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Pablo D'Erasmo, Enrique G. Mendoza, Distributional Incentives in an Equilibrium Model of Domestic Sovereign Default, Journal of the European Economic Association, Volume 14, Issue 1, 1 February 2016, Pages 7–44, https://doi.org/10.1111/jeea.12168
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Abstract
Europe's debt crisis resembles historical episodes of outright default on domestic public debt about which little research exists. This paper proposes a theory of domestic sovereign default based on distributional incentives affecting the welfare of risk-averse debt and nondebtholders. A utilitarian government cannot sustain debt if default is costless. If default is costly, debt with default risk is sustainable, and debt falls as the concentration of debt ownership rises. A government favoring bond holders can also sustain debt, with debt rising as ownership becomes more concentrated. These results are robust to adding foreign investors, redistributive taxes, or a second asset.
1. Introduction
The seminal study by Reinhart and Rogoff (2011) identified 68 episodes in which governments defaulted outright (i.e., by means other than inflation) on their domestic creditors in a cross-country database going back to 1750. These domestic defaults occurred via mechanisms such as forcible conversions, lower coupon rates, unilateral reductions of principal, and suspensions of payments. Reinhart and Rogoff also documented that domestic public debt accounts for a large fraction of total government debt in the majority of countries (about two thirds on average) and that domestic defaults were associated with periods of severe financial turbulence, which often included defaults on external debt, banking system collapses, and full-blown economic crises. Despite these striking features, the authors also found that domestic sovereign default is a “forgotten history” that remains largely unexplored in economic research.
The ongoing European debt crisis also highlights the importance of studying domestic sovereign default. In particular, four features of this crisis make it more akin to a domestic default than to the typical external default that dominates the literature on public debt default. First, countries in the Eurozone are highly integrated, with the majority of their public debt denominated in their common currency and held by European residents. Hence, from a European standpoint, default by one or more Eurozone governments means a suspension of payments to “domestic” agents, instead of external creditors. Second, domestic public-debt/GDP ratios are high in the Eurozone in general, and very large in the countries at the epicenter of the crisis (Greece, Ireland, Italy, Spain, and Portugal). Third, the Eurozone's common currency and common central bank rule out the possibility of individual governments resorting to inflation as a means to lighten their debt burden without an outright default. Fourth, and perhaps most important from the standpoint of the theory proposed in this paper, European-wide institutions such as the European Central Bank (ECB) and the European Commission are weighting the interests of both creditors and debtors in assessing the pros and cons of sovereign defaults by individual countries, and creditors and debtors are aware of these institutions' concern and of their key role in influencing expectations and default risk.1Hall and Sargent (2014) document a similar situation in the process by which the US government handled the management of its debt in the aftermath of the Revolutionary War.
Table 1 shows that the Eurozone's fiscal crisis has been characterized by rapid increases in public debt ratios and sovereign spreads that coincided with rising government expenditure ratios. The table also shows that debt ownership, as proxied by Gini coefficients of wealth distributions, is unevenly distributed in the seven countries listed, with mean and median Gini coefficients of around two thirds.2
. | Gov. debt . | Gov. exp. . | Spreads . | Gini wealth . | ||||
---|---|---|---|---|---|---|---|---|
Moment (%) . | Avg. . | 2011 . | Avg. . | “Crisis peak” . | Avg. . | “Crisis peak” . | . | |
France | 34.87 | 62.72 | 23.40 | 24.90 | 0.08 | 1.04 | 0.73 | |
Germany | 33.34 | 52.16 | 18.80 | 20.00 | – | – | 0.67 | |
Greece | 84.25 | 133.09 | 18.40 | 23.60 | 0.37 | 21.00 | 0.65 | |
Ireland | 14.07 | 64.97 | 16.10 | 20.50 | 0.11 | 6.99 | 0.58 | |
Italy | 95.46 | 100.22 | 19.40 | 21.40 | 0.27 | 3.99 | 0.61 | |
Portugal | 35.21 | 75.83 | 20.00 | 22.10 | 0.20 | 9.05 | 0.67 | |
Spain | 39.97 | 45.60 | 17.60 | 21.40 | 0.13 | 4.35 | 0.57 | |
Avg. | 48.17 | 76.37 | 19.10 | 21.99 | 0.22 | 7.74 | 0.64 | |
Median | 35.21 | 64.97 | 18.80 | 21.40 | 0.17 | 5.67 | 0.65 |
. | Gov. debt . | Gov. exp. . | Spreads . | Gini wealth . | ||||
---|---|---|---|---|---|---|---|---|
Moment (%) . | Avg. . | 2011 . | Avg. . | “Crisis peak” . | Avg. . | “Crisis peak” . | . | |
France | 34.87 | 62.72 | 23.40 | 24.90 | 0.08 | 1.04 | 0.73 | |
Germany | 33.34 | 52.16 | 18.80 | 20.00 | – | – | 0.67 | |
Greece | 84.25 | 133.09 | 18.40 | 23.60 | 0.37 | 21.00 | 0.65 | |
Ireland | 14.07 | 64.97 | 16.10 | 20.50 | 0.11 | 6.99 | 0.58 | |
Italy | 95.46 | 100.22 | 19.40 | 21.40 | 0.27 | 3.99 | 0.61 | |
Portugal | 35.21 | 75.83 | 20.00 | 22.10 | 0.20 | 9.05 | 0.67 | |
Spain | 39.97 | 45.60 | 17.60 | 21.40 | 0.13 | 4.35 | 0.57 | |
Avg. | 48.17 | 76.37 | 19.10 | 21.99 | 0.22 | 7.74 | 0.64 | |
Median | 35.21 | 64.97 | 18.80 | 21.40 | 0.17 | 5.67 | 0.65 |
Notes: Authors’ calculations are based on OECD statistics, Eurostat, ECSB, and Davies et al. (2009). “Gov. debt” refers to Total General Government Net Financial Liabilities (avg. 1990–2007); “Gov. exp.” corresponds to government purchases in National Accounts (avg. 2000–2007); “Spreads” corresponds to the difference between interest rates of the given country and Germany for bonds of similar maturity (avg 2000–2007). For a given country |$i$|, they are computed as |${{( {1 + r^i } )} / {( {1 + r^{Ger} }) - 1}}$| “Crisis peak” refers to the maximum value observed during 2008–2012 using data from Eurostat. “Gini wealth” refers to Gini wealth coefficients for 2000 from Davies et al. (2009), Appendix V.
. | Gov. debt . | Gov. exp. . | Spreads . | Gini wealth . | ||||
---|---|---|---|---|---|---|---|---|
Moment (%) . | Avg. . | 2011 . | Avg. . | “Crisis peak” . | Avg. . | “Crisis peak” . | . | |
France | 34.87 | 62.72 | 23.40 | 24.90 | 0.08 | 1.04 | 0.73 | |
Germany | 33.34 | 52.16 | 18.80 | 20.00 | – | – | 0.67 | |
Greece | 84.25 | 133.09 | 18.40 | 23.60 | 0.37 | 21.00 | 0.65 | |
Ireland | 14.07 | 64.97 | 16.10 | 20.50 | 0.11 | 6.99 | 0.58 | |
Italy | 95.46 | 100.22 | 19.40 | 21.40 | 0.27 | 3.99 | 0.61 | |
Portugal | 35.21 | 75.83 | 20.00 | 22.10 | 0.20 | 9.05 | 0.67 | |
Spain | 39.97 | 45.60 | 17.60 | 21.40 | 0.13 | 4.35 | 0.57 | |
Avg. | 48.17 | 76.37 | 19.10 | 21.99 | 0.22 | 7.74 | 0.64 | |
Median | 35.21 | 64.97 | 18.80 | 21.40 | 0.17 | 5.67 | 0.65 |
. | Gov. debt . | Gov. exp. . | Spreads . | Gini wealth . | ||||
---|---|---|---|---|---|---|---|---|
Moment (%) . | Avg. . | 2011 . | Avg. . | “Crisis peak” . | Avg. . | “Crisis peak” . | . | |
France | 34.87 | 62.72 | 23.40 | 24.90 | 0.08 | 1.04 | 0.73 | |
Germany | 33.34 | 52.16 | 18.80 | 20.00 | – | – | 0.67 | |
Greece | 84.25 | 133.09 | 18.40 | 23.60 | 0.37 | 21.00 | 0.65 | |
Ireland | 14.07 | 64.97 | 16.10 | 20.50 | 0.11 | 6.99 | 0.58 | |
Italy | 95.46 | 100.22 | 19.40 | 21.40 | 0.27 | 3.99 | 0.61 | |
Portugal | 35.21 | 75.83 | 20.00 | 22.10 | 0.20 | 9.05 | 0.67 | |
Spain | 39.97 | 45.60 | 17.60 | 21.40 | 0.13 | 4.35 | 0.57 | |
Avg. | 48.17 | 76.37 | 19.10 | 21.99 | 0.22 | 7.74 | 0.64 | |
Median | 35.21 | 64.97 | 18.80 | 21.40 | 0.17 | 5.67 | 0.65 |
Notes: Authors’ calculations are based on OECD statistics, Eurostat, ECSB, and Davies et al. (2009). “Gov. debt” refers to Total General Government Net Financial Liabilities (avg. 1990–2007); “Gov. exp.” corresponds to government purchases in National Accounts (avg. 2000–2007); “Spreads” corresponds to the difference between interest rates of the given country and Germany for bonds of similar maturity (avg 2000–2007). For a given country |$i$|, they are computed as |${{( {1 + r^i } )} / {( {1 + r^{Ger} }) - 1}}$| “Crisis peak” refers to the maximum value observed during 2008–2012 using data from Eurostat. “Gini wealth” refers to Gini wealth coefficients for 2000 from Davies et al. (2009), Appendix V.
Taken together, the history of domestic defaults and the risk of similar defaults in Europe pose two important questions. What accounts for the existence of domestic debt ratios exposed to default risk? Can the concentration of the ownership of government debt be a determinant of domestic debt exposed to default risk?
This paper aims to answer these questions by proposing a framework for explaining domestic sovereign defaults driven by distributional incentives. This framework is motivated by the key fact that a domestic default entails substantial redistribution across domestic agents, with all of these agents, including government debtholders, entering in the payoff function of the sovereign. This is in sharp contrast to what standard models of external sovereign default assume, particularly those based on the classic work of Eaton and Gersovitz (1981).
We propose a tractable two-period model with heterogeneous agents and noninsurable aggregate risk in which domestic default can be optimal for a government responding to distributional incentives. A fraction |$\gamma$| of agents are low-wealth (|$L$|) agents who do not hold government debt, and a fraction |$1 - \gamma$| are high-wealth (|$H$|) agents who hold the debt. The government finances the gap between exogenous stochastic expenditures and endogenous taxes by issuing non–state-contingent debt, retaining the option to default. In our benchmark case, the government is utilitarian, so the social welfare function assigns the weights |$\gamma$| and |$1 - \gamma$| to the welfare of |$L$| and |$H$| agents, respectively.
If the government is utilitarian and default is costless, the model cannot support an equilibrium with debt. This is because for any given level of debt that could have been issued in the first period, the government always attains the second period's socially efficient levels of consumption allocations and redistribution by choosing to default, and if default in period 2 is certain the debt market collapses in the first period. An equilibrium with debt under a utilitarian government can exist if default entails an exogenous cost in terms of disposable income. When default is costly, repayment becomes optimal if the amount of period-2 consumption dispersion that the competitive equilibrium with repayment supports yields higher welfare than the default equilibrium net of default cost.
Alternatively, we show that an equilibrium with debt can be supported if the government's payoff function displays a “political” bias in favor of bondholders, even if default is costless. In this case, the government's weight on |$H$|-type agents is higher than the actual fraction of these agents in the distribution of bond holdings. In this extension, the debt is an increasing function of the concentration of debt ownership, instead of decreasing as in the utilitarian case. This is because incentives to default become weaker as the government's weight on |$L$|-type agents falls increasingly below |$\gamma$|. The model with political bias also yields the interesting result that agents who do not hold public debt may prefer a government that weighs bond holders more than a utilitarian government. This is because the government with political bias has weaker default incentives, and can thus sustain higher debt at lower default probabilities, which relaxes a liquidity constraint affecting agents who do not hold public debt by improving tax smoothing.
We also explore three other important extensions of the model to show that the main result of the benchmark model—namely, the existence of equilibria with domestic public debt exposed to default risk—is robust. We examine extensions opening the economy so that a portion of the debt is held by foreign investors, introducing taxation as another instrument for redistributive policy and adding a second asset as an alternative vehicle for saving.
This work is related to various strands of the extensive literature on public debt. First, studies on public debt as a self-insurance mechanism and a vehicle that alters consumption dispersion in heterogeneous agents models without default include Aiyagari and McGrattan (1998), Golosov and Sargent (2012), Azzimonti etal. (2014), Floden (2001), and Heathcote (2005).3
A second strand is the literature on external sovereign default in the line of the Eaton and Gersovitz (1981) model (e.g., Aguiar and Gopinath 2006; Arellano 2008; Pitchford and Wright 2012; Yue 2010). 4 Also in this literature, and closely related, Aguiar and Amador (2013) analyze the interaction between public debt, taxes, and default risk, and Lorenzoni and Werning (2013) study the dynamics of debt and interest rates in a model in which default is driven by insolvency and debt issuance follows a fiscal rule.
A third strand is the literature on political economy and sovereign default, which also focuses mostly on external default (e.g., Amador 2003; Dixit and Londregan 2000; D' Erasmo 2011; Guembel and Sussman 2009; Hatchondo et al. 2009; Tabellini 1991). A few studies such as those of Alesina and Tabellini (1990) and Aghion and Bolton (1990) focus on political economy aspects of government debt in a closed economy, including default, and Aguiar et al. (2013) examine optimal policy in a monetary union subject to self-fulfilling debt crises.
A fourth important strand of the literature focuses on the consequences of default on domestic agents, the role of secondary markets, discriminatory versus nondiscriminatory default, and the role of domestic debt in providing liquidity (see Basu 2009; Broner et al. 2010; Broner and Ventura 2011; Brutti 2011; Di Casola and Sichlimiris 2014; Gennaioli et al. 2014; Guembel and Sussman 2009; Mengus 2014).5 As in most of these studies, default in our setup is nondiscriminatory, because the government cannot discriminate across any of its creditors when it defaults. Our analysis differs in that default is driven by distributional incentives.6
Finally, there is also a newer literature that is closer to our work in that it studies the tradeoffs between distributional incentives to default on domestic debt and the use of debt in infinite-horizon models with heterogeneous agents (see, in particular, D'Erasmo and Mendoza 2014; Dovis et al. 2014). In D'Erasmo and Mendoza, debt is determined by a fiscal rule, whereas in this paper, we model public debt as an optimal choice and derive analytical expressions that characterize equilibrium prices and the solution of the government's problem. Our work differs from that of Dovis et al. in that they assume complete domestic asset markets, so their analysis abstracts from the role of public debt in providing social insurance, although the non–state-contingent nature of public debt plays a central role in the distributional incentives we examine here and in the endogenous default costs studied in D'Erasmo and Mendoza (2014). In addition, Dovis etal. focus on the solution to a Ramsey problem that supports equilibria in which default is not observed along the equilibrium path, whereas in our work default is an equilibrium outcome.7
2. Model Environment
Consider a two-period economy inhabited by a continuum of agents with aggregate unit measure. Agents differ in their initial wealth position, which is characterized by their holdings of government debt at the beginning of the first period. The government is represented by a social planner with a utilitarian payoff who issues one-period, non–state-contingent debt, levies lump-sum taxes, and has the option to default. Government debt is the only asset available in the economy and is entirely held by domestic agents.
2.1. Household Preferences and Budget Constraints
All agents receive a nonstochastic endowment |$y$| each period and pay lump-sum taxes |$\tau_t$|, which are uniform across agents. Taxes and newly issued government debt are used to pay for government consumption |$g_t$| and repayment of outstanding government debt. The initial supply of outstanding government bonds at |$t = 0$| is denoted by |$B_0$|. Given |$B_0$|, the initial wealth distribution is defined by a fraction |$\gamma$| of households that are the |$L$|-type individuals with initial bond holdings |$b_0^L$|, and a fraction |$(1 - \gamma)$| that are the |$H$| types and hold |$b_0^H \gt b_0^L$|. These initial bond holdings satisfy market clearing: |$\gamma b_0^L + (1 - \gamma )b_0^H = B_0$|, which given |$b_0^H \gt b_0^L$| implies that |$b_0^H \gt B_0$| and |$b_0^L \lt B_0$|.
2.2. Government
At the beginning of |$t = 0$|, the government has outstanding debt |$B_0$| and can issue one-period, non–state-contingent discount bonds |$B_1 \in \mathcal {B} \equiv [0,\infty )$| at the price |$q_0 \ge 0$|. Each period, the government collects lump-sum revenues |$\tau _t$| and pays for |$g_t$|. Since |$g_0$| is known at the beginning of the first period, the relevant uncertainty with respect to government expenditures is for |$g_1$|, which is characterized by a well-defined probability distribution function with mean |$\mu _g$|. We do not restrict the sign of |$\tau _t$|, so |$\tau _t \lt 0$| represents lump-sum transfers.
The government has the option to default at |$t = 1$|. The default decision is denoted by |$d_1 \in \left\{ {0,1} \right\}$| where |$d_1 = 0$| implies repayment. The government evaluates the values of repayment and default as a benevolent planner with a social welfare function. In the rest of this section we focus on the case of a standard utilitarian social welfare function: |$\gamma u(c_1^L ) + (1 - \gamma )u(c_1^H )$|. The government, however, cannot discriminate across the two types of agents when setting taxation, debt, and default policies.
3. Equilibrium
The analysis of the model's equilibrium proceeds in three stages. First, we characterize the households' optimal savings problem and determine their payoff (or value) functions, taking as given the government debt, taxes, and default decision. Second, we study how optimal government taxes and the default decision are determined. Third, we examine the optimal choice of debt issuance that internalizes the outcomes of the first two stages.
3.1. Households' Problem
Given that income and taxes are homogeneous across agents, it has to be the case at equilibrium that |$b_1^H \gt b_1^L$|, which therefore implies that |$H$| types are never credit constrained. In contrast, whether |$L$| types are credit constrained or not depends on parameter values. This is less likely to happen the higher |$b_0^L ,B_0 \,\, \text {or}\,\,B_1 $|. Whenever the |$L$| types are constrained, the |$H$| types are the marginal investor and their Euler equation can be used to derive the equilibrium price. For the remainder of the paper we focus on equilibria in which |$L$| types are constrained |$(\text {i}.\text {e}.,b_0^L = b_1^L = 0)$|, to capture the feature of heterogeneous agents models with incomplete markets that a fraction of agents is always credit constrained endogenously, and public debt has the social benefit that it contributes to reduce the tightness of this constraint.
A similar expression can be obtained for the “risk-free” price (i.e., the bond price that arises in a model with full commitment) to show that the risk premium is nonnegative, and it is strictly positive if there is default in equilibrium. The premium is increasing in |$B_1 $| since the default set is increasing in |$B_1 $|. We also show in Section A.4 of the Online Appendix that the spread is a multiple of |$1/\beta (y - g_0 + (\gamma /(1 - \gamma ))B_0 )$|. As a result, the total date-0 resources available for consumption of the |$H$| types |$(y - g_0 + ({\gamma \mathord{\left/ {{{(1 - \gamma )}}} \right.}})B_0 )$| have a first-order negative effect on default risk spreads. This is because, as this measure of income rises, the marginal utility of date-0 consumption of |$H$| types falls, which pushes up bond prices. Changes in |$\gamma $| have a similar impact on this ratio also pushing spreads downward. However, as |$\gamma $| increases, default incentives also strengthen, as the welfare of debtholders is valued less. Thus, in principle, the response of spreads to increases in the concentration of debt ownership is ambiguous.
3.2. Government's Problem
3.2.1. Government Default Decision at |$t = 1$|.

Clearly, the maximum welfare under repayment is attained when |$\varepsilon = 0$|, which is also the efficient amount of consumption dispersion |$\varepsilon ^{SP}$|.13 Given that the only policy instruments the government can use, other than the default decision, are non–state-contingent debt and lump-sum taxes, it is straightforward to conclude that default is always optimal. This is because default produces identical allocations in a decentralized equilibrium as the socially efficient ones, since default produces zero-consumption dispersion with consumption levels |$c^L = c^H = y - g_1 $|. This outcome is invariant to the values of |$B_1$|, |$g_1$|, and |$\gamma$|. This result also implies that the model without default costs cannot support equilibria with domestic debt subject to default risk because default is always optimal.
The outcome is very different when default is costly. With |$\varphi (g_1 ) \gt 0$|, default still yields zero-consumption dispersion but at lower levels of consumption and therefore utility, since consumption allocations in the default state become |$c^L = c^H = (1 - \varphi (g_1 ))y - g_1 $|. This does not alter the result that the first-best social optimum is |$\varepsilon ^{SP} = 0$|, but what changes is that default can no longer support the consumption allocations of the first best. Hence, there is now a threshold amount of consumption dispersion in the decentralized equilibrium, |${\widehat {\varepsilon}} (\gamma )$|, which varies with |$\gamma $| and such that for |$\varepsilon \ge {\widehat {\varepsilon}} (\gamma )$| default is again optimal, but for lower |$\varepsilon $| repayment is now optimal. This is because when |$\varepsilon $| is below the threshold, repayment produces a level of social welfare higher than the one that default yields. Figure 1 also illustrates this scenario.
3.2.2. Government Debt Decision at |$t = 0$|.
If both types of agents were unconstrained in their bonds' choice, so that in particular |$\mu ^L = 0$|, and if there is no change in the risk of default (or assuming commitment to remove default risk entirely), so that |$E_{g_1 } [\Delta d\Delta W_1 ] = 0$|, then the optimality condition simplifies to |$u^{\prime}(c_0^H ) = u^{\prime}(c_0^L )$|. Hence, in this case, the social planner issues debt to equalize marginal utilities of consumption across agents at date 0, which requires simply setting |$B_1 $| to satisfy |$q(B_1 ,\gamma )B_1 = B_0 $|.
If |$H$|-type agents are unconstrained and |$L$|-type are constrained (|$({\text{i}}{\text{.e}}.,\,\mu _L \gt 0)$|), which is the scenario we are focusing on, and still assuming no change in default risk or a government committed to repay, the optimality condition reduces to |$u^{\prime}(c_0^H ) = u^{\prime}(c_0^L ) + (\eta \mu ^L )/q(B_1 ,\gamma )$|. Since |$\eta \lt 0$|, this result implies |$c_0^L \lt c_0^H $| because |$u^{\prime}(c_0^L ) \gt u^{\prime}(c_0^H )$|. Thus, the government's debt choice sets |$B_1 $| as needed to maintain an optimal, positive level of consumption dispersion. Moreover, since optimal consumption dispersion is positive, we can also ascertain that |$B_0 \gt q(B_1 ,\gamma )B_1 $|, which, using the government budget constraint, implies that the government runs a primary surplus at |$t = 0$|. The government borrows resources, but less than it would need to eliminate all consumption dispersion (which requires zero primary balance).
The intuition for the optimality of issuing debt can be presented in terms of tax smoothing and savings: date-0 consumption dispersion without debt issuance would be |$B_0 /(1 - \gamma )$|, but this is more dispersion than what the government finds optimal because by choosing |$B_1 \gt 0$| the government provides tax smoothing (i.e., reduces date-0 taxes) for everyone, which in particular eases the |$L$|-type agents' credit constraint and provides also a desired vehicle of savings for |$H$| types. Thus, positive debt increases consumption of |$L$| types (since |$({\text{since}} \, c_{0}^{L} = y - g_0 - B_0 + q(B_1 ,\gamma )$|) and reduces consumption of |$H$| types (since |$({\text{since}} \, c_{0}^{H} = y - g_0 + (\gamma/(1-\gamma)) (B_0 - q(B_1 ,\gamma )B_1))$|). However, issuing debt (assuming repayment) also increases consumption dispersion at |$t = 1$|, since debt is then paid with higher taxes on all agents, although |$H$| agents collect also the debt repayment. Thus, the debt is being chosen optimally to trade off the social costs and benefits of reducing (increasing) date-0 consumption and increasing (reducing) date-1 consumption for rich (poor) agents.
In the presence of default risk and if default risk changes near the optimal debt choice, the term |$E_{g_1 } [\Delta d\Delta W_1 ]$| enters in the government's optimality condition with a positive sign, which means that the optimal gap in the date-0 marginal utilities of the two agents widens even more. Hence, the government's optimal choice of consumption dispersion for |$t = 0$| is greater than without default risk, and the expected dispersion for |$t = 1$| is lower, because in some states of the world, the government will choose to default and consumption dispersion would then drop to zero. Moreover, the debt Laffer curve now plays a central role in the government's weakened incentives to borrow because as default risk rises, the price of bonds drops to zero faster and the resources available to reduce date-0 consumption dispersion peak at lower debt levels. In short, default risk reduces the government's ability to use non–state-contingent debt to reduce consumption dispersion.
3.3. Competitive Equilibrium with Optimal Debt and Default Policy
For a given value of |$\gamma $|, a competitive equilibrium with optimal debt and default policy is a pair of household value functions |$\upsilon ^i (B_1 ,\gamma )$| and decision rules |$b^i (B_1 ,\gamma )$| for |$i = L,H$|, a government bond pricing function |$q_0 (B_1 ,\gamma )$|, and a set of government policy functions |$\tau _0 (B_1 ,\gamma ),\,\tau _1^{d \in \{ 0,1\} } (B_1 ,g_1 ,\gamma ),\,d(B_1 ,g_1 ,\gamma ),\,B_1 (\gamma )$| such that:
given the pricing function and government policy functions, |$\upsilon ^i (B_1 ,\gamma )$| and |$b_1 ^i (B_1 ,\gamma )$| solve the households' problem;
|$q_0 (B_1 ,\gamma )$| satisfies the market-clearing condition of the bond market (equation (10));
the government default decision |$d(B_1 ,g_1 ,\gamma )$| solves problem (12);
taxes |$\tau _0 (B_1 ,\gamma )$| and |$\tau _1^d (B_1 ,g_1 ,\gamma )$| are consistent with the government budget constraints;
the government debt policy |$B_1 (\gamma )$| solves problem (16).
4. Quantitative Analysis
In this section, we study the model's quantitative predictions based on a calibration using European data. The goal is to show whether a reasonable set of parameter values can produce an equilibrium with debt subject to default risk and to study how the properties of this equilibrium change with the model's key parameters. Since the two-period model is not well suited to account for the time-series dynamics of the data, we see the results more as an illustration of the potential relevance of the model's argument for explaining domestic default rather than as an evaluation of the model's general ability to match observed public debt dynamics.14
4.1. Calibration
The model is calibrated to annual frequency, and most of the parameter values are set so that the model matches moments from European data. The calibrated parameter values are summarized in Table 2. The details of the calibration are available in Section A.5 of the Online Appendix. Note also that we assume a log-normal process for |$g_1 $|, so that |$\ln (g_1 )\sim N((1 - \rho _g )\ln (\mu _g ) + \rho _g \ln (g_0 ),\,\sigma _e^2 /(1 - \rho _g^2 ))$| and the cost of default takes the following functional form: |$\varphi (g_1 ) = \varphi _0 + (\bar g_1 - g_1 )/y$|.
Parameter . | . | Value . |
---|---|---|
Discount factor | |$\beta$| | 0.96 |
Risk aversion | |$\sigma$| | 1.00 |
Average income | |$y$| | 0.79 |
Low household wealth | |$b_0^L$| | 0.00 |
Average government consumption | |$\mu_g$| | 0.18 |
Autocorrel. G | |$\rho_g$| | 0.88 |
Std. dev. error | |$\sigma_e$| | 0.017 |
Initial government debt | |$B_0$| | 0.35 |
Output cost default | |$\varphi_0$| | 0.004 |
Parameter . | . | Value . |
---|---|---|
Discount factor | |$\beta$| | 0.96 |
Risk aversion | |$\sigma$| | 1.00 |
Average income | |$y$| | 0.79 |
Low household wealth | |$b_0^L$| | 0.00 |
Average government consumption | |$\mu_g$| | 0.18 |
Autocorrel. G | |$\rho_g$| | 0.88 |
Std. dev. error | |$\sigma_e$| | 0.017 |
Initial government debt | |$B_0$| | 0.35 |
Output cost default | |$\varphi_0$| | 0.004 |
Notes: Government expenditures, income, and debt values are derived using data from France, Germany, Greece, Ireland, Italy, Spain, and Portugal.
Parameter . | . | Value . |
---|---|---|
Discount factor | |$\beta$| | 0.96 |
Risk aversion | |$\sigma$| | 1.00 |
Average income | |$y$| | 0.79 |
Low household wealth | |$b_0^L$| | 0.00 |
Average government consumption | |$\mu_g$| | 0.18 |
Autocorrel. G | |$\rho_g$| | 0.88 |
Std. dev. error | |$\sigma_e$| | 0.017 |
Initial government debt | |$B_0$| | 0.35 |
Output cost default | |$\varphi_0$| | 0.004 |
Parameter . | . | Value . |
---|---|---|
Discount factor | |$\beta$| | 0.96 |
Risk aversion | |$\sigma$| | 1.00 |
Average income | |$y$| | 0.79 |
Low household wealth | |$b_0^L$| | 0.00 |
Average government consumption | |$\mu_g$| | 0.18 |
Autocorrel. G | |$\rho_g$| | 0.88 |
Std. dev. error | |$\sigma_e$| | 0.017 |
Initial government debt | |$B_0$| | 0.35 |
Output cost default | |$\varphi_0$| | 0.004 |
Notes: Government expenditures, income, and debt values are derived using data from France, Germany, Greece, Ireland, Italy, Spain, and Portugal.
We abstain from setting a calibrated value for |$\gamma $| and instead show results for |$\gamma \in [0,1]$|. Data from the United States and Europe suggest that the empirically relevant range for |$\gamma $| is [0.55, 0.85], and hence, when taking a stance on a particular value of |$\gamma $| is useful, we use |$\gamma = 0.7$|, which is the midpoint of the plausible range.15
4.2. Results
We examine the quantitative results in the same order in which the backward solution algorithm works. We start with the second period's utility of households under repayment and default. We then move to the first period and examine the equilibrium bond prices. Finally, we study the optimal government debt issuance |$B_1$| for a range of values of |$\gamma$|.
4.2.1. Second-Period Default Incentives for Given |$(B_1 ,g_1 ,\gamma )$|.
Figure 2 shows two intensity plots of the social welfare gain of default for the ranges of values of |$B_1$| and |$\gamma$| in the vertical and horizontal axes, respectively. Panel (i) is for a low value of government purchases, |${\underline g}_1$|, set three standard deviations below |$\mu _g$|, and panel (ii) is for a high value |$\bar g_1$| set three standard deviations above |$\mu _g$|. Figure A.2 in the Online Appendix shows the default decision rules that correspond to these two plots. The intensity of the color or shading in these plots indicates the magnitude of the welfare gain according to the legend shown to the right of each. The regions shown in white color and marked as “No Equilibrium Zone” represent values of |$(B_1 ,\gamma )$| for which the debt market collapses and no equilibrium exists. In this zone, there is no equilibrium because, at the given |$\gamma$|, the government chooses to default on the given |$B_1$| for all values of |$g_1$|.17

Social welfare gain of default exists. |$\bar \alpha (B_1 ,g_1 ,\gamma )$|.
The area in which the social welfare gains of default are well defined in these intensity plots illustrates two of the key mechanisms driving the government's distributional incentives to default. First, fixing |$\gamma$|. The area in which, the welfare gain of default is higher at higher levels of debt, or conversely the gain of repayment is lower. Second, keeping |$B_1$| constant, the welfare gain of default is also increasing in |$\gamma$| (i.e., higher concentration of debt ownership increases the welfare gain of default). This implies that lower concentration of debt ownership is sufficient to trigger default at higher levels of debt.18 For example, for a debt of 20% of GDP |$(B_1 = 0.20)$| and |$g_1 = \bar g_1$|, social welfare is higher under repayment if |$0 \le \gamma \le 0.10$|, but it becomes higher under default if |$0.10 \le \gamma \le 0.6$|, and for higher |$\gamma$|, there is no equilibrium because the government prefers default not only for |$g_1 = \bar g_1$| but for all possible |$g_1$|. If, instead, the debt is 35% of GDP, then social welfare is higher under default for all the values of |$\gamma$| for which an equilibrium exists.
The two panels in Figure 2 differ in that panel (ii) displays a well-defined transition from a region in which repayment is socially optimal |$(\bar \alpha (B_1 ,g_1 ,\gamma ) \lt 0)$| to one in which default is optimal (|$(\bar \alpha (B_1 ,g_1 ,\gamma ) \gt 0)$|) but in panel (i) the social welfare gain of default is never positive, so repayment is always optimal. This reflects the fact that higher |$g_1$| also weakens the incentives to repay.
4.2.2. Bond Prices for Given (|$(B_1 ,\gamma )$|).
Figure 3 shows |$q_0 (B_1 ,\gamma )$| as a function of |$\gamma$| for three values of |$B_1 (B_L \lt B_M \lt B_H )$| and a comparison with the prices from the model with the government committed to repay |$q^{RF}$|. The bond price functions are truncated when the equilibrium does not exist.

Figure 3 illustrates the following three key features of public debt prices discussed in Section 3.
The equilibrium price is decreasing in|$B_1$|for given|$\gamma$|(the pricing functions shift downward as|$B_1$|rises). This follows from a standard demand-and-supply argument: for a given |$\gamma$|, as the government borrows more, the price at which households are willing to demand the additional debt falls and the interest rate rises. This effect is present even without uncertainty, but it is stronger in the presence of default risk.19
Default risk reduces the price of bonds below the risk-free price and thus induces a risk premium. Prices are either identical or nearly identical for the values shown for |$B_1$| when |$\gamma \le 0.5$| since the probability of default is either zero or very close to zero. As |$\gamma$| increases above 0.5, however, the risk premium becomes nontrivial and bond prices subject to default risk fall sharply below the risk-free prices.
Bond prices are a nonmonotonic function of|$\gamma$|. When default risk is sufficiently low, bond prices are increasing in |$\gamma$|, but eventually they become a steep decreasing function of |$\gamma$|. Whether bond prices are increasing or decreasing in |$\gamma$| depends on the relative strength of a demand composition effect versus the effect of increasing |$\gamma$| on default incentives. The composition effect results from the fact that, as |$\gamma$| increases, |$H$|-type agents become a smaller fraction of the population and wealthier in per capita terms, and therefore, a higher |$q_0 (B_1 ,\gamma )$| is needed to clear the market. However, higher concentration of debt ownership strengthens distributional incentives to default, which pushes for lower bond prices. This second effect starts to dominate for |$\gamma \gt 0.5$|, producing bond prices that fall sharply as |$\gamma$| rises, whereas for lower |$\gamma$|, the composition effect dominates and prices rise gradually with |$\gamma$|.20
4.2.3. Optimal Debt Choice and Competitive Equilibrium.
Given the solutions for household decision rules, tax policies, bond pricing function, and default decision rule, we finally solve for the government's optimal choice of debt issuance in the first period (i.e., the optimal |$B_1$| that solves problem (16)) for a range of values of |$\gamma$|. Given this optimal debt, we can go back and identify the equilibrium values of the rest of the model's endogenous variables that are associated with the optimal debt choices.
Figure 4 shows the four main components of the equilibrium: panel (i) plots the optimal first-period debt issuance in the model with default risk, |$B_1 ^*(\gamma )$|, and in the case when the government is committed to repay so that the debt is risk free, |$B_1 ^{RF} (\gamma )$|; panel (ii) shows the equilibrium debt prices that correspond to the optimal debt of the same two economies; panel (iii) shows the default spread (the difference in the inverses of the bond prices); and panel (iv) shows the probability of default. Since the government that has the option to default can still choose a debt level for which it prefers to repay in all realizations of |$g_1$|, we identify with a square in panel (i) the equilibria in which |$B_1 ^* (\gamma )$| has a positive default probability. This is the case for all but the smallest value of |$\gamma$| considered (|$(\gamma = 0.05)$|), in which the government sets |$B_1^* (\gamma )$| at 20% of GDP with zero default probability.

It is evident from panel (i) of Figure 4 that optimal debt falls as |$\gamma$| increases in both the economy with default risk and the economy with a government committed to repay. This occurs because in both cases the government seeks to reallocate consumption across agents and across periods by altering the product |$q(B_1 ,\gamma )B_1$| optimally, and in doing this, the government internalizes the response of bond prices to its choice of debt. As |$\gamma$| rises, this response is influenced by the stronger default incentives and demand composition effect. At equilibrium, the latter dominates in this quantitative experiment, because panel (ii) shows that the equilibrium bond prices rise with |$\gamma$|. Hence, the government internalizes that as |$\gamma$| rises, the demand composition effect strengthens the demand for bonds, pushing bond prices higher, and as a result, it can actually attain a higher |$q(B_1 ,\gamma )B_1$| by choosing lower |$B_1$|. This is a standard Laffer curve argument: in the upward sloping segment of this curve, increasing debt increases the amount of resources that the government acquires by borrowing in the first period.
Although the Laffer curve argument and the demand composition effect explain why both |$B_1 ^* (\gamma )$| and |$B_1^{RF} (\gamma )$| are decreasing in |$\gamma$|, default risk is not innocuous. As panel (i) shows, the optimal |$B_1$| choices of the government that cannot commit to repay are lower than those of the government that can. This reflects the fact that the government optimally chooses smaller debt levels once it internalizes the effect of default risk on the debt Laffer curve and its distributional implications. The negative relationship between |$B_1$| and |$\gamma$| is in line with the empirical evidence on the negative relationship between public debt ratios and wealth Gini coefficients at relatively high levels of inequality noted in the Introduction and documented in Section A.1 of the Online Appendix.
Panels (iii) and (iv) show that, in contrast with standard models of external default, in this model, the default spread is neither similar to the probability of default nor does it have a monotonic relationship with it.21 Both the spread and the default probability start at zero for |$\gamma = 0.05$| because |$B_1 ^* (0.05)$| has zero default probability. As |$\gamma$| increases up to 0.5, both the spread and the default probability of the optimal debt choice are similar in magnitude and increase together, but for the regions where the default probability is constant (for |$\gamma \gt 0.5$|, the spread falls with |$\gamma$|.22 These results are in line with the findings of the theoretical analysis in Section 3.
The determination of the optimal debt choice and the relationship among the four panels of Figure 4 can be illustrated further as follows. Define a default-threshold value of |$\gamma$|, |$\hat \gamma (B_1 ,g_1 )$|, as the one such that the government is indifferent between defaulting and repaying for a given |$(B_1 ,g_1 )$|. The government chooses to default if |$\gamma \ge \hat \gamma$|. Figure 5 shows the optimal debt choice |$B_1 ^* (\gamma )$| together with curves representing |$\hat \gamma (B_1 ,g_1 )$| for several realizations of |$g_1$|. The curves for the lowest |$(\underline {g} )$|, highest |$(\bar g)$|, and mean |$(\mu _g )$| realizations are identified with labels.

Default threshold, debt policy, and equilibrium default. |${\underline {g} }$| and |${\bar g}$| are the smallest and largest possible realizations of |$g_1$| in the Markov process of government expenditures, which are set to |$- / + 3$| standard deviations off the mean respectively. The dotted lines correspond to a set of selected thresholds for different values of |$g_1$|.
Figure 5 shows that, because of the stronger default incentives at higher |$\gamma$| and higher realizations of |$g_1$|, the default-threshold curves are decreasing in |$B_1$| and |$g_1$|. There are, therefore, two key “border curves”. First, for pairs |$(B_1 ,\gamma )$| below |$\hat \gamma (B_1 ,\bar g_1 )$|, repayment can be expected to occur for sure because the government will repay even if the highest realization of |$g_1$| is observed. Second, for pairs |$(B_1 ,\gamma )$| above |$\hat \gamma (B_1 ,\underline {g}_1 )$|, default can be expected to occur for sure because the government will choose default even if the lowest realization of |$g_1$| is observed.
It follows from the previous example that, for equilibria with debt exposed to default risk to exist, the optimal debt choice |$B_1^* (\gamma )$| must lie in between the two borders (if it is below |$\hat \gamma (B_1 ,\bar g_1 )$| the debt is issued at zero default risk, and if it is above |$\hat \gamma (B_1 ,\underline {g} _1 )$|, there is no equilibrium). Moreover, the probability of default is implicitly determined as the cumulative probability of the value of |$g_1$| corresponding to the highest debt-threshold curve that |$B_1 ^* (\gamma )$| reaches. This explains why the default probability in panel (iv) of Figure 4 shows constant segments as |$\gamma$| rises above 0.5. As Figure 5 shows, for |$\gamma \le 0.5$|, the optimal debt is relatively invariant to increases in |$\gamma$|, starting from a level that is actually in the region of risk-free debt and then moving into the region exposed to default risk. In this segment, the optimal debt falls slightly, and the probability of default rises gradually as |$\gamma$| rises. For |$\gamma$| from 0.5 to 0.6, the optimal debt falls but along the same default threshold curve (not shown in the plot), and hence the default probability remains constant at about 0.007. For |$\gamma \gt 0.6$|, the debt choice falls gradually but always along the default-threshold curve associated with a default probability of 0.015.
These findings suggest that the optimal debt is being chosen seeking to sell the “most debt” that can be issued while keeping default risk low. In turn, the most debt that is optimal to issue responds to the incentives to reallocate consumption across agents and across periods internalizing the dependence of the debt Laffer curve on the debt choice. In fact, for all values of |$B_1^* (\gamma )$| that are exposed to nontrivial risk of default (those corresponding to |$\gamma \ge 0.5$|), |$B_1^* (\gamma )$| coincides with the maximum point of the corresponding debt Laffer curve (see Figure A.6 of the Online Appendix). Hence, the optimal debt yields the maximum resources to the government that it can procure given its inability to commit to repay. Setting debt higher is suboptimal because default risk reduces bond prices sharply, resulting in a lower amount of resources, and setting it lower is also suboptimal because then default risk is low and extra borrowing generates more resources since bond prices fall little.
5. Extensions
This section summarizes the results of four important extensions of the model. First, a political bias case in which the social welfare function assigns weights to agents that deviate from the fraction of |$L$| and |$H$| types observed in the economy; second, an economy in which risk-neutral foreign investors can buy government debt; third, a case in which proportional distortionary taxes on consumption are used as an alternative tool for redistributive policy; and fourth, a case in which agents have access to a second asset as a vehicle for saving.
5.1. Biased Welfare Weights
Assume now that the weights of the government's payoff function differ from the utilitarian weights |$\gamma$| and |$1 - \gamma$|. This can be viewed as a situation in which, for political reasons, the government's welfare weights are biased in favor of one group of agents. The government's welfare weights on |$L$|- and |$H$|-type households are denoted |$\omega$| and |$(1 - \omega )$|, respectively, and we refer to |$\omega$| as the government's political bias.
The socially efficient planner chooses its optimal consumption dispersion |${\varepsilon ^{SP} }$| as the value of |$\varepsilon$| that maximizes the aforementioned expression. Since as of |$t = 1$|, the only instrument the government can use to manage consumption dispersion relative to what the decentralized allocations support is the default decision, it will repay only if doing so allows it to get closer to |${\varepsilon ^{SP} }$| than by defaulting.
This condition implies that optimal consumption dispersion for the planner is 0 only if |$\omega = \gamma$|. For |$\omega \gt \gamma$|, the planner likes consumption dispersion to favor |$L$| types so that |$c_1^L \gt c_1^H$|, and the opposite holds for |$\omega \lt \gamma$|.
The key difference of political bias versus the model with a utilitarian government is that the former can support equilibria with debt subject to default risk even without default costs. Assuming |$\varphi (g_1 ) = 0$|, there are two possible scenarios depending on the relative size of |$\gamma$| and |$\omega$|. First, if |$\omega \ge \gamma$|, the planner again always chooses default as in the setup of Section 2. This is because for any decentralized consumption dispersion |$\varepsilon \gt 0$| the consumption allocations feature |$c^H \gt c^L$|, whereas the planner's optimal consumption dispersion requires |$c^H \le c^L$|, and hence, |$\varepsilon ^{SP}$| cannot be implemented. Default brings the planner the closest it can get to the payoff associated with |$\varepsilon ^{SP}$|, and hence, it is always chosen. In the second scenario, |$\omega \lt \gamma$| (i.e., the political bias assigns more (less) weight to |$H (L)$| types than the fraction of each type of agents that actually exists). In this case, the model can support equilibria with debt even without default costs. In particular, there is a threshold consumption dispersion |${\hat \varepsilon }$| such that default is optimal for |$\varepsilon \ge \hat \varepsilon$|, where |${\hat \varepsilon }$| is the value of |$\varepsilon$| at which |$W_1^{d = 0} \left( {\varepsilon ,g_1 ,\gamma ,\omega } \right)$| and |$W_1^{d = 1} (g_1 )$| intersect. For |$\varepsilon \lt \hat \varepsilon$| repayment is preferable because |$W_1^{d = 0} \left( {\varepsilon ,g_1 ,\gamma ,\omega } \right) \gt W_1^{d = 0} (g_1 )$|. Thus, without default costs, equilibria for which repayment is optimal require two conditions: (a) that the government's political bias favors bondholders |$(\omega \lt \gamma )$|, and (b) that the debt holdings chosen by private agents do not produce consumption dispersion in excess of |${\hat \varepsilon }$|.
Figure 6 illustrates the main quantitative predictions of the model with political bias. The scenario with |$\omega = \gamma$|, shown in blue, corresponds to the utilitarian case of Section 4, and the other two scenarios correspond to high and low values of |$\omega$||$(\omega _L = 0.25\,{\text{and}}\,\omega _E = 0.45,\,\,{\text{respectively}})$|.23

Equilibrium of the model with political bias for different values of |$\omega$|.
Figure 6 shows that the optimal debt level is increasing in |$\gamma$|. This is because the incentives to default grow weaker and the repayment zone widens as |$\gamma$| increases for a fixed value of |$\omega$|. Moreover, the demand composition effect of higher |$\gamma$| is still present, so along with the lower default incentives, we still have the increasing per capita demand for bonds of |$H$| types. These two effects combined drive the increase in the optimal debt choice of the government. It is also interesting to note that in the |$\omega _L$| and |$\omega _H$| cases, the equilibrium exists for all values of |$\gamma$| (even those that are lower than |$\omega$|). Without default costs, each curve would be truncated exactly where |$\gamma$| equals either |$\omega _L$| or |$\omega _H$|, but since these simulations retain the default costs used in the utilitarian case, there can still be equilibria with debt for lower values of |$\gamma$| (as explained earlier).
In this model with political bias, the government is still aiming to optimize debt by focusing on the resources it can reallocate across periods and agents, which are still determined by the debt Laffer curve, and internalizing the response of bond prices to debt choices.24 This relationship, however, behaves very differently than in the benchmark model because now higher optimal debt is carried at decreasing default probabilities, which leads the planner internalizing the price response to choose higher debt, whereas in the benchmark model, lower optimal debt was carried at increasing equilibrium default probabilities, which led the planner internalizing the price response to choose lower debt.
In the empirically relevant range of |$\gamma$|, and for values of |$\omega$| lower than that range (since |$\omega _L = 0.25$| and |$\omega _H = 0.45$|, whereas the relevant range of |$\gamma$| is [0.55, 0.85]), this model can sustain sign|$\omega _H = 0.45$| ificantly higher debt ratios than the model with utilitarian payoff, and those ratios are close to the observed European median. At the lower end of that range of |$\gamma$|, a government with |$\omega _H$| chooses a debt ratio of about 25%, whereas a government with |$\omega _L$| chooses a debt ratio of about 35%.
The behavior of equilibrium bond prices (panel (ii)) with either |$\omega _L = 0.25$| or |$\omega _H = 0.45$| differs markedly from the utilitarian case. In particular, the prices no longer display an increasing, convex shape, instead they are (for most values of |$\gamma$|) a decreasing function of |$\gamma$|. This occurs because the higher supply of bonds that the government finds optimal to provide offsets the demand composition effect that increases individual demand for bonds as |$\gamma$| rises. At low values of |$\gamma$|, the government chooses lower debt levels (panel (i)) in part because the default probability is higher (panel (iv)), which also results in higher spreads (panel (iii)). However, as |$\gamma$| rises and repayment incentives strengthen (because |$\omega$| becomes relatively smaller than |$\gamma$|), the probability of default falls to zero, the spreads vanish, and debt levels increase. The price remains relatively flat because, again, the higher debt supply offsets the demand composition effect.
The political bias extension yields an additional interesting result: for a sufficiently concentrated distribution of bond holdings (high |$\gamma$|, |$L$|-type agents prefer that the government weights the bondholders more than a utilitarian government (i.e., there are values of |$\gamma$| and |$\omega$| for which, comparing equilibrium payoffs under a government with political bias versus a utilitarian government, |$\upsilon ^L (B_1 ,\omega ,\gamma ) \gt \upsilon ^L (B_1 ,\gamma ))$|). To illustrate this result, Figure 7 plots the equilibrium payoffs in the political-bias model for the two types of agents as |$\omega$| varies for two values of |$\gamma (\gamma _L = 0.15\;{\text{and}}\;\gamma _H = 0.85)$|. The payoffs for the |$L$| and |$H$| types are in panels (i) and (ii), respectively. The vertical lines identify the payoffs that would be attained with a utilitarian government (which by construction coincide with those under political bias when |$\omega = \gamma$|).

Welfare as a function of political bias for different values of |$\gamma$|.
Panel (ii) shows that the payoff of |$ H $| types is monotonically decreasing in |$\omega$| because |$ H $| types always prefer the higher debt levels attained by low |$\omega$| governments, since debt enhances their ability to smooth consumption at a lower risk of default. In contrast, panel (i) shows that the payoff of |$ L $| types is nonmonotonic in |$\omega$| and has a well-defined maximum. In the |$\gamma = \gamma _L$| case, the maximum point is at |$\omega = \gamma _L$|, which corresponds to the equilibrium under the utilitarian government, but when |$\gamma = \gamma _H$|, the maximum point is at about |$\omega = 0.75$|, which is smaller than |$\gamma _H$|. Thus, in this case, ownership of public debt is sufficiently concentrated for the agents that do not hold it to prefer a government that chooses |$B_1$|, weighting the welfare of bond holders by more than the utilitarian government. This occurs because with |$\gamma _H$|, the utilitarian government has strong incentives to default, and thus, the equilibrium supports low debt, but |$L$|-type agents would be better off if the government could sustain more debt, which the government with |$\omega = 0.75$| can do because it weights the welfare of bondholders more and thus has weaker default incentives. The |$L$|-type agents desire more debt because they are liquidity constrained (i.e., |$(\text {i}.\text {e}., \,\mu ^L \gt 0$|) and higher debt improves the smoothing of taxation and thus makes this constraint less tight.
These results yield an important political economy implication: Under a majority voting electoral system in which candidates are represented by values of |$\omega$|, it can be the case that majorities of either |$L$| or |$H$| types elect governments with political bias |$\omega \lt \gamma$|. This possibility is captured in Figure 7. When the actual distribution of bond holdings is given by |$\gamma _H$|, the majority of voters are |$L$| types, and thus, it follows from panel (i) that the government represented by |$\omega$| at the maximum point (around 0.75) is elected. In this case, agents who do not hold government bonds vote for a government that favors bondholders (i.e., |$L$| types are weighed at 0.75 instead of 0.85 in the government's payoff function). When the distribution of bond holdings is given by |$\gamma _L$|, the majority of voters are |$H$| types and the electoral outcome is determined in panel (ii). Since the payoff of |$H$|-type agents is decreasing in |$\omega$|, they elect the government at the lower bound of |$\omega$|. Hence, under both |$\gamma _L$| and |$\gamma _H$|, a candidate with political bias beats the utilitarian candidate. This result is not general, however, because we cannot rule out the possibility that there could be a |$\gamma \ge 0.5$| such that the maximum point of the |$L$|-type payoff is where |$\omega = \gamma$|, and hence, the utilitarian government is elected under majority voting.
5.2. International Investors
Although a large fraction of sovereign debt in Europe is in the hands of domestic households (for the countries in Table 1, 60% is the median and 75% the average), the fraction in the hands of foreign investors is not negligible. For this reason, we extend the benchmark model to incorporate foreign investors and move from a closed to an open economy. In particular, we assume that there is a pool of international investors modeled in the same way as in the Eaton—Gersovitz class of external default models: risk-neutral lenders with an opportunity cost of funds equal to an exogenous, world-determined real interest rate |${\bar r}$|. As is common practice, we assume that these bonds are pari passu contracts, which rules out the possibility for the government to discriminate among borrowers when choosing to default. This also maintains the symmetry with the baseline model, in which the government was not allowed to default on a particular set of domestic households.
Since foreign lenders are the marginal investors of sovereign debt, in this model, the price of the bond is given by |$q(B_1 ,\gamma ) = (1 - p(B_1 ,\gamma ))/(1 + \bar r)$| where |$p(B_1 ,\gamma )$| is the default probability. More precisely, |$p(B_1 ,\gamma ) = E_{g_1 } [d(B_1 ,\gamma ,g_1 )]$|. While the arbitrage condition is functionally identical to the one of the Eaton—Gersovitz models, they embody different mechanisms. The two are similar in indicating that, because of risk neutrality, risk premia are equal to default probabilities. However, there is a critical difference in how these probabilities are determined. In Eaton—Gersovitz models, they follow from the values of continuation versus default of a representative agent, whereas in our model, they are determined by comparing those values for a utilitarian social welfare function, which in turn depend on the dispersion of individual payoffs of default versus repayment (and on the welfare weights). Hence, concentration of debt ownership affects default probabilities via changes in the relative magnitudes of individual payoffs of default versus repayment.
We do not restrict the value of |$[\gamma b_1^L + (1 - \gamma )b_1^H ]$| to be less than or equal to |$B_1$| so |$B_1^f$| could be positive or negative. When |$B_1^f \gt 0$| the country is a net external borrower, because the bonds issued by the government are less than the domestic demand for them, and when |$B_1^f \lt 0$|.
The problems of the agents and the government remain identical to those described in Section 3. Of course, agents understand that there is a new pricing equation and that market-clearing conditions incorporate the foreign demand. We solve the model numerically using the same parameter values of the benchmark model. We set |$\varphi ^f = 0.25$| to match the average fraction of foreign debt observed in our sample of European countries and |${\bar r}$| to 2% to match the average real interest rate in Germany in the 2000/2007 period.
Figure 8 shows how the planner’s welfare gain of default varies with |$\gamma$| and |${B_1 }$| for different levels of government expenditures (|$g_1 = \underline {g} _1$| and |$g_1 = \bar g_1$|). The no-equilibrium region, which exists for the same reasons as before, is shown in white. In line with the characteristics of default incentives of the benchmark model, within the region where the equilibrium is well defined, for a given |$\gamma$|, the planner’s value of default increases monotonically with the level of debt |${B_1 }$|. However, we observe that, contrary to the benchmark case, in the economy with foreign lenders, conditional on |${B_1 }$|, the welfare gain of default has an inverted-U shape in the |$\gamma$| dimension. This is, for a given |${B_1 }$|, the value of |$\alpha$| decreases with |$\gamma$|, for low |$\gamma$| reaches a minimum point, and then increases with |$\gamma$|. This also determines a bell-shaped No Equilibrium Zone. The intuition for this result is simple and derives from the decision rules of domestic agents (see Figure A.12 of the Online Appendix for the corresponding plot). For a given level of |${B_1 }$|, when |$\gamma$| is below |$\gamma \lt 0.25$| for |$B_1 = B_M$|, the country is on average a foreign borrower (i.e., |$B^f \gt 0$|). This implies that a default generates a direct increase in domestic resources equal to the forgone debt payments to foreign lenders. In this region, both |$L$|-type and |$H$|-type agents are at the borrowing limit. As |$\gamma$| increases, the country becomes a net saver in foreign markets. Increases in |$\gamma$| are associated with an increasing portion of domestic debt in the hands of |$H$| types. This reduces the benefit of a default on foreign lenders. However, as in the model without foreign lenders, domestic consumption dispersion increases. For midrange |$\gamma$|, the first effect dominates the second, and repayment is the preferred option. As |$\gamma$| increases even further, the dispersion in domestic consumption increases to points where default is again the optimal alternative for the government. In this region, the main driver of domestic default is redistribution among domestic agents as in our benchmark economy.

Planner’s welfare gain of default |$\bar \alpha \left( {B_1 ,g_1 ,\gamma } \right)$|.
Figure 9 shows the comparison of the equilibrium functions in the economy with foreign lenders versus the benchmark economy. As we described before, the introduction of foreign lenders and the possibility of a “foreign” default constraint debt values and result in lower debt levels than in the benchmark, for values of |$\gamma$| lower than 0.75 (see panel (i)). This is also reflected in higher default probabilities and spreads in the economy with foreign lenders than in the benchmark (see panels (iii) and (iv)). By construction, the upper bound on the price of the economy with foreign lenders is |$(1 + r)^{ - 1}$|, so the distributive effect that negative real interest rates have in the benchmark economy dissipate (see panel (ii)). This induces the government to take on more risk and redistribute via debt issuance in the economy with foreign lenders than in the benchmark.

5.3. Redistributive Taxation (Partial Default)
In the benchmark model, issuing debt in the first period and defaulting in the second are the only tools the government can use to reduce consumption dispersion. We now examine how the model’s predictions change by adding an alternative tool for redistribution. In particular, we introduce a proportional consumption tax that is invariant across periods and realizations of |$g_1$|.26
These expressions are similar to those presented in Section 4, except that the terms that include public debt in the date-0 difference and the date-1 difference under repayment are divided by |$(1 + \tau ^c )$|. Consumption taxes reduce consumption dispersion in both instances. Intuitively, the consumption tax plays a role akin to inflation in reducing the real value of public debt. Hence, this tax always reduces consumption dispersion in the first period (as long as there is a primary deficit) and always reduces dispersion in the second period under repayment. In fact, for assuming repayment, this tax is a better mechanism for redistribution because debt can only reduce dispersion at date 0 at the expense of increasing it at date one. Moreover, the tax’s ability to redistribute is not hampered by the debt Laffer curve that hampers the ability to redistribute with debt because of default risk. The tax acts in fact as a defacto partial default in both periods. As a result, if we allow it to be chosen optimally, letting |$\tau ^c$| go to infinity would be optimal because it completely removes consumption dispersion in all periods at no cost. The purpose of this analysis, however, is to see how the existence of an alternative redistribution tool affects the results we have obtained for debt and default risk, rather than focus on the optimal use of the consumption tax.27
We solve the model numerically using the parameters of the benchmark calibration. For the value of the consumption tax, we use |$\tau ^c = 0.16$| which is the value estimated for the Eurozone in Mendoza et al. (2014). We also solve the model for |$\tau ^c = 0.32$| and |$\tau ^c = 0.48$|, which are two and three times larger than the data proxy.28Figure 10 presents the comparison of the equilibrium of the benchmark economy with the three economies with taxes. This figure shows that the main result of the paper (the ability to support the existence of public debt with positive default risk) is robust to incorporating distributive taxes.

Comparison equilibrium benchmark vs proportional consumption taxes.
As consumption taxes increase, default incentives grow weaker (see lower default probabilities in panel (iv)), which is natural because the tax reduces consumption dispersion, and hence it reduces the need for the government to use default to lower dispersion. As a result, the government can sustain higher debt levels (see panel (iv)), although the effect is not very large (debt increases by at most 5 percentage points for |$\gamma = 0.6$| and a change in the consumption tax from 0% to 48%). This effect is negligible when default incentives are weak to start with, which occurs at low values of |$\gamma$|. The properties of the price function observed in the benchmark case (i.e., increasing in |$\gamma$| for a given level of default risk) are sustained in the economy with consumption taxes (see panel (ii)). In line with the reduction in default probabilities, the model with consumption taxes also displays lower spreads than the benchmark (see panel (iii)).
The findings in this section suggest that the existence of other tools for redistribution contributes to support higher debt levels at lower default frequencies. The results are also interesting as an illustration of what happens if we combine de jure outright default with partial, de facto default (the latter commonly takes place via inflation, but as noted earlier the consumption tax in this setup plays the same role as inflation in reducing the real debt burden).
5.4. A Second Asset
We now review the implications of adding a risk-free asset that agents can use as a vehicle of savings, in addition to public debt. In particular, agents have access to a nonstochastic production technology |$y_t^i = z(k_t^i )^\theta$| with |$0 \lt \theta \lt 1$|, where |$y_t^i$| is total output and |$k_t^i$| is capital for agent of type |$i$| in period |$t$|, respectively.29 The initial aggregate level of capital is denoted by |$K_0$|. |$L$|-type agents are now endowed with |$b_0^L$| and |$k_0^L$| units of public debt and capital, respectively, whereas |$H$|-type agents have endowments given by |$b_0^H = (B_0 - \gamma b_0^L )/(1 - \gamma )$| and |$k_0^H = (K_0 - \gamma k_0^L )/(1 - \gamma )$|. Capital depreciates at rate |$\delta$|. At period 0, agents choose how much of their savings they want to allocate to public bonds |$b_1^i \ge 0$| and capital |$k_1^i \ge 0$|.
Thus, the default spread of this model (i.e., the gap between the yield on government bonds and the marginal productivity of capital) is determined by default risk, weighted by marginal utility in each state of the world, since agents are risk averse. From this perspective, the second asset has a similar effect as introducing the foreign investor because it introduces an opportunity cost of funds, but with the difference that this cost is now endogenous and falls as capital investment rises.
Adding the second asset also affects the government’s debt and default choices. In particular, the government now has to consider the initial distributions of both bonds and capital across agents, and the effects of its choices on the individual capital and bond decisions |$\left\{ {b_1^i ,k_1^i } \right\}$|, which depend on |$B_1$| and default risk. Default is more costly because via its effects on capital allocations it can cause higher consumption dispersion than before.
To make this case as close as possible to the benchmark model for the quantitative analysis, we assume |$k_0^i = K_0$| (i.e., all initial heterogeneity is in initial bond holdings). Still, this results in heterogeneous bond and capital holdings for the second period. We set |$z$| to normalize GDP to 1, |$1,\theta = 0.33$|, and |$\delta = 0.10$| (standard values), and set |$K_0$| so the capital-to-output ratio is equal to 2. All other parameters are the same as in the benchmark.
Agents in this setup have the option to switch from debt to capital as default risk rises, which adversely affects the capacity of the government to issue debt. However, the weaker default incentives because of the additional adverse consumption dispersion effects enhance the ability to issue debt. In line with these arguments, we found that for low enough |$\gamma$|, so that default is a zero-probability event, both models have nearly identical results. As |$\gamma$| rises up to |$\gamma \lt 0.6$|, the model with two assets sustains higher optimal debt than the benchmark (by at most 4 percentage points), but at higher values of |$\gamma$| the opposite is true (debt in the benchmark is at most 2 percentage points higher). Thus, optimal debt levels do not differ by wide margins, even though the effects of adding the second asset are noticeable.
The arbitrage condition connecting bond prices and capital returns also has interesting quantitative implications for spreads and bond prices. First, since reallocation of savings from debt to capital reduces the marginal product of capital, the default risk spread rises as |$\gamma$| rises, and this happens even though the probability of default, which rises with |$\gamma$|, is actually lower than in the benchmark. Second, bond prices rise at a much lower rate as |$\gamma$| rises than in the benchmark, and, in fact, are never higher than 1 even with the highest values of |$\gamma$| considered (i.e., bond yields are never negative). In this dimension, the quantitative results are very different in the setup with two assets, although they are qualitatively similar.
6. Conclusions
This paper proposes a framework in which domestic sovereign default and public debt subject to default risk emerge as an equilibrium outcome. In contrast to standard models of sovereign default on external debt, this model highlights the role of the domestic distribution of public debt ownership and the distributional effects of default across domestic agents in shaping the government’s default incentives. These are features common to both the historical episodes of outright domestic default documented by Reinhart and Rogoff (2011) and the ongoing European debt crisis.
In this environment, the distribution of public debt across private agents interacts with the government’s optimal default, debt issuance and tax decisions. Distributional incentives alone cannot support equilibria with debt, because default is always optimal in the second period, and hence, the debt market cannot function in the first period. We also showed that equilibria with debt exposed to default risk can exist if we introduce either exogenous default costs or government preferences biased in favor of bond holders.
The main finding of the paper (i.e., that distributional incentives to default tempered by default costs can support equilibria with debt exposed to default risk) is robust to three other important extensions of the model: adding foreign investors who can hold a portion of the debt issued by the government, introducing consumption taxes as an alternative tool for redistributing resources across agents, and adding a second asset as a vehicle for saving.
This paper is a first attempt at developing a blueprint for research into models of domestic sovereign default driven by distributional incentives, and their interaction with agent heterogeneity and incomplete insurance markets. It has two main limitations, both byproducts of the two-period life horizon. First, it takes as given a distribution of bond ownership across agents. Second, it does not capture endogenous costs of default that would result from losing the benefits of public debt as a vehicle for self-insurance and liquidity-provision (see Aiyagari and McGrattan 1998). In further work (D’Erasmo and Mendoza 2014) we develop a framework that takes both issues into account.
Acknowledgments
We acknowledge the generous support of the National Science Foundation under awards 1325122 and 1324740. Comments by Jesse Schreger, Fabrizio Perri, Vincenzo Quadrini, and by participants at presentations at Columbia University, the 2014 Winter Meetings of the Econometric Society, and the NBER Conference on Sovereign Debt and Financial Crises are also gratefully acknowledged. We thank Jingting Fan for excellent research assistance. The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of Philadelphia or the Federal Reserve System. Mendoza is a Research Associate at NBER.
References
The analogy with a domestic default is imperfect, however, because the Eurozone is not a single country, and in particular, there is no fiscal entity with tax and debt-issuance powers over all the members.
In Section A.1 of the Online Appendix, we present a more systematic analysis of the link between debt and inequality and show that government debt is increasing in inequality when inequality is low but decreasing for high levels of inequality.
A related literature initiated by Aiyagari et al. (2002) studies optimal taxation and public debt dynamics with aggregate uncertainty and incomplete markets, but in a representative-agent environment. Pouzo and Presno (2014) extended this framework to incorporate default and renegotiation.
See Panizza et al. (2009), Aguiar and Amador (2014), and Tomz and Wright (2012) for recent reviews of the sovereign debt literature. Some studies in this area have examined models that include tax and expenditure policies, as well as settings with foreign and domestic lenders, but always maintaining the representative agent assumption (e.g., Cuadra et al. 2010; Vasishtha 2010; and more recently Dias et al. 2012) have examined the benefits of debt relief from the perspective of a global social planner with utilitarian preferences.
Motivated by the recent financial crisis and extending the theoretical work of Gennaioli et al. (2014), a set of recent papers focuses on the interaction between sovereign debt and domestic financial institutions, such as Sosa-Padilla (2012), Bocola (2014), Boz et al. (2014), and Perez (2015).
Andreasen et al. (2011), Ferriere (2014), and Jeon and Kabukcuoglu (2014) study environments in which domestic income heterogeneity plays a central role in the determination of external defaults.
See also Golosov and Sargent (2012) who study debt dynamics without default risk in a similar environment.
This formulation is analogous to the stepwise default cost as a function of income proposed by Arellano (2008) and now widely used in the external default literature, and it also captures the idea of asymmetric costs of tax collection (see Barro 1979; Calvo 1988). Note, however, that for the model to support equilibria with debt under a utilitarian government all we need is |$\varphi (g_1 ) \gt 0$|. The additional structure is useful for the quantitative analysis and for making it easier to compare the model with the standard external default models. In external default models, the nonlinear cost makes default more costly in “good” states, which alters default incentives to make default more frequent in “bad” states, and it also contributes to support higher debt levels.
Notice in particular that the payoff in case of default does not depend on the level of individual debt holdings |$(b_1^i )$|, reflecting the fact that the government cannot discriminate across households when it defaults.
Utility in the case of default equals |$u\left( {y(1 - \varphi (g_1 )) - g_1 } \right)$|, and is independent of |$b_1^i$|.
Note also that from the agents’ perspective, their bond decisions do not affect |$d_1 (B_1 ,g_1 ,\gamma )$|.
It is straightforward to show that revenue |$R(B_1 ) = q_0 (B_1 ,\gamma )B_1 $| follows a Laffer curve in the |$[0,B_1^{\max } ]$| interval, where |$B_1^{\max } $| is the upper bound of debt such that the government chooses default for any realization of |$g_1$||$q_0 (B_1^{\max } ,\gamma ) = 0$| and thus |$R(0) = 0$|. Since with |$R^{\prime}(0) = q(0,\gamma) \gt 0$|, and |$R( {B_1 ^{\max } } ) = 0$| with |$R^{\prime}(B_1^{\max } ) = q_0^{\prime} (B_1^{\max } ,\gamma )B_1^{\max } \lt 0$|, it follows by Rolle’s theorem that |$R(B_1 )$| has at least one local maximum in |$(0,B_1^{\max } )$|.
Recall also that we defined the relevant range of decentralized consumption dispersion for |$ \varepsilon \gt 0, $|, so welfare under repayment is decreasing in |$\varepsilon$| over the relevant range.
We solve the model following a similar backward-recursive strategy as in the theoretical analysis. First, taking as given a set of values |$\{ B_1 ,\gamma \} $|, we solve for the equilibrium pricing and default functions by iterating on (|$(q_0 ,b_1^i )$|), and the default decision rule |$d_1 $| until the date-0 bond market clears when the date-1 default decision rule solves the government’s optimal default problem (12). Then, in the second stage, we complete the solution of the equilibrium by finding the optimal choice of |$B_1 $| that solves the government’s date-0 optimization problem (16). It is important to recall that, as explained earlier, for given values of |$B_1 $| and |$\gamma $|, an equilibrium with debt will not exist if either the government finds it optimal to default on |$B_1 $| for all realizations of |$g_1 $| or if at the given |$B_1 $| the consumption of |$L$| types is nonpositive. In these cases, there is no finite price that can clear the debt market.
In the United States, the 2010 Survey of Consumer Finances indicates that only 12% of households hold savings bonds and 50.4% have retirement accounts (which are very likely to include government bonds). These figures would suggest values of |$\gamma $| ranging from 0.5 to 0.88. In Europe, comparable statistics are not available for several countries, but Davies et al. (2009) document that the wealth distribution is highly concentrated with Gini coefficients ranging between 0.55 and 0.85. In our model, since |$b_0^L = 0$|, the Gini coefficient of wealth is equal to |$\gamma $|.
These calculations are straightforward given that, in the equilibria, we solve for |$b_1^L = 0$|, and hence |$b_1^H = B_1 /(1 - \gamma )$|. The same formula would apply, however, even if these conditions do not hold, using instead the policy functions |$b_1^i (B_1 ,g_1 ,\gamma )$| that solve the households’ problems for any given pair of functions |$d_1 (B_1 ,g_1 ,\gamma )$| and |$q_0 (B_1 ,\gamma )$|, including the ones that are consistent with the government’s default decision and equilibrium in the bond market.
There is another potential “No Equilibrium Zone” that could arise if the given |$(B_1 ,\gamma )$| would yield |$c_0^L \le 0$| at the price that induces market clearing, and so the government would not supply that particular |$B_1$|. This happens for low levels of |$B_1$| relative to |$B_0$|. To determine if |$c_0^L \le 0$| at some |$(B_1 ,\gamma )$| we need |$q_0 (B_1 ,\gamma )$|, since combining the budget constraints of the |$L$| types and the government yields |$c_0^L = y - g_0 - B_0 + q_0 B_1$|. Hence, to evaluate this condition we take the given |$B_1$| and use the |$H$|-types’ Euler equation and the market clearing condition to solve for |$q_0 (B_1 ,\gamma )$|, and then determine if |$y - g_0 - B_0 + q_0 B_1 \le 0$|; if this is true, then |$(B_1 ,\gamma )$| is in the lower No Equilibrium Zone.
Note that the cross-sectional variance of initial debt holdings is given by |${\text{Var}}(b) = B^2 (\gamma )/(1 - \gamma )$| when |$b_0^L = 0$|. This implies that the cross-sectional coefficient of variation is equal to |$CV(b) = (\gamma )(1 - \gamma )$|, which is increasing in |$\gamma$| for |$\gamma \le 1/2$|.
Sections A.3 and A.4 of the Online Appendix provide proofs showing that |$q^{\prime}(B_1 ,\gamma ) \lt 0$| in the log-utility case. In Figure 3, the scale of the vertical axis is too wide to make the fall in |$q(B_1 ,\gamma )$| as |$B_1$| rises visible for |$\gamma \lt 0.5$|.
Sections A.3 and A.4 of the Online Appendix provide further details, including an analysis of the bond demand decision rules that validates the intuition provided here.
In the standard models, the two are similar and are a monotonic function of each other because of the arbitrage condition of a representative risk-neutral lender.
As we explained before, this is derived from the composition effect that strengthens the demand for bonds and results in increasing prices (with default risk and without) as |$\gamma$| increases. This result disappears in the extension of the model that introduces foreign lenders (see Section 5).
Note that along the blue curve of the utilitarian case both |$\omega$| and |$\gamma$| effectively vary together because they are always equal to each other, whereas in the other two plots |$\omega$| is fixed and |$\gamma$| varies. For this reason, the line corresponding to the |$\omega_L$| case intersects the benchmark solution when |$\gamma = 0.25$|, and the one for |$\omega_H$| intersects the benchmark when |$\gamma = 0.45$|.
When choosing |$B_1$|, the government takes into account that higher debt increases disposable income for |$L$|-type agents in the initial period but it also implies higher taxes in the second period (as long as default is not optimal). Thus, the government is willing to take on more debt when |$\omega$| is lower.
This assumption is made only to approximate the quantitative predictions of the model with the data but is not crucial for any of the results presented in what follows.
Using this tax as opposed to other taxes simplifies the solution of the model because aggregate consumption (as a function of |$y$| and |$g_t$|) is pinned down by the resource constraint, and thus, it is known before solving the agents’ problem. As a result, the algorithm does not need to iterate on the level of transfers to close the government budget constraint, although introducing a wealth tax would require it. An income tax would require not only iterating on aggregate transfers but also extending the model to consider endogenous labor supply because the benchmark model assumes no income heterogeneity, which makes proportional income tax close to a lump-sum tax.
The consumption tax as modeled lacks important tradeoffs that would make it suboptimal to set it infinitely large. In particular, it is nondistortionary because there is no labor supply choice, and because, since the tax is constant, it does not distort savings decisions. The tax is also assumed to be known with perfect foresight, whereas uncertainty about inflation would have an effect similar to a partial risk of default that would affect demand for bonds and bond prices.
In the tax rate estimates in Mendoza et al. (2014), the highest consumption tax is observed in Finland at 24%.
The curvature in production allows us to obtain a well-defined portfolio choice of bonds and capital for private agents.
Supporting Information
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Author notes
The editor in charge of this paper was Fabrizio Zilibotti.