Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 806, April 2004--Screen Reader Version*
International Finance Discussion Papers numbers 797-807 were presented on November 14-15, 2003 at the second conference sponsored by the International Research Forum on Monetary Policy sponsored by the European Central Bank, the Federal Reserve Board, the Center for German and European Studies at Georgetown University, and the Center for Financial Studies at the Goethe University in Frankfurt.
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We propose an integrated treatment of the problems of optimal monetary and fiscal policy, for an economy in which prices are sticky (so that the supply-side effects of tax changes are more complex than in standard fiscal analyses) and the only available sources of government revenue are distorting taxes (so that the fiscal consequences of monetary policy must be considered alongside the usual stabilization objectives). Our linear-quadratic approach allows us to nest both conventional analyses of optimal monetary stabilization policy and analyses of optimal tax-smoothing as special cases of our more general framework. We show how a linear-quadratic policy problem can be derived which yields a correct linear approximation to the optimal policy rules from the point of view of the maximization of expected discounted utility in a dynamic stochastic general-equilibrium model. Finally, in addition to characterizing the optimal dynamic responses to shocks under an optimal policy, we derive policy rules through which the monetary and fiscal authorities may implement the optimal equilibrium. These take the form of optimal targeting rules, specifying an appropriate target criterion for each authority.
Keywords: Loss function, output gap, tax smoothing, targeting rules
JEL Classifications: E52, E61, E63
We propose an integrated treatment of the problems of optimal monetary and fiscal policy, for an economy in which prices are sticky and the only available sources of government revenue are distorting taxes. The integrated treatment allows us to consider how familiar characterizations of optimal monetary policy must be generalized to take account of the consequences of alternative monetary policies for the government budget, and to consider fiscal shocks as one of the types of disturbances to which monetary policy may need to respond. It also allows us to show how conventional characterizations of optimal tax policy can be generalized to the case in which economic activity depends not solely upon supply-side incentives, but on aggregate (nominal) demand as well.
We show how a linear-quadratic policy problem can be derived that yields a correct linear approximation to the optimal policy rules from the point of view of the maximization of expected discounted utility. This requires that we take account of the effects of stabilization policy (i.e., of the variances of endogenous variables) on the average levels of consumption and hours worked; but we show that such effects can be incorporated into the quadratic objective, so that we need not consider nonlinearities in the constraints on our policy problem.
We show that a quadratic loss function can be derived that consists of a weighted average of two terms each period: squared deviations of the inflation rate from an optimal rate of zero, and squared deviations of log output from a target output level that varies over time as a function of exogenous disturbances to preferences, technology, and the government’s exogenous fiscal constraints. Thus consideration of the effects of tax distortions does not introduce any additional stabilization objectives beyond the ones (inflation stabilization and output-gap stabilization) considered in conventional treatments of monetary stabilization policy; both monetary and fiscal instruments should be used to stabilize inflation and the (appropriately defined) output gap. However, allowing for the distortions associated with raising government revenue can affect the weights on these objectives, and the proper definition of the target rate of output.
We consider how the optimal responses to shocks vary depending on the degree of price stickiness, and show, in a calibrated example, that the optimal responses that would be derived under the assumption of complete price flexibility are quite different than those that are optimal if prices are even slightly sticky; this indicates that allowing for price stickiness is quite important in exercises of this kind. In particular, optimal policy involves much greater stability of the inflation rate if prices are sticky, while shocks should instead have permanent effects on the level of government debt and on tax rates, even in the presence of nominal government debt.
Finally, we derive targeting rules through which the monetary and fiscal authorities may implement the optimal equilibrium. An optimal targeting rule for monetary policy in the case of distorting taxes still has the form of a flexible inflation targeting rule, as in the literature that ignores the fiscal consequences of monetary policy, but the output gap should modify the inflation target in a different way. We also obtain a targeting rule for the fiscal authority, and this too requires the fiscal authority to base policy on the projected consequences of alternative government budgets for future inflation.
While there are by now substantial literatures seeking to characterize optimal monetary and fiscal policy respectively, the two literatures have largely developed in isolation, and upon apparently contradictory foundations. The modern literature on dynamically optimal fiscal policy often abstracts from monetary aspects of the economy altogether, and so implicitly allows for no useful role for monetary policy. When monetary policy is considered within the theory of optimal fiscal policy, it is most often in the context of models with flexible prices; in these models, monetary policy matters only (i) because the level of nominal interest rates (and hence the opportunity cost of holding money) determines the size of certain distortions that result from the attempt to economize on money balances, and (ii) because the way in the price level varies in response to real disturbances determines the state-contingent real payoffs on (riskless) nominally-denominated government debt, which may facilitate tax-smoothing in the case that explicitly state-contingent debt is not available. The literature on optimal monetary policy has instead been mainly concerned with quite distinct objectives for monetary stabilization policy, namely the minimization of the distortions that result from prices or wages that do not adjust quickly enough to clear markets. At the same time, this literature typically ignores the fiscal consequences of alternative monetary policies; the characterizations of optimal monetary policy that are obtained are thus strictly correct only for a world in which lump-sum taxes are available.
Here we wish to consider the way in which the conclusions reached in each of these two familiar literatures must be modified if one takes simultaneous account of the basic elements of the policy problems addressed in each literature. On the one hand, we wish to consider how conventional conclusions with regard to the nature of an optimal monetary policy rule must be modified if one recognizes that the government's only sources of revenue are distorting taxes, so that the fiscal consequences of monetary policy matter for welfare. And on the other hand, we wish to consider how conventional conclusions with regard to optimal tax policy must be modified if one recognizes that prices do not instantaneously clear markets, so that output determination depends on aggregate demand, in addition to the supply-side factors stressed in the conventional theory of optimal taxation.
A number of recent papers have also sought to jointly consider optimal monetary and fiscal policy, in the context of models with sticky prices; important examples include Correia et al., (2001), Schmitt-Grohé and Uribe (2001), and Siu (2001). Our approach differs from those taken in these papers, however, in several respects. First, we model price stickiness in a different way than in any of these papers, namely, by assuming staggered pricing of the kind introduced by Calvo (1983). This particular form of price stickiness has been widely used both in analyses of optimal monetary policy in models with explicit microfoundations (e.g., Goodfriend and King, 1997; Clarida et al., 1999; Woodford, 2003) and in the empirical literature on optimizing models of the monetary transmission mechanism (e.g., Rotemberg and Woodford, 1997; Gali and Gertler, 1999; Sbordone, 2002).
Perhaps more importantly, we obtain analytical results rather than purely numerical ones. To obtain these results, we propose a linear-quadratic approach to the characterization of optimal monetary and fiscal policy, that allows us to nest both conventional analyses of optimal monetary policy, such as that of Clarida et al. (1999), and analyses of optimal tax-smoothing in the spirit of Barro (1979), Lucas and Stokey (1983), and Aiyagari et al. (2002) as special cases of our more general framework. We show how a linear-quadratic policy problem can be derived which yields a correct linear approximation to the optimal policy rules from the point of view of the maximization of expected discounted utility in a dynamic stochastic general-equilibrium model, building on the work of Benigno and Woodford (2003) for the case of optimal monetary policy when lump-sum taxes are available.
Finally, we do not content ourselves with merely characterizing the optimal dynamic responses of our policy instruments (and other state variables) to shocks under an optimal policy, given one assumption or another about the nature and statistical properties of the exogenous disturbances to our model economy. Instead, we also wish to derive policy rules that the monetary and fiscal authorities may reasonably commit themselves to follow, as a way of implementing the optimal equilibrium. In particular, we seek to characterize optimal policy in terms of optimal targeting rules for monetary and fiscal policy, of the kind proposed in the case of monetary policy by Svensson (1999), Svensson and Woodford (2003), and Giannoni and Woodford (2002, 2003). The rules are specified in terms of a target criterion for each authority; each authority commits itself to use its policy instrument each period in whatever way is necessary in order to allow it to project an evolution of the economy consistent with its target criterion. As discussed in Giannoni and Woodford (2002), we can derive rules of this form that are not merely consistent with the desired equilibrium responses to disturbances, but that in addition (i) imply a determinate rational-expectations equilibrium, so that there are not other equally possible (but less desirable) equilibria consistent with the same policy; and (ii) bring about optimal responses to shocks regardless of the character of and statistical properties of the exogenous disturbances in the model.
Here we describe our assumptions about the economic environment and pose the optimization problem that jointly optimal monetary and fiscal policies are intended to solve. The approximation method that we use to characterize the solution to this problem is then presented in the following section. Further details of the derivation of the structural equations of our model of nominal price rigidity can be found in Woodford (2003, chapter 3).
The goal of policy is assumed to be the maximization of the level of expected utility of a representative household. In our model, each household seeks to maximize
We assume a common technology for the production of all goods, in which (industry-specific) labor is the only variable input,
We can furthermore express the relative quantities demanded of the differentiated goods each period as a function of their relative prices. This allows us to write the utility flow to the representative household in the form where
The producers in each industry fix the prices of their goods in monetary units for a random interval of time, as in the model of staggered pricing introduced by Calvo (1983). We let be the fraction of prices that remain unchanged in any period. A supplier that changes its price in period chooses its new price to maximize
The function , defined in the appendix, indicates the after-tax nominal profits of a supplier with price in an industry with common price when the aggregate price index is equal to , aggregate demand is equal to , and sales revenues are taxed at rate Profits are equal to after-tax sales revenues net of the wage bill, and the real wage demanded for labor of type is assumed to be given by
Each of the suppliers that revise their prices in period choose the same new price Under our assumed functional forms, the optimal choice has a closed-form solution
The price index then evolves according to a law of motion
We abstract here from any monetary frictions that would account for a demand for central-bank liabilities that earn a substandard rate of return; we nonetheless assume that the central bank can control the riskless short-term nominal interest rate 6 which is in turn related to other financial asset prices through the arbitrage relation
Our abstraction from monetary frictions, and hence from the existence of seignorage revenues, does not mean that monetary policy has no fiscal consequences, for interest-rate policy and the equilibrium inflation that results from it have implications for the real burden of government debt. For simplicity, we shall assume that all public debt consists of riskless nominal one-period bonds. The nominal value of end-of-period public debt then evolves according to a law of motion
Rational-expectations equilibrium requires that the expected path of government surpluses must satisfy an intertemporal solvency condition
Under the standard (Ramsey) approach to the characterization of an optimal policy commitment, one chooses among state-contingent paths from some initial date onward that satisfy (11), (12), and (15) for each given initial government debt and price dispersion so as to maximize (5). Such a optimal plan requires commitment, insofar as the corresponding optimal plan for some later date , given the conditions obtaining at that date, will not involve a continuation of the optimal plan. This failure of time consistency occurs because the constraints on what can be achieved at date consistent with the existence of a rational-expectations equilibrium, depend on the expected paths of inflation, output and taxes at later dates; but in the absence of a prior commitment, a planner would have no motive at those later dates to choose a policy consistent with the anticipations that it was desirable to create at date
However, the degree of advance commitment that is necessary to bring about an optimal equilibrium is of only a limited sort. Let
In the first stage, values of the endogenous variables , where , and state-contingent commitments for the following period, are chosen, subject to a set of constraints stated in the appendix, including the requirement that the choices for each possible state of the world . These variables are chosen so as to maximize the objective where we define the functional
The optimization problem in stage two of this reformulation of the Ramsey problem is of the same form as the Ramsey problem itself, except that there are additional constraints associated with the precommitted values for the elements of Let us consider a problem like the Ramsey problem just defined, looking forward from some period except under the constraints that the quantities must take certain given values, where This constrained problem can similarly be expressed as a two-stage problem of the same form as above, with an identical stage two problem to the one described above. Stage two of this constrained problem is thus of exactly the same form as the problem itself. Hence the constrained problem has a recursive form. It can be decomposed into an infinite sequence of problems, in which in each period , are chosen to maximize , subject to the constraints of the stage one problem, given the predetermined state variables and the precommitted values
Our aim here is to characterize policy that solves this constrained optimization problem (stage two of the original Ramsey problem), i.e., policy that is optimal from some date onward given precommitted values for . Because of the recursive form of this problem, it is possible for a commitment to a time-invariant policy rule from date onward to implement an equilibrium that solves the problem, for some specification of the initial commitments A time-invariant policy rule with this property is said by Woodford (2003, chapter 7) to be ``optimal from a timeless perspective.''10 Such a rule is one that a policymaker that solves a traditional Ramsey problem would be willing to commit to eventually follow, though the solution to the Ramsey problem involves different behavior initially, as there is no need to internalize the effects of prior anticipation of the policy adopted for period 11 One might also argue that it is desirable to commit to follow such a rule immediately, even though such a policy would not solve the (unconstrained) Ramsey problem, as a way of demonstrating one's willingness to accept constraints that one wishes the public to believe that one will accept in the future.
In fact, we shall here characterize the solution to this problem (and similarly, derive optimal time-invariant policy rules) only for initial conditions near certain steady-state values, allowing us to use local approximations in characterizing optimal policy.12 We establish that these steady-state values have the property that if one starts from initial conditions close enough to the steady state, and exogenous disturbances thereafter are small enough, the optimal policy subject to the initial commitments remains forever near the steady state. Hence our local characterization would describe the long run character of Ramsey policy, in the event that disturbances are small enough, and that deterministic Ramsey policy would converge to the steady state.13 Of greater interest here, it describes policy that is optimal from a timeless perspective in the event of small disturbances.
We first must show the existence of a steady state, i.e., of an optimal policy (under appropriate initial conditions) that involves constant values of all variables. To this end we consider the purely deterministic case, in which the exogenous disturbances each take constant values and for all and assume an initial real public debt . We wish to find an initial degree of price dispersion and initial commitments such that the solution to the ``stage two'' problem defined above involves a constant policy each period, in which is equal to the initial real debt and is equal to the initial price dispersion. We show in the appendix that the first-order conditions for this problem admit a steady-state solution of this form, and we verify below that the second-order conditions for a local optimum are also satisfied.
Regardless of the initial public debt we show that (zero inflation), and correspondingly that (zero price dispersion). Note that our conclusion that the optimal steady-state inflation rate is zero generalizes the result of Benigno and Woodford (2003) for the case in which taxes are lump-sum at the margin. We may furthermore assume without loss of generality that the constant values of and are chosen (given the initial government debt ) so that in the optimal steady state, and each period.14 The associated steady-state tax rate is given by
We next wish to characterize the optimal responses to small perturbations of the initial conditions and small fluctuations in the disturbance processes around the above values. To do this, we compute a linear-quadratic approximate problem, the solution to which represents a linear approximation to the solution to the ``stage two'' policy problem, using the method introduced in Benigno and Woodford (2003). An important advantage of this approach is that it allows direct comparison of our results with those obtained in other analyses of optimal monetary stabilization policy. Other advantages are that it makes it straightforward to verify whether the second-order conditions hold that are required in order for a solution to our first-order conditions to be at least a local optimum,15 and that it provides us with a welfare measure with which to rank alternative sub-optimal policies, in addition to allowing computation of the optimal policy.
We begin by computing a Taylor-series approximation to our welfare measure (5), expanding around the steady-state allocation defined above, in which for each good at all times and at all times.16 As a second-order (logarithmic) approximation to this measure, we obtain17
Under the Calvo assumption about the distribution of intervals between price changes, we can relate the dispersion of prices to the overall rate of inflation, allowing us to rewrite (17) as
We note that when there is a non-zero linear term in (18), which means that we cannot expect to evaluate this expression to second order using only an approximate solution for the path of aggregate output that is accurate only to first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with approximations to the structural equations that are accurate only to first order. Rotemberg and Woodford (1997) avoid this problem by assuming an output subsidy (i.e., a value ) of the size needed to ensure that Here we do not wish to make this assumption, because we assume that lump-sum taxes are unavailable, in which case would be possible only in the case of a particular initial level of government assets Furthermore, we are more interested in the case in which government revenue needs are more acute than that would imply.
Benigno and Woodford (2003) propose an alternative way of dealing with this problem, which is to use a second-order approximation to the aggregate-supply relation to eliminate the linear terms in the quadratic welfare measure. In the model that they consider, where taxes are lump-sum (and so do not affect the aggregate supply relation), a forward-integrated second-order approximation to this relation allows one to express the expected discounted value of output terms as a function of purely quadratic terms (except for certain transitory terms that do not affect the ``stage two'' policy problem). In the present case, the level of distorting taxes has a first-order effect on the aggregate-supply relation (see equation (22) below), so that the forward-integrated relation involves the expected discounted value of the tax rate as well as the expected discounted value of output. However, as shown in the appendix, a second-order approximation to the intertemporal solvency condition (15 )19 provides another relation between the expected discounted values of output and the tax rate and a set of purely quadratic terms. These two second-order approximations to the structural equations that appear as constraints in our policy problem can then be used to express the expected discounted value of output terms in (18) in terms of purely quadratic terms.
In this manner, we can rewrite (18) as
In order for this linear-quadratic problem to have a bounded solution (which then approximates the solution to the exact problem), we must verify that the quadratic objective (20) is convex. We show in the appendix that so that the objective is convex, as long as the steady-state tax rate and share of government purchases in the national product are below certain positive bounds. We shall here assume that these conditions are satisfied, i.e., that the government's fiscal needs are not too severe. Note that in this case, our quadratic objective turns out to be of a form commonly assumed in the literature on monetary policy evaluation; that is, policy should seek to minimize the discounted value of a weighted sum of squared deviations of inflation from an optimal level (here zero) and squared fluctuations in an ``output gap'' where the target output level depends on the various exogenous disturbances in a way discussed in the appendix. It is also perhaps of interest to note that a ``tax smoothing'' objective of the kind postulated by Barro (1979) and Bohn (1990) does not appear in our welfare measure as a separate objective. Instead, tax distortions are relevant only insofar as they result in ``output gaps'' of the same sort that monetary stabilization policy aims to minimize.
We turn next to the form of the log-linear constraints in the approximate policy problem. A first-order Taylor series expansion of (11) around the zero-inflation steady state yields the log-linear aggregate-supply relation
It is useful to write this approximate aggregate-supply relation in terms of the welfare-relevant output gap . Equation (21) can be equivalently be written as
The effects of the various exogenous disturbances in on the ``cost-push'' term are explained in the appendix. It is worth noting that under certain conditions is unaffected by some disturbances. In the case that the cost-push term is given by
Even when if there are no government purchases (so that ) and no fiscal shocks (meaning that and ), then the term is again of the form (24), but with as discussed in Benigno and Woodford (2003). Hence in this case neither taste or technology shocks have ``cost-push'' effects. The reason is that in this ``isoelastic'' case, if neither taxes nor the wage markup ever vary, the flexible-price equilibrium value of output and the efficient level vary in exactly the same proportion in response to each of the other types of shocks; hence inflation stabilization also stabilizes the gap between actual output and the efficient level. Another special case is the limiting case of linear utility of consumption ( ); in this case, is again of the form (24 ), for a different value of In general, however, when and all of the disturbances shift the flexible-price equilibrium level of output (under a constant tax rate) and the efficient level of output to differing extents, resulting in ``cost-push'' contributions from all of these shocks.
The other constraint on possible equilibrium paths is the intertemporal government solvency condition. A log-linear approximation to (15) can be written in the form
We note that the only reason why it should not be possible to completely stabilize both inflation and the output gap from some date onward is if the sum is non-zero. The composite disturbance therefore completely summarizes the information at date about the exogenous disturbances that determines the degree to which stabilization of inflation and output is not possible; and under an optimal policy, the state-contingent evolution of the inflation rate, the output gap, and the real public debt depend solely on the evolution of the single composite disturbance process
This result contrasts with the standard literature on optimal monetary stabilization policy, in which (in the absence of a motive for interest-rate stabilization, as here) it is instead the cost-push term that summarizes the extent to which exogenous disturbances require that fluctuations in inflation and in the output gap should occur. Note that in the case that there are no government purchases and no fiscal shocks, corresponds simply to (24). Thus, for example, it is concluded (in a model with lump-sum taxes) that there should be no variation in inflation in response to a technology shock (Khan et al., 2002; Benigno and Woodford, 2003). But even in this simple case, the fiscal stress is given by an expression of the form
Finally, we wish to consider optimal policy subject to the constraints that and take given (precommitted) values. Again, only log-linear approximations to these constraints matter for a log-linear approximate characterization of optimal policy. As discussed in the appendix, the corresponding constraints in our approximate model are precommitments regarding the state-contingent values of and
To summarize, our approximate policy problem involves the choice of state-contingent paths for the endogenous variables from some date onward so as to minimize the quadratic loss function (20), subject to the constraint that conditions (23) and (25) be satisfied each period, given an initial value and subject also to the constraints that and equal certain precommitted values (that may depend on the state of the world in period ). We shall first characterize the state-contingent evolution of the endogenous variables in response to exogenous shocks, in the rational-expectations equilibrium that solves this problem. We then turn to the derivation of optimal policy rules, commitment to which should implement an equilibrium of this kind.
In considering the solution to the problem of stabilization policy just posed, it may be useful to first consider the simple case in which prices are fully flexible. This is the limiting case of our model in which with the consequence that in (20), and that in (23). Hence our optimization problem reduces to the minimization of
This last equation implies that unexpected inflation must equal the innovation in the fiscal stress,
We thus obtain simple conclusions about the determinants of fluctuations in inflation, output and the tax rate under optimal policy. Unexpected inflation variations occur as needed in order to prevent taxes from ever having to be varied in order to respond to variations in fiscal stress, as in the analyses of Bohn (1990) and Chari and Kehoe (1999). This allows a model with only riskless nominal government debt to achieve the same state-contingent allocation of resources as the government would choose to bring about if it were able to issue state-contingent debt, as in the model of Lucas and Stokey (1983).
Because taxes do not have to adjust in response to variations in fiscal stress, as in the tax-smoothing model of Barro (1979), it is possible to ``smooth'' them across states as well as over time. However, the sense in which it is desirable to ``smooth'' tax rates is that of minimizing variation in the gap , rather than variation in the tax rate itself.24 In other words, it is really the ``tax gap'' that should be smoothed. Under certain special circumstances, it will not be optimal for tax rates to vary in response to shocks; these are the conditions, discussed above, under which shocks have no cost-push effects, so that there is no change in . For example, if there are no government purchases and there is no variation in the wage markup, this will be the case. But more generally, all disturbances will have some cost-push effect, and result in variations in There will then be variations in the tax rate in response to these shocks under an optimal policy. However, there will be no unit root in the tax rate, as in the Barro (1979) model of optimal tax policy. Instead, as in the analysis of Lucas and Stokey (1983), the optimal fluctuations in the tax rate will be stationary, and will have the same persistence properties as the real disturbances (specifically, the persistence properties of the composite cost-push shock).
Variations in fiscal stress will instead require changes in the tax rate, as in the analysis of Barro (1979), if we suppose that the government issues only riskless indexed debt, rather than the riskless nominal debt assumed in our baseline model. (Again, for simplicity we assume that only one-period riskless debt is issued.) In this case the objective function (20) and the constraints (25) and (29) remain the same, but b the real value of private claims on the government at the beginning of period , is now a predetermined variable. This means that unexpected inflation variations are no longer able to relax the intertemporal government solvency condition. In fact, rewriting the constraint (25) in terms of we see that the path of inflation is now completely irrelevant to welfare.
The solution to this optimization problem is now less trivial, as complete stabilization of the output gap is not generally possible. The optimal state-contingent evolution of output and taxes can be determined using a Lagrangian method, as in Woodford (2003, chapter 7). The Lagrangian for the present problem can be written as
The resulting first-order condition with respect to is
An analytical solution to these equations is easily given. Using equation (29) to substitute for in the forward-integrated version of (25), then equations (31) and (32) to substitute for as a function of the path of and finally using (33) to replace all terms of the form (for ) by we obtain an equation that can be solved for The solution is of the form
In this case, unlike that of nominal debt, inflation is not affected by a pure fiscal shock (or indeed any other shock) under the optimal policy, but instead the output gap and the tax rate are. Note also that in the above solution, the multiplier the output gap, and the tax rate all follow unit root processes: a temporary disturbance to the fiscal stress permanently changes the level of each of these variables, as in the analysis of the optimal dynamics of the tax rate in Barro (1979) and Bohn (1990). However, the optimal evolution of the tax rate is not in general a pure random walk as in the analysis of Barro and Bohn. Instead, the tax gap is an IMA(1,1) process, as in the local analysis of Aiyagari et al. (2002); the optimal tax rate may have more complex dynamics, in the case that exhibits stationary fluctuations. In the special case of linear utility ( ), and both the output gap and the tax gap follow random walks (as both co-move with ). If the only disturbances are fiscal disturbances ( and ), then there are also no fluctuations in in this case, so that the optimal tax rate follows a random walk.
More generally, we observe that optimal policy ``smooths'' the value (in units of marginal utility) of additional government revenue in period , so that it follows a random walk. This is the proper generalization of the Barro tax-smoothing result, though it only implies smoothing of tax rates in fairly special cases. We find a similar result in the case that prices are sticky, even when government debt is not indexed, as we now show.
We turn now to the characterization of the optimal responses to shocks in the case that prices are sticky (). The optimization problem that provides a first-order characterization of optimal responses in this case is that of choosing processes from date onward to minimize (20), subject to the constraints (23) and (25) for each , together with initial constraints of the form
The first-order condition with respect to is given by
In the unique bounded solution to these equations, the dynamics of government debt and of the shadow value of government revenue are again of the form
We now discuss the optimal response of the variables to a disturbance to the level of fiscal stress. The laws of motion just derived for government debt and the Lagrange multiplier imply that temporary disturbances to the level of fiscal stress cause a permanent change in the level of both the Lagrange multiplier and the public debt. This then implies a permanent change in the level of output as well, which in turn requires (since inflation is stationary) a permanent change in the level of the tax rate. Since inflation is proportional to the change in the Lagrange multiplier, the price level moves in proportion to the multiplier, which means a temporary disturbance to the fiscal stress results in a permanent change in the price level, as in the flexible-price case analyzed in the previous section. Thus in this case, the price level, output gap, government debt, and tax rate all have unit roots, combining features of the two special cases considered in the previous section.29 Both price level and are random walks. They jump immediately to new permanent level in response to change in fiscal stress. In the case of purely transitory (white noise) disturbances, government debt also jumps immediately to a new permanent level. Given the dynamics of the price level and government debt, the dynamics of output and tax rate then jointly determined by the aggregate-supply relation and the government budget constraint.
We further find that the degree to which fiscal stress is relieved by a price-level jump (as in the flexible-price, nominal-debt case) as opposed to an increase in government debt and hence a permanently higher tax rate (as in the flexible-price, indexed-debt case) depends on the degree of price stickiness. We illustrate this with a numerical example. We calibrate a quarterly model by assuming that and in accordance with the estimates of Rotemberg and Woodford (1997). We furthermore assume an elasticity of substitution among alternative goods of , an overall level of steady-state distortions a steady-state tax rate of and a steady-state debt level (debt equal to 60 percent of a year's GDP). Given the assumed degree of market power of producers (a steady-state gross price markup of 1.11) and the assumed size of the tax wedge, the value corresponds to a steady-state wage markup of If we assume that there are no government transfers in the steady state, then the assumed level of tax revenues net of debt service would finance steady-state government purchases equal to a share of output.
Let us suppose that the economy is disturbed by an exogenous increase in transfer programs , equal to one percent of aggregate output, and expected to last only for the current quarter. Figure 1 shows the optimal impulse response of the government debt to this shock (where quarter zero is the quarter of the shock), for each of 7 different values for the slope of the short-run aggregate-supply relation, maintaining the values just stated for the other parameters of the model. The solid line indicates the optimal response in the case of our baseline value for , based on the estimates of Rotemberg and Woodford; the other cases represent progressively greater degrees of price flexibility, up to the limiting case of fully flexible prices (the case ). Figures 2 and 3 similarly show the optimal responses of the tax rate and the inflation rate to the same disturbance, for each of the same 7 cases.30
We see that the volatility of both inflation and tax rates under optimal policy depends greatly on the degree of stickiness of prices. Table 1 reports the initial quarter's response of the inflation rate, and the long-run response of the tax rate, for each of the 7 cases. The table also indicates for each case the implied average time (in weeks) between price changes, where is the fraction of prices unchanged for an entire quarter implied by the assumed value of .31 We first note that our baseline calibration implies that price changes occur only slightly less frequently than twice per year, which is consistent with survey evidence.32 Next, we observe that even were we to assume an aggregate-supply relation several times as steep as the one estimated using U.S. data, our conclusions with regard to the size of the optimal responses of the (long-run) tax rate and the inflation rate would be fairly similar. At the same time, the optimal responses with fully flexible prices are quite different: the response of inflation is 80 times as large as under the baseline sticky-price calibration (implying a variance of inflation 6400 times as large), while the long-run tax rate does not respond at all in the flexible-price case.33 But even a small degree of stickiness of prices makes a dramatic difference in the optimal responses; for example, if prices are revised only every five weeks on average, the variance of inflation is reduced by a factor of more than 200, while the optimal response of the long-run tax rate to the increased revenue need is nearly the same size as under the baseline degree of price stickiness. Thus we find, as do Schmitt-Grohé and Uribe (2001) in the context of a calibrated model with convex costs of price adjustment, that the conclusions of the flexible-price analysis are quite misleading if prices are even slightly sticky. Under a realistic calibration of the degree of price stickiness, inflation should be quite stable, even in response to disturbances with substantial consequences for the government's budget constraint, while tax rates should instead respond substantially (and with a unit root) to variations in fiscal stress.
We can also compare our results with those that arise when taxes are lump-sum. In this case, and the first-order condition (36) requires that The remaining first-order conditions reduce to
Note that the familiar case from the literature on monetary stabilization policy does not result simply from assuming that sources of revenue that do not shift the aggregate-supply relation are available ; it is also important that the sort of tax that does shift the AS relation (like the sales tax here) is not available. We could nest both the standard model and our present baseline case within a single, more general framework by assuming that revenue can be raised using either the sales tax or a lump-sum tax, but that there is an additional convex cost (perhaps representing ``collection costs'', assumed to reduce the utility of the representative household but not using real resources) of increases in either tax rate. The standard case would then appear as the limiting case of this model in which the collection costs associated with the sales tax are infinite, while those associated with the lump-sum tax are zero; the baseline model here would correspond to an alternative limiting case in which the collection costs associated with the lump-sum tax are infinite, while those associated with the sales tax are zero. In intermediate cases, we would continue to find that fiscal stress affects the optimal evolution of both inflation and the output gap, as long as there is a positive collection cost for the lump-sum tax. At the same time, the result that the shadow value of additional government revenue follows a random walk under optimal policy (which would still be true) will not in general imply, as it does here, that the price level should also be a random walk; for the perfect co-movement of and that characterizes optimal policy in our baseline case will not be implied by the first-order conditions except in the case that there are no collection costs associated with the sales tax. Nonetheless, the price level will generally contain a unit root under optimal policy, even if it will not generally follow a random walk.
We also obtain results more similar to those in the standard literature on monetary stabilization policy if we assume (realistically) that it is not possible to adjust tax rates on such short notice in response to shocks as is possible with monetary policy. As a simple way of introducing delays in the adjustment of tax policy, suppose that the tax rate has to be fixed in period . In this case, the first-order conditions characterizing optimal responses to shocks are the same as above, except that (36) is replaced by
We now wish to characterize the policy rules that the monetary and fiscal authorities can follow in order to bring about the state-contingent responses to shocks described in the previous section. One might think that it suffices to solve for the optimal state-contingent paths for the policy instruments. But in general this is not a desirable approach to the specification of a policy rule, as discussed in Svensson (2003) and Woodford (2003, chapter 7). A description of optimal policy in these terms would require enumeration of all of the types of shocks that might be encountered later, indefinitely far in the future, which is not feasible in practice. A commitment to a state-contingent instrument path, even when possible, also may not determine the optimal equilibrium as the locally unique rational-expectations equilibrium consistent with this policy; many other (much less desirable) equilibria may also be consistent with the same state-contingent instrument path.
Instead, we here specify targeting rules in the sense of Svensson (1999, 2003) and Giannoni and Woodford (2003). These targeting rules are commitments on the part of the policy authorities to adjust their respective instruments so as to ensure that the projected paths of the endogenous variables satisfy certain target criteria. We show that under an appropriate choice of these target criteria, a commitment to ensure that they hold at all times will determine a unique non-explosive rational-expectations equilibrium, in which the state-contingent evolution of inflation, output and the tax rate solves the optimization problem discussed in the previous section. Moreover, we show that it is possible to obtain a specification of the policy rules that is robust to alternative specifications of the exogenous shock processes.
We apply the general approach of Giannoni and Woodford (2002), which allows the derivation of optimal target criteria with the properties just stated. In addition, Giannoni and Woodford show that such target criteria can be formulated that refer only to the projected paths of the target variables (the ones in terms of which the stabilization objectives of policy are defined -- here, inflation and the output gap). Briefly, the method involves constructing the target criteria by eliminating the Lagrange multipliers from the system of the system of first-order conditions that characterize the optimal state-contingent evolution, regardless of character of the (additive) disturbances. We are left with linear relations among the target variables, that do not involve the disturbances and with coefficients independent of the specification of the disturbances, that represent the desired target criteria.
Recall that the first-order conditions that characterize the optimal state-contingent paths in the problem considered in the previous section are given by (34) - (37). As explained in the previous section, the first three of these conditions imply that the evolution of inflation and of the output gap must satisfy (38) - (39) each period. We can solve (38) - (39) for the values of implied by the values of that are observed in an optimal equilibrium. We can then replace in these two relations by the multiplier implied in this way by observed values of Finally, we can eliminate from these two relations, to obtain a necessary relation between and , given and given by
The remaining first-order condition from the previous section, not used in the derivation of (41), is (37). By similarly using the solutions for implied by observations of to substitute for the multipliers in this condition, one obtains a further target criterion
Moreover, this equilibrium solves the above first-order conditions for a particular specification of the initial lagged multipliers , which are inferred from the initial values in the way just explained. Hence this equilibrium minimizes expected discounted losses (20) given and subject to constraints on initial outcomes of the form
We have established a pair of target criteria with the property that if they are expected to be jointly satisfied each period, the resulting equilibrium involves the optimal responses to shocks. This result in itself, however, does not establish which policy instrument should be used to ensure satisfaction of which criterion. Since the variables referred to in both criteria can be affected by both monetary and fiscal policy, there is not a uniquely appropriate answer to that question. However, the following represents a relatively simple example of a way in which such a regime could be institutionalized through separate targeting procedures on the part of monetary and fiscal authorities.
Let the central bank be assigned the task of maximizing social welfare through its adjustment of the level of short-term interest rates, taking as given the state-contingent evolution of the public debt , which depends on the decisions of the fiscal authority. Thus the central bank treats the evolution of the public debt as being outside its control, just like the exogenous disturbances and simply seeks to forecast its evolution in order to correctly model the constraints on its own policy. Here we do not propose a regime under which it is actually true that the evolution of the public debt would be unaffected by a change in monetary policy. But there is no inconsistency in the central bank's assumption (since a given bounded process will continue to represent a feasible fiscal policy regardless of the policy adopted by the central bank), and we shall show that the conduct of policy under this assumption does not lead to a suboptimal outcome, as long as the state-contingent evolution of the public debt is correctly forecasted by the central bank.
The central bank then seeks to bring about paths for from date onward that minimize (20), subject to the constraints (23) and (25) for each , together with initial constraints of the form (43) - (44), given the evolution of the processes . The first-order conditions for this optimization problem are given by (34), (35) and (37) each period, which in turn imply that (41) must hold each period, as shown above. One can further show that a commitment by the central bank to ensure that (41) holds each period determines the equilibrium evolution that solves this problem, in the case of an appropriate (self-consistent) choice of the initial constraints (43) - (44). Thus (41) is an optimal target criterion for a policy authority seeking to solve the kind of problem just posed; and since the problem takes as given the evolution of the public debt, it is obviously a more suitable assignment for the central bank than for the fiscal authority. The kind of interest-rate reaction function that can be used to implement a ``flexible inflation target'' of this kind is discussed in Svensson and Woodford (2003) and Woodford (2003, chapter 7).
Correspondingly, let the fiscal authority be assigned the task of choosing the level of government revenue each period that will maximize social welfare, taking as given the state-contingent evolution of output , which it regards as being determined by monetary policy. (Again, it need not really be the case that the central bank ensures a particular state-contingent path of output, regardless of what the fiscal authority does. But again, this assumption is not inconsistent with our model of the economy, since it is possible for the central bank to bring about any bounded process that it wishes, regardless of fiscal policy, in the case that prices are sticky.) If the fiscal authority regards the evolution of output as outside its control, its objective reduces to the minimization of
Forward integration of (23) implies that
The first-order condition for the optimal choice of is then simply (42), which we find is indeed a suitable target criterion for the fiscal authority. The decision rule implied by this target criterion is seen to be
We have shown that it is possible to jointly analyze optimal monetary and fiscal policy within a single framework. The two problems, often considered in isolation, turn out to be more closely related than might have been expected. In particular, we find that variations in the level of distorting taxes should be chosen to serve the same objectives as those emphasized in the literature on monetary stabilization policy: stabilization of inflation and of a (properly defined) output gap. A single output gap can be defined that measures the total distortion of the level of economic activity, resulting both from the stickiness of prices (and the consequent variation in markups) and from the supply-side effects of tax distortions. It is this cumulative gap that one wishes to stabilize, rather than the individual components resulting from the two sources; and both monetary policy and tax policy can be used to affect it. Both monetary policy and tax policy also matter for inflation determination in our model, because of the effects of the tax rate on real marginal cost and hence on the aggregate-supply relation. Indeed, we have exhibited a pair of robustly optimal targeting rules for the monetary and fiscal authorities respectively, under which both authorities consider the consequences of their actions for near-term inflation projections in determining how to adjust their instruments.
And not only should the fiscal authority use tax policy to serve the traditional goals of monetary stabilization policy; we also find that the monetary authority should take account of the consequences of its actions for the government budget. In the present model, that abstracts entirely from transactions frictions, these consequences have solely to do with the implications of alternative price-level and interest-rate paths for the real burden of interest payments on the public debt, and not any contribution of seignorage to government revenues. Nonetheless, under a calibration of our model that assumes a debt burden and a level of distorting taxes that would not be unusual for an advanced industrial economy, taking account of the existence of a positive shadow value of additional government revenue (owing to the non-existence of lump-sum taxes) makes a material difference for the quantitative characterization of optimal monetary policy. In fact, we have found that the crucial summary statistic that indicates the degree to which various types of real disturbances should be allowed to affect short-run projections for either inflation or the output gap is not the degree to which these disturbances shift the aggregate-supply curve for a given tax rate (i.e., the extent to which they represent ``cost-push'' shocks), but rather the degree to which they create fiscal stress (shift the intertemporal government solvency condition).
Our conclusion that monetary policy should take account of the requirements for government solvency does not imply anything as strong as the result of Chari and Kehoe (1999) for a flexible-price economy with nominal government debt, according to which surprise variations in the inflation rate should be used to completely offset variations in fiscal stress, so that tax rates need not vary (other than as necessary to stabilize the output gap). We find that in the case of even a modest degree of price stickiness -- much less than what seems to be consistent with empirical evidence for the U.S. -- it is not optimal for inflation to respond to variations in fiscal stress by more than a tiny fraction of the amount that would be required to eliminate the fiscal stress (and that would be optimal with fully flexible prices); instead, a substantial part of the adjustment should come through a change in the tax rate. But the way in which the acceptable short-run inflation projection should be affected by variations in the projected output gap is substantially different in an economy with only distorting taxes than would be the case in the presence of lump-sum taxation. For with distorting taxes, the available tradeoff between variations in inflation and in the output gap depends not only on the way these variables are related to one another through the aggregate-supply relation, but also on the way that each of them affects the government budget.
In this section, we derive equation (11) in the main text and we define the variables and In the Calvo model, a supplier that changes its price in period chooses a new price to maximize
Here Dixit-Stiglitz monopolistic competition implies that the individual supplier faces a demand curve each period of the form
We note that supplier 's profits are a concave function of the quantity sold since revenues are proportional to and hence concave in , while costs are convex in . Moreover, since is proportional to the profit function is also concave in . The first-order condition for the optimal choice of the price is the same as the one with respect to hence the first-order condition with respect to is both necessary and sufficient for an optimum.
For this first-order condition, we obtain
Using the isoelastic functional forms given in the text, we obtain a closed-form solution to (49), given by
Under the standard (Ramsey) approach to the characterization of an optimal policy commitment, one chooses among state-contingent paths from some initial date onward that satisfy
We now show that the optimal plan (Ramsey problem) can be obtained as the solution to a two-stage optimization problem. To this purpose, let
The two-stage optimization problem is the following. In the first stage, values of the endogenous variables , where , and state-contingent commitments for the following period, are chosen so as to maximize an objective defined below. In the second stage, the equilibrium evolution from period onward is chosen to solve the maximization problem that defines the value function , given the state of the world and the precommitted values for associated with that state.
In defining the objective for the first stage of this equivalent formulation of the Ramsey problem, it is useful to let denote the value of that solves (56) for given values of and , and to let denote the real primary surplus defined by (59) in the case of given values of and . We also define the functional relationships
Then in the first stage, and are chosen so as to maximize
(i) and satisfy (57);
(ii) the values
(iii) the value
(iv) the choices for each possible state of the world .
These constraints imply that the objective is well-defined, and that values are chosen for which the stage two problem will be well-defined, whichever state of the world is realized in period Furthermore, in the case of any stage-one choices consistent with the above constraints, and any subsequent evolution consistent with the constraints of the stage-two problem, (66) implies that (56) is satisfied in period while (62) implies that (58) is satisfied in period Constraint (i) above implies that (57) is also satisfied in period Finally, the constraints of the stage-two problem imply that both (56), (57) and (58) are satisfied in each period thus the state-contingent evolution that solves the two-stage problem is a rational-expectations equilibrium. Conversely, one can show that any possible rational-expectations equilibrium satisfies all of these constraints.
One can then reformulate the Ramsey problem, replacing the set of requirements for rational-expectations equilibrium by the stage-one constraints plus the stage-two constraints. Since no aspect of the evolution from period onward, other than the specification of , affects the stage-one constraints, the optimization problem decomposes into the two stages defined above, where the objective (63) corresponds to the maximization of in the first stage.
The optimization problem in stage two of this reformulation of the Ramsey problem is of the same form as the Ramsey problem itself, except that there are additional constraints associated with the precommitted values for the elements of Let us consider a problem like the Ramsey problem just defined, looking forward from some period except under the constraints that the quantities must take certain given values, where This constrained problem can similarly be expressed as a two-stage problem of the same form as above, with an identical stage two problem to the one described above. The stage one problem is also identical to stage one of the Ramsey problem, except that now the plan chosen in stage one must be consistent with the given values so that conditions (64), (65) and (67) are now added to the constraints on the possible choices of in stage one. (The stipulation that implies that the constraint set remains non-empty despite these additional restrictions.)
Stage two of this constrained problem is thus of exactly the same form as the problem itself. Hence the constrained problem has a recursive form. It can be decomposed into an infinite sequence of problems, in which in each period , are chosen to maximize , given the predetermined state variables and the precommitted values subject to the constraints that
(ii) the precommitted values are fulfilled, i.e.,
(iii) the choices for each possible state of the world .
Our aim in the paper is to provide a local characterization of policy that solves this recursive optimization, in the event of small enough disturbances, and initial conditions that are close enough to consistency with the steady state characterized in the next section of this appendix.
Here we show the existence of a steady state, i.e., of an optimal policy (under appropriate initial conditions) of the `recursive policy problem just defined that involves constant values of all variables. We now consider a deterministic problem in which the exogenous disturbances , , , , , each take constant values , , , and for all , and we start from initial conditions . (The value of is arbitrary, subject to an upper bound discussed below.) We wish to find an initial degree of price dispersion and initial commitments such that the recursive (or ``stage two'') problem involves a constant policy , each period, in which is equal to the initial real debt and is equal to the initial price dispersion.
We thus consider the problem of maximizing
We introduce Lagrange multipliers through corresponding to constraints (72) through (77) respectively. We also introduce multipliers dated corresponding to the constraints implied by the initial conditions ; the latter multipliers are normalized in such a way that the first-order conditions take the same form at date as at all later dates. The first-order conditions of the maximization problem are then the following. The one with respect to is
We search for a solution to these first-order conditions in which , and at all times. A steady-state solution of this kind also requires that the Lagrange multipliers take constant values. We furthermore conjecture the existence of a solution in which as stated in the text. Note that such a solution implies that and . Using these substitutions, we find that (the steady-state version of) each of the first-order conditions (78) - (85) is satisfied if the steady-state values satisfy
We have thus verified that a constant solution to the first-order conditions exists. With a method to be explained below, we check that this solution is indeed at least a local optimum. Note that as asserted in the text, this deterministic steady state involves zero inflation, and a steady-state tax rate
We derive here equations (17) and (18) in the main text, taking a second-order approximation to (equation (60)) following the treatment in Woodford (2003, chapter 6). We start by approximating the expected discounted value of the utility of the representative household
First we note that
The first term in (89) can be approximated using a second-order Taylor expansion around the steady state defined in the previous section as
We may similarly approximate by
We next take a second-order expansion of the law of motion (55) for the price index, obtaining
We note further that a second-order approximation to the identity yields
This can be expressed compactly in the form
Note that in a first-order approximation, (100) can be written as simply
We now derive a second-order approximation to the intertemporal government solvency condition. We use the definition
We recall now that the primary surplus is defined as
Note that in the first-order approximation we can simply write (110) as
In particular, we obtain that and that
We have further defined , the desired level of output, as
Then an optimal policy from a timeless perspective is a policy from date onward that minimizes the quadratic loss function (114) subject to the constraints implied by the linear structural relations (101) and (111) holding in each period given the initial values and subject also to the constraints that certain predetermined values for and (or alternatively, for and for ) be achieved.40 We note that under the assumption that , , which implies that . Moreover, if
The AS equation (101) can be written as
The flow budget constraint (111) can be solved forward to yield the intertemporal solvency condition
The coefficients , , , are defined as
The coefficients and of section 5 are defined as
We now show that there is a determinate equilibrium if policy is conducted so as to ensure that the two target criteria
Because is lower triangular, its eigenvalues are the four diagonal elements: 0, 0, 1, and Hence there is exactly one eigenvalue outside the unit circle, and equilibrium is determinate (but possesses a unit root). Because of the triangular form of the matrix, one can also easily solve explicitly for the elements of the left eigenvector
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1. Published in Mark Gertler and Kenneth Rogoff, eds., NBER Macroeconomics Annual 2003, Cambridge: MIT Press, http://mitpress.mit.edu. Reprinted with permission of The MIT Press. We would like to thank Stefania Albanesi, Marios Angeletos, Albert Marcet, Ramon Marimon, seminar participants at New York University, Rutgers University, Universitat Pompeu Fabra, the NBER Macroeconomics Annual conference, and the editors for helpful comments, Brad Strum and Vasco Curdia for research assistance, and the National Science Foundation for research support through a grant to the NBER. Return to text
2. We might alternatively assume specialization across households in the type of labor supplied; in the presence of perfect sharing of labor income risk across households, household decisions regarding consumption and labor supply would all be as assumed here. Return to text
3. The government is assumed to need to obtain an exogenously given quantity of the Dixit-Stiglitz aggregate each period, and to obtain this in a cost-minimizing fashion. Hence the government allocates its purchases across the suppliers of differentiated goods in the same proportion as do households, and the index of aggregate demand is the same function of the individual quantities as is of the individual quantities consumed , defined in (2). Return to text
8. See Woodford (2003, chapter 2) for derivation of this condition from household optimization together with market clearing. The condition should not be interpreted as an a priori constraint on possible government policy rules, as discussed in Woodford (2001). However, when we consider the problem of choosing an optimal plan from the among the possible rational-expectations equilibria, this condition must be imposed among the constraints on the set of equilibria that one may hope to bring about. Return to text
9. In our notation for the value function denotes not simply the vector of disturbances in period , but all information in period about current and future disturbances. This corresponds to the disturbance vector referred to earlier in the case that the disturbance vector follows a Markov process. Return to text
11. For example, in the case of positive initial nominal government debt, the optimal policy would involve a large inflation in period in order to reduce the pre-existing debt burden, but a commitment not to respond similarly to the existence of nominal government debt in later periods. Return to text
12. Local approximations of the same sort are often used in the literature in numerical characterizations of Ramsey policy. Strictly speaking, however, such approximations are valid only in the case of initial commitments near enough to the steady-state values of these variables, and the optimal (Ramsey) policy need not involve values of near the steady-state values, even in the absence of random disturbances. Return to text
13. Benigno and Woodford (2003) gives an example of an application in which Ramsey policy does converge asymptotically to the steady state, so that the solution to the approximate problem approximates the response to small shocks under the Ramsey policy, at dates long enough after . We cannot make a similar claim in the present application, however, because of the unit root in the dynamics associated with optimal policy. Return to text
15. Benigno and Woodford (2003) show that these conditions can fail to hold, so that a small amount of arbitrary randomization of policy is welfare-improving, but argue that the conditions under which this occurs in their model are not empirically plausible. Return to text
16. Here the elements of are assumed to be and so that a value of zero for this vector corresponds to the steady-state values of all disturbances. The perturbations and are not defined to be logarithmic so that we do not have to assume positive steady-state values for these variables. Return to text
18. Specifically, we use the notation as shorthand for where in each case hats refer to log deviations from the steady-state values of the various parameters of the policy problem. We treat as an expansion parameter, rather than because (12) implies that deviations of the inflation rate from zero of order only result in deviations in the dispersion measure from one of order We are thus entitled to treat the fluctuations in as being only of second order in our bound on the amplitude of disturbances, since if this is true at some initial date it will remain true thereafter. Return to text
19. Since we are interested in providing an approximate characterization of the ``stage two'' policy problem, in which a precommitted value of appears as a constraint, it is actually a second-order approximation to that constraint that we need. However, this latter constraint has the same form as (15); the difference is only in which quantities in the relation are taken to have predetermined values. Return to text
21. If we restrict attention to bounded paths for the endogenous variables, then a path satisfies (25) in each period if and only if it satisfies the flow budget constraint (26) in each period. Return to text
22. Note that this preference can be justified in terms of our model, in the case that is positive though extremely small. For there will then be a very small positive value for , implying that reduction of the expected discounted value of inflation is preferred to the extent that this does not require any increase in the expected discounted value of squared output gaps. Return to text
23. The result relies upon the fact that the distortions created by the monetary frictions are minimized in the case of a zero opportunity cost of holding money each period, as argued by Friedman (1969). Neither the existence of effects of nominal interest rates on supply costs (so that an interest-rate term should appear in the aggregate-supply relation (29)) nor the contribution of seignorage revenues to the government budget constraint make any difference to the result, since unexpected changes in revenue needs can always be costlessly obtained through unexpected inflation, while any desired shifts in the aggregate-supply relation to offset cost-push shocks can be achieved by varying the tax rate. Return to text
24. A number of authors (e.g., Chari et al., 1991, 1994; Hall and Krieger, 2000; Aiyagari et al., 2002) have found that in calibrated flexible-price models with state-contingent government debt, the optimal variation in labor tax rates is quite small. Our results indicate this as well, in the case that real disturbances have only small cost-push effects, and we have listed earlier various conditions under which this will be the case. But under some circumstances, optimal policy may involve substantial volatility of the tax rate, and indeed, more volatility of the tax rate than of inflation. This would be the case if shocks occur that have large cost-push effects while having relatively little effect on fiscal stress. Return to text
26. It should be recalled that in order for policy to be optimal from a timeless perspective, the state-contingent initial commitment must be chosen in a way that conforms to the state-contingent commitment regarding that will be chosen in all later periods, so that the optimal policy can be implemented by a time-invariant rule. Hence it is convenient to present the first-order conditions in a time-invariant form. Return to text
27. In the only such solution, the variables , and are all permanently affected by shocks, even when the disturbances are all assumed to be stationary (and bounded) processes. Hence a bounded solution exists only under the assumption that random disturbances occur only in a finite number of periods. However, our characterization of optimal policy does not depend on a particular bound on the number of periods in which there are disturbances, or which periods these are; in order to allow disturbances in a larger number of periods, we must assume a tighter bound on the amplitude of disturbances, in order for the optimal paths of the endogenous variables to remain within a given neighborhood of the steady-state values. Aiyagari et al. (2002) discuss the asymptotic behavior of the optimal plan in the exact nonlinear version of a problem similar to this one, in the case that disturbances occur indefinitely. Return to text
28. The initial condition for is in turn chosen so that the solution obtained is consistent with the initial constraint Under policy that is optimal from a timeless perspective, this initial commitment is in turn chosen in a self-consistent fashion, as discussed further in section 5. Note that the specification of does not affect our conclusions about the optimal responses to shocks, emphasized in this section. Return to text
29. Schmitt-Grohé and Uribe (2001) similarly observe that in a model with sticky prices, the optimal response of the tax rate is similar to what would be optimal in a flexible-price model with riskless indexed government debt. Return to text
30. In figure 1, a response of 1 means a one percent increase in the value of from 60 percent to 60.6 percent of a year's GDP. In figure 2, a response of 1 means a one percent decrease in , from 20 percent to 20.2 percent. In figure 3, a response of 1 means a one percent per annum increase in the inflation rate, or an increase of the price level from 1 to 1.0025 over the course of a quarter (given that our model is quarterly). The responses reported in Table 1 are measured in the same way. Return to text
31. We have used the relation between and for a continuous-time version of the Calvo model in order to express the degree of price stickiness in terms of an average time between price changes. Return to text
32. The indicated average time between price changes for the baseline case is shorter than that reported in Rotemberg and Woodford (1997), both because we here assume a slightly larger value of implying a smaller value of and because of the continuous-time method used here to convert into an implied average time interval. Return to text
33. The tax rate does respond in the quarter of the shock in the case of flexible prices, but with the opposite sign to that associated with optimal policy under our baseline calibration. Under flexible prices, as discussed above, the tax rate does not respond to variations in fiscal stress at all. Because the increase in government transfers raises the optimal level of output for reasons explained in the appendix, the optimal tax rate actually falls, in order to induce equilibrium output to increase; and under flexible prices, this is the optimal response of Return to text
36. As stated in the text, in our notation for the value function denotes not simply the vector of disturbances in period , but all information in period about current and future disturbances. Return to text
38. This is true for any tax rate at which exceeds Fixing any value our Inada conditions imply that this inequality holds for all small enough values of . And if the inequality holds for some then by continuity it must hold for an open interval of values of Return to text
39. In fact, there must exist at least two such solutions, since the Inada conditions also imply that for all close enough to 1. These multiple solutions correspond to a ``Laffer curve'' result, under which two distinct tax rates result in the same equilibrium level of government revenues. We select the lower-tax, higher-output solution as the one around which we compute our Taylor-series expansions; this is clearly the higher-utility solution. Return to text
40. The constraint associated with a predetermined value for can be neglected, in a first-order characterization of optimal policy, because the variable does not appear in the first-order approximation to the aggregate-supply relation. Return to text
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