is implemented. The magnitude of their losses is substantially higher than the gains of the poorest percentiles. For the top 1% of the population, it amounts to a loss of % of consumption. The richer the agent is, the higher the loss. This is explained by the fact that the income of rich agents depends mainly on capital income whereas poor agents rely more on labor income. When the level of public debt is lower, poor agents benefit from the decrease in taxes, higher wages and smaller crowding out of capital. But as the interest rate decreases, rich agents are worse off 13. This illustrates the importance of interest rate effects when studying public debt policies in incomplete markets economies. Moreover, this particular analysis underlines the importance of having a realistic reproduction of the wealth distribution and especially of its fat right tail. 13 The same type of effects are discussed in Ball and Mankiw (1996) and Floden (2001). 13

14 Consumption gains in (%) held by bottom Population 1% 5% 10% 20% 30% 40% 50% Average Recessions Expansions Consumption gains in (%) held by top Population 1% 5% 10% 20% 30% 40% 50% Average Recessions Expansions Table 3: Welfare gains in percent consumption of being at the optimal level of public debt for bottom and top percentiles of the population Consumption gains in (%) held by Population Bottom: 10% Top: 10% Employed and β=low Employed and β=middle Employed and β=high Unemployed and β=low Unemployed and β=middle Unemployed and β=high Table 4: Welfare gains in percent consumption of being at the optimal level of public debt for different types of agents Table 4 describes the average consumption gains for specific types of agents in the economy. Precisely, these results distinguish the idiosyncratic elements due to the labor market status from the discount factor heterogeneity and thus disentangle composition effects. Some fraction of the consumption gain or loss of agents depends on their expectations about future wages and interest rates. For instance, an impatient and unemployed agent in the bottom percentiles would hardly benefit or suffer today from a wage change as her income is not wage dependent in our economic specification. Nevertheless, she would benefit or suffer from a change in policy as she expects her future wage income to change. She also expects her capital income to vary as the interest rate will adjust, but as she is unemployed and poor this is less harmful to her. The table also shows that for the bottom 10%, the less patient the agent, the higher her consumption gain is. Impatient agents save less and their income is biased toward their wage income whereas patient agents save more on average even for bottom percentiles of the population. Therefore, when the level of public debt is reduced the benefit is lower for patient agents. Now, comparing the bottom percentile employed and unemployed agents, we note that employed agents gain more from the change in policy except for the most patient agents. Impatient and moderately patient agents are on average poorer than patient agents and their income is biased toward the labor income. Thus, those agents benefit from the wage increase due to the change in policy whereas unemployed agents current income is not wage dependent. However, for employed and patient agents, who are on average richer than unemployed and patient 14

15 agents, the change in policy is less beneficial as a decrease in the interest rate matters more. Turning now to the top 10% percentile of the population, we note that for those agents, regardless of their status on the labor market, the more patient they are, the higher the loss is. This is because patient agents save more than impatient agents and are consequently more concerned by the movement of the interest rate. We also note that the loss is relatively higher among unemployed agents since they do not benefit from the current increase in their wage income Business cycle effects Next, we report the results of our model along the business cycle. To generate statistics along the cycle, we compute separately the average values of our aggregate variables either when the economy is in expansion or when it is in recession over the whole sequence of simulation. Table 3 documents these computations. For the poorest percentiles of the population, the consumption gain to be at the optimal level is higher in recessions than on average and then than in expansions. This is intuitive because when we account only for recessions, a lower level of public debt will be more beneficial to a poor individual because it will raise the wage rate. As she is wealth poor, she will be more concerned with wage movements than interest rate movements. Now, when we consider the bottom 20% of the population and beyond, the consumption gain is higher in expansions than in recessions. As noted above, the new agents in the bottom 20% when compared to the bottom 10% are mostly employed, more patient and therefore richer. For those agents, a policy change, from the benchmark level to the optimal level of debt, accounting only for recessions is costly in terms of capital income: not only the interest rate is lower because of the productivity shock but it decreases further after the policy. In expansions, this cost is lower. The percent change in the after-tax interest rate amounts to 4.935% in recessions and to 3.627% in expansions (whereas the percent change in the after-tax wage rate amounts to 3.791% in recessions and 3.761% in expansions) for the same policy change as above. Those individuals gain from the policy change because of higher wages but gain less in recessions because of lower interest rates, thus explaining the reversal in consumption gains between recessions and expansions. For top percentiles of the population, the losses are more important in recessions than on average and then than on expansions. Reducing the level of public debt will lower the interest rate. This is particularly harmful for a rich individual who is more concerned with the interest return on her wealth than with wage income. Finally, in the first column of Table 2, we outline that a policy that changes the level of public debt from the benchmark level to the optimal level generates consumption gains that are moderately higher in expansions than in recessions over the whole population. In expansions this gain amounts to 0.255% of consumption whereas in recessions it amounts to 0.251%. The explanation of this result derives from the effects underlined above. 15

16 3.4 Higher unemployment rate and longer unemployment spell In this section, we consider alternative calibrations of the labor market in the cycle that can serve as robustness exercises. We modify only the labor market features and leave the rest of the calibration unchanged. We either increase the duration of an unemployment spell or the unemployment rate in the business cycle with respect to their values in the benchmark calibration. For instance, if we consider a symmetric 10% (percent change) increase in the duration of unemployment, we would set the duration of an unemployment spell to be 2.75 quarters in recessions and 1.65 quarters in expansions. Similarly, for a 10% (percent change) increase in the unemployment rate, we would set the unemployment rate to be 11% in recessions and 4.4% in expansions. Figure 3 reports the results of this investigation. The left figure represents the symmetric increase in the unemployment duration whereas the right figure represents the symmetric increase in the unemployment rate. Raising either the unemployment duration or the unemployment rate increases the optimal level of public debt. In our simulations, after a 40% (percent change) increase in the unemployment duration, the optimal level of debt is 42.5% on a quarterly basis instead of 20% in the benchmark economy, which corresponds to a 112.5% (percent) change. With a 40% (percent change) increase in the unemployment rate, the optimal level of public debt is 45%, which corresponds to a 125% (percent) change. Changing those parameters affect the precautionary saving behavior of agents as the labor market becomes more risky in the cycle. Therefore, a higher level of public debt becomes optimal. (%) Change in the Optimal Level of Public Debt (%) Change in the Optimal Level of Public Debt (%) Change in the Unemployment Duration (%) Change in the Unemployment Rate Figure 3: Change in the optimal level of public debt after a change in business cycle variables We also consider an economy where both the unemployment rate and the unem- 16

17 ployment duration are changed. We investigate a symmetric (percent change) increase of 40% of these parameters: the unemployment rate is 14% in recessions and 5.6% in expansions whereas the unemployment duration is 3.5 quarters in recessions and 2.1 quarters in expansions. We now find the optimal level of debt to be 80% on a quarterly basis, which corresponds to a 300% (percent change) increase with respect to the optimal level in the benchmark case. As we could have expected after the previous experiment, longer unemployment spells and higher unemployment rates raise the optimal level of debt. This alternative calibration purposely strengthens the effect of employment fluctuation along the cycle: unemployed agents have a harder time finding a job in this economy as compared to the benchmark economy and employed agents face a higher risk of losing their jobs. In recessions, this is amplified. The precautionary motive is stronger here than in the benchmark case and thus the costs of reducing the level of public debt are higher. Because of that, agents settle for a higher level of public debt. Even though, the changes in the unemployment rate and duration with respect to the benchmark economy are not comparable, the decomposition in the previous experiment helps in disentangling the contribution of each change. We also find that the welfare decomposition across the population and the cycle in this economy is qualitatively similar to our description in the benchmark economy. The quantitative differences are explained by the change in policy to a higher level of public debt due to the variations in the labor market parameters. 3.5 Optimal level of debt without aggregate fluctuations In this section, we focus on a setup without aggregate fluctuations. We call this model the idiosyncratic risk economy for contrast with the aggregate risk economy. There are several ways of eliminating business cycles 14 to derive a comparable long-run idiosyncratic model. We use the method described in Imrohoroglu (1989). To derive the model without aggregate risk, 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 economy with aggregate fluctuations. All other calibrated parameters are kept to their benchmark values, especially time preference rates and the risk aversion parameter 15. This model is similar to the aggregate risk model in many ways: higher levels of debt raise the interest rate 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 the interest rate is higher in this model. Figure 4 depicts the welfare profile we find in this idiosyncratic risk model. The optimal level of public debt is 10% of output on a quarterly basis and the consumption gain of being at this level instead of the benchmark level of 267% is 0.286%. The optimal level is now lower than in the aggregate risk economy. In the aggregate risk economy, as the unemployment rate and the unemployment duration increase during recessions, the precautionary motive is stronger. Moreover, the interest rate and the 14 For a survey, see Barlevy (2004). 15 For greater details on the model without aggregate risk and its calibration, see appendix D. 17

18 Consumption Gain (%) Quarterly Debt to GDP Ratio Figure 4: Welfare gains in percent consumption in a model without aggregate fluctuations level of physical capital are smaller in recessions. Therefore, the cost of precautionary saving is higher. That is why the need for public debt is more important in the aggregate risk economy. In the idiosyncratic risk model, prices and unemployment rates do not fluctuate and, for that reason, the level of risk is lower. Both the precautionary motive and the cost of precautionary saving are reduced and therefore agents save less. As a result, the interest rate is at a higher level here and the economy can settle for a lower optimal level of public debt. To emphasize the role of prices in determining the optimal level of public debt, we also experiment with a small open economy variant of the idiosyncratic risk model above. This extension is straightforward: for any ratio of public debt considered, we impose that prices are equal to their levels in the benchmark case, i.e. when the debt over GDP ratio is 267%. Doing this, the capital market does not clear as in the closed economy case and we assume that international flows are able to clear the markets (outside the scope of this model). While prices are constant and starting from the baseline ratio of public debt, a lower level of public debt leads to lower distortionary taxes. This is welfare enhancing. But now, there is no cost in terms of precautionary savings associated with a lower level of public debt because prices, and especially the interest rate, do not change. Thus, now the optimal level of public debt should be somewhat below the optimal level found in the previous baseline model. Our simulations reveal that the level of debt that would be optimal is, as expected, clearly below the optimal level in the closed economy model 16, because of the absence of a precautionary saving 16 Our limited set of simulations is not able to pin down a level of debt that would be optimal in this 18

19 effect. Finally, consumption gains of reducing the level of public debt are now many times higher than in the baseline idiosyncratic case, again because there are no costs in terms of precautionary saving to do so. 4 Conclusions This paper stresses the importance of aggregate risk in the determination of the optimal level of public debt in an incomplete markets economy where households face both idiosyncratic and aggregate risk. We determine the optimal level of public debt and quantify the impact of several setups that alter the properties of the business cycle ranging from longer unemployment spells and larger unemployment rates to a setting without aggregate risk. We also point out that welfare gains of public debt policies differ considerably across the population and along the business cycle. Precautionary saving and price variations play a significant role in explaining our results. Aggregate and idiosyncratic risk, credit constraints and the incompleteness of markets lead agents to engage in precautionary saving in order to smooth consumption. The correlation between aggregate risk and idiosyncratic risk changes the motive for precautionary saving along the cycle. Moreover, price fluctuations and most notably variations in the interest rate change the cost of precautionary saving. One limitation of this paper is the long-run approach used to compare alternative public debt policies that resemble the steady state comparisons performed in models without aggregate risk. Although the idea of full comparisons including transitional dynamics is immediately appealing, we lose on the side of simplicity and technical feasibility. Nevertheless, this paper helps in setting bounds on the interaction between aggregate fluctuations and the optimal level of public debt. Another limitation is the exogenous nature of the policies imposed on the economy in the sense that agents do not expect policy changes. A recent and active literature on the time-consistent determination of public policies 17 sheds a very promising light on this issue. model. Decreasing the level of public debt remains optimal as far as we tried. 17 See for instance Krusell (2002); Hassler et al. (2003); Klein and Rios-Rull (2003); Hassler et al. (2005); Klein et al. (2005, 2008). 19

20 Acknowledgements We are indebted to Yann Algan, Antoine d Autume, Fabrice Collard, Wouter J. Den Haan, Jean-Olivier Hairault, Michel Juillard, Per Krusell, Franck Portier and Thomas Weitzenblum 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. Appendix A Computational strategy A.1 Solving the model We solve the model by 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 and that limiting this set to the mean of the wealth distribution might be sufficient: 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, β; z, Γ). 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 use the following prediction rules: log( K ) = a 0 + a 1 log( K), if z = z g log( K ) = c 0 + c 1 log( K), if z = z b where K and K denote respectively the average stock of capital of the next period and of the current period. Thus, the strategy is the following: 1. As the aggregate variables are not constant, even in the limit, in an economy with aggregate fluctuations, we approximate the steady state of this economy by averaging aggregate variables over long periods of time. We first make a guess for the average long-run interest rate 18. From this, we can deduce the long-run GDP and using the debt over GDP ratio and the public expenses ratio, we can derive the level of public debt and public expenses in this economy given our guess of the long-run interest rate. From the long-run budget constraint of the government, we then derive the long-run tax rate for the economy. Given this tax rate, we execute steps (2), (3) and (4) below. 2. Given a set of parameter values (a 0, a 1, c 0, c 1 ) for the law of motion, we solve the individual problem. To solve the individual problem, we iterate on the Euler equation: 18 Further details about this are given in the section A.2 below. 20