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Default, reputation, and balanced-budget rules
Review of Economic Dynamics 7 (2004) 382–405 www.elsevier.com/locate/red
Default, reputation, and balanced-budget rules David R. Stockman Economics Department, University of Delaware, Newark, DE 19716, USA Received 4 December 2001; revised 4 June 2003
Abstract Is a balanced-budget rule compatible with a government honoring its debt obligations? According to the conventional explanation, governments honor their debt obligations to maintain a good reputation for future borrowing. The ability to borrow is desirable because it allows for greater tax smoothing. However, a balanced-budget rule limits the ability to smooth taxes, rendering a large class of competitive equilibria not compatible with a government honoring its debt obligations. The reputation model predicts default as the equilibrium outcome under a balanced-budget restriction. Insofar as this prediction is falsified by empirical observation, mechanisms other than reputation must be at work. 2003 Elsevier Inc. All rights reserved. JEL classification: E61; H21; H63
1. Introduction Is a balanced-budget rule compatible with a government honoring its debt obligations? A balanced-budget rule is a restriction on fiscal policy currently used by some governments and may be adopted by more. For example, many US states have some form of balancedbudget rule, and the US Congress has considered passing a balanced-budget amendment for the federal government. One also finds a balanced-budget rule in the debate on the monetary union in Europe. However, many economic models suggest that optimal fiscal policy calls for the government to smooth taxes by running surpluses and deficits financed through borrowing and lending.1 Why would a government limit its ability to smooth taxes by adopting a balanced-budget restriction? Advocates for a balanced-budget restriction E-mail address: [email protected]. 1 For example, tax smoothing results financed through borrowing and lending are obtained in Barro (1979),
Lucas and Stokey (1983), and Mankiw (1987). 1094-2025/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.red.2003.09.002
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argue that the political system results in excessive spending and that a balanced-budget amendment will help control this tendency.2 Opponents argue that the constraint will inhibit stabilizing fiscal action and increase the amplitude of the business cycle. Prescott (1977) illustrates a time inconsistency problem of debt that suggests the government has an incentive to renege on its previous debt obligations. He argues that if only distortionary taxes are available to service the debt, these interest payments result in a loss of welfare. If the government were to default on its outstanding debt, the distortionary taxes would be lower and welfare would be increased. However, households anticipate this and choose not hold government debt, which results in government debt equaling zero in equilibrium. This runs counter to empirical evidence: households hold government debt and governments honor their debt obligations. Why do households lend to governments and why does the government not default? The conventional explanation for why a government honors its debt obligations is that it wishes to maintain a good reputation for future borrowing (e.g., Chari and Kehoe, 1993). The ability to borrow is desirable because debt serves as a buffer to help smooth distortionary taxes over time resulting in higher economic welfare. Given that the government has not defaulted on its debt in the past, the household is willing to lend to the government. If the government were to default on its debt, it would damage its reputation as a reliable borrower. The household might no longer hold debt, distortionary taxes would fluctuate more and welfare would be lower. Here, I would like to focus attention on the potential incompatibility of a no-default competitive equilibrium where the government honors its debt obligations and a balancedbudget restriction. Why is there an incompatibility? The reputation argument for not defaulting depends on an effective deterrent: if the government ever defaults, it may never borrow again. This repercussion of default is undesirable if there is great need to smooth taxes and debt is a useful tax-smoothing instrument. The inconsistency occurs because a balanced-budget rule lessens the deterrent by limiting debt’s usefulness to smooth taxes. This coupling of balanced-budget rules and default provides a novel perspective from which to investigate the merits of a balanced-budget restriction.3 The principal results of this paper are: • A no-default competitive equilibrium is incompatible with a balanced-budget restriction if the return on debt is positive. • A no-default competitive equilibrium may be compatible with a balanced-budget restriction if there is sufficient tax smoothing through state-contingent returns on debt (this requires a negative return on debt in some states). However, enormous variation in the state-contingent returns is required (e.g., ±25% and higher). 2 In Europe, the argument has been made that a monetary union without fiscal convergence would be unstable. Without strict fiscal guidelines, there will be excessive deficits (perhaps politically motivated) that are not consistent with long-term solvency. See Corsetti and Roubini (1992). 3 In related work, Schmitt-Grohé and Uribe (1997) show that balanced-budget rules may result in sunspot equilibria. Both the possibility of sunspot equilibria and increase in the incentive to default are independent of concerns about stabilization policy and excessive spending. Consideration of these issues is necessary to gain a broader understanding of balanced-budget rules.
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• Adding capital to the model greatly exacerbates the sustainability problem under a balanced-budget restriction. In this case, state-contingent capital income taxes render state-contingent returns on debt superfluous and any no-default competitive equilibrium where the implicit risk-free return on debt is positive is incompatible with a balanced-budget restriction. In summary, the reputation model predicts default as the likely equilibrium outcome under a balanced-budget restriction. Insofar as this prediction is falsified by empirical observation, mechanisms other than reputation must be at work. The rest of the paper is organized as follows. Section 2 contains a brief review of Chari and Kehoe (1993) who formalize the reputation argument in a game-theoretic framework. In Section 3, I give sufficient conditions under which a large class of competitive equilibria without default is not consistent with a balanced-budget restriction. Next, I characterize Ramsey allocations both with and without a balanced-budget restriction and all stationary competitive equilibria with and without a balanced-budget restriction in Section 4. Numerical examples in Section 5 illustrate what competitive equilibria without default are consistent with a balanced-budget restriction. In Section 6, I discuss the effect of adding capital to the model and conclude in Section 7.
2. Chari and Kehoe model The model consists of a large number of infinitely-lived identical households. Output is produced by a constant-returns-to-scale production function using the only factor of production, labor. Output can be used for either private consumption (c(g t )) or government consumption (gt ). Government consumption {gt }∞ t =0 is exogenous and may be stochastic. The state of the economy at time t is a realization of government expenditures up to and t := (g , g , . . . , g including time t: g t := (g0 , . . . , gt ). For a given g t with t 1, let g−1 0 1 t −1 ) t t t be the first t − 1 values of g so g ≡ (g−1 , gt ). If gt is stochastic, it may take on one of a finite number of possible values {γ1 , γ2 , . . . , γN }. The probability of g t is given by µ(g t ). Units of labor and output are chosen such that one unit of labor provides one unit of output. Each household has preferences over state-contingent consumption and labor streams {c(g t ), l(g t )} given by ∞ t t t , (1) β U c g ,l g E0 t =0
with discount factor 0 < β < 1. The felicity function, U , is assumed to be bounded, strictly concave, increasing in consumption, and decreasing in labor.4 The representative household’s objective is to maximize (1) by choice of {c(g t ), l(g t ), b(g t +1 )}, subject to (2) q g t , gt +1 b g t , gt +1 1 − τ g t l g t + 1 − δ(g t ) b g t , c(g t ) + gt+1 4 I further assume the U satisfies the Inada conditions lim c→0 Uc = +∞ and liml→l¯ Ul = −∞, where l¯ represents a time endowment and therefore an upper bound on labor as well.
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taking the tax on labor income τ (g t ), default rates δ(g t ), prices of government debt q(g t , gt +1 ), and initial debt b(g 0 ) as given. Government debt is state-contingent and q(g t , gt +1 ) gives the price at g t in units of c(g t ) per unit payoff in state (g t , gt +1 ). An t t t t allocation is understood to be x = {x(g t )}∞ t =0 where x(g ) = {c(g ), l(g ), b(g )}. The government selects a policy of state-contingent default rates, bond prices, and tax rates on labor while taking its expenditures as given to satisfy a sequence of budget constraints t gt + 1 − δ g t b g t = τ g t l g t + q g , gt +1 b g t +1 ,
(3)
gt+1
for t = 0, 1, . . . , ∞. A policy for the government is denoted by π = {π(g t )}∞ t =0 where π(g t ) = {δ(g t ), q(g t ), τ (g t )}. The government is assumed to be benevolent (the government’s preferences are identical to the representative household’s). The default rate δ(g t ) is restricted to be between 0 and 1, where δ(g t ) = 0 implies that the government honor its debt completely, and δ(g t ) = 1 corresponds to complete default. Chari and Kehoe call a pair (π, x) attainable under commitment if it satisfies both the government budget constraint and household optimization. Chari and Kehoe call a competitive equilibrium a sustainable equilibrium if the government honors its debt obligation in the absence of a commitment technology. They characterize the entire set of sustainable outcomes by identifying the worst sustainable equilibrium and using revert-to-autarky plans. The autarky plan is defined as follows: (i) regardless of what has happened in the past, the government chooses q(g t ) = 0 and δ(g t ) = 1 and sets τ a (g t ) equal to that given by the solution to max U (c(g t ), l(g t )) subject to c(g t ) + gt = l(g t ), τ a (g t )l(g t ) = gt , and (1 + Ul (g t )/Uc (g t ))l(g t ) = gt , (ii) the household allocation rule sets debt equal to zero (regardless of what has happened in the past) and chooses consumption and labor to maximize utility subject to the given tax rates. Let [ca (g t ), l a (g t )] denote these choices. The autarky level of utility at time s in state g s is ∞ β t U ca g t , l a g t g s . W a g s := E
(4)
t =s
The revert-to-autarky plans specify that for a given candidate policy and allocation (π, x), the government and the agent continue with (π, x) as long as there have been no deviations in the past. If there has been any deviation, switch to the autarky plan. The main characterization result from their paper is the following. Proposition 1. An arbitrary pair (π, x) is an outcome of a sustainable equilibrium if and only if
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(i) the pair is attainable under commitment and (ii) for every s and g s , the following holds: ∞ t t s t β U c g , l g | g W a gs . E t =s
Proof. See Chari and Kehoe (1993). ✷ This proposition provides a convenient way to check whether or not a particular competitive equilibrium under commitment is sustainable. Note that deterministic government spending is a covered as well by this proposition. By utilizing this characterization of sustainable equilibria, Chari and Kehoe go on to demonstrate that if government expenditures follows a Markov chain and some tax-smoothing is desirable, the Ramsey outcomes are sustainable for sufficiently high discount factors. 3. Balanced-budget rules and default In this section, I provide sufficient conditions for a competitive equilibrium under a balanced-budget restriction to be unsustainable.5 This set of competitive equilibria is quite large and reasonable. It may include the Ramsey equilibrium under a balanced-budget restriction. There are three steps in the argument: (1) Show that a consumption-labor allocation in a competitive equilibrium under a balanced-budget restriction with the gross return on bonds greater than one is the same for a competitive equilibrium with no debt and a sequence of positive lump-sum transfers. (2) Show that under a balanced-budget restriction with zero debt, a consumption-labor allocation with strictly positive lump-sum transfers provides lower utility than the consumption-labor allocation with no lump-sum transfers. (3) Show that the autarky equilibrium corresponds to the equilibrium under a balancedbudget restriction, zero debt, and zero lump-sum transfers. To facilitate defining a balanced-budget restriction, I model bonds as selling at face value and earning interest. Let R(g t ) be the gross return on bonds held from period t − 1 t to t in state g t .6 I assume that the balanced-budget restriction takes the form in state g−1 bt = b¯ > 0 for all t. The household seeks to maximize ∞ t t t , β U c g ,l g E0 t =0 5 I use competitive equilibrium interchangeably with what Chari and Kehoe refer to as a policy-allocation pair attainable under commitment. 6 Chari and Kehoe restrict b 0 in their model. With bonds selling at face value, the restriction is now Rb 0 (which is the same provided the gross return R > 0).
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by choice of {c(g t ), l(g t ), b(g t )} subject to t c g t + b g t 1 − τ g t l g t + R g t b g−1 taking as given {R(g t ), τ (g t )} and initial condition b−1 (with b−1 = b¯ > 0). The government’s budget constraint is ¯ gt + R g t b¯ = τ g t l g t + b. (5) Let Uc (g t ) := Uc (c(g t ), l(g t )) and Ul (g t ) := Ul (c(g t ), l(g t )). A competitive equilibrium under a balanced-budget restriction is characterized by c g t + gt = l g t , (6) t t t t (7) Uc g = E βUc g , gt +1 R g , gt +1 g , t t t (8) −Ul g /Uc g = 1 − τ g , t t t t ¯ (9) c g = 1 − τ g l g + R g − 1 b, for all t, and the transversality condition limj →∞ E[β t +j Uc (g t +j )b¯ | g t ] = 0 for all t and g t . ¯ t ), gt , τ¯ (g t ), R(g t ), b¯t } be a competitive equilibrium with Definition 1. Let E := {c(g ¯ t ), l(g ¯bt = b¯ > 0, R(g t ) 1 for all t, and R(g t ) > 1 for some g t with µ(g t ) > 0. Next, I show that a consumption-labor allocation under a balanced-budget restriction with b¯ > 0 can be supported under a balanced-budget restriction with b¯ = 0 and a suitably chosen set of lump-sum transfers. Proposition 2. Given a competitive equilibrium E, there exists a sequence of lump¯ t ), gt , τ¯ (g t ), R(g t ), b¯ T (g t ), T (g t )} is a ¯ t ), l(g sum transfers T (g t ), such that E T := {c(g T t t ¯ competitive equilibrium with b (g ) = 0, T (g ) 0 for all g t and T (g t ) > 0 for some g t with µ(g t ) > 0. Proof. See Appendix C.
✷
Next, I restrict the utility function so that both consumption and leisure are normal goods and assume that the competitive equilibrium E is never on the bad side of the Laffer curve nor at the top of the Laffer curve. ¯ → R is strictly concave in [c, −l] Assumption 1. The felicity function U (c, l) : R+ × [0, l] and satisfies Uc > 0, Ul < 0, Ucc − Ucl Uc /Ul < 0, and Ull − Ucl Ul /Uc < 0. Assumption 2. Given a competitive equilibrium E and corresponding competitive ¯ t )] solves ¯ t ), l(g equilibrium E T , for each g t , [c(g max U (c, l), c,l
subject to c + gt = l,
t Ul (c, l) gt + T g = 1 + l. Uc (c, l)
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Assumption 3. Given a competitive equilibrium E and corresponding competitive equilibrium E T , for each g t , there exists open and connected set T (g t ) such that [0, T (g t )] ⊂ T (g t ) and for all T ∈ T (g t ) there exists a solution to max U (c, l), subject to c + gt = l,
Ul (c, l) l. gt + T = 1 + Uc (c, l)
(10) (11)
Let c(gt , T ) and l(gt , T ) denote this solution. The next proposition demonstrates that both consumption and labor are decreasing in the lump-sum transfer. Proposition 3. Under Assumptions 1 and 3, ∂c(g, T ) ∂l(g, T ) = < 0, for T ∈ T . ∂T ∂T Proof. See Appendix C.
✷
Let E denote the set of all competitive equilibria E that satisfy Assumptions 1–3. Next, I prove that a competitive equilibrium with zero debt and strictly positive lump-sum transfers provides strictly less utility than a competitive equilibrium with zero debt and no lump-sum transfers, i.e., positive lump-sum transfers financed by a distortionary tax are dead-weight losses. It follows then that no competitive equilibrium that satisfies Assumptions 1–3 is sustainable. Proposition 4. Given a competitive equilibrium E ∈ E and corresponding competitive ˆ t )] maximize U (c, l) ˆ t ), l(g equilibrium E T defined in Proposition 2, for each g t let [c(g subject to
Then
c + gt = l,
(12)
gt = (1 + Ul /Uc )l,
(13)
c 0, l¯ l.
(14) (15)
ˆ l ), gt , τˆ (g t )} := {c(g E ˆ t ), l(g
is a competitive equilibrium where ˆ t )) t t Ul (c(g ˆ t ), l(g τˆ g = 1 + lˆ g ˆ t )) Uc (c(g ˆ t ), l(g
with
E0
∞ t =0
∞ 0 > W 0 := E0 β t U cˆt , lˆt =: W β t U c¯t , l¯t . t =0
(16)
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Proof. See Appendix C.
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✷
Proposition 5. No equilibrium E ∈ E is sustainable. Proof. See Appendix C.
✷
The intuition for this result is as follows. In the Chari–Kehoe model, the government does not default on its debt to maintain its good reputation as a borrower for future borrowing. The ability to issue debt is desirable because it allows the government to smooth taxes. In this model, debt is useful for smoothing taxes in two ways: (1) state-contingent returns that provide “insurance” to the household, and (2) altering the level of debt. A balanced-budget restriction shuts down channel (2), and the condition that R(g t ) 1 eliminates channel (1). Consequently, the threat of not being able to borrow in the future if the government ever defaults on its debt is not an effective deterrent under a balancedbudget restriction with non-negative interest rates since both channels are inoperative.7 What is the importance of a positive interest rate? A negative interest rate would result in the interest payments acting like a positive lump-sum tax which would allow a lower labor tax rate when compared to the autarky equilibrium. It is therefore possible for utility under a balanced-budget rule with positive debt to be greater than the autarky utility level. For this reason, in the next section I look at competitive equilibria under a balanced-budget restriction more generally and Ramsey allocations under a balanced-budget restriction. What is important to note is that anything that impedes the use of these two channels diminishes the desirable tax-smoothing feature of debt. 4. Ramsey allocations and competitive equilibria In this section, I characterize the Ramsey allocation and competitive equilibria under a balanced-budget restriction and discuss the solution method used to solve the numerical examples in the next section. I also briefly describe the much simpler Ramsey allocation and competitive equilibria with Rb 0. 4.1. Balanced-budget Ramsey The Ramsey problem under a balanced-budget restriction is much more difficult to solve than the Ramsey with Rb 0. The balanced-budget restriction imposes a form of market incompleteness on the Ramsey problem. The following proposition describes the program that gives the Ramsey allocation. Proposition 6. The Ramsey allocation under a balanced-budget restriction is the solution to the following optimization problem: maximize (1) by choice of {c(g t ), l(g t )} subject to 7 As long as one channel is operative, the competitive equilibrium may be sustainable. For example, there are equilibria in Chari and Kehoe that are sustainable where R(g t ) 1 and R(g t ) > 1 for some t, but this will
necessarily involve changing the level of the debt.
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c g t + gt = l g t , ¯ + U t +1 lt +1 , Uct b¯ = Et β Uct +1 (ct +1 + b) l g0 + (R0 − 1)b¯ = 1 + Ul0 /Uc0 l0 .
(17) (18) (19)
Proof. Similar to Stockman (2001). ✷ Let σt denote the multiplier on (18) and γ be the multiplier on (19). With this characterization of the constraints facing the Ramsey allocation under a balanced-budget restriction, the first-order conditions for this program are given by the following for t 1: t 0 = Uct + Ult (1 + σt −1 ) + (σt −1 − σt ) Ucc + Ulct b¯ t (20) + σt −1 Ucc + Uclt c g t + Ullt + Ulct l g t , (17) and (18). For t = 0, first-order conditions are (17)–(19) and 0 0 = Uc0 + Ul0 − σ0 Ucc + Ulc0 b¯ + γ 1 + Ul0 /Uc0 0 0 2 0 + γ l0 Uc0 Ulc0 + Ucc − Ul0 Ucc Uc . + Ucl0
(21)
The problem is recursive from for t 1 with the state being given by [gt , σt −1 ]. In fact, ¯ under a balanced-budget restriction, the allocation at t = 0 is determined by given R0 b, the budget constraint and resource constraint alone. With this observation, I can recast the Ramsey problem into a recursive dynamic programming framework which greatly facilitates the calculation of expected utility since this is captured by the value function. Since the Ramsey problem under a balanced-budget restriction is recursive, one can solve for the allocation using dynamic programming. The natural state is [R, g] and the control is [c, l, R , c , l ]. The Bellman equation is given by (22) V (R, g) = max U (c, l) + βE V (R , g ) | R, g subject to
Uc (c, l) = E βUc (c , l )R | R, g , ¯ Uc (c, l) c + b¯ + Ul (c, l)l = Uc (c, l)R b, c + g = l, ¯ Uc (c , l ) c + b¯ + Ul l = Uc (c , l )R b,
c +g =l ,
(23) (24) (25) (26) (27)
and the given stochastic process for g. I solve this Bellman equation numerically.8 8 I parametrically approximate the value function V (R, g) with Chebyshev polynomials as described in Judd (1998) or Ljungqvist and Sargent (2000). Let W (R, a(g)) be an N -degree Chebyshev polynomial with a(g) being an N × 1 vector of parameters (one vector for each value of g). For initial conditions on the Chebyshev polynomial, I use the consumption and leisure values from the log-linear approximation in a second-order approximation of the utility function. This is described in Appendix A. I then iterate on Bellman’s equation using the collocation method until convergence a(g) → a ∗ (g).
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4.2. Competitive equilibria under a balanced-budget rule A competitive equilibrium under a balanced-budget restriction is characterized by the following equations: (28) c g t + gt = l g t , t +1 t t Uc = Et βUc R(g , gt +1 ) , (29) t t t t t t ¯ Uc c g + b¯ + Ul l g = Uc R g b, (30) and the law of motion for government expenditures. The state vector is [R(g t ), gt ]. Assuming {gt } follows a stationary AR(1) process and log-linearizing this system, we get
t +1 t R R =A , (31) Et gˆt +1 gˆ t where xˆt denotes a percent deviation from steady state. Let λi be the eigenvalues of A. I show in Appendix B that |λi | < 1 for i = 1, 2 indicating local indeterminacy.9 An equilibrium is then given by stochastic processes {ψt +1 , et +1 } with Et [ψt +1 ] = Et [et +1 ] = 0, and
t +1 t R R ψt +1 =A + . (32) gˆt +1 gˆt et +1 Since {gt } is exogenous, the process {et +1 } is exogenous. The process {ψt +1 } is not uniquely determined in the model. The conditional mean being zero is the only restriction put on the ψt +1 process. Another way of thinking about the Ramsey problem is as a choice of {ψt +1 }. In fact, the log-linear approximate solution to the Ramsey equilibrium selects ψt +1 = η∗ et +1 for some constant η∗ . In terms of the non-linear system, we have the following: t
βUc R g t , gt +1 , gt +1 R g t , gt +1 Uc R g , gt Et = (33) ¯ gt +1 − g¯ ρ(gt − g) or
t
Uc R g , gt zt +1 βUc R g t , gt +1 , gt +1 R g t , gt +1 = + . ¯ et +1 gt +1 − g¯ ρ(gt − g)
(34)
The shock {et +1} is exogenous, but {zt +1 } is only restricted by Et [zt +1 ] = 0. Again, the Ramsey problem can be viewed as one of choosing this stochastic process to maximize expected utility, and to a first-order approximation zt +1 = η∗ et +1 under Ramsey. Without introducing extrinsic uncertainty and retaining a Markovian structure, the z shock takes the form zt +1 = Z(Rt , gt , et +1 ). 9 This indeterminacy is related to the results of Schmitt-Grohé and Uribe (1997) with the capital’s share parameter equal to zero.
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This restriction still does not impose enough structure for a pragmatic numerical exploration. Another level of restrictions might be zt +1 = et +1 Z(Rt , gt ). This implies a conditional variance Et [zt2+1 ] = Z(Rt , gt )2 σe2 . Using the above observation on the first-order approximation of the Ramsey allocation, I will limit Z(R, g) to constant functions in the numerical examples in the next section, i.e., a single parameter family that covers the Ramsey allocation to a first-order approximation.10 Letting prime variable denote next period values, and given z = ηe , one can solve for R = R(R, g, g ; η). With this law of motion for R , I can calculate [c, l] recursively. Let ∞ t t t V (R, g; η) := E0 , β U c R g , gt , l R g , gt t =0
then
V (R, g; η) = U c(R, g), l(R, g) + E βV R(R, g, g ; η), g g .
This functional equation can be solved again by approximating V by a Chebyshev polynomial. No iteration is necessary because no maximization is done. 4.3. Ramsey and competitive equilibria with Rb 0 In this subsection, I characterize both Ramsey and stationary competitive equilibria with Rb 0 and briefly describe the numerical solution method. This exposition differs from Chari–Kehoe insofar as the initial level of debt is positive.11 The next proposition describes the program whose solution is the Ramsey allocation. Proposition 7. The Ramsey allocation with Rb 0 is the solution to the following optimization problem: given R0 b−1 > 0, maximize (1) by choice of {c(g t ), l(g t ), bt } subject to (35) c g t + gt = l g t , t +1 t +1 t Uc bt = Et β Uc (ct +1 + bt +1 ) + Ul lt +1 , (36) t t t t (37) Uc c g + bt + Ul l g 0, 0 0 g0 + R0 b−1 = 1 + Ul /Uc l0 + b0 . (38) 10 This is the closest analogy to the type of competitive equilibria explored by Chari and Kehoe. 11 Chari and Kehoe set b −1 = 0 and write “. . . if the inherited debt at date zero given by b−1 is positive,
the government will default and the Ramsey problem is unchanged. Since we require debt be non-negative the assumption that b−1 = 0 is without loss of generality.” This sets δ0 = 1 without the possibility of it being otherwise and results in some blurring of the difference between δ0 = 1 and b−1 = 0. I will be making the distinction and let the government choose the initial default rate while taking the initial level of debt as given. In the model without commitment, setting the initial default rate to one should be perceived as default by the agents and have potential adverse effects on future borrowing.
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Proof. Similar to that given in Stockman (2001). ✷ The first-order conditions for this program for t 1 are given by t 0 = Uct + Ult (1 + σt −1 ) + (σt −1 − σt ) Ucc + Uclt bt t + σt −1 Ucc + Uclt c g t + Ullt + Ulct l g t , γt = σt − σt −1 , l g t = c g t + gt , 0 = γt Uct c g t + bt + Ult l g t , Uct bt = Et β Uct +1 (ct +1 + bt +1 ) + Ult +1 lt +1 ,
(39) (40) (41) (42) (43)
taking σ0 as given. Given a solution to this system of difference equations, one then solves the t = 0 first-order conditions. This illustrates that the Ramsey allocation is stationary for t 1. In the numerical examples in the next section, I assume government consumption is i.i.d. and takes on two values g 1 and g 2 with g0 = g 2 , g 1 < g 2 , and Prob[g = gi ] = µi . Under this assumption, I can solve the Ramsey problem using a slight modification of the Chari–Kehoe method. First, the restrictions put on stationary competitive equilibria with the restriction Rb 0 are the following: H (c0 , l0 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) = Uc (c0 , l0 )R0 b−1 , (44) (45) H (c1 , l1 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) 0, (46) H (c2 , l2 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) 0, c0 + g0 = l0 ,
(47)
1
c1 + g = l1 ,
(48)
c2 + g = l2 ,
(49)
2
where H is defined by H (ci , li ) ≡ Uc (ci , li )ci + Ul (ci , li )li ,
(50)
for i = 0, 1, 2. Again, this differs from Chari–Kehoe insofar as the initial level of debt is positive. Let b−1 > 0 (assuming R0 = β −1 ) be an initial condition. The Ramsey problem is to maximize U (c0 , l0 ) + β µ1 U (c1 , l1 ) + µ2 U (c2 , lc ) /(1 − β), subject to (44)–(49) by choice of [ci , li ]2i=0 . Debt in a stationary equilibria is given by β[µ1 H (c1 , l1 ) + µ2 H (c2, l2 )]/(1 − β) , b gi = Uc (ci , li )
(51)
and the total gross return to debt is given by H (ci , li ) + β[µ1 H (c1, l1 ) + µ2 H (c2 , l2 )]/(1 − β) . R gi b gi = Uc (ci , li )
(52)
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Note that when b−1 = 0 (as in Chari and Kehoe) the Ramsey allocation at t = 0 coincides with the allocation at later dates when g = g 2 . Stationary equilibria can be characterized by the values of surplus functions: H i = H (ci , li ), i = 0, 1, 2. These values must satisfy the above constraint on the Ramsey allocation. In terms of these equilibria being sustainable, I first check the level of expected utility for t 1. To do this note that given surplus levels H 1 and H 2 , one can solve for [c(H i ), l(H i )] for i = 1 and 2 using the market clearing conditions. This implies U (c(H i ), l(H i )) = U (H i ). To systematically investigate the sustainability of all stationary competitive equilibria, I grid [H 1 , H 2 ] that satisfy the Ramsey constraints, calculate the expected utility for t 1. If this tail is sustainable, I then solve for [c0 , l0 ] given [H 1 , H 2 ] using (44) and (47), and calculate expected utility at t = 0. If this value is greater than autarky utility, the competitive equilibrium is sustainable. A remark on sustainable utilities. For a given parameterization, the Ramsey allocation might not be sustainable while other non-autarky allocations might be.12 This might seem counter-intuitive given that the Ramsey allocation maximizes utility. The potential sustainability problem with Ramsey is that while taking into account the optimal trade-off between current and future utility, the Ramsey allocation might place the economy on a future trajectory such that the utility along this tail is not high enough compared to autarky.
5. Numerical examples In Section 3, we saw that a competitive equilibrium is not sustainable if R(g t ) 1 for all t with a strict inequality holding for some t. When R(g t ) < 1, state-contingent debt may be providing insurance to the household that lessens the need for high taxes when g is high. This insurance may raise utility enough for the competitive equilibrium to be sustainable. In this section, I provide numerical examples to illustrate the impact of a balanced-budget restriction on the sustainability of the Ramsey allocation and utility levels of other stationary competitive equilibria. I also illustrate the degree to which R < 1 for the competitive equilibrium to be sustainable. Preferences are given by 1−α U (c, l) = cα l¯ − l . Government spending is i.i.d. and takes on two value g 1 and g 2 with equal probability. Initial conditions are g0 = g 2 and R0 = β −1 .13 Both the range of parameter values and benchmark values are recorded in Table 1. Utility levels are normalized as a ratio to autarky utility as in Chari and Kehoe. The discount factor, volatility of government spending and initial level of debt are varied. The volatility of government expenditures ranges from 5% to 50% and the level of initial debt is varied from 0.5 to 4.0 which corresponds to roughly 10% to 80% of output in the 12 Chari and Kehoe (1993) illustrate this point. 13 This numerical example is similar to that in Chari and Kehoe and meant for illustrative purposes only.
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Table 1 Benchmark and range of parameters values considered in simulations Preferences
β ∈ [0.90, 0.91, . . . , 0.98] l¯ = 10
α = 1/2
Government
g¯ = 1.0 b¯ ∈ [0.5, 1.0, . . . , 4.0] g ∈ {g 1 , g 2 }
σg ∈ [0.05, 0.10, . . . , 0.5]
Benchmark values
β = 0.95 σg = 0.5
Prob(g = gi ) = 0.5 b = 1.0
model. The impact of a balanced-budget restriction on sustainable utilities is illustrated in Figs. 1–3. We see, as expected, that the range of sustainable utilities is smaller under a balanced-budget restriction. Perhaps surprising, this range under a balanced-budget restriction is not that much smaller. This suggests either (1) the insurance channel of debt through state-contingent returns is quantitatively more important for welfare than the ability to change the level of debt or (2) state-contingent returns are to a large extent substitutable for altering the level of debt. All of the intuitive findings of Chari and Kehoe are preserved under a balanced-budget restriction: higher β and σ make sustainability more likely. In addition, we see that a higher initial level of debt b−1 makes sustainability less likely. For sustainable utility levels, we know that the gross return on debt must be less than 1 during some t. How much variability in the gross return is needed for sustainable utilities? The range of state-contingent returns required for the competitive equilibrium under a
Fig. 1. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the discount factor. Other parameters: σ = 0.5 and b¯ = 1.0.
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Fig. 2. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the standard deviation of government spending. Other parameters: β = 0.95 and b¯ = 1.0.
Fig. 3. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the initial level of debt. Other parameters: β = 0.95 and σ = 0.5.
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balanced-budget restriction to be sustainable along with those determined by Ramsey under a balanced-budget restriction for different discount factors is plotted in Fig. 4. In a sustainable equilibrium, the return on debt R(g) is high when g is low and R(g) is low when g is high. It is precisely this feature that makes state-contingent returns desirable. If these returns are too compressed, the competitive equilibrium will not be sustainable (these would correspond to the area labeled “Not Sustainable”). We see that substantial variability in the gross return on debt is needed for sustainability. Isolating two values on the cusp of sustainability, in Figs. 5 and 6 we see the amount of state-contingent returns being utilized by Ramsey under a balanced-budget restriction with β = 0.93 (sustainable) and β = 0.92 (unsustainable). For β = 0.93, the range of state-contingent returns is R ∈ [0.69, 1.47]. For β = 0.92, the range of state-contingent returns is R ∈ [0.7, 1.48]. For β = 0.93, utility levels are sustainable from autarky up to Ramsey under a balanced-budget restriction. The return on debt for the sustainable competitive equilibrium that delivers the autarky utility level is in Fig. 7. We still see substantial state-contingent returns on debt with R ∈ [0.82, 1.34]. Note this competitive equilibrium is the worst sustainable equilibrium where the government does not default and it requires variation in the gross returns on debt of about ±25% from the mean (1/β = 1.075) to do it.
Fig. 4. Range of state-contingent returns needed for sustainability. For sustainable equilibria, the level of state contingent returns fluctuates between the upper and lower shaded areas. The upper and lower shaded regions illustrates range for R(g 1 ) (a high return on debt occurs when government spending is low) and R(g 2 ) (a low return on debt when government spending is high). The returns in the region labeled “Not Sustainable” are too compressed to be sustainable.
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Fig. 5. Return on debt from sustainable Ramsey with a balanced-budget restriction: β = 0.93, σ = 0.5, b = 1.0.
Fig. 6. Return on debt from unsustainable Ramsey with a balanced-budget restriction: β = 0.92, σ = 0.5, b = 1.0.
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Fig. 7. Return on debt from the sustainable competitive equilibrium under balanced-budget restriction that delivers the autarky utility level: β = 0.93, σ = 0.5, b = 1.0. As a benchmark, the implicit risk-free returns under the autarky equilibrium with β = 0.93 and σ = 0.5 are 1.0078 and 1.1427.
6. Adding capital to the model Extending the model to include capital greatly exacerbates the sustainability problem under a balanced-budget restriction.14 As shown in Zhu (1992) and Chari, Christiano, and Kehoe (1994) in a model with labor and capital income taxation, there is indeterminacy in the tax plans that support the Ramsey allocation. This result carries over into competitive equilibria under a balanced-budget restriction. Essentially, one can use either statecontingent capital income taxes or state-contingent returns on debt to decentralize a competitive equilibrium allocation. This suggests that if state-contingent capital income taxes are available, the state-contingent returns to debt are superfluous, and default would be more likely. Take the standard business cycle model with capital as in Chari et al. (1994). Output is produced using a constant returns to scale production function yt = F (kt , lt ). Capital income is taxed at rate θt , so the government budget constraint is given by gt + (Rt − 1)b¯ = τt wt lt + θt rtk − δ kt , 14 I thank an anonymous referee for suggesting this possibility.
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where δ is the depreciation rate, rtk the rental rate of capital, and wt the real wage. A competitive equilibrium is defined in the usual way. The indeterminacy of taxes carries over to competitive equilibrium under a balanced-budget restriction. Proposition 8. Let {c¯t , l¯t , k¯t g¯t } and {τ¯t , θ¯t , R t , r¯tk , w¯ t } be a competitive equilibrium. Then t , r¯tk , w¯ t } is a competitive if {θˆt , R t } satisfy {c¯t , l¯t , k¯t g¯t } and {τ¯t , θˆt , R t +1 , Et Uc c¯t +1 , l¯t +1 R t +1 = Et Uc c¯t +1 , l¯t +1 R (53) t +1 t +1 ¯ ¯ ¯ ˆ Et Uc c¯t +1 , lt +1 θt +1 F k − δ = Et Uc (c¯t +1 , lt +1 )θt +1 F k − δ , (54) t t − 1 b¯ = τ¯t w¯ t l¯t + θˆt F k − δ k¯t , g¯t + R (55) where F tk = Fk (k¯t , l¯t ). Proof. Similar to Zhu (1992) or Chari et al. (1994). ✷ This means given an allocation from a competitive equilibrium under a balanced-budget restriction, there is more than one set of state-contingent taxes on capital income and returns on debt that will support this allocation. In particular, one can use non-statecontingent returns on debt and set t +1 = R
Uc (c¯t , l¯t ) . Et [βUc (c¯t +1 , l¯t +1 )]
The following proposition gives sufficient conditions for a competitive equilibrium under a balanced-budget restriction to be unsustainable. Proposition 9. Any competitive equilibrium under a balanced-budget restriction with positive debt is not sustainable if Uc (c¯t , l¯t ) 1, Et [βUc (c¯t +1 , l¯t +1 )] for all t with a strict inequality for some t. Proof. The reasons are similar to Proposition 5. ✷ Note that we do not need the competitive equilibrium to involve risk-free debt with a positive return, but only the much weaker condition that the implicit risk-free return on debt be positive.
7. Conclusion Balanced-budget rules limit the effectiveness of debt to smooth taxes, but it is precisely this feature of debt that keeps a government from defaulting. In this paper, I have demonstrated that if the return on debt is positive, then a no-default competitive equilibrium is inconsistent with a balanced-budget restriction. State-contingent returns on a fixed
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level of debt allows for some tax smoothing and this may be enough to reconcile the inconsistency. However, enormous variation in the real ex post return to debt (in excess of ±25%) is needed to have the tax-smoothing benefits outweigh the benefit of default according to numerical examples. Moreover, adding capital to the model makes the state-contingent returns on debt unnecessary to smooth taxes because state-contingent capital income taxes can serve this function. In this case, a balanced-budget restriction is inconsistent with a competitive equilibrium with positive debt under even weaker conditions (a positive implicit risk-free return to debt). Two additional lessons emerge from this analysis. First, balanced-budget rules that include “escape clauses” in times of unusually high government spending (e.g., a war) or the establishment of a separate capital account for capital expenditures allow for more tax-smoothing over time. These features may be extremely important in providing the government an incentive to preserve its reputation as a good borrower by not defaulting. Second, taking the reputation model seriously, adoption of a balanced-budget restriction would predict default as the equilibrium outcome. Insofar as this outcome is not observed, the reputation argument cannot be the complete answer to the question: Why don’t governments default on their debt? Reputation effects may be important, but other equally important mechanisms must be at work.
Acknowledgments This paper is a substantially revised chapter from my PhD dissertation in economics at University of Chicago. I thank Larry Christiano, Michael Chwe, Lars Hansen, Tom Sargent, an anonymous referee, and seminar participants at the University of Delaware, Williams College, SUNY–Albany, and Congressional Budget Office for helpful comments. All remaining errors are mine.
Appendix A. Calculating utility under a balanced-budget rule The competitive equilibria under a balanced-budget restriction are Markovian in x = [R, g] . Chari and Kehoe deal with equilibria that are Markovian in [g] alone. This makes the comparisons of utility somewhat easier due to the fact that g takes on a finite number of values. Under a balanced-budget restriction one needs to calculate expected utility for each g and every possible R. I approximate the Ramsey value function using Chebyshev polynomials using a projection method called collocation. For initial coefficient values for the Chebyshev polynomial that approximates the value function, I use the consumption and leisure values from the log-linear approximation in a second-order approximation of the utility function. Given the value of expected utility for different initial levels of g0 and R0 , one solves for the polynomial coefficient that sets the value function approximation equal to these values. t gˆt ] be the state vector. Given a log-linear approximation to the equilibrium Let xˆt = [R satisfies: Et [xˆt +1 ] = Axˆt
or xˆt +1 = Axˆt + Cet +1 ,
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where C := [η 1] . I can write
cˆt = Γ xˆ t . lˆt Then a second-order Taylor approximation of utility at time t is given by U c g t , l g t ≈ U (c∗ , l ∗ ) + DU (c∗ , l ∗ )Γ xˆ t + (1/2)xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt . Then it is easy to show that Σxt = A Σxt −1 A + CΣe C ,
E0 [xˆt ] = At xˆ0 , where
Σxt := E0 xˆt xˆt ,
Σe := E0 et et ,
with Σx0 = x0 x0 . Using these expressions, I can get a simple expression for E0 [U (c(g t ), l(g t ))]: E0 U c g t , l g t ≈ U (c∗ , l ∗ ) + DU (c∗ , l ∗ )Γ At xˆ0 + (1/2)E0 xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt . I can express the last conditional expectation as E0 xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt = E0 tr xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt = E0 tr Γ D 2 U (c∗ , l ∗ )Γ xˆt xˆt = tr E0 Γ D 2 U (c∗ , l ∗ )Γ xˆt xˆt = tr Γ D 2 U (c∗ , l ∗ )Γ E0 xˆt xˆt = tr Γ D 2 U (c∗ , l ∗ )Γ Σxt , where tr(·) is the trace operator. So expected utility is provided by ∞ t t t E0 β U c g ,l g t =0
≈
∞ U (c∗ , l ∗ ) t + β DU (c∗ , l ∗ )Γ At xˆ0 + (1/2) tr Γ D 2 U (c∗ , l ∗ )Γ Σxt . 1−β t =0
Appendix B. Stability of steady state under a balanced-budget rule Let A be the matrix in (31). Using (28) and (30) we can write t + ηcg gˆt , t + ηlg gˆt . cˆt = ηcR R lˆt = ηlR R Using the same argument as in Proposition 3, it is true that ηcR c¯ = ηlR l¯ < 0 as long as b¯ > 0. Then one can show that the eigenvalues of A are given by λ1 = γR /(γR + Uc ),
λ2 = ρ,
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where γR = ηcR c(U ¯ cc + Ucl ). Then |λ2 | < 1 if g is stationary and 0 < λ1 < 1 if γR > 0. If Ucc + Ucl < 0 then γR < 0. If Ucl < 0 then Ucc + Ucl < 0 since Ucc < 0. A positive revenue requirement (τ ∗ > 0) implies in steady state −Ul /Uc = (1 − τ ∗ ) < 1
or
− Uc /Ul > 1.
So if Ucl > 0 we have Ucc + Ucl < Ucc − Ucl Uc /Ul < 0. The last inequality follows from the assumption that consumption and leisure are normal goods.
Appendix C. Proofs
Proof of Proposition 2. Since E is a competitive equilibrium under a balanced-budget ¯ Given restriction with b¯ > 0 we have (6)–(9) satisfied. We set T (g t ) = [R(g t ) − 1]b. t t t the assumption on interest rates, T (g ) 0 for all g and T (g ) > 0 for some g t with µ(g t ) > 0. Then by direct inspection, E T satisfies (6)–(8), and the appropriately modified version of (9) to include lump-sum transfers (dropping b¯ T (g t ) since it equals zero): c¯ g t = 1 − τ¯ g t l¯ g t + T g t . (C.1) Therefore E T is a competitive equilibrium. ✷ Proof of Proposition 3. Define G(c, l) := (1 + Ul /Uc )l and
c+g−l F (c, l, g, T ) := . G(c, l) − g − T In equilibrium, F (c, l, g, T ) ≡ 0. By Assumption 1, Uc [Ull − Ucl Ul /Uc ] < 0,
−Ul [Ucc − Ucl Uc /Ul ] < 0.
Combining these gives 0 > Uc [Ull − Ucl Ul /Uc ] − Ul [Ucc − Ucl Uc /Ul ] = Gc + Gl . This allows us to use the implicit function theorem which gives
−1
∂c(g,T ) ∂c(g,T )
1 1 −1 1 0 1 − Gl ∂g ∂T = − = ∂l(g,T ) ∂l(g,T ) Gc Gl −1 −1 1 + Gc G + G c l ∂g ∂T
1 . 1
This establishes the equality of the two partial derivatives. Since Gc + Gl < 0, the sign of the derivatives is negative as well. ✷ satisfies (6)–(9) and is Proof of Proposition 4. Again, direct inspection reveals E t (g t ). Since therefore a competitive equilibrium. If T (g ) = 0 we have U (g t ) = U ∂c(g, T ) ∂l(g, T ) = < 0, ∂T ∂T
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we have for T > 0, ∂c(g, T ) ∂l(g, T ) ∂l(g, T ) ∂U (c(g, T ), l(g, T )) = Uc + Ul = (Uc + Ul ) ∂T ∂T ∂T ∂T ∂l(g, T ) < 0, = Uc (1 + Ul /Uc ) ∂T since (1 + Ul /Uc ) > 0 because (1 + Ul /Uc )l = g + T > 0, i.e., a positive revenue requirement. This establishes that for each g t such that T (g t ) > 0, we have U cˆ g t , lˆ g t > U c¯ g t , l¯ g t . (g t ) > U (g t ) for some g t with µ(g t ) > 0, (g t ) U (g t ) for all g t with U This establishes U which gives (16). ✷ Proof of Proposition 5. Recall Chari and Kehoe define the autarky plans as follows. For any history, the government sets δ a (g t ) = 1 and q a (g t ) = 0. The tax rate τ a (gt ) is given by τa (gt ) = 1 + Ula (gt )/Uca (gt ), where [ca (gt ), l a (gt )] solves max U (c, l) subject to c + gt = l and gt = (1 + Ul /Uc )l. The household’s allocation rules specifies for every history, hold zero debt and choose consumption and labor to maximize utility given the autarky tax rates. Autarky utility is given by ∞ a t a t a t . (C.2) β U c g ,l g W0 := E0 t =0
0 = W a , where W 0 is defined in (16). Let {c(g ¯ t )} denote the Note first that W ¯ t ), l(g 0 allocation in E ∈ E and ∞ t t t ¯ . W 0 := E0 β U c¯ g , l g t =0
We know from Proposition 3 that there exists a competitive equilibrium E T with a sequence of non-negative transfers T (g t ) 0 for all g t and T (g t ) > 0 for some g t with µ(g t ) > 0 that decentralizes this allocation with zero debt. Let ∞ t t T t . W 0 := E0 β U c¯ g , l¯ g t =0
0 = W a , and therefore E ∈ E is not From Proposition 4, we have W 0 = W T0 < W 0 sustainable. ✷
References Barro, R.J., 1979. On the determination of public debt. Journal of Political Economy 87, 940–971. Chari, V.V., Christiano, L.J., Kehoe, P.J., 1994. Optimal fiscal policy in a business cycle model. Journal of Political Economy 102, 617–652. Chari, V.V., Kehoe, P.J., 1993. Sustainable plans and mutual default. Review of Economic Studies 60, 175–195.
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Corsetti, G., Roubini, N., 1992. Tax smoothing discretion versus balanced budget rules in the presence of politically motivated fiscal deficits: the design of optimal fiscal rules for Europe after 1992. Discussion paper 682. Center for Economic Policy Research. Judd, K., 1998. Numerical Methods in Economics. MIT Press. Lucas, R.E. Jr., Stokey, N.L., 1983. Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55–93. Ljungqvist, L., Sargent, T., 2000. Recursive Macroeconomic Theory. MIT Press. Mankiw, N.G., 1987. The optimal collection of seigniorage: theory and evidence. Journal of Monetary Economics 20, 327–341. Prescott, E.C., 1977. Should control theory be used for economic stabilization? Brunner, K. (Ed.), Optimal Policies, Control Theory and Technology Exports. In: Carnegie–Rochester Conference Series on Public Policy, vol. 7. North-Holland, Amsterdam, pp. 13–38. Schmitt-Grohé, S., Uribe, M., 1997. Balanced-budget rules and aggregate instability. Journal of Political Economy 105, 976–1000. Stockman, D.R., 2001. Balanced-budget rules: optimal policies and welfare consequences. Review of Economic Dynamics 4, 438–459. Zhu, X., 1992. Optimal fiscal policy in a stochastic growth model. Journal of Economic Theory 58, 250–289.
Default, reputation, and balanced-budget rules David R. Stockman Economics Department, University of Delaware, Newark, DE 19716, USA Received 4 December 2001; revised 4 June 2003
Abstract Is a balanced-budget rule compatible with a government honoring its debt obligations? According to the conventional explanation, governments honor their debt obligations to maintain a good reputation for future borrowing. The ability to borrow is desirable because it allows for greater tax smoothing. However, a balanced-budget rule limits the ability to smooth taxes, rendering a large class of competitive equilibria not compatible with a government honoring its debt obligations. The reputation model predicts default as the equilibrium outcome under a balanced-budget restriction. Insofar as this prediction is falsified by empirical observation, mechanisms other than reputation must be at work. 2003 Elsevier Inc. All rights reserved. JEL classification: E61; H21; H63
1. Introduction Is a balanced-budget rule compatible with a government honoring its debt obligations? A balanced-budget rule is a restriction on fiscal policy currently used by some governments and may be adopted by more. For example, many US states have some form of balancedbudget rule, and the US Congress has considered passing a balanced-budget amendment for the federal government. One also finds a balanced-budget rule in the debate on the monetary union in Europe. However, many economic models suggest that optimal fiscal policy calls for the government to smooth taxes by running surpluses and deficits financed through borrowing and lending.1 Why would a government limit its ability to smooth taxes by adopting a balanced-budget restriction? Advocates for a balanced-budget restriction E-mail address: [email protected]. 1 For example, tax smoothing results financed through borrowing and lending are obtained in Barro (1979),
Lucas and Stokey (1983), and Mankiw (1987). 1094-2025/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.red.2003.09.002
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argue that the political system results in excessive spending and that a balanced-budget amendment will help control this tendency.2 Opponents argue that the constraint will inhibit stabilizing fiscal action and increase the amplitude of the business cycle. Prescott (1977) illustrates a time inconsistency problem of debt that suggests the government has an incentive to renege on its previous debt obligations. He argues that if only distortionary taxes are available to service the debt, these interest payments result in a loss of welfare. If the government were to default on its outstanding debt, the distortionary taxes would be lower and welfare would be increased. However, households anticipate this and choose not hold government debt, which results in government debt equaling zero in equilibrium. This runs counter to empirical evidence: households hold government debt and governments honor their debt obligations. Why do households lend to governments and why does the government not default? The conventional explanation for why a government honors its debt obligations is that it wishes to maintain a good reputation for future borrowing (e.g., Chari and Kehoe, 1993). The ability to borrow is desirable because debt serves as a buffer to help smooth distortionary taxes over time resulting in higher economic welfare. Given that the government has not defaulted on its debt in the past, the household is willing to lend to the government. If the government were to default on its debt, it would damage its reputation as a reliable borrower. The household might no longer hold debt, distortionary taxes would fluctuate more and welfare would be lower. Here, I would like to focus attention on the potential incompatibility of a no-default competitive equilibrium where the government honors its debt obligations and a balancedbudget restriction. Why is there an incompatibility? The reputation argument for not defaulting depends on an effective deterrent: if the government ever defaults, it may never borrow again. This repercussion of default is undesirable if there is great need to smooth taxes and debt is a useful tax-smoothing instrument. The inconsistency occurs because a balanced-budget rule lessens the deterrent by limiting debt’s usefulness to smooth taxes. This coupling of balanced-budget rules and default provides a novel perspective from which to investigate the merits of a balanced-budget restriction.3 The principal results of this paper are: • A no-default competitive equilibrium is incompatible with a balanced-budget restriction if the return on debt is positive. • A no-default competitive equilibrium may be compatible with a balanced-budget restriction if there is sufficient tax smoothing through state-contingent returns on debt (this requires a negative return on debt in some states). However, enormous variation in the state-contingent returns is required (e.g., ±25% and higher). 2 In Europe, the argument has been made that a monetary union without fiscal convergence would be unstable. Without strict fiscal guidelines, there will be excessive deficits (perhaps politically motivated) that are not consistent with long-term solvency. See Corsetti and Roubini (1992). 3 In related work, Schmitt-Grohé and Uribe (1997) show that balanced-budget rules may result in sunspot equilibria. Both the possibility of sunspot equilibria and increase in the incentive to default are independent of concerns about stabilization policy and excessive spending. Consideration of these issues is necessary to gain a broader understanding of balanced-budget rules.
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• Adding capital to the model greatly exacerbates the sustainability problem under a balanced-budget restriction. In this case, state-contingent capital income taxes render state-contingent returns on debt superfluous and any no-default competitive equilibrium where the implicit risk-free return on debt is positive is incompatible with a balanced-budget restriction. In summary, the reputation model predicts default as the likely equilibrium outcome under a balanced-budget restriction. Insofar as this prediction is falsified by empirical observation, mechanisms other than reputation must be at work. The rest of the paper is organized as follows. Section 2 contains a brief review of Chari and Kehoe (1993) who formalize the reputation argument in a game-theoretic framework. In Section 3, I give sufficient conditions under which a large class of competitive equilibria without default is not consistent with a balanced-budget restriction. Next, I characterize Ramsey allocations both with and without a balanced-budget restriction and all stationary competitive equilibria with and without a balanced-budget restriction in Section 4. Numerical examples in Section 5 illustrate what competitive equilibria without default are consistent with a balanced-budget restriction. In Section 6, I discuss the effect of adding capital to the model and conclude in Section 7.
2. Chari and Kehoe model The model consists of a large number of infinitely-lived identical households. Output is produced by a constant-returns-to-scale production function using the only factor of production, labor. Output can be used for either private consumption (c(g t )) or government consumption (gt ). Government consumption {gt }∞ t =0 is exogenous and may be stochastic. The state of the economy at time t is a realization of government expenditures up to and t := (g , g , . . . , g including time t: g t := (g0 , . . . , gt ). For a given g t with t 1, let g−1 0 1 t −1 ) t t t be the first t − 1 values of g so g ≡ (g−1 , gt ). If gt is stochastic, it may take on one of a finite number of possible values {γ1 , γ2 , . . . , γN }. The probability of g t is given by µ(g t ). Units of labor and output are chosen such that one unit of labor provides one unit of output. Each household has preferences over state-contingent consumption and labor streams {c(g t ), l(g t )} given by ∞ t t t , (1) β U c g ,l g E0 t =0
with discount factor 0 < β < 1. The felicity function, U , is assumed to be bounded, strictly concave, increasing in consumption, and decreasing in labor.4 The representative household’s objective is to maximize (1) by choice of {c(g t ), l(g t ), b(g t +1 )}, subject to (2) q g t , gt +1 b g t , gt +1 1 − τ g t l g t + 1 − δ(g t ) b g t , c(g t ) + gt+1 4 I further assume the U satisfies the Inada conditions lim c→0 Uc = +∞ and liml→l¯ Ul = −∞, where l¯ represents a time endowment and therefore an upper bound on labor as well.
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taking the tax on labor income τ (g t ), default rates δ(g t ), prices of government debt q(g t , gt +1 ), and initial debt b(g 0 ) as given. Government debt is state-contingent and q(g t , gt +1 ) gives the price at g t in units of c(g t ) per unit payoff in state (g t , gt +1 ). An t t t t allocation is understood to be x = {x(g t )}∞ t =0 where x(g ) = {c(g ), l(g ), b(g )}. The government selects a policy of state-contingent default rates, bond prices, and tax rates on labor while taking its expenditures as given to satisfy a sequence of budget constraints t gt + 1 − δ g t b g t = τ g t l g t + q g , gt +1 b g t +1 ,
(3)
gt+1
for t = 0, 1, . . . , ∞. A policy for the government is denoted by π = {π(g t )}∞ t =0 where π(g t ) = {δ(g t ), q(g t ), τ (g t )}. The government is assumed to be benevolent (the government’s preferences are identical to the representative household’s). The default rate δ(g t ) is restricted to be between 0 and 1, where δ(g t ) = 0 implies that the government honor its debt completely, and δ(g t ) = 1 corresponds to complete default. Chari and Kehoe call a pair (π, x) attainable under commitment if it satisfies both the government budget constraint and household optimization. Chari and Kehoe call a competitive equilibrium a sustainable equilibrium if the government honors its debt obligation in the absence of a commitment technology. They characterize the entire set of sustainable outcomes by identifying the worst sustainable equilibrium and using revert-to-autarky plans. The autarky plan is defined as follows: (i) regardless of what has happened in the past, the government chooses q(g t ) = 0 and δ(g t ) = 1 and sets τ a (g t ) equal to that given by the solution to max U (c(g t ), l(g t )) subject to c(g t ) + gt = l(g t ), τ a (g t )l(g t ) = gt , and (1 + Ul (g t )/Uc (g t ))l(g t ) = gt , (ii) the household allocation rule sets debt equal to zero (regardless of what has happened in the past) and chooses consumption and labor to maximize utility subject to the given tax rates. Let [ca (g t ), l a (g t )] denote these choices. The autarky level of utility at time s in state g s is ∞ β t U ca g t , l a g t g s . W a g s := E
(4)
t =s
The revert-to-autarky plans specify that for a given candidate policy and allocation (π, x), the government and the agent continue with (π, x) as long as there have been no deviations in the past. If there has been any deviation, switch to the autarky plan. The main characterization result from their paper is the following. Proposition 1. An arbitrary pair (π, x) is an outcome of a sustainable equilibrium if and only if
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(i) the pair is attainable under commitment and (ii) for every s and g s , the following holds: ∞ t t s t β U c g , l g | g W a gs . E t =s
Proof. See Chari and Kehoe (1993). ✷ This proposition provides a convenient way to check whether or not a particular competitive equilibrium under commitment is sustainable. Note that deterministic government spending is a covered as well by this proposition. By utilizing this characterization of sustainable equilibria, Chari and Kehoe go on to demonstrate that if government expenditures follows a Markov chain and some tax-smoothing is desirable, the Ramsey outcomes are sustainable for sufficiently high discount factors. 3. Balanced-budget rules and default In this section, I provide sufficient conditions for a competitive equilibrium under a balanced-budget restriction to be unsustainable.5 This set of competitive equilibria is quite large and reasonable. It may include the Ramsey equilibrium under a balanced-budget restriction. There are three steps in the argument: (1) Show that a consumption-labor allocation in a competitive equilibrium under a balanced-budget restriction with the gross return on bonds greater than one is the same for a competitive equilibrium with no debt and a sequence of positive lump-sum transfers. (2) Show that under a balanced-budget restriction with zero debt, a consumption-labor allocation with strictly positive lump-sum transfers provides lower utility than the consumption-labor allocation with no lump-sum transfers. (3) Show that the autarky equilibrium corresponds to the equilibrium under a balancedbudget restriction, zero debt, and zero lump-sum transfers. To facilitate defining a balanced-budget restriction, I model bonds as selling at face value and earning interest. Let R(g t ) be the gross return on bonds held from period t − 1 t to t in state g t .6 I assume that the balanced-budget restriction takes the form in state g−1 bt = b¯ > 0 for all t. The household seeks to maximize ∞ t t t , β U c g ,l g E0 t =0 5 I use competitive equilibrium interchangeably with what Chari and Kehoe refer to as a policy-allocation pair attainable under commitment. 6 Chari and Kehoe restrict b 0 in their model. With bonds selling at face value, the restriction is now Rb 0 (which is the same provided the gross return R > 0).
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by choice of {c(g t ), l(g t ), b(g t )} subject to t c g t + b g t 1 − τ g t l g t + R g t b g−1 taking as given {R(g t ), τ (g t )} and initial condition b−1 (with b−1 = b¯ > 0). The government’s budget constraint is ¯ gt + R g t b¯ = τ g t l g t + b. (5) Let Uc (g t ) := Uc (c(g t ), l(g t )) and Ul (g t ) := Ul (c(g t ), l(g t )). A competitive equilibrium under a balanced-budget restriction is characterized by c g t + gt = l g t , (6) t t t t (7) Uc g = E βUc g , gt +1 R g , gt +1 g , t t t (8) −Ul g /Uc g = 1 − τ g , t t t t ¯ (9) c g = 1 − τ g l g + R g − 1 b, for all t, and the transversality condition limj →∞ E[β t +j Uc (g t +j )b¯ | g t ] = 0 for all t and g t . ¯ t ), gt , τ¯ (g t ), R(g t ), b¯t } be a competitive equilibrium with Definition 1. Let E := {c(g ¯ t ), l(g ¯bt = b¯ > 0, R(g t ) 1 for all t, and R(g t ) > 1 for some g t with µ(g t ) > 0. Next, I show that a consumption-labor allocation under a balanced-budget restriction with b¯ > 0 can be supported under a balanced-budget restriction with b¯ = 0 and a suitably chosen set of lump-sum transfers. Proposition 2. Given a competitive equilibrium E, there exists a sequence of lump¯ t ), gt , τ¯ (g t ), R(g t ), b¯ T (g t ), T (g t )} is a ¯ t ), l(g sum transfers T (g t ), such that E T := {c(g T t t ¯ competitive equilibrium with b (g ) = 0, T (g ) 0 for all g t and T (g t ) > 0 for some g t with µ(g t ) > 0. Proof. See Appendix C.
✷
Next, I restrict the utility function so that both consumption and leisure are normal goods and assume that the competitive equilibrium E is never on the bad side of the Laffer curve nor at the top of the Laffer curve. ¯ → R is strictly concave in [c, −l] Assumption 1. The felicity function U (c, l) : R+ × [0, l] and satisfies Uc > 0, Ul < 0, Ucc − Ucl Uc /Ul < 0, and Ull − Ucl Ul /Uc < 0. Assumption 2. Given a competitive equilibrium E and corresponding competitive ¯ t )] solves ¯ t ), l(g equilibrium E T , for each g t , [c(g max U (c, l), c,l
subject to c + gt = l,
t Ul (c, l) gt + T g = 1 + l. Uc (c, l)
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Assumption 3. Given a competitive equilibrium E and corresponding competitive equilibrium E T , for each g t , there exists open and connected set T (g t ) such that [0, T (g t )] ⊂ T (g t ) and for all T ∈ T (g t ) there exists a solution to max U (c, l), subject to c + gt = l,
Ul (c, l) l. gt + T = 1 + Uc (c, l)
(10) (11)
Let c(gt , T ) and l(gt , T ) denote this solution. The next proposition demonstrates that both consumption and labor are decreasing in the lump-sum transfer. Proposition 3. Under Assumptions 1 and 3, ∂c(g, T ) ∂l(g, T ) = < 0, for T ∈ T . ∂T ∂T Proof. See Appendix C.
✷
Let E denote the set of all competitive equilibria E that satisfy Assumptions 1–3. Next, I prove that a competitive equilibrium with zero debt and strictly positive lump-sum transfers provides strictly less utility than a competitive equilibrium with zero debt and no lump-sum transfers, i.e., positive lump-sum transfers financed by a distortionary tax are dead-weight losses. It follows then that no competitive equilibrium that satisfies Assumptions 1–3 is sustainable. Proposition 4. Given a competitive equilibrium E ∈ E and corresponding competitive ˆ t )] maximize U (c, l) ˆ t ), l(g equilibrium E T defined in Proposition 2, for each g t let [c(g subject to
Then
c + gt = l,
(12)
gt = (1 + Ul /Uc )l,
(13)
c 0, l¯ l.
(14) (15)
ˆ l ), gt , τˆ (g t )} := {c(g E ˆ t ), l(g
is a competitive equilibrium where ˆ t )) t t Ul (c(g ˆ t ), l(g τˆ g = 1 + lˆ g ˆ t )) Uc (c(g ˆ t ), l(g
with
E0
∞ t =0
∞ 0 > W 0 := E0 β t U cˆt , lˆt =: W β t U c¯t , l¯t . t =0
(16)
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Proof. See Appendix C.
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✷
Proposition 5. No equilibrium E ∈ E is sustainable. Proof. See Appendix C.
✷
The intuition for this result is as follows. In the Chari–Kehoe model, the government does not default on its debt to maintain its good reputation as a borrower for future borrowing. The ability to issue debt is desirable because it allows the government to smooth taxes. In this model, debt is useful for smoothing taxes in two ways: (1) state-contingent returns that provide “insurance” to the household, and (2) altering the level of debt. A balanced-budget restriction shuts down channel (2), and the condition that R(g t ) 1 eliminates channel (1). Consequently, the threat of not being able to borrow in the future if the government ever defaults on its debt is not an effective deterrent under a balancedbudget restriction with non-negative interest rates since both channels are inoperative.7 What is the importance of a positive interest rate? A negative interest rate would result in the interest payments acting like a positive lump-sum tax which would allow a lower labor tax rate when compared to the autarky equilibrium. It is therefore possible for utility under a balanced-budget rule with positive debt to be greater than the autarky utility level. For this reason, in the next section I look at competitive equilibria under a balanced-budget restriction more generally and Ramsey allocations under a balanced-budget restriction. What is important to note is that anything that impedes the use of these two channels diminishes the desirable tax-smoothing feature of debt. 4. Ramsey allocations and competitive equilibria In this section, I characterize the Ramsey allocation and competitive equilibria under a balanced-budget restriction and discuss the solution method used to solve the numerical examples in the next section. I also briefly describe the much simpler Ramsey allocation and competitive equilibria with Rb 0. 4.1. Balanced-budget Ramsey The Ramsey problem under a balanced-budget restriction is much more difficult to solve than the Ramsey with Rb 0. The balanced-budget restriction imposes a form of market incompleteness on the Ramsey problem. The following proposition describes the program that gives the Ramsey allocation. Proposition 6. The Ramsey allocation under a balanced-budget restriction is the solution to the following optimization problem: maximize (1) by choice of {c(g t ), l(g t )} subject to 7 As long as one channel is operative, the competitive equilibrium may be sustainable. For example, there are equilibria in Chari and Kehoe that are sustainable where R(g t ) 1 and R(g t ) > 1 for some t, but this will
necessarily involve changing the level of the debt.
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c g t + gt = l g t , ¯ + U t +1 lt +1 , Uct b¯ = Et β Uct +1 (ct +1 + b) l g0 + (R0 − 1)b¯ = 1 + Ul0 /Uc0 l0 .
(17) (18) (19)
Proof. Similar to Stockman (2001). ✷ Let σt denote the multiplier on (18) and γ be the multiplier on (19). With this characterization of the constraints facing the Ramsey allocation under a balanced-budget restriction, the first-order conditions for this program are given by the following for t 1: t 0 = Uct + Ult (1 + σt −1 ) + (σt −1 − σt ) Ucc + Ulct b¯ t (20) + σt −1 Ucc + Uclt c g t + Ullt + Ulct l g t , (17) and (18). For t = 0, first-order conditions are (17)–(19) and 0 0 = Uc0 + Ul0 − σ0 Ucc + Ulc0 b¯ + γ 1 + Ul0 /Uc0 0 0 2 0 + γ l0 Uc0 Ulc0 + Ucc − Ul0 Ucc Uc . + Ucl0
(21)
The problem is recursive from for t 1 with the state being given by [gt , σt −1 ]. In fact, ¯ under a balanced-budget restriction, the allocation at t = 0 is determined by given R0 b, the budget constraint and resource constraint alone. With this observation, I can recast the Ramsey problem into a recursive dynamic programming framework which greatly facilitates the calculation of expected utility since this is captured by the value function. Since the Ramsey problem under a balanced-budget restriction is recursive, one can solve for the allocation using dynamic programming. The natural state is [R, g] and the control is [c, l, R , c , l ]. The Bellman equation is given by (22) V (R, g) = max U (c, l) + βE V (R , g ) | R, g subject to
Uc (c, l) = E βUc (c , l )R | R, g , ¯ Uc (c, l) c + b¯ + Ul (c, l)l = Uc (c, l)R b, c + g = l, ¯ Uc (c , l ) c + b¯ + Ul l = Uc (c , l )R b,
c +g =l ,
(23) (24) (25) (26) (27)
and the given stochastic process for g. I solve this Bellman equation numerically.8 8 I parametrically approximate the value function V (R, g) with Chebyshev polynomials as described in Judd (1998) or Ljungqvist and Sargent (2000). Let W (R, a(g)) be an N -degree Chebyshev polynomial with a(g) being an N × 1 vector of parameters (one vector for each value of g). For initial conditions on the Chebyshev polynomial, I use the consumption and leisure values from the log-linear approximation in a second-order approximation of the utility function. This is described in Appendix A. I then iterate on Bellman’s equation using the collocation method until convergence a(g) → a ∗ (g).
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4.2. Competitive equilibria under a balanced-budget rule A competitive equilibrium under a balanced-budget restriction is characterized by the following equations: (28) c g t + gt = l g t , t +1 t t Uc = Et βUc R(g , gt +1 ) , (29) t t t t t t ¯ Uc c g + b¯ + Ul l g = Uc R g b, (30) and the law of motion for government expenditures. The state vector is [R(g t ), gt ]. Assuming {gt } follows a stationary AR(1) process and log-linearizing this system, we get
t +1 t R R =A , (31) Et gˆt +1 gˆ t where xˆt denotes a percent deviation from steady state. Let λi be the eigenvalues of A. I show in Appendix B that |λi | < 1 for i = 1, 2 indicating local indeterminacy.9 An equilibrium is then given by stochastic processes {ψt +1 , et +1 } with Et [ψt +1 ] = Et [et +1 ] = 0, and
t +1 t R R ψt +1 =A + . (32) gˆt +1 gˆt et +1 Since {gt } is exogenous, the process {et +1 } is exogenous. The process {ψt +1 } is not uniquely determined in the model. The conditional mean being zero is the only restriction put on the ψt +1 process. Another way of thinking about the Ramsey problem is as a choice of {ψt +1 }. In fact, the log-linear approximate solution to the Ramsey equilibrium selects ψt +1 = η∗ et +1 for some constant η∗ . In terms of the non-linear system, we have the following: t
βUc R g t , gt +1 , gt +1 R g t , gt +1 Uc R g , gt Et = (33) ¯ gt +1 − g¯ ρ(gt − g) or
t
Uc R g , gt zt +1 βUc R g t , gt +1 , gt +1 R g t , gt +1 = + . ¯ et +1 gt +1 − g¯ ρ(gt − g)
(34)
The shock {et +1} is exogenous, but {zt +1 } is only restricted by Et [zt +1 ] = 0. Again, the Ramsey problem can be viewed as one of choosing this stochastic process to maximize expected utility, and to a first-order approximation zt +1 = η∗ et +1 under Ramsey. Without introducing extrinsic uncertainty and retaining a Markovian structure, the z shock takes the form zt +1 = Z(Rt , gt , et +1 ). 9 This indeterminacy is related to the results of Schmitt-Grohé and Uribe (1997) with the capital’s share parameter equal to zero.
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This restriction still does not impose enough structure for a pragmatic numerical exploration. Another level of restrictions might be zt +1 = et +1 Z(Rt , gt ). This implies a conditional variance Et [zt2+1 ] = Z(Rt , gt )2 σe2 . Using the above observation on the first-order approximation of the Ramsey allocation, I will limit Z(R, g) to constant functions in the numerical examples in the next section, i.e., a single parameter family that covers the Ramsey allocation to a first-order approximation.10 Letting prime variable denote next period values, and given z = ηe , one can solve for R = R(R, g, g ; η). With this law of motion for R , I can calculate [c, l] recursively. Let ∞ t t t V (R, g; η) := E0 , β U c R g , gt , l R g , gt t =0
then
V (R, g; η) = U c(R, g), l(R, g) + E βV R(R, g, g ; η), g g .
This functional equation can be solved again by approximating V by a Chebyshev polynomial. No iteration is necessary because no maximization is done. 4.3. Ramsey and competitive equilibria with Rb 0 In this subsection, I characterize both Ramsey and stationary competitive equilibria with Rb 0 and briefly describe the numerical solution method. This exposition differs from Chari–Kehoe insofar as the initial level of debt is positive.11 The next proposition describes the program whose solution is the Ramsey allocation. Proposition 7. The Ramsey allocation with Rb 0 is the solution to the following optimization problem: given R0 b−1 > 0, maximize (1) by choice of {c(g t ), l(g t ), bt } subject to (35) c g t + gt = l g t , t +1 t +1 t Uc bt = Et β Uc (ct +1 + bt +1 ) + Ul lt +1 , (36) t t t t (37) Uc c g + bt + Ul l g 0, 0 0 g0 + R0 b−1 = 1 + Ul /Uc l0 + b0 . (38) 10 This is the closest analogy to the type of competitive equilibria explored by Chari and Kehoe. 11 Chari and Kehoe set b −1 = 0 and write “. . . if the inherited debt at date zero given by b−1 is positive,
the government will default and the Ramsey problem is unchanged. Since we require debt be non-negative the assumption that b−1 = 0 is without loss of generality.” This sets δ0 = 1 without the possibility of it being otherwise and results in some blurring of the difference between δ0 = 1 and b−1 = 0. I will be making the distinction and let the government choose the initial default rate while taking the initial level of debt as given. In the model without commitment, setting the initial default rate to one should be perceived as default by the agents and have potential adverse effects on future borrowing.
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Proof. Similar to that given in Stockman (2001). ✷ The first-order conditions for this program for t 1 are given by t 0 = Uct + Ult (1 + σt −1 ) + (σt −1 − σt ) Ucc + Uclt bt t + σt −1 Ucc + Uclt c g t + Ullt + Ulct l g t , γt = σt − σt −1 , l g t = c g t + gt , 0 = γt Uct c g t + bt + Ult l g t , Uct bt = Et β Uct +1 (ct +1 + bt +1 ) + Ult +1 lt +1 ,
(39) (40) (41) (42) (43)
taking σ0 as given. Given a solution to this system of difference equations, one then solves the t = 0 first-order conditions. This illustrates that the Ramsey allocation is stationary for t 1. In the numerical examples in the next section, I assume government consumption is i.i.d. and takes on two values g 1 and g 2 with g0 = g 2 , g 1 < g 2 , and Prob[g = gi ] = µi . Under this assumption, I can solve the Ramsey problem using a slight modification of the Chari–Kehoe method. First, the restrictions put on stationary competitive equilibria with the restriction Rb 0 are the following: H (c0 , l0 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) = Uc (c0 , l0 )R0 b−1 , (44) (45) H (c1 , l1 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) 0, (46) H (c2 , l2 ) + β µ1 H (c1 , l1 ) + µ2 H (c2 , l2 ) /(1 − β) 0, c0 + g0 = l0 ,
(47)
1
c1 + g = l1 ,
(48)
c2 + g = l2 ,
(49)
2
where H is defined by H (ci , li ) ≡ Uc (ci , li )ci + Ul (ci , li )li ,
(50)
for i = 0, 1, 2. Again, this differs from Chari–Kehoe insofar as the initial level of debt is positive. Let b−1 > 0 (assuming R0 = β −1 ) be an initial condition. The Ramsey problem is to maximize U (c0 , l0 ) + β µ1 U (c1 , l1 ) + µ2 U (c2 , lc ) /(1 − β), subject to (44)–(49) by choice of [ci , li ]2i=0 . Debt in a stationary equilibria is given by β[µ1 H (c1 , l1 ) + µ2 H (c2, l2 )]/(1 − β) , b gi = Uc (ci , li )
(51)
and the total gross return to debt is given by H (ci , li ) + β[µ1 H (c1, l1 ) + µ2 H (c2 , l2 )]/(1 − β) . R gi b gi = Uc (ci , li )
(52)
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Note that when b−1 = 0 (as in Chari and Kehoe) the Ramsey allocation at t = 0 coincides with the allocation at later dates when g = g 2 . Stationary equilibria can be characterized by the values of surplus functions: H i = H (ci , li ), i = 0, 1, 2. These values must satisfy the above constraint on the Ramsey allocation. In terms of these equilibria being sustainable, I first check the level of expected utility for t 1. To do this note that given surplus levels H 1 and H 2 , one can solve for [c(H i ), l(H i )] for i = 1 and 2 using the market clearing conditions. This implies U (c(H i ), l(H i )) = U (H i ). To systematically investigate the sustainability of all stationary competitive equilibria, I grid [H 1 , H 2 ] that satisfy the Ramsey constraints, calculate the expected utility for t 1. If this tail is sustainable, I then solve for [c0 , l0 ] given [H 1 , H 2 ] using (44) and (47), and calculate expected utility at t = 0. If this value is greater than autarky utility, the competitive equilibrium is sustainable. A remark on sustainable utilities. For a given parameterization, the Ramsey allocation might not be sustainable while other non-autarky allocations might be.12 This might seem counter-intuitive given that the Ramsey allocation maximizes utility. The potential sustainability problem with Ramsey is that while taking into account the optimal trade-off between current and future utility, the Ramsey allocation might place the economy on a future trajectory such that the utility along this tail is not high enough compared to autarky.
5. Numerical examples In Section 3, we saw that a competitive equilibrium is not sustainable if R(g t ) 1 for all t with a strict inequality holding for some t. When R(g t ) < 1, state-contingent debt may be providing insurance to the household that lessens the need for high taxes when g is high. This insurance may raise utility enough for the competitive equilibrium to be sustainable. In this section, I provide numerical examples to illustrate the impact of a balanced-budget restriction on the sustainability of the Ramsey allocation and utility levels of other stationary competitive equilibria. I also illustrate the degree to which R < 1 for the competitive equilibrium to be sustainable. Preferences are given by 1−α U (c, l) = cα l¯ − l . Government spending is i.i.d. and takes on two value g 1 and g 2 with equal probability. Initial conditions are g0 = g 2 and R0 = β −1 .13 Both the range of parameter values and benchmark values are recorded in Table 1. Utility levels are normalized as a ratio to autarky utility as in Chari and Kehoe. The discount factor, volatility of government spending and initial level of debt are varied. The volatility of government expenditures ranges from 5% to 50% and the level of initial debt is varied from 0.5 to 4.0 which corresponds to roughly 10% to 80% of output in the 12 Chari and Kehoe (1993) illustrate this point. 13 This numerical example is similar to that in Chari and Kehoe and meant for illustrative purposes only.
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Table 1 Benchmark and range of parameters values considered in simulations Preferences
β ∈ [0.90, 0.91, . . . , 0.98] l¯ = 10
α = 1/2
Government
g¯ = 1.0 b¯ ∈ [0.5, 1.0, . . . , 4.0] g ∈ {g 1 , g 2 }
σg ∈ [0.05, 0.10, . . . , 0.5]
Benchmark values
β = 0.95 σg = 0.5
Prob(g = gi ) = 0.5 b = 1.0
model. The impact of a balanced-budget restriction on sustainable utilities is illustrated in Figs. 1–3. We see, as expected, that the range of sustainable utilities is smaller under a balanced-budget restriction. Perhaps surprising, this range under a balanced-budget restriction is not that much smaller. This suggests either (1) the insurance channel of debt through state-contingent returns is quantitatively more important for welfare than the ability to change the level of debt or (2) state-contingent returns are to a large extent substitutable for altering the level of debt. All of the intuitive findings of Chari and Kehoe are preserved under a balanced-budget restriction: higher β and σ make sustainability more likely. In addition, we see that a higher initial level of debt b−1 makes sustainability less likely. For sustainable utility levels, we know that the gross return on debt must be less than 1 during some t. How much variability in the gross return is needed for sustainable utilities? The range of state-contingent returns required for the competitive equilibrium under a
Fig. 1. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the discount factor. Other parameters: σ = 0.5 and b¯ = 1.0.
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Fig. 2. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the standard deviation of government spending. Other parameters: β = 0.95 and b¯ = 1.0.
Fig. 3. Non-negative debt with and without a balanced-budget restriction: sustainable utilities vs. the initial level of debt. Other parameters: β = 0.95 and σ = 0.5.
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balanced-budget restriction to be sustainable along with those determined by Ramsey under a balanced-budget restriction for different discount factors is plotted in Fig. 4. In a sustainable equilibrium, the return on debt R(g) is high when g is low and R(g) is low when g is high. It is precisely this feature that makes state-contingent returns desirable. If these returns are too compressed, the competitive equilibrium will not be sustainable (these would correspond to the area labeled “Not Sustainable”). We see that substantial variability in the gross return on debt is needed for sustainability. Isolating two values on the cusp of sustainability, in Figs. 5 and 6 we see the amount of state-contingent returns being utilized by Ramsey under a balanced-budget restriction with β = 0.93 (sustainable) and β = 0.92 (unsustainable). For β = 0.93, the range of state-contingent returns is R ∈ [0.69, 1.47]. For β = 0.92, the range of state-contingent returns is R ∈ [0.7, 1.48]. For β = 0.93, utility levels are sustainable from autarky up to Ramsey under a balanced-budget restriction. The return on debt for the sustainable competitive equilibrium that delivers the autarky utility level is in Fig. 7. We still see substantial state-contingent returns on debt with R ∈ [0.82, 1.34]. Note this competitive equilibrium is the worst sustainable equilibrium where the government does not default and it requires variation in the gross returns on debt of about ±25% from the mean (1/β = 1.075) to do it.
Fig. 4. Range of state-contingent returns needed for sustainability. For sustainable equilibria, the level of state contingent returns fluctuates between the upper and lower shaded areas. The upper and lower shaded regions illustrates range for R(g 1 ) (a high return on debt occurs when government spending is low) and R(g 2 ) (a low return on debt when government spending is high). The returns in the region labeled “Not Sustainable” are too compressed to be sustainable.
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Fig. 5. Return on debt from sustainable Ramsey with a balanced-budget restriction: β = 0.93, σ = 0.5, b = 1.0.
Fig. 6. Return on debt from unsustainable Ramsey with a balanced-budget restriction: β = 0.92, σ = 0.5, b = 1.0.
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Fig. 7. Return on debt from the sustainable competitive equilibrium under balanced-budget restriction that delivers the autarky utility level: β = 0.93, σ = 0.5, b = 1.0. As a benchmark, the implicit risk-free returns under the autarky equilibrium with β = 0.93 and σ = 0.5 are 1.0078 and 1.1427.
6. Adding capital to the model Extending the model to include capital greatly exacerbates the sustainability problem under a balanced-budget restriction.14 As shown in Zhu (1992) and Chari, Christiano, and Kehoe (1994) in a model with labor and capital income taxation, there is indeterminacy in the tax plans that support the Ramsey allocation. This result carries over into competitive equilibria under a balanced-budget restriction. Essentially, one can use either statecontingent capital income taxes or state-contingent returns on debt to decentralize a competitive equilibrium allocation. This suggests that if state-contingent capital income taxes are available, the state-contingent returns to debt are superfluous, and default would be more likely. Take the standard business cycle model with capital as in Chari et al. (1994). Output is produced using a constant returns to scale production function yt = F (kt , lt ). Capital income is taxed at rate θt , so the government budget constraint is given by gt + (Rt − 1)b¯ = τt wt lt + θt rtk − δ kt , 14 I thank an anonymous referee for suggesting this possibility.
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where δ is the depreciation rate, rtk the rental rate of capital, and wt the real wage. A competitive equilibrium is defined in the usual way. The indeterminacy of taxes carries over to competitive equilibrium under a balanced-budget restriction. Proposition 8. Let {c¯t , l¯t , k¯t g¯t } and {τ¯t , θ¯t , R t , r¯tk , w¯ t } be a competitive equilibrium. Then t , r¯tk , w¯ t } is a competitive if {θˆt , R t } satisfy {c¯t , l¯t , k¯t g¯t } and {τ¯t , θˆt , R t +1 , Et Uc c¯t +1 , l¯t +1 R t +1 = Et Uc c¯t +1 , l¯t +1 R (53) t +1 t +1 ¯ ¯ ¯ ˆ Et Uc c¯t +1 , lt +1 θt +1 F k − δ = Et Uc (c¯t +1 , lt +1 )θt +1 F k − δ , (54) t t − 1 b¯ = τ¯t w¯ t l¯t + θˆt F k − δ k¯t , g¯t + R (55) where F tk = Fk (k¯t , l¯t ). Proof. Similar to Zhu (1992) or Chari et al. (1994). ✷ This means given an allocation from a competitive equilibrium under a balanced-budget restriction, there is more than one set of state-contingent taxes on capital income and returns on debt that will support this allocation. In particular, one can use non-statecontingent returns on debt and set t +1 = R
Uc (c¯t , l¯t ) . Et [βUc (c¯t +1 , l¯t +1 )]
The following proposition gives sufficient conditions for a competitive equilibrium under a balanced-budget restriction to be unsustainable. Proposition 9. Any competitive equilibrium under a balanced-budget restriction with positive debt is not sustainable if Uc (c¯t , l¯t ) 1, Et [βUc (c¯t +1 , l¯t +1 )] for all t with a strict inequality for some t. Proof. The reasons are similar to Proposition 5. ✷ Note that we do not need the competitive equilibrium to involve risk-free debt with a positive return, but only the much weaker condition that the implicit risk-free return on debt be positive.
7. Conclusion Balanced-budget rules limit the effectiveness of debt to smooth taxes, but it is precisely this feature of debt that keeps a government from defaulting. In this paper, I have demonstrated that if the return on debt is positive, then a no-default competitive equilibrium is inconsistent with a balanced-budget restriction. State-contingent returns on a fixed
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level of debt allows for some tax smoothing and this may be enough to reconcile the inconsistency. However, enormous variation in the real ex post return to debt (in excess of ±25%) is needed to have the tax-smoothing benefits outweigh the benefit of default according to numerical examples. Moreover, adding capital to the model makes the state-contingent returns on debt unnecessary to smooth taxes because state-contingent capital income taxes can serve this function. In this case, a balanced-budget restriction is inconsistent with a competitive equilibrium with positive debt under even weaker conditions (a positive implicit risk-free return to debt). Two additional lessons emerge from this analysis. First, balanced-budget rules that include “escape clauses” in times of unusually high government spending (e.g., a war) or the establishment of a separate capital account for capital expenditures allow for more tax-smoothing over time. These features may be extremely important in providing the government an incentive to preserve its reputation as a good borrower by not defaulting. Second, taking the reputation model seriously, adoption of a balanced-budget restriction would predict default as the equilibrium outcome. Insofar as this outcome is not observed, the reputation argument cannot be the complete answer to the question: Why don’t governments default on their debt? Reputation effects may be important, but other equally important mechanisms must be at work.
Acknowledgments This paper is a substantially revised chapter from my PhD dissertation in economics at University of Chicago. I thank Larry Christiano, Michael Chwe, Lars Hansen, Tom Sargent, an anonymous referee, and seminar participants at the University of Delaware, Williams College, SUNY–Albany, and Congressional Budget Office for helpful comments. All remaining errors are mine.
Appendix A. Calculating utility under a balanced-budget rule The competitive equilibria under a balanced-budget restriction are Markovian in x = [R, g] . Chari and Kehoe deal with equilibria that are Markovian in [g] alone. This makes the comparisons of utility somewhat easier due to the fact that g takes on a finite number of values. Under a balanced-budget restriction one needs to calculate expected utility for each g and every possible R. I approximate the Ramsey value function using Chebyshev polynomials using a projection method called collocation. For initial coefficient values for the Chebyshev polynomial that approximates the value function, I use the consumption and leisure values from the log-linear approximation in a second-order approximation of the utility function. Given the value of expected utility for different initial levels of g0 and R0 , one solves for the polynomial coefficient that sets the value function approximation equal to these values. t gˆt ] be the state vector. Given a log-linear approximation to the equilibrium Let xˆt = [R satisfies: Et [xˆt +1 ] = Axˆt
or xˆt +1 = Axˆt + Cet +1 ,
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where C := [η 1] . I can write
cˆt = Γ xˆ t . lˆt Then a second-order Taylor approximation of utility at time t is given by U c g t , l g t ≈ U (c∗ , l ∗ ) + DU (c∗ , l ∗ )Γ xˆ t + (1/2)xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt . Then it is easy to show that Σxt = A Σxt −1 A + CΣe C ,
E0 [xˆt ] = At xˆ0 , where
Σxt := E0 xˆt xˆt ,
Σe := E0 et et ,
with Σx0 = x0 x0 . Using these expressions, I can get a simple expression for E0 [U (c(g t ), l(g t ))]: E0 U c g t , l g t ≈ U (c∗ , l ∗ ) + DU (c∗ , l ∗ )Γ At xˆ0 + (1/2)E0 xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt . I can express the last conditional expectation as E0 xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt = E0 tr xˆt Γ D 2 U (c∗ , l ∗ )Γ xˆt = E0 tr Γ D 2 U (c∗ , l ∗ )Γ xˆt xˆt = tr E0 Γ D 2 U (c∗ , l ∗ )Γ xˆt xˆt = tr Γ D 2 U (c∗ , l ∗ )Γ E0 xˆt xˆt = tr Γ D 2 U (c∗ , l ∗ )Γ Σxt , where tr(·) is the trace operator. So expected utility is provided by ∞ t t t E0 β U c g ,l g t =0
≈
∞ U (c∗ , l ∗ ) t + β DU (c∗ , l ∗ )Γ At xˆ0 + (1/2) tr Γ D 2 U (c∗ , l ∗ )Γ Σxt . 1−β t =0
Appendix B. Stability of steady state under a balanced-budget rule Let A be the matrix in (31). Using (28) and (30) we can write t + ηcg gˆt , t + ηlg gˆt . cˆt = ηcR R lˆt = ηlR R Using the same argument as in Proposition 3, it is true that ηcR c¯ = ηlR l¯ < 0 as long as b¯ > 0. Then one can show that the eigenvalues of A are given by λ1 = γR /(γR + Uc ),
λ2 = ρ,
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where γR = ηcR c(U ¯ cc + Ucl ). Then |λ2 | < 1 if g is stationary and 0 < λ1 < 1 if γR > 0. If Ucc + Ucl < 0 then γR < 0. If Ucl < 0 then Ucc + Ucl < 0 since Ucc < 0. A positive revenue requirement (τ ∗ > 0) implies in steady state −Ul /Uc = (1 − τ ∗ ) < 1
or
− Uc /Ul > 1.
So if Ucl > 0 we have Ucc + Ucl < Ucc − Ucl Uc /Ul < 0. The last inequality follows from the assumption that consumption and leisure are normal goods.
Appendix C. Proofs
Proof of Proposition 2. Since E is a competitive equilibrium under a balanced-budget ¯ Given restriction with b¯ > 0 we have (6)–(9) satisfied. We set T (g t ) = [R(g t ) − 1]b. t t t the assumption on interest rates, T (g ) 0 for all g and T (g ) > 0 for some g t with µ(g t ) > 0. Then by direct inspection, E T satisfies (6)–(8), and the appropriately modified version of (9) to include lump-sum transfers (dropping b¯ T (g t ) since it equals zero): c¯ g t = 1 − τ¯ g t l¯ g t + T g t . (C.1) Therefore E T is a competitive equilibrium. ✷ Proof of Proposition 3. Define G(c, l) := (1 + Ul /Uc )l and
c+g−l F (c, l, g, T ) := . G(c, l) − g − T In equilibrium, F (c, l, g, T ) ≡ 0. By Assumption 1, Uc [Ull − Ucl Ul /Uc ] < 0,
−Ul [Ucc − Ucl Uc /Ul ] < 0.
Combining these gives 0 > Uc [Ull − Ucl Ul /Uc ] − Ul [Ucc − Ucl Uc /Ul ] = Gc + Gl . This allows us to use the implicit function theorem which gives
−1
∂c(g,T ) ∂c(g,T )
1 1 −1 1 0 1 − Gl ∂g ∂T = − = ∂l(g,T ) ∂l(g,T ) Gc Gl −1 −1 1 + Gc G + G c l ∂g ∂T
1 . 1
This establishes the equality of the two partial derivatives. Since Gc + Gl < 0, the sign of the derivatives is negative as well. ✷ satisfies (6)–(9) and is Proof of Proposition 4. Again, direct inspection reveals E t (g t ). Since therefore a competitive equilibrium. If T (g ) = 0 we have U (g t ) = U ∂c(g, T ) ∂l(g, T ) = < 0, ∂T ∂T
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we have for T > 0, ∂c(g, T ) ∂l(g, T ) ∂l(g, T ) ∂U (c(g, T ), l(g, T )) = Uc + Ul = (Uc + Ul ) ∂T ∂T ∂T ∂T ∂l(g, T ) < 0, = Uc (1 + Ul /Uc ) ∂T since (1 + Ul /Uc ) > 0 because (1 + Ul /Uc )l = g + T > 0, i.e., a positive revenue requirement. This establishes that for each g t such that T (g t ) > 0, we have U cˆ g t , lˆ g t > U c¯ g t , l¯ g t . (g t ) > U (g t ) for some g t with µ(g t ) > 0, (g t ) U (g t ) for all g t with U This establishes U which gives (16). ✷ Proof of Proposition 5. Recall Chari and Kehoe define the autarky plans as follows. For any history, the government sets δ a (g t ) = 1 and q a (g t ) = 0. The tax rate τ a (gt ) is given by τa (gt ) = 1 + Ula (gt )/Uca (gt ), where [ca (gt ), l a (gt )] solves max U (c, l) subject to c + gt = l and gt = (1 + Ul /Uc )l. The household’s allocation rules specifies for every history, hold zero debt and choose consumption and labor to maximize utility given the autarky tax rates. Autarky utility is given by ∞ a t a t a t . (C.2) β U c g ,l g W0 := E0 t =0
0 = W a , where W 0 is defined in (16). Let {c(g ¯ t )} denote the Note first that W ¯ t ), l(g 0 allocation in E ∈ E and ∞ t t t ¯ . W 0 := E0 β U c¯ g , l g t =0
We know from Proposition 3 that there exists a competitive equilibrium E T with a sequence of non-negative transfers T (g t ) 0 for all g t and T (g t ) > 0 for some g t with µ(g t ) > 0 that decentralizes this allocation with zero debt. Let ∞ t t T t . W 0 := E0 β U c¯ g , l¯ g t =0
0 = W a , and therefore E ∈ E is not From Proposition 4, we have W 0 = W T0 < W 0 sustainable. ✷
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