Government Debt and Optimal Monetary and Fiscal Policy

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1 Government Debt and Optimal Monetary and Fiscal Policy Klaus Adam Manneim University and CEPR - preliminary version - June 7, 21 Abstract How do di erent levels of government debt a ect te optimal conduct of monetary and scal policies? Tis paper studies a standard monetary policy model wit nominal rigidities and monopolistic competition and adds to it a scal autority tat issues nominal non-state contingent debt, levies distortionary labor income taxes and determines te level of public goods provision. In tis setting, adverse tecnology socks cause government tax revenues to fall. In te absence of outstanding government debt, te budget is optimally balanced troug spending reductions alone, leaving tax rates, in ation and government debt uncanged. Wit positive government debt, tax rates are iger to start wit and te revenue losses correspondingly larger. It is ten optimal to allow for a persistent increase in government debt and income tax rates and for temporarily iger in ation. Since government debt gives rise to budget and tax risk, it becomes optimal to reduce government debt towards zero over time. Te optimal speed of debt reduction can be quantitatively signi cant, especially wen tecnology movements are largely unpredictable in nature. Keyword: Ramsey optimal policy, second-order approximation, noncontingent debt, sticky prices JEL Class No: E63, E61 1 Introduction Following te nancial crisis in 27 and te ensuing Great Recession, governments in many OECD economies ave implemented expansionary scal policy measures in addition to o ering rescue packages of unprecedented size to te nancial sector. Tese decisions ave lead to a considerable increase in te level of government indebtedness, triggering in some countries even fears about te sustainability of public nances. Figure 1 illustrates tis debt increase, depicting te evolution of te central governments liabilities 1

2 Figure 1: Central Government Debt/GDP: History & OECD Forecast in relation to GDP for a selected group of OECD economies. Debt levels strongly increased over te period and te OECD forecasts for te years 21 and 211 sow tat debt levels are expected to increase even furter. Naturally tese developments raise te question wic normative implications follow from te large build-up of government debt for te conduct of monetary and scal policy? Is it optimal to allow keep debt levels at tese elevated levels? Sould stabilization attempts in te future depend on te fact tat government debt levels are iger to start wit next time an adverse sock its te economy? To provide an answer to tese questions, te present paper analyzes a stylized dynamic equilibrium model and determines ow te optimal conduct of monetary and scal policy depends on te level of accumulated government debt. Te model analyzed in tis paper tereby considers tree government instruments tat are generally considered relevant for te conduct of monetary and scal policy, namely (1) monetary policy de ned as control of te sort-term nominal interest rate, (2) scal policy in te form of spending decisions on public goods, and (3) a scal nancing decision determining weter to use labor income taxes or government debt as means to nance current expenditure, were government debt is assumed to be nominal and non-state contingent. Te paper determines ow tese tools sould be used as stabilization instruments in response to tecnology socks and weter tis sould depend on te level of outstanding government debt. In addition, it determines te economy s long-run outcomes under optimal monetary and 2

3 scal policy. Te economic environment considered in tis paper features tree important distortions. First, rms are assumed to possess monopoly power in product markets wic allows tem to carge a mark-up over marginal cost. Tis causes output to generally fall sort of its rst best level. Second, scal policy as to use distortionary labor income taxes to nance public goods provision and interest payments on outstanding debt. Public spending and outstanding debt tus ave additional adverse labor supply and output e ects. Finally, nominal rigidities in te price of nal goods prevent prices from fully adjusting in response to economic disturbances and policy measures. 1 Eac of tese distortions as important implications for te optimal conduct of policy. Firms monopoly power and te fact tat te government can levy distortionary income taxes only make it optimal to reduce government spending below wat is suggested by te allocation rule for public and private consumption in te rst best outcome. It is tereby optimal to reduce public goods provision more, te iger is te outstanding government debt/gdp ratio. Furtermore, as in Scmitt-Groé and Uribe (24), nominal rigidities prevent te government from using price level canges as an important source of state-contingent taxation in te presence of nominal government debt. As a result, government debt optimally follows a near random walk, as in Barro (1979) and Aiyagari et al. (22). However, unlike in tese papers, te standard deviations of te innovation to tis near random walk crucially depends on te level of outstanding government debt. Tis is result emerges because te present paper considers a model wit endogenous spending decisions in wic tecnology socks are te underlying driving force, wile te papers mentioned above assume government spending to evolve according to an exogenous stocastic process. Te latter as important implications for optimal debt dynamics. In particular, wen te government as no debt outstanding, it is optimal to balance any revenue sortfall from an adverse tecnology sock exclusively troug a corresponding reduction in government spending. Tax rates and debt can ten remain uncanged in response to socks. Yet, for positive (or negative) debt levels, taxes and debt levels respond by permanent movements and te size of tese movements increases in te level of outstanding debt. Speci cally, larger initial government debt levels result in larger debt increases following an adverse tecnology sock and require correspondingly larger tax increases. Larger government debt tus gives rise to larger budget risk. And as is sown in te paper, tis provides incentives to reduce government debt over time to zero, so tat debt dynamics deviate from 1 Nominal rigidities, owever, also allow monetary policy to a ect real interest rates and tereby te real allocations in te economy. 3

4 te random walk beavior in a second order approximation to optimal policy. Te quantitative strengt of tis force can be signi cant, e.g., it can be optimal to reduce te debt/gdp ratio by about.6% eac year, but tis depends on weter or not te variance of tecnology socks is largely due to predictable or unpredictable movements. Unpredictable components tereby provide stronger incentives for debt reduction. Te economic model in tis paper is related to earlier work by Adam and Billi (28, 29). It extends tese earlier settings by allowing for distortionary income taxation and for government debt dynamics at te same time. Moreover, it considers fully optimal stabilization policies wile previous work was concerned wit time-consistent (or discretionary) policymaking and te design of institutions tat would allow overcoming te distortions generated by te lack of commitment. Te paper is organized as follows. Section 2 describes te economic model and derives te implementability conditions summarizing optimal private sector beavior. After determining te rst best allocation in section 3, section 4 describes te optimal policy problem and te numerical solution strategy. It also derives analytical results regarding te deterministic steady state outcomes associated wit optimal monetary and scal policy. Te quantitative implication of government debt for te deterministic steady state outcomes is analyzed in section 5, wile section 4 determines te impulse responses of te economy to tecnology socks and sows ow tese depend on te outstanding level of government debt. Section 7 discusses te implications government debt for te incentives to reduce or increase debt over time. It sows tat te quantitative implications arising from second order terms can be sizable, especially wen te variance of te tecnology process is largely due to unpredictable components. A conclusion summarizes. 2 Description of te Economic Model Te next sections adapt te sticky price model presented in Adam and Billi (28) to te empirically more relevant setting wit distortionary income taxes and credible government debt. Besides presenting te model ingredients, tis section derives te implementability constraints caracterizing optimal private sector beavior, i.e., derives te optimality conditions determining ouseolds consumption and labor supply decisions and rms price setting decisions. 4

5 2.1 Private Sector Tere is a continuum of identical ouseolds wit preferences given by " 1 # X E t u(c t ; t ; g t ) t= (1) were c t denotes consumption of an aggregate consumption good, t 2 [; 1] denotes te labor supply, and g t public goods provision by te government in te form of aggregate consumption goods and t a sock to te discount factor. Trougout te paper we impose te following conditions. Condition 1 u(c; ; g) is separable in c,, and g; and u c >, u cc <, u <, u, u g >, u gg <. Eac ouseold produces a di erentiated intermediate good. Demand for tat good is given by! ep t y t d were y t denotes (private and public) demand for te aggregate good, e P t is te price of te good produced by te ouseold, and P t is te price of te aggregate good. Te demand function d() satis es P t d(1) P e t =P t ) (1) = were 2 ( 1; 1) is te price elasticity of demand for te di erentiated goods. Importantly, te previously stated assumptions about te demand function are consistent wit optimizing individual beavior wen private and public consumption goods Dixit-Stiglitz aggregates of te goods produced by di erent ouseolds. Te ouseold cooses P e t and ten ires te necessary amount of labor e t to satisfy te resulting product demand, i.e.,! z tt e ep t = y t d (2) were z t is an aggregate tecnology sock wic follows an exogenous stocastic process and as unconditional mean z = 1 : P t z t+1 = (1 z ) + z z t + " z;t+1 were " z;t+1 iin(; 2 ) 5

6 Following Rotemberg (1982) we introduce sluggis nominal price adjustment by assuming tat rms face quadratic resource costs for adjusting prices according to 2 ( Pt e 1) 2 ep t 1 were >. Te ow budget constraint of te ouseold is ten given by " ept P t c t +B t = R t 1 B t 1 +P t y t d( e # P t ) w t e t P t P t 2 ( Pt e 1) 2 +P t w t t (1 t ) ep t 1 (3) were B t denotes nominal government bonds tat pay B t R t in period t + 1, w t is te real wage paid in a competitive labor market, and t is a labor income tax. 2 Altoug nominal government bonds are te only available nancial instrument, adding complete nancial markets for claims between ouseolds would make no di erence for te analysis: since ouseolds ave identical incomes in a symmetric price setting equilibrium, tere exists no incentive to actually trade suc claims. One sould note tat we also abstract from money oldings. Tis sould be interpreted as te casless limit of an economy wit money, see Woodford (1998). Money tus imposes only a lower bound on te gross nominal interest rate, i.e., R t 1 (4) eac period. Abstracting from money entails tat we ignore seigniorage revenues generated in te presence of positive nominal interest rates. Given te size of tese revenues in relation to GDP in industrialized economies, tis does not seem to be an important omission for te analysis conducted ere. 3 Finally, we impose a no Ponzi sceme constraint on ouseold beavior, i.e., "! # t+j 1 lim E Y 1 t B t+j (5) j!1 i= Te ouseold s problem consists of coosing state-contingent processes {c t ; t ; e t ; P e t ; B t } 1 t= so as to maximize (1) subject to (2), (3), and (5) taking as given {y t ; P t ; w t ; R t ; g t ; t } 1 t= as well as te exogenous stocastic productivity process fz t g 1 t=. 2 Considering income or consumption taxes, instead, would be equivalent to a labor income tax plus a lump sum tax (on pro ts). 3 As empasized by Leeper (1991), owever, seigniorage may neverteless be an important marginal source of revenue. R i 6

7 Using equation (2) to substitute e t in (3) and letting te Lagrange multiplier on (3) be given by t t e "t =P t, te rst order conditions of te ouseold s problem are ten equations (2), (3), and (5) olding wit equality and also were u c;t = t (6) u ;t = t w t (1 t ) (7) R t t = E t (8) t+1 t+1 = t y t d(r t ) + r t y t d w t (r t ) y t d r t (r t ) ( t 1) t z t r t 1 r t 1 + E t t+1 ( r t+1 t+1 1) r t+1 t+1 r t r 2 t P r t = e t P t denotes te relative price. Furtermore, tere is te transversality constraint lim j!1 E t t+j u c;t+j B t+j P t+j wic as to old at eac contingency. 2.2 Government = (9) Te government consists of two autorities. First, tere is a monetary autority wic controls te nominal interest rates on sort-term nominal bonds troug open market operations. Since we consider a casless limit economy, te open market operations are in nitesimally small allowing us to abstract from seigniorage revenue. Second, tere is a scal autority deciding on te level of government expenditures, labor income taxes and on debt policy. Government expenditures consist of spending related to te provision of public goods g t and interest payments on outstanding debt. Te level of public goods provision is a coice variable of te government. Te government nances current expenditures by raising labor income taxes and by issuing new debt so tat its budget constraint is given by B t P t + t w t t = g t + R t 1 t B t 1 P t 1 (1) Te government can credibly commit to repay its debt. Te government debt instruments are assumed to be nominal and not state-contingent, consistent wit te type of debt typically issued by governments around te 7

8 globe. Tese features imply, owever, tat monetary policy decisions a ect te government budget troug two cannels: rst, te nominal interest rate policy of monetary autority in uences directly te nominal return te government as to o er on its instruments; second, nominal interest rate decisions also a ect te price level and tereby te real value of outstanding government debt. Tus, to te extent tat te monetary policy can a ect te real interest rate or te price level, it will a ect te government budget, as is te case in Diaz-Gimenez et al. (28). In wat follows we assume tat government debt and tax policies are suc tat te no-ponzi constraint (5) and te transversality constraint (9) are bot satis ed. 2.3 Rational Expectations Equilibrium In a symmetric equilibrium te relative price is given by r t = 1 for all t. Te private sectors optimality conditions can ten be condensed into a (non-linear) Pillips curve u c;t ( t 1) t = u c;tz t t u ;t u c;t (1 t ) + E t [u c;t+1 ( t+1 1) t+1 ] (11) z t and a consumption Euler equation u c;t = E t Using (6) and (7) and de ning R t u c;t+1 t+1 (12) b t = B t P t te government budget constraint can be expressed as b t t 1 t u ;t u c;t t = g t + R t 1 t b t 1 (13) De nition 1 (Rational Expectations Equilibrium) Given te initial outstanding debt level (R 1 b 1 ), a Rational Expectations Equilibrium (REE) consists of a sequence of government policies fr t 1; t ; g t ; b t g 1 t= and private sector coices fc t ; t ; t g 1 t= satisfying equations (11) and (12), te market clearing condition c t + 2 ( t 1) 2 + g t = z t t (14) and te government budget constraint (13), as well as te no-ponzi constraint (5) and te transversality condition (9). 8

9 3 First Best Allocation Te rst best allocation, wic takes into account only ouseold preferences and te constraints imposed by te production tecnology, satis es u g;t = u c;t = u ;t z t It tus turns out to be optimal to equate te marginal utilities of private and public consumption to te marginal disutility of work were te latter is scaled by labor productivity. Tis simple allocation rule is optimal because it is equally costly to produce te public and te private consumption goods. 4 Optimal Monetary and Fiscal Policy Tis section describes te monetary and scal policy problem. It is important to note tat - due to te existence of a number of important economic distortions - policy can generally not acieve te welfare maximizing allocation determined in te previous section. First, market power by rms generally implies tat wages fall sort of teir marginal product, so tat labor supply and terefore output is too low relative to te optimal allocation. 4 Second, te requirement to nance government expenditure and interest payments on outstanding government debt wit distortionary income taxes additionally depresses labor supply and output. Tird, te presence of nominal rigidities may prevent te price system from providing te appropriate scarcity signals. Monetary an scal policy will seek to minimize te e ects of all tese distortions. As we will see below, tis will involve reducing government consumption below its rst best level so as to reduce te averse labor supply consequences of income taxes. Te optimal policy problem (Ramsey problem) wic takes into account te existence of all tese distortion is given by " 1 # X max E t u(c t ; t ; g t ) (15) fc t; t; t;r t1; t;g t;b t=p tg 1 t= t= s.t.: Equations (11); (12); (13); (14) for all t R 1 b 1 given 4 Tis assumes non-negative income tax rates, as are required wen government debt is non-negative. 9

10 Te Lagrangian of te problem is max min fc t; t; t;r t1; t;g t;b tg 1 t= ft 1;2 t ;3 t ;4 t g 1 t= E 2 P 1 t= t u(c t ; t ; g t ) u + t 1 u c;t ( t 1) c;tz t t t u ;t t u c;t+1 ( t+1 1) t+1 + t 2 uc;t t R t u c;t+1 t t t 3 z t t c t 2 ( t 1) 2 g t + t t 4 b t u ;t R t 1 t u c;t t g t 1 t t b t 1 u c;t(1 t) z t! (16) Te rst order necessary conditions for te Lagrangian problem are derived in appendix A.1. Te appendix sows tat te nonlinear solution to tese FOCs take te form y t = g(x t ; ) (17) were y t = c t ; t ; t ; R t ; t ; g t ; 1 t ; 2 t ; 3 t ; 4 t denote te decision variables and x t = z t ; 1 t ; 2 t ; b t 1 ; R t 1 te state variables. Te parameter in equation (17) indicates tat te solution depends on te standard deviation of te tecnology socks. Te state variables evolve according to z t+1 = z + z z t + z; " z;t+1 (18) 1 t+1 = 1 t (19) 2 t+1 = t 2 (2) b t = t u ;t t + g t + R t 1 b t 1 t u c;t t 1 (21) R t = R t (22) Te state variables i t (i = 1; 2) denote te lagged Lagrange multipliers associated wit te forward-looking constraints in (16). At time zero, tese states assume initial values i = (i = 1; 2). As is well known, tis gives rise to transitory non-stationary components in te solution to te optimal policy problem, even in te absence of socks. Speci cally, in te initial period te policymaker may nd it optimal to generate surprise in ation so as to erode te real value of any outstanding government debt. Likewise, te policymaker may nd it optimal to transitorily increase taxes. In wat follows, I wis to abstract from tese non-stationary deterministic components of optimal policy and focus instead on a time-invariant deterministic long-run outcome. Tis outcome will be called te Ramsey steady state. Tecnically, te time-invariance is acieved by setting te time zero values of i (i = 1; 2) equal to teir steady state value rater tan to zero. Economically, tis amounts to imposing an initial commitment on te policymaker not to generate surprise movements in taxes, government spending, 1

11 or nominal interest rates in period zero. Tis is standard practice in te optimal taxation literature, e.g., Cari et al. (1991). 4.1 Steady State: Analytical Results Te model possesses a continuum of Ramsey steady states, eac of wic is associated wit a di erent level of government debt. 5 To see tis note tat te rst order condition of te optimal policy problem (16) wit respect to bonds is given by = 4 t E t 4 t+1 R t t+1 In a deterministic steady state, te Euler equation (12) implies R = 1 so tat te rst order condition for bonds stated above imposes no restrictions on te steady state outcomes: tere exists an indeterminacy of one dimension. Despite te existence of a continuum of steady states, tese steady states sare a number of common properties. As sown in appendix A.3 all Ramsey steady states satisfy = 1 (23) R = 1 (24) Equation (23) sows tat it is optimal to implement price stability in te absence of socks. Tis olds independently of te level of outstanding government debt and sows tat it is suboptimal to use in ation in steady state wit te objective to reduce te real value of outstanding government debt. Equation (24) gives te nominal interest rate consistent wit price stability. Since < 1, nominal interest rates are positive. Using te steady state real interest rate derived above, te government budget constraint can be written as wit ex given by t w t t = g t + ex (25) ex = ( 1 1)b and denoting te interest rate payments on outstanding government debt. From an economic point of view, interest payments involve just income redistribution, but in an environment wit distortionary taxation suc redistribution is costly to provide and as real consequences. Appendix A.3 also 5 Te di erent steady states also imply di erent values for te initial state variables i, as explained in te previous sections. Tis is te case because te incentives to generate surprise movements in policy vary wit te initial debt level. 11

12 sows tat is optimal to ave u u g (26) wit a strict inequality if government debt is positive (or at least not too negative). Equation (26) demonstrates tat it is optimal to reduce public spending to a level below tat suggested by consumer preferences and tecnology, i.e., below te rst best allocation rule determined in section 3. Te economic rationale for restraining spending on public goods provision can be seen from te following equation wic is also derived in appendix A.3: 1 + u = g + ex u c (27) It sows tat tere exists a wedge between te marginal utility of (private) consumption and te disutility of labor. 6 Tis wedge consists of two components: rst, te monopoly power of rms leads to te price mark-up 1+, and second, te need to nance public expenditure and interest rate payments troug distortionary taxation leads to additional term in equation (27). Reducing public spending below its rst best level reduces te required labor tax rates and terefore elps reducing te wedge between te marginal utility of private consumption and te marginal utility of leisure. Te previous arguments also suggest tat in an economy wit a iger stock of real government debt, i.e. a iger level for ex, tere are stronger incentives to reduce public consumption below its rst best level (ceteris paribus) because taxes are ig due to a ig interest rate burden. Interestingly, equation (27) sows tat te steady state distortions could be entirely eliminated if te government ad accumulated a su ciently large amount of claims against te private sector, so tat interest income allows to (1) o set te monopoly distortions via negative labor income tax rates and (2) to pay for (te rst best level of) public goods provision. 4.2 Numerical Solution and Model Calibration Tis section explains ow one can determine locally approximate solutions for te optimal policies of te stocastic version of te model. Since tere exists a continuum of deterministic steady states, one as to exogenously x one of te steady state dimensions. Tis is done by xing te initial real value of outstanding government bonds b 1 inerited from te past (wic may be negative in case te government as accumulated claims against te private sector). One can ten determine te steady state values for te remaining variables tat solve te system of rst order conditions of problem 6 Tis is true wenever ex is not too negative, i.e., wenever te government as not accumulated a too ig level of claims against te private sector. 12

13 (16). 7 Finally, one can determine rst order - and wen required - second order accurate approximations to te optimal non-linear policy functions (17) and te state transition equations (39)-(43) using perturbation tecniques. Furter details are provided in appendix A.2. For te numerical exercises te following preference speci cation is considered, wic satis es condition 1 and is consistent wit balanced growt u(c t ; t ; g t ) = log (c t ) 1+' t! 1 + ' +! g log (g t ) (28) wit! >,! g and te parameter ' denoting te inverse of te Frisc labor supply elasticity. Te model is calibrated as summarized in table 1 below, following Adam and Billi (28). Te quarterly discount factor is cosen to matc te average ex-post U.S. real interest rate, 3:5%, during te period 1983:1-22:4. Te value for te elasticity of demand implies a gross mark-up equal to 1.2. Te elasticity of labor e ort is assumed to be one (' = 1) and te values of! and! g are cosen suc tat in te Ramsey steady state witout government debt, agents work 2% of teir time and it is optimal to spend 2% of total output on public goods. Appendix A.4 provides details on ow te parameters ave to be cosen to acieve tis. Te price stickiness parameter is selected suc tat te loglinearized version of te Pillips curve (11) is consistent wit te estimates of Sbordone (22), as in Scmitt-Groé and Uribe (24). Te quarterly standard deviation of te tecnology socks is.6% and te socks ave a quarterly persistence equal to z = :95. Parameter De nition Assigned Value quarterly discount factor = :9913 price elasticity of demand = 6 degree of price stickiness = 17:5 1/elasticity of labor supply ' = 1 utility weigt on labor e ort! = 19:792 utility weigt on public goods! g = :2656 tecnology sock process persistence z = :95 quarterly s.d. tecnology sock innovation = :6% Table 1: Baseline Calibration 7 Tis assumes tat te maximum tax revenue tat can be raised according to te model s La er curve is su cient to pay for te interest payments te assumed outstanding debt level, see te discussion in te next section. 13

14 Steady state tau debt/gdp ratio Figure 2: Steady State Tax Rates 5 Steady State Implications of Government Debt Tis section explores te quantitative implications of di erent government debt levels for te deterministic steady state outcomes and welfare. Wile qualitative implications ave been discussed in section 4.1, tis section sows tat government debt as enormously a ects steady welfare. Using te calibration from te previous section, table 2 below reports te steady state outcomes for private consumption, ours worked, government consumption and taxes for alternative initial debt levels. Te last column of te table lists te welfare equivalent consumption variation tat is required eac period to make agents in te zero debt steady state as well o as in te considered alternative debt scenarios. 8 Te outcomes for a government debt level of 1% and 2% of GDP sow tat iger debt requires considerably 8 Let (c ZC; ZD; g ZD) denote te allocation in te zero debt steady state and (c A; A; g A) te allocation in some alternative steady state. We ten determine te increase in consumption v requires in te zero steady state tat implies tat agents in tat steady state enjoy te same welfare as in te alternative steady state,i.e., log (c ZD (1 + v)) 1+' ZD! +!g log (gzd) 1 + ' 1+' A = log (c A)! +!g log (ga) 1 + ' 14

15 .15.1 consumption equivalent welfare debt/gdp ratio Figure 3: Steady State Welfare Implications of Government Debt Steady tax revenue debt/gdp ratio Figure 4: La er Curve 15

16 iger income tax rates. Tese distort downward labor supply and tereby private consumption. Public consumption also decreases to avoid an even furter increase in tax rates. Te welfare implications of debt are large by any conventional standards and amount to 5.6% and 11% of consumption eac period, respectively. Te table also reports te steady state outcome under a scenario wit large negative government debt. Te level of government claims against te private sector as tereby been cosen suc tat te interest income earned by te government allows to pay for te rst best level of public consumption and to o set te adverse labor supply e ects of monopolistic competition via a negative income tax rate. Suc a policy eliminates all steady state distortion in te economy, i.e., acieves te rst best allocation, and gives rise to a welfare increase of 7% of consumption eac period. Monopoly power by rms and te requirement to raise government revenue troug distortionary income taxes tus give rise to large distortions in te economy, even if government debt is zero. priv. cons ours gov. cons. taxes welfare equiv. (c) () (g) () cons. variation Zero debt %.% 1% debt/gdp % -5.58% Cange wrt zero debt -2.61% -2.78% -3.47% +16.8% 2% debt/gdp % -11.% Cange wrt zero debt -5.25% -5.62% -7.2% +33.3% First best steady state -176% debt/gdp % +7.6% Cange wrt zero debt +25% +26.5% +32.5% n.a. Table 2: Steady State E ect of Government Debt Overall, te e ects of di erent steady state debt/gdp ratios on te allocations and tax rates are surprisingly linear. Figure 2 provides as an example for tis outcome: it sows te steady state tax rate as a function of te 16

17 steady state debt/gdp ratio. Te same olds true for te consumption equivalent welfare losses, wic are sown in gure 3. Yet, since utility convex, te utility gains from consumption increases are smaller tan te utility gains from consumption losses. Te relationsip between te debt/gdp ratio and steady state utility is tus convex, wit increasing convexity at ig debt/gdp ratios. Te model feature one important non-linearity resulting from te La er curve. Figure 4 depicts te steady state tax revenue as a function of te debt/gdp ratio (wic is almost linearly related to te steady state tax rate). At some point, tax revenue ceases to increase wit te tax rate, so tat tere exists a maximum sustainable deterministic steady state debt/gdp ratio. For te present calibration tis ratio is fairly ig and just above 1% of GDP. 6 Optimal Stabilization Policy Tis section studies te optimal monetary and scal policy response to large adverse tecnology socks. It tereby focuses on te optimal dynamics around two alternative deterministic Ramsey steady states, one were te initial level of government debt level is zero and one were it equals 1% of GDP. As will be sown te level of government debt as important implications for te conduct of stabilization policy. Figure 5 depicts te impulse responses to a negative tecnology sock under optimal monetary and scal policy. 9 Te sock size equals 3 unconditional standard deviations and implies tat tecnology temporarily drops by about 5.7%, tereafter slowly reverts to steady state. 1 Under bot debt scenarios, output as well as private and public consumption approximately drop by te same amount as tecnology. 11 Te initial impact for output and private consumption is sligtly more contained wen debt equals 1%, wile te drop in government consumption is stronger on impact. Notable di erences across te two scenarios emerge, owever, wen considering te optimal response of in ation, nominal interest rates, taxes, and government debt. Wen government debt is zero, tere is no response of in ation, taxes or debt watsoever. Te reduction in tax revenue induced by a negative tecnology sock is completely o set by a reduction in government spending. 12 Nominal interest rates rst increase and ten slowly revert back to base- 9 Te gure presents te rst order accurate impulse repsonse. Second order e ects are discussed in te next section. 1 Te assumed sock process implies tat te economy spends less tat.13% of te quarters in states wit suc or worse tecnology levels. 11 Te response is sligtly more muted because te negative wealt e ect of tecnology socks implies tat labor supply expands somewat. 12 Tis continues to be true wen looking at second order approximate impulse responses. 17

18 c (%dev) g (%dev) y (%dev) Pi (%) tau b/y (%) 5 zero debt 1% debt R(%) Figure 5: Impulse Response to a Negative Tecnology Sock (-3 std. dev.) 18

19 line, so as to induces a pat for real interest rates tat makes te observed consumption pat consistent wit te absence of in ation. Wen government debt is zero, te situation di ers notably. Wit large positive debt levels, it is suboptimal to o set te drop in tax revenue following an adverse tecnology socks by a corresponding drop in government spending. Tis is so because large part of te steady state tax revenue is used to pay interest on debt, so tat an enormous government spending reduction would be required to balance te budget. Tis is suboptimal if marginal utility of government consumption is very ig, as is actually te case because te steady state level is already fairly low, see te discussion in section 5. Wile government spending falls more on impact wen debt is ig, te reduction is insu cient to balance te budget. Te government is tus forced to issue additional debt and to increase taxes to nance it. As a result, te debt level and te tax rate move permanently iger. Tis is an example of Barro s (1979) tax smooting result, but emerging ere in a setting wit endogenous spending decisions and endogenous interest rates. Wit ig initial debt, te monetary autority optimally lowers nominal interest rates on impact rater tan increasing tem, as is optimal in te absence of debt. Tis reduces real interest rates and implies a less severe collapse in output and consumption tan oterwise in te rst period. It also generates some small amount of in ation, wic elps to reduce te real value of outstanding government debt. Yet, as is known from te work of Scmitt-Groé and Uribe (24), it is suboptimal in te presence of even small amounts of nominal rigidities to bring about large price level canges, i.e., to use nominal bonds a state contingent source of taxation. Tis sows up ere once more in te form rater small movements of in ation in response to fairly large sized socks. Te rst order accurate impulse response dynamics sown in gure 5 work symmetrically for positive and negative tecnology socks. Yet, as sould be clear from te impulse response analysis, te larger te amount of outstanding government debt, te larger are te budget risks associated wit tecnology socks. Terefore, altoug up to rst order, te government debt/gdp ratio evolves locally like a random walk, te fact tat budget risk increases wit debt levels sould provide incentives to lower government debt over time in te presence of socks. 13 Tis issue is investigated in te next section, wic analyzes second order accurate optimal equilibrium dynamics. 7 Te Optimal Evolution of Debt over Time Using te approac detailed in appendix A.2, tis section presents te results from a second order accurate approximation of te debt/gdp dynamics un- 13 In te absence of socks, it is optimal to old debt constant over time. 19

20 .6.4 Annual drift in te debt/gdp ratio debt/gdp ratio Figure 6: Te Optimal Speed of Debt Reduction (baseline sock parameterization) optimal annual drift in debt/gdp ratio ro=.95 ro=.5 ro= debt/gdp ratio Figure 7: Te Optimal Speed of Debt Reduction: Alternative Sock Processes 2

21 c (%dev) g (%dev) y (%dev) % debt, 2nd order, ro =.5 1% debt, 1st order, ro = Pi (%) R(%) tau b/y (%) Figure 8: Comparison of First and Second Order Accuarate Impulse Responses 21

22 der optimal monetary and scal policy. It evaluates to wat extent te budget risk considerations discussed in te previous section provide incentives to reduce government debt over time. Suc risk consideration cannot be captured by te rst order approximation analyzed in te previous section. As before, consideration is restricted to a local analysis around some pre-speci ed deterministic Ramsey steady state. Te analysis tus ignores te incentives for debt reduction arising from global constraints suc as borrowing constraints. As sown in Aiyagari et al. (22), suc global constraints can provide additional incentives for debt reduction. Te point ere is to sow tat potentially important incentives for debt reduction exist even wen restricting consideration to a local analysis. Figure 6 depicts te optimal drift in te debt/gdp ratio as a function of assumed steady state debt/gdp ratio. Speci cally, te gure reports te constant in te state transition law for bonds emerging from a second order approximate solution to optimal monetary and scal policy and scales it by te GDP level. It sows tat debt sould be reduced towards zero wenever it is positive, and tat it sould be increased wenever it is negative. Tis suggests tat over time debt optimally converges to zero. 14 Quantitatively, te drift term is not very large for te baseline parameterization, but interestingly te optimal speed of te debt drift is non-monotone in te initial debt/gdp ratio. Speci cally, as te debt/gdp ratio increases, te optimal speed of debt reduction rst rises but ten falls. It is actually not surprising, tat tis relationsip between te debt level and te optimal speed of debt reduction can be non-monotone. Wile budget risk increases wit te debt level, te cost of repaying debt equally rise. And since ig debt levels imply ig steady state taxes, te cost of fast repayment may rise faster tan te bene ts of repayment, wic results in a non-monotone relationsip between tese two variables. Figure 6 also sows tat for su ciently negative debt levels te government actually reduces te speed at wic it decumulates its claims against te private sector. Wile budget risk increases as debt becomes more negative, te utility consequences of tis risk actually decrease, so tat it is not clear wic e ect dominates. Tis is so because for su ciently large negative debt levels, te government can actually implement te rst best equilibrium in te absence of socks, as as been sown in section 5. Utility is tus at locally wit respect to tax canges at tis point. It turns out tat te quantitative importance of debt reduction is robust to many parameter canges. Signi cant canges only occur wen te unpredictable components of tecnology socks increases. To illustrate tis 14 Tis di ers from te results in Aiyagari et al. (22) were debt converged to a large negative value. Unlike in Aiyagari et al, te present setup does not allow for lump sum rebates of government revenue. 22

23 nding, gure 7 depicts te optimal debt drift under less persistent tecnology sock processes. Te standard deviation of te innovation to te tecnology process is tereby adjusted so as to keep te overall unconditional standard deviation of tecnology socks uncanged wen compared to te baseline speci cation. As te gure sows, it may ten be optimal to reduce te debt to GDP ratio by as muc as.6% per year. Tis sows tat second order e ects can easily be quantitatively important. Figure 8 depicts te rst and second order accurate impulse responses to a negative tecnology socks wen te sock persistence is given by z = :5. 15 It sows tat important di erences emerge between a rst and second order approximation for te optimal response of taxes and government debt. Especially, te optimal evolution of government debt deviates signi cantly from random walk beavior. 8 Conclusions Tis paper as sown tat te recent increase in government debt as important implications for te optimal conduct of monetary and scal policy. Higer government debt requires lower public spending on average and exposes te scal budget to increased risk from tecnology socks. Hig debt makes it optimal in response to adverse tecnology socks to cut public spending more strongly, to increase income taxes and to lower nominal interest rates (or increase tem less strongly). In te absence of outstanding government debt, taxes sould remain uncanged, public spending needs to fall by less and nominal interest rates sould increase. Budget risk considerations can provide quantitatively important incentives to reduce government debt over time towards zero, but optimal speed of debt reduction is not necessarily monotone in te level of accumulated government debt. Optimal policy suggest, owever, tat te debt dynamics sould deviate in important ways from random walk beavior and sould optimally converge to zero over time. 15 Te standard deviation of te tecnology sock innovation is again adjusted so as to keep te unconditional standard deviation of tecnology socks uncanged compared to te baseline parameterization. 23

24 A Appendix A.1 FOCs and Solution Strategy Te rst order conditions of problem (16) wit respect to te decision variables (c t ; t ; t ; R t ; t ; g t ; b t ) are c t : = u c;t + t 1 1 t ucc;t ( t 1) t t 1! t 1 u c;t z t u cc;t u ;t t (u c;t ) 2 (1 t ) z t + 2 t u cc;t R t t : = u ;t + 3 t z t 4 t 2 t u cc;t t 3 t + 4 t t u cc;t u ;t t 1 t (u c;t ) 2 u cc;t z t t 1 + +! u ;t u c;t (1 t ) (29) t 1 u c;t z t u ;t u ;t t u c;t (1 t ) z t u c;t (1 t ) z t t 1 (u ;t t + u ;t ) (3) 1 t u c;t t : = 1 t 1 t uc;t (2 t 1) + 2 t u c;t ( t ) 2 t 3 ( t 1) + t 4 R t 1 b t 1 ( t ) 2 (31) z t R t : = t : = 2 t 1 t u c;t t+1 4 (R t ) 2 b t E t (32) t+1 u c;t z t u;t 1 t u c;t z t (1 t ) 2 t 4 1 u ;t (1 t ) 2 t (33) u c;t g t : = u g;t 3 t 4 t (34) R t b t : = t 4 E t t+1 4 (35) t+1 Te derivatives wit respect to te rst tree Lagrange multipliers are given by: u c;t z t u c;t ( t 1) t t u ;t u c;t (1 t ) E t u c;t+1 ( t+1 1) t+1 = (36) u c;t R t z t E t u c;t+1 t+1 = (37) z t t c t 2 ( t 1) 2 g t = (38) 24

25 We can ten de ne 5 state variables and te corresponding 5 transition equations: z t+1 = z + z z t + z; " z;t+1 (39) 1 t+1 = 1 t (4) 2 t+1 = t 2 (41) b t = t u ;t t + g t + R t 1 b t 1 t u c;t t 1 (42) R t = R t (43) were te fort equation is te derivative of te Lagrangian (16) wit respect to te last multiplier and te i t denote te lagged Lagrange multipliers associated wit te forward looking constraints in (16). In te last equation, R t denotes bot te future state variable and te current decision variable. As we sow below, optimal policies are going to be a function of tese state variables. Te state transition equations (39)-(43), te FOCs (29)-(35), and te implementability constraints (36)-(38) form a system of 5 state transition equations and 1 additional equations of te form E t f(x t+1 ; y t+1 ; x t ; y t ) = (44) wic is of te form analyzed in Gomme and Klein (21). As argued in Gomme and Klein, a nonlinear solution to tis system of nonlinear expectational di erence equations is given by nonlinear decision functions of te form y t = g(x t ; ) (45) were y t = c t ; t ; t ; R t ; t ; g t ; 1 t ; 2 t ; 3 t ; 4 t denote te decision variables (note tat tis list does not contain b t ) and were x t = z t ; 1 t ; 2 t ; b t 1 ; R t 1 denotes te state vector. Te parameter tereby denotes te standard deviation of te tecnology sock innovation. Te nonlinear state transition is tereby described by equations (39)-(43). A.2 Local Approximation of Optimal Dynamics Let (x; y) denote a deterministic steady state of te nonlinear equation system (44) solving f(x; y; x; y) = A second order approximation of te non-linear solution (45) around tis steady state is given by a decision function ey t = k y + F ex t I 5 ex t Eext 25

26 and a state transition function ex t+1 = k x + P ex t I 1 ex t Gext + " t+1 were te tilde indicates tat a variable is expressed in terms of deviations from its steady state. Te approximation is taken wit respect to te expansion variable around = and te values for (k y ; F; E) and (k x ; P; G) can be computed using te code provided by Gomme and Klein (21). Te rst order accurate dynamics of te model can be obtained by te same equations wen setting k y ; k x ; E; G all equal to zero. A.3 Ramsey Steady State To simplify matters we start by eliminating taxes and te government budget constraint from te Lagrangian (16). Note tat te FOCs (6) and (7) imply u ;t u c;t = (1 t )w t = w t t w t and from steady state version of te government budget (25) we ave t w t = g t + ex t Substituting te latter equation into te former gives te following expression for te real wage w t = u ;t + g t + ex u c;t t wic allows expressing te Pillips curve witout reference to taxes. Te simpli ed constant debt version of te Lagrangian (16) is ten max min fc t; t; t;r t1;g tg 1 t= ft 1;2 t ;3 t g t= + t 1 t u c;t ( t 1) t u c;tz t t + t t 2 uc;t uc;t+1 R t t+1 + t t 3 t c t 2 ( t 1) 2 g t 1X t u(c t ; t ; g t ) (46) u;t u c;t [u c;t+1 ( t+1 1) t+1 ] g t+ex t! 26

27 Te FOCs consist of te tree constraints and c t :u c;t + 1 t 1 t 1 ucc;t ( t 1) t 1 t +t 2 u cc;t t :u ;t R t 2 t 1 1 t u c;t u cc;t u cc;t z t t 1 + g t + ex t t 3 = (47) t u;t g t + ex u;t t + g t + ex u c;t t u c;t ( t ) 2 t : t 1 t 1 1 (2t 1) + t 2 uc;t 1 ( t ) 2 u c;t + 3 t = (48) 3 t ( t 1) = (49) R t :t 2 (R t ) 2 = (5) g t :u g;t + t 1 u c;t 3 t = (51) We now impose steady state conditions by dropping time subscripts. From (5) 2 = so tat (49) gives and from (12) one obtains = 1 R = 1 Using tese results and imposing tem on te steady state version of te derivative of (46) wit repect to 1 t one obtains u u c = 1 + g + ex (52) wic is equation (27) in te main text. Since u < and u c > te previous equation implies 1 + g + ex < (53) In te steady state equations (47), (48) and (51) simplify to u c 1 u cc 1 + g + ex 3 = (54) u 1 g + ex u + u c + 3 = (55) u g + 1 u c 3 = (56) 27

28 were 3 > denotes te marginal utility of relaxing te resource constraint. Te previous FOCs indicate te alternative possible uses of additional resources, namely private consumption (equation (54)), leisure (equation (55)) and public consumption (equation (56)). Since public consumption is only one of tree possible uses of resources, it must be te case tat 3 u g in te optimum. Equation (56) terefore implies tat 1. Combining equations (54) and (56) to eliminate 3 ten gives u c = u g + 1 ucc 1 + g+ex (1 1 ) From 1 and (53) ten follows tat i u g + 1 ucc 1 + g+ex u c = (1 1 ) u g + 1 u cc 1 + g + ex u g and since u c > u, we ave u g > u, as claimed in te main text. A.4 Utility Parameters and Ramsey Steady State Here we sow ow te utility parameters! and! g are determined by te Ramsey steady state values. Let variables witout subscripts denote teir steady state values and consider te Ramsey steady state wit constant debt from appendix A.3. Since = 1 te Pillips curve constraint in (46) implies wic delivers 1 +! ' c 1+ g + ex g+ex i = (57)! = ' (58) c and allows to determine te steady state values of u and u. Adding up equations (54) and (55) ten delivers and (55) gives 1 = 3 = It ten follows from (56) u + u c g+ex u + 1! g = g 3 u c + u + ucc 1 + g+ex g + ex u + u c 1 u c i 28

29 References Adam, K., and R. Billi (28): Monetary Conservatism and Fiscal Policy, Journal of Monetary Economics, 55, (29): Distortionary Fiscal Policy and Monetary Policy Goals, University of Manneim Mimeo. Aiyagari, S. R., A. Marcet, T. J. Sargent, and J. Seppälä (22): Optimal Taxation Witout State-Contingent Debt, Journal of Political Economy, 11(6), Barro, R. J. (1979): On te Determination of Public Debt, Journal of Political Economy, 87 (5), Cari, V., L. J. Cristiano, and P. J. Keoe (1991): Optimal Fiscal and Monetary Policy: Some Recent Results, Journal of Money Credit and Banking, 23, Díaz-Giménez, J., G. Giovannetti, R. Marimon, and P. Teles (28): Nominal Debt as a Burden on Monetary Policy, Review of Economic Dynamics, 11, Gomme, P., and P. Klein (21): Second-Order Approximation of Dynamic Models Witout te Use of Tensors, University of Western Ontario Mimeo. Leeper, E. M. (1991): Equilibria under Active and Passive Monetary and Fiscal Policies, Journal of Monetary Economics, 27, Rotemberg, J. J. (1982): Sticky Prices in te United States, Journal of Political Economy, 9, Sbordone, A. (22): Prices and Unit Labor Costs: A New Test of Price Stickiness, Journal of Monetary Economics, 49, Scmitt-Groé, S., and M. Uribe (24): Optimal Fiscal and Monetary Policy under Sticky Prices, Journal of Economic Teory, 114(2), Woodford, M. (1998): Doing Witout Money: Controlling In ation in a Post-Monetary World, Review of Economic Dynamics, 1,