INTRODUCTION
Central banks and other policy makers are commonly modeled as agents whose objective is to minimize some loss function subject to constraints that contain forward-looking rational expectations [see Woodford (2002), Clarida, Galí, and Gertler (1999), or the Taylor (1999) conference volume, for example]. But of course Kydland and Prescott (1977) showed that in the absence of some commitment mechanism optimal policies were time-inconsistent. The modern policy literature addresses this time-consistency problem either by simply assuming that the policymaker can commit to not reoptimize or by explicitly modeling the strategic interactions that occur between the various economic agents in the model.1
An alternative approach is to assume that the policy maker sets policy according to a “timeless perspective” (Woodford, 1999). According to the timelsss perspective the policymaker optimizing today behaves as it would have chosen to if it had optimized at a time far in the past. Useful discussions and analyses of timelessly optimal policies can be found in Giannoni and Woodford (2002) and Jensen and McCallum (2002).
See Holly and Hughes-Hallett (1989), Amman and Kendrick (1999), Chen and Zadrozney (2002), and Blake (2004), for other approaches.
Several methods for solving for optimal commitment policies and for Markov-perfect Stackelberg-Nash equilibria (optimal discretionary policies in what follows) have been developed. For commitment, available algorithms include those developed in Currie and Levine (1985, 1993) and Söderlind (1999), which are based on Hamiltonian and Lagrangian methods, respectively, and Backus and Driffill (1986), which transforms the system to a dynamic programming problem to obtain the commitment policy. For discretion, two popular algorithms are those developed by Oudiz and Sachs (1985) and Backus and Driffill (1986). Both Oudiz and Sachs (1985) and Backus and Driffill (1986) solve for optimal discretionary rules using dynamic programming, but Oudiz and Sachs (1985) solve for the stationary feedback rule by taking the limit as t→−∞ of a finite horizon problem, whereas Backus and Driffill (1986) solve the asymptotic problem directly.3
Söderlind (1999) provides a popular implementation of the Backus and Driffill (1986) algorithm. Krusell, Quadrini, and Rios-Rull (1997) provide an alternative, guess-and-verify, approach to solving for time-consistent policies.
A feature common to all of these solution methods is that they require the optimization constraints to be written in state-space form. An attractive feature of the state-space form is that it provides a compact and efficient representation of an economic model in a form that contains only first-order dynamics. However, reliance on the state-space form also has disadvantages. In particular, use of the state-space representation forces the distinction between predetermined and nonpredetermined variables and often requires considerable effort to manipulate the model in the required form. Both of these requirements, but especially the latter, mean that these solution methods come with “overhead” costs that can be considerable.
In this paper, we develop algorithms to solve for optimal commitment policies and optimal discretionary policies that differ from existing algorithms in that they allow the constraints to be written in structural form rather than in state-space form. As a broadbrush characterization, for small-scale models that have only a few nonpredetermined variables, it is generally not too difficult to express the model in state-space form, in which case solving the model using a state-space solution method may be convenient. However, even for models that are only modestly complex it is often much easier to write the model in structural form, and the algorithms presented here have been developed with these models in mind.4
Dennis (2004a) shows how to solve for optimal simple rules when the optimization constraints are written in structural form.
Although the relative strength of these algorithms is that they can be easily applied to models that are difficult to manipulate into a state-space form, to illustrate their use we apply them to several, reasonably simple, New Keynesian models and compare their properties to existing solution methods. For this comparison, we take the models developed by Galí and Monacelli (2005) and Erceg, Henderson, and Levin (2000) and solve for optimal commitment policies and optimal discretionary policies for a range of policy regimes. Although the algorithms developed in this paper exploit less model structure than the state-space methods, they are still reasonably efficient. To demonstrate that they can usefully solve larger models, we also apply them to the model developed by Fuhrer and Moore (1995).5
Dennis and Söderström (Forthcoming) apply the methods developed in this paper to the macro-policy model developed by Orphanides and Wieland (1998), which is somewhat more complicated than the Fuhrer and Moore (1995) model.
The remainder of this paper is structured as follows. The following section introduces the components of the policy maker's optimization problem, describing the objective function and the equations that constrain the optimization process. Section 3 shows how to solve for optimal commitment rules. Section 4 turns to discretion. The discretionary problem is formulated in terms of an unconstrained optimization, which contrasts with the more usual dynamic programming or Lagrangian-based approaches. In Section 5, we employ two New Keynesian models to calculate the computation times of a range of solution methods. Section 5 also discusses situations for which convergence problems can be encountered when solving for optimal policies. Section 6 concludes while appendices contain technical details.
THE OPTIMIZATION PROBLEM
Let yt be an (n×1) vector of endogenous variables and xt be a (p×1) vector of policy instruments. For convenience, the variables in yt and xt represent deviations from nonstochastic steady-state values. The policy maker sets xt to minimize the loss function6
The omission of terms interacting yt and xt in equation (1) is without loss of generality. When the constraints are in structural form it is always possible in include xt within yt with W containing the relevant penalty terms.
where β∈(0, 1) is the discount factor and W (n × n) and Q (p × p) are symmetric, positive semidefinite, matrices containing policy preferences, or policy tastes. Here E0 represents the mathematical expectations operator conditional on period 0 information. Loss functions like equation (1) are widely employed in the literature because, with linear constraints, they lead to linear decision rules. Furthermore, as is now well documented, this loss function can represent a second-order approximation to a representative agent's utility function [see Diaz-Gimenez (1999), for example] and in certain situations it collapses to an expression containing just output and inflation (Woodford, 2002).
The policy maker minimizes (1) subject to the following system of dynamic constraints7
Binder and Pesaran (1995) show how a wide class of models, which can contain quite general lead-lag structures and expectations formed with varying information sets, can be written in terms of equation (2).
where
is an (s×1, s≤n) vector of innovations and the matrices A0, A1, A2, A3, A4 , and A5 contain the model's structural parameters, which are conformable with yt, xt, and vt, as necessary. The dating convention in equation (2) is such that every element in
has a known value as of the beginning of period t. A0 is assumed to be nonsingular.
It is not necessary to explicitly include the matrix A5 in equation (2) because it is always possible to include vt within yt. However, its presence allows us to easily accommodate models where the innovations are scaled by structural parameters without having to expand the state vector or redefine the innovations. An important feature of equation (2) is that it contains not just the contemporaneous policy instruments but also the expectation of next period's instrument vector, which is useful for models that contain an interest rate term structure, for example.8
COMMITMENT
Under commitment the policymaker optimizes once and never reoptimizes. The assumption is that either the policy maker can adhere to its chosen policy or that the discount factor is sufficiently large to allow the chosen policy to be supported as a reputation equilibrium [see Currie and Levine (1993, Chapter 5)]. To solve for optimal commitment policies, we observe that from the definition of rational expectations we can write
where the expectation errors,
and
, are martingale difference sequences. Substituting equations (3) and (4) into equation (2) gives
To solve the optimization problem from the standpoint of period t0, we form the Lagrangian
and take derivatives with respect to xt, yt, and λt, giving9
Because W and Q are symmetric positive semidefinite and the policy constraints are convex the Lagrangian is also convex and the first-order conditions are sufficient for locating the minimum.
where
. The fact that the policy maker behaves differently in the first period than it does in subsequent periods in reflected in the fact that equations (11) and (12) apply in the initial period (t=t0), whereas equations (8) and (9) apply in all subsequent periods. However, the recursive nature of the system can be restored by employing equations (8)–(10), for all t≥t0, but with the initial conditions
and
.10
More precisely, the constraints on
are
. Because
is not necessarily zero, the restriction in the text,
, is sufficient but stronger than necessary.
Given
and
, equations (8)–(10), can be solved in a number of ways. One possibility is to write them in second-order form
and to apply an undetermined-coefficient technique, such as these developed by Binder and Pesaran (1995), McCallum (1999), or Uhlig (1999). An alternative approach is to write equations (8)–(10) in first-order form
where
and to solve the system using an eigenvalue method, such as Anderson and Moore (1985), Klein (2000), King and Watson (2002), Christiano (2002), or Sims (2002). Notice that the vectors in equations (15) have been allocated such that the predetermined variables,
and
, enter at the top of the system, the “stable” block, whereas the jump variables, λt, yt, and xt, enter at the bottom, the “unstable” block. The order of the variables within these vectors is irrelevant because by construction every element in
and
has a known value at the beginning of period t. Although the order of the variables within the system is an issue for many solution methods, here one simply orders vectors, which makes these Euler equations particularly easy to solve.
Both undetermined-coefficient methods and eigenvalue methods return solutions in the form
The optimal commitment rule can be seen in equation (16) to depend on the vector of lagged Lagrange multipliers. These Lagrange multipliers enter the rule to ensure that today's policy validates how private sector expectations were formed in the past.
Remark 1: The Euler equations for optimal commitment policy, equations (8)–(10), do not depend on the covariance matrix of the shocks, Ω. Consequently, optimal commitment policies are certainty equivalent.
Remark 2: Let
,
and, in obvious notation, equation (16) be written as
. Then, the loss function can be written as
, where
(see Appendix A.2).
Remark 3: A consequence of Remark 2 is that
=tr(KΦ) where Φ is the unconditional variance-covariance matrix of
and
(see Appendix A.3).
DISCRETION
In this section, we consider discretion and develop a numerical procedure that solves for Markov-perfect Stackelberg-Nash equilibria in which the policy maker optimizing today is the Stackelberg leader and private sector agents and future policymakers are Stackelberg followers. Broadly, the innovations in this section are that we allow expected future instruments to enter the optimization constraints and that we base the algorithm on a framework in which the optimization constraints are written in structural form, not in state-space form. The outcome is a solution procedure that is easy to apply and that eliminates the need to represent the optimization constraints in state-space form. The solution method also returns the Euler equations associated with the optimal discretionary policy, not just the decisions rules, which makes it particularly well suited for studying “targeting rules” (Svensson, 2003).
Because the system's state variables are
and vt, in the equilibrium that we seek the endogenous variables and the policy instruments will all be functions of these variables. Assume, then, that a stationary solution to the policy maker's optimization problem exists and is given by
It is the time-invariant matrices H1, H2, F1, and F2 that we seek.
Substituting equations (17) and (18) into equation (2) gives
where
The matrix D embeds how future policy makers respond to movements in yt and the policy maker setting xt today allows for D, rather than just A0, when calculating the impact of its policy decisions. Thus, future policy makers are followers with respect to current policy makers.
Using equations (17) and (18), we can express the loss function in terms of H1, H2, F1, F2, yt and the instrument vector, xt as (see Appendix B.1)
where
Although a Lagrangian could be used, it is just as simple to substitute equation (19) into equation (21) and transform what was a constrained optimization problem into an unconstrained optimization problem. Following this substitution we have11
This transformation requires that D have full rank, which is invariably satisfied because A0 has full rank.
Differentiating equation (23) with respect to xt and setting the resulting derivative equal to zero gives
Equation (24) can alternatively be expressed in terms of endogenous variables as
which is useful when a “targeting rule” rather than an “instrument rule” (Svensson, 2003) is of interest. Solving equation (24) for xt produces
which when inserted back into equation (19) leads to
Of course, both P and D are implicit functions of H1, H2, F1, and F2. Thus, equations (26) and (27) must be solved for a fixed point to obtain the desired solution. The numerical procedure is as follows
Step 1. Initialize H1, H2, F1, and F2.
Step 2. Solve for D according to equation (20) and for P from equation (22), iterating “backward through time” using a method such as the doubling algorithm.12
A description of the doubling algorithm along with other methods for solving matrix Sylvester equations can be found in Anderson, Hansen, McGrattan, and Sargent (1996).
Step 3. Update H1, H2, F1, and F2 according to
Step 4. Iterate over Steps 2 and 3 until convergence.
As should be clear from these four steps, the procedure is easy to implement, requiring no more than standard matrix operations. It is worth noting that the procedure does not require the predetermined variables to be separated from the non-predetermined variables, thereby avoiding the matrix partitioning required by state-space methods.
Remark 4: The feedback parameter matrices, F1 and F2, are independent of Ω and thus optimal discretionary policies are certainty equivalent. Moreover, P and D, and hence H1, H2, F1, and F2, are independent of the economy's initial state.
Remark 5: Let
and
For given F1, F2, and H2 matrices, and an H1 matrix whose spectral radius is less than one, the transition equation for zt is
, where H and G are formed from F1, F2, H2, and H1 in a straightforward way. Then equation (1) can be written as
, where
(see Appendix B.2).
Remark 6: As a consequence of Remark 5,
, where Φ is the unconditional covariance matrix of zt (see Appendix B.3).
COMPARING SOLUTION METHODS13 All simulations were conducted in Gauss 6.0.17 using a Pentium IV 2.6Ghz processor with 1.5GB RAM.
All simulations were conducted in Gauss 6.0.17 using a Pentium IV 2.6Ghz processor with 1.5GB RAM.
The previous two sections showed how to solve for optimal commitment policies and optimal discretionary policies when the optimization problem is formulated with the constraints in structural form. This contrasts with the approach taken in Oudiz and Sachs (1985), Currie and Levine (1985, 1993), Backus and Driffill (1986), and Söderlind (1999), which is to minimize
subject to
where
combines the predetermined variables, p1t, and the nonpredetermined variables,
. The key difference between this formulation and that employed in this paper lies largely in how the constraints are expressed, structural form, equation (2), or state-space form, equation (33).
In this section we apply the algorithms developed in Sections 3 and 4 to two optimization-based New Keynesian models and compare their computation times to state-space solution methods. Because optimal commitment policies and optimal discretionary policies are often studied in the monetary policy literature, the models that we analyze are drawn from that literature. We study the small open-economy model developed by Galí and Monacelli (2005) and closed-economy model developed by Erceg, Henderson, and Levin (2000) (EHL), both are representative of the models used to examine monetary policy issues. For discretion, we compare the algorithm developed in Section 4 to those developed in Oudiz and Sachs (1985) and Backus and Driffill (1986). For commitment, we compare the algorithm presented in Section 3 to those developed by Backus and Driffill (1986) and Söderlind (1999). Subsequently, we apply our procedures to the larger Fuhrer and Moore (1995) model and discuss situations in which convergence problems can sometimes be encountered when solving for optimal policies.
Solving for optimal commitment policies using the approach of Section 3 requires solving a rational expectations (RE) model and there are numerous techniques available to perform this task. We take the opportunity to compare two RE solution methods, the “brute force” undetermined-coefficient method presented in Binder and Pesaran (1995) and the QZ-decomposition method presented in Klein (2000). For the latter, we used both the real QZ decomposition and the complex QZ decomposition14
Routines to perform the real QZ decomposition and the complex QZ decomposition in Gauss are available from Paul Söderlind's Web site. These excellent routines were employed in this study.
Söderlind's (1999) method of solving for commitment equilibria also utilizes the real QZ decomposition.
A Small-Scale Open-Economy Model
In the open-economy model developed by Galí and Monacelli (2005), households consume an aggregate of domestically produced and imported goods. The law-of-one-price applies to imported goods. Domestic firms are monopolistically competitive and subject to Calvo-style price rigidities (Calvo, 1983). Financial assets are internationally tradable and nominal uncovered interest parity holds. The law-of-one-price together with the definition for consumer price inflation implies that the real exchange rate and the terms-of-trade are perfectly correlated, whereas nominal uncovered interest parity coupled with the law-of-one-price implies that real uncovered interest parity holds.
For our purposes, the relevant model equations are16
The structural form and the state-space form used for the Galí -Monacelli model are contained in a technical appendix that is available on request.
where yt denotes the output gap, πt denotes domestic goods' inflation,
denotes consumer price inflation rate, qt denotes the real exchange rate (an increase in qt represents a real depreciation), it and
denote domestic and foreign nominal interest rates, respectively,
denotes foreign inflation, and gt, ut, and εt are white noise shocks.17
We follow Galí and Monacelli (2005) and set η=1, α=0.4, β=0.99,
σ=1,
ωα=1+α(2−α)(ση−1), and
. We condition on the foreign variables and normalize their values to zero.
The central bank sets the nominal interest rate, it, to minimize the loss function
where the policy preference parameters, λ and ν, are restricted to be nonnegative. In the experiments that follow, we consider a range of policy preference parameters, allowing λ to take on the values 0, 1, and 3 and ν to take on the values 0,
, and 1.
Solution times
We first assume that policy is set with discretion. Table 1 reports the time it takes to solve the Galí-Monacelli model using the algorithm in Section 4 and the algorithms in Oudiz and Sachs (1985), Backus and Driffill (1986), and Söderlind (1999).18
The model was solved 100,000 times using each algorithm from which average solution times were computed. The convergence criteria were set so that the algorithms all returned solutions that were identical to 13 significant figures.
The first point to note is that all of the algorithms are able to quickly obtain the optimal discretionary rule for each parameter combination considered. At the same time, the results indicate that the Oudiz-Sachs algorithm is the fastest for this model, but only marginally; the difference between the fastest algorithm (Oudiz-Sachs) and the slowest algorithm (Section 4) is just 0.00012 second on average. It takes slightly longer to solve the model when the constraints are in structural form because the structural form does not exploit as much of the model's structure as the state-space form does. No convergence problems were encountered with any of the algorithms.
Turning to the commitment solution methods, Table 2 shows that for the algorithms that use recursive methods—Binder-Pesaran and Backus-Driffill—the solution time is increasing in λ and ν. The algorithms that obtain the commitment policy using the real QZ decomposition are both quicker in general than the recursive methods and have solution times that are largely invariant to the loss function's parameterization. As with discretion, use of the structural form weighs on the computation time slightly. Table 2 also shows that for this model the Backus-Driffill algorithm cannot obtain the solution when ν=0, regardless of the value for λ.19
The source of this difficulty is a singularity in the transformation the algorithm uses to rotate the dynamic programming solution into state/co-state space.
This instability differs from the standard form of instability encountered in rational expectations models, which is that the system contains more unstable eigenvalues than there are nonpredetermined variables.
A Small-Scale Closed-Economy Model
The second model that we examine is that developed by Erceg, Henderson, and Levin (2000) (EHL). EHL assume that both prices and wages are subject to Calvo-contracts, but the model is otherwise a standard New Keynesian business cycle model with monopolistically competitive firms. When log-linearized about the model's Pareto optimal equilibrium, the relevant equations are:21
The structural form and the state-space form used for the EHL model are contained in a technical appendix that is available on request.
where gt denotes the output gap, it denotes the short-term nominal interest rate, πt denotes price inflation,
denotes the real wage, mplt denotes the marginal product of labor, wt denotes wage inflation, mrst denotes the marginal rate of substitution between consumption and leisure, and
,
, and
denote the Pareto optimal real interest rate, real wage, and level of output, respectively. Finally, xt, zt, and qt, denote a technology shock, a leisure preference shock, and a consumption preference shock.22
We parameterize the model according to Erceg, Henerson, and Levin (2000), setting σ=χ=1.5, α=0.3,
, ζw=ζp=0.75, ρ=0.95, β=0.99,
,
,
,
,
,
,
,
, Λ=α+χln+(1−α)σlc, λmpl=α/(1−α), λmrs=σlc+χln/(1−α), κp=(1−ζp)(1−βζp)/ζp,
,
,
,
,
,
,
.
For the purposes of this section, the central bank's loss function is assumed to be
with the nominal interest rate, it, serving as the central bank's policy instrument.
Solution times
Table 3 reports the time taken to solve the model under discretion for different parameterizations of the policy objective function.23
The model was solved 100,000 times using each algorithm from which average solution times were computed. The convergence criteria were set so that the algorithms all returned solutions that were identical to 12 significant figures.
As with the Galí-Monacelli model, none of the algorithms had any difficulty solving the model for any of the parameterizations considered. Aside from two notable outliers, the Oudiz-Sachs algorithm is again the most efficient while the algorithm developed in Section 4 is typically the slowest, taking on average 1.5 hundreds of a second longer the obtain the solution. Although all of the algorithms are notably slower when solving the EHL model than when solving the Galí-Monacelli model, the relative inefficiency of the structural form method is more apparent here because the EHL model contains a larger number of non-predetermined variables, whose lagged values are all treated as candidate state variables by the algorithm.
Table 4 reports computation times for the EHL model when solving for optimal commitment policies; several interesting results emerge. First, the computation time is longer when a recursive method is used, such as the Backus-Driffill algorithm or Binder and Pesaran's RE solution method. Second, optimal commitment policies can generally be obtained more quickly than optimal discretionary policies, even if a recursive method is used to solve for the optimal commitment policy. Third, as earlier, the Backus-Driffill algorithm is unable to solve for the optimal commitment policy when ν=0, but now none of the other algorithms have problems.
A Larger Model
The final model that we consider is the well-known model developed by Fuhrer and Moore (1995). The particular specification that we analyze comes from Fuhrer (1997, Table IV). The model equations are given by
where yt represents the output gap, pt represent the price level, wt represents the nominal wage, πt represents inflation,
represents the real wage, ρt represents the real yield on a 10-year bond, and it, the short-term nominal interest rate, serves as the central bank's policy instrument.
Unlike the optimization-based models analyzed earlier, which each contained only two endogenous state variables, the Fuhrer-Moore model contains seven endogenous state variables, and it also contains a relatively large number of non-predetermined variables. Despite its larger size, the Fuhrer-Moore model can easily be written in second-order structural form and solved using the algorithms developed in Sections 3 and 4. As earlier, the central bank's loss function takes the form
Table 5 reports the time taken (in hundreds of seconds) to solve the Fuhrer-Moore model using the algorithms presented in Sections 3 and 4 for different values of λ and ν. The discount factor, β, is set to 0.99.24
The model was solved 10,000 times using each algorithm from which average solution times were computed.
Table 5 underscores the point that it is much more efficient to solve for optimal commitment policies the real QZ decomposition than it is to use the Binder-Pesaran solution method, which employs recursive iterations, and that optimal commitment policies can be obtained more quickly than optimal discretionary policies. Nevertheless, averaging across the parameterizations for which the solution could be obtained, it takes less than one-tenth of a second to solve for the optimal discretionary policy. Perhaps the most notable feature of Table 5, however, is that when λ=ν=0 the optimal commitment policy is obtained when using the real QZ decomposition but not (to sufficient accuracy) when using Binder-Pesaran. Moreover, for this parameterization of the loss function, the optimal discretionary policy could not be obtained.
Convergence Issues
For the three models analyzed here, one or more of the solution algorithms could not obtain the optimal (commitment/discretionary) policy for some parameterizations of the loss function. Looking closely at the offending parameterizations, it is notable that they are all parameterizations for which no penalty is placed on the policy instruments, that is cases where the central bank is not concerned about interest rate smoothing/stabilization. Similar instances are documented in Svensson (2000) and Dennis and Söderström (Forthcoming), and, together, these occurrence highlight the fact that the properties of the solution algorithms, particularly those for obtaining discretionary policies, have not been explored. In fact, when discussing their algorithm for obtaining optimal discretionary policies, Oudiz and Sachs (1985, p. 311) comment “We do not know of any general result concerning the convergence of this process. However in our empirical applications we have not run into major problems.” Similarly, Söderlind (1999, p. 819), notes that “The general properties of this algorithm are unknown. Practical experience suggests that it is often harder to find the discretionary equilibrium than the commitment equilibrium. It is unclear if this is due to the algorithm.”
The approach taken in this paper provides insights as to why, and under what circumstances, convergence problems may be encountered when solving for optimal discretionary policies. For this purpose, the key equations in the algorithm are
where equations (35) and (36) are taken directly from Sections 4 and (37) is obtained by substituting equation (28) into equation (30). By inspection, difficulties obtaining the optimal discretionary policy can arise when:
- The discount factor is close to one and the H1 matrix associated with the optimal discretionary policy has a spectral radius close to one. In this situation, it take a long time for recursive methods to obtain the fixed point P. If this problem occurs, it may be useful to solve for P using a nonrecursive method, such as the Hessenberg-Schur algorithm described in Anderson et al. (1996).
- The iterations produce a D matrix that is singular. Because A2 and A4 are typically sparse and A0 has full rank, a singularity in D is unlikely to occur.
- The iterations produce a matrix that is singular. This will occur if both Q and PD-1A3 equal null matrices, that is, if the loss function does not penalize movements in the policy instrument(s) and the target variables are contemporaneously unaffected by movements in the policy instruments. When both PD-1A3 and Q equal null matrices the optimal discretionary policy is not uniquely determined by setting the derivative of the loss function with respect to the policy instrument(s) to zero, see equation (24).
Among these three situations, the first will likely increase the computation time, the second is very unlikely to occur, and the third accounts for the problems encountered above, in Svensson (2000), and in Dennis and Sö derström (Forthcoming), which all involve a loss function that does not penalize the policy instrument. In fact, the assumption that Q is positive definite—rather than just positive semi-definite—which would preclude this problem, is standard in linear-quadratic control theory (Rustem and Zarrop, 1979; Anderson et al. 1996, p. 175). We did not restrict Q to be positive definite in our analysis because policies where the central bank does not smooth or stabilize interest rates are of broad interest and can be obtained for many models.
CONCLUSION
This paper has presented new algorithms to solve for optimal commitment rules and optimal discretionary rules in rational expectation models. These algorithms differ from those developed by Oudiz and Sachs (1985), Backus and Driffill (1986), and Söderlind (1999) in that they do not require the optimization constraints to be written in state-space form. This paper has also extended the class of models that can be analyzed, allowing expectations of future policy instruments to enter the optimization constraints, which can be useful when solving models that contain an interest rate term structure, and showed how to derive the Euler equation for the optimal discretionary policy, which is needed to study targeting rules.
After setting up the optimization problem in Section 2, the approaches taken to solve for optimal commitment policies and optimal discretionary policies were described in Sections 3 and 4, respectively. Section 5 took two optimization-based New Keynesian models and used a range of solution algorithms to solve them for optimal commitment policies and optimal discretionary policies for a variety of policy objective functions. For these models it was feasible to analytically derive state-space representations. Although the relative advantage of the solution methods developed in this paper is that they save the user from having to manipulate a model into state-space form, computational experiments on these small-scale models showed that the state-space solution methods are generally more efficient, precisely because they exploit more of the model's structure. Nevertheless, the methods developed here can be employed to solve larger models and for small models they are efficient enough to be used in estimation exercises, which may involve a model being solved many times within a hill-climbing routine [see Dennis (2004b) for an application]. Section 5 also solved the Fuhrer-Moore model, a model that can be much more quickly expressed in structural form than in a state-space form.
The paper also has discussed some of the computational problems that can be encountered when solving for optimal policies and showed why, in practice, placing a small penalty on movements in the policy instrument(s), invariably overcomes these problems. These experiments also showed that it is generally more efficient to solve for optimal commitment policies by solving the underlying rational expectations model using the real QZ-decomposition rather than either the complex QZ-decomposition or the Binder-Pesaran algorithm.
APPENDIX A
A.1. MATHEMATICAL PRELIMINARY
Assume that 0<β<1 and that the spectral radius of the matrix
is less than one, then the infinite series
is convergent. Consequently,
, and S can be found as the fixed point of
.
A.2. ESTABLISHING REMARK 2
Let
and
From equation (27) the recursive equilibrium law of motion for zt can be expressed as
. Given this law of motion, we have
which using the result regarding convergent infinite series in Appendix A.1 can be simplified to
where
. Finally, again using the law of motion for zt we obtain
as required.
A.3. ESTABLISHING REMARK 3
From A.2 the loss function under commitment can be written as
where
, and the economy's evolution is given by
From Appendix A.2, the unconditional variance-covariance matrix for zt is the fixed point of
from which the unconditional variance-covariance matrix for
, Φ, can be obtained. Multiplying (A.1) through by (1−β) gives
Now employing (A.3) we have
Provided the spectral radius of H is less than one,
will remain bounded even in the limit as β↑1. Therefore,
. However, defining
so that
only the elements in Σ associated with yt and xt (i.e., Φ) are relevant. Consequently, we have
, as required.
APPENDIX B
B.1. DERIVING EQUATION (21)
We show that
where
.
To establish this equality we exploit the properties of convergent geometric series, and the requirement that, in equilibrium, yt+j=H1yt+j−1+H2vt+j, and xt+j=F1yt+j−1+F2vt+j, ∀ j>0. First write
The first term on the RHS of (B.1) is in the appropriate form.We will treat the second and third terms on the RHS separately, beginning with the second term. We can write
Assuming that the spectral radius of H1 is less than one, the result in Appendix A.1 leads to
where
.
Turning to the third term on the RHS of (B.1)
Again, provided the spectral radius of H1 is less than one, the result in Appendix A.1 gives
. Therefore,
Substituting (B.2) and (B.3) back into (B.1) gives
Finally, let
giving
or
as required.
B.2. ESTABLISHING REMARK 5
Let
, then the policy objective function, (B.1), can be written as
where
and
Employing (B.5) in (B.4) gives
Now, exploiting the properties of convergent geometric series from Appendix A.1, we have
or
where
, as necessary.
B.3. ESTABLISHING REMARK 6
In this appendix we establish that
. Recall that the policy objective function can be written as
where
The components of the unconditional variance-covariance matrices for yt and xt are given by
Scaling the policy objective function by (1−β) gives
Exploiting the properties of the trace operator results in
Employing (B.6) and (B.7) gives
Because the spectral radius of H1 is less than one, P remains bounded even as β↑1. Thus,
and
, which gives us the result that
, where
I would like to thank the associate editor and two anonymous referees for comments. I would also like to thank Kirk Moore for research assistance. The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of San Francisco or the Federal Reserve System.
