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Balanced budget vs. Tax smoothing in a small open economy: A welfare comparison
Journal of Macroeconomics 31 (2009) 438–463
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Balanced budget vs. Tax smoothing in a small open economy: A welfare comparison q Constantine Angyridis * Department of Economics, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3
a r t i c l e
i n f o
Article history: Received 9 June 2006 Accepted 30 August 2008 Available online 17 September 2008
JEL classification: E62 H21
a b s t r a c t The objective of this paper is to investigate the effect of lending and borrowing constraints on the dynamics of public debt and optimal taxation policy in the context of a general equilibrium model with tax smoothing. The results from the numerical simulation of the model show significant welfare gains, provided that the policymaker is allowed to borrow and lend in order to smooth taxes across time instead of maintaining a balanced budget at all times. Moreover, for a specific process for asset prices, it is also shown that if the government can issue state-contingent debt then overall welfare can be further improved substantially. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Balanced budget Complete markets Incomplete markets Welfare analysis
1. Introduction Is a series of continuous budget deficits necessarily bad? Besides being a clear sign of fiscal prudence, does a balanced budget rule also improve social welfare? How should tax rates and, consequently, public debt respond to innovations in government expenditures? Theoretical advancements on solving stochastic intertemporal optimization problems with implementability constraints,1 have reinvigorated research on optimal debt issue and the implied structure of taxation policy.2 The present paper applies this new technique in the context of a small open economy, in which the social planner (the government) is faced with stochastic expenditures. The objective is to study the welfare implications if the policymaker adopts a fiscal policy plan that deviates from a strict balanced budget rule. It is shown that there are significant welfare gains to be made if the government equates revenues with expenditures in a present value sense, instead of every period, such that its intertemporal infinite horizon budget constraint is satisfied. q The author would like to thank Arman Mansoorian, Leo Michelis, Xiaodong Zhu, the co-editor and an anonymous referee for their valuable comments and suggestions. All remaining errors are my own. * Tel.: +1 416 979 5000x7725; fax: +1 416 598 5916. E-mail address: [email protected] 1 Standard dynamic programming techniques are applicable under the condition that the set of feasible current actions available to a social planner depend only on past variables. Implementability constraints imposed in contract and optimal fiscal policy problems usually depend on plans for future variables, thus constraining the set of current feasible actions available to the social planner. Extending the work of Kydland and Prescott (1980) for a general class of contract problems involving incentive constraints, Marcet and Marimon (1999) show how one can compensate for this lack of recursivity by expanding the state space to include a new variable that depends on past Lagrange multipliers. 2 For example, see Aiyagari et al. (2002); Angeletos (2002); Buera and Nicolini (2001) and Marcet and Scott (2003).
0164-0704/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2008.08.003
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
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Stockman (2001) has performed a comprehensive analysis of the welfare effects of balanced budget restrictions in the context of a closed economy with exogenous growth. Using a similar framework to the one used in Chari et al. (1994, 1995), he considered the Ramsey problem for a model calibrated to the US economy, placing restrictions on the amount of debt that the government can issue. In his setting, he found substantial welfare losses associated with the balanced-budget restriction: depending on preference parameterization, consumption would have to increase from 0.61% up to 1.45% at each date and for each state in order to make the representative household indifferent between fiscal policy regimes. Stockman’s study builds on the work by Lucas and Stokey (1983) who considered a general equilibrium model with complete markets, exogenous Markov government expenditures and state-contingent taxes. The Lucas and Stokey framework emphasizes the role of state-contingent debt as an ‘‘insurance” against bad times, that allows the government to smooth tax distortions across both time and states of nature. As a result, in times of temporary increases in government expenditures, the level of public debt falls. In addition, Lucas and Stokey show that the serial correlations of optimal tax rates are closely related to those for government expenditures. In their model, tax rates are smooth only to the extent that they exhibit a smaller variance than a balanced budget would imply. Aiyagari et al. (2002) reconsider the Lucas and Stokey economy under the restriction that the government cannot issue state contingent debt; but rather it can borrow and lend at an endogenous risk-free rate. In their attempt to replicate Barro (1979) classic ‘‘tax smoothing” result in a general equilibrium environment, they show that this restriction (i.e., market incompleteness) imposes additional implementability constraints with respect to the equilibrium allocation, beyond the single implementability constraint imposed under complete markets. In particular, these constraints require the allocation to be such that at each date the present value of the budget surplus evaluated at current period Arrow prices be known one period ahead. Under the condition that an ad hoc limit is imposed on the government’s asset holdings, the authors show that the Ramsey equilibrium exhibits features shared by both Barro’s and Lucas and Stokey’s economies, but the dynamics of debt and taxes actually resemble more closely Barro’s ‘‘tax-smoothing” result.3 Recently, several authors have reported tax smoothing behavior for various small open economies. Among others, Strazicich (1997) using annual time-series data over the period 1929–1990, fails to reject the tax-smoothing hypothesis for the Canadian federal government. Cashin et al. (1998) find that the behavior of the central government of India is consistent with the tax-smoothing hypothesis for the period 1951–1952 to 1996–1997. A similar result is obtained by Cashin et al. (1999) for Pakistan during the period 1956–1995, but the hypothesis is rejected for Sri Lanka during 1964–1997. Strazicich (2002) considers a panel of 19 industrial countries over the period 1955–1988 and finds evidence in support of tax smoothing as a theory of public debt. Adler (2006) finds that Barro’s model can explain about 60 percent of the variability in the Swedish central government budget surplus for the period 1952–1999. These empirical studies point to the need to analyze the tax smoothing motive in the context of a small open economy facing exogenous asset prices. This provides one motivation for the present paper. We consider an asymmetric small open economy with respect to accessibility to financial markets: households are allowed to perform transactions involving statecontingent financial claims, while the government is restricted to borrow and lend only at a constant risk-free interest rate.4 A possible justification for this assumed financial asymmetry might be the inherent ‘‘moral hazard” that is present if the government is allowed to issue state-contingent debt or bonds. The return on these bonds would have been linked to the level of macroeconomic variables, such as the inflation rate or the Debt/GDP ratio. However, these variables are under the strong influence of the government’s actions, thus damaging the marketability of these bonds among private agents. Another justification for such an asymmetry follows from the work of Sleet (2004) and Sleet and Yeltekin (2006), who demonstrate that if the government has access to a complete set of contingent claims markets but is unable to either commit to future debt repayments or truthfully reveal private information regarding its spending needs, then the market for public debt becomes endogenously incomplete. Studying the implications of the coexistence of complete and incomplete markets in a small open economy is new in the literature on optimal taxation. 5 Furthermore, if one wishes to maintain the representative agent framework, the simplest environment to discuss the implications of this asymmetry is that of a small open economy.6 In our setting, the government is required to determine the optimal fiscal policy plan as of time zero that maximizes the welfare of the representative household, given a stream of stochastic government expenditures. These expenditures are 3 Barro (1979) demonstrated that debt policy between 1916 and 1976 in the U.S. and United Kingdom was consistent with the predictions of his theory. Sargent and Velde (1995) argue that the behavior of British debt during the 18th century conforms with Barro’s theory, since it closely resembles a martingale with a drift. Extending Barro’s model to allow for stochastic variation in interest and growth rates, Lloyd-Ellis et al. (2001) show that the US debt policy is also in agreement with the theory of tax smoothing during the transition period from the large budget deficits of the 1980’s to the high surpluses in the mid 1990’s. Finally, based on the persistence properties of public debt and its positive response to an innovation in government expenditures, Marcet and Scott (2003) argue that the behavior of the US public debt can be accounted for by a model in which the government issues only one-period risk-free bonds. 4 The existence of a market for riskless government bonds is redundant from the point of view of the households, since it does not alter their trading opportunities: households can always adjust their portfolio of state-contingent securities in such a way to obtain the same return as that of a government bond. 5 The literature on optimal fiscal policy for a small open economy focuses exclusively on factor income taxation, incorporates capital stock accumulation and assumes away uncertainty. Recent studies include Atkeson et al. (1999); Chari and Kehoe (1999); Correia (1996) and Razin and Sadka (1995). 6 The existing literature has considered different degrees of market completeness between the public and private sectors of a closed economy. For instance, Aiyagari et al. (2002) assume that the government faces ad-hoc debt and asset limits that are more stringent than those faced by the representative household. On the other hand, Schmitt-Grohé and Uribe (2004) consider a stochastic production economy with sticky product prices in which households can acquire two types of final assets: fiat money and one-period state-contingent nominal assets. In contrast, the government can print money and issue nominal non-state contingent bonds.
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financed by taxing labor income and issuing debt. In contrast to Aiyagari et al. analysis, here the price of government bonds is assumed to be exogenously determined. Another contribution of the present paper is to provide an extensive numerical analysis of the tax smoothing motive and welfare gains with alternative assumptions regarding the stochastic process for government expenditures. For the incomplete markets case, we consider three stochastic processes: (i) serially uncorrelated, (ii) serially correlated, and (iii) serially correlated expenditures whose volatility is adjusted to be equal to that of uncorrelated expenditures. We study the implications of each process for the optimal fiscal policy as the government’s lending and borrowing constraints are relaxed. The comparison of the various cases is based on the persistence and other descriptive statistics of the equilibrium tax rates, as well as the distribution of the government’s asset and debt holdings. It is shown that providing more flexibility to the government in terms of the amounts it can borrow and lend, leads to smoother tax rates across time and higher social welfare compared to a balanced budget policy regime. In other words, as the government’s lending and borrowing range becomes wider, the persistence of the equilibrium tax rates increases and their volatility declines for all types of government expenditures considered. As a consequence, welfare increases. These findings are consistent with the arguments against balanced-budget rules in closed economies, as in Schmitt-Grohé and Uribe (1997) and Stockman (2001). Our results demonstrate also that one does not have to restrict access to complete markets for both sectors of the economy (private and public) in order to retain the random walk property of tax rates. An additional implication is that minimizing tax distortions is the most significant factor influencing the decision of the fiscal authority. In Aiyagari et al. the government has to consider the effect of debt issue on interest rates. By abstracting completely from interest rate effects, we still obtain the tax smoothing result, which implies that the quantitative results of Aiyagari et al. are relevant even for small open economies. Further, using a particular stochastic process for the prices of the state-contingent bonds and complete financial markets for the government, we show that overall welfare significantly improves relative to the balanced budget and incomplete markets cases. In addition, compared to the incomplete markets case, the resulting welfare gains increase as the persistence of the government expenditures shocks rises. These results can be interpreted as the ‘‘cost” of the moral hazard problem that was used to justify the assumed asymmetry between the public and private sectors of the economy. 7 Finally, it should be noted that the improvement in welfare when the government shifts from maintaining a balanced budget to issuing state-contingent debt is slightly higher than the one reported by Stockman (2001) in the context of a closed economy, although admittedly the differences in model specification do not permit a completely satisfactory comparison of the two models.8 The rest of the paper is organized as follows. Section 2 describes the theoretical model for the small open economy and defines the Ramsey problem that maximizes the representative household’s utility. Section 3 provides details regarding the parameterization of the model for the numerical analysis. Section 4 reports the quantitative results and compares the welfare results for different types of government expenditures and borrowing and lending constraints for the government. Section 5 performs welfare analysis under the assumptions of complete financial markets for the government and a specific process for asset prices, and compares the results to the case of incomplete financial markets and balanced government budget. Section 6 concludes the paper. 2. The model Consider a small open economy with a representative agent and no population growth. As in Lucas and Stokey’s closed economy model, the small open economy produces a single non-storable final good, using only labor, such that one unit of labor yields one unit of output. The time endowment in each period t is fixed at T ¼ 100, and it is allocated to labor, nt , and leisure, lt .9 An event st takes place in each period t. Let S be an infinite set of all the possible events. At any point in time t, the state of the economy can then be described by the state variable st ¼ ðs0 ; s1 ; . . . ; st Þ. This vector can also be thought of as representing the history of events up to and including period t. Hence, the history at any period can be described by st ¼ ðst ; st1 Þ. The probability density function as of period 0 of any particular history st taking place is denoted by pðst Þ. The representative household’s income consists of the sum of its net-of-tax labor earnings ð1 st Þnt and matured stateh contingent securities bt . These are one-period forward Arrow securities that pay a unit of output if a particular state of nature is realized and nothing otherwise. Total income is allocated to consumption expenditures ct and the purchase of financial h claims btþ1 at the exogenously determined prices phtþ1 for each possible event stþ1 . The asset prices are measured in units of time-t output and are assumed to be determined by the international financial markets.
7
I am grateful to an anonymous referee for suggesting this insightful interpretation. Schmitt-Grohé and Uribe (1997) and Stockman (2004) define a balanced budget rule as one in which the government holds the real level of debt constant. In this case, the choice of this level will matter when comparing the impact of state vs. non-state contingent debt as long as the government’s initial g g indebtedness b0 > 0. If b0 ¼ 0, then the particular choice does not matter. In the present paper, a balanced budget rule means that the government is not allowed to issue debt. We would like to thank an anonymous referee for this insightful comment. 9 I follow Aiyagari et al. in adopting a rescaled version of the commonly time endowment constraint used in the literature that sets T ¼ 1. This is done for technical reasons, since it facilitates the computational scheme of the paper. 8
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The objective of the household is to maximize, as of time 0, expected lifetime utility with respect to the choice variables h h ct ; lt ; nt ; btþ1 taking as given b0 , the government’s tax rate st and asset prices phtþ1 :
max
1
fct ;lt ;nt ;bhtþ1 gt¼0
W ¼ E0
1 X
bt uðct ; lt Þ;
0
ð1Þ
h
ð2Þ
t¼0
subject to:
ct þ
Z
h
phtþ1 btþ1 dstþ1 6 ð1 st Þnt þ bt ;
8t P 0
lt þ nt ¼ 100; h b0 ;
s
h t ; ptþ1
8t P 0
ð3Þ
given:
Expressions (2) and (3) represent the household’s flow budget constraint and time endowment constraint, respectively. The period utility uðÞ is strictly increasing in ct and lt , strictly concave and satisfies the Inada conditions. The first order conditions for this intertemporal optimization problem are
ul;t ¼ 1 st ; uc;t
8t P 0
phtþ1 ¼ bpðstþ1 jst Þ
uc;tþ1 ; uc;t
ð4Þ
8t P 0; 8st ; stþ1 2 S
ð5Þ
along with the household’s flow budget constraint (2) satisfied with equality. Condition (4) equates the marginal rate of substitution between leisure and consumption with the net return to the household from supplying one extra unit of labor. The right-hand side of expression (5) represents the one-period asset pricing kernel (or discount factor) for a state-contingent financial security purchased at time t and maturing in the following period. The government is faced with an exogenous stream of stochastic and wasteful expenditures denoted by g t . Following Aiyagari et al., we assume g t 2 ½g min ; g max ; 8t P 0. Government expenditures can be financed either by taxing labor income g at the time-varying flat rate st or by issuing bonds btþ1 at time t which mature at time t þ 1. In contrast to the financial claims h g btþ1 purchased by the private sector, btþ1 are assumed to be nonstate-contingent; that is, they yield a rate of return that is independent of the state of nature in the economy. The government’s flow budget constraint can be written as g
g
pgt btþ1 ¼ bt þ g t st nt ; where, in addition,
pgt ¼
Z
pgt
8t P 0
ð6Þ 10
is the price of a government bond measured in units of time-t output.
phtþ1 dstþ1 ;
8t P 0
We assume that
ð7Þ
which ensures that there are no arbitrage opportunities in the international bonds market. For simplicity, we assume pgt ¼ b; 8t P 0, so that the government borrows and lends at a constant risk-free price equal to b. Finally, in order to ensure a sustainable and meaningful optimal fiscal policy, the following asset and debt constraints are imposed on the government: g
B 6 bbtþ1 6 B;
8t P 0
ð8Þ
where B and B denote the government’s maximum level of indebtedness and maximum level of asset holdings, respectively. These limits could be of two types: ‘‘natural” limits and ‘‘ad hoc” limits; see Aiyagari (1994). The ‘‘natural” debt limit represents the maximum level of indebtedness for which debt can be paid back with probability 1, under some tax policy. Since the tax policy depends on the stochastic process for g t , the natural debt limit is itself a random variable. The ‘‘natural” asset limit represents the minimum level of assets beyond which the government has no incentive to accumulate additional assets, since it can finance all of its expenditures entirely through the interest on its assets, even if g t takes its highest possible value in all periods.11 In contrast, a debt or asset limit is called ‘‘ad hoc” if it is more stringent than a natural one. In what follows, we define the concepts of feasible allocations and competitive equilibrium, and use them to characterize the government or Ramsey problem. g
h
Definition 1. Given b0 ; b0 , a stochastic process for government expenditures fg t g, and a stochastic process for stateh contingent financial claims prices fphtþ1 g satisfying (7), a feasible allocation is a stochastic process fct ; nt ; g t ; btþ1 g satisfying the economy’s flow budget constraint
10 The existence of a market of riskless government bonds has no effect on the trading opportunities of the representative household. To see this, consider the g simultaneous purchase of btþ1 units of output of Arrow securities corresponding to each possible state of nature. This transaction will yield next period a payoff g of btþ1 units of output regardless of stþ1 . However, this payoff is identical to that of a government bond. 11 In general, natural debt and asset limits are difficult to compute. In the present model we can derive an analytic expression for the natural asset limit by g g considering the government’s flow budget constraint (6) with zero tax revenues at the maximum level of government expenditure. Letting btþ1 ¼ bt ¼ B=b and solving for B implies that natural asset limit for the government is B ¼ g max ðb=ð1 bÞÞ.
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
Z g g h h ct þ g t ¼ nt ðbt bbtþ1 Þ þ bt phtþ1 btþ1 dstþ1 ;
ð9Þ h
g
g
whose time t elements are measurable with respect to ðst ; b0 ; b0 Þ. A government policy fst ; btþ1 g is a stochastic process whose h g time t elements are measurable with respect to ðst ; b0 ; b0 Þ. h
g
Definition 2. Given b0 ; b0 , a stochastic process fg t g, and a stochastic processes fphtþ1 g satisfying (7), a competitive equilibrium is a feasible allocation and a government policy that solve the household’s optimization problem and satisfy the government’s budget constraints (6) and (8). The government’s or Ramsey problem is to maximize (1) over competitive equilibria. We follow the primal approach as in Atkinson and Stiglitz (1980) in order to characterize the competitive equilibrium with distortionary taxes. The basic idea is to recast the problem of choosing the optimal fiscal policy as a problem of choosing feasible allocations that can be supported as a competitive equilibrium for some choice of taxes and bond issue. Then, combining the first order conditions for the government’s problem with those of the household’s one can determine the optimal tax rates. In the present context, the Ramsey problem, can be formally stated as follows:
max
ct ;lt ;nt ;bgtþ1 ;bhtþ1
f
1
gt¼0
W ¼ E0
1 X
bt uðct ; lt Þ
ð10Þ
t¼0
subject to:
1 X uc;tþj ctþj ul;tþj ntþj bj ; 8t P 0 uc;t j¼0 ul;t g g bbtþ1 ¼ bt þ g t 1 nt ; 8t P 0 uc;t uc;t ; 8t P 1 pht ¼ bpðst jst1 Þ uc;t1 h
bt ¼ Et
g
B 6 bbtþ1 6 B; lt þ nt ¼ 100; h g b0 ; b0
8t P 0 8t P 0
ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ
given:
Eq. (11) represents the household’s present value budget constraint as of time t. It is derived by combining conditions (4) and h (5) with (2), and using forward substitution along with the transversality condition for btþ1 given by h
lim bTþ1 Et uc;tþTþ1 btþTþ1 ¼ 0:
T!1
This conditional expectation must hold for any t with probability one. The reason for this is that the transversality condition is in the nature of a first-order necessary condition ‘‘at infinity” and is consistent with Bellman’s principle of optimality for any arbitrary t. Eq. (12) is derived by solving first order condition (4) from the household’s problem for st and substituting it into the government’s flow budget constraint (6). Eq. (13) corresponds to first order condition (5) from the household’s problem. Finally, Eqs. (14) and (15) are the government’s asset and debt limit constraint (8) and the household’s time endowment constraint (3), respectively. The Ramsey allocation can be characterized using the Lagrange approach. However, since the system (10)–(15) is nonrecursive, we apply the technique of Marcet and Marimon (1999) in order to transform the problem into a recursive one, and then formulate the Lagrangian for the recursive problem. To this end, we first eliminate constraint (15) by solving it for lt and substituting it into (10). Next, let the multipliers bt /t ; bt kt ; bt h1t and bt h2t be attached to constraints (11), (12) and the left and right sides of (14), and let bt lt be the multiplier of constraint (13) which is rewritten as:
pht uc;t1 bpðst jst1 Þuc;t ¼ 0: Also, define a pseudo-multiplier ct as
ct ¼ /t þ ct1 ; 8t P 0
ð16Þ
with c 1 ¼ 0. Using these transformations, the Lagrangian function for the recursive Ramsey problem can be written as
uc;t ct ul;t nt h bt uðct ; 100 nt Þ þ ðct ct1 Þbt ct uc;t t¼0 1 X g u g g g l;t ^ht uc;t1 buc;t ; ^ t ½p bt l þkt bbtþ1 bt g t þ 1 nt þ h1t B bbtþ1 þ h2t bbtþ1 B þ E0 uc;t t¼1
L ¼ E0
1 X
ð17Þ
^ht pht =pðst jst1 Þ and l ^ t lt =pðst1 Þ. Appendix A discusses the solution of the Ramsey problem assuming that the where p instantaneous utility function of the representative household is separable in consumption and leisure:
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
uðct ; lt Þ ¼ tðct Þ þ Hðlt Þ;
8t P 0:
443
ð18Þ
Appendix B provides a detailed description of the computational scheme for this problem. 3. Numerical simulations of the model This section states the specific functional forms and parameter values that we use in the numerical simulations of the model. Following Aiyagari et al. the instantaneous utility function of the representative household is assumed to be
! 1r r1 c1 1 lt 2 1 t : uðct ; lt Þ ¼ þg 1 r1 1 r2
Government expenditures are assumed to follow the stochastic process
g tþ1 ¼ ð1 qÞg þ qg t þ
egtþ1 sg
;
ð19Þ
where g is the unconditional long-run mean of the process g t 2 ½g min ; g max ; 8t P 0; q is its persistence (autocorrelation) parameter, egtþ1 is an i.i.d. Nð0; 1Þ random variable, and sg is a nonrandom scale factor. Notice that the truncation of (19) prevents g t from being negative. ^htþ1 is assumed to be The stochastic process for p
ep ^htþ1 ¼ b 1 þ tþ1 ; p sp
ð20Þ
^htþ1 2 ½p ^hmin ; p ^hmax ; 8 t P 0. The specification in where eptþ1 is i.i.d. Nð0; 1Þ and sp is a nonrandom scale factor. We also assume p (20) ensures that the no-arbitrage condition (7) is satisfied. Multiplying both sides of (20) with pðstþ1 jst Þ, integrating over stþ1 and using the distributional assumption for eptþ1 yields (7) with pgt ¼ b; 8t P 0. The conditional expectations in the first order conditions of the Ramsey Problem were parameterized using the family of approximating functions mapping R4þ into Rþ :
wðai ; xt Þ ¼ expðPn ðxt ÞÞ; where P n denotes a polynomial of degree n with coefficient parameters ai , and xt is a 4 1 vector of the state variables: g 0 ^ht ; g t ; see Appendix A. In the numerical simulations we estimate the function wðai ; xt Þ by nonlinear least xt ¼ bt ; ct1 ; p squares. To ensure that all variables in the nonlinear least squares regressions were of similar orders of magnitudes, we followed Aiyagari et al. in applying to each state variable separately the function u : ðk; kÞ#ð1; 1Þ, where k and k are prespecified lower and upper bounds for the argument k of function u. The latter is defined as
uðkÞ ¼ 2
kk kk
1;
which implies that the wðÞ functions used to parameterize the conditional expectations that appear in the first order conditions of the Ramsey problem are of the form
wðai ; xt Þ ¼ expðPn ðuðxt ÞÞÞ: In order to carry out the simulations we set the following parameter values. Following Aiyagari et al., the preference paramg h eters were set at: r1 ¼ 0:5; r2 ¼ 2 and g ¼ 1, the initial indebtedness of the government and the household at b0 ¼ b0 ¼ 0, the long-run mean of government spending g ¼ 30; b ¼ 0:95; sg ¼ 0:40 and sp ¼ 800. We consider three types of government expenditures: (i) serially uncorrelated ðq ¼ 0:00Þ, (ii) serially correlated ðq ¼ 0:75Þ, and (iii) serially correlated expenditures whose volatility is adjusted to be equal to the one of uncorrelated expenditures. The reason for this adjustment is that the unadjusted correlated expenditures are more volatile compared to the uncorrelated expenditures. We verify this in our numerical simulations in Appendix B where the standard deviation of g is 2.50 in cases (i) and (iii), and 3.73 in case (iii). Adjusting the variance of the correlated expenditures, allows us to study the effect of the persistence in g on the equilibrium tax rate when comparing case (i) with case (iii). In all three cases for g, every realization of (19) is constrained to lie in the interval [20,40], and every realization of (20) is ^htþ1 before constrained to lie in the interval [0.92,0.98]. These bounds were also used for the transformed variables g t and p g using them as inputs in the parameterized expectations estimations. For btþ1 the bounds were set at B=b and B=b, while the ones for ct1 were set at [40, 100]. Finally, for each of the three types of stochastic processes for government expenditures, we consider three pairs of debt and asset limits ðB; BÞ: (0,0), which corresponds to the ‘‘balanced budget” case, (10,10) and (50,50).12 For each pair of limits and process for g, the Ramsey problem is solved numerically. The resulting 12 These limits are more stringent than the ad hoc limits (1000,1000) used by Aiyagari et al. in their numerical experiments. The reason is that, in the present model and its parameterization the natural debt limit turns out to be 760. As Aiyagari et al. show in their paper, when B is equal to the natural debt limit, the optimal policy is to accumulate assets and set the tax rate equal to zero. I experimented with an increasing range for ðB; BÞ. The result was that for a sufficiently wide range, the government finds it optimal to accumulate as many assets as it is allowed by B, while thetax rate gets close to zero. Given this outcome, I chose to consider a smaller range for ðB; BÞ because it yields equilibrium tax rates that are consistent with tax smoothing.
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
series of equilibrium tax rates are then compared in terms of their persistence and the implications of their dynamics for the welfare of the representative household. 4. Quantitative results In this section we report the quantitative predictions of the model for different government expenditure processes with alternative borrowing and lending constraints. These numerical results will be with regards to three important issues pertaining to our analysis. The first issue is the degree of tax smoothing achieved by the government for the various cases considered. The second issue is the distribution of government debt and asset holdings. The third issue is the welfare ranking for these different cases. 4.1. Tax smoothing Our quantitative results will be better understood if we first look at the series for the endogenous and exogenous variables generated from a representative simulation, as displayed in Figs. 1–4. Fig. 1 plots the series for the three types of gov^ht , and the two ernment expenditures, g t , and the common series for the price of state-contingent financial claims, p disturbance terms ðegt ; ept Þ. This figure allows a visual comparison of the series generated under the different specifications for g t . The volatility-adjusted correlated expenditures clearly exhibit more persistence compared to the uncorrelated expenditures and they also vary less relative to the volatility-unadjusted correlated g t . Figs. 2–4 consists of three panels of graphs (a–c) corresponding to the three ðB; BÞ pairs: (0,0), (10,10), (50,50), each in combination with the three types of g corresponding to: q ¼ 0:00; q ¼ 0:75 and q ¼ 0:75 with adjusted variance. Each panel plots the entire allocation series for consumption, leisure, tax rate, debt issue, budget surplus, the sum ct þ g t þ lt and the household’s financial wealth wt defined as
wt
Z
h
h
phtþ1 btþ1 dstþ1 bt ;
which, following from the budget constraint (2), reduces to wt ¼ ð1 st Þnt ct .
Gov. Expenditures: ρ = 0.00
Gov. Expenditures: ρ = 0.75
40
40
35
35
30
30
25
25
20
2000 4000 6000 8000 10000 Gov. Expenditures: ρ = 0.75 - Adj. Var(g)
40
20
8000
10000
2000
4000
6000
8000
10000
2000
4000
6000
8000
10000
0.952
30
0.95
25
0.948 0.946 2000
4000
ε
g
6000
8000
10000
5
5
0
0
-5
4000 6000 Asset Prices
0.954
35
20
2000
2000
4000
6000
8000
10000
-5
εp
Fig. 1. Government expenditures, asset prices and error terms.
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a
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Fig. 2(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:00.
Regarding the variable ct þ g t þ lt , notice that in the closed economy model of Aiyagari et al. this variable is equal to the time endowment ðT ¼ 100Þ in every period. This contrasts with the corresponding constraint for the current open economy model as given by Eq. (9). As shown in Figs. 2–4, making the ðB; BÞ range wider causes the equilibrium allocation series for leisure, tax revenue, the variable ct þ g t þ lt and household wealth to become less volatile, while there is minimal change with respect to consumption.13 In contrast, the government surplus and public debt series become more volatile as the range ðB; BÞ widens. These results are common for all three specifications of (19). Furthermore, relaxing the government’s borrowing and lending constraints causes the equilibrium tax rates to become increasingly smoother and persistent, regardless of the underlying serial correlation properties of g. We can show this quantitatively by comparing estimates of the autoregressive parameter, b, from the regression model
st ¼ a þ bst1 þ ut ;
ð21Þ
13 This is due to first order condition (33) of the Ramsey problem (see Appendix A). This condition implies that once c0 is determined, ct evolves as a function ^h Þ1=r1 ; 8t P 1, while the series for p ^ h remains common across the various cases for g and ðB; BÞ. of the ratio ðb=p
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b
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4 3.8
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3.6 3.4
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8000
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90 80
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4000
6000
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4000
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Fig. 2(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:00.
where a is constant and ut is a least-squares residual that is assumed to be orthogonal to st1 . The coefficient b measures the persistence in s. The closer the estimated values of a and b are to 0 and 1, respectively, the more ‘‘tax smoothing” is achieved by the government. Table 1 presents the first two unconditional moments of s, the estimated intercept terms and slope coefficients, and the R2 statistics. In general, as the government’s borrowing and lending range becomes wider, the estimated value of the coefficient of the lagged equilibrium tax rate increases and gets closer to 1. In other words, providing the government with more flexibility in terms of the amounts that it can borrow and lend leads to an increase in the persistence of the equilibrium tax rate. Furthermore, raising the asset and debt limits causes the estimated value of the intercept term in (21) to decline and move closer to 0. At the same time, the goodness of fit of the autoregression improves with the value of the R2 statistic increasing and getting closer to 1. When the ðB; BÞ range is sufficiently narrow, the estimated persistence of the equilibrium tax rates, the estimated intercept term and the goodness of fit of Eq. (21) are strongly influenced by the serial correlation properties of the underlying process for government expenditures. For example, in the period-by-period balanced budget case with serially uncorrelated government expenditures, the equilibrium tax rates do not exhibit any persistence ðb ¼ 0:000927Þ, the estimated intercept ða ¼ 0:310521Þ is essentially equal to the average tax rate (0.310812), while the value of the R2 statistic is zero. On the other
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Leisure
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60
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120 0
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90 80
2000
4000
6000
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4000
6000
8000
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Fig. 2(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:00.
hand, when government expenditures are serially correlated, the equilibrium tax rates ‘‘inherit” their persistence: the estimated autoregressive parameter b takes on a value close to q ¼ 0:75ðb ¼ 0:745718Þ. Predictably, the fit of the autoregression improves relative to the uncorrelated case ðR2 ¼ 0:556206Þ, while the estimated value of a falls ða ¼ 0:079023Þ. These results are not surprising, since the optimal tax rate under a balanced budget rule is a function of g alone and, therefore, must have correlation properties similar to the ones for government expenditures.14 The variation of the average tax rate across the different cases reflects the government’s borrowing and lending behavior as the ðB; BÞ range becomes wider. Table 2 presents statistics that help describe this behavior. These statistics are the gov-
14 Aiyagari et al. consider only the case of serially uncorrelated government expenditures. Comparing their results from estimating (21) with the results reported in Table 1 for the corresponding case ðq ¼ 0:00Þ when ðB; BÞ ¼ ð50; 50Þ, suggests that their model yields equilibrium tax rates that are closer to the predictions of Barro’s tax smoothing theory. Their estimated a and bcoefficients are closer to 0 (0.0031 vs. 0.0252) and 1 (0.9888 vs. 0.9172), respectively, while the estimated std ½s is of similar magnitude (0.0191 vs. 0.0113). However, one has to take into account that theborrowing and lending range considered by Aiyagari et al. is significantly wider, which provides the government in their case the additional flexibility needed in achieving a higher degree of tax smoothing. Increasing the asset and debt limits in my case beyond the (50,50) range, at least marginally, generates tax rates that become closer in resembling arandom walk. Finally, it should be noted that a comparison of my results with those of Aiyagari et al. based on a formal statistical test is hindered by the fact that they do not report the standard errors of their estimates for (21).
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a
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Leisure
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3.6 3.4
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90 80
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4000
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Fig. 3(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:75.
ernment’s borrowing frequency (the percentage of time it borrows), the percentage of time the asset and debt limit constraints are binding, as well as the average asset and debt levels. When the government is restrained in terms of the amounts it can borrow and lend, it tends to be on average a lender rather than a borrower. This results in the average tax rate being lower compared to the balanced-budget case. In contrast, when the government’s borrowing and lending range becomes wider, the frequency by which it borrows increases, leading to a rise in the average debt level. The higher level of debt causes the average tax rate to increase relative to the case of the narrow ðB; BÞ range. The observed pattern in the government’s borrowing frequency can be interpreted in terms of a precautionary savings motive on its part. If the maximum amount that it is allowed to borrow is only 10, the government has a strong incentive to accumulate assets when its expenditures are low, i.e. in the ‘‘good times”. The reason is that the government knows that during the ‘‘bad times”, when its expenditures are high, it will have to raise taxes to finance its spending because of its limited ability in terms of the amount of debt it can issue. However, raising taxes creates a distortion in the economy. In order to minimize it, the government optimally chooses to accumulate sufficiently large amounts of assets during the ‘‘good times” given the constraint that its asset limit is 10. The interest earned from holding these assets helps finance expenditures during the ‘‘bad times”, thus reducing the amount by which taxes would have to increase otherwise. On the other hand, when its borrowing and lending range becomes wider (i.e., ðB; BÞ ¼ ð50; 50Þ), the government enjoys more flexibility in satisfying its
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3.6 3.4
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4000
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Fig. 3(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:75.
precautionary savings motive: it can now accumulate more assets during the ‘‘good times”, but it can also borrow more during the ‘‘bad times”. With the debt limit set at 50, there is less pressure for the government to accumulate assets compared to the ðB; BÞ ¼ ð10; 10Þ case, resulting in an increase in its borrowing frequency. Fig. 5 displays impulse response functions for leisure, tax rate, tax revenue, budget deficit, public debt and household wealth to a one unit positive innovation in government expenditures. Panel (a) corresponds to the case ðB; BÞ ¼ ð10; 10Þ, while Panel (b) refers to the case ðB; BÞ ¼ ð50; 50Þ. Each panel presents the impulse response functions of the various endogenous variables for each of the three specifications of (19) discussed above.15 As shown in Fig. 5, the increase in government expenditures causes the tax rate to rise. However, the impact response of s is smaller relative to the balanced budget case. This is shown by the fact that the resulting increase in tax revenues is not sufficient to finance the higher g entirely, leading the government to incur a budget deficit. The shortfall in revenues is financed through debt issue. Also, in the present model, the higher tax rate results to an increase in leisure. In combination with the higher tax rate, the reduction in the labor supply causes the representative household’s wealth to decline.
15 Recall that consumption evolves as a function of a ratio that is parametric to both the household and the government (see Footnote 13). For this reason, impulse response functions for consumption and the variable ct þ g t þ lt are not displayed.
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Fig. 3(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:75.
These impulse response functions are qualitatively similar to the ones reported by Aiyagari et al. with the exception of leisure: in our model a positive shock in government expenditures causes leisure to rise, while in their model leisure falls. The reason behind this result is related to the following two effects. First, a positive innovation in government expenditures leads the government to raise the tax rate, which results in a reduction of the net wage and labor supply (the ‘‘net wage effect”). Second, the increase in government expenditures leads to an increase in borrowing, which raises the interest rate. This results to an increase in labor supply through the intertemporal substitution of leisure (the ‘‘interest rate effect”). In Aiyagari et al., the second effect is the dominant one. As a result, in their case leisure falls and labor supply rises. In the present small open economy framework, however, the interest rate effect is absent since consumption is determined purely by the (exogenous) stochastic process for the prices of the state-contingent claims. As a consequence, a positive innovation in government expenditures leads to a reduction in labor supply and an increase in leisure. As the ðB; BÞ range increases, the impact response of leisure, tax rate, tax revenue and household wealth becomes smaller, while the response of the budget deficit and public debt becomes larger. This is the case for all three specifications for g t and it reflects the fact that the increased borrowing and lending capacity of the government allows it to better distribute the distortionary cost of taxation over time.
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Consumption
Leisure
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Fig. 4(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:75 [Adjusted Var (g)].
4.2. Government asset and debt holdings Fig. 6 displays the distribution of the government’s asset and debt holdings for the two ðB; BÞ unbalanced budget pairs. Along with Table 2, these histograms provide a concise summary of how the government engages in precautionary savings. When the borrowing and lending range is narrow (Fig. 6a, the distribution of asset/debt holdings is skewed for all three types of government expenditures. The asset limit constraint is binding less frequently when q ¼ 0:00 compared to the two cases of serially correlated expenditures. This indicates that the government’s precautionary savings motive is stronger when q ¼ 0:75. A possible explanation for this result may be related to the more ‘‘permanent” nature of a negative government spending shock in the latter case. That is, a high level of g in the current period is likely to be accompanied by high expenditure levels in the following periods that will eventually prompt the government to raise the tax rate. This provides the government with a strong incentive to accumulate assets that will contribute in minimizing the resulting distortion. With serially uncorrelated expenditures, it is less likely that a negative spending shock will be followed by high levels of g in subsequent periods. As a consequence, the government feels less pressure to save during the ‘‘good times”, resulting in the asset limit constraint binding less frequently in this case. Fig. 6b shows that widening the government’s borrowing and lending range causes a dramatic shift in the distribution of its asset/debt holdings. With its incentive to accumulate assets weakened, the government borrows more resulting in the
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b
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Leisure
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Fig. 4(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:75 [Adjusted Var (g)].
distribution of its asset/debt holdings to no longer be skewed towards its asset limit, but rather be fairly symmetric and ceng tered around the initial level of indebtedness b0 ¼ 0. These results have important implications regarding the degree of tax smoothing achieved by the government. As shown in Table 1, the government’s borrowing and lending behavior has an impact on the volatility of the equilibrium tax rates as well. In general, as the ðB; BÞ range increases from (0,0) to (10,10), and then to (50,50), the standard deviation of the tax rate std ½s declines and the tax rates become smoother for all three types of government expenditures. Further, for a given ðB; BÞ range, a larger variability in g leads to a higher standard deviation in s: when q ¼ 0:75, std ½s is higher compared to the cases of the serially uncorrelated and volatility-adjusted serially correlated government expenditures. For example, consider the range (50,50): std ½s is 0.02956 when q ¼ 0:75 (unadjusted), 0.020537 when q ¼ 0:75 (adjusted), and 0.011302 when q ¼ 0:00. Finally, adjusting the variance of serially correlated expenditures still yields equilibrium tax rates that exhibit higher volatility relative to those obtained for serially uncorrelated expenditures. For instance, in the case of the range (10,10), std ½s is 0.034535 when q ¼ 0:75 (adjusted), and 0.014661 when q ¼ 0:00. This reflects the more ‘‘permanent” nature of the correlated spending shocks discussed earlier. This evidence makes clear that allowing the government to borrow and lend leads to smoother tax rates compared to the case of maintaining a balanced budget, regardless of the serial correlation properties of the underlying process for govern-
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Fig. 4(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:75 [Adjusted Var (g)].
ment expenditures. It also suggests that there should be welfare gains for the representative household from the ‘‘tax smoothing” achieved by the government. In what follows, we consider how significant these welfare gains are. 4.3. Welfare analysis Table 3 reports the computed welfare values associated with all cases under study. The separability assumption regarding the representative household’s instantaneous utility allows us to distinguish between the welfare gains in terms of expected utility of consumption and leisure. Regardless of the serial correlation properties and volatility of government expenditures, welfare clearly improves from moving away from the period-by-period balanced budget constraint not only in terms of the total, but also in terms of consumption and leisure separately. As shown in the table, when the government is not permitted to borrow or lend the case that yields the highest welfare is that of the volatility-adjusted serially correlated g: total expected utility equals 305.7612. This is because the tax rate is influenced by the properties of government expenditures, which, in this case, are persistent and have low variance. In contrast, the lowest welfare in the balanced budget case is attained when q ¼ 0:00: total expected utility is equal to 305.6575. The complete lack of persistence in s outweighs the positive effect of the lower variability in g, thus ranking this case below that
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Table 1 Autoregressions of the tax rate ðB; BÞ
q ¼ 0:00 E½s std ½s a b R2
q ¼ 0:75 E½s std ½s a b R2
q ¼ 0:75 (Adj. Var(g)) E½s std½s a b R2
(0,0)
(10,10)
(50,50)
0.310812 (4.95083E-05) 0.02612 (1.68775E-05) 0.310521 (0.000353928) 0.000927 (0.001123558) 0.000121 (2.22609E-05)
0.309281 (0.00033999) 0.014661 (0.000292308) 0.15626 (0.005429174) 0.493288 (0.018086705) 0.275826 (0.016017117)
0.310808 0.011302 0.025241 0.917216 0.853218
0.310778 (0.000108633) 0.038987 (4.57851E-05) 0.079023(0.000206123) 0.745718 (0.000670511) 0.556206 (0.00099916)
0.308767 (0.000273902) 0.034535 (0.000136991) 0.050973 (0.000823633) 0.834713 (0.002795309) (0.004591606) 0.69759
0.311822 (0.000651786) 0.02956 (0.000111808) 0.026443 (0.000310484) 0.914964 (0.00118193) 0.837378 (0.002126377)
(7.99937E-05) 0.31077 (3.46008E-05) 0.026126 0.077718 (0.000218999) (0.000709507) 0.749917 0.562498 (0.001061168)
0.308885 (0.00028298) 0.022795 (0.000116778) 0.047133 (0.00101784) 0.847138 (0.003424574) 0.71893 (0.005703914)
0.310625 0.020537 0.029746 0.904042 0.817707
(0.000975966) (0.00024071) (0.003124988) (0.010758395) (0.015567273)
(0.00054263) (9.3308E-05) (0.000326213) (0.001208596) (0.002168645)
Note: Standard errors in parentheses.
Table 2 Government Borrowing and Lending Behavior Borrowing Frequency (%)
Binding Asset Limit (%)
Binding Debt Limit (%)
Average Assets
Average Debt
37.1973 (2.81602) 27.2277 (1.99281) 29.1935 (2.42449)
12.9266 (1.72029) 20.6868 (2.14007) 21.5661 (2.33242)
2.0023 (0.28077) 2.8820 (1.94100) 2.5380 (0.34823)
5.1368 (0.19662) 6.2217 (0.16964) 6.063754 (0.19566)
3.5538 (0.12213) 4.1580 (0.08128) 3.8108 (0.10803)
51.4797 (3.30006) 53.2361 (2.41591) 50.7788 (2.85682)
1.5885 (0.61585) 0.4309 (0.30992) 0.2686 (0.20543)
0.1719 (0.04936) 0.3952 (0.05956) 0.0015 (0.00073)
12.8915 (0.903050) 10.9416 (0.55749) 8.1180 (0.54192)
12.9978 (0.73473) 13.5692 (0.38339) 8.1909 (0.28579)
ðB; BÞ ¼ ð10; 10Þ
q ¼ 0:00 q ¼ 0:75 q ¼ 0:75 (Adj. Var(g)) ðB; BÞ ¼ ð50; 50Þ
q ¼ 0:00 q ¼ 0:75 q ¼ 0:75 (Adj. Var(g))
Note: Standard errors in parentheses.
of the volatility-unadjusted correlated expenditures in terms of welfare. In this case, total expected utility is equal to 305.7144. Once the period-by-period balanced budget constraint is relaxed, total utility and its components invariably increase. Providing the government with the ability to borrow and lend allows it to maintain persistent and smooth tax rates, thus reducing the distortionary effect of taxation. This leads to higher expected utility of consumption and leisure. The separability between consumption and leisure in the utility function facilitates a welfare comparison in terms of actual consumption and leisure of the representative household. Table 4 shows the required increase in consumption per period that will make the representative household indifferent between the balanced-budget policy regime and the one in which the government can borrow or lend.16 When q ¼ 0:00 and the government is allowed to borrow and lend in the range (10,10), the increase in consumption is approximately 0.2377% in all periods. For the volatility unadjusted and adjusted serially correlated government expenditures, this number drops to 0.1304% and 0.1155%, respectively. The magnitudes of the resulting welfare gains reflect the relative ranking of each case in terms of welfare when the government is forced to maintain a period-by-period balanced budget. As discussed earlier, the case of serially uncorrelated expenditures yields the lowest utility under the balanced budget constraint compared to the other two cases. Therefore, with q ¼ 0:00 there is plenty of room for improvement in welfare. On the other hand, in the case of serially correlated expenditures, the tax rate is already quite persistent and the (10,10) range is not wide enough to generate larger welfare gains. When the government is allowed to borrow and lend in the (50,50) range, the resulting welfare gains relative to the balanced-budget policy regime increase further: 0.5592% when q ¼ 0:00, 0.4064% when q ¼ 0:75, and 0.5368% for the vol16
The calculation of the reported welfare gains is described in Appendix B.
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a
x 10
-3
Leisure
x 10
8
6
6
4
4
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2
2 0 0
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10 15 Tax Revenue
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25
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20
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20
25
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0.2 0 0 0
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Debt 1 0 -0.2 0.5 -0.4 -0.6 0
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15
Fig. 5(a). ðB; BÞ ¼ ð10; 10Þ - Impulse response functions.
atility-adjusted correlated g. These values indicate that the strong persistence of the spending shocks in the serially correlated cases hinders the government’s effort in smoothing taxes further. This can be seen from Table 1 by comparing the volatility of s in these two cases relative to the serially uncorrelated one. Reducing std ½s further, and thus improving welfare relative to the balanced budget case, would require that the ðB; BÞ range becomes even wider: less for the volatility-adjusted case and more for the unadjusted one. The reason is that the former type of expenditures exhibits lower variability compared to the latter and, therefore, the government would find a narrower ðB; BÞ range sufficient for smoothing taxes. Finally, as part of a preliminary sensitivity analysis, we considered different values for the two preference parameters. In particular, we set r1 ¼ 0:6 and r2 ¼ 2:2. Although in either case the results obtained are quantitatively different, qualitatively they are similar in nature to the ones for the benchmark parameterization.17 5. A comparison of incomplete with complete markets In this section we compare the welfare gains in the incomplete markets case with the complete markets case in which the government is also allowed to issue state-contingent debt. In order to facilitate computations in the incomplete markets case and be able to derive an analytic solution for the equilibrium allocation in the complete markets case,18 we make the simplifying assumption that the normalized asset prices are constant
^htþ1 ¼ b: p
ð22Þ
Eq. (22) is a special case of Eq. (20) with sp ! 1. In the complete markets case, the household’s problem is the same as before, described by expressions (1)–(3), and we assume that the utility function is given by Eq. (18). The government’s flow budget constraint (6) is replaced by
17
These results are available from the author upon request. Þ, as it is the If I were to use (20) for the complete markets case, then I would have to guess two multipliers (see multipliers U and K below) instead of one ðc case under incomplete markets. This would make the numerical scheme described in Appendix B computationally much more intensive. 18
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
b
x 10
-3
Leisure
x 10
8
6
6
4
4
-3
Tax Rate
2
2 0 0
0
5
10 15 Tax Revenue
20
25
0.6
1
0.4
0.5
0
5
10 15 Budget Deficit
20
25
0
5
10 15 Household Wealth
20
25
0
5
20
25
0.2 0 0 0
5
10
15
20
25
Debt 1 0 -0.2 0.5 -0.4 -0.6 0
0
5
10
15
20
25
10
15
Fig. 5(b). ðB; BÞ ¼ ð50; 50Þ - Impulse response functions.
Z
g
g
phtþ1 btþ1 dstþ1 ¼ bt þ g t st nt ;
8t P 0
ð23Þ
which states that the government finances its expenditures using a labor income tax at the time-varying flat rate st and the g sale of state-contingent bonds btþ1 at the exogenously determined prices phtþ1 for each possible event stþ1 . As before, these asset prices are measured in units of time-t output and are assumed to be determined by the international financial markets. In this case, the Ramsey problem becomes
max1 E0
fct ;nt gt¼0
1 X
bt ½tðct Þ þ Hð100 nt Þ
ð24Þ
t¼0
subject to:
t0 ðc0 Þbh0 ¼ E0
1 X
bt ½t0 ðct Þct H0 ð100 nt Þnt ;
ð25Þ
t¼0
pht ¼ bpðst jst1 Þ
t0 ðc0 Þbg0 ¼ E0
t0 ðct Þ ; 8t P 1 t0 ðct1 Þ
1 X
bt f½t0 ðct Þ H0 ð100 nt Þnt t0 ðct Þg t g;
ð26Þ ð27Þ
t¼0 h
g
and b0 ; b0 given. Constraint (25) is the time-0 version of the household’s present value budget constraint (11). The right-hand side of constraint (26) is the one-period asset pricing kernel. Eq. (27) is the government’s present value budget constraint. It is derived by substituting (4) and (5) into (23), and using forward substitution along with the transversality condition on g btþ1 . Given (22), optimality condition (26) reduces to t0 ðct Þ ¼ t0 ðct1 Þ, which implies that the household achieves perfect consumption smoothing: ct ¼ c; 8t P 0. Using this result, the above Ramsey problem reduces to maximizing (24) subject to
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 30
rho = 0.00
rho = 0.75
rho = 0.75 (Adj. Var(g))
Asset/Debt Holdings (%)
25
20
15
10
5
0 [-10,-9) [-9,-8) [-8,-7) [-7,-6) [-6,-5) [-5,-4) [-4,-3) [-3,-2) [-2,-1)
[-1,0)
(0,1]
(1,2]
(2,3]
(3,4]
(4,5]
(5,6]
(6,7]
(7,8]
(8,9]
(9,10]
Asset and Debt Limit Range Fig. 6a. Histogram of asset/debt holdings – range (10,10).
18
16
rho = 0.00
rho = 0.75
rho = 0.75 (Adj. Var(g))
(10,15]
(20,25]
(30,35]
Asset/Debt Holdings (%)
14
12
10
8
6
4
2
0 [-50,-45) [-45,-40) [-40,-35) [-35,-30) [-30,-25) [-25,-20) [-20,-15) [-15,-10) [-10,-5)
[-5,0)
(0,5]
(5,10]
(15,20]
(25,30]
(35,40]
(40,45]
(45,50]
Asset and Debt Limit Range Fig. 6b. Histogram of asset/debt holdings – range (50,50).
(25) and (27). Attaching multipliers U and K to constraints (25) and (27), respectively, the first order conditions with respect to nt ; c; U and K are given by:
H0 ð100 nt Þð1 U KÞ H00 ð100 nt Þnt ðU þ KÞ Kt0 ðcÞ ¼ 0; 8t P 0 " # 1 X t0 ðcÞ 1 g t 00 h 00 0 00 þ U t ðcÞb0 ðt ðcÞc þ t ðcÞÞ þ Kt ðcÞ b0 þ E0 b ðg t nt Þ ¼ 0; 1b 1b t¼0
ð28Þ
; 8t P 0. along with (25) and (27). Since c; U and K are constant, for condition (28) to hold it must be the case that nt ¼ n ; 8t P 0. Condition (4) from the household’s problem then implies that st ¼ s In order to facilitate the implementation of a welfare comparison between the current complete markets case, and the incomplete markets case, we keep the same parameterization and numerical scheme discussed in Section 3. The only excep-
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Table 3 Welfare Values ðB; BÞ
(0,0)
(10,10)
(50,50)
291.4932853 (0.013817863) 14.16421793 (0.000202164) 305.6575032 (0.013941728)
291.8835986 (0.013832602) 14.16777129 (0.000205043) 306.0513699 (0.013958878)
292.4060292 (0.013874024) 14.17696593 (0.000204682) 306.5829951 (0.014001694)
291.548406 (0.013820154) 14.16601672 (0.000567954) 305.7144227 (0.013959256)
291.7623963 (0.013827545) 14.16818736 (0.000572013) 305.9305836 (0.013970694)
292.2104451 (0.01383665) 14.17700934 (0.000571063) 306.3874544(0.013977294)
291.596178 (0.013822958) 14.16508113 (0.000392043) 305.7612591 (0.013958741)
291.7853859(0.013834417) 14.16732935 (0.000394198) 305.9527152 (0.013972074)
292.4711116 (0.013849848) 14.17892195 (0.000393016) 306.6500336 (0.013984458)
q ¼ 0:00 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 (Adj. Var(g) Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility Note: Standard errors in parentheses.
Table 4 Welfare Gains as a % Increase in Consumption per Period From
To
q ¼ 0:00
q ¼ 0:75
q ¼ 0:75 (Adj. Var(g)
(0,0)
(10,10)
(0,0)
(50,50)
0.237773% (1.296632E05) 0.559157% (6.715276E05)
0.130437% (1.301744E05) 0.406405% (1.277812E05)
0.115509% (8.557878E06) 0.536777% (4.917141E06)
Note: Standard errors in parentheses.
Table 5 Welfare Values Fiscal Policy Regime
Balanced Budget
Incomplete Markets
Complete Markets
291.6063904 (0.001365166) 14.1652645 (0.000155885) 305.7716549 (0.001316528)
293.0216887 (0.000227834) 14.18384096 (0.000163744) 307.2055297 (0.000236365)
293.7756740 (0.009132014) 14.18469800(7.99113E-05) 307.9603720 (0.009052161)
291.4581813 (0.00136542) 14.16529552 (0.000548679) 305.6234768 (0.001246853)
292.6766109 (0.000568209) 14.18057163 (0.000560944) 306.8571825 (0.000622881)
293.7719290 (0.031419772) 14.18472600 (0.000274935) 307.9566550 (0.031144854)
q ¼ 0:00 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility Note: Standard errors in parentheses.
tion is the scale factor sp in (20), whose value is raised to 75,000.19 Finally, for the incomplete markets case, we consider only the range ðB; BÞ ¼ ð50; 50Þ, which is the most favorable range from a welfare point of view. Table 5 reports the computed welfare values associated with three cases: (i) complete markets, (ii) incomplete markets and (iii) balanced-budget with serially uncorrelated and serially correlated (volatility unadjusted) government expenditures. Compared to the corresponding cases in Table 3, it is clear that shutting down one source of uncertainty - the one associated ^htþ1 – generates welfare gains in terms of consumption and leisure in both the incomplete markets and balanced-budwith p get cases. Nonetheless, both are dominated in terms of total welfare by the complete markets case. These results hold for both types of government expenditures. Table 6 demonstrates that when q ¼ 0:00, the consumption expenditures of the representative household have to increase by 0.453850% in all periods in order to make it indifferent between the incomplete and complete markets cases. This increment in consumption should be 1.324432% in all periods in order to result in indifference between complete markets and a balanced budget. For the representative household to be indifferent between a balanced budget at all times and a government that is allowed to issue state-contingent debt, when g is serially correlated, its consumption expenditures have to
19
^ht is approximately equal to b in all periods. This is a large enough number to ensure that p
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
459
Table 6 Welfare Gains as a % Increase in Consumption per Period From
To
q ¼ 0:00
q ¼ 0:75
Balanced Budget Incomplete Markets
Complete Markets Complete Markets
1.324432% (0.000529) 0.453850% (0.000551)
1.412866% (0.001892) 0.662165% (0.001899)
Note: Standard errors in parentheses.
increase by 1.412866% in all periods. This increment in consumption spending has to fall to 0.662165% in all periods to result in indifference between the incomplete and complete markets cases. Overall, the effect of assumption (22) about asset prices on the welfare gains of moving away from a fiscal policy regime that restricts the government to maintain a balanced budget at all times towards a regime that allows it to borrow and lend roughly 50% of its GDP, is stronger when q ¼ 0:00. In this case, consumption should increase by 0.870582% in all periods in order for the representative household to be indifferent between the incomplete markets and balanced-budget cases. With serially correlated government expenditures this increment in consumption is only 0.750701% in all periods. This is consistent with the result obtained in Section 4 where we allowed for substantial variation in consumption expenditures. The above results imply that the improvement in welfare relative to a balanced budget from the existence of state-contingent public debt is smaller when q ¼ 0:75 compared to the case of serially uncorrelated government expenditures. Finally, comparing the complete and incomplete markets cases in the present context, the welfare gains of state-contingent government debt are higher when government expenditures are serially correlated. As it was shown in the previous section, the welfare gains from allowing the government to issue nonstate-contingent debt fall when the degree of persistence in government expenditures increases. Hence, it is not surprising that the welfare gains from complete relative to incomplete markets increase in the present case compared to the serially uncorrelated one. 6. Conclusions This paper considered a small open economy in which only households have access to complete financial markets. On the other hand, the government is constrained to borrow and lend at a constant and exogenously determined risk-free interest rate. Allowing the policymaker to borrow and lend large amounts yields a Ramsey equilibrium that replicates Barro’s ‘‘taxsmoothing” result regarding the conduct of optimal fiscal policy in the presence of uncertainty. Considering three different stochastic processes for government expenditures, it is shown that relaxing the policymaker’s borrowing and lending constraints allows more flexibility to smooth tax rates across time, thus reducing the distortionary effect of taxation and improving social welfare. This is in contrast to a fiscal policy regime, in which the policymaker is constrained to maintain a balanced budget at all times. Allowing the government to issue state-contingent debt and given a specific stochastic process for the prices of the statecontingent financial claims, it is shown that overall welfare improves relative to the incomplete markets and balanced budget cases. Furthermore, compared to the incomplete markets case, the resulting welfare gains appear to be higher as the persistence in the stochastic process for government expenditures increases. It would be interesting to work out several extensions of the current model. For instance, the welfare comparison between the complete markets case with the incomplete markets and balanced-budget cases was based on assumption (22) about asset prices. This assumption allowed us to derive an analytic solution for the equilibrium allocation in the complete markets case. Future work should appropriately modify this numerical scheme and solve the complete markets model under assumption (20). This will enable us to get a sense of how the welfare gains resulting from issuing state-contingent debt depend on the assumed asset pricing process. Furthermore, given the overall focus of the paper on welfare, sensitivity analysis is especially important regarding the exogenous process for the price of state-contingent claims (20). It would also be interesting to consider preferences such as the ones discussed in Barro and Sala-i-Martin (2004, p. 423), that are consistent with the empirical regularities observed in the growth literature, as well as technology shocks in addition to stochastic government expenditures as in Marcet and Scott (2003). The basic model of this paper can be extended by endogenizing the choice of government expenditures and adding a maturity structure in the issue of public debt. These modifications will not only add more realistic features to the present setup, but will also facilitate a quantitative exercise in which the model is calibrated to an actual economy. Finally, an interesting question that needs to be addressed is whether the welfare loss resulting from a balanced budget rule is higher or lower in a small open economy compared to a closed one. Appendix A. Solving the Ramsey problem This appendix provides a description of the computational scheme involved in solving the Ramsey problem numerically. The decision variables at the initial period t ¼ 0 are
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 g
h
y0 ¼ ½c0 ; n0 ; l0 ; b1 ; b1 ; c0 ; /0 ; k0 ; h10 ; h20 0 ; and for t P 1
h i0 g h ^ t ; kt ; h1t ; h2t ; yt ¼ ct ; nt ; lt ; btþ1 ; btþ1 ; ct ; /t ; l h h g ^ht ; g t Þ, the state variables at t ¼ 0 are where btþ1 is a random variable measurable with respect to ðst ; b0 ; b0 Þ. Letting st ¼ ðp
h i0 g h x0 ¼ b0 ; b0 ; c1 ; s0 ;
^h0 ¼ 0 and, by definition, c1 ¼ 0. On the other hand, at t P 1: where p
h i0 g h xt ¼ bt ; bt ; ct1 ; ct1 ; st : h
h
Next, we argue that bt should not be included in the vector of state variables, xt . Constraint (11) implies that bt is determined by a time invariant function DðÞ:20
g h bt ¼ D bt ; ct1 ; ct1 ; st ;
8t P 1:
In other words, the stock of state-contingent financial claims held by the household at time t P 1 depends on the same set of state variables as yt . Therefore, it is not necessary to include it as an additional state variable when solving for yt . Based h on this, the implementation of the algorithm does not require solving for btþ1 .21 For the quantitative component of the paper, the instantaneous utility function of the representative household is assumed to be additively separable in consumption and leisure:
uðct ; lt Þ ¼ tðct Þ þ Hðlt Þ;
8t P 0: g
^ t ; kt and ct are Given this assumption, the first order conditions for the Ramsey problem with respect to ct ; nt ; btþ1 ; l
"
t00 ðc0 ÞH0 ð100 n0 Þ ^ 1p ^h1 ¼ 0; þ bt00 ðc0 ÞE0 l t0 ðc0 Þ2
t0 ðc0 Þ c0 ðc0 k0 Þn0 "
t0 ðct Þ ct ðct kt Þnt
#
# t00 ðct ÞH0 ð100 nt Þ
t0 ðct Þ2
H0 ð100 nt Þ þ kt þ ðct kt Þ kt h1t þ h2t ¼ Et ktþ1 ; ^ht 0 ðct1 Þ p
^ tþ1 p ^htþ1 ¼ 0; ^ t þ bt00 ðct ÞEt l bt00 ðct Þl
0 H ð100 nt Þ H00 ð100 nt Þnt ¼ 0; t0 ðct Þ
8t P 0
0
t
¼ bt ðct Þ; 8t P 1 H0 ð100 nt Þ ¼ þ gt 1 nt ; 8t P 0 0 t ðct Þ 0 H ð100 nt Þ h h bt ¼ ct nt þ bEt btþ1 ; 8t P 1 t0 ðct Þ
g bbtþ1
ð29Þ
g bt
8t P 0
8t P 1
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
In addition, the standard Kuhn–Tucker conditions apply:
g h1t B bbtþ1 ¼ 0; g h2t bbtþ1 B ¼ 0;
h1t P 0; 8t P 0
ð36Þ
h2t P 0; 8t P 0:
ð37Þ
A useful result can be derived from the first order condition with respect to the state-contingent bonds purchased by the h household, btþ1 , which is given by:
btþ1 pðstþ1 jst Þ/tþ1 ¼ 0;
8t P 0; 8st ; stþ1 2 S:
ð38Þ
This implies that
/tþ1 ¼ 0;
8t P 0:
ð39Þ
Hence, by making use of definition (16), we obtain
ctþ1 ¼ ct ¼ c; 8t P 0:
20
ð40Þ
For a formal argument, see Proposition 1 in Marcet and Scott (2003). This does not imply that the portfolio of state-contingent financial claims purchased by the household in each period is indeterminate. Once the Ramsey problem in its current form is solved and the policy rules for all endogenous variables as a function of the state variablesbecome known, I can use (11) to derive h bt for each period. However, this is computationally a very expensive task to perform. 21
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
In other words, ct remains constant in all periods and can be removed from xt ; 8t P 0. Using a more compact notation than previously, the state and decision variables that are the focus of the computational component in the paper can be summarized as follows:
0 g ^ t ; kt ; h1t ; h2t ; yt ¼ ct ; nt ; lt ; btþ1 ; ct ; /t ; l
8t P 0
and
g 0 xt ¼ bt ; ct1 ; st ;
8t P 0
where
^ht ; g t Þ; 8t P 0 st ¼ ðp ct ¼ c; 8t P 0 ; /0 ¼ c /t ¼ 0; c1 ¼
^h0 p
8t P 1 ^ 0 ¼ 0: ¼ c1 ¼ l
Appendix B. Computational scheme This Appendix discusses the numerical scheme used in solving the Ramsey problem outlined in Section 2 and Appendix A. The proposed algorithm is an appropriate adaptation to the present context of the Parameterized Expectations Algorithm (PEA), as it is described in den Haan and Marcet (1990). According to this algorithm the agents’ conditional expectations about functions of future variables are replaced by an approximating function involving the state variables of the system and coefficients on each of these variables. The approximating function is used to generate T realizations of the endogenous variables of the model, where T is a very large number. These realizations are used as observations in a series of nonlinear least squares regressions which are used to reestimate the coefficients of the approximating function. Then the new set of coefficients obtained are used to generate a new series of length T for the endogenous variables. Iterations continue until the regression coefficients obtained from the use of successive sets of coefficients for the approximating function converge up to a prespecified tolerance level. The steps involved are the following: Step 1. Choose appropriate values for the parameters of the model. Assuming that the exogenous state variables of the ^ht , follow particular stochastic processes, generate a series of realizations for each of them. These series model, g t and p have length T, where T is a large number, and are drawn only once. j , where subscript j is an integer used to denote the current Step 2. Make a guess for the value of the pseudo-multiplier c; c outer-loop iteration step. Step 3. Replace the conditional expectations included in first-order conditions 29, 30 and 32 of the Ramsey Problem by approximating functions wðaki ; xt Þ, where aki denotes a fixed vector of parameters, subscript i represents the current inner-loop iteration step, and k ¼ 1; 2 corresponds to the particular conditional expectation replaced by a function wð; xt Þ. Formally, expressions 29, 30 and 32 can be compactly rewritten as: 0
v ðct Þ ct ðct kt Þnt
" # v00 ðct ÞH0 ð100 nt Þ v0 ðc
2
tÞ
^ t þ bv00 ðct Þw a1i ; xt ¼ 0; bv00 ðct Þl
8t P 0
ð41Þ
and
kt h1t þ h2t ¼ w a2i ; xt ;
8t P 0
ð42Þ
^ 0 ¼ 0. where l ~i ; c j ÞgTt¼0 , that solves the sys~i ¼ a1i ; a2i 0 , obtain a long series of the endogenous variables of the model fyt ða Step 4. Letting a tem of Eqs. 15, 31, 33, 34, 36, 37, 41 and 42.22 Towards this end, we have to distinguish between four possible cases, depending on whether: (i) constraint (14) is binding, and (ii) the allocation derived corresponds to the initial period. g (a) Unconstrained Case for t ¼ 0: In this case, B < bb1 < B and from the Kuhn–Tucker conditions (36) and (37) it follows that h10 ¼ h20 ¼ 0. Then, from (42) we obtain k0 . First-order conditions (31) and (41) consist a system of two (nonlinear) equations in two unknowns: c0 and n0 . Once n0 has been calculated, the time endowment constraint g (15) provides us with l0 . Finally, making use of c0 and n0 , we compute b1 from (34). g (b) Constrained Case for t ¼ 0: In this case, either b1 ¼ B=b or B=b, which by making use of (36) and (37) implies that h10 > 0 and h20 ¼ 0 or h20 > 0 and h10 ¼ 0, respectively. Eqs. (41), (31) and (34) consist a system of three equations in three unknowns: c0 ; n0 and k0 . Once n0 has been calculated, the time endowment constraint (15) yields l0 . Finally, given the multiplier k0 , first-order condition (42) provides us with either h10 or h20 . 22
h
As it was explained in Appendix A, since bt is not a variable of immediate interest, first-order condition (35) is omitted.
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 g
(c) Unconstrained Case for t P 1: In this case, B < bbtþ1 < B and from the Kuhn–Tucker conditions (36) and (37) it follows that h1t ¼ h2t ¼ 0. From (42), we then obtain kt . Given ct1 calculated from the previous period, first-order condition (33) provides us with ct . Eqs. (31) and (41) then consist a system of two equations in two unknowns: nt ^ t . Given the optimal supply of labor, the time endowment constraint (15) yields lt . Finally, by substituting ct and l g and nt into (34), we compute btþ1 . g (d) Constrained Case for t P 1: In this case, either btþ1 ¼ B=b or B=b, which by making use of (36) and (37) implies that h1t > 0 and h2t ¼ 0 or h2t > 0 and h1t ¼ 0, respectively. As in the previous case, given ct1 calculated from the previous period, ct can be derived from first-order condition (33). Eqs. (31), (34) and (41) consist then a system of three ^ t ; nt and kt . Once nt has been calculated, we obtain lt from the time endowment conequations in three unknowns: l straint (15). Finally, given the multiplier kt , first-order condition (42) provides us with either h1t or h2t . Step 5. Given the series of the endogenous variables of the model calculated in the previous step, compute the variables that ^ tþ1 p ^htþ1 and ktþ1 ; 8t P 0. Next, appear inside the conditional expectations in expressions (29), (30) and (32): that is, l perform nonlinear least squares regressions of these variables on the corresponding wð; xt Þ functions. Define the ~i Þ. result of these regressions as Sða ~i according to the following scheme: Step 6. Using a predetermined relaxation parameter d 2 ð0; 1, update a
~iþ1 ¼ ð1 dÞa ~i þ dSða ~i Þ: a ~iþ1 a ~i k < , where is a small positive number. Denote the fixed point obtained by a ~0 . Step 7. Iterate until ka Step 8. Consider H histories, each history being P periods long. Generate corresponding series for the exogenous state vari~0 , repeat Steps 3 and 4 to derive the allocations and corresponding tax ^ht ) and, using a ables of the model (g t and p rates. Next, compute the following expression: g
dj ¼ b0
X P 1 H X j Þnh;p ðc j Þ g h;p ; bp ½sh;p ðc H h¼1 p¼0
j until jdj j < e, which is an approximation to the government’s present value budget constraint.23Keep iterating on c where e is a small positive number. adjustment in the variance of the serially correlated g is performed by multiplying the ðegtþ1 =sg Þ term in (19) by ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pThe 1 q2 , where q ¼ 0:75. Then, the variance of g for the uncorrelated and correlated cases is the same and equal to 1=s2g . In Step 1 of our algorithm, we set T ¼ 10; 000. In Step 8, we set H ¼ 99 and P ¼ 101. The exact same sequences of shocks in Steps 1 and 8 are used for all three pairs of debt and asset limits. In order to facilitate the computation of standard errors for all the results reported in Tables 1–4, we perform in each case N ¼ 100 simulations of length T ¼ 10; 000 each. Finally, note that the average standard deviation of the simulated g series is (standard errors in parentheses): 2.500956 (0.00152536) when q ¼ 0:00; 3:7319 ð0:004384799Þ when q ¼ 0:75, and 2.500825 (0.003302303) when we adjust the volatility of serially correlated government expenditures. On the other hand, ^h series is 0.0012 (3.051E-09). the average standard deviation of the simulated p In the ‘‘balanced budget” case, we set B ¼ B and both limits equal to 1E 08 for computational simplicity. This number is essentially zero and allows us to use the computational scheme described above without having to write a separate code. It should be noted, however, that the balanced budget case in this model is easy to solve since the dynamic model separates into a sequence of static unconnected optimization problems. Given the chosen parameterization, the government’s budget constraint reduces to: 1 3
1 2
ct 2 lt þ ð100 g t Þct 2 lt þ lt 100 ¼ 0: For t ¼ 0, Step 4(b) of the above numerical scheme applies. For t P 1; ct is determined by (33) in Appendix A and the government’s budget constraint becomes a cubic equation in leisure. From the three implied tax rates we pick the lowest one. Both methods yield virtually identical results.24 The calculation of the reported welfare gains is based on performing in every case N ¼ 100 simulations of length T ¼ 10; 000 each. For each individual simulation, the calculation of the welfare gains is performed as follows. Consider two fiscal policy regimes: A and B. The separability assumption allows us to write the expected welfare in either regime as W i ¼ W ci þ W li , where i ¼ A; B. W ci denotes the expected utility of consumption and W li the expected utility fA ¼ W f c þ W l , such that: of leisure. Assume that W A < W B . Let W A A
f A ¼ W B: W
ð43Þ
f c is given by: By definition, W A 23
The latter can be obtained by using forward substitution on (6) and imposing a transversality condition on the government’s debt as T ! 1 to obtain: g
b0 ¼ E0
1 X
bt ðst nt g t Þ
t¼0 24
I would like to thank an anonymous referee for pointing this issue to me.
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
f c ¼ E0 W A
1 X t¼0
bt
! ec t ðAÞ1r1 1 : 1 r1
463
ð44Þ
Let f denote the increase in consumption in all periods that is required in order to make the representative household indifferent between the two fiscal policy regimes. Setting ~ct ðAÞ ¼ ð1 þ fÞct ðAÞ in (44) and solving (43) for f yields:
31=ð1r1 Þ 1 þ ð1 r1 Þð1 bÞ W cB þ W lB W lA 5 f¼4 1: 1 þ ð1 r1 Þð1 bÞW cA 2
In calculating the expected utility of consumption and leisure, we consider H ¼ 99 histories, each history being P ¼ 101 periods long. For each history, we generate series for government expenditures and asset prices, and derive the corresponding allocation according to Step 4 of our numerical scheme. Finally, we compute the following expression:
! " !# X r1 1r P H X P c1 1 lh;p 2 1 1 H X 1 X h;p p p þ ; b b g 1 r1 H h¼1 p¼0 H h¼1 p¼0 1 r2 which is an approximation to the representative household’s expected utility. References Adler, J., 2006. The tax-smoothing hypothesis: Evidence from Sweden, 1952–1999. Scandinavian Journal of Economics 108 (1), 81–95. Aiyagari, S.R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109 (3), 659–684. Aiyagari, S.R., Marcet, A., Sargent, T.J., Seppala, J., 2002. Optimal taxation without state-contingent debt. Journal of Political Economy 110, 1220–1254. Angeletos, G.M., 2002. Fiscal policy with non-contingent debt and the optimal maturity structure. Quarterly Journal of Economics 117 (3), 1105–1131. Atkeson, A., Chari, V.V., Kehoe, P.J., 1999. Taxing capital income: A bad idea. Federal Reserve Bank of Minneapolis Quarterly Review 23, 3–17. Atkinson, A.B., Stiglitz, J., 1980. Lectures on Public Economics. McGraw-Hill, New York. Barro, R.J., 1979. On the determination of public debt. Journal of Political Economy 87, 940–971. Barro, R.J., Sala-i-Martin, X., 2004. Economic Growth, second ed. MIT Press. Buera, F., Nicolini, J.P., 2001. Optimal Maturity of Government Debt without State Contingent Bonds. University of Chicago, Mimeo. Cashin, P., Olekalns, N., Sahay, R., 1998. Tax Smoothing in a Financially Repressed Economy: Evidence from India, IMF Working Paper No. WP/98/122, International Monetary Fund, Washington, DC. Cashin, P., Haque, N., Olekalns, N., 1999. Spend Now, Pay Later? Tax Smoothing and Fiscal Sustainability in South Asia, IMF Working Paper No. WP/99/63, International Monetary Fund, Washington, DC. Chari, V.V., Kehoe, P.J., 1999. Optimal fiscal and monetary policy. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, 1. North Holland, Amsterdam, pp. 1671–1745. 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., Christiano, L.J., Kehoe, P.J., 1995. Policy analysis in business cycle models. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Correia, I.H., 1996. Dynamic optimal taxation in small open economies. Journal of Economic Dynamics and Control 20, 691–708. den Haan, W.J., Marcet, A., 1990. Solving the stochastic growth model by parameterizing expectations. Journal of Business and Economic Statistics 8, 31–34. Kydland, F.E., Prescott, E.C., 1980. Dynamic optimal taxation, rational expectations and optimal control. Journal of Economic Dynamics and Control 2, 79–91. Lloyd-Ellis, H., Zhan, S., Zhu, X., (2001). Tax Smoothing with Stochastic Interest Rates: A Re-assessment of Clinton’s Fiscal Legacy, Center for Research on Economic Fluctuations and Employment, UQAM, Working Paper No. 125. Lucas Jr., F.E., Stokey, N.L., 1983. Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55–93. Marcet, A., Marimon, R., 1999. Recursive Contracts. Universitat Pompeu Fabra, Barcelona, Mimeo. Marcet, A., Scott, A., 2003. Debt and Deficit Fluctuations and the Structure of Bond Markets. Universitat Pompeu Fabra, CREI and CEPR, Mimeo. Razin, A., Sadka, E., 1995. The status of capital income taxation in the open economy. FinanzArchiv 52, 21–32. Sargent, T.J., Velde, F., 1995. Macroeconomic features of the french revolution. Journal of Political Economy 103 (3), 474–518. Schmitt-Grohé, S., Uribe, M., 1997. balanced-budget rules, distortionary taxes, and aggregate instability. Journal of Political Economy 105, 976–1000. Schmitt-Grohé, S., Uribe, M., 2004. Optimal fiscal and monetary policy under sticky prices. Journal of Economic Theory 114, 198–230. Sleet, C., 2004. Optimal taxation with private government information. Review of Economic Studies 71, 1217–1239. Sleet, C., Yeltekin, S., 2006. Optimal taxation with endogenously incomplete debt markets. Journal of Economic Theory 127, 36–73. Stockman, D.R., 2001. balanced-budget rules: Welfare loss and optimal policies. Review of Economic Dynamics 4, 438–459. Stockman, D.R., 2004. Default, reputation, and balanced-budget rules. Review of Economic Dynamics 7, 382–405. Strazicich, M.C., 1997. Does tax smoothing differ by the level of government? Time series evidence from Canada and The United States. Journal of Macroeconomics 19 (2), 305–326. Strazicich, M.C., 2002. International evidence of tax smoothing in a panel of industrial countries. Applied Economics 34, 2325–2331.
Contents lists available at ScienceDirect
Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro
Balanced budget vs. Tax smoothing in a small open economy: A welfare comparison q Constantine Angyridis * Department of Economics, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3
a r t i c l e
i n f o
Article history: Received 9 June 2006 Accepted 30 August 2008 Available online 17 September 2008
JEL classification: E62 H21
a b s t r a c t The objective of this paper is to investigate the effect of lending and borrowing constraints on the dynamics of public debt and optimal taxation policy in the context of a general equilibrium model with tax smoothing. The results from the numerical simulation of the model show significant welfare gains, provided that the policymaker is allowed to borrow and lend in order to smooth taxes across time instead of maintaining a balanced budget at all times. Moreover, for a specific process for asset prices, it is also shown that if the government can issue state-contingent debt then overall welfare can be further improved substantially. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Balanced budget Complete markets Incomplete markets Welfare analysis
1. Introduction Is a series of continuous budget deficits necessarily bad? Besides being a clear sign of fiscal prudence, does a balanced budget rule also improve social welfare? How should tax rates and, consequently, public debt respond to innovations in government expenditures? Theoretical advancements on solving stochastic intertemporal optimization problems with implementability constraints,1 have reinvigorated research on optimal debt issue and the implied structure of taxation policy.2 The present paper applies this new technique in the context of a small open economy, in which the social planner (the government) is faced with stochastic expenditures. The objective is to study the welfare implications if the policymaker adopts a fiscal policy plan that deviates from a strict balanced budget rule. It is shown that there are significant welfare gains to be made if the government equates revenues with expenditures in a present value sense, instead of every period, such that its intertemporal infinite horizon budget constraint is satisfied. q The author would like to thank Arman Mansoorian, Leo Michelis, Xiaodong Zhu, the co-editor and an anonymous referee for their valuable comments and suggestions. All remaining errors are my own. * Tel.: +1 416 979 5000x7725; fax: +1 416 598 5916. E-mail address: [email protected] 1 Standard dynamic programming techniques are applicable under the condition that the set of feasible current actions available to a social planner depend only on past variables. Implementability constraints imposed in contract and optimal fiscal policy problems usually depend on plans for future variables, thus constraining the set of current feasible actions available to the social planner. Extending the work of Kydland and Prescott (1980) for a general class of contract problems involving incentive constraints, Marcet and Marimon (1999) show how one can compensate for this lack of recursivity by expanding the state space to include a new variable that depends on past Lagrange multipliers. 2 For example, see Aiyagari et al. (2002); Angeletos (2002); Buera and Nicolini (2001) and Marcet and Scott (2003).
0164-0704/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2008.08.003
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
439
Stockman (2001) has performed a comprehensive analysis of the welfare effects of balanced budget restrictions in the context of a closed economy with exogenous growth. Using a similar framework to the one used in Chari et al. (1994, 1995), he considered the Ramsey problem for a model calibrated to the US economy, placing restrictions on the amount of debt that the government can issue. In his setting, he found substantial welfare losses associated with the balanced-budget restriction: depending on preference parameterization, consumption would have to increase from 0.61% up to 1.45% at each date and for each state in order to make the representative household indifferent between fiscal policy regimes. Stockman’s study builds on the work by Lucas and Stokey (1983) who considered a general equilibrium model with complete markets, exogenous Markov government expenditures and state-contingent taxes. The Lucas and Stokey framework emphasizes the role of state-contingent debt as an ‘‘insurance” against bad times, that allows the government to smooth tax distortions across both time and states of nature. As a result, in times of temporary increases in government expenditures, the level of public debt falls. In addition, Lucas and Stokey show that the serial correlations of optimal tax rates are closely related to those for government expenditures. In their model, tax rates are smooth only to the extent that they exhibit a smaller variance than a balanced budget would imply. Aiyagari et al. (2002) reconsider the Lucas and Stokey economy under the restriction that the government cannot issue state contingent debt; but rather it can borrow and lend at an endogenous risk-free rate. In their attempt to replicate Barro (1979) classic ‘‘tax smoothing” result in a general equilibrium environment, they show that this restriction (i.e., market incompleteness) imposes additional implementability constraints with respect to the equilibrium allocation, beyond the single implementability constraint imposed under complete markets. In particular, these constraints require the allocation to be such that at each date the present value of the budget surplus evaluated at current period Arrow prices be known one period ahead. Under the condition that an ad hoc limit is imposed on the government’s asset holdings, the authors show that the Ramsey equilibrium exhibits features shared by both Barro’s and Lucas and Stokey’s economies, but the dynamics of debt and taxes actually resemble more closely Barro’s ‘‘tax-smoothing” result.3 Recently, several authors have reported tax smoothing behavior for various small open economies. Among others, Strazicich (1997) using annual time-series data over the period 1929–1990, fails to reject the tax-smoothing hypothesis for the Canadian federal government. Cashin et al. (1998) find that the behavior of the central government of India is consistent with the tax-smoothing hypothesis for the period 1951–1952 to 1996–1997. A similar result is obtained by Cashin et al. (1999) for Pakistan during the period 1956–1995, but the hypothesis is rejected for Sri Lanka during 1964–1997. Strazicich (2002) considers a panel of 19 industrial countries over the period 1955–1988 and finds evidence in support of tax smoothing as a theory of public debt. Adler (2006) finds that Barro’s model can explain about 60 percent of the variability in the Swedish central government budget surplus for the period 1952–1999. These empirical studies point to the need to analyze the tax smoothing motive in the context of a small open economy facing exogenous asset prices. This provides one motivation for the present paper. We consider an asymmetric small open economy with respect to accessibility to financial markets: households are allowed to perform transactions involving statecontingent financial claims, while the government is restricted to borrow and lend only at a constant risk-free interest rate.4 A possible justification for this assumed financial asymmetry might be the inherent ‘‘moral hazard” that is present if the government is allowed to issue state-contingent debt or bonds. The return on these bonds would have been linked to the level of macroeconomic variables, such as the inflation rate or the Debt/GDP ratio. However, these variables are under the strong influence of the government’s actions, thus damaging the marketability of these bonds among private agents. Another justification for such an asymmetry follows from the work of Sleet (2004) and Sleet and Yeltekin (2006), who demonstrate that if the government has access to a complete set of contingent claims markets but is unable to either commit to future debt repayments or truthfully reveal private information regarding its spending needs, then the market for public debt becomes endogenously incomplete. Studying the implications of the coexistence of complete and incomplete markets in a small open economy is new in the literature on optimal taxation. 5 Furthermore, if one wishes to maintain the representative agent framework, the simplest environment to discuss the implications of this asymmetry is that of a small open economy.6 In our setting, the government is required to determine the optimal fiscal policy plan as of time zero that maximizes the welfare of the representative household, given a stream of stochastic government expenditures. These expenditures are 3 Barro (1979) demonstrated that debt policy between 1916 and 1976 in the U.S. and United Kingdom was consistent with the predictions of his theory. Sargent and Velde (1995) argue that the behavior of British debt during the 18th century conforms with Barro’s theory, since it closely resembles a martingale with a drift. Extending Barro’s model to allow for stochastic variation in interest and growth rates, Lloyd-Ellis et al. (2001) show that the US debt policy is also in agreement with the theory of tax smoothing during the transition period from the large budget deficits of the 1980’s to the high surpluses in the mid 1990’s. Finally, based on the persistence properties of public debt and its positive response to an innovation in government expenditures, Marcet and Scott (2003) argue that the behavior of the US public debt can be accounted for by a model in which the government issues only one-period risk-free bonds. 4 The existence of a market for riskless government bonds is redundant from the point of view of the households, since it does not alter their trading opportunities: households can always adjust their portfolio of state-contingent securities in such a way to obtain the same return as that of a government bond. 5 The literature on optimal fiscal policy for a small open economy focuses exclusively on factor income taxation, incorporates capital stock accumulation and assumes away uncertainty. Recent studies include Atkeson et al. (1999); Chari and Kehoe (1999); Correia (1996) and Razin and Sadka (1995). 6 The existing literature has considered different degrees of market completeness between the public and private sectors of a closed economy. For instance, Aiyagari et al. (2002) assume that the government faces ad-hoc debt and asset limits that are more stringent than those faced by the representative household. On the other hand, Schmitt-Grohé and Uribe (2004) consider a stochastic production economy with sticky product prices in which households can acquire two types of final assets: fiat money and one-period state-contingent nominal assets. In contrast, the government can print money and issue nominal non-state contingent bonds.
440
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
financed by taxing labor income and issuing debt. In contrast to Aiyagari et al. analysis, here the price of government bonds is assumed to be exogenously determined. Another contribution of the present paper is to provide an extensive numerical analysis of the tax smoothing motive and welfare gains with alternative assumptions regarding the stochastic process for government expenditures. For the incomplete markets case, we consider three stochastic processes: (i) serially uncorrelated, (ii) serially correlated, and (iii) serially correlated expenditures whose volatility is adjusted to be equal to that of uncorrelated expenditures. We study the implications of each process for the optimal fiscal policy as the government’s lending and borrowing constraints are relaxed. The comparison of the various cases is based on the persistence and other descriptive statistics of the equilibrium tax rates, as well as the distribution of the government’s asset and debt holdings. It is shown that providing more flexibility to the government in terms of the amounts it can borrow and lend, leads to smoother tax rates across time and higher social welfare compared to a balanced budget policy regime. In other words, as the government’s lending and borrowing range becomes wider, the persistence of the equilibrium tax rates increases and their volatility declines for all types of government expenditures considered. As a consequence, welfare increases. These findings are consistent with the arguments against balanced-budget rules in closed economies, as in Schmitt-Grohé and Uribe (1997) and Stockman (2001). Our results demonstrate also that one does not have to restrict access to complete markets for both sectors of the economy (private and public) in order to retain the random walk property of tax rates. An additional implication is that minimizing tax distortions is the most significant factor influencing the decision of the fiscal authority. In Aiyagari et al. the government has to consider the effect of debt issue on interest rates. By abstracting completely from interest rate effects, we still obtain the tax smoothing result, which implies that the quantitative results of Aiyagari et al. are relevant even for small open economies. Further, using a particular stochastic process for the prices of the state-contingent bonds and complete financial markets for the government, we show that overall welfare significantly improves relative to the balanced budget and incomplete markets cases. In addition, compared to the incomplete markets case, the resulting welfare gains increase as the persistence of the government expenditures shocks rises. These results can be interpreted as the ‘‘cost” of the moral hazard problem that was used to justify the assumed asymmetry between the public and private sectors of the economy. 7 Finally, it should be noted that the improvement in welfare when the government shifts from maintaining a balanced budget to issuing state-contingent debt is slightly higher than the one reported by Stockman (2001) in the context of a closed economy, although admittedly the differences in model specification do not permit a completely satisfactory comparison of the two models.8 The rest of the paper is organized as follows. Section 2 describes the theoretical model for the small open economy and defines the Ramsey problem that maximizes the representative household’s utility. Section 3 provides details regarding the parameterization of the model for the numerical analysis. Section 4 reports the quantitative results and compares the welfare results for different types of government expenditures and borrowing and lending constraints for the government. Section 5 performs welfare analysis under the assumptions of complete financial markets for the government and a specific process for asset prices, and compares the results to the case of incomplete financial markets and balanced government budget. Section 6 concludes the paper. 2. The model Consider a small open economy with a representative agent and no population growth. As in Lucas and Stokey’s closed economy model, the small open economy produces a single non-storable final good, using only labor, such that one unit of labor yields one unit of output. The time endowment in each period t is fixed at T ¼ 100, and it is allocated to labor, nt , and leisure, lt .9 An event st takes place in each period t. Let S be an infinite set of all the possible events. At any point in time t, the state of the economy can then be described by the state variable st ¼ ðs0 ; s1 ; . . . ; st Þ. This vector can also be thought of as representing the history of events up to and including period t. Hence, the history at any period can be described by st ¼ ðst ; st1 Þ. The probability density function as of period 0 of any particular history st taking place is denoted by pðst Þ. The representative household’s income consists of the sum of its net-of-tax labor earnings ð1 st Þnt and matured stateh contingent securities bt . These are one-period forward Arrow securities that pay a unit of output if a particular state of nature is realized and nothing otherwise. Total income is allocated to consumption expenditures ct and the purchase of financial h claims btþ1 at the exogenously determined prices phtþ1 for each possible event stþ1 . The asset prices are measured in units of time-t output and are assumed to be determined by the international financial markets.
7
I am grateful to an anonymous referee for suggesting this insightful interpretation. Schmitt-Grohé and Uribe (1997) and Stockman (2004) define a balanced budget rule as one in which the government holds the real level of debt constant. In this case, the choice of this level will matter when comparing the impact of state vs. non-state contingent debt as long as the government’s initial g g indebtedness b0 > 0. If b0 ¼ 0, then the particular choice does not matter. In the present paper, a balanced budget rule means that the government is not allowed to issue debt. We would like to thank an anonymous referee for this insightful comment. 9 I follow Aiyagari et al. in adopting a rescaled version of the commonly time endowment constraint used in the literature that sets T ¼ 1. This is done for technical reasons, since it facilitates the computational scheme of the paper. 8
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
The objective of the household is to maximize, as of time 0, expected lifetime utility with respect to the choice variables h h ct ; lt ; nt ; btþ1 taking as given b0 , the government’s tax rate st and asset prices phtþ1 :
max
1
fct ;lt ;nt ;bhtþ1 gt¼0
W ¼ E0
1 X
bt uðct ; lt Þ;
0
ð1Þ
h
ð2Þ
t¼0
subject to:
ct þ
Z
h
phtþ1 btþ1 dstþ1 6 ð1 st Þnt þ bt ;
8t P 0
lt þ nt ¼ 100; h b0 ;
s
h t ; ptþ1
8t P 0
ð3Þ
given:
Expressions (2) and (3) represent the household’s flow budget constraint and time endowment constraint, respectively. The period utility uðÞ is strictly increasing in ct and lt , strictly concave and satisfies the Inada conditions. The first order conditions for this intertemporal optimization problem are
ul;t ¼ 1 st ; uc;t
8t P 0
phtþ1 ¼ bpðstþ1 jst Þ
uc;tþ1 ; uc;t
ð4Þ
8t P 0; 8st ; stþ1 2 S
ð5Þ
along with the household’s flow budget constraint (2) satisfied with equality. Condition (4) equates the marginal rate of substitution between leisure and consumption with the net return to the household from supplying one extra unit of labor. The right-hand side of expression (5) represents the one-period asset pricing kernel (or discount factor) for a state-contingent financial security purchased at time t and maturing in the following period. The government is faced with an exogenous stream of stochastic and wasteful expenditures denoted by g t . Following Aiyagari et al., we assume g t 2 ½g min ; g max ; 8t P 0. Government expenditures can be financed either by taxing labor income g at the time-varying flat rate st or by issuing bonds btþ1 at time t which mature at time t þ 1. In contrast to the financial claims h g btþ1 purchased by the private sector, btþ1 are assumed to be nonstate-contingent; that is, they yield a rate of return that is independent of the state of nature in the economy. The government’s flow budget constraint can be written as g
g
pgt btþ1 ¼ bt þ g t st nt ; where, in addition,
pgt ¼
Z
pgt
8t P 0
ð6Þ 10
is the price of a government bond measured in units of time-t output.
phtþ1 dstþ1 ;
8t P 0
We assume that
ð7Þ
which ensures that there are no arbitrage opportunities in the international bonds market. For simplicity, we assume pgt ¼ b; 8t P 0, so that the government borrows and lends at a constant risk-free price equal to b. Finally, in order to ensure a sustainable and meaningful optimal fiscal policy, the following asset and debt constraints are imposed on the government: g
B 6 bbtþ1 6 B;
8t P 0
ð8Þ
where B and B denote the government’s maximum level of indebtedness and maximum level of asset holdings, respectively. These limits could be of two types: ‘‘natural” limits and ‘‘ad hoc” limits; see Aiyagari (1994). The ‘‘natural” debt limit represents the maximum level of indebtedness for which debt can be paid back with probability 1, under some tax policy. Since the tax policy depends on the stochastic process for g t , the natural debt limit is itself a random variable. The ‘‘natural” asset limit represents the minimum level of assets beyond which the government has no incentive to accumulate additional assets, since it can finance all of its expenditures entirely through the interest on its assets, even if g t takes its highest possible value in all periods.11 In contrast, a debt or asset limit is called ‘‘ad hoc” if it is more stringent than a natural one. In what follows, we define the concepts of feasible allocations and competitive equilibrium, and use them to characterize the government or Ramsey problem. g
h
Definition 1. Given b0 ; b0 , a stochastic process for government expenditures fg t g, and a stochastic process for stateh contingent financial claims prices fphtþ1 g satisfying (7), a feasible allocation is a stochastic process fct ; nt ; g t ; btþ1 g satisfying the economy’s flow budget constraint
10 The existence of a market of riskless government bonds has no effect on the trading opportunities of the representative household. To see this, consider the g simultaneous purchase of btþ1 units of output of Arrow securities corresponding to each possible state of nature. This transaction will yield next period a payoff g of btþ1 units of output regardless of stþ1 . However, this payoff is identical to that of a government bond. 11 In general, natural debt and asset limits are difficult to compute. In the present model we can derive an analytic expression for the natural asset limit by g g considering the government’s flow budget constraint (6) with zero tax revenues at the maximum level of government expenditure. Letting btþ1 ¼ bt ¼ B=b and solving for B implies that natural asset limit for the government is B ¼ g max ðb=ð1 bÞÞ.
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
Z g g h h ct þ g t ¼ nt ðbt bbtþ1 Þ þ bt phtþ1 btþ1 dstþ1 ;
ð9Þ h
g
g
whose time t elements are measurable with respect to ðst ; b0 ; b0 Þ. A government policy fst ; btþ1 g is a stochastic process whose h g time t elements are measurable with respect to ðst ; b0 ; b0 Þ. h
g
Definition 2. Given b0 ; b0 , a stochastic process fg t g, and a stochastic processes fphtþ1 g satisfying (7), a competitive equilibrium is a feasible allocation and a government policy that solve the household’s optimization problem and satisfy the government’s budget constraints (6) and (8). The government’s or Ramsey problem is to maximize (1) over competitive equilibria. We follow the primal approach as in Atkinson and Stiglitz (1980) in order to characterize the competitive equilibrium with distortionary taxes. The basic idea is to recast the problem of choosing the optimal fiscal policy as a problem of choosing feasible allocations that can be supported as a competitive equilibrium for some choice of taxes and bond issue. Then, combining the first order conditions for the government’s problem with those of the household’s one can determine the optimal tax rates. In the present context, the Ramsey problem, can be formally stated as follows:
max
ct ;lt ;nt ;bgtþ1 ;bhtþ1
f
1
gt¼0
W ¼ E0
1 X
bt uðct ; lt Þ
ð10Þ
t¼0
subject to:
1 X uc;tþj ctþj ul;tþj ntþj bj ; 8t P 0 uc;t j¼0 ul;t g g bbtþ1 ¼ bt þ g t 1 nt ; 8t P 0 uc;t uc;t ; 8t P 1 pht ¼ bpðst jst1 Þ uc;t1 h
bt ¼ Et
g
B 6 bbtþ1 6 B; lt þ nt ¼ 100; h g b0 ; b0
8t P 0 8t P 0
ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ
given:
Eq. (11) represents the household’s present value budget constraint as of time t. It is derived by combining conditions (4) and h (5) with (2), and using forward substitution along with the transversality condition for btþ1 given by h
lim bTþ1 Et uc;tþTþ1 btþTþ1 ¼ 0:
T!1
This conditional expectation must hold for any t with probability one. The reason for this is that the transversality condition is in the nature of a first-order necessary condition ‘‘at infinity” and is consistent with Bellman’s principle of optimality for any arbitrary t. Eq. (12) is derived by solving first order condition (4) from the household’s problem for st and substituting it into the government’s flow budget constraint (6). Eq. (13) corresponds to first order condition (5) from the household’s problem. Finally, Eqs. (14) and (15) are the government’s asset and debt limit constraint (8) and the household’s time endowment constraint (3), respectively. The Ramsey allocation can be characterized using the Lagrange approach. However, since the system (10)–(15) is nonrecursive, we apply the technique of Marcet and Marimon (1999) in order to transform the problem into a recursive one, and then formulate the Lagrangian for the recursive problem. To this end, we first eliminate constraint (15) by solving it for lt and substituting it into (10). Next, let the multipliers bt /t ; bt kt ; bt h1t and bt h2t be attached to constraints (11), (12) and the left and right sides of (14), and let bt lt be the multiplier of constraint (13) which is rewritten as:
pht uc;t1 bpðst jst1 Þuc;t ¼ 0: Also, define a pseudo-multiplier ct as
ct ¼ /t þ ct1 ; 8t P 0
ð16Þ
with c 1 ¼ 0. Using these transformations, the Lagrangian function for the recursive Ramsey problem can be written as
uc;t ct ul;t nt h bt uðct ; 100 nt Þ þ ðct ct1 Þbt ct uc;t t¼0 1 X g u g g g l;t ^ht uc;t1 buc;t ; ^ t ½p bt l þkt bbtþ1 bt g t þ 1 nt þ h1t B bbtþ1 þ h2t bbtþ1 B þ E0 uc;t t¼1
L ¼ E0
1 X
ð17Þ
^ht pht =pðst jst1 Þ and l ^ t lt =pðst1 Þ. Appendix A discusses the solution of the Ramsey problem assuming that the where p instantaneous utility function of the representative household is separable in consumption and leisure:
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
uðct ; lt Þ ¼ tðct Þ þ Hðlt Þ;
8t P 0:
443
ð18Þ
Appendix B provides a detailed description of the computational scheme for this problem. 3. Numerical simulations of the model This section states the specific functional forms and parameter values that we use in the numerical simulations of the model. Following Aiyagari et al. the instantaneous utility function of the representative household is assumed to be
! 1r r1 c1 1 lt 2 1 t : uðct ; lt Þ ¼ þg 1 r1 1 r2
Government expenditures are assumed to follow the stochastic process
g tþ1 ¼ ð1 qÞg þ qg t þ
egtþ1 sg
;
ð19Þ
where g is the unconditional long-run mean of the process g t 2 ½g min ; g max ; 8t P 0; q is its persistence (autocorrelation) parameter, egtþ1 is an i.i.d. Nð0; 1Þ random variable, and sg is a nonrandom scale factor. Notice that the truncation of (19) prevents g t from being negative. ^htþ1 is assumed to be The stochastic process for p
ep ^htþ1 ¼ b 1 þ tþ1 ; p sp
ð20Þ
^htþ1 2 ½p ^hmin ; p ^hmax ; 8 t P 0. The specification in where eptþ1 is i.i.d. Nð0; 1Þ and sp is a nonrandom scale factor. We also assume p (20) ensures that the no-arbitrage condition (7) is satisfied. Multiplying both sides of (20) with pðstþ1 jst Þ, integrating over stþ1 and using the distributional assumption for eptþ1 yields (7) with pgt ¼ b; 8t P 0. The conditional expectations in the first order conditions of the Ramsey Problem were parameterized using the family of approximating functions mapping R4þ into Rþ :
wðai ; xt Þ ¼ expðPn ðxt ÞÞ; where P n denotes a polynomial of degree n with coefficient parameters ai , and xt is a 4 1 vector of the state variables: g 0 ^ht ; g t ; see Appendix A. In the numerical simulations we estimate the function wðai ; xt Þ by nonlinear least xt ¼ bt ; ct1 ; p squares. To ensure that all variables in the nonlinear least squares regressions were of similar orders of magnitudes, we followed Aiyagari et al. in applying to each state variable separately the function u : ðk; kÞ#ð1; 1Þ, where k and k are prespecified lower and upper bounds for the argument k of function u. The latter is defined as
uðkÞ ¼ 2
kk kk
1;
which implies that the wðÞ functions used to parameterize the conditional expectations that appear in the first order conditions of the Ramsey problem are of the form
wðai ; xt Þ ¼ expðPn ðuðxt ÞÞÞ: In order to carry out the simulations we set the following parameter values. Following Aiyagari et al., the preference paramg h eters were set at: r1 ¼ 0:5; r2 ¼ 2 and g ¼ 1, the initial indebtedness of the government and the household at b0 ¼ b0 ¼ 0, the long-run mean of government spending g ¼ 30; b ¼ 0:95; sg ¼ 0:40 and sp ¼ 800. We consider three types of government expenditures: (i) serially uncorrelated ðq ¼ 0:00Þ, (ii) serially correlated ðq ¼ 0:75Þ, and (iii) serially correlated expenditures whose volatility is adjusted to be equal to the one of uncorrelated expenditures. The reason for this adjustment is that the unadjusted correlated expenditures are more volatile compared to the uncorrelated expenditures. We verify this in our numerical simulations in Appendix B where the standard deviation of g is 2.50 in cases (i) and (iii), and 3.73 in case (iii). Adjusting the variance of the correlated expenditures, allows us to study the effect of the persistence in g on the equilibrium tax rate when comparing case (i) with case (iii). In all three cases for g, every realization of (19) is constrained to lie in the interval [20,40], and every realization of (20) is ^htþ1 before constrained to lie in the interval [0.92,0.98]. These bounds were also used for the transformed variables g t and p g using them as inputs in the parameterized expectations estimations. For btþ1 the bounds were set at B=b and B=b, while the ones for ct1 were set at [40, 100]. Finally, for each of the three types of stochastic processes for government expenditures, we consider three pairs of debt and asset limits ðB; BÞ: (0,0), which corresponds to the ‘‘balanced budget” case, (10,10) and (50,50).12 For each pair of limits and process for g, the Ramsey problem is solved numerically. The resulting 12 These limits are more stringent than the ad hoc limits (1000,1000) used by Aiyagari et al. in their numerical experiments. The reason is that, in the present model and its parameterization the natural debt limit turns out to be 760. As Aiyagari et al. show in their paper, when B is equal to the natural debt limit, the optimal policy is to accumulate assets and set the tax rate equal to zero. I experimented with an increasing range for ðB; BÞ. The result was that for a sufficiently wide range, the government finds it optimal to accumulate as many assets as it is allowed by B, while thetax rate gets close to zero. Given this outcome, I chose to consider a smaller range for ðB; BÞ because it yields equilibrium tax rates that are consistent with tax smoothing.
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series of equilibrium tax rates are then compared in terms of their persistence and the implications of their dynamics for the welfare of the representative household. 4. Quantitative results In this section we report the quantitative predictions of the model for different government expenditure processes with alternative borrowing and lending constraints. These numerical results will be with regards to three important issues pertaining to our analysis. The first issue is the degree of tax smoothing achieved by the government for the various cases considered. The second issue is the distribution of government debt and asset holdings. The third issue is the welfare ranking for these different cases. 4.1. Tax smoothing Our quantitative results will be better understood if we first look at the series for the endogenous and exogenous variables generated from a representative simulation, as displayed in Figs. 1–4. Fig. 1 plots the series for the three types of gov^ht , and the two ernment expenditures, g t , and the common series for the price of state-contingent financial claims, p disturbance terms ðegt ; ept Þ. This figure allows a visual comparison of the series generated under the different specifications for g t . The volatility-adjusted correlated expenditures clearly exhibit more persistence compared to the uncorrelated expenditures and they also vary less relative to the volatility-unadjusted correlated g t . Figs. 2–4 consists of three panels of graphs (a–c) corresponding to the three ðB; BÞ pairs: (0,0), (10,10), (50,50), each in combination with the three types of g corresponding to: q ¼ 0:00; q ¼ 0:75 and q ¼ 0:75 with adjusted variance. Each panel plots the entire allocation series for consumption, leisure, tax rate, debt issue, budget surplus, the sum ct þ g t þ lt and the household’s financial wealth wt defined as
wt
Z
h
h
phtþ1 btþ1 dstþ1 bt ;
which, following from the budget constraint (2), reduces to wt ¼ ð1 st Þnt ct .
Gov. Expenditures: ρ = 0.00
Gov. Expenditures: ρ = 0.75
40
40
35
35
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2000 4000 6000 8000 10000 Gov. Expenditures: ρ = 0.75 - Adj. Var(g)
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0.952
30
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25
0.948 0.946 2000
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ε
g
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5
5
0
0
-5
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0.954
35
20
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4000
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-5
εp
Fig. 1. Government expenditures, asset prices and error terms.
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a
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Fig. 2(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:00.
Regarding the variable ct þ g t þ lt , notice that in the closed economy model of Aiyagari et al. this variable is equal to the time endowment ðT ¼ 100Þ in every period. This contrasts with the corresponding constraint for the current open economy model as given by Eq. (9). As shown in Figs. 2–4, making the ðB; BÞ range wider causes the equilibrium allocation series for leisure, tax revenue, the variable ct þ g t þ lt and household wealth to become less volatile, while there is minimal change with respect to consumption.13 In contrast, the government surplus and public debt series become more volatile as the range ðB; BÞ widens. These results are common for all three specifications of (19). Furthermore, relaxing the government’s borrowing and lending constraints causes the equilibrium tax rates to become increasingly smoother and persistent, regardless of the underlying serial correlation properties of g. We can show this quantitatively by comparing estimates of the autoregressive parameter, b, from the regression model
st ¼ a þ bst1 þ ut ;
ð21Þ
13 This is due to first order condition (33) of the Ramsey problem (see Appendix A). This condition implies that once c0 is determined, ct evolves as a function ^h Þ1=r1 ; 8t P 1, while the series for p ^ h remains common across the various cases for g and ðB; BÞ. of the ratio ðb=p
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b
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Fig. 2(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:00.
where a is constant and ut is a least-squares residual that is assumed to be orthogonal to st1 . The coefficient b measures the persistence in s. The closer the estimated values of a and b are to 0 and 1, respectively, the more ‘‘tax smoothing” is achieved by the government. Table 1 presents the first two unconditional moments of s, the estimated intercept terms and slope coefficients, and the R2 statistics. In general, as the government’s borrowing and lending range becomes wider, the estimated value of the coefficient of the lagged equilibrium tax rate increases and gets closer to 1. In other words, providing the government with more flexibility in terms of the amounts that it can borrow and lend leads to an increase in the persistence of the equilibrium tax rate. Furthermore, raising the asset and debt limits causes the estimated value of the intercept term in (21) to decline and move closer to 0. At the same time, the goodness of fit of the autoregression improves with the value of the R2 statistic increasing and getting closer to 1. When the ðB; BÞ range is sufficiently narrow, the estimated persistence of the equilibrium tax rates, the estimated intercept term and the goodness of fit of Eq. (21) are strongly influenced by the serial correlation properties of the underlying process for government expenditures. For example, in the period-by-period balanced budget case with serially uncorrelated government expenditures, the equilibrium tax rates do not exhibit any persistence ðb ¼ 0:000927Þ, the estimated intercept ða ¼ 0:310521Þ is essentially equal to the average tax rate (0.310812), while the value of the R2 statistic is zero. On the other
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c
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Fig. 2(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:00.
hand, when government expenditures are serially correlated, the equilibrium tax rates ‘‘inherit” their persistence: the estimated autoregressive parameter b takes on a value close to q ¼ 0:75ðb ¼ 0:745718Þ. Predictably, the fit of the autoregression improves relative to the uncorrelated case ðR2 ¼ 0:556206Þ, while the estimated value of a falls ða ¼ 0:079023Þ. These results are not surprising, since the optimal tax rate under a balanced budget rule is a function of g alone and, therefore, must have correlation properties similar to the ones for government expenditures.14 The variation of the average tax rate across the different cases reflects the government’s borrowing and lending behavior as the ðB; BÞ range becomes wider. Table 2 presents statistics that help describe this behavior. These statistics are the gov-
14 Aiyagari et al. consider only the case of serially uncorrelated government expenditures. Comparing their results from estimating (21) with the results reported in Table 1 for the corresponding case ðq ¼ 0:00Þ when ðB; BÞ ¼ ð50; 50Þ, suggests that their model yields equilibrium tax rates that are closer to the predictions of Barro’s tax smoothing theory. Their estimated a and bcoefficients are closer to 0 (0.0031 vs. 0.0252) and 1 (0.9888 vs. 0.9172), respectively, while the estimated std ½s is of similar magnitude (0.0191 vs. 0.0113). However, one has to take into account that theborrowing and lending range considered by Aiyagari et al. is significantly wider, which provides the government in their case the additional flexibility needed in achieving a higher degree of tax smoothing. Increasing the asset and debt limits in my case beyond the (50,50) range, at least marginally, generates tax rates that become closer in resembling arandom walk. Finally, it should be noted that a comparison of my results with those of Aiyagari et al. based on a formal statistical test is hindered by the fact that they do not report the standard errors of their estimates for (21).
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a
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Leisure
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Fig. 3(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:75.
ernment’s borrowing frequency (the percentage of time it borrows), the percentage of time the asset and debt limit constraints are binding, as well as the average asset and debt levels. When the government is restrained in terms of the amounts it can borrow and lend, it tends to be on average a lender rather than a borrower. This results in the average tax rate being lower compared to the balanced-budget case. In contrast, when the government’s borrowing and lending range becomes wider, the frequency by which it borrows increases, leading to a rise in the average debt level. The higher level of debt causes the average tax rate to increase relative to the case of the narrow ðB; BÞ range. The observed pattern in the government’s borrowing frequency can be interpreted in terms of a precautionary savings motive on its part. If the maximum amount that it is allowed to borrow is only 10, the government has a strong incentive to accumulate assets when its expenditures are low, i.e. in the ‘‘good times”. The reason is that the government knows that during the ‘‘bad times”, when its expenditures are high, it will have to raise taxes to finance its spending because of its limited ability in terms of the amount of debt it can issue. However, raising taxes creates a distortion in the economy. In order to minimize it, the government optimally chooses to accumulate sufficiently large amounts of assets during the ‘‘good times” given the constraint that its asset limit is 10. The interest earned from holding these assets helps finance expenditures during the ‘‘bad times”, thus reducing the amount by which taxes would have to increase otherwise. On the other hand, when its borrowing and lending range becomes wider (i.e., ðB; BÞ ¼ ð50; 50Þ), the government enjoys more flexibility in satisfying its
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b
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Fig. 3(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:75.
precautionary savings motive: it can now accumulate more assets during the ‘‘good times”, but it can also borrow more during the ‘‘bad times”. With the debt limit set at 50, there is less pressure for the government to accumulate assets compared to the ðB; BÞ ¼ ð10; 10Þ case, resulting in an increase in its borrowing frequency. Fig. 5 displays impulse response functions for leisure, tax rate, tax revenue, budget deficit, public debt and household wealth to a one unit positive innovation in government expenditures. Panel (a) corresponds to the case ðB; BÞ ¼ ð10; 10Þ, while Panel (b) refers to the case ðB; BÞ ¼ ð50; 50Þ. Each panel presents the impulse response functions of the various endogenous variables for each of the three specifications of (19) discussed above.15 As shown in Fig. 5, the increase in government expenditures causes the tax rate to rise. However, the impact response of s is smaller relative to the balanced budget case. This is shown by the fact that the resulting increase in tax revenues is not sufficient to finance the higher g entirely, leading the government to incur a budget deficit. The shortfall in revenues is financed through debt issue. Also, in the present model, the higher tax rate results to an increase in leisure. In combination with the higher tax rate, the reduction in the labor supply causes the representative household’s wealth to decline.
15 Recall that consumption evolves as a function of a ratio that is parametric to both the household and the government (see Footnote 13). For this reason, impulse response functions for consumption and the variable ct þ g t þ lt are not displayed.
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c
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Fig. 3(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:75.
These impulse response functions are qualitatively similar to the ones reported by Aiyagari et al. with the exception of leisure: in our model a positive shock in government expenditures causes leisure to rise, while in their model leisure falls. The reason behind this result is related to the following two effects. First, a positive innovation in government expenditures leads the government to raise the tax rate, which results in a reduction of the net wage and labor supply (the ‘‘net wage effect”). Second, the increase in government expenditures leads to an increase in borrowing, which raises the interest rate. This results to an increase in labor supply through the intertemporal substitution of leisure (the ‘‘interest rate effect”). In Aiyagari et al., the second effect is the dominant one. As a result, in their case leisure falls and labor supply rises. In the present small open economy framework, however, the interest rate effect is absent since consumption is determined purely by the (exogenous) stochastic process for the prices of the state-contingent claims. As a consequence, a positive innovation in government expenditures leads to a reduction in labor supply and an increase in leisure. As the ðB; BÞ range increases, the impact response of leisure, tax rate, tax revenue and household wealth becomes smaller, while the response of the budget deficit and public debt becomes larger. This is the case for all three specifications for g t and it reflects the fact that the increased borrowing and lending capacity of the government allows it to better distribute the distortionary cost of taxation over time.
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a
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Fig. 4(a). ðB; BÞ ¼ ð0; 0Þ q ¼ 0:75 [Adjusted Var (g)].
4.2. Government asset and debt holdings Fig. 6 displays the distribution of the government’s asset and debt holdings for the two ðB; BÞ unbalanced budget pairs. Along with Table 2, these histograms provide a concise summary of how the government engages in precautionary savings. When the borrowing and lending range is narrow (Fig. 6a, the distribution of asset/debt holdings is skewed for all three types of government expenditures. The asset limit constraint is binding less frequently when q ¼ 0:00 compared to the two cases of serially correlated expenditures. This indicates that the government’s precautionary savings motive is stronger when q ¼ 0:75. A possible explanation for this result may be related to the more ‘‘permanent” nature of a negative government spending shock in the latter case. That is, a high level of g in the current period is likely to be accompanied by high expenditure levels in the following periods that will eventually prompt the government to raise the tax rate. This provides the government with a strong incentive to accumulate assets that will contribute in minimizing the resulting distortion. With serially uncorrelated expenditures, it is less likely that a negative spending shock will be followed by high levels of g in subsequent periods. As a consequence, the government feels less pressure to save during the ‘‘good times”, resulting in the asset limit constraint binding less frequently in this case. Fig. 6b shows that widening the government’s borrowing and lending range causes a dramatic shift in the distribution of its asset/debt holdings. With its incentive to accumulate assets weakened, the government borrows more resulting in the
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b
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Fig. 4(b). ðB; BÞ ¼ ð10; 10Þ q ¼ 0:75 [Adjusted Var (g)].
distribution of its asset/debt holdings to no longer be skewed towards its asset limit, but rather be fairly symmetric and ceng tered around the initial level of indebtedness b0 ¼ 0. These results have important implications regarding the degree of tax smoothing achieved by the government. As shown in Table 1, the government’s borrowing and lending behavior has an impact on the volatility of the equilibrium tax rates as well. In general, as the ðB; BÞ range increases from (0,0) to (10,10), and then to (50,50), the standard deviation of the tax rate std ½s declines and the tax rates become smoother for all three types of government expenditures. Further, for a given ðB; BÞ range, a larger variability in g leads to a higher standard deviation in s: when q ¼ 0:75, std ½s is higher compared to the cases of the serially uncorrelated and volatility-adjusted serially correlated government expenditures. For example, consider the range (50,50): std ½s is 0.02956 when q ¼ 0:75 (unadjusted), 0.020537 when q ¼ 0:75 (adjusted), and 0.011302 when q ¼ 0:00. Finally, adjusting the variance of serially correlated expenditures still yields equilibrium tax rates that exhibit higher volatility relative to those obtained for serially uncorrelated expenditures. For instance, in the case of the range (10,10), std ½s is 0.034535 when q ¼ 0:75 (adjusted), and 0.014661 when q ¼ 0:00. This reflects the more ‘‘permanent” nature of the correlated spending shocks discussed earlier. This evidence makes clear that allowing the government to borrow and lend leads to smoother tax rates compared to the case of maintaining a balanced budget, regardless of the serial correlation properties of the underlying process for govern-
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Fig. 4(c). ðB; BÞ ¼ ð50; 50Þ q ¼ 0:75 [Adjusted Var (g)].
ment expenditures. It also suggests that there should be welfare gains for the representative household from the ‘‘tax smoothing” achieved by the government. In what follows, we consider how significant these welfare gains are. 4.3. Welfare analysis Table 3 reports the computed welfare values associated with all cases under study. The separability assumption regarding the representative household’s instantaneous utility allows us to distinguish between the welfare gains in terms of expected utility of consumption and leisure. Regardless of the serial correlation properties and volatility of government expenditures, welfare clearly improves from moving away from the period-by-period balanced budget constraint not only in terms of the total, but also in terms of consumption and leisure separately. As shown in the table, when the government is not permitted to borrow or lend the case that yields the highest welfare is that of the volatility-adjusted serially correlated g: total expected utility equals 305.7612. This is because the tax rate is influenced by the properties of government expenditures, which, in this case, are persistent and have low variance. In contrast, the lowest welfare in the balanced budget case is attained when q ¼ 0:00: total expected utility is equal to 305.6575. The complete lack of persistence in s outweighs the positive effect of the lower variability in g, thus ranking this case below that
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Table 1 Autoregressions of the tax rate ðB; BÞ
q ¼ 0:00 E½s std ½s a b R2
q ¼ 0:75 E½s std ½s a b R2
q ¼ 0:75 (Adj. Var(g)) E½s std½s a b R2
(0,0)
(10,10)
(50,50)
0.310812 (4.95083E-05) 0.02612 (1.68775E-05) 0.310521 (0.000353928) 0.000927 (0.001123558) 0.000121 (2.22609E-05)
0.309281 (0.00033999) 0.014661 (0.000292308) 0.15626 (0.005429174) 0.493288 (0.018086705) 0.275826 (0.016017117)
0.310808 0.011302 0.025241 0.917216 0.853218
0.310778 (0.000108633) 0.038987 (4.57851E-05) 0.079023(0.000206123) 0.745718 (0.000670511) 0.556206 (0.00099916)
0.308767 (0.000273902) 0.034535 (0.000136991) 0.050973 (0.000823633) 0.834713 (0.002795309) (0.004591606) 0.69759
0.311822 (0.000651786) 0.02956 (0.000111808) 0.026443 (0.000310484) 0.914964 (0.00118193) 0.837378 (0.002126377)
(7.99937E-05) 0.31077 (3.46008E-05) 0.026126 0.077718 (0.000218999) (0.000709507) 0.749917 0.562498 (0.001061168)
0.308885 (0.00028298) 0.022795 (0.000116778) 0.047133 (0.00101784) 0.847138 (0.003424574) 0.71893 (0.005703914)
0.310625 0.020537 0.029746 0.904042 0.817707
(0.000975966) (0.00024071) (0.003124988) (0.010758395) (0.015567273)
(0.00054263) (9.3308E-05) (0.000326213) (0.001208596) (0.002168645)
Note: Standard errors in parentheses.
Table 2 Government Borrowing and Lending Behavior Borrowing Frequency (%)
Binding Asset Limit (%)
Binding Debt Limit (%)
Average Assets
Average Debt
37.1973 (2.81602) 27.2277 (1.99281) 29.1935 (2.42449)
12.9266 (1.72029) 20.6868 (2.14007) 21.5661 (2.33242)
2.0023 (0.28077) 2.8820 (1.94100) 2.5380 (0.34823)
5.1368 (0.19662) 6.2217 (0.16964) 6.063754 (0.19566)
3.5538 (0.12213) 4.1580 (0.08128) 3.8108 (0.10803)
51.4797 (3.30006) 53.2361 (2.41591) 50.7788 (2.85682)
1.5885 (0.61585) 0.4309 (0.30992) 0.2686 (0.20543)
0.1719 (0.04936) 0.3952 (0.05956) 0.0015 (0.00073)
12.8915 (0.903050) 10.9416 (0.55749) 8.1180 (0.54192)
12.9978 (0.73473) 13.5692 (0.38339) 8.1909 (0.28579)
ðB; BÞ ¼ ð10; 10Þ
q ¼ 0:00 q ¼ 0:75 q ¼ 0:75 (Adj. Var(g)) ðB; BÞ ¼ ð50; 50Þ
q ¼ 0:00 q ¼ 0:75 q ¼ 0:75 (Adj. Var(g))
Note: Standard errors in parentheses.
of the volatility-unadjusted correlated expenditures in terms of welfare. In this case, total expected utility is equal to 305.7144. Once the period-by-period balanced budget constraint is relaxed, total utility and its components invariably increase. Providing the government with the ability to borrow and lend allows it to maintain persistent and smooth tax rates, thus reducing the distortionary effect of taxation. This leads to higher expected utility of consumption and leisure. The separability between consumption and leisure in the utility function facilitates a welfare comparison in terms of actual consumption and leisure of the representative household. Table 4 shows the required increase in consumption per period that will make the representative household indifferent between the balanced-budget policy regime and the one in which the government can borrow or lend.16 When q ¼ 0:00 and the government is allowed to borrow and lend in the range (10,10), the increase in consumption is approximately 0.2377% in all periods. For the volatility unadjusted and adjusted serially correlated government expenditures, this number drops to 0.1304% and 0.1155%, respectively. The magnitudes of the resulting welfare gains reflect the relative ranking of each case in terms of welfare when the government is forced to maintain a period-by-period balanced budget. As discussed earlier, the case of serially uncorrelated expenditures yields the lowest utility under the balanced budget constraint compared to the other two cases. Therefore, with q ¼ 0:00 there is plenty of room for improvement in welfare. On the other hand, in the case of serially correlated expenditures, the tax rate is already quite persistent and the (10,10) range is not wide enough to generate larger welfare gains. When the government is allowed to borrow and lend in the (50,50) range, the resulting welfare gains relative to the balanced-budget policy regime increase further: 0.5592% when q ¼ 0:00, 0.4064% when q ¼ 0:75, and 0.5368% for the vol16
The calculation of the reported welfare gains is described in Appendix B.
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a
x 10
-3
Leisure
x 10
8
6
6
4
4
-3
Tax Rate
2
2 0 0
0
5
10 15 Tax Revenue
20
25
0.6
1
0.4
0.5
0
5
10 15 Budget Deficit
20
25
0
5
10 15 Household Wealth
20
25
0
5
20
25
0.2 0 0 0
5
10
15
20
25
Debt 1 0 -0.2 0.5 -0.4 -0.6 0
0
5
10
15
20
25
10
15
Fig. 5(a). ðB; BÞ ¼ ð10; 10Þ - Impulse response functions.
atility-adjusted correlated g. These values indicate that the strong persistence of the spending shocks in the serially correlated cases hinders the government’s effort in smoothing taxes further. This can be seen from Table 1 by comparing the volatility of s in these two cases relative to the serially uncorrelated one. Reducing std ½s further, and thus improving welfare relative to the balanced budget case, would require that the ðB; BÞ range becomes even wider: less for the volatility-adjusted case and more for the unadjusted one. The reason is that the former type of expenditures exhibits lower variability compared to the latter and, therefore, the government would find a narrower ðB; BÞ range sufficient for smoothing taxes. Finally, as part of a preliminary sensitivity analysis, we considered different values for the two preference parameters. In particular, we set r1 ¼ 0:6 and r2 ¼ 2:2. Although in either case the results obtained are quantitatively different, qualitatively they are similar in nature to the ones for the benchmark parameterization.17 5. A comparison of incomplete with complete markets In this section we compare the welfare gains in the incomplete markets case with the complete markets case in which the government is also allowed to issue state-contingent debt. In order to facilitate computations in the incomplete markets case and be able to derive an analytic solution for the equilibrium allocation in the complete markets case,18 we make the simplifying assumption that the normalized asset prices are constant
^htþ1 ¼ b: p
ð22Þ
Eq. (22) is a special case of Eq. (20) with sp ! 1. In the complete markets case, the household’s problem is the same as before, described by expressions (1)–(3), and we assume that the utility function is given by Eq. (18). The government’s flow budget constraint (6) is replaced by
17
These results are available from the author upon request. Þ, as it is the If I were to use (20) for the complete markets case, then I would have to guess two multipliers (see multipliers U and K below) instead of one ðc case under incomplete markets. This would make the numerical scheme described in Appendix B computationally much more intensive. 18
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
b
x 10
-3
Leisure
x 10
8
6
6
4
4
-3
Tax Rate
2
2 0 0
0
5
10 15 Tax Revenue
20
25
0.6
1
0.4
0.5
0
5
10 15 Budget Deficit
20
25
0
5
10 15 Household Wealth
20
25
0
5
20
25
0.2 0 0 0
5
10
15
20
25
Debt 1 0 -0.2 0.5 -0.4 -0.6 0
0
5
10
15
20
25
10
15
Fig. 5(b). ðB; BÞ ¼ ð50; 50Þ - Impulse response functions.
Z
g
g
phtþ1 btþ1 dstþ1 ¼ bt þ g t st nt ;
8t P 0
ð23Þ
which states that the government finances its expenditures using a labor income tax at the time-varying flat rate st and the g sale of state-contingent bonds btþ1 at the exogenously determined prices phtþ1 for each possible event stþ1 . As before, these asset prices are measured in units of time-t output and are assumed to be determined by the international financial markets. In this case, the Ramsey problem becomes
max1 E0
fct ;nt gt¼0
1 X
bt ½tðct Þ þ Hð100 nt Þ
ð24Þ
t¼0
subject to:
t0 ðc0 Þbh0 ¼ E0
1 X
bt ½t0 ðct Þct H0 ð100 nt Þnt ;
ð25Þ
t¼0
pht ¼ bpðst jst1 Þ
t0 ðc0 Þbg0 ¼ E0
t0 ðct Þ ; 8t P 1 t0 ðct1 Þ
1 X
bt f½t0 ðct Þ H0 ð100 nt Þnt t0 ðct Þg t g;
ð26Þ ð27Þ
t¼0 h
g
and b0 ; b0 given. Constraint (25) is the time-0 version of the household’s present value budget constraint (11). The right-hand side of constraint (26) is the one-period asset pricing kernel. Eq. (27) is the government’s present value budget constraint. It is derived by substituting (4) and (5) into (23), and using forward substitution along with the transversality condition on g btþ1 . Given (22), optimality condition (26) reduces to t0 ðct Þ ¼ t0 ðct1 Þ, which implies that the household achieves perfect consumption smoothing: ct ¼ c; 8t P 0. Using this result, the above Ramsey problem reduces to maximizing (24) subject to
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 30
rho = 0.00
rho = 0.75
rho = 0.75 (Adj. Var(g))
Asset/Debt Holdings (%)
25
20
15
10
5
0 [-10,-9) [-9,-8) [-8,-7) [-7,-6) [-6,-5) [-5,-4) [-4,-3) [-3,-2) [-2,-1)
[-1,0)
(0,1]
(1,2]
(2,3]
(3,4]
(4,5]
(5,6]
(6,7]
(7,8]
(8,9]
(9,10]
Asset and Debt Limit Range Fig. 6a. Histogram of asset/debt holdings – range (10,10).
18
16
rho = 0.00
rho = 0.75
rho = 0.75 (Adj. Var(g))
(10,15]
(20,25]
(30,35]
Asset/Debt Holdings (%)
14
12
10
8
6
4
2
0 [-50,-45) [-45,-40) [-40,-35) [-35,-30) [-30,-25) [-25,-20) [-20,-15) [-15,-10) [-10,-5)
[-5,0)
(0,5]
(5,10]
(15,20]
(25,30]
(35,40]
(40,45]
(45,50]
Asset and Debt Limit Range Fig. 6b. Histogram of asset/debt holdings – range (50,50).
(25) and (27). Attaching multipliers U and K to constraints (25) and (27), respectively, the first order conditions with respect to nt ; c; U and K are given by:
H0 ð100 nt Þð1 U KÞ H00 ð100 nt Þnt ðU þ KÞ Kt0 ðcÞ ¼ 0; 8t P 0 " # 1 X t0 ðcÞ 1 g t 00 h 00 0 00 þ U t ðcÞb0 ðt ðcÞc þ t ðcÞÞ þ Kt ðcÞ b0 þ E0 b ðg t nt Þ ¼ 0; 1b 1b t¼0
ð28Þ
; 8t P 0. along with (25) and (27). Since c; U and K are constant, for condition (28) to hold it must be the case that nt ¼ n ; 8t P 0. Condition (4) from the household’s problem then implies that st ¼ s In order to facilitate the implementation of a welfare comparison between the current complete markets case, and the incomplete markets case, we keep the same parameterization and numerical scheme discussed in Section 3. The only excep-
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
Table 3 Welfare Values ðB; BÞ
(0,0)
(10,10)
(50,50)
291.4932853 (0.013817863) 14.16421793 (0.000202164) 305.6575032 (0.013941728)
291.8835986 (0.013832602) 14.16777129 (0.000205043) 306.0513699 (0.013958878)
292.4060292 (0.013874024) 14.17696593 (0.000204682) 306.5829951 (0.014001694)
291.548406 (0.013820154) 14.16601672 (0.000567954) 305.7144227 (0.013959256)
291.7623963 (0.013827545) 14.16818736 (0.000572013) 305.9305836 (0.013970694)
292.2104451 (0.01383665) 14.17700934 (0.000571063) 306.3874544(0.013977294)
291.596178 (0.013822958) 14.16508113 (0.000392043) 305.7612591 (0.013958741)
291.7853859(0.013834417) 14.16732935 (0.000394198) 305.9527152 (0.013972074)
292.4711116 (0.013849848) 14.17892195 (0.000393016) 306.6500336 (0.013984458)
q ¼ 0:00 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 (Adj. Var(g) Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility Note: Standard errors in parentheses.
Table 4 Welfare Gains as a % Increase in Consumption per Period From
To
q ¼ 0:00
q ¼ 0:75
q ¼ 0:75 (Adj. Var(g)
(0,0)
(10,10)
(0,0)
(50,50)
0.237773% (1.296632E05) 0.559157% (6.715276E05)
0.130437% (1.301744E05) 0.406405% (1.277812E05)
0.115509% (8.557878E06) 0.536777% (4.917141E06)
Note: Standard errors in parentheses.
Table 5 Welfare Values Fiscal Policy Regime
Balanced Budget
Incomplete Markets
Complete Markets
291.6063904 (0.001365166) 14.1652645 (0.000155885) 305.7716549 (0.001316528)
293.0216887 (0.000227834) 14.18384096 (0.000163744) 307.2055297 (0.000236365)
293.7756740 (0.009132014) 14.18469800(7.99113E-05) 307.9603720 (0.009052161)
291.4581813 (0.00136542) 14.16529552 (0.000548679) 305.6234768 (0.001246853)
292.6766109 (0.000568209) 14.18057163 (0.000560944) 306.8571825 (0.000622881)
293.7719290 (0.031419772) 14.18472600 (0.000274935) 307.9566550 (0.031144854)
q ¼ 0:00 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility
q ¼ 0:75 Expected Utility of Consumption Expected Utility of Leisure Total Expected Utility Note: Standard errors in parentheses.
tion is the scale factor sp in (20), whose value is raised to 75,000.19 Finally, for the incomplete markets case, we consider only the range ðB; BÞ ¼ ð50; 50Þ, which is the most favorable range from a welfare point of view. Table 5 reports the computed welfare values associated with three cases: (i) complete markets, (ii) incomplete markets and (iii) balanced-budget with serially uncorrelated and serially correlated (volatility unadjusted) government expenditures. Compared to the corresponding cases in Table 3, it is clear that shutting down one source of uncertainty - the one associated ^htþ1 – generates welfare gains in terms of consumption and leisure in both the incomplete markets and balanced-budwith p get cases. Nonetheless, both are dominated in terms of total welfare by the complete markets case. These results hold for both types of government expenditures. Table 6 demonstrates that when q ¼ 0:00, the consumption expenditures of the representative household have to increase by 0.453850% in all periods in order to make it indifferent between the incomplete and complete markets cases. This increment in consumption should be 1.324432% in all periods in order to result in indifference between complete markets and a balanced budget. For the representative household to be indifferent between a balanced budget at all times and a government that is allowed to issue state-contingent debt, when g is serially correlated, its consumption expenditures have to
19
^ht is approximately equal to b in all periods. This is a large enough number to ensure that p
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
459
Table 6 Welfare Gains as a % Increase in Consumption per Period From
To
q ¼ 0:00
q ¼ 0:75
Balanced Budget Incomplete Markets
Complete Markets Complete Markets
1.324432% (0.000529) 0.453850% (0.000551)
1.412866% (0.001892) 0.662165% (0.001899)
Note: Standard errors in parentheses.
increase by 1.412866% in all periods. This increment in consumption spending has to fall to 0.662165% in all periods to result in indifference between the incomplete and complete markets cases. Overall, the effect of assumption (22) about asset prices on the welfare gains of moving away from a fiscal policy regime that restricts the government to maintain a balanced budget at all times towards a regime that allows it to borrow and lend roughly 50% of its GDP, is stronger when q ¼ 0:00. In this case, consumption should increase by 0.870582% in all periods in order for the representative household to be indifferent between the incomplete markets and balanced-budget cases. With serially correlated government expenditures this increment in consumption is only 0.750701% in all periods. This is consistent with the result obtained in Section 4 where we allowed for substantial variation in consumption expenditures. The above results imply that the improvement in welfare relative to a balanced budget from the existence of state-contingent public debt is smaller when q ¼ 0:75 compared to the case of serially uncorrelated government expenditures. Finally, comparing the complete and incomplete markets cases in the present context, the welfare gains of state-contingent government debt are higher when government expenditures are serially correlated. As it was shown in the previous section, the welfare gains from allowing the government to issue nonstate-contingent debt fall when the degree of persistence in government expenditures increases. Hence, it is not surprising that the welfare gains from complete relative to incomplete markets increase in the present case compared to the serially uncorrelated one. 6. Conclusions This paper considered a small open economy in which only households have access to complete financial markets. On the other hand, the government is constrained to borrow and lend at a constant and exogenously determined risk-free interest rate. Allowing the policymaker to borrow and lend large amounts yields a Ramsey equilibrium that replicates Barro’s ‘‘taxsmoothing” result regarding the conduct of optimal fiscal policy in the presence of uncertainty. Considering three different stochastic processes for government expenditures, it is shown that relaxing the policymaker’s borrowing and lending constraints allows more flexibility to smooth tax rates across time, thus reducing the distortionary effect of taxation and improving social welfare. This is in contrast to a fiscal policy regime, in which the policymaker is constrained to maintain a balanced budget at all times. Allowing the government to issue state-contingent debt and given a specific stochastic process for the prices of the statecontingent financial claims, it is shown that overall welfare improves relative to the incomplete markets and balanced budget cases. Furthermore, compared to the incomplete markets case, the resulting welfare gains appear to be higher as the persistence in the stochastic process for government expenditures increases. It would be interesting to work out several extensions of the current model. For instance, the welfare comparison between the complete markets case with the incomplete markets and balanced-budget cases was based on assumption (22) about asset prices. This assumption allowed us to derive an analytic solution for the equilibrium allocation in the complete markets case. Future work should appropriately modify this numerical scheme and solve the complete markets model under assumption (20). This will enable us to get a sense of how the welfare gains resulting from issuing state-contingent debt depend on the assumed asset pricing process. Furthermore, given the overall focus of the paper on welfare, sensitivity analysis is especially important regarding the exogenous process for the price of state-contingent claims (20). It would also be interesting to consider preferences such as the ones discussed in Barro and Sala-i-Martin (2004, p. 423), that are consistent with the empirical regularities observed in the growth literature, as well as technology shocks in addition to stochastic government expenditures as in Marcet and Scott (2003). The basic model of this paper can be extended by endogenizing the choice of government expenditures and adding a maturity structure in the issue of public debt. These modifications will not only add more realistic features to the present setup, but will also facilitate a quantitative exercise in which the model is calibrated to an actual economy. Finally, an interesting question that needs to be addressed is whether the welfare loss resulting from a balanced budget rule is higher or lower in a small open economy compared to a closed one. Appendix A. Solving the Ramsey problem This appendix provides a description of the computational scheme involved in solving the Ramsey problem numerically. The decision variables at the initial period t ¼ 0 are
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 g
h
y0 ¼ ½c0 ; n0 ; l0 ; b1 ; b1 ; c0 ; /0 ; k0 ; h10 ; h20 0 ; and for t P 1
h i0 g h ^ t ; kt ; h1t ; h2t ; yt ¼ ct ; nt ; lt ; btþ1 ; btþ1 ; ct ; /t ; l h h g ^ht ; g t Þ, the state variables at t ¼ 0 are where btþ1 is a random variable measurable with respect to ðst ; b0 ; b0 Þ. Letting st ¼ ðp
h i0 g h x0 ¼ b0 ; b0 ; c1 ; s0 ;
^h0 ¼ 0 and, by definition, c1 ¼ 0. On the other hand, at t P 1: where p
h i0 g h xt ¼ bt ; bt ; ct1 ; ct1 ; st : h
h
Next, we argue that bt should not be included in the vector of state variables, xt . Constraint (11) implies that bt is determined by a time invariant function DðÞ:20
g h bt ¼ D bt ; ct1 ; ct1 ; st ;
8t P 1:
In other words, the stock of state-contingent financial claims held by the household at time t P 1 depends on the same set of state variables as yt . Therefore, it is not necessary to include it as an additional state variable when solving for yt . Based h on this, the implementation of the algorithm does not require solving for btþ1 .21 For the quantitative component of the paper, the instantaneous utility function of the representative household is assumed to be additively separable in consumption and leisure:
uðct ; lt Þ ¼ tðct Þ þ Hðlt Þ;
8t P 0: g
^ t ; kt and ct are Given this assumption, the first order conditions for the Ramsey problem with respect to ct ; nt ; btþ1 ; l
"
t00 ðc0 ÞH0 ð100 n0 Þ ^ 1p ^h1 ¼ 0; þ bt00 ðc0 ÞE0 l t0 ðc0 Þ2
t0 ðc0 Þ c0 ðc0 k0 Þn0 "
t0 ðct Þ ct ðct kt Þnt
#
# t00 ðct ÞH0 ð100 nt Þ
t0 ðct Þ2
H0 ð100 nt Þ þ kt þ ðct kt Þ kt h1t þ h2t ¼ Et ktþ1 ; ^ht 0 ðct1 Þ p
^ tþ1 p ^htþ1 ¼ 0; ^ t þ bt00 ðct ÞEt l bt00 ðct Þl
0 H ð100 nt Þ H00 ð100 nt Þnt ¼ 0; t0 ðct Þ
8t P 0
0
t
¼ bt ðct Þ; 8t P 1 H0 ð100 nt Þ ¼ þ gt 1 nt ; 8t P 0 0 t ðct Þ 0 H ð100 nt Þ h h bt ¼ ct nt þ bEt btþ1 ; 8t P 1 t0 ðct Þ
g bbtþ1
ð29Þ
g bt
8t P 0
8t P 1
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
In addition, the standard Kuhn–Tucker conditions apply:
g h1t B bbtþ1 ¼ 0; g h2t bbtþ1 B ¼ 0;
h1t P 0; 8t P 0
ð36Þ
h2t P 0; 8t P 0:
ð37Þ
A useful result can be derived from the first order condition with respect to the state-contingent bonds purchased by the h household, btþ1 , which is given by:
btþ1 pðstþ1 jst Þ/tþ1 ¼ 0;
8t P 0; 8st ; stþ1 2 S:
ð38Þ
This implies that
/tþ1 ¼ 0;
8t P 0:
ð39Þ
Hence, by making use of definition (16), we obtain
ctþ1 ¼ ct ¼ c; 8t P 0:
20
ð40Þ
For a formal argument, see Proposition 1 in Marcet and Scott (2003). This does not imply that the portfolio of state-contingent financial claims purchased by the household in each period is indeterminate. Once the Ramsey problem in its current form is solved and the policy rules for all endogenous variables as a function of the state variablesbecome known, I can use (11) to derive h bt for each period. However, this is computationally a very expensive task to perform. 21
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C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
In other words, ct remains constant in all periods and can be removed from xt ; 8t P 0. Using a more compact notation than previously, the state and decision variables that are the focus of the computational component in the paper can be summarized as follows:
0 g ^ t ; kt ; h1t ; h2t ; yt ¼ ct ; nt ; lt ; btþ1 ; ct ; /t ; l
8t P 0
and
g 0 xt ¼ bt ; ct1 ; st ;
8t P 0
where
^ht ; g t Þ; 8t P 0 st ¼ ðp ct ¼ c; 8t P 0 ; /0 ¼ c /t ¼ 0; c1 ¼
^h0 p
8t P 1 ^ 0 ¼ 0: ¼ c1 ¼ l
Appendix B. Computational scheme This Appendix discusses the numerical scheme used in solving the Ramsey problem outlined in Section 2 and Appendix A. The proposed algorithm is an appropriate adaptation to the present context of the Parameterized Expectations Algorithm (PEA), as it is described in den Haan and Marcet (1990). According to this algorithm the agents’ conditional expectations about functions of future variables are replaced by an approximating function involving the state variables of the system and coefficients on each of these variables. The approximating function is used to generate T realizations of the endogenous variables of the model, where T is a very large number. These realizations are used as observations in a series of nonlinear least squares regressions which are used to reestimate the coefficients of the approximating function. Then the new set of coefficients obtained are used to generate a new series of length T for the endogenous variables. Iterations continue until the regression coefficients obtained from the use of successive sets of coefficients for the approximating function converge up to a prespecified tolerance level. The steps involved are the following: Step 1. Choose appropriate values for the parameters of the model. Assuming that the exogenous state variables of the ^ht , follow particular stochastic processes, generate a series of realizations for each of them. These series model, g t and p have length T, where T is a large number, and are drawn only once. j , where subscript j is an integer used to denote the current Step 2. Make a guess for the value of the pseudo-multiplier c; c outer-loop iteration step. Step 3. Replace the conditional expectations included in first-order conditions 29, 30 and 32 of the Ramsey Problem by approximating functions wðaki ; xt Þ, where aki denotes a fixed vector of parameters, subscript i represents the current inner-loop iteration step, and k ¼ 1; 2 corresponds to the particular conditional expectation replaced by a function wð; xt Þ. Formally, expressions 29, 30 and 32 can be compactly rewritten as: 0
v ðct Þ ct ðct kt Þnt
" # v00 ðct ÞH0 ð100 nt Þ v0 ðc
2
tÞ
^ t þ bv00 ðct Þw a1i ; xt ¼ 0; bv00 ðct Þl
8t P 0
ð41Þ
and
kt h1t þ h2t ¼ w a2i ; xt ;
8t P 0
ð42Þ
^ 0 ¼ 0. where l ~i ; c j ÞgTt¼0 , that solves the sys~i ¼ a1i ; a2i 0 , obtain a long series of the endogenous variables of the model fyt ða Step 4. Letting a tem of Eqs. 15, 31, 33, 34, 36, 37, 41 and 42.22 Towards this end, we have to distinguish between four possible cases, depending on whether: (i) constraint (14) is binding, and (ii) the allocation derived corresponds to the initial period. g (a) Unconstrained Case for t ¼ 0: In this case, B < bb1 < B and from the Kuhn–Tucker conditions (36) and (37) it follows that h10 ¼ h20 ¼ 0. Then, from (42) we obtain k0 . First-order conditions (31) and (41) consist a system of two (nonlinear) equations in two unknowns: c0 and n0 . Once n0 has been calculated, the time endowment constraint g (15) provides us with l0 . Finally, making use of c0 and n0 , we compute b1 from (34). g (b) Constrained Case for t ¼ 0: In this case, either b1 ¼ B=b or B=b, which by making use of (36) and (37) implies that h10 > 0 and h20 ¼ 0 or h20 > 0 and h10 ¼ 0, respectively. Eqs. (41), (31) and (34) consist a system of three equations in three unknowns: c0 ; n0 and k0 . Once n0 has been calculated, the time endowment constraint (15) yields l0 . Finally, given the multiplier k0 , first-order condition (42) provides us with either h10 or h20 . 22
h
As it was explained in Appendix A, since bt is not a variable of immediate interest, first-order condition (35) is omitted.
462
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463 g
(c) Unconstrained Case for t P 1: In this case, B < bbtþ1 < B and from the Kuhn–Tucker conditions (36) and (37) it follows that h1t ¼ h2t ¼ 0. From (42), we then obtain kt . Given ct1 calculated from the previous period, first-order condition (33) provides us with ct . Eqs. (31) and (41) then consist a system of two equations in two unknowns: nt ^ t . Given the optimal supply of labor, the time endowment constraint (15) yields lt . Finally, by substituting ct and l g and nt into (34), we compute btþ1 . g (d) Constrained Case for t P 1: In this case, either btþ1 ¼ B=b or B=b, which by making use of (36) and (37) implies that h1t > 0 and h2t ¼ 0 or h2t > 0 and h1t ¼ 0, respectively. As in the previous case, given ct1 calculated from the previous period, ct can be derived from first-order condition (33). Eqs. (31), (34) and (41) consist then a system of three ^ t ; nt and kt . Once nt has been calculated, we obtain lt from the time endowment conequations in three unknowns: l straint (15). Finally, given the multiplier kt , first-order condition (42) provides us with either h1t or h2t . Step 5. Given the series of the endogenous variables of the model calculated in the previous step, compute the variables that ^ tþ1 p ^htþ1 and ktþ1 ; 8t P 0. Next, appear inside the conditional expectations in expressions (29), (30) and (32): that is, l perform nonlinear least squares regressions of these variables on the corresponding wð; xt Þ functions. Define the ~i Þ. result of these regressions as Sða ~i according to the following scheme: Step 6. Using a predetermined relaxation parameter d 2 ð0; 1, update a
~iþ1 ¼ ð1 dÞa ~i þ dSða ~i Þ: a ~iþ1 a ~i k < , where is a small positive number. Denote the fixed point obtained by a ~0 . Step 7. Iterate until ka Step 8. Consider H histories, each history being P periods long. Generate corresponding series for the exogenous state vari~0 , repeat Steps 3 and 4 to derive the allocations and corresponding tax ^ht ) and, using a ables of the model (g t and p rates. Next, compute the following expression: g
dj ¼ b0
X P 1 H X j Þnh;p ðc j Þ g h;p ; bp ½sh;p ðc H h¼1 p¼0
j until jdj j < e, which is an approximation to the government’s present value budget constraint.23Keep iterating on c where e is a small positive number. adjustment in the variance of the serially correlated g is performed by multiplying the ðegtþ1 =sg Þ term in (19) by ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pThe 1 q2 , where q ¼ 0:75. Then, the variance of g for the uncorrelated and correlated cases is the same and equal to 1=s2g . In Step 1 of our algorithm, we set T ¼ 10; 000. In Step 8, we set H ¼ 99 and P ¼ 101. The exact same sequences of shocks in Steps 1 and 8 are used for all three pairs of debt and asset limits. In order to facilitate the computation of standard errors for all the results reported in Tables 1–4, we perform in each case N ¼ 100 simulations of length T ¼ 10; 000 each. Finally, note that the average standard deviation of the simulated g series is (standard errors in parentheses): 2.500956 (0.00152536) when q ¼ 0:00; 3:7319 ð0:004384799Þ when q ¼ 0:75, and 2.500825 (0.003302303) when we adjust the volatility of serially correlated government expenditures. On the other hand, ^h series is 0.0012 (3.051E-09). the average standard deviation of the simulated p In the ‘‘balanced budget” case, we set B ¼ B and both limits equal to 1E 08 for computational simplicity. This number is essentially zero and allows us to use the computational scheme described above without having to write a separate code. It should be noted, however, that the balanced budget case in this model is easy to solve since the dynamic model separates into a sequence of static unconnected optimization problems. Given the chosen parameterization, the government’s budget constraint reduces to: 1 3
1 2
ct 2 lt þ ð100 g t Þct 2 lt þ lt 100 ¼ 0: For t ¼ 0, Step 4(b) of the above numerical scheme applies. For t P 1; ct is determined by (33) in Appendix A and the government’s budget constraint becomes a cubic equation in leisure. From the three implied tax rates we pick the lowest one. Both methods yield virtually identical results.24 The calculation of the reported welfare gains is based on performing in every case N ¼ 100 simulations of length T ¼ 10; 000 each. For each individual simulation, the calculation of the welfare gains is performed as follows. Consider two fiscal policy regimes: A and B. The separability assumption allows us to write the expected welfare in either regime as W i ¼ W ci þ W li , where i ¼ A; B. W ci denotes the expected utility of consumption and W li the expected utility fA ¼ W f c þ W l , such that: of leisure. Assume that W A < W B . Let W A A
f A ¼ W B: W
ð43Þ
f c is given by: By definition, W A 23
The latter can be obtained by using forward substitution on (6) and imposing a transversality condition on the government’s debt as T ! 1 to obtain: g
b0 ¼ E0
1 X
bt ðst nt g t Þ
t¼0 24
I would like to thank an anonymous referee for pointing this issue to me.
C. Angyridis / Journal of Macroeconomics 31 (2009) 438–463
f c ¼ E0 W A
1 X t¼0
bt
! ec t ðAÞ1r1 1 : 1 r1
463
ð44Þ
Let f denote the increase in consumption in all periods that is required in order to make the representative household indifferent between the two fiscal policy regimes. Setting ~ct ðAÞ ¼ ð1 þ fÞct ðAÞ in (44) and solving (43) for f yields:
31=ð1r1 Þ 1 þ ð1 r1 Þð1 bÞ W cB þ W lB W lA 5 f¼4 1: 1 þ ð1 r1 Þð1 bÞW cA 2
In calculating the expected utility of consumption and leisure, we consider H ¼ 99 histories, each history being P ¼ 101 periods long. For each history, we generate series for government expenditures and asset prices, and derive the corresponding allocation according to Step 4 of our numerical scheme. Finally, we compute the following expression:
! " !# X r1 1r P H X P c1 1 lh;p 2 1 1 H X 1 X h;p p p þ ; b b g 1 r1 H h¼1 p¼0 H h¼1 p¼0 1 r2 which is an approximation to the representative household’s expected utility. References Adler, J., 2006. The tax-smoothing hypothesis: Evidence from Sweden, 1952–1999. Scandinavian Journal of Economics 108 (1), 81–95. Aiyagari, S.R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109 (3), 659–684. Aiyagari, S.R., Marcet, A., Sargent, T.J., Seppala, J., 2002. Optimal taxation without state-contingent debt. Journal of Political Economy 110, 1220–1254. Angeletos, G.M., 2002. Fiscal policy with non-contingent debt and the optimal maturity structure. Quarterly Journal of Economics 117 (3), 1105–1131. Atkeson, A., Chari, V.V., Kehoe, P.J., 1999. Taxing capital income: A bad idea. Federal Reserve Bank of Minneapolis Quarterly Review 23, 3–17. Atkinson, A.B., Stiglitz, J., 1980. Lectures on Public Economics. McGraw-Hill, New York. Barro, R.J., 1979. On the determination of public debt. Journal of Political Economy 87, 940–971. Barro, R.J., Sala-i-Martin, X., 2004. Economic Growth, second ed. MIT Press. Buera, F., Nicolini, J.P., 2001. Optimal Maturity of Government Debt without State Contingent Bonds. University of Chicago, Mimeo. Cashin, P., Olekalns, N., Sahay, R., 1998. Tax Smoothing in a Financially Repressed Economy: Evidence from India, IMF Working Paper No. WP/98/122, International Monetary Fund, Washington, DC. Cashin, P., Haque, N., Olekalns, N., 1999. Spend Now, Pay Later? Tax Smoothing and Fiscal Sustainability in South Asia, IMF Working Paper No. WP/99/63, International Monetary Fund, Washington, DC. Chari, V.V., Kehoe, P.J., 1999. Optimal fiscal and monetary policy. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, 1. North Holland, Amsterdam, pp. 1671–1745. 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., Christiano, L.J., Kehoe, P.J., 1995. Policy analysis in business cycle models. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Correia, I.H., 1996. Dynamic optimal taxation in small open economies. Journal of Economic Dynamics and Control 20, 691–708. den Haan, W.J., Marcet, A., 1990. Solving the stochastic growth model by parameterizing expectations. Journal of Business and Economic Statistics 8, 31–34. Kydland, F.E., Prescott, E.C., 1980. Dynamic optimal taxation, rational expectations and optimal control. Journal of Economic Dynamics and Control 2, 79–91. Lloyd-Ellis, H., Zhan, S., Zhu, X., (2001). Tax Smoothing with Stochastic Interest Rates: A Re-assessment of Clinton’s Fiscal Legacy, Center for Research on Economic Fluctuations and Employment, UQAM, Working Paper No. 125. Lucas Jr., F.E., Stokey, N.L., 1983. Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55–93. Marcet, A., Marimon, R., 1999. Recursive Contracts. Universitat Pompeu Fabra, Barcelona, Mimeo. Marcet, A., Scott, A., 2003. Debt and Deficit Fluctuations and the Structure of Bond Markets. Universitat Pompeu Fabra, CREI and CEPR, Mimeo. Razin, A., Sadka, E., 1995. The status of capital income taxation in the open economy. FinanzArchiv 52, 21–32. Sargent, T.J., Velde, F., 1995. Macroeconomic features of the french revolution. Journal of Political Economy 103 (3), 474–518. Schmitt-Grohé, S., Uribe, M., 1997. balanced-budget rules, distortionary taxes, and aggregate instability. Journal of Political Economy 105, 976–1000. Schmitt-Grohé, S., Uribe, M., 2004. Optimal fiscal and monetary policy under sticky prices. Journal of Economic Theory 114, 198–230. Sleet, C., 2004. Optimal taxation with private government information. Review of Economic Studies 71, 1217–1239. Sleet, C., Yeltekin, S., 2006. Optimal taxation with endogenously incomplete debt markets. Journal of Economic Theory 127, 36–73. Stockman, D.R., 2001. balanced-budget rules: Welfare loss and optimal policies. Review of Economic Dynamics 4, 438–459. Stockman, D.R., 2004. Default, reputation, and balanced-budget rules. Review of Economic Dynamics 7, 382–405. Strazicich, M.C., 1997. Does tax smoothing differ by the level of government? Time series evidence from Canada and The United States. Journal of Macroeconomics 19 (2), 305–326. Strazicich, M.C., 2002. International evidence of tax smoothing in a panel of industrial countries. Applied Economics 34, 2325–2331.