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Political Disagreement, Lack of Commitment and the Level of Debt*

Davide Debortoli, University of California San Diego
Ricardo Nunes, Federal Reserve Board

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

We analyze how public debt evolves when successive policymakers have different policy goals and cannot make credible commitments about their future policies. We consider several cases to be able to disentangle and quantify the respective effects of imperfect commitment and political disagreement. Absent political turnover, imperfect commitment drives the long-run level of debt to zero. With political disagreement, debt is a sizeable fraction of GDP and increasing in the degree of polarization among parties, no matter the degree of commitment. The frequency of political turnover does not produce quantitatively relevant effects. These results are consistent with much of the existing empirical evidence. Finally, we find that in the presence of political disagreement the welfare gains of building commitment are lower.

Keywords: Time-consistency, political disagreement

JEL classification: C61, E61, E62, P16


1  Introduction

1.1  Motivation

In the fiscal policy literature, there is not a clear theoretical understanding of the forces driving the observed patterns of public debt. This paper explores how debt evolves when governments cannot make credible commitments about future policies and when policymakers with different policy goals alternate in office. We consider several cases to be able to disentangle and quantify the respective effects of imperfect commitment and political disagreement.

As it is well known, the evolution of debt matters in a world where the provision of public goods has to be financed by raising distortionary taxes.1 In this context, as shown e.g. in the works of Barro (1979), Lucas and Stokey (1983) and Aiyagari et al. (2002), debt is used to smooth over time the deadweight losses associated with such distortions. These models can account for many aspects of the evolution of debt for many countries. However, these theories do not provide a complete explanation of some basic and stylized facts, like why public debt is a sizeable fraction of GDP in many developed countries and why there is a substantial variation in the debt/GDP ratio across countries with similar economic conditions.2

In macroeconomic models, the optimal (second-best) allocations are usually characterized as the solution to a Ramsey problem. It is assumed that the same planner is always in charge and that he can commit to future policies, maximizing the welfare of an infinitely lived representative agent.3 Under these assumptions and with complete financial markets, as shown by Lucas and Stokey (1983), the long-run level of debt crucially depends on the initial conditions.4 Countries starting with high debt will have high debt forever, and countries with low debt will have low debt forever. Since initial conditions are exogenous to the model and empirically difficult to determine, such a theory can not explain what induces countries to accumulate debt.

Policymaking in practice departs from the idealized environment described in Lucas and Stokey (1983) in many dimensions. In this work, we investigate how imperfect commitment and disagreement among successive policymakers can provide an incentive to accumulate debt. There are important reasons to think that these two forces may considerably affect the behavior of debt.

First, the role of commitment is related to the time-inconsistency problem in optimal policy choices, as illustrated in the seminal works of Kydland and Prescott (1977) and Barro and Gordon (1983). In our context, the solution under full-commitment is time inconsistent because a planner, at a given point in time, is willing to abandon his previous plans to manipulate the interest rate. For example, if the planner needs to issue debt, he has an incentive to reduce the interest rate. Hence, the planner is willing to lower current taxes, in order to foster current consumption. Because of a smoothing motive, this leads to an increase in the demand for savings and thus to a reduction in the interest rate. As a consequence, because of the lower tax revenues, in a one-time deviation from the full-commitment solution, the planner runs deficits and accumulates debt.5 Therefore, it seems worth exploring how debt evolves when the planner cannot make credible commitments about his future policies.6 We thus check whether a positive long-run level of debt may be the outcome of the optimal policy under the no-commitment assumption and other imperfect commitment settings.

Second, some studies in the political economy literature (see e.g. Alesina and Tabellini (1990) and Persson and Svensson (1989)), have emphasized how the presence of political disagreement may provide incentives to accumulate an inefficient level of debt. In a world characterized by political disagreement, the assumption of full-commitment seems unrealistic. Due to this reason, the literature assumed that governments always lack commitment. However, it would still be reasonable to assume that governments may have commitment during their tenures, but cannot commit on behalf of their successors, who have different objectives. In this paper, we consider a framework with political disagreement among successive policymakers, where commitment plays an important role in the strategic game between policymakers and private agents.7 In this context, the incumbent policymaker makes different choices depending on his ability to commit while staying in office. This allows us to explicitly analyze the effects of commitment in a world with political disagreement among successive policymakers.

We build on the simple Lucas and Stokey (1983) model, introducing endogenous government expenditure, which has to be financed by raising a proportional income tax and/or by issuing debt. We develop a framework that allows us to disentangle and quantify the effects of imperfect commitment, frequency of turnover and political disagreement in a dynamic context. In this respect, our contribution is methodological. Our framework can be used to analyze the effects of commitment in a wide set of infinite-horizon optimal policy problems, where policymakers with different objectives alternate in office. In other words, the methodology developed here allows us to integrate the analysis about the time-inconsistency of optimal policy choices, typical of the dynamic macroeconomic literature, into a political economy model. By doing so, we are able to measure the implications of building commitment in the presence of political disagreement.

From an economic point of view, the main contribution and findings of our analysis are the following. First, abstracting from political disagreement, we study the optimal fiscal policy under the no-commitment assumption. Under a wide set of initial conditions and parameterizations, we find that debt goes to zero in the long-run. Perhaps surprisingly, this means that there is a striking difference in the behavior of debt in a one-time deviation from commitment and in the no-commitment (time-consistent) solution. As we will discuss later, reducing debt over time is the only way the planner with no-commitment can favorably affect the interest rate.

Second, we study the behavior of debt in cases where the planner has access to a commitment technology, but under some circumstances, say because of political pressures, big shocks etc., he may renege on his past promises. This is what we call the loose commitment setting. Because of the striking difference in the behavior of debt between the full-commitment and the no-commitment cases, it seems worth checking how debt evolves under loose commitment. We find that in this last case the level of debt still converges to zero in the long-run. This suggests that the steady-state dependency on initial conditions found in Lucas and Stokey (1983) is not robust to small deviations from the full-commitment case. In addition, our results suggest that departing from the full-commitment assumption cannot help explaining why the level of debt is a sizeable fraction of GDP.

Third, we also find that debt is increased in periods when the planner reneges on his past promises and reduced over the periods of commitment.

This result is interesting since it suggests that the simple expectation that the planner may surprise the economy at a future date induces him to commit to reduce debt over time.

Fourth, we investigate one case where the imperfect commitment assumption is natural, i.e. when successive planners have different policy goals. We find that in the presence of political disagreement, debt is a sizeable fraction of GDP, regardless of the commitment assumptions. In our numerical exercises, political disagreement seems to be the main driving force for accumulating deficits. On the contrary, the effects of imperfect commitment and political turnover have a small impact on the level of debt. Our predictions are consistent with most of the existing empirical evidence. Indeed, while there is a large consensus on the positive relationship between the degree of political polarization and debt accumulation, the empirical findings about the effects of the frequency of political turnover are less clear-cut. More importantly, our results suggest that when testing empirically the effects of political instability on the level of debt, it is important to control both for measures of polarization among parties and measures of political turnover, rather than using any of them as a generic indicator of political instability.

Finally, when analyzing welfare implications, we find that the gains from commitment are lower in the presence of political disagreement than in a no-disagreement case. From an intuitive point of view, this happens because in the absence of political disagreement governments with more commitment will maximize overall social welfare. However, with political disagreement a better commitment technology can be used by each party to maximize specific groups' welfare.

1.2  Related literature

Krusell et al. (2006) analyze the time-consistent solution of the otherwise standard Lucas and Stokey (1983) model, where government expenditures are exogenous. The authors find as a solution a multiplicity of steady-states and discontinuous policy functions, where debt adjusts for one or two periods and then remains constant. Their main finding is that under no-commitment the equilibrium is close to the solution under commitment. In our paper, we also build on the Lucas and Stokey (1983) model, but consider the case where government expenditure is endogenous. The presence of this additional instrument in the hands of the policymaker widens the set of his feasible choices. In section 3, we extensively discuss how this makes a difference. We obtain continuous policy functions, and we find that in the absence of commitment debt goes to zero. This result is surprising because it is usually the case that in a one-time deviation from commitment debt increases.

In the literature, several papers have analyzed the effects of lack of commitment on debt in monetary economies. When nominal debt is present, the monetary authority usually has an incentive to raise the price level to reduce the real value of the outstanding debt. The first period of the full-commitment solution reveals such incentives, since debt is eroded in real terms. Martin (2006a) and Diaz-Gimenez et al. (2006) analyze monetary economies under discretion where the cash-in-advance constraint is key to determine the level of debt. They find that the steady-state level of debt can be positive, negative or zero depending on the parametrization of the utility function. If it is easy (difficult) for households to substitute cash goods then government holds assets (debt).8 As in Krusell et al. (2006) we focus on a real economy without a cash-in-advance constraint. Since in most countries central banks are independent and committed to price stability, we believe that focusing on a real economy is a reasonable assumption. Our result that debt converges to zero is not due to the presence of nominal bonds nor it is achieved with surprise inflation.

Some studies in the political economy literature, like Alesina and Tabellini (1990), have analyzed how policy decisions are formulated when policymakers with different political views alternate in office. Azimonti-Renzo (2004), as we do here, extends the previous works to an infinite horizon problem, but in a context where commitment about future policy does not affect private agents' choices. The author considers a fiscal policy model with balanced budget, and public but no private capital. Instead, we focus on the effects of political disagreement on the level of government debt. Our main contribution with respect to this literature is to study optimal policy where commitment plays a role in the strategic interactions between agents and the policymakers. Moreover, we solve the problem under different commitment settings. We indeed consider the case where parties cannot commit at all, but we also assume that parties can credibly commit for the future, in case they are reappointed in office. This allows to disentangle and quantify the effects of imperfect commitment, political disagreement and frequency of political turnover on the level of debt. Finally, it allows to measure the welfare gains from commitment in the presence of political disagreement.

In recent work, Song et al. (2006) and Battaglini and Coate (2008) study the evolution of debt in a dynamic political economy framework, and provide an explanation for the presence of a long-run positive level of debt. They consider models with political conflicts over public goods redistribution, either across generations or across geographical districts. In these works, however, the interest rate is exogenous and the commitment problem arises because of repeated voting. In our work, we instead study an infinite horizon problem, where the disagreement is about the composition of a public good, while considering a simpler voting mechanism. More importantly, we analyze a case where policy choices are time-inconsistent because of the policymaker's incentive to manipulate the interest rate, which would be present even in the absence of repeated voting or political turnover. In such context, we study the strategic interactions between policymakers with different objectives alternating in office.

The paper is organized as follows: in section 2 we introduce the model and, as a benchmark for our analysis, we recover the solution under full-commitment. In section 3, we describe the solution under no-commitment, i.e. the time-consistent solution. In section 4, we illustrate the behavior of debt under the less extreme assumption of loose commitment. In section 5, we study the joint implications of political disagreement and imperfect commitment and we compare our findings with the existing empirical literature. Finally, we discuss welfare implications. Section 6 concludes.


2  The model

We build our analysis on a simple model, as in Lucas and Stokey (1983), where time-inconsistency issues arise. For the time being, we abstract from uncertainty and political disagreement between successive governments.9 We consider an economy where labor is the only factor of production, and technology is linear, and output can be used either for private consumption $ c_{t}$ or for public consumption $ g_{t}$. The economy's aggregate budget constraint is therefore

$\displaystyle c_{t}+g_{t}=1-x_{t} \hspace{1cm} \forall t=0,1,2,...$ (1)

The public good is provided by a benevolent government and financed through a proportional tax $ (\tau)$ on labor income and by issuing a one-period bond $ b^{G}_{t}$ with price $ p_{t}$. At any point in time, the government budget constraint is

$\displaystyle g_{t}+b^{G}_{t-1} =\tau_{t}(1-x_{t})+p_{t}b^{G}_{t} .$ (2)

In a decentralized equilibrium, given taxes, prices and the quantities of public expenditure, the representative household chooses consumption, savings and leisure by solving the following problem:

$\displaystyle \max_{\left\{ c_{t},x_{t},b^{P}_{t}\right\} _{t=0}^{\infty}}$ $\displaystyle \sum _{t=0}^{\infty}\beta^{t}u(c_{t},x_{t},g_{t})$    
$\displaystyle s.t. \hspace{1cm}$ $\displaystyle c_{t}+p_{t}b^{P}_{t}=(1-x_{t})(1-\tau_{t})+b^{P} _{t-1},\hspace{1cm} \forall t=0,1,2,...$ (3)

where $ p_{t}$ is the price at time $ {t}$ of private bond holdings ($ b^{P}_{t} $), paying one unit of consumption at time t+1.

The household's first order conditions are

$\displaystyle \frac{u_{x,t}}{u_{c,t}}$ $\displaystyle =(1-\tau_{t})$ (4)
$\displaystyle p_{t}$ $\displaystyle =\beta\frac{u_{c,t+1}}{u_{c,t}},$ (5)

together with the budget constraint (3). Equation (4) and (5) represent the equilibrium condition in the labor market and the bond market, respectively.

In what follows, we analyze the problem of the government and characterize its solution under the assumption of full-commitment. This will serve as a benchmark for our discussion in subsequent sections.

2.1  The case of full-commitment

If the government has full-commitment, for a given initial level of debt ($ b_{-1}$), it solves the following problem

  $\displaystyle \max_{\left\{ c_{t},g_{t},b_{t}\right\} _{t=0}^{\infty} } \sum _{t=0}^{\infty}\beta^{t} u(c_{t},1-c_{t}-g_{t},g_{t})$    
$\displaystyle s.t.\hspace{1cm}$ $\displaystyle c_{t}u_{c,t}+\beta u_{c,t+1}b_{t}=(c_{t}+g_{t} )u_{x,t}+b_{t-1}u_{c,t},\hspace{1cm} \forall t=0,1,2,...$ (6)

where we made use of the household's optimality conditions (3)-(5), the resource constraint (1) and the market clearing condition $ b^{P}_{t}+b^{G}_{t}=0$, to substitute for taxes, public expenditure, leisure and government debt. We rule out Ponzi schemes, by imposing the transversality condition

$\displaystyle \lim_{T\rightarrow\infty}\beta^{T}u_{c,T}b_{T}=0.$ (7)

For our purposes it is worth recalling some features of the resulting equilibrium. As discussed in Lucas and Stokey (1983), in the full-commitment case after an initial jump, all the allocations, including the amount of debt, reach their steady-state level, and remain constant from then on. This is because, apart from $ t=0$, all the periods are identical and the government is willing to smooth private and public consumption over time. However, the steady-state allocations depend on the initial condition $ b_{-1} $. In other words, countries starting with high debt will have high debt forever, and countries with low debt will have low debt forever. Because of this dependency on initial conditions, which are exogenous to the model and empirically difficult to determine, this theory cannot explain why countries accumulate debt to start with. Moreover, it cannot explain why the level of debt is so different across countries with similar economic conditions.

The first-period allocations are different, because of the time-inconsistency problem typical of this setting. The government, when making its plans at period $ t=0$, would like to use taxes and public expenditure to manipulate the bond price. This is because of the following. For a generic $ t>0$, current consumption influences both $ p_{t}$ and $ p_{t-1} $. As a consequence, if the government uses taxes and public expenditure to increase the price of the bond $ p_{t}$, other things equal, it also decreases $ p_{t-1}$. At an optimum, it turns out that the costs of such a procedure offset the benefits. However, at $ t=0$ things are different, because consumers' savings and previous prices ($ p_{-1}$) are given. Therefore, if the government inherits a positive level of debt, it can benefit from an increase in the price of the bond without incurring any additional cost. For example, by setting its policies such that current consumption is higher than in the future, the government is able to foster the demand for savings, thus selling bonds at a more convenient price.10 These incentives to increase initial consumption prevail whenever the government is allowed to make a new plan. This is why the solution to this problem is in general time-inconsistent.

Figure 1:  Debt dynamics under full-commitment

Data for Figure 1 immediately follows.

Note:  The figure plots, for different level of initial debt, the level of consumption in the first period (solid line) and the steady-state level of consumption (dashed line). The reported values correspond to the calibration specified in table A-2.


Data for Figure 1

Probability
Consumption at steady state
Initial Consumption
0.00
0.24682
0.24682
0.25
0.24361
0.25940
0.50
0.24087
0.27259
0.75
0.23853
0.28527
1.00
0.23648
0.29707

To explain better the mechanism described above, in figure 1 we plot the level of consumption at $ t=0$ ($ c_{0}$) and the steady-state level of consumption ($ c_{ss}$), for a given positive initial level of debt ( $ b_{-1}\geq0$), under the full-commitment assumption.11 We can see that the higher is debt, the bigger is the difference between current and future consumption, and thus the higher is the drop in the interest rate. This happens because the higher is debt the larger is the base on which the improved interest rate is applied. As a consequence, the higher is the inherited level of debt, the greater is the willingness to manipulate the interest rate.

Now we can look at the behavior of debt in the first period, by looking at the government budget constraint in equation (2). On the one hand, the tax cut necessary to foster initial consumption reduces the tax revenues of the government. On the other hand, the resulting lower interest rate allows the government to sell bonds at a higher price. Whether $ b_{0}>b_{-1}$ depends on the composite effect of these two forces. In figure 2, we plot the level of debt chosen in the first period (and thus the steady-state level of debt), as a function of $ b_{-1}$. For low levels of $ b_{-1}$, the government accumulates debt. However, if the initial level of debt is large enough, the increase in bond prices applies to a larger base. As a consequence, the tax cut can be self-financed and the level of debt can also decrease.

Figure 2:  Debt dynamics under full-commitment

In Figure 2, the vertical axis refers to bonds in steady-state, and the horizontal axis refers to initial bonds. There is a 45 degree line, and another line plotting the policy function. The lines intersect each other two times: when bonds equal zero, and when bonds are around 0.35. The policy function is above the 45 degree line for positive debt levels with the exception of the region to the right of the 0.35 point.

Note:  The figure plots the steady-state level of debt (bs s), that is the level of debt prevailing from the first period on, as a function of the initial debt (b0). The reported values correspond to the parameterization specified in table A-2.



3  The time-consistent solution

In this section, we analyze the problem of a benevolent planner which, as opposed to the case of the previous section, does not have access to a commitment technology. More precisely, we consider the case in which the current planner cannot make credible promises about his future actions. We keep the assumption that the planner can credibly commit to repay his loans.12 In what follows, we also assume that reputation mechanisms are not operative, focusing only on Markov-Perfect equilibria, as defined for instance in Klein et al. (2004).

In this case the problem of the planner is

  $\displaystyle V(b_{t-1})=\max_{\left\{ c_{t},g_{t},b_{t}\right\} }u(c_{t},1-c_{t} -g_{t},g_{t})+\beta V(b_{t})$ (8)
  $\displaystyle s.t. \hspace{1cm}c_{t}u_{c,t}+\beta u_{c}(\Psi(b_{t}))b_{t}=(c_{t} +g_{t})u_{x,t}+b_{t-1}u_{c,t}.$ (9)

The function $ \Psi(b_{t})$ in constraint (9) determines the quantity of consumption the consumer expects for period $ t+1$ as a function of the debt level $ b$ outstanding at the beginning of next period. This represents the main difference with respect to the full-commitment case. Since the current planner cannot make credible commitments about his future actions, the future stream of consumption is not under his direct control. By taking as given the policy $ \Psi(b_{t})$ of his successor (or himself in the next period), the current planner can only influence future consumption through his current debt policy. Being the function $ \Psi(b_{t})$ unknown, the solution of this problem relies on solving a fixed point problem in $ \Psi(b_{t})$.13

We can now look at the first order conditions of the associated Lagrangian, and in particular at the generalized Euler equation (GEE)

$\displaystyle \gamma_{t}(u_{cc,t+1}\Psi_{b,t}b_{t}+u_{c,t+1})=u_{c,t+1}\gamma_{t+1} ,$ (10)

where $ \gamma_{t}$ indicates the Lagrange multiplier attached to constraint (9).14$ ^{,}$15 The inspection of the previous equation allows us to describe the behavior of the economy in a (deterministic) steady-state. In particular, for the GEE to be satisfied in steady-state, it must be that

$\displaystyle \gamma u_{cc}\Psi_{b}b=0 .$ (11)

We can identify three different cases in which such relationship holds, as illustrated in figure 3. This figure, together with the steady-states implied by eq. (11), gives a qualitative representation of the transition dynamics obtained in our numerical experiments.

Figure 3:  Debt dynamics in the time-consistent Case

In Figure 3, the vertical axis refers to bonds next period, and the horizontal axis refers to current bonds. There is a 45 degree line, and another line plotting the policy function. The lines intersect each other three times: when bonds equal zero, when bonds are at a negative level (point A), and a third point with very negative debt levels (point B). The policy function is below the 45 degree line for positive debt levels and between point A and point B.

Note:  The figure is a qualitative representation of debt equilibrium dynamics resulting from our numerical experiments.


First, we have the case in which $ \gamma=0$. This means that constraint (9) is not binding, and we are at an unconstrained optimum. From an economic point of view, this is saying that the planner can avoid to raise distortionary taxes and can finance his public expenditure through the interest payments received on his outstanding assets. This represents the first-best solution.16

Second, we have the case in which $ \Psi_{b}=0$. This can happen when a marginal change in the level of debt does not induce any change in the equilibrium level of private consumption. This case cannot be ruled out. However, given the presence of distortionary taxation, this is not due to Ricardian equivalence. On the contrary, when a planner inherits a higher level of debt, he has to raise more distortionary taxes. Because of the bigger distortions created, by a substitution effect, this will reduce hours worked and private consumption. An increase in debt also creates a wealth effect that decreases hours worked and increases private consumption.

Both the wealth and substitution effects lead to a reduction in hours worked as debt increases. The composite effect on private consumption can be understood by examining the aggregate resource constraint. By differentiating equation (1) with respect to debt ($ b$) it holds

$\displaystyle \frac{\partial{c}}{\partial{b}}+\frac{\partial{g}}{\partial{b}}=-\frac {\partial{x}}{\partial{b}}.$ (12)

In a model where public expenditure is exogenous, the effects on consumption must be equal to the ones on hours worked. As a consequence, in such case, $ \Psi_{b}$ cannot be zero. But in our framework, there is another way for the planner to cope with the higher burden created by the higher debt. That is, by reducing the amount of public good provision. As a result, it is possible that a marginal change in the level of debt does not produce any effect on the level of equilibrium consumption (i.e. $ \Psi_{b}=0$) as long as the effects on leisure ($ x$) and public expenditure ($ g$) exactly offset each other.

Finally, we have a steady-state, associated with a level of debt equal to zero. When debt is zero, the government does not have any incentive to manipulate the interest rate. At this point, policymakers' commitment is irrelevant and thus debt remains constant at a zero level, as in the full-commitment case.

We now turn to explain the transition dynamics of the model. Under full-commitment, after the initial period debt is constant. With no-commitment the pictures change significantly and this is due temptation to influence the interest rate not only in the first period, as in the full-commitment case, but in every period. As illustrated in Figure 3 we find that, in the (more relevant) cases in which the government initially holds a positive amount of debt or relatively small amount of assets, the economy will converge to the steady-state with zero debt.

As explained for the full-commitment case, whenever a government inherits a positive amount of debt, it has the incentive to use the instruments at its disposal to reduce the interest rate payments or, equivalently, to increase the selling price of bonds, as given by (5). To do so, the demand for savings should increase, which will happen if current consumption increases more than future consumption. A government with full-commitment could promise the desired level of future consumption regardless of the debt level, as long as the allocation is feasible. In the no-commitment case this is no longer true. The government can only influence future actions through the state variables, which in our case is debt. The higher the inherited debt, the higher will be the incentive in the next period to increase consumption again, in order to manipulate bond prices. Therefore, to face favorable bond prices, the current government needs to leave a lower debt to its successor. If it does not do so, the successor will raise consumption even more, and the anticipated positive consumption growth would harm the current bond price. It follows that debt is reduced until a level of zero debt is reached. At this point, the incentive to manipulate the interest rate vanishes. A symmetric argument also explains why a government that starts with assets, but to the right of the point where $ \Psi_{b}=0$, would instead reduce the asset holdings to manipulate the bond price, until the zero debt level is reached.

The mechanism that we explained above relies on the temptation that every government has to manipulate the bond price. If a government reduces debt, then tomorrow's government will face a smaller temptation to manipulate the bond price, and consequently consumption will be lower than today's. However, there is a second effect. As we mentioned before, when debt is lowered, the government can afford to lower taxes. As a consequence, leisure decreases, output increases and the economy can increase both private and public consumption. According to this effect, if tomorrow's government has lower debt then it will increase private consumption. Notice that this second effect goes in an opposite direction of the first one. At the point $ \Psi _{b}=0$ the two effects exactly cancel out. To the left of $ \Psi_{b}=0$ the second effect dominates, i.e. when assets are accumulated (debt is reduced) consumption increases. The amount of debt at which $ \Psi_{b}=0$ depends on the marginal rate of substitution between private and public consumption and between consumption and leisure.17 Under our baseline calibration, as it can be seen in Figure 3 the point where $ \Psi_{b}=0$ is associated with government asset holdings ($ b<0$). In this case, the steady-state with $ \Psi_{b}=0$ is unstable, while the steady state with $ b=0$ is stable.18

From a theoretical point of view, it is also possible to have $ \Psi_{b}=0$ at a point where debt is positive. In that case, such steady-state with positive debt is stable, while the steady-state with $ b=0$ is unstable. In other words, whenever the government starts with debt it would converge to the point $ \Psi_{b}=0$. And, whenever the government starts with assets it would accumulate further assets, until public expenditures can be financed only through the associated interest payments. In our numerical exercises, we found that for calibrations implying a plausible level of public expenditure the case depicted in Figure 3 is the relevant one. In particular, one can obtain that the steady-state with zero debt is unstable only when the steady-state public expenditures are unreasonably low.19 In what follows we abstract from considering these cases and focus on the case where the steady-state with $ b=0$ is stable.

To provide a more concrete description of the behavior of our economy, we solve the model numerically by assuming the following functional form for the utility function:20

$\displaystyle u(c,x,g)=(1-\phi_{g})\left[ \phi_{c}\frac{c^{1-\sigma_{c}}-1}{1-\sigma_{c} }+(1-\phi_{c})\frac{x^{1-\sigma_{x}}-1}{1-\sigma_{x}}\right] +\phi_{g} \frac{g^{1-\sigma_{g}}-1}{1-\sigma_{g}},$ (13)

where $ \phi_{c}$ and $ \phi_{g}$ denote the preference weights on private and public consumption.

We use a standard calibration for an annualized model of the US economy in order to match long-run ratios of our variables. Table A-2 summarizes the parameter values.21

The evolution of the allocations over time is illustrated in figures 4 and 5 where, for comparison, we also display the solution under full-commitment. For a given level of initial debt, we can observe a decreasing pattern of private consumption and an increasing interest rate.22 This is achieved by lowering taxation and increasing public consumption over time.

Figure 4:  Commitment vs. no-commitment: time pattern of allocations

Data for Figure 4 immediately follows.

Note:  The figure plots the equilibrium allocations over time, giving an initial condition of b = .16 which is roughly 50% of GDP under our parameterization. The interest rate (lower-left panel) for the full-commitment case (continuous line) has to be referred to the right-hand scale.


Data for Figure 4

Time
Output:
Discretion
Output:
Commitment
Taxes:
Discretion
Taxes:
Commitment
Private Consumption:
Discretion
Private Consumption:
Commitment
Public Consumption:
Discretion
Public Consumption:
Commitment
Leisure:
Discretion
Leisure:
Commitment
0
0.3127
0.3241
0.2398
0.1598
0.2484
0.2547
0.0643
0.0694
0.6873
0.6759
1
0.3130
0.3168
0.2395
0.2490
0.2483
0.2447
0.0647
0.0720
0.6870
0.6832
2
0.3133
0.3168
0.2391
0.2490
0.2482
0.2447
0.0651
0.0720
0.6867
0.6832
3
0.3135
0.3168
0.2388
0.2490
0.2481
0.2447
0.0654
0.0720
0.6865
0.6832
4
0.3137
0.3168
0.2385
0.2490
0.2481
0.2447
0.0657
0.0720
0.6863
0.6832
5
0.3139
0.3168
0.2382
0.2490
0.2480
0.2447
0.0660
0.0720
0.6861
0.6832
6
0.3141
0.3168
0.2379
0.2490
0.2479
0.2447
0.0662
0.0720
0.6859
0.6832
7
0.3143
0.3168
0.2377
0.2490
0.2479
0.2447
0.0664
0.0720
0.6857
0.6832
8
0.3145
0.3168
0.2374
0.2490
0.2478
0.2447
0.0667
0.0720
0.6855
0.6832
9
0.3146
0.3168
0.2372
0.2490
0.2478
0.2447
0.0669
0.0720
0.6854
0.6832
10
0.3148
0.3168
0.2369
0.2490
0.2477
0.2447
0.0670
0.0720
0.6852
0.6832
11
0.3149
0.3168
0.2367
0.2490
0.2477
0.2447
0.0672
0.0720
0.6851
0.6832
12
0.3151
0.3168
0.2365
0.2490
0.2477
0.2447
0.0674
0.0720
0.6849
0.6832
13
0.3152
0.3168
0.2363
0.2490
0.2476
0.2447
0.0675
0.0720
0.6848
0.6832
14
0.3153
0.3168
0.2361
0.2490
0.2476
0.2447
0.0677
0.0720
0.6847
0.6832
15
0.3154
0.3168
0.2359
0.2490
0.2476
0.2447
0.0678
0.0720
0.6846
0.6832
16
0.3155
0.3168
0.2357
0.2490
0.2475
0.2447
0.0680
0.0720
0.6845
0.6832
17
0.3156
0.3168
0.2355
0.2490
0.2475
0.2447
0.0681
0.0720
0.6844
0.6832
18
0.3157
0.3168
0.2354
0.2490
0.2475
0.2447
0.0682
0.0720
0.6843
0.6832
19
0.3158
0.3168
0.2352
0.2490
0.2475
0.2447
0.0683
0.0720
0.6842
0.6832
20
0.3159
0.3168
0.2350
0.2490
0.2474
0.2447
0.0684
0.0720
0.6841
0.6832
21
0.3160
0.3168
0.2349
0.2490
0.2474
0.2447
0.0685
0.0720
0.6840
0.6832
22
0.3160
0.3168
0.2347
0.2490
0.2474
0.2447
0.0686
0.0720
0.6840
0.6832
23
0.3161
0.3168
0.2346
0.2490
0.2474
0.2447
0.0687
0.0720
0.6839
0.6832
24
0.3162
0.3168
0.2344
0.2490
0.2474
0.2447
0.0688
0.0720
0.6838
0.6832
25
0.3163
0.3168
0.2343
0.2490
0.2474
0.2447
0.0689
0.0720
0.6837
0.6832
26
0.3163
0.3168
0.2342
0.2490
0.2473
0.2447
0.0690
0.0720
0.6837
0.6832
27
0.3164
0.3168
0.2340
0.2490
0.2473
0.2447
0.0691
0.0720
0.6836
0.6832
28
0.3165
0.3168
0.2339
0.2490
0.2473
0.2447
0.0692
0.0720
0.6835
0.6832
29
0.3165
0.3168
0.2338
0.2490
0.2473
0.2447
0.0692
0.0720
0.6835
0.6832
30
0.3166
0.3168
0.2337
0.2490
0.2473
0.2447
0.0693
0.0720
0.6834
0.6832
31
0.3166
0.3168
0.2336
0.2490
0.2473
0.2447
0.0694
0.0720
0.6834
0.6832
32
0.3167
0.3168
0.2335
0.2490
0.2473
0.2447
0.0694
0.0720
0.6833
0.6832
33
0.3167
0.3168
0.2334
0.2490
0.2473
0.2447
0.0695
0.0720
0.6833
0.6832
34
0.3168
0.3168
0.2333
0.2490
0.2472
0.2447
0.0696
0.0720
0.6832
0.6832
35
0.3168
0.3168
0.2332
0.2490
0.2472
0.2447
0.0696
0.0720
0.6832
0.6832
36
0.3169
0.3168
0.2331
0.2490
0.2472
0.2447
0.0697
0.0720
0.6831
0.6832
37
0.3169
0.3168
0.2330
0.2490
0.2472
0.2447
0.0697
0.0720
0.6831
0.6832
38
0.3170
0.3168
0.2329
0.2490
0.2472
0.2447
0.0698
0.0720
0.6830
0.6832
39
0.3170
0.3168
0.2328
0.2490
0.2472
0.2447
0.0698
0.0720
0.6830
0.6832
40
0.3171
0.3168
0.2327
0.2490
0.2472
0.2447
0.0699
0.0720
0.6829
0.6832
41
0.3171
0.3168
0.2326
0.2490
0.2472
0.2447
0.0699
0.0720
0.6829
0.6832
42
0.3171
0.3168
0.2325
0.2490
0.2472
0.2447
0.0700
0.0720
0.6829
0.6832
43
0.3172
0.3168
0.2324
0.2490
0.2472
0.2447
0.0700
0.0720
0.6828
0.6832
44
0.3172
0.3168
0.2324
0.2490
0.2472
0.2447
0.0701
0.0720
0.6828
0.6832
45
0.3173
0.3168
0.2323
0.2490
0.2471
0.2447
0.0701
0.0720
0.6827
0.6832
46
0.3173
0.3168
0.2322
0.2490
0.2471
0.2447
0.0702
0.0720
0.6827
0.6832
47
0.3173
0.3168
0.2321
0.2490
0.2471
0.2447
0.0702
0.0720
0.6827
0.6832
48
0.3174
0.3168
0.2321
0.2490
0.2471
0.2447
0.0702
0.0720
0.6826
0.6832
49
0.3174
0.3168
0.2320
0.2490
0.2471
0.2447
0.0703
0.0720
0.6826
0.6832
50
0.3174
0.3168
0.2319
0.2490
0.2471
0.2447
0.0703
0.0720
0.6826
0.6832
51
0.3175
0.3168
0.2318
0.2490
0.2471
0.2447
0.0703
0.0720
0.6825
0.6832
52
0.3175
0.3168
0.2318
0.2490
0.2471
0.2447
0.0704
0.0720
0.6825
0.6832
53
0.3175
0.3168
0.2317
0.2490
0.2471
0.2447
0.0704
0.0720
0.6825
0.6832
54
0.3175
0.3168
0.2316
0.2490
0.2471
0.2447
0.0705
0.0720
0.6825
0.6832
55
0.3176
0.3168
0.2316
0.2490
0.2471
0.2447
0.0705
0.0720
0.6824
0.6832
56
0.3176
0.3168
0.2315
0.2490
0.2471
0.2447
0.0705
0.0720
0.6824
0.6832
57
0.3176
0.3168
0.2315
0.2490
0.2471
0.2447
0.0706
0.0720
0.6824
0.6832
58
0.3177
0.3168
0.2314
0.2490
0.2471
0.2447
0.0706
0.0720
0.6823
0.6832
59
0.3177
0.3168
0.2313
0.2490
0.2471
0.2447
0.0706
0.0720
0.6823
0.6832
60
0.3177
0.3168
0.2313
0.2490
0.2471
0.2447
0.0706
0.0720
0.6823
0.6832
61
0.3177
0.3168
0.2312
0.2490
0.2471
0.2447
0.0707
0.0720
0.6823
0.6832
62
0.3178
0.3168
0.2312
0.2490
0.2471
0.2447
0.0707
0.0720
0.6822
0.6832
63
0.3178
0.3168
0.2311
0.2490
0.2471
0.2447
0.0707
0.0720
0.6822
0.6832
64
0.3178
0.3168
0.2311
0.2490
0.2470
0.2447
0.0708
0.0720
0.6822
0.6832
65
0.3178
0.3168
0.2310
0.2490
0.2470
0.2447
0.0708
0.0720
0.6822
0.6832
66
0.3178
0.3168
0.2310
0.2490
0.2470
0.2447
0.0708
0.0720
0.6822
0.6832
67
0.3179
0.3168
0.2309
0.2490
0.2470
0.2447
0.0708
0.0720
0.6821
0.6832
68
0.3179
0.3168
0.2309
0.2490
0.2470
0.2447
0.0709
0.0720
0.6821
0.6832
69
0.3179
0.3168
0.2308
0.2490
0.2470
0.2447
0.0709
0.0720
0.6821
0.6832
70
0.3179
0.3168
0.2308
0.2490
0.2470
0.2447
0.0709
0.0720
0.6821
0.6832
71
0.3179
0.3168
0.2307
0.2490
0.2470
0.2447
0.0709
0.0720
0.6821
0.6832
72
0.3180
0.3168
0.2307
0.2490
0.2470
0.2447
0.0709
0.0720
0.6820
0.6832
73
0.3180
0.3168
0.2307
0.2490
0.2470
0.2447
0.0710
0.0720
0.6820
0.6832
74
0.3180
0.3168
0.2306
0.2490
0.2470
0.2447
0.0710
0.0720
0.6820
0.6832
75
0.3180
0.3168
0.2306
0.2490
0.2470
0.2447
0.0710
0.0720
0.6820
0.6832
76
0.3180
0.3168
0.2305
0.2490
0.2470
0.2447
0.0710
0.0720
0.6820
0.6832
77
0.3181
0.3168
0.2305
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
78
0.3181
0.3168
0.2304
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
79
0.3181
0.3168
0.2304
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
80
0.3181
0.3168
0.2304
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
81
0.3181
0.3168
0.2303
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
82
0.3181
0.3168
0.2303
0.2490
0.2470
0.2447
0.0711
0.0720
0.6819
0.6832
83
0.3182
0.3168
0.2303
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
84
0.3182
0.3168
0.2302
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
85
0.3182
0.3168
0.2302
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
86
0.3182
0.3168
0.2301
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
87
0.3182
0.3168
0.2301
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
88
0.3182
0.3168
0.2301
0.2490
0.2470
0.2447
0.0712
0.0720
0.6818
0.6832
89
0.3182
0.3168
0.2300
0.2490
0.2470
0.2447
0.0713
0.0720
0.6818
0.6832
90
0.3183
0.3168
0.2300
0.2490
0.2470
0.2447
0.0713
0.0720
0.6817
0.6832
91
0.3183
0.3168
0.2300
0.2490
0.2470
0.2447
0.0713
0.0720
0.6817
0.6832
92
0.3183
0.3168
0.2299
0.2490
0.2470
0.2447
0.0713
0.0720
0.6817
0.6832
93
0.3183
0.3168
0.2299
0.2490
0.2470
0.2447
0.0713
0.0720
0.6817
0.6832
94
0.3183
0.3168
0.2299
0.2490
0.2470
0.2447
0.0713
0.0720
0.6817
0.6832
95
0.3183
0.3168
0.2299
0.2490
0.2470
0.2447
0.0714
0.0720
0.6817
0.6832
96
0.3183
0.3168
0.2298
0.2490
0.2470
0.2447
0.0714
0.0720
0.6817
0.6832
97
0.3183
0.3168
0.2298
0.2490
0.2470
0.2447
0.0714
0.0720
0.6817
0.6832
98
0.3184
0.3168
0.2298
0.2490
0.2470
0.2447
0.0714
0.0720
0.6816
0.6832
99
0.3184
0.3168
0.2297
0.2490
0.2470
0.2447
0.0714
0.0720
0.6816
0.6832

Figure 5:  Commitment vs. no-commitment: time pattern of debt

Data for Figure 5 immediately follows.

Note:  The figure plots the evolution of debt over time, giving an initial condition of b = .16 which is roughly 50% of GDP under our parameterization. The solid line corresponds to the full-commitment case, while the dashed line corresponds to the no-commitment case.


Data for Figure 5

Time
Discretion
Commitment
1
0.160
0.160
2
0.155
0.171
3
0.151
0.171
4
0.147
0.171
5
0.143
0.171
6
0.139
0.171
7
0.136
0.171
8
0.133
0.171
9
0.129
0.171
10
0.126
0.171
11
0.124
0.171
12
0.121
0.171
13
0.118
0.171
14
0.116
0.171
15
0.113
0.171
16
0.111
0.171
17
0.109
0.171
18
0.107
0.171
19
0.104
0.171
20
0.102
0.171
21
0.100
0.171
22
0.099
0.171
23
0.097
0.171
24
0.095
0.171
25
0.093
0.171
26
0.092
0.171
27
0.090
0.171
28
0.089
0.171
29
0.087
0.171
30
0.086
0.171
31
0.084
0.171
32
0.083
0.171
33
0.081
0.171
34
0.080
0.171
35
0.079
0.171
36
0.078
0.171
37
0.076
0.171
38
0.075
0.171
39
0.074
0.171
40
0.073
0.171
41
0.072
0.171
42
0.071
0.171
43
0.070
0.171
44
0.069
0.171
45
0.068
0.171
46
0.067
0.171
47
0.066
0.171
48
0.065
0.171
49
0.064
0.171
50
0.063
0.171