Reducing Government Debt in the Presence of Inequality

Transcription

1 Reducing Government Debt in the Presence of Inequality Sigrid Röhrs and Christoph Winter July 18, 2014 Abstract What are the welfare effects of government debt? In particular, what are the welfare consequences of government debt reductions? We answer these questions with the help of an incomplete markets economy with production. Households are subject to uninsurable income shocks. We make several contributions. First, when comparing stationary equilibria, we find that quantitatively sizable welfare gains can be achieved by reducing public debt, contrary to what the literature has suggested. Second, when we also consider the transition between stationary equilibria, we find that the welfare costs of a debt reduction outweigh the benefits of being in a stationary equilibrium with lower debt. Third, by targeting the skewed wealth and earnings distribution of the US economy in our calibration, we identify inequality as a major driver of the welfare effects. Our results have important implications for the design of debt reduction policies. Since the skewed wealth distribution generates a large fraction of borrowing-constrained households, the public debt reduction should be non-linear, such that the tax burden is postponed into the future. Key words: Government Debt, Borrowing Limits, Incomplete Markets, Crowding Out JEL classification: E2, H6, D52 Acknowledgements: We would like to thank Alexander Bick, Timo Boppart, Nicola Fuchs-Schündeln, Wouter den Haan, John Hassler, Marcus Hagedorn, Kenneth Judd, Leo Kaas, Timothy Kehoe, Nobuhiro Kiyotaki, Felix Kübler, Alex Michaelides, Dirk Niepelt, Karl Schmedders, Kjetil Storesletten, Iván Werning as well as participants of various seminars and especially Fabrizio Zilibotti for many useful comments. Röhrs would like to thank the University of Zurich for financial support (Forschungskredit Nr ). Winter gratefully acknowledges financial support from the European Research Council (ERC Advanced Grant IPCDP ) and the National Centre of Competence in Research Financial Valuation and Risk Management (NCCR FINRISK). All remaining errors are our own. Parts of this project were previously circulated under the title Wealth Inequality and the Optimal Level of Government Debt. Goethe University Frankfurt, Chair of Macroeconomics and Development, Office HOF 3.42, Grüneburgplatz 1, D Frankfurt am Main (Germany), +49 (069) , [email protected] (corresponding author) University of Zurich, Department of Economics, Office SOF-G-21, Schönberggasse 1, 8001 Zurich (Switzerland), +41 (044) , [email protected]

2 1 Introduction Many countries - including the United States - have experienced a dramatic surge in their public debt levels in the aftermath of the financial crisis. Drastic measures, including tax increases, need to be applied to bring government debt back to its pre-crisis level. 1 However, history has shown that austerity measures are very unpopular and that their political implementation is difficult. As suggested by Reinhart and Rogoff (2009), historically, many countries have chosen to default on their sovereign debt even at moderate debt/gdp ratios in order to avoid painful austerity policies. Motivated by these observations, we study the welfare effects of government debt. We conduct our analysis with the help of an incomplete markets framework in the tradition of Aiyagari (1994). Households are subject to idiosyncratic productivity shocks. These shocks are uninsurable because insurance markets are absent. Households can self-insure against adverse shocks by accumulating precautionary savings or by borrowing. In our framework, borrowing is limited. This restricts the ability of households to self-insure. This framework is ideal for our purposes for two reasons. First, government debt plays a non-trivial role if markets are incomplete and if borrowing is restricted. As it is well-known from the seminal work by Woodford (1990) and Aiyagari and McGrattan (1998), government debt relaxes borrowing constraints and therefore helps to complete markets by facilitating precautionary saving. Issuing government debt might thus be an effective way to improve risk-sharing and aggregate welfare (Flodén, 2001, Shin, 2006, Albanesi, 2008). Second, the fact that markets are incomplete and risk-sharing is limited generates a non-trivial distribution of assets and consumption. One contribution of this project is to show that the degree of inequality implied by the model is crucial. Important papers discussing the welfare effects of government debt in an incomplete markets environment with production are Aiyagari and McGrattan (1998), Flodén (2001) and Desbonnet and Weitzenblum (2011). In order to generate a distribution of earnings and assets that resembles the skewed distributions of earnings and wealth in the US economy, we follow Castañeda, Díaz-Giménez, and Ríos-Rull (2003) in our calibration of the stochastic process that governs the evolution of idiosyncratic earning shocks. We find that the debt/gdp ratio that leads to the highest stationary equilibrium welfare is negative, i.e. when the government holds assets, not debt. This result is remarkable, given that the previous literature, which uses the Aiyagari (1994) framework, most notably Aiyagari and McGrattan (1998), Flodén (2001) and Desbonnet and Weitzenblum (2011), has concluded that debt/gdp ratio that maximizes social welfare in a stationary state is positive, not negative. In order to fully appreciate our finding and to understand how our outcome is influenced by inequality, it is useful to consider the impact of borrowing constraints in greater detail. If borrowing constraints are binding, lowering government debt crowds in private capital. This is because households that face binding borrowing constraints will not decrease their savings one-to-one in response to a decrease in debt, and the Ricardian Equivalence breaks down. This implies that the demand for private bonds will not meet the supply of private bonds issued by the firm. We say that public debt crowds in the capital stock, and therefore also production and output. As a result, the equilibrium interest rate that clears 1 See Chen and Imrohoroglu (2012) for an overview over several proposals to reduce government debt that are discussed for the US. 1

3 the private bond market will decrease, and the marginal product of labor will increase. Moreover, if taxation is distortionary instead of lump-sum, lower debt/gdp ratios are associated with even stronger positive effects on capital accumulation and output, since less government debt means lower taxes and therefore reduces the inefficiencies caused by distortionary taxes. To the extent that more capital and output translates into higher average consumption of goods and leisure, a lower debt/gdp ratio is welfare-improving. 2 Besides the impact on average consumption, the change in the interest rate and the wage rate resulting from crowding in and lower distortionary taxes give rise to other welfare effects in a world in which markets are incomplete and in which there is heterogeneity among households. Firstly, a lower interest rate makes self-insurance more difficult for private households, since saving yields a lower return (Aiyagari and McGrattan, 1998). Put differently, the price of the riskless production factor (capital) decreases, while the price of the risky factor (labor) increases. Lower debt/gdp ratios therefore have a negative effect on aggregate welfare through this insurance effect. Changes in market clearing prices that affect welfare when markets are incomplete are also key in Gottardi, Kajii, and Nakajima (2010) and Davila et al. (2011). These papers analyze whether the laissez-faire outcome is constrained efficient if markets are incomplete. 3 Second, government debt also affects the distribution of consumption via the composition of income, because households that receive capital income lose and households that mainly rely on labor income win. This income composition counteracts the negative effect of the insurance channel on aggregate welfare, since the households that profit the most from the increase in wages are the consumption-poor. Which of these three effects dominates the development of social (Utilitiarian) welfare depends crucially on the distribution of wealth and income. Since in the US, both wealth and income are very unequally distributed across the population, and because households that are consumption-poor also hold no wealth or are even in debt, the positive welfare effects associated with lower debt/gdp ratios more than offset the insurance effect. Instead, Aiyagari and McGrattan (1998) and Flodén (2001) conclude that the opposing effects almost cancel out, leading to only weak overall welfare effects of government debt. In light of our results that lower debt/gdp ratios are welfare-improving when we consider stationary equilibria, the fact that debt reductions face so much political resistance is puzzling at the first glance. 4 It should be noted that so far we have ignored the costs that occur over the transition from a stationary equilibrium with a high debt/gdp ratio to the new one that is characterized by a low debt/gdp ratio. As a second contribution of our project, we therefore also incorporate the welfare effects of government debt reductions occurring over the transition into our analysis. We are interested in whether the losses occurring over the transition offset the benefits of being in a stationary equilibrium with a lower debt/gdp ratio. In particular, we want to understand how the curvature of the debt/gdp path and the time span affect these costs. Both curvature and time horizon are important policy parameters. For this reason, we focus on a small but realistic unanticipated reduction of the debt/gdp ratio from 0.66, 2 Using the terminology of Flodén (2001), this is the level effect. 3 Gottardi, Kajii, and Nakajima (2010) and Davila et al. (2011) find that, depending on the structure of uncertainty, constrained efficiency requires a higher level of capital compared to the competitive equilibrium outcome when markets are incomplete. 4 Protests can be fierce, as documented by Ponticelli and Voth (2011), who found that fiscal consolidation increase the likelihood of social unrest. 2

4 the long-run average of the US, to 0.6. This reduction is financed by increasing the (linear) personal income tax rate, therefore keeping the distortions on asset and labor supply constant. We find that the government can minimize transitional costs by following a concave debt/gdp path, where the reduction is slow initially and accelerates over time. Extending the time horizon during which the debt reduction is implemented further helps to reduce the transitional welfare costs. However, for all cases that we compute, we find that the transitional welfare costs associated with reducing government debt more than offset the long-run benefits of living in a stationary equilibrium with a lower debt/gdp ratio. In order to understand the determinants of the transitional welfare effects, it is again important to consider the role of inequality. When choosing the curvature of the debt/gdp path, the government faces the following trade-off. On the one hand, a convex debt/gdp path offers the advantage of minimizing intertemporal distortions arising through asset income taxation. This is because a convex debt/gdp path implies that taxes are high at the beginning, when assets are pre-determined. On the other hand, households who are initially at the borrowing constraint prefer a concave debt/gdp path, under which the tax burden is initially low and increasing over time. The fact that in our calibration, consistent with the US economy, a substantial fraction of the population holds no assets or is in debt, explains why the concave path is superior to a convex or a linear path. 5 Our results contribute to the literature by showing that inequality is important for welfare effects of government debt in the transition and in stationary equilibrium. The previous literature has largely ignored transitional welfare effects altogether (see e.g. Aiyagari and McGrattan, 1998 and Flodén, 2001). An exception is the work by Desbonnet and Weitzenblum (2011), who however do not match the large degree of wealth inequality observed in the data, which makes their stationary equilibrium results very different from ours. Moreover, they only study linear debt/gdp paths. Their focus is on computing the debt/gdp ratio that maximizes total (transitional plus stationary equilibrium) welfare, whereas we are interested in analyzing the welfare consequences of debt reduction policies. Our results about the importance of inequality for the transitional costs of debt reductions and inequality is consistent with the findings of D Erasmo and Mendoza (2012) who study the link between domestic inequality and the likelihood to default on domestic debt in an incomplete markets model similar to ours. They show that inequality increases the probability to default. In earlier work, Alesina and Perotti (1996) have established a link between inequality and social unrest. Apart from the aforementioned work by Aiyagari and McGrattan (1998) and Flodén (2001), our project is also related to other papers that study the impact of fiscal policy in quantitative models with incomplete markets. Domeij and Heathcote (2004) and Conesa, Kitao, and Krueger (2009) study the welfare effects of abolishing capital taxation. Açikgöz (2013) studies the long run characteristics of the optimal Ramsey policy under incomplete markets (including government debt). In line with our findings, he concludes debt/gdp that maximizes welfare in stationary equilibrium is negative. 5 As a special case, we also consider the policy where the government unexpectedly raises the asset income tax in the initial period only. We find that in this case, government debt reductions lead to an overall welfare gain, which is however much smaller than the associated increase in stationary equilibrium welfare, implying that the transitional phase is still welfare reducing. The reason is the existence of borrowing constraints, which prevent some poor households from immediately benefiting from the higher income lower debt/gdp will bring to them in the future. 3

5 Moreover, he finds that the debt/gdp ratio that maximizes overall welfare is positive, which confirms our results that the costs of debt reductions over the transition are large. In this project, we show that inequality is the important driver of the welfare effects of government debt, both in the transition as well as in stationary equilibrium. Ludwig and Krüger (2013) compute the optimal tax progressivity and education subsidies in a model with endogenous human capital formation. Heathcote (2005) analyses the impact of tax changes on aggregate consumption. Gomes, Michaelides, and Polkovnichenko (2013) quantify the crowding out effect on the capital stock of a 10 percent increase in government debt in an incomplete market economy with aggregate risk. Angeletos and Panousi (2009) and Challe and Ragot (2010) study the effects of changes in government expenditures in incomplete market models. Oh and Reis (2012) study the impact of government transfers. Gomes, Michaelides, and Polkovnichenko (2012) aim at quantifying the fiscal costs of the recent financial crisis in US. Desbonnet and Weitzenblum (2011) and Azzimonti, de Francisco, and Quadrini (2012) also exploit the fact that government debt facilitates self-insurance, as we do in our work. The explicit characterization of the welfare consequences of debt reductions during transition and in stationary state is not the main objective of any of the aforementioned papers. The remainder of the paper is structured as follows. We present the baseline model in the next section. In section 3 we discuss the calibration of the model. Section 4 shows the quantitative results. Section 5 concludes. 2 The Baseline Model The economy we consider is a neoclassical growth model with incomplete markets where households face uninsurable income shocks, as in Aiyagari (1994). The economy consists of three sectors: households, firms and a government. In the following, we describe the three sectors in greater detail. We start by describing the bonds that households in our economy use to accumulate savings. 2.1 Supply and Demand for Bonds Households self-insure against income fluctuations by saving in one-period risk-free bonds. Bonds are issued by firms and the government, as in Aiyagari and McGrattan (1998) and Flodén (2001). Bonds issued by firms are claims to physical capital. We abstract from aggregate risk, which implies that claims to physical capital and government bonds are perfect substitutes and thus yield the same return, r t. 6 Different from Aiyagari and McGrattan (1998) and Flodén (2001), we also allow households to borrow up to a certain limit. We view this as an important modification, given that the fraction of households that actually borrow in the data is substantial Household Sector The economy is populated by a continuum of ex-ante identical, infinitely lived households with total mass of one. Households maximize their expected utility by making a series of consumption, c t, labor, 6 In Gomes, Michaelides, and Polkovnichenko (2012, 2013), government bonds and private capital are imperfect substitutes due to aggregate uncertainty. 7 Borrowing by households can be interpreted as bonds that are issued to other households ( IOUs ) or to the government. 4

6 l t, and savings, a t+1, choices subject to a budget constraint and a borrowing limit on assets. In period t = 0, before any uncertainty has realized, their expected utility is given by U({c t, 1 l t } t=1,2,... ) = E 0 t=0 β t u(c t, 1 l t ) where β is the subjective discount factor. The per-period utility function, u(.), is assumed to be strictly increasing, strictly concave and continuously differentiable. Additionally, the first derivative is assumed to satisfy the following limiting (Inada) conditions: lim c(c, 1 l) c 0 =, lim c(c, 1 l) = 0 c lim 1 l(c, 1 l) l 0 = Household productivity is subject to a shock, ɛ, that follows a Markov process with transition matrix π(ɛ ɛ). A household faces the following per-period budget constraint: c t + a t+1 = y t + a t where a t+1 denotes the bond holdings of a household. y t is the household s (after-tax) income. Notice that a t+1 may also be negative, in which case the household borrows. Borrowing is restricted: a t+1 a The fact that we impose an exogenous borrowing constraint that is tighter than the natural borrowing limit is important. With a natural borrowing constraint and lump sum taxes, government debt would be neutral (Ricardian equivalence holds). The tax system in our economy consists of proportional taxes on asset income, τ k,t, and labor income, τ l,t, and a lump sum transfer, χ t. Figure 1 in Golosov and Sargent (2012) suggests that affine taxes are a very good approximation to the US tax code. We conceptualize the tax rates as a baseline component equal to the labor income tax, τ t = τ l,t, and a premium on the asset income tax, ζ, such that τ k,t = ζτ t. This way of defining the tax rates will be useful later during our policy experiments where we change only the baseline component, τ t, keeping the premium on the asset income tax fixed. To focus on government debt and the timing of the (linear) taxes, we assume that transfers/output remains constant over time. Furthermore, we assume that only non-negative asset income is taxed. In other words, there are no proportional subsidies if households are in debt. More precisely, we define the tax on asset income τ k,t, as follows: { τk,t if a 0 τ k,t (a) = 0 if a < 0 The after-tax interest rate is therefore given by r t = (1 τ k,t (a))r t. The after-tax wage rate is given by w t = (1 τ l,t )w t where w t is the price of labor in the economy. After-tax income is thus given by: y t = w t ɛl t + r t a t + χ t 5

7 The optimization problem of a household is given by the following functional equation: { } W t (a, ɛ) = max c,l,a u(c, 1 l) + β ɛ π(ɛ ɛ)w t+1 (a, ɛ ) s.t. c + a = w t ɛl + (1 + r t )a + χ t (2) a a The household s problem is time-dependent because we do not only study steady-states but also transitions between steady-states. 8 (1) 2.3 Firm Sector We assume that the aggregate production technology which is operated by a representative firm to produce output, Y t, using aggregate capital, K t, and aggregate labor, N t, as inputs is given as follows: Y t = F (K t, X t N t ) where X t denotes exogenous labor-augmenting technological progress. This technology is assumed to grow exogenously at a constant rate X t+1 = (1 + g)x t. For simplicity we normalize initial technology to X 0 = 1, such that: X t = (1 + g) t The presence of technological progress implies that households, on average, become richer over time. Technological progress thus increases the propensity of households to borrow. The aggregate production function, F, is assumed to have the standard properties, in particular constant returns to scale. This ensures that in competitive equilibrium, the number of firms is indeterminate and we can assume the existence of a representative firm, without loss of generality. 2.4 Government Sector The government has to finance government spending, G t, and the total transfers to households, T r t, by issuing new government bonds, B t+1, and levying taxes on positive asset and labor income. We assume that government spending/output is constant. Furthermore, the government services its debt, B t, and makes interest payments, r t B t. The government budget constraint is thus given by: G t + r t B t + T r t = B t+1 B t + τ l w t N t + τ k r t  t (3) where Ât A t is the tax base for the asset income tax. As explained above taxes are only levied on positive financial income (no proportional transfers from the government for indebted people) and thus the tax base is defined as:  t = adθ t (ɛ, a) a 0 8 Notice that as long we focus on transition between steady-states, the dynamic programming problem of the household falls into the class of stationary dynamic programming problems for which the principle of optimality is satisfied. We numerically show that our economy converges to a new steady-state. 6

8 where θ t (ɛ, a) denotes the distribution of households over income and asset states. Aggregate transfers have to equal the sum of all individual transfers: χ t dθ t (ɛ, a) = T r t 2.5 Competitive Equilibrium Using the characterization of the three sectors we can now define the competitive equilibrium. Definition 1. Competitive Equilibrium: Given a transition matrix π, a government policy {B t, τ t, G} t=0, and an initial distribution of the idiosyncratic productivity shocks and of the asset holdings θ 0 (ɛ, a) a recursive competitive equilibrium is defined by a law of motion Γ, factor prices {r t, w t } t=0, a sequence of value functions {W t } t=0 and a sequence of policy functions {c t, l t, a t} t=0 such that 1. Households solve the utility maximization problem as defined in equation (1). 2. Competitive firms maximize profits, such that factor prices are given by w t = F L (K t, X t N t ) (4) r t = F K (K t, X t N t ) δ (5) 3. The government budget constraint as defined in equation (3) holds. 4. Factor and goods markets have to clear: Labor market: N t = ɛl t dθ t (ɛ, a) Asset market: A t+1 = a tdθ t (ɛ, a) = K t+1 + B t+1 Goods market: c t dθ t (ɛ, a) + G t + I t = F (K t, X t N t ) where investment I is given by I K (1 δ)k 5. Rational expectations of households about the law of motion of the distribution of shocks and asset holdings, Γ reflect the true law of motion, as given by θ t+1 (a, ɛ) = Γ[θ t (a, ɛ)] (6) where θ t (a, ɛ) denotes the joint distribution of asset holdings and productivity shocks. 7

9 2.6 Welfare Measure In order to be able to compare the welfare effects of different government policies, we have to define a welfare criterion. Following the previous literature as for example, Aiyagari and McGrattan (1998) and Flodén (2001), we compute the aggregate value function: Ω t = W t (a, ɛ)dθ t (a, ɛ) This criterion can either be interpreted as (1) a Utilitarian social welfare function where every individual has the same weight for the planner, (2) a steady-state ex ante welfare of an average consumer before realizing income shocks and initial asset holdings or (3) the probability limit of the utility of an infinitely lived dynasty where households utilities are altruistically linked to each other. In order to facilitate the interpretation, we compute the average welfare change in consumption equivalent units, i.e. the consumption that needs to be given to each household in order to make households indifferent on average between two specific policies. 9 More details are provided in the Appendix. 3 Calibration We calibrate the model such that it is consistent with long run features of the US economy. The resulting allocation serves as a benchmark for our welfare calculations. The parameter values that result from our calibration procedure are shown in Table 1. Parameter values that are adopted from the existing literature are given in Table 2. In the following, we discuss the rationale behind our parameter choices in greater detail. 3.1 Utility Function and Production Technology We assume that preferences can be represented by a constant relative risk aversion utility function: u(c, 1 l) = (cη (1 l) 1 η ) 1 µ 1 µ µ determines the curvature of the utility function. We set µ to 2. This implies a coefficient of risk aversion of 1 µ + ηµ = 1.3, which is well in the range (between 1 and 3) commonly chosen in the literature. η denotes the share of consumption in the utility function. We calibrate η such that the average share of time worked is 0.3. This results in η = This choice implies an aggregate Frisch elasticity of This is broadly in line with the outcome of other macro models in which the Frisch elasticity of the overall population is considered, but an order of magnitude larger than the Frisch elasticity estimated using micro data from prime age workers More precisely, this measure provides the percentage increase in benchmark consumption at every date and state (with leisure at every date and state held fixed at benchmark values) that leads to the same welfare (under the benchmark policy) as under the new policy. 10 For our choice of the utility function, the Frisch elasticity is given by (1 µ + ηµ)/µ (T l)/ l, where T denotes the time endowment (normalized to 1 in our case) and l denotes the fraction of time spend at work, in our case The debate on whether micro and macro elasticities are consistent is ongoing. See Keane and Rogerson (2011) for a summary. 8

10 We assume that the aggregate technology is given by a Cobb-Douglas production function: F (K, XN) = K α (XN) 1 α Initial technology is normalized to X 0 = 1, such that X t = (1 + g) t. We set g = 0.02, which implies that our economy grows at a rate of 2 percent per year. The parameter α, which denotes the share of capital in total production, is set to 0.3. This implies a labor share of 0.7. The discount factor β is chosen such that the model reproduces a wealth-output ratio of 3.1 (cf. Cooley and Prescott (1995) or Ábrahám and Cárceles-Poveda (2010)). The resulting β is equal to The annual depreciation rate δ is set to 7 percent, which is a common value in the literature (see e.g. Trabandt and Uhlig (2009)). Table 1: Calibrated Parameter Values Parameter Value Target Data Model Discount factor, β 0.96 Capital to output ratio Weight of consumpt. in the util. funct., η 0.31 Average labor supply Borrowing constraint a 0.3 % of HH with no assets or debt Gov. spending/gdp, G 0.15 gov. budget constraint clearing Taxes and Government Debt Following Trabandt and Uhlig (2009), we set the labor income tax rate, τ l, to 0.28, and we set the asset income tax rate, τ k, to This implicitly defines the baseline component of our tax system τ l = τ = 0.28 and the premium on the asset income tax ζ = τ k τ = Lump-sum transfers/gdp, χ, are set to 0.083, also in accordance with Trabandt and Uhlig (2009). Similar values are reported by Mendoza, Razin, and Tesar (1994) for an earlier time period, as well. Following Aiyagari and McGrattan (1998), we use a debt/gdp ratio of Aiyagari and McGrattan (1998) calculate this figure as the average of the sum of US federal and state debt divided by gross domestic product over the postwar period. Table 2: Parameters Set Exogenously Parameter Value Capital s share, α 0.3 Growth rate, g 0.02 Debt/GDP, B 0.66 Labor tax, τ l 0.28 Capital tax, τ k 0.36 Transfers/GDP, χ Curvature µ 2 9

11 Finally, government spending/gdp, G, is set such that the government s budget constraint clears, given all other parameters. 3.3 Income Process We calibrate the vector of income states, s, and the transition matrix, Π, such that the distribution of earnings and net worth generated by the model are consistent with the data. Disciplining the model such that it is consistent with the skewed distribution of earnings and wealth observable in the US economy is key for assessing the welfare-maximizing government debt/gdp ratio. In particular, we are interested in the share of (consumption-) poor households in the economy, who matter more in our Utilitarian welfare criterion, given the concavity of the utility function. In addition, matching inequality is also important in order to determine the factor income composition of the poor, which in turn determines whether poor households gain or lose from changes in government debt. We compute the distribution of earnings and net financial assets from the 2007 Survey of Consumer Finances (SCF) (see Table 3 and 4). We define net financial assets as net worth excluding housing and other durable goods, following Ábrahám and Cárceles-Poveda (2010). Earnings are defined as labor earnings (wages and salaries) plus a fraction of business income before taxes, excluding government transfers. 12 This definition is close to the concept of earnings that is implied by our model as well. Table 3: Distributional Properties at Benchmark Stationary Economy Q1 Q2 Q3 Q4 Q5 Gini Net financial assets Data 1.60% 0.10% 1.64% 8.29% 91.57% 0.90 Baseline Economy 1.57% 0.88% 3.92% 7.23% 89.54% 0.83 Aiyagari/McGrattan s Economy 3.24% 10.07% 16.96% 25.71% 44.03% 0.41 Earnings Data 0.40% 3.19% 12.49% 23.33% 61.39% 0.62 Baseline Economy 0.00% 2.38% 12.58% 22.73% 62.31% 0.65 Aiyagari/McGrattan s Economy 1.21% 9.70% 16.18% 26.85% 46.07% 0.45 Remarks: Quintiles (Q1-Q5) denote net financial assets (resp. earnings) of a group in percent of total net financial assets (resp. earnings). The entries in data are computed from the 2007 SCF. See main text for precise definitions. Notice that earnings can be negative due to the fact that labor earnings also contain part of the gains (or losses) of small enterprises. Table 3 and 4 show that both earnings and net financial assets are very unequally distributed in the data. The richest 20 percent of the population hold more than 90 percent of all financial assets, net of debt. The distribution of earnings is less skewed. Households in the top quintile earn around 60 percent 12 Unfortunately, it is not declared exactly in the SCF how much of the business income is actually labor and how much is capital income. We take business income from sole proprietorship or a farm to be labor earnings, whereas we define business income from other businesses or investments, net rent, trusts, or royalties as capital income. 10

12 Table 4: Upper Percentiles of Wealth Distribution at Benchmark upper 10% upper 5% upper 1% Net financial assets Data 79.64% 66.83% 39.09% Benchmark Calibration 70.58% 47.03% 13.53% Remarks: The table shows the percent of net financial assets held by the wealthiest 10% (upper 10%), 5% (upper 5%) and 1% (upper 1%). of the total earnings. It is well known that for a standard parameterization of the earnings process, incomplete markets models in the tradition of Aiyagari (1994) generate too little inequality compared to the data (see e.g. Quadrini and Ríos-Rull (1997)). We follow Castañeda, Díaz-Giménez, and Ríos-Rull (2003) and calibrate the vector of income states s and the transition matrix Π to match the Lorenz curves of US earnings and wealth as found in our analysis of the 2007 SCF. In the model, wealth inequality (and consumption inequality) are the result of households optimal decisions with respect to consumption and saving. Household use assets in order to smooth consumption over time and in order to save against uninsurable income shocks. Households decision making in turn depends on preference parameters and the specification of the earnings process. In our calibration, we modify the uninsurable income shocks such that the equilibrium asset allocation resembles the observable wealth inequality in the US 13 Tables 3 and 4 show that our calibration tracks the observable wealth and earnings distribution very closely, with the exception of the top decile of the wealth distribution. Here, our earnings process generates less inequality than the data, but much more than a standard AR(1) process. We find the following vector of income states: s = {0.055, 0.551, 1.195, 7.351} It should be noted that the highest income state is more than 130 times as high as the lowest income state. Furthermore, we get the following transition matrix for the income states: Π = As can be seen from the transition matrix, there is a 10 percent probability of moving from the highest income state today to the lowest income state tomorrow. This generates a strong saving motive for income-rich households, leading to the high degree of wealth inequality that we also observe in the data. The same mechanism is also present in the transition matrix found by Castañeda, Díaz-Giménez, and Ríos-Rull (2003). 13 An alternative approach, which is due to Krusell and Smith (1998), would be to introduce preference heterogeneity. Krusell and Smith (1998) assume that some households are more patient than others, and thus also accumulate more savings. 11

13 3.4 Borrowing Limit We calibrate the ad-hoc borrowing limit to match the percentage of households with negative or zero financial assets in the 2007 SCF (24 percent). We find a borrowing limit of a = Results We are now ready to compute the welfare effects of reductions in government debt. We distinguish between transition (in the following also labeled as short-run) and stationary equilibrium (in the following also labeled as long-run). We proceed in two steps. First, we analyze the welfare consequences of government debt reductions in stationary equilibrium. This is done by comparing stationary equilibria that are characterized by different debt/gdp ratios, keeping all other parameters constant. In order to keep the budget of the government balanced, we adjust taxes such that the relative distortions between capital and labor (ζ) remains constant. We find that in the long-run, debt reductions are associated with welfare gains. In a second step, we also incorporate the welfare effects that occur over the transition between a high-debt stationary equilibrium to a low-debt equilibrium. We find that the costs of debt reductions occurring over the transition can be substantial, depending on the policy. The short-run costs of debt reductions can easily outweigh the long-run gains. 4.1 Effects of Public Debt in Stationary Equilibrium In this section, we compute the long-run welfare maximizing debt/gdp ratio in our economy. Our analysis is motivated by the fact that in the textbook neoclassical model with lump-sum taxes, government debt is neutral and therefore has no impact on welfare. In a seminal paper, Aiyagari and McGrattan (1998) showed that in the presence of borrowing constraints and in the absence of complete markets, government debt has non-trivial welfare effects. Two quantitative results from their paper have been very influential for the following research. First, Aiyagari and McGrattan (1998) argue that the debt/gdp ratio that maximizes long-run welfare is around 2/3, which is also the historical average of debt/gdp in the US economy. Second, they also find that welfare effects of deviations from the long-run historical average are small. We show that these two conclusions drastically change in our calibration. In Figure 1, we plot the aggregate welfare changes for stationary equilibria for different public debt/gdp ratios, relative to the benchmark in which debt amounts to 2/3 of GDP. Figure 1 conveys a clear message: according to our calibration, stationary equilibria with lower debt/gdp ratios offer more aggregate welfare, compared to the benchmark. In our economy, the debt/gdp ratio maximizing long-run welfare is 0.8. Moreover, reducing debt/gdp relative to the long-run average results in large welfare gains, up to 0.6 percent of consumption per year for the average household in our economy. In the following, we analyze in greater detail the driving forces behind the long-run welfare effects of changes in government debt in an Aiyagari (1994) style incomplete markets model. In particular, we want to understand the driving forces that explain the differences between Aiyagari and McGrattan (1998) and our baseline calibration. The long-run welfare effects of government debt result from the interaction between public debt and private capital. The interplay between public debt and private capital accumulation is a consequence of 12

14 1 0.8 Our Model Aiyagari/McGrattan Welfare Change in Consumption Equivalents (in %) Public Debt/GDP Figure 1: Comparing Welfare Between Different Stationary Equilibria. In this exercise we plot the welfare change in consumption equivalent units implied by our baseline calibration (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa), relative to the benchmark in which public debt amounts to 2/3 of GDP (green diamond and vertical line). Two cases: (1) Our Baseline Economy (blue crosses);(2) Aiyagari and McGrattan s economy (black circles). the fact that the borrowing constraint a is exogenous and fixed. Because of this, an increase in government debt crowds out private capital accumulation by effectively relaxing the borrowing limit (Aiyagari and McGrattan), a fact which facilitates precautionary savings. Notice that not only households who are currently at the constraint profit from the effective relaxation of the borrowing limit, but also those households that fear to be constrained in the future. For our baseline calibration, the effect of public debt on private capital (private savings minus public debt) can be seen in Figure 2, where we depict assets (i.e. aggregate private savings) and the capital stock, relative to GDP. 14 Higher public debt/gdp ratios decrease capital. This means that the increase in assets demanded by households cannot compensate for the increase in asset supplied by the government. The reverse also holds. If government debt is reduced, the capital stock is crowded in. Households do not reduce their asset holdings as much as the government reduces its debt. As a result, the capital stock increases. If the capital stock in the baseline calibration is below its efficient level, crowding in of capital increases welfare, all other things equal. In a recent paper, Davila et al. (2011) show that in an environment in which markets are incomplete, inequality, in particular the nature of the income risk as well as the composition of income for the consumption-poor, are important determinants for the constrained efficient capital stock. Davila et al. (2011) show that if the earnings process is calibrated as in Castañeda, Díaz-Giménez, and Ríos-Rull (2003) - the method that we employ in this paper - the constrained efficient capital stock is much higher than its laissez-faire level. This finding is in line with our result that government debt reductions increase the private capital stock and raise welfare. 14 We study the relationship between public debt and borrowing limits in a separate paper, see Röhrs and Winter (2013). In the other project, we also allow for endogenous borrowing limits that result from a limited commitment problem. 13

15 Capital/Benchmark GDP Public Debt/GDP Assets/Benchmark GDP Public Debt/GDP 2 2 Income before taxes (GDP) Income after taxes Public Debt/GDP Public Debt/GDP Figure 2: Capital, Assets, Income Before Taxes, Income After Taxes. This figure shows the changes in selected aggregate economic variables (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa). In the benchmark public debt amounts to 2/3 of GDP (green diamond and vertical line). All variables relative to GDP in the benchmark. There are three channels, through which government debt reductions affect welfare in the long-run. which we describe each in greater detail in the following. 15 Level effect: Since government debt/gdp ratios lead to more private capital, they are associated with more production (see the lower panel in Figure 2). To the extent that more production translates into more consumption, crowding-in of private capital is welfare-enhancing. In general, we expect the level effect to be non-linear. Flodén (2001) finds that the level effect is concave around the benchmark debt/gdp ratio. Since we use a different calibration of the earnings process which is closer to Davila et al. (2011), we would expect that lower debt/gdp ratios (and hence more private capital), relative to the benchmark, are welfare-enhancing for a while. We will get back to this point later. Insurance effect: A decrease in public debt means that households effectively face tighter borrowing 15 We adopt the labels of Flodén (2001), who distinguishes a level effect, an uncertainty effect and an inequality effect. By re-labeling the inequality effect into income composition effect, we would like to stress the origin of inequality in the model, namely the fact the composition of income between households differs. 14

16 limits. Precautionary saving becomes more difficult. The economy accumulates more private capital. As a consequence, we observe a decrease in the interest rate r and an increase in the wage rate w (see Figure 3). This, by itself, makes it more difficult for households to self-insure against adverse productivity shocks. Consequently, the fraction of households which are at the borrowing constraint, as plotted in Figure 4, rises. Interest Rate Public Debt/GDP Wage Rate Public Debt/GDP Figure 3: Interest Rate and Wage Rate. This figure shows the changes in equilibrium prices for capital and labor (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa). In the benchmark public debt amounts to 2/3 of GDP (green diamond and vertical line). In addition, a fall in the interest rate and an increase in the wage rate increases uncertainty in total income, all other things equal. This is because the weight of the uncertain income component, namely labor income is increased relative to capital income, which is certain in our economy. As a result, uncertainty about consumption is amplified as well, and households experience an ex-ante welfare loss. Income composition effect: A reduction in government debt/gdp implies that the interest rate r falls, while the wage rate w increases. As a consequence, asset owners experience, on average, a loss in their income if there is more government debt, while those households who primarily depend on labor income, experience a gain. We call this the income composition effect. It is important to notice that the income composition effect would also exist if markets were complete and if idiosyncratic shocks were fully insurable. This distinguishes the income composition effect from the uncertainty channel that was previously outlined. In our environment, in which markets are incomplete, the impact of the income composition effect on aggregate welfare is even larger. This is because in our calibration, wealth is primarily held in order to self-insure against uninsurable income fluctuations. Households who own little wealth, and thus depend heavily on their labor income, are those that have been hit by a sequence of bad shocks. These households are also consumption-poor and thus matter more in the Utiliterian welfare criterion, given the concavity of the utility function. In sum, a reduction in government debt affects aggregate welfare in the long-run via a level effect, an insurance channel and through the composition of individual income. The impact of the level effect 15

17 20 Our Model Baseline using Aiyagari/McGrattan income process Fraction of Constrained Households (in %) Public Debt/GDP Figure 4: Fraction of Households at the Borrowing Constraint. This figure shows the changes in the fraction of constrained households (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa). In the benchmark public debt amounts to 2/3 of GDP (green diamond and vertical line). Two cases: Our Model (blue crosses); Baseline using Aiyagari/McGrattan income process (red squares). on aggregate welfare is ambiguous, depending on the efficient level of capital. The income composition effect implies that lower debt/gdp increase aggregate welfare, whereas the insurance effect implies that welfare decreases for lower debt/gdp ratios. The relative strength of each channel depends on the degree of wealth and income inequality in the economy. Why is the welfare function non-monotone? Interestingly, the welfare function reveals that lowering debt/gdp beyond 0.8 does not increase welfare anymore. 16 At the first glance, this result is perhaps surprising, given that stationary equilibria with smaller debt/gdp ratios should ceteris paribus require lower tax rates, given that the debt service is smaller. We argue that the non-linear interaction between the level effect, the insurance effect and the incomecomposition effect can explain the shape of the welfare function. The interaction between the insurance effect and the income-composition effect can be illustrated by analyzing in greater detail the welfare effects of different population subgroups. More specifically, we divide the population into three groups by applying the following definition: 1. Poor: Households with zero or negative assets. 2. Rich: Group of households who together own 70 percent of total assets. 3. Middle class: Households who do not belong to either of the previous categories. 16 Actually, welfare starts to decline for debt/gdp ratios below -1. Welfare effects from lowering debt/gdp ratios beyond -1 can be obtained from authors upon request. 16

18 The poor do not receive asset income and finance their consumption with labor income only. The rich instead depend primarily on capital income. Finally, the middle class receives income from both labor and capital income. We keep the definition constant in all of the following experiments. 40 Welfare 80 Group Size Welfare Change in Consumption Equivalents (in %) Public Debt/GDP Group Size in % of Total Population Poor Middle Class Rich Public Debt/GDP Figure 5: Group Size and Welfare Change of Wealth Groups. This figure shows the changes in group size and welfare of a group (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa). In the benchmark public debt amounts to 2/3 of GDP (green diamond and vertical line). Three groups: (1) the poor have no assets or are in debt (black circles), (2) the rich own 70% of assets as a group (red squares), (3) the middle class is the rest of the households that are neither rich nor poor (blue crosses). At the benchmark equilibrium with debt/gdp of 66 percent, around 25 percent of all households are poor according to our definition. About 2 percent are rich and the rest belongs to the middle class. The relative group-sizes are not invariant to changes in the debt/gdp ratio, as the first panel of Figure 5 makes apparent. The lower the debt/gdp ratios, the more the number of poor households grow, at the expense of the number of middle-income households. 17 Intuitively, lower debt/gdp ratios discourage saving through depressing interest rates. This illustrates the insurance effect: lower debt/gdp ratios hamper the ability of households to self-insurance, and increase the fraction of households at the borrowing constraint. The group-specific welfare functions are shown in the second panel of Figure 5. The between-group differences are enormous. In particular, the welfare function of the rich is almost exactly the mirror image of the welfare function of the poor: poor households gain in the long-run as we move to lower debt/gdp ratios, while rich households lose. This reflects the income-composition effect. Poor households, who by definition do not possess assets, profit from the higher wage rate that prevails at lower debt/gdp ratios. Rich households instead suffer from the decrease in the interest rate. The fact that welfare is not monotonically increasing for lower debt/gdp ratios can thus be partly explained by the counteracting forces of the insurance effect (which leads to an impoverishment of a large part of the population if debt is reduced) and the income-composition effect (which increases the 17 Reducing government debt thus increases wealth inequality (but reduces consumption inequality) in the long-run, a result that was already emphasized by Flodén (2001). 17

19 welfare of the poor). The Contribution of Inequality. In the following, we want to shed more light on the relative contribution of inequality in explaining the differences between the welfare effects of government debt in our baseline case relative to the seminal paper by Aiyagari and McGrattan (1998) Welfare Change in Consumption Equivalents (in %) Baseline Economy Baseline with 1) Aiyagari/McGrattan income process Baseline with 1) and 2) Aiyagari/McGrattan borrowing limit -0.8 Baseline with 1), 2) and 3) Aiyagari/McGrattan common tax rate on both factors Baseline with 1)-3) and 4) Aiyagari/McGrattan risk aversion Aiyagari/McGrattan Public Debt/GDP Figure 6: Comparing Welfare in Different Economies. In this exercise we plot the welfare change in consumption equivalent units implied by our model (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa), relative to the benchmark in which public debt amounts to 2/3 of GDP (yellow diamond and vertical line). We consecutively changed parameters for the baseline economy: 1) Aiyagari/McGrattan s income process (AR(1) process with persistence of ρ = 0.6 and variance of σ = 0.3); 2) Borrowing limit=0; 3) One common tax rate on capital and labor; 4) Aiyagari/McGrattan s risk aversion (µ = 1.5) As can be seen from Figure 6, using the AR(1) earnings process from Aiyagari and McGrattan (1998) instead of our calibration procedure substantially attenuates the welfare effects of government debt. Clearly, as Table 4 shows, altering the income process leads to a sharp drop in the level of inequality generated by our model. Notice that in this step, as well as in all other steps that follow, we recalibrate the economy to be consistent with all other targets of our baseline calibration. Another difference between our setup and the one of Aiyagari and McGrattan (1998) is that we allow for borrowing, while Aiyagari and McGrattan (1998) impose a zero-borrowing constraint. Prohibiting borrowing in our model additionally decreases wealth inequality (which is obvious, given that there are no households in debt anymore, see Table 5). Interestingly, Figure 6 shows that this measure further mutes the welfare effects of government debt. Figure 6 further reveals that changing the earnings process and imposing a tighter borrowing limit together accounts for almost half of the differences between our welfare function and the one computed by Aiyagari and McGrattan (1998). All other checks, such as implementing the risk-aversion of Aiyagari and McGrattan (1998), their tax distortions etc., do not alter the welfare function substantially. 18 Importantly, Table 5 shows that changing these parameters 18 We also use a different growth rate and a different depreciation rate compared to Aiyagari and McGrattan (1998). This accounts for the gap between specification 4 and Aiyagari and McGrattan (1998). 18

20 Table 5: Distributional properties of baseline and other economies Q1 Q2 Q3 Q4 Q5 Gini Net financial assets Data -1.60% 0.10% 1.64% 8.29% 91.57% 0.9 Baseline Economy -1.57% 0.88% 3.92% 7.23% 89.54% 0.83 Baseline Economy with 1) 2.24% 9.63% 16.94% 26.08% 45.11% 0.43 Baseline with 1) and 2) 3.30% 10.15% 17.05% 25.70% 43.79% 0.41 Baseline with 1), 2) and 3) 3.60% 10.48% 17.23% 25.63% 43.06% 0.4 Baseline with 1)-3) and 4) 3.53% 10.40% 17.17% 25.63% 43.28% 0.4 Aiyagari/McGrattan 3.24% 10.07% 16.96% 25.71% 44.03% 0.41 Earnings Data -0.40% 3.19% 12.49% 23.33% 61.39% 0.62 Baseline 0.00% 2.38% 12.58% 22.73% 62.31% 0.62 Baseline with 1) 2.88% 10.98% 16.82% 26.18% 43.15% 0.41 Baseline with 1) and 2) 2.96% 11.05% 16.84% 26.11% 43.04% 0.41 Baseline with 1), 2) and 3) 2.78% 10.87% 16.79% 26.22% 43.34% 0.41 Baseline with 1)-3) and 4) 2.27% 10.49% 16.58% 26.40% 44.26% 0.43 Aiyagari/McGrattan 1.21% 9.70% 16.18% 26.85% 46.07% 0.45 * Quintiles (Q1-Q5) denote wealth (resp. earnings) of a group in percent of total wealth (resp. earnings). ** Last column denotes percent of population with no or negative assets. 1) Aiyagari/McGrattan s income process used for this economy (AR(1) process with persistence of ρ = 0.6 and variance of σ = 0.3) 2) Borrowing limit=0 for this economy 3) One common tax rate on capital and labor (same value as in Aiyagari/McGrattan) for this economy 4) Aiyagari/McGrattan s risk aversion for this economy (µ = 1.5). does not influence inequality either. Hence, this confirms our result that accounting for inequality is particularly important when calculating the welfare effects of government debt. Decomposition of Total Welfare Change. We now decompose the total welfare effect of public debt in its subcomponents level, insurance and income composition effect, following Flodén (2001). We then use these subcomponents to better understand the role that the income process (and therefore also inequality) plays for the total welfare effects of government debt. In order to do so, we do this decomposition for our baseline calibration using two different earnings processes, once with our earnings process and once using the AR(1) process of Aiyagari and McGrattan (1998). We recalibrate the economy to be consistent with the targets of our baseline calibration. Figure 7 shows the results. The following interesting pattern are visible. First of all, the insurance and the income composition effect show the expected sign. The income composition effect is positive for lower debt/gdp ratios, implying that rich households lose but poor households win in the long-run if the government reduces its debt. The insurance effect is negative, because lower debt/gdp ratios make it more difficult for households to self-insure. It is perhaps not surprising that both effects are weaker under the income process used by Aiyagari and McGrattan (1998), at least for debt/gdp ratios that are not too far away from the long-run average, given that the Aiyagari and McGrattan (1998) income process produces far less inequality. 19

21 Second of all, it is striking to see that, independently of the calibration, both the income composition and the insurance effect exactly offset each other, so that the overall welfare effect is entirely driven by the level effect. Again, this is true for debt/gdp ratios in the neighborhood of For ratios below zero, the income composition effect is stronger than the insurance effect, which means that the level effect is not the sole driver of the total welfare change. The third interesting observation is that level effect is clearly positive for our income process, whereas it is basically zero for the other one. This suggests that the average stationary consumption of goods and leisure is higher for lower debt/gdp ratios when there is more inequality. As a consequence, the overall stationary welfare function is steeper for lower debt/gdp ratios when there is more inequality, a fact that we described in the previous paragraph. 10 Benchmark Economy Aiyagari/McGrattan (1998) earnings process 10 Welfare Change in Consumption Equivalents (in %) Welfare Change in Consumption Equivalents (in %) Public Debt/GDP Public Debt/GDP Zero line Total Welfare Change Level Effect Insurance Effect Income Composition Effect Figure 7: Welfare Decompositon. We plot the welfare change in consumption equivalent units implied by our baseline calibration (on the ordinate, left panel) and the baseline calibration where we replace our earnings process with the AR(1) process of Aiyagari and McGrattan (1998) (on the ordinate, right panel) for various stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa), relative to the benchmark in which public debt amounts to 2/3 of GDP (yellow diamond and vertical line). We decompose total welfare (black solid line) into a level effect (blue line with crosses), an income decomposition effect (green line with circles) and an insurance effect (red line with squares). 4.2 Welfare Over the Transition Path Our previous analysis has shown that reducing government debt is welfare-improving in the long-run. From this finding, the following question emerges: given that less government debt implies more welfare, why do policies that aim at reducing government debt often face so much resistance in practice? 19 The fact that so far, we have ignored the short-run costs of reducing government debt, could partially explain this apparent contradiction. Moving from a stationary equilibrium characterized by a higher 19 Ponticelli and Voth (2011), for example, document that fiscal consolidation increase the likelihood of social unrest. 20

22 debt/gdp ratio to a stationary equilibrium with a lower debt/gdp ratio means that the government has to increase taxes. 20 In this section, we study to what extent the short-run costs stemming from higher taxes outweigh the long-run gains from living in an environment characterized by a lower debt/gdp ratio. 21 Structure. In order to keep the problem manageable, we impose several restrictions. Firstly, we calculate the welfare effects of a reduction in government debt from the long-run average debt/gdp of 0.66 to Secondly, we only adjust the proportional part of the tax rate, and keep the transfer (lumpsum) component constant. And thirdly, we increase taxes such that the relative distortions between capital and labor (ζ) remains constant. Our experiment can be interpreted as a change in personal income taxes which includes both income from labor and assets. We leave it to future research to study the optimal tax mix that implements the overall welfare-optimizing government debt/gdp ratio. Here, we are interested in computing the optimal curvature of the debt/gdp path between 0.66 and 0.60, which (through the government budget constraint) implies a specific curvature of the income tax path. This relatively small reduction is chosen because we believe it is realistic. All other results that we present in the remainder of this section also hold for a larger reduction. 22 Our experiments. We generate different curvatures by defining the debt path as a shape-preserving (quadratic) spline of the following form (see also Judd, 1998, p.232): { s left (t) = 2 3 s(t) : + s 0 t + s s0 2(ξ t 0) t2 if t [0, ξ] s right (t) = s left (ξ) + s (t ξ) + s Tp s 2(T p ξ) (t ξ)2 if t [ξ, T p ] where t is the time period, T p is the length of the whole policy period, ξ is a middle point, s left (t) is the spline at the left hand side of the middle point, s right (t) is the spline at the right hand side of the middle point, s 0 is the slope of the spline at the initial point, s is the slope of the spline at the middle point and s Tp is the slope at the end point. We study two different time spans, namely T p [15, 25]. The final stationary equilibrium value for the debt/gdp ratio is set to 0.60, i.e. s right (T p ) = The slope at the middle point, s, follows from the definition of the middle point and two other slopes: ) ( ) ξs0 + (T p ξ) s Tp s = 2 ( s left (ξ) 2 3 This leaves us with three parameters to define: ξ, s 0 and s Tp. Judd (1998) provides an algorithm to choose the middle point, ξ, such that the spline passes through initial and end point given s 0 and s Tp (Schumaker Shape-Preserving Interpolation). We choose s 0 and s Tp such that we obtain monotonously decreasing and differently curved lines for the debt paths. 20 In the following. we do not consider reductions in government expenditures, as government expenditures do not affect enter the utility of households/nor the production sector. Therefore, the welfare effects of debt reductions would be trivial in our context. 21 In a related paper, Desbonnet and Weitzenblum (2011) also compute the overall welfare effects of government debt, consisting of short-run (transition) and long-run (stationary) welfare, with the aim of finding the welfare-maximizing debt/gdp ratio. the Since debt reductions are costly in the short-run, they conclude that the government should increase debt with respect to the long-run average. 22 Because the welfare function in Figure 1is relatively steep around the long-run average of 0.66, a reduction to a debt/gdp ratio below 0.60 implies smaller stationary welfare gains and therefore larger overall (stationary+transition) losses. Results for a reduction from 0.66 to zero can be obtained from the authors upon request. T p 21

23 0.67 Paths of Public Debt Considered 0.34 Tax Paths in Response Public Debt/GDP Convex Concave Tax Rate Time Time Figure 8: Paths of Public Debt and Taxes Considered. Here we plot the paths of public debt/gdp ratio that we consider for our policy experiment (left panel) and the implied tax rate (right panel). More precisely, we choose one parameter, γ, to define both slopes in the following way: s 0 = 2/3 T p γ s Tp = 2/3 1 T p γ We consider the following values for the parameter, γ [ 5, 5 2, 5 3, 5 4, 1, 4 5, 3 5, 2 5], which yields the debt and tax paths shown in Figure 8. In addition, we also add two corner cases, one in which debt is reduced in the first period only and one in which debt reduction takes place in the last period only. Note that values of γ > 1 produce a concave path for debt (blue lines in Figure 8), meaning a more rapid adjustment of government debt in the beginning and more slowly adjustment towards the end. As a result, the tax rate is decreasing over time. Correspondingly, values of γ < 1 produce a convex path for debt (red lines in Figure 8), meaning that debt is reduced more slowly in the beginning and faster towards the end. The tax rate that implements this path of the debt/gdp ratio is increasing over time. The case of γ = 1 produces a linear path for the debt/gdp ratio such that debt is reduced by the same percentage of GDP each period (black line in Figure 8). Optimal curvature of the debt/gdp path. Figure 9 shows the development of overall welfare, i.e. the combination of short-run and long-run, for different slopes of the debt/gdp ratio. Two findings are worth mentioning. First, all curvatures lead to a reduction in overall welfare, which implies that the short-run costs of reducing debt/gdp outweighs the long-run gains from living in a stationary equilibrium with a debt/gdp ratio of Second, it is better to reduce debt slowly at the beginning, and faster towards the end. Note that there are important differences for different subgroups in the population, as shown in Appendix C. 22

24 0 Shape of Debt Path Convex Linear Concave fare change in consum mption equiv valents Welf Periods 25 Periods Figure 9: Welfare over the Transition. In this exercise we plot the welfare change in consumption equivalent units when including the transition path implied by our model for reduction of the debt/gdp ratio to 0.60 (on the ordinate) with different curvatures of the transition path (on the abscissa), relative to the benchmark stationary equilibrium in which public debt amounts to 2/3 of GDP. Welfare for the linear transition path is highlighted (in green). In order to understand why this shape of the tax path helps the government to reduce the welfare costs of debt reductions, it is important to consider the role of borrowing limits. Suppose that households could borrow and save freely. Assume further that only labor income was taxed and that general equilibrium effects on prices were absent. In this environment, households would work more in periods in which the tax burden is low, and less in periods in which the tax burden is high. Households would use borrowing and saving in order to smooth out their consumption across periods. The presence of borrowing constraints makes this policy partially infeasible. In particular, borrowing constraints hamper the adjustment to a decreasing tax path, in which taxes are initially high and decreasing over time. In this case, households would prefer to work less initially and borrow more (or save less) in order to sustain their consumption. In later periods, when the tax burden is relatively low, they would like to work more and repay their debt. If borrowing is restricted, a decreasing tax path induces additional welfare costs for those households, since their consumption is lower than optimal in periods in which the constraint is binding. In our framework, we do not only tax labor income, but also asset income. It is well-known from the literature that studies optimal capital income taxes that taxing capital initially, when it is predetermined, is efficient in the sense that it avoids distortions. Considering asset income taxes alone would imply that the cost-minimizing tax plan is decreasing over time. Therefore, the government faces a trade-off between choosing a decreasing tax path in order to avoid distorting the supply of assets, and an increasing tax path in order to facilitate labor tax smoothing. Our finding that tax paths which imply an increasing tax rate reduce welfare by less than tax paths over which the tax rate is decreasing indicate that the second motive is stronger. Clearly, the net effect depends crucially on the degree of inequality in the economy, in particular on the fraction of households that is affected by borrowing constraints. Given the textbook results on optimal capital taxation under full commitment (see e.g. Ljungqvist 23

25 and Sargent (2004) Ch. 15), it is tempting to conclude that this trade-off between efficiency and borrowing constraints could be avoided by just taxing asset income in the first period. According to the textbook, this policy does not generate distortions since assets are pre-determined and the tax change is unanticipated. Moreover, it should not be harmful to those at the borrowing constraint, since, by definition, they do not hold assets anyway and are therefore not affected by the tax increase. As we will see below, this logic is not quite right. Borrowing constraints turn out to matter, even in this case. Before we analyze this proposal in greater detail, we turn to the impact of the time span on the welfare effects of debt reductions. Extending the time horizon Figure 9 also reveals that extending the time span during which the tax increases from 15 to 25 periods is welfare-improving. This is intuitive, given that a longer time horizon implies that the per-period tax burden is lower, which makes tax smoothing easier for households. However, extending the number of periods does not affect ordering of the preferred policies, nor does it affect our overall conclusion that the short-run costs more than offset the long-run gain. One period asset income tax change. We now analyze the effects of an unanticipated increase in the asset income tax rate, which takes place in the first period only. This policy exploits the fact that assets are pre-determined, and therefore reduces government debt without generating distortions. In addition, the tax change does not directly affect households at the borrowing constraint, given that only those households with positive assets are taxed. However, as we will argue later in this subsection, this policy indirectly affects households at the constraint through price and tax changes implied by the general equilibrium feature of our model. The welfare effects of a one-period increase in the asset income tax are plotted in Figure 10. Interestingly, the combined welfare effect consisting of short-run and long-run are positive only for small debt reductions. For large debt reductions, it is negative. In all cases, the overall welfare effects are well below the long-run gains. Comparing the overall welfare effect with the long-run welfare changes alone therefore suggests that the short-run costs of reducing government debt are substantial, despite the fact that the government manages to raise the amount of tax revenues needed to reduce government debt without distorting households optimal decisions. In the following, we analyze the channels which have adverse effects on welfare. In Figure 11, we plot the sum of discounted utilities for the average household in different subpopulations along the transition after the government reduces public debt/gdp from 0.66 to We distinguish between rich, middle-class and poor households. The definition of our subgroups is identical to Section 4.1. The average welfare of rich households is below the benchmark welfare in all periods. Poor households instead, i.e. those with zero or negative financial assets, benefit in all periods, implying that their average welfare is higher than at the benchmark. Middle-class households, i.e. those with some assets, also benefit, with the exception of the first periods in the beginning of the transition. Interestingly, the average welfare of both poor and middle-class households is lower during the transition, relative to the welfare that is achieved in the final stationary equilibrium. This explains why average welfare of the total population, including the transition, is lower than welfare in the new stationary equilibrium, recall 24

26 Welfare Change in Consumption Equivalents Public Debt/GDP Welfare in Stationary Equilibrium with Changes in Both Asset and Labor Income Tax Welfare in Stationary Equilibrium with Changes in the Asset Income Tax Only Welfare including Transition with Changes in Asset Income Tax Only Figure 10: Welfare over the Transition with a 1-Period Policy. In this exercise we plot the welfare change in consumption equivalent units including the transition path (on the ordinate) for different stationary equilibria that differ with respect to the public debt/gdp ratio (on the abscissa), relative to the benchmark stationary equilibrium in which public debt amounts to 2/3 of GDP. The reduction in the debt/gdp ratio is financed by a 1 period increase in the asset income tax. -25 Poor -25 Middle class -25 Rich Value Function Value Function Value Function Time Periods Time Periods Time Periods New Stationary Equilibrium Including Transition Path Calibrated Benchmark Stationary Equilibrium New Stationary Equilibrium Figure 11: Value functions of Different Subgroups. In this figure, we plot the path of value functions of poor households (left panel), middle class households (middle panel) and rich households (right panel) for a reduction of the public debt/gdp ratio to 0.60 financed by a 1 period increase in the asset income tax rate. Figure 10. Clearly, households that own assets (the rich and the middle-class) suffer a welfare loss because they bear the higher asset income tax. Higher taxes today imply a lower debt/gdp ratio and, from the government s budget constraint, lower asset income taxes in the future. This can be seen from the top 25

27 0.8 5 Public Debt/GDP Time Periods Asset Income Tax Time Periods Capital/Benchmark GDP Time Periods After-Tax Labor Income Time Periods Aggregate Labor Supply Time Periods Wage Rate Time Periods New Stationary Equilibrium Including Transition Path Calibrated Benchmark Stationary Equilibrium Figure 12: Path of Variables after 1 Period Policy. In this figure, we plot the path of variables for a particular 1 period policy. Public debt/gdp is reduced to 0.60 financed by a 1 period increase in the asset income tax. panel of Figure 12. Because there is wealth mobility, implying that there is a positive probability that are rich today are less wealthy tomorrow, our policy experiment implies redistribution of expected net worth from rich to poor households. Therefore, the increase in asset income taxation constitutes a negative wealth shock for all asset owners. As a consequence, wealthy households work harder, which increases the aggregate supply of labor, see the second panel of Figure 12. The path of aggregate variables depicted in Figure 12, in particular of the aggregate capital stock, also implies that the interest rate declines as we move from the benchmark equilibrium to the new stationary equilibrium where debt/gdp is equal to This exerts an additional negative welfare effect on asset owners. In turn, poor households suffer from a decline in wages on impact, which is caused by increase in labor supply from asset owners. However, as can be seen from the third panel of Figure 12, wages soon start to rise to a level that is higher than the one at the benchmark. Interestingly, welfare of poor households increases almost one-to-one with wages (recall the left panel of Figure 12). There is one exception, however: wages are initially below the benchmark, whereas welfare along the transition is always above the benchmark. This suggests that welfare along the transition is determined by factors other than wages. One such factor is the future development of the asset income tax, which also affects the current poor, since there is a certain probability that they will become wealthy in the future. Hence, the future decline in the 26

28 Table 6: Increase in Asset Income Tax Required to Finance Reductions of Public/Debt to GDP New Debt/GDP ratio Asset Income Tax % % 0 449% asset income tax increases the expected wealth of households, even if they do not own assets today. This explains why the welfare of the poor is higher compared the benchmark, despite the fact that wages are lower. In essence, welfare of poor households is on average higher than at the benchmark because they increase their consumption of goods and leisure, anticipating their future increase in income, which is due to higher wages and lower taxes. Notice that this is only possible for those poor households who are not at the borrowing constraint. Since this is the case for a substantial fraction of the population, the average welfare of poor households is lower during the transition, relative to the new stationary equilibrium. Therefore, we conclude that borrowing constraints are a major determinant of the welfare effects of government debt reductions, independently of the kind of tax instrument that the government uses. In the remainder of this subsection, we would like to stress that our experiments regarding the one-period increase in the asset income tax rate is not only of theoretical interest, but is also relevant for public policy. During the recent European debt crisis, a popular proposal has been to introduce an unanticipated wealth tax. Examples include Bach (2012), International Monetary Fund (2013) or Deutsche Bundesbank (2014). There is also an historical example for the introduction of an unexpected wealth tax: the La Patrimoniale, implemented in Italy in Our experiment can be interpreted as a wealth tax if the tax rate on asset income exceeds one. Table 6 shows that even small debt/gdp reductions require a tax rate on asset income above one. In order to reduce debt/gdp from 0.66 to 0.6, for example, a tax rate of almost 450% is needed. The findings from this section suggest that the welfare effects of an unexpected introduction of wealth taxes could have small positive effects on aggregate welfare, as long as the tax rate is modest. It is important to note that our results are derived from an economy which is calibrated according to US data, and may be different for other countries. Majority voting vs. Utilitarian welfare criterion. We now analyze which of the debt reduction policies are politically feasible. We assume that a certain policy is feasible if it is supported by a majority of households. Analyzing the political support of debt reduction policies is a topical question, given that many developing countries will have to implement debt reduction plans in the near future. Reducing government debt has been proven difficult in the past. Our results are mixed. As Figure 13 shows, 23 At that time, 0.6% of the wealth saved in bank accounts were taxed. See Frankfurter Allgemeine Zeitung, July 20, 2011 (http : // was ist bloss in italien los html), retrieved on June 10,

29 Majority Voting over the Transition Majority Voting 1 PeriodPolicies Fra ction of Pro o-reform Vo oters Periods 25 Periods Convex Linear Concave Shape of Debt Path ro-reform Voters Fra action of Pr Public Debt / Figure 13: Majority Voting. In this figure, we plot the fraction of pro-reform voters for different policies. On the left hand side, the pro-reform voters for multi-period policies with differently shaped debt paths are shown. Public debt/gdp in this case is reduced to 0.6 financed by the asset and labor income tax. On the right hand side, the pro-reform voters for 1-period policies ending at different public debt/gdp ratios are shown. The reductions in debt are financed by the asset income tax only in this case. only policies where the government increases asset income in the first period turn out to be politically feasible. The explanation for this is as follows. We know from our previous findings that poor households are better-off under this policy, relative to the benchmark. Since there are many poor households in our economy, this policy is supported the majority of the population. All other debt reduction strategies that we have analyzed previously by jointly increasing labor and capital taxes such that the relative distortion ζ between the two remains constant find very little support, if at all. This can be seen from the left panel of Figure 13. Perhaps surprising is the fact that the debt paths that are preferred if we apply the Utilitarian welfare criterion, namely concave paths, do worse than convex ones, once we look at majority voting do worse than convex ones. The reason for this switch is the fact that the Utilitarian welfare criterion is mainly driven by those households that have a high marginal utility of consumption, i.e. those at the borrowing constraint. Instead, the support of a policy depends on the fraction of households that gain with respect to the benchmark. This means that under a concave path of debt/gdp, everybody loses with respect to the benchmark. If the government instead reduces debt/gdp by following a convex path, some households achieve a higher utility with respect to the benchmark, namely the poor that are not at the borrowing constrained. However, in the convex scenario, those households at the borrowing constrained experience such a large welfare loss that aggregate welfare is smaller than under the concave scenario, despite the fact that number of households that suffer during the transition is smaller. We therefore conclude that debt reductions are only politically feasible if the government can generate enough revenues by increasing asset income taxes unexpectedly in the first period. Clearly, the fact that this policy occurs unexpectedly is important. One might object that this is not possible, given the political implementation lag that necessarily arises in a democracy. We therefore also conducted experiments where we allowed for an implementation lag, and find that this substantially reduces the support. 24 This suggests that other mechanisms need to be found in order to compensate those that 24 Results are available upon request. 28

30 lose during the transition, i.e. through higher redistribution. We leave this to future research. 5 Conclusion In this paper, we analyzed the welfare effects of government debt reductions, both in stationary equilibrium and over the transition. By comparing stationary equilibria, we found that quantitatively sizable welfare gains can be achieved by reducing government debt. However, these welfare gains disappear once the transition is incorporated. A key insight from our analysis is that both the welfare effects in the long-run and in the short-run depend on the distribution of wealth and income. By lowering debt, the government raises the amount of capital available for production, which in turn increases the equilibrium wage rate and thus benefits those households who depend heavily on labor income. During the transition towards a stationary equilibrium with a lower debt/gdp ratio, the existence of borrowing constraints prevents poor households who do not own assets from consumptionsmoothing. Constrained households are forced to reduce their consumption, because taxes increase during the transition. Therefore, reducing public debt does not benefit the poor as much as it would if borrowing constraints were absent. Households who own assets suffer because the debt reduction depresses the interest rate. At the aggregate level, this explains why the transitional welfare costs more than outweigh the long-run gains that can be achieved in the new stationary equilibrium with a lower debt/gdp ratio. Our results have important implications for the design of debt reduction strategies. The government can somehow limit the transitional welfare costs by following a non-linear path to reduce government debt. Along this path, the reduction in public debt (and therefore also the increase in the tax rate) is initially small and increasing over time, a strategy which is preferred by those households at the constrained. In the paper, we analyzed a debt reduction that decreased debt/gdp from 0.66 to It is worth noting that our results would also go through for larger debt reductions. Given the shape of the welfare function, larger debt reductions would yield a smaller increment of the long-run, welfare gain in the new stationary state. Since larger debt reductions produce larger transitional costs, it follows that the overall welfare loss would be even larger. An interesting application of our framework is the Euro area, where currently many proposals to curb government debt are discussed. One of these proposals is to outsource all debt of the individual member countries which exceeds the threshold of so-called Maastricht criteria to a special fund. From this fund, a certain fraction will have to be repaid during a fixed number of predetermined years, by a joint effort of all Euro zone countries. 25 According to our results, which were obtained for the U.S. economy, the debt repayment scheme should be non-linear. We leave it for future research to calibrate our model to the Euro area and to check whether our findings are robust. In the following, we want to conclude by stating other interesting extensions of our framework. First of all, borrowing constraints play an important role in our analysis. They govern to what extent 25 This proposal, also labeled Schuldentilgungsfond, is due to, inter alia, the German Council of Economic Advisors. A time horizon of 20 to 25 years was put forward. See Frankfurter Allgemeine Zeitung, html, retrieved July 13,

31 households are indebted, a crucial determinant of the overall welfare effects. Moreover, the response of households to changes in distorting taxes also depends on their ability to borrow. We assumed that borrowing limits are exogenous, and do not change if public policy changes. In an extension to this project, we relax this assumption and endogenize borrowing constraints by assuming limited commitment (Röhrs and Winter, 2013). Our results suggest that endogenous borrowing constraints attenuate the negative effects of government debt. Another interesting extension could be the introduction of aggregate shocks as an additional motive to smooth taxes, along the lines of Barro (1979), Lucas and Stokey (1983), Aiyagari et al. (2002) or Heathcote (2005). In our paper, we followed the approach of Castañeda, Díaz-Giménez, and Ríos-Rull (2003) to generate wealth and earnings inequality. An alternative approach would be to assume stochastic shocks to households discount factors, as proposed by Krusell and Smith (1998). This approach is likely to lead to different welfare effects of government debt, and we leave it for future research to analyze whether the conclusions of our paper are robust to alternatives ways to generate inequality. The contribution of our project is to highlight the importance of inequality, and to compare our calibration to the calibration of Aiyagari and McGrattan (1998). The closed-economy assumption of our model also deserves further justification, given that it is crucial for our results. If all US bonds were held abroad, then there would be no general equilibrium effect on aggregate prices within the US, which were crucial for our welfare effects. While it is true that a substantial share of US government debt is owned by foreigners, it is also the case that there is empirical evidence showing that changes in government debt have a sizable impact on aggregate prices (see e.g. Laubach (2009)). Last but not least, we restricted our attention to affine taxes. It would be also interesting to allow for more sophisticated forms of non-linear tax schedules, as for example in Conesa, Kitao, and Krueger (2009). In recent work, Heathcote, Storesletten, and Violante (2014) analyzes the optimal degree of tax progressivity. We leave it for future research to study optimal debt reduction policies using progressive taxation. 30

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35 Appendix A: Detrended Formulation of the Households Maximization Problem In our model, there is a balanced growth path along which variables will be growing at the rate of technology growth. To find the stationary equilibrium of the model or to compute the transition from one stationary equilibrium to another it is useful to first detrend variables with respect to this exogenous productivity growth component to obtain a formulation where variables are constant in the balanced growth equilibrium. (This procedure was also used in the earlier literature, for example by Aiyagari and McGrattan, 1998 and Flodén, 2001). Denote a detrended variable by tilde : x = x Y. The present value of lifetime utility (for a Cobb-Douglas can then be denoted as follows: U({ c t } t=1,2,..., {1 l t } t=1,2... ) = E 0 t=0 β t Y η(1 µ) t u( c t, 1 l t ) Now using the fact that Y t = Y 0 (1 + g) t, where Y 0 is output in period 0, we can write: U({ c t } t=1,2,..., {1 l t } t=1,2... ) = Y η(1 µ) 0 E 0 where β = β (1 + g) η(1 µ). = Y η(1 µ) 0 E 0 β t (1 + g) tη(1 µ) u( c t, 1 l t ) t=0 β t u( c t, 1 l t ) Similarly, we can find a detrended version of the household budget constraint by dividing it by Y t : t=0 c t Y t + Y t+1 Y t a t+1 Y t+1 = w t Y t ɛl t + (1 + r t ) a Y t + χ t Y t c t + (1 + g)ã t+1 = w t ɛl t + (1 + r t )ã t + χ t Also the borrowing constraint can be detrended: ã t+1 ã t The resulting recursive formulation in detrended variables is given by: W t (ã, ɛ; θ) = max ã, c,l Y η(1 µ) 0 u( c, 1 l) + β ɛ π(ɛ ɛ)w t+1 (ã, ɛ ; θ ) s.t. c + (1 + g)ã = wɛl + (1 + r)ã + χ ã ã θ = Γ[θ] B: Definition of the Consumption Equivalent Welfare Change A new stationary equilibrium versus the benchmark. The consumption equivalent welfare change for the average household is defined as the percentage change in consumption that the household must incur in the old situation in order to be indifferent between staying in the old situation and being in a 34

36 new stationary equilibrium with different policies for debt and taxes. Let the old stationary equilibrium be denoted by the subscript old and be characterized by a (detrended) debt level b old = B old Y old and a resulting density θ old. In our computations this point of comparison will always be the benchmark equilibrium with b old = 2 3. Let the new situation be denoted by the subscript new and characterized by the debt level b new b old and a resulting density θ new. Using this notation, the consumption equivalent change for the average household, x old new, is defined as follows: W old (ã, ɛ; x old new )dθ old (ã, ɛ) = W new (ã, ɛ)dθ new (ã, ɛ) E 0 t=0 E 0 t=0 β t ( (c old (ã, ɛ)(1 + x old new )) η (1 l old (ã, ɛ)) 1 η) 1 µ 1 µ β t ( c new (ã, ɛ) η (1 l new (ã, ɛ)) 1 η) 1 µ Solving this equation for x old new we obtain: 1 µ dθ new (ã, ɛ) β Et=0 t=0 t (c new(ã,ɛ) η (1 l new(ã,ɛ)) 1 η ) 1 µ 1 µ dθ new (ã, ɛ) x old new = β Et=0 t=0 t (c old(ã,ɛ) η (1 l old (ã,ɛ)) 1 η ) 1 µ 1 µ dθ old (ã, ɛ) ( ) 1 Wnew η(1 µ) (ã, ɛ)dθ new (ã, ɛ) = 1 Wold (ã, ɛ)dθ old (ã, ɛ) dθ old (ã, ɛ) = 1 η(1 µ) If x old x new is positive, the average household would prefer changing to the new equilibrium, even without being compensated. If instead x old x new is negative, compensation in terms of x old x new consumption units is required in order to make the household indifferent. 1 Comparing the transitional path to a new stationary equilibrium and the benchmark. When we include the transitional path into our considerations, we compare welfare of the average household in period 0 when a change in policy is announced and when staying at the benchmark. Let the detrended debt level in period 0 be denoted by b 0 and the density by θ 0. We assume that households are surprised by the new policy. Similar to above we will denote the old situation with the subscript old and the new situation with the subscript new. The only difference is that we now use subscripts t (since we are not always in the stationary equilibrium) and the initial debt level b 0 and density θ 0 are the same in both situations. The consumption equivalent welfare change for the average household x old new is again defined as the percentage change in consumption in situation old that makes the household indifferent between staying in old and going to new: W old,t=0 (ã, ɛ; x old new )dθ 0 (ã, ɛ) = W new,t=0 (ã, ɛ)dθ 0 (ã, ɛ) E 0 t=0 E 0 t=0 β t ( (c old,t (ã, ɛ)(1 + x old new )) η (1 l old,t (ã, ɛ)) 1 η) 1 µ 1 µ β t ( c new,t (ã, ɛ) η (1 l new,t (ã, ɛ)) 1 η) 1 µ 1 µ dθ 0 (ã, ɛ) dθ 0 (ã, ɛ) = 35

37 Solving this equation for x old new we obtain: x old new = ( ) 1 Wnew,t=0 η(1 µ) (ã, ɛ)dθ 0 (ã, ɛ) 1 Wold,t=0 (ã, ɛ)dθ 0 (ã, ɛ) C: Welfare for Wealth Groups 0 Shape of Debt Path Convex Linear Concave valents mption equiv e in consum fare change Welf Poor Middle Class Rich Total Figure 14: Welfare for Wealth Groups. In this exercise we plot the welfare change in consumption equivalent units including the transition path (on the ordinate) for different paths of public debt/gdp ratio (on the abscissa), relative to the benchmark stationary equilibrium in which public debt amounts to 2/3 of GDP. The reduction in debt is financed by a tax on both capital and labor income. 36