Using a Projection Method to Analyze Inflation Bias in a Micro-Founded Model

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Using a Projection Method to Analyze Inflation Bias in a Micro-Founded Model Gary S. Anderson, Jinill Kim, and Tack Yun 2010-18 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Using a Projection Method to Analyze Inflation Bias in a Micro-Founded Model Gary S. Anderson, Jinill Kim, and Tack Yun Federal Reserve Board Abstract Since Kydland and Prescott (1977) and Barro and Gordon (1983), most studies of the problemoftheinflationbiasassociatedwithdiscretionarymonetarypolicyhaveassumed a quadratic loss function. We depart from the conventional linear-quadratic approachto the problem in favor of a projection method approach. We investigate the size of the inflationbiasthatarisesinamicrofoundednonlinearenvironmentwithCalvopricesetting. The inflation bias is found to lie between 1% and 6% for a reasonable range of parameter values, when the bias is defined as the steady-state deviation of the discretionary inflation rate from the optimal inflation rate under commitment. Keywords: Inflation Bias, Discretionary Monetary Policy, Projection Methods, JEL: E31, E52, C61, C63 1. Introduction Since Kydland and Prescott (1977) initiated the literature of rules versus discretion, improvement upon discretionary equilibria by reducing inflation bias has long been a researchthemeinpolicycirclesaswellasacademia,includingBarroandGordon(1983), Clarida, Gali and Gertler (1999), M. King (1997) and Woodford (2003). In most of the existing papers on the inflation bias, the one-period loss function assigned to the central bank is quadratic in inflation and the level of output relative to its target. It is well known that Rotemberg and Woodford (1997) and Benigno and Woodford (2006) have providedamicrofoundationfortheuseofsuchalossfunctionbyshowingthatthissimple quadratic function can be derived as the second-order approximation to the non-linear social welfare function in a Calvo model. However, as discussed in Woodford (2003), such a derivation does not hold under discretion unless the steady-state level of output under flexible prices is sufficiently close toitsefficientlevel;thesepapersapproximatethemodelaroundthedeterministicsteady 1The authors appreciate comments by Andrew Levin, Edward Nelson, David Lopez-Salido, Victor R´ios-Rull, Alex Wolman, and participants at 2008 SCE Conference, as well as encouragements from MikeWoodford. Theviewsinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbe interpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemoranyother personassociatedwiththeFederalReserveSystem. Preprint submitted to Elsevier Wednesday 17th February, 2010

statewithzeroinflation,buttheoptimalallocationunderdiscretionleadstoanunknown positive inflation under monopolistic distortion. In light of this observation, this paper does not follow the conventional linear-quadratic approach to studying the inflation bias induced by discretion. Instead, we use a projection method to analyze the inflation bias in a microfounded non-linear model with a Calvo price-setting environment. In our model, since the optimal inflation rate under commitment is zero, the inflation bias is defined as the (optimal) discretionary inflation rate. To do so, we characterize a set of conditions for the optimal allocation under discretion without any approximations. We then use Chebyshev polynomials to approximate policy functions that link inflation and output to a set of state variables, thereby converting optimization conditions into a set of non-linear equations for the coefficients of Chebyshev polynomials. The results on inflation bias based on the global projection method are compared with those based on the linear-quadratic approximation method. Wewouldliketonotethatperturbationmethodscanbemodifiedandusedtoanalyze thisproblem. Forexample,DotseyandHornstein(2003)andKlein,KrusellandR´ios-Rull (2008) have employed a perturbation method, with an iterative procedure to compute numerical solutions: Dotsey and Hornstein (2003) solve an optimal discretion problem with an iteration of the linear-quadratic approximation, while Klein, Krusell and R´ios- Rull (2008) apply a perturbation procedure to a nonlinear Generalized Euler Equation. These methods can be used to compute the optimal inflation rate at a deterministic steady state. But we have chosen to use the projection method since this method can conveniently be extended to a stochastic setting with technology shocks. Our paper is not the only one to analyze the discretionary equilibrium in a nonlinear Calvomodel. WolmanandVanZandweghe(2008)useafixed-pointalgorithmtosolvefor theoptimalpolicyinstrumentandinvestigatewhethermultipleMarkovPerfectEquilibria can arise in the Calvo model—as compared to the results of King and Wolman (2004) for the Taylor pricing contract. In addition, Adam and Billi (2007) work on optimal discretion in a model that is linear in every aspect except for the zero lower bound for the nominal interest rate. Therestofthispaperisorganizedasfollows. Insection2,wedescribeadiscretionary equilibrium in the Calvo (1983) pricing model where the planner is not allowed to make any commitment about his or her future behavior. Section 3 contains numerical results based on the projection method. In section 4, we conclude. 2. Economic Structure and Discretionary Equilibrium This section describes the economic structure in our model and the discretionary equilibrium of the planner’s problem. 2.1. Economic Structure The economy is populated by households and firms. 2.1.1. Households At period 0, the preference ordering of the representative household is summarized by X ∞ " C1−σ−1 υH1+χ # E βt t − t , (1) 0 1−σ 1+χ t=0 2

where C denotes consumption, H denotes hours worked. The parameter β denotes the t t time-discountfactor,σ measuresthedegreeofrelativeriskaversion,χcontrolsthelabor supplyelasticity,andυplaystheroleoffixingthesteady-statelevelforlabor. Households purchasedifferentiatedgoodsinretailmarketsandcombinethemintoasinglecomposite goodusingtheDixit-Stiglitzaggregator,andutilsofhouseholdsdependupontheamount of the composite good. The demand curve for each good z can be derived from the following cost-minimization problem: Z 1 (cid:18)Z 1 (cid:15)−1 (cid:19) (cid:15)− (cid:15) 1 min P t (z)C t (z)dz s.t. C t = C t (z) (cid:15) dz , (cid:15)>1, (2) 0 0 where P (z) represents the nominal price of good z and C (z) is its demand. The firstt t order condition for this cost-minimization problem yields the demand curve for firm z: (cid:18) P (z) (cid:19)−(cid:15) C (z)= t C . (3) t P t t The parameter (cid:15) represents the elasticity of demand, and the aggregate price level P is t (cid:18)Z 1 (cid:19) 1− 1 (cid:15) P = P1−(cid:15)(z)dz . (4) t t 0 The household’s dynamic budget constraint at period t is given by (cid:20) (cid:21) B B W C +E Q t+1 = t +(1−τ ) tH +Ξ −T , (5) t t t,t+1P P W P t t t t+1 t t where B is the nominal payoff at period t+1 of the bond-portfolio held at period t, t+1 W isnominalwage,andΞ istherealdividendincome,T isthereallump-sumtax,and t t t τ denotes a constant employment tax rate (or subsidy when negative) that is applied W to labor income. In addition, Q is the stochastic discount factor used for computing t,t+1 the real value at period t of one unit of the consumption good at period t+1. Hence, if R represents the risk-free nominal (gross) rate of interest at period t, the absence of t arbitrage in equilibrium leads to (cid:20) (cid:21) 1 Q =E t,t+1 . (6) R P t P t t t+1 The representative household maximizes (1) subject to the flow budget constraints (5) in each period. The first-order conditions are given by W υCσHχ =(1−τ ) t, (7) t t W P t (cid:18) C (cid:19)σ Q =β t , (8) t,t+1 C t+1 and substitution of (8) into (6) yields the consumption Euler equation: (cid:20)(cid:18) C (cid:19)σ P (cid:21) βR E t t =1. (9) t t C P t+1 t+1 3

2.1.2. Firms Each firm produces a differentiated good z using a constant returns to scale production function: Y (z)=A H (z), (10) t t t where Y (z) is the output of firm z, and H (z) denotes the hours hired by the firm and t t A is an exogenous aggregate productivity shock at period t. Firms set prices as in the t sticky price model of Calvo (1983). Specifically, each period a fraction of firms (1−α) are allowed to change prices, whereas the other fraction, α, keeps prices the same. Let P∗ be the new price charged by a firm resetting its price. Then, resetting firms choose a t newoptimalpriceinordertomaximizethefollowingexpecteddiscountedsumofprofits: X ∞ αkE " Q (cid:18) (1−τ ) P t ∗ − W t+k (cid:19)(cid:18) P t ∗ (cid:19)−(cid:15) Y # , (11) t t,t+k P P A P P t+k t+k t+k t+k t+k k=0 where τ denotes the amount of proportional revenue tax (or subsidy when negative). P Differentiating this expression with respect to P∗ gives rise to the first-order condition: t X ∞ αkE (cid:20) Q (cid:18) (1−τ ) P t ∗ − (cid:15) W t+k (cid:19) P(cid:15) Y (cid:21) =0. (12) t t,t+k P P (cid:15)−1A P t+k t+k t+k t+k t+k k=0 Furthermore, the Calvo type staggering transforms equation (4) into h i 1 P = (1−α)(P∗)1−(cid:15)+αP1−(cid:15) 1−(cid:15) . (13) t t t−1 Next, we will show that the profit maximization condition (12) can be rewritten in a recursive way. In order to see this, note that substituting (8) into (12) and then rearranging, we have (1−τ ) X ∞ (αβ)kE "(cid:18) Y t+k (cid:19)(cid:18) P t+k (cid:19)(cid:15)−1 # P t ∗ (14) P t Cσ P P k=0 t+k t t = (cid:15) X ∞ (αβ)kE (cid:20)(cid:18) W t+k Y t+k (cid:19)(cid:18) P t+k (cid:19)(cid:15)(cid:21) . (cid:15)−1 t A P Cσ P k=0 t+k t+k t+k t It is now useful to define two variables, F and S , as follows. t t F = (1−τ ) X ∞ (αβ)kE "(cid:18) Y t+k (cid:19)(cid:18) P t+k (cid:19)(cid:15)−1 # , (15) t P t Cσ P k=0 t+k t S = (cid:15) X ∞ (αβ)kE (cid:20)(cid:18) W t+k Y t+k (cid:19)(cid:18) P t+k (cid:19)(cid:15)(cid:21) . t (cid:15)−1 t A P Cσ P k=0 t+k t+k t+k t We then have the following recursive representations of the two variables F and S : t t F =(1−τ ) Y t +αβE (cid:2) Π(cid:15)−1F (cid:3) , (16) t P Cσ t t+1 t+1 t 4

(cid:18) (cid:19)(cid:18) (cid:19) S = (cid:15) W t Y t +αβE (cid:2) Π(cid:15) S (cid:3) , (17) t (cid:15)−1 P A Cσ t t+1 t+1 t t t with two terminal conditions, " T ! # " T ! # lim (αβ)T E Y Π(cid:15)−1 F =0, lim (αβ)T E Y Π(cid:15) S =0, t t+k t+T t t+k t+T T→∞ T→∞ k=1 k=1 where Π = P /P . We now substitute the definitions of F and S specified above in t t t−1 t t (15) into the profit maximization condition (14), to yield P∗ S t = t. (18) P F t t In addition, substituting equation (18) into (13) leads to (cid:18) S (cid:19)1−(cid:15) 1=(1−α) t +αΠ(cid:15)−1. (19) F t t We have thus expressed the profit maximization condition (14) and the price level definition (13) in terms of F and S with their intertemporal evolution equations (16) and t t (17). 2.1.3. Social Resource Constraint In any model with staggered price setting, relative prices can differ across firms. Furthermore, if firms have different relative prices, there are distortions that create a wedge between the aggregate output measured in terms of production factor inputs and the aggregate demand measured in terms of the composite goods. In order to see the relative price distortions, let us aggregate individual outputs: Z 1(cid:18) P (z) (cid:19)−(cid:15) A H =Y t dz, t t t P 0 t R1 where H = H (z)dz. By defining a measure of relative price distortion as t 0 t Z 1(cid:18) P (z) (cid:19)−(cid:15) ∆ = t dz, (20) t P 0 t the aggregate production function can be written as follows: A Y = tH . (21) t ∆ t t In order to obtain a law of motion for the measure of relative price distortion described above, note that the Calvo-type staggering allows one to rewrite the measure of relative price distortions specified in equation (20) as (cid:18) P∗(cid:19)−(cid:15) ∆ =(1−α) t +αΠ(cid:15)∆ . (22) t P t t−1 t 5

Then, substituting (13) into (22), one can derive an expression for how the measure of relative price distortions evolves over time: (cid:18) 1−αΠ(cid:15)−1(cid:19) (cid:15)− (cid:15) 1 ∆ =(1−α) t +αΠ(cid:15)∆ . (23) t 1−α t t−1 Finally, the aggregate market clearing condition is given by Y =C , (24) t t so the social resource constraint in period t is therefore given by A tH =C . (25) ∆ t t t 2.2. The Planner’s Problem under Discretion In this section, following Woodford (2003), we interpret a planner’s problem without commitmentasanoptimalplanningproblem. Inhisbook(p. 465),theoptimalallocation under discretion is defined as “a procedure under which at each time that an action is to be taken, the central bank evaluates the economy’s current state and hence its possible future paths from now on, and chooses the optimal current actions in the light ofthisanalysis,withnoadvancecommitmentaboutfutureactions,exceptthattheywill similarly be the ones that seem best in whatever state may be reached in the future.” Before proceeding, it is worth discussing implementability constraints, which restrict the feasible allocations of the social planner. First, the household budget constraint is not included as a constraint for the optimal allocation problem because of the lumpsum tax. Second, the size of the employment subsidy rate determines whether the profit maximizationconditionisbindingornotasanimplementabilityconstraintintheoptimal allocation problem. In order to gain some insights about the role of the employment subsidy, we describe the equilibrium conditions for the flexible price model and then compare them with those for the first-best equilibrium. Since α=0 corresponds to the flexible-price model, it follows from (12) that the profit maximization condition for the flexible-price model turns out to be W f,t =(1−τ ) (cid:0) 1−(cid:15)−1(cid:1) A . (26) P P t f,t where W and P are the nominal wage rate and the price level in the flexible price f,t f,t model. Combining (7) with (26), we can see that the relationship between MRS and MPL in the flexible price model is given by υCσ Hχ =(1−Φ)A , (27) f,t f,t t where C and H denote consumption and labor input in the flexible-price model f,t f,t and Φ measures the overall distortion in the steady-state output level as a result of taxes/subsidies and market power: Φ=1−(1−τ )(1−τ ) (cid:0) 1−(cid:15)−1(cid:1) . P W 6

We can see from (27) that when we set Φ = 0, the flexible price model can achieve the efficient level of output—which would be attained at the perfectly competitive equilibrium. We now characterize the planner’s problem under discretion, which is similar to the setup of Adam and Billi (2007) except for their imposition of the zero lower bound and our more disaggregated nonlinear constraints. The government at period 0 chooses a set of decision rules for { C , H , F , S , Π , ∆ }∞ in order to maximize t t t t t t t=0 ( ) C1−σ−1 υH1+χ V (∆ ,A )=max t − t +βE [V (∆ ,A )] , (28) t t−1 t 1−σ 1+χ t t+1 t t+1 subject to the following equilibrium conditions in each period t=0, ···, ∞: A C = tH , (29) t ∆ t t F =(1−τ ) A t H t +αβE (cid:2) Π (∆ ,A )(cid:15)−1F (∆ ,A ) (cid:3) , (30) t P ∆ Cσ t t+1 t t+1 t+1 t t+1 t t υH1+χ S = t +αβE [Π (∆ ,A )(cid:15)S (∆ ,A )], (31) t (1−τ )(1−(cid:15)−1)∆ t t+1 t t+1 t+1 t t+1 W t (cid:18) 1−αΠ(cid:15)−1(cid:19) (cid:15)− (cid:15) 1 ∆ =(1−α) t +αΠ(cid:15)∆ , (32) t 1−α t t−1 (cid:18) 1−αΠ(cid:15)−1(cid:19) 1− 1 (cid:15) S =F t . (33) t t 1−α Here, the absence of commitment leads us to express the values of period t + 1 of the planner’s choice variables in terms of the values at period t + 1 of state variables such as F (∆ ,A ), S (∆ ,A ), and Π (∆ ,A ); that is, fut+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 ture variables are taken as given by the planner, instead of chosen optimally as under commitment. In addition, we allow for the possibility that the value function on the right-hand side differs from that on the left-hand side while the system is away from a stationary equilibrium.2 The same principle is applied to the notation of functions F (∆ ,A ), S (∆ ,A ), Π (∆ ,A ) so that we do not record t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 F(∆ ,A ), S(∆ ,A ), and Π(∆ ,A ). In our numerical implementation of t+1 t+1 t+1 t+1 t+1 t+1 theprojectionmethod,however,weemploytheassumptionthatfunctionalformsofthese functions are invariant over time. 3. Projection Methods and Numerical Results This section starts with a description of a projection method to obtain numerical solutions for the discretionary equilibrium. We will also present our numerical results regarding the size of optimal inflation under discretion that are compared with those from a linear-quadratic approximation analysis (e.g. Woodford, 2003). 2Thecharacterizationofoptimalpolicyconditionsandourapplicationoftheprojectionmethodare describedintheappendix. 7

3.1. Projection Method with Homotopy Procedure: A Nontechnical Guide We employ a projection method to compute numerical solutions that approximate the nonlinear dynamic system of implementablity conditions of the planner’s problem and its first-order conditions. In order to deal with a feature of the generalized Euler equation that future variables should be expressed as functions of current variables, we approximate policy functions by a set of Chebyshev polynomials because functional forms of derivatives of Chebyshev polynomials are analytically known. We also adopt a homotopyproceduretoimproveonourinitialguessesforthenonlinearsolution. Ouruse ofahomotopyprocedureismotivatedbyourfindingthatinthecourseofobtainingvalid solutions over the relevant range of state variables, it was important to have flexibility in setting and readjusting the range of these variables.3 In our computation, we begin by characterizing the full set of dynamic equilibrium conditionsinanon-linearstate-spacerepresentation. Inordertodothis,wedefineanew function Γ in order to collect the policy functions of endogenous variables as follows: Γ(s )=Γ(∆(s ),Π(s ),C(s ),F(s ),H(s ),S(s ),φ (s ),φ (s ),φ (s ),φ (s ),φ (s )) t t t t t t t 1 t 2 t 3 t 4 t 5 t where the realized values at period t of these functions are determined by the following state at period t: s =[∆ A ]0. Here, functions (φ (s ), φ (s ), φ (s ), φ (s ) and t t−1 t 1 t 2 t 3 t 4 t φ (s )) represent Lagrange multipliers of 5 constraints of the planner’s problem (29)– 5 t (33). Given the specification of the function Γ, the equilibrium conditions lead to a systemofequationssatisfyingN(Γ())=0whereN(Γ())=0representsoursystemof11 equations: equations (29)–(33) and (36)–(41) for endogenous variables. We also assume that the logarithm of the aggregate productivity disturbance follows an AR(1) process: a =ρa +θ , t t−1 t where a = logA and the mean-zero Gaussian white noise, θ , is identically and indet t t pendently distributed over time. Turning to the solution method, we adopt a projection method to approximate the functions. Furthermore,sinceweallowforrandomtechnologyshocks,weexpresseachof the functions, Γ , as a linear combination of an outer product of orthogonal polynomials i in ∆ and A : t−1 t Γˆ (∆ ,A )= X k1 X k2 µ ϕ (∆ ,A ) i t−1 t j1j2 j1j2 t−1 t j1=0 j2 3Judd(1998)providesanexpositionabouthomotopycontinuationmethodsasapartofhisdiscussion onnumericalsolutionstononlinearequations. Theideabehindcontinuationmethodsistoexamineand solve a series of problems, beginning with a problem for which we know the solution and ending with theproblemofinterest. Thereasonwhyacontinuationmethodisofourinterestisthatwecanobtain an analytic solution for the discretionary equilibrium in the Calvo pricing model when fiscal policy eliminates the distortion associated with monopolistic competition in retail goods markets. We then takethiscaseasaproblemwhosesolutionisknown. Withthisknownsolutionasastartingpoint, we can solve a series of problems until we reach solution under our target parameter values. In addition, ourprocedureiscomparabletoalinearhomotopyamonghisexamples. Adescriptionofouruseofthe homotopymethodcanbefoundintheappendix. 8

where ϕ (∆ ,A ) express the value at period t of the product of j th order Chebyj1j2 t−1 t 1 shev polynomial for ∆ and j th order Chebyshev polynomial for A .4 We then use a t 2 t collocationmethodtodeterminethecoefficientµ . Inparticular,weemployNewton’s j1j2 methodtofindµ suchthat N(Γˆ(∆ ,A ) )=0 ateachpoint(∆ , A ) ofChebyj1j2 t−1 t l t−1 t l shev nodes {(∆ , A ) }M with the use of Gauss-Hermite integration for computing t−1 t l l=1 the expected values of future variables. 3.2. Some Implementation Issues Initially, we investigated adapting existing code for solving the problem. We have located freely available FORTRAN code from Judd (1992) and MATLAB code from Gapen and Cosimano (2005). We found that the code was very useful for benchmarking and validation but difficult to modify to solve our particular problem. For the problem at hand, we have found that—to obtain convergence for a given degree of approximation—it is important to start with a narrow range of values of ∆ t−1 in the definition of the Chebyshev polynomials, and then gradually extend the range. Thus, our code systematically adjusts the range of Chebyshev polynomials from narrow to wide, for a given set of parameters and a given degree of approximation. We have implemented the projection method software in Java. The object oriented nature of Java and the availability of the open-source Eclipse IDE (Geer, 2005) for Java greatly facilitated developing the software.5 Furthermore, because of the notoriously slow “for” loops in MATLAB, the Java code runs much faster than it would have if we had used generic MATLAB routines.6 The Java code can run on both Linux and Windowsmachines. WecurrentlyuseMathematicaasauser-interfacetotheJavaCode. Both JBENDGE and JMulTi are based on the JStatCom which provides a standardized application interface which we hope to adopt in the future.7 There are a number of improvements in the code that could be addressed in the future. Weenvisiondevelopingagenericopensourcetool,butcurrentlythecodedepends on Mathematica; we would like to develop a Dynare interface. The program uses a simple operator overloading while it would be preferable to use more efficient automatic differentiation techniques. 3.3. Numerical Results To determine the optimal inflation rate under discretion, we must assign numerical values to the parameters. Although we experimented with many different values, the 4Chebyshevpolynomialsareasequenceoforthogonalpolynomialsimportantinapproximationtheory. The Chebyshev nodes are the roots of the Chebyshev polynomials. Chebyshev polynomial approximationsthatinterpolateattheChebyshevnodesprovideanapproximationthatisclosetothepolynomial ofthebestapproximationtoacontinuousfunctionunderthemaximumnorm. 5Unliketraditionalapproachestodevelopingsoftware,anIntegratedDevelopmentEnvironment(IDE) brings all of the programmers tools into one convenient place. In the past, programmers had to edit files, save the files out, run the compiler, then the linker, build the application then run it through a debugger. Today’sIDEsbringeditor,compiler,linkeranddebuggerintooneplacetoincreaseprogrammerproductivity. MATLABalsoprovidesanobjectorientedcapability,butobjectcreationandmethod invocationaremuchslowerforobjectorientedMATLABthanforJava. 6MATLABcodeconsistingofvector-basedoperationscanbefasterthanJavacode,butitwouldbe difficult to construct a set of programs implementing a projection method that relies solely on these typesofoperations. 7For more information on JBENDGE and JMulTi, see Winschel (2008) and Lu¨tkepohl and Kr¨atzig (2004),respectively. 9

Figure1: Phasediagramforrelativepricedistortion 1.02 Inflation (= Π) t Distortion (= ∆) t 45o Line 1.015 1.01 1.005 1 1 1.005 1.01 1.015 1.02 Lagged Distortion (=∆ ) t−1 Note: This figure expresses the current level of relative price distortion as a function of its lagged level in ordertodemonstratehowthemeasureoftherelativepricedistortiondistortionconvergestoitssteady-state level. benchmark parameter values are taken from Yun (2005). For example, we assumed that utility is logarithmic in consumption (σ = 1) and quadratic in labor (χ = 1). We also set (cid:15) = 11, α = 0.75, and β = 0.99. We depart from Yun (2005) along one dimension: there is no subsidy nullifying the monopolistic distortion so the degree of monopolistic distortion is kept at Φ = (cid:15)−1 in the benchmark. Table 1 summarizes our benchmark parameter values. Using this benchmark specification, we solve the model via a projection method contemplating values for the relative price distortion in the range of (1,1.2). Figure 1 illustrates the solution of this discretionary equilibrium. The solid line represents the values of ∆ as a function of ∆ , shown for a narrow range of (1,1.02) to focus on t t−1 the area around the steady state. This line crosses the 45-degree line (dotted) at around 1.0026, which is the steady-state value for the dispersion measure. At this steady state, the value of Π¯ is about 1.0054 (dashed line). In terms of the annualized rate for net inflation, this steady-state inflation rate corresponds to 2.2%.8 Since the results in Figure 1 are based on a global solution method, it would be instructivetoprovidesomemeasureoferrorintheapproximation. Asaheuristicmeasure, 8Wedefinetheannualizedinflationrateinpercentas100× ` Π¯4−1 ´ . 10

Table1: CalibrationofParameters Parameter Value Definition σ 1 Relative risk-aversion coefficient χ 1 Inverse of labor supply elasticity β 0.99 Time-discount factor (cid:15) 11 Demand elasticity α 0.75 Probability of fixing prices in each period Φ 0.091 Degree of distortion ρ 0.95 AR(1) coefficient of the logarithm of labor productivity σ 0.01 Standard deviation of technology shock θ wecomparetwowaysofcomputingtherelativepricedistortion. Oneistheapproximate solutionforthedistortionasreportedinFigure1,andtheotheristheright-handsideof (32)withinflationsettothevaluesreportedinthisfigure. Wethencomputetherelative differencebetweenthesetwowaysofcomputingthesizeofrelativepricedistortion. Over the full collocation range of distortion, (1,1.2), the maximum percentage difference is on the order of 10−8. Itiswidelyknownthatthesizeoftheinflationbiasdependsonthedegreeofmonopolistic competition, since imperfect competition makes equilibrium flexible-price output lower than the socially efficient output. In our benchmark specification used in Figure 1, the elasticity of substitution ((cid:15)) determines how monopolistically competitive the economy is, and the size of distortion (Φ) is equal to its reciprocal. Figure 2 shows the size of inflation bias changes as we change the markup by varying (cid:15). Our benchmark of (cid:15) = 11 corresponds to the markup of 1.1. As we decrease (cid:15), the markup and inflation bias increase. When we choose (cid:15)=4, the markup is 1.33, and the inflation bias is about 6% annually. To bring out how other parameters affect the size of the inflation bias, we do some comparative statics with the model as shown in Table 2. First, according to the results based on the projection method using our benchmark parameter values, increasing α increases the inflation bias for values of α below 0.85. Increasing α decreases steady 11

Figure2: Impactofmonopolisticdistortionontheinflationbias 6 5 4 3 2 1 0 1.05 1.1 1.15 1.2 1.25 1.3 etaR noitalfnI annual % Monopolistic Distortion (= ε/(ε−1)) Note: The monopolistic distortion means the markup of firms at the deterministic steady-state with zero inflation rate. This figure depicts how changes in the size of the monopolistic distortion in retail goods marketaffectsontheinflationbias. state inflation for values of α above 0.85.9 Second, the smaller the curvature parameters (σ and χ), the bigger the inflation bias. When the utility function moves closer to being linear in consumption and labor, the size of the inflation bias increases significantly. 3.4. Comparison with the Linear-Quadratic Approximations We now compare our results with those from the conventional linear-quadratic approach (e.g. Woodford, 2003). The inflation bias expression that emerges from this approach is (cid:18) (cid:19) κλ Φ π¯ = , (34) (1−β)λ+κ2 σ+χ where κ = (1−α)(1−αβ)(σ+χ)/α and λ = κ/(cid:15). This formula produces an inflation bias of 1.6% per annum, under our benchmark calibration including the size of monopolistic competition.10 Compared to the size of the inflation bias based on the projection 9WethankAlexWolmanforpointingoutthisnon-monotonicity. 10Monopolisticcompetitionisanindispensablefeatureinoursteady-stateanalysisoftheinflationbias. Evenwithoutthisdeterministicinflationbias,thereisadifferencebetweendiscretionandcommitment inastochasticenvironmentthatcanariseintheabsenceofmonopolisticcompetition. Thisstabilization biashasbeenstudiedinarecentpaperbyS¨oderstr¨ometal. (2005). Insuchamodel,thedeterministic steadystateisclosetotheefficientequilibrium,solinear-quadraticapproximationswouldbevalid. 12

Figure3: Impactoftotaldistortionontheinflationbias 6 5 4 3 2 1 0 0 0.05 0.1 Size of Distortion (= Φ) etaR noitalfnI annual % Nonlinear Model LQ Approximation Note: The “total distortion” refers to the wedge between the marginal rate of substitution between consumption and leisure and the marginal product of labor at the steady-state with zero inflation rate. This figure depicts the dependence of the inflation bias on the size of the total steady-state distortion, Φ = 0 correspondingtotheabsenceofdistortion. method(2.2%),thelinear-quadraticapproachunderestimatestheinflationbiasbyabout a third. Figure3depictshowmuchthelinear-approximationunderestimatestheinflationbias as we alter the value of Φ by varying the value of τ or τ . Note that the range of Φ in W P thisfigurecoversourbenchmarkparametrizationofΦ=0.091. Thesolidlinerepresents thesizeofinflationbiasunderourprojectionmethod. Themagnitudeoftheinflationbias increasesfasterthanlinearlywithrespecttotheamountofmonopolisticdistortion. The dashedlinerepresentsthelevelofinflationbiasunderthelinear-quadraticapproximation and becomes tangent to the solid line as Φ gets close to 0. The difference between the two lines increases as monopolistic distortion moves the economy farther away from the efficient outcome. Finally, we compare the results of the sensitivity analysis that are obtained from the twoapproaches. Inparticular,equation(34)impliesthatsincethediscountfactorisclose to unity, we can approximate the inflation bias under the linear-quadratic approach by usingπ¯ ≈Φ/[(cid:15)(σ+χ)]. Accordingtothisformula, theinflationbiasisinverselyrelated to σ, χ, and (cid:15) (given the size of the total distortion), while the bias is approximately proportional to the size of monopolistic distortion (given the size of the markup). These predictions of the linear-quadratic approach on the sensitivity analysis are confirmed 13

by numerical results in the final column of Table 2. Furthermore, as noted earlier, the nonlinear projection method indicates that the smaller the curvature parameters (σ and χ), the bigger the inflation bias. The linear-quadratic and nonlinear projection approaches therefore appear to produce a similar relationship between parameter values andthesizeoftheinflationbias. Basedonthesenumericalresults,onemightarguethat thesimplicityandtransparencyofthelinear-quadraticapproachwouldbeveryhelpfulin relating the size of the inflation bias to the values for the parameters, though the linearquadraticapproachunderestimatesthesizeofinflationbias. Butitshouldbenotedthat such an interpretation could potentially be misleading. For example, when the discount factoris lessthanunity, the expressionforπ¯ indicates thatanincrease inα wouldimply a decrease in π¯. However, according to the results based on the projection method using our benchmark parameter values, the size of inflation bias is not a monotone function of α. 4. Conclusion Wehavedemonstratedhowaprojectionmethodcanbeusedtocomputetheinflation bias in a full nonlinear version of the Calvo model. The annual inflation bias is between 1% and 6% under plausible parameter values. In a recent paper, Schmitt-Grohe and Uribe (2009) report that the optimal inflation rate under commitment predicted by leading theories of monetary nonneutrality ranges from minus the real rate of interest to numbers insignificantly above zero. They also argue that the zero bound on nominal interest rates does not represent an impediment to setting inflation targets near or below zero. Meanwhile, our results indicate that the optimal inflation rate turns out to be substantially higher than zero in the absence of commitment. In particular, we expect that the larger the “degree” of commitment, the smaller the size of the inflation bias. It would thus be interesting to see how the change in the “degree” of discretion affects the size of the inflation bias. In this vein, the format of Debortoli and Nunes (2007) provides an interesting starting point because they have modelled an imperfect commitment setting in which there is a continuum of loose commitment possibilities ranging from full commitment to full discretion.11 In addition, we note that it would be possible to use the same projection method to analyze the effects of loose commitment on the inflation bias. 11Schaumburg and Tambalotti (2007) also discussed intermediate cases between discretion and commitmentusingalinear-quadraticmodel. 14

Table2: SensitivityAnalysis Parameter Values Numerical Results α σ χ (cid:15) Price Nonlinear LQ Distortion Solution Solution 0.5 1 1 11 1.001 1.004 1.004 α 0.75 1 1 11 1.003 1.005 1.004 0.95 1 1 11 1.093 1.003 1.003 0.75 0.16 1 11 1.048 1.016 1.007 σ 0.75 1 1 11 1.003 1.005 1.004 0.75 5 1 11 1.001 1.003 1.001 0.75 1 0.25 11 1.027 1.013 1.008 χ 0.75 1 1 11 1.003 1.005 1.004 0.75 1 4.75 11 1.001 1.002 1.001 (cid:15) 0.75 1 1 11 1.003 1.005 1.004 0.75 1 1 21 1.001 1.001 1.001 Note: Thelasttwocolumnsrepresentquarterly(gross)inflationforeachsetofparametervalues. Specifically, the nonlinear solution corresponds to Π¯ and the linear-quadratic solution represents (1+π¯). In addition, thepricedistortionmeasures∆¯,where∆¯ denotesthesteady-stateleveloftherelativepricedistortion. The sensitivityanalysisfor(cid:15)iscarriedoutbysettingthevalueofΦatitsbenchmark. 15

References Adam, K. and R. Billi (2007) “Discretionary Monetary Policy and the Zero Lower Bound on Nominal Interest Rates,”Journal of Monetary Economics, 54, 728-752. Barro, R.andD.Gordon(1983)“APositiveTheoryofMonetaryPolicyinaNatural Rate Model,”Journal of Political Economy, 91, 589–610. Benigno, P. and M. Woodford. (2006) “Optimal Taxation in an RBC Model: A Linear-QuadraticApproach,”JournalofEconomicDynamicsandControl,30,1445-1489. Calvo, G. (1983) “Staggered Prices in a Utility Maximizing Framework”Journal of Monetary Economics, 12, 383-398. Clarida, R., J. Gali, and M. Gertler (1999) “The Science of Monetary Policy: A New Keynesian Perspective,”Journal of Economic Literature, 37, 1661-1707. Debortoli, D. and R. Nunes. (2007) “Loose Commitment,”Federal Reserve Board, International Finance Discussion Paper 916. Dotsey M. and A. Hornstein. (2003) “Should a Monetary Policy Maker Look at Money?”Journal of Monetary Economics, 50, 547-579. Gapen, M. and T. Cosimano. (2005) “Solving Ramsey Problems with Nonlinear Projection Methods,”Studies in Nonlinear Dynamics & Econometrics 9. Geer, D. (2005) “Eclipse Becomes the Dominant Java IDE.”Computer, 38, 16-18. Judd,K.(1992)“ProjectionMethodsforSolvingAggregateGrowthModels,”Journal of Economic Theory, 58, 410-452 Judd, K. (1998) Numerical Methods in Economics, MIT Press, Cambridge, MA. Klein, P., P. Krushell, and J. V. R´ios-Rull. (2008) “Time-Consistent Public Policy,”Review of Economic Studies, 75(3), 789-808. King, M. (1997) “Changes in UK Monetary Policy: Rules and Discretion in Practice,”Journal of Monetary Economics, 39, 81-97. King, R. and A. Wolman (2004) “Monetary Discretion, Pricing Complementarity, and Dynamic Multiple Equilibria,”Quarterly Journal of Economics, 119, 1513-1553. Kydland,F.andE.Prescott(1977)“RulesratherthanDiscretion: TheInconsistency of Optimal Plans,”Journal of Political Economy, 85, 473-491. Schaumburg, E. and A. Tambalotti (2007) “An investigation of the gains from commitment in monetary policy,”Journal of Monetary Economics, 54, 302-324. Schmitt-Grohe,S.andM.Uribe(2009)“TheOptimalRateofInflation,”Forthcoming in Handbook of Monetary Economics. Wolman, A. and W. Van Zandweghe (2008) “Discretionary Monetary Policy in the Calvo Model,” Unpublished Manuscript. Woodford, M. (2003) Interest and Prices, Princeton University Press, Princeton, NJ. Yun,T.(2005)“OptimalMonetaryPolicywithRelativePriceDistortions,”American Economic Review, 95, 89-108. 16

Appendices This appendix provides additional detail about the model specification and our solution technique. Section Appendix A presents a full description of the Lagrangian of the government’s planning problem when the planner cannot make commitment about his or herfuturebehavior. SectionAppendixBprovidesadditionaldetailaboutourimplementation of the residual function for the projection method. Section Appendix C describes how we use homotopy methods to obtain solutions. Appendix A. Lagrangian In the presence of technology shocks, the Lagrangian of this problem can be written as C1−σ−1 υH1+χ £ = t − t +βE [V (∆ ,A )] 1−σ 1+χ t t t+1 (cid:20) (cid:21) A H +φ t t −C 1t ∆ t t (cid:20) (cid:21) A H −φ (1−τ ) t t +αβE [L(∆ ,A )]−F 2t P ∆ Cσ t t t+1 t t t " # υ(1−τ )H1+χ −φ P t +αβE [M(∆ ,A )]−S 3t (1−Φ)∆ t t t+1 t t " (cid:18) 1−αΠ(cid:15)−1(cid:19) (cid:15)− (cid:15) 1 # +φ (1−α) t +αΠ(cid:15)∆ −∆ 4t 1−α t t−1 t " (cid:18) 1−αΠ(cid:15)−1(cid:19) 1− 1 (cid:15) # −φ F t −S . 5t t 1−α t where auxiliary functions L(∆ ,A ) and M(∆ ,A ) are defined as t t+1 t t+1 L(∆ ,A ) = Π(cid:15)−1F , (A.1) t t+1 t+1 t+1 M(∆ ,A ) = Π(cid:15) S . (A.2) t t+1 t+1 t+1 An noted earlier, the absence of commitment leads us to express the values of period t+1 of the planner’s choice variables in terms of the values at period t+1 of state variables such as F (∆ ,A ), S (∆ ,A ), and Π (∆ ,A ). In order t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 tosimplifythecharacterizationofthefirst-orderconditionsoftheplanner’sproblem, we introduce two new functions L(∆ ,A ) and M(∆ ,A ) as composite functions of t t+1 t t+1 F (∆ ,A ), S (∆ ,A ), and Π (∆ ,A ) respectively. t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 Having described the optimal policy problem under discretion, the first-order conditions can be summarized as follows: A H 1+σ t tφ =φ Cσ, (A.3) ∆ C 2t 1t t t t υ(1+χ) υ∆ CσHχ+A φ + φ CσHχ =φ A Cσ, (A.4) t t t t 2t (1−Φ) 3t t t 1t t t 17

(cid:18) 1−αΠ(cid:15)−1(cid:19) 1− 1 (cid:15) φ =φ t , (A.5) 2t 5t 1−α φ =−φ , (A.6) 3t 5t (cid:18) 1−αΠ(cid:15)−1(cid:19) (cid:15)− 1 1 ! −1 (cid:18) 1−αΠ(cid:15)−1(cid:19) (cid:15) − − (cid:15) 1 (cid:15) t −Π ∆ φ = t F φ , (A.7) 1−α t t−1 4t 1−α 1−α t 5t A H υH1+χ t t φ +φ t −φ +αβE [Π (∆ ,A )(cid:15)φ ]= ∆2Cσ 2t 3t(1−Φ)∆2 4t t t+1 t t+1 4t+1 t t t A H φ t t +αβE [φ L (∆ ,A )+φ M (∆ ,A )], (A.8) 1t ∆2 t 2t 1 t t+1 3t 1 t t+1 t where φ , φ , φ , φ , and φ are Lagrange multipliers for (29), (30), (31), (32), and 1t 2t 3t 4t 5t (33) respectively, and τ is assumed to be zero for simplicity. P Appendix B. Projection Method with Collocation We will approximate 11 policy functions by using Chebyshev polynomials as follows: X k1 X k2 Γ (∆ ,a )= ω ϕ (∆ ,a ), i=C,H,∆,Π,S,F,φ ,φ ,φ ,φ ,φ i t−1 t j1j2 j1j2 t−1 t 1 2 3 4 5 j1=1j2=1 where the function ϕ is defined as 2(∆−∆ ) 2(a−a ) ϕ (∆ ,a )=T ( min −1)T ( min −1) j1j2 t−1 t j1−1 ∆ −∆ j2−1 a −a max min max min Here T (x) denotes jth order Chebyshev polynomials. In this appendix, we use the j logarithm of labor productivity as an argument of policy functions. Having determined functional forms of approximate policy functions, we will determine a nonlinear system of equations for weights of 11 approximate policy functions. Specifically we use 11 equilibrium conditions to define 11 residual functions as follows. Each equilibrium condition generates a residual function as can be seen below: exp(a )Γ (s ) R = t H t −Γ (s ) 1 Γ (s ) C t ∆ t R =Γ (s )1−σ+αβE [L(s )]−Γ (s ) 2 C t t t+1 F t νΓ (s )1+χΓ (s ) R = H t ∆ t +αβE [M(s )]−Γ (s ) 3 1−Φ t t+1 S t 1−αΓ (s )(cid:15)−1 R 4 =Γ ∆ (s t )−(1−α)( 1− Π α t )(cid:15)− (cid:15) 1 −αΓ Π (s t )(cid:15)∆ t−1 1−αΓ (s )(cid:15)−1 R 5 =Γ F (s t )( 1− Π α t )1− 1 (cid:15) −Γ S (s t ) 18

R =σΓ (s )−Γ (s )σΓ (s )+1 6 φ2 t C t φ1 t R =Γ (s )exp(a )+νΓ (s )σΓ (s )χ(Γ (s )+ζΓ (s ))−Γ (s )σΓ (s )exp(a ) 7 φ2 t t C t H t ∆ t φ3 t C t φ1 t t (cid:18) 1−αΓ (s )(cid:15)−1(cid:19) 1− 1 (cid:15) R =Γ (s )− Π t Γ (s ) 8 φ2 t 1−α φ5 t R =Γ (s )+Γ (s ) 9 φ3 t φ5 t (cid:18)(cid:16) (cid:17) 1 (cid:19) R = (cid:15) 1−αΓΠ(st)1−(cid:15) (cid:15)−1 −Γ (s )Γ (s ) Γ (s ) 10 1−α Π t ∆ t φ4 t (cid:16) (cid:17)−(cid:15) + 1 1−αΓΠ(st)(cid:15)−1 (cid:15)−1 Γ (s )Γ (s ) 1−α 1−α F t φ5 t R = ΓC(st)1−σ Γ (s )+Γ (s )ν(1+χ)ΓH(st)1+χ −Γ (s ) 11 Γ∆(st) φ2 t φ3 t (1−Φ)Γ∆(st) φ4 t +αβE [Γ (s )(cid:15)Γ (s )]−Γ (s )Γ (s )−Γ (s ) t Π t+1 φ4 t+1 φ1 t C t ∆ t −αβE [Γ (s )L (s )+Γ (s )M (s )] t φ2 t 1 t+1 φ3 t 1 t+1 where ζ = 1+χ, functions L(s ) and M(s ) are defined as L(s) = Γ (s )(cid:15)Γ (s ) and 1−Φ t t Π t F t M(s)=Γ (s )(cid:15)Γ (s ). Hence,thepartialderivativesofthesetwofunctionswithrespect Π t S t to ∆ can be written as t L (s )=((cid:15)−1)Γ (s )(cid:15)−2Γ (s )∂Γ (s )+Γ (s )(cid:15)−1∂Γ (s ) 1 t Π t F t Π t π t F t M (s )=(cid:15)Γ (s )(cid:15)−1Γ (s )∂Γ (s )+Γ (s )(cid:15)∂Γ (s ) 1 t Π t S t Π t Π t S t The derivatives of policy functions ∂Γ (s ), ∂Γ (s ) and ∂Γ (s ) can be derived as Π t F t S t follows: ∂Γ (s )= X k1 X k2 2ω j1j2 T0 ( 2(∆−∆ min ) −1)T ( 2(a−a min ) −1) i t ∆ −∆ j1−1 ∆ −∆ j2−1 a −a max min max min max min j1=1j2=1 for Π, F, and S and where T0 (x) denotes the derivative of the jth order Chebyshev j1−1 polynomials. The derivatives of the Chebyshev polynomials are easy to compute using the following relation: T0(x)=j U (x) j j−1 for j = 1, ···, ∞ and sequences of two polynomials {T (x)}∞ and {U (x)}∞ are j j=0 j j=0 recursively defined as U (x)=2xU (x)−U (x) U (x)=2x U (x)=1 j+1 j j−1 1 0 T (x)=xT (x)−T (x) T (x)=x T (x)=1. j+1 j j−1 1 0 We now move onto the characterization of the integrals appearing in the residual functions. Fortunately, the expectation operator only involves three terms. αβE [L(s )] t t+1 αβE [M(s )] t t+1 αβE [Γ (s )L (s )+Γ (s )M (s )] t φ2 t 1 t+1 φ3 t 1 t+1 19

It is assumed in the paper that the technology shock follows a normal distribution with mean zero and standard deviation σ . Hence, we standardize the shock by using θ = θ t σ z wherez ∼N(0,1). Theintegralofanexpressioninvolvingourapproximatedpolicy θ t t functions I(∆,a,W,z) is then approximated using the following finite sum Z ∞ exp(−z2) X kz √ I(∆,a,W,z) √ 2 dz = I(∆,a,W, 2z )m l l 2 −∞ l=1 wherem andz areGauss-HermitequadratureweightsandpointsandW isthesetthat l l includes all weights of 11 approximate policy functions. Given this approximation of the integrals, conditional expectations of functions L(s ) and M(s ) can be written as t+1 t+1 E [M(s )]= X kz Mˆ(∆,a,W, √ 2z )m ;E [L(s )]= X kz Lˆ(∆,a,W, √ 2z )m t t+1 l l t t+1 l l l=1 l=1 where functions Mˆ(s,ω,z) and Lˆ(s,ω,z) are defined as Mˆ(s,ω,z) =Γ (Γ (∆,a),ρ a+z)Γ (Γ (∆,a),aρ +z) Π ∆ a F ∆ a Lˆ(s,ω,z) =Γ (Γ (∆,a),aρ +z)Γ (Γ (∆,a),aρ +z) Π ∆ a S ∆ a We will consider collocation. Under orthogonal collocation, we choose k ×k zeros j1 j2 of ϕ (∆,a) and then substitute them into residual functions. Since all of 11 residual j1j2 functions should become zero for each point of (∆ ,a ), it means that R(∆ ,a ) j1 j2 j1 j2 = 0 holds where (∆ ,a ) represents a collocation point among k ×k zeros of 11×1 j1 j2 j1 j2 ϕ (∆,a) and R(∆ ,a ) represents a vector function that contain residual functions: j1j2 j1 j2 R(∆ ,a )=[R (∆ ,a ),···R (∆ ,a )]0. j1 j2 1 j1 j2 11 j1 j2 In addition, the set of zeros of Chebyshev polynomials can be written as follows. ∆ −∆ a −a ∆ =(z +1) max min +∆ ; a =(z +1) max min +a j1 j1 2 min j2 j2 2 min wherez andz aredefinedasz =cos((2j1−1)π)andz =cos((2j2−1)π). Asaresult, j1 j2 j1 2k1 j2 2k2 wehaveanonlinearsystemofequationsforweightsofapproximatepolicyfunctions. We then use Newton’s method in order to find a numerical solution to this nonlinear system of equation. Finallywediscusshowwechooserangesoftheaggregateproductivityandtherelative price distortion. Following Judd (1992), the maximum of log productivity is set equal to the long-run value of a that would occur if θ = 2σ for all t: a = 2σ /(1−ρ).12 θ max θ The minimum of log productivity is the negative of maximum of log productivity. The minimum value of the relative price distortion is 1. However, it is hard to make an appropriate choice of the maximum of the relative price distortion. In particular, this issue is closely related to our application of homotopy method that will be explained in the next section. 12Judd actually used 3σ, but, for our benchmark parameter settings, the use of a smaller range had noimpactontheresults. 20

Appendix C. Projection Method with Homotopy Procedure In the course of developing our model, we created routines for several heuristics for solving the nonlinear-equation system determining the collocation-polynomial weights. This section characterizes these heuristics using the homotopy formalism described in Judd (1998). To summarize the basic idea, we begin by solving the model for a set of parameters that makes the model easy to solve. We use this solution to facilitate the solution of a “nearby” model that has parameters set closer to the parameter settings that we are reallyinterestedin. Werepeatthisprocess,solvingasequenceofsimilarmodelsenroute to solving the model with our benchmark parametrization. It might prove useful, in general, to use information provided by perturbation solutionsasabasisforinitialweightsforthecollocatingpolynomials. However,inourmodel wefoundthefollowingheuristicseasilyimplementableandcapableofreliablyproducing accurate approximations with the appropriate dynamic properties. Although we did not do so, in the general case it seems likely to be worthwhile to investigate how to reliably exploit high order perturbation solutions to provide initial weights for projection calculations. Therearetwodistinctphasesinthissolutionprocess. Inthefirstphase, wesolvethe model using 0th order polynomials, i.e. constant functions varying the parameters from theeasyvaluestothebenchmarkvalues.13 Thisphaseemploystheparameterhomotopy described in Algorithm 1 below.14 In the second phase, we increase the order of the Chebyshev polynomials. Prior to this phase, we are collocating with constant functions, and the range of the Chebyshev polynomials plays no role. In the second phase, the specific range of the Chebyshev polynomials can have a dramatic effect on the solvability of the system. Fortunately, experience with our model supports the following conjecture: Conjecture 1 Let M correspond to a basis consisting entirely of 0th order Chebyshev 0 polynomials. Suppose we can solve the 0th order problem so that there exists W∗ such 0 that W∗ =N(Λˆ(s ,Υ,I,M ),W∗) 0 t 0 0 then ∃γ >03W∗ =N(Λˆ(s ,Υ,I(γ),M),W∗)6=∅ t 0 (cid:4) Thus, when the Chebyshev polynomial domain is small enough, the nonlinear system we must solve is similar to the system for W . As a result, Newton’s method will 0 also converge for the problem with higher order polynomials when the domain of the 13Inourmodel,sincesettingτP =0,τW = (cid:15)− 1 1 leadstoanon-distortedsteadystatewith∆¯ =Π¯ =1 thisparametrizationiseasytosolve. 14Wealsouseparameterhomotopytogenerategraphsofthesteadystatevaluesofvariablevisavis parameters. 21

Chebyshev polynomials is small enough. Consequently, when necessary, we can employ a homotopy on the range to extend the range from a small range to the full range. Although, in practice, the algorithms rarely solve the problem on such a small domain, the existence of such a domain guarantees that the algorithms will terminate. We have automated this heuristic using the algorithms described in Algorithm 2 below. We express a projection method using the function Λˆ(s ,Υ,I,M) where s is the t t state at period t, Υ is the set of parameters, I is the set of ranges for the Chebyshev polynomials,andM isthesetofordersfortheChebyshevpolynomials. WeuseNewton’s method to update the Chebyshev polynomials’ weights in the following way: (cid:26) W , if Newton method converges; N(Λˆ(s ,Υ,I,M),W )= i+1 t i ∅, if Newton method fails where N represents an application of Newton’s method and W is a set of weights. i We now describe the two types of homotopy. In the case of the range homotopy, we predetermineasetofrangesoftheChebyshevpolynomials. Forexample,weuseatensor product of finite number of bounded and closed intervals: I = [l ,u ]⊗···⊗[l ,u ] 1 1 m m wheremrepresentsthenumberofstatevariablesandeachintervalspecifiestheminimum and maximum of a state variable. In order to implement the range homotopy, we define a nested range for each range [l , u ] by using a parameter γ: k k l +u l +u µ(γ,[l ,u ])=[l +(1−γ) k k,u −(1−γ) k k] for k =1,··· ,m k k k 2 k 2 where µ(γ,[l ,u ]) denotes the nested range for each interval [l ,u ]. Hence, we have a k k k k new set of ranges of the Chebyshev polynomials: I(γ) = µ([l ,u ])⊗···⊗µ([l ,u ]). 1 1 m m As a result, a particular Newton method step can be described as follows: (cid:26) W , if Newton method converges; N(Λˆ(s ,Υ,I(γ),M),W ,γ)= i+1 t i ∅, if Newton method fails. In applying a homotopy, we seek a W and a sequence {γ }N such that there is no 0 n n=0 failure of Newton’s method for each value of {γ }N . Thus, any algorithm applying a n n=0 homotopymethodmustimplementstrategiesforadjustingthevalueofγ whenNewton’s method fails. In our code, we choose a recursive updating rule for the value of γ. For example, suppose that we failed the Newton’s method at γ = γ . In this case, we shrink n the value of γ by setting a new value of γ as follows: γ1 = ν γ0, where γ1 denotes the n n n n n new trial value of γ, γ0 is the old trial value of γ at the last round of Newton’s method n and ν is a shrink factor that is a positive constant between 0 and 1. If the Newton’s method does not fail, we use the current value of γ as a new value of γ . The following n n pseudo-code characterizes algorithms that work for our model. 22

Algorithm 1 Parameter Homotopy Procedures 1: procedure MoveLowerEndToUpper(N(Λˆ(s t ,·,I,M),W),Υ∗,{Υ 0 ,W}) 2: Q:=true 3: Υ:=Υ 0 4: W =W 0 5: while Q do 6: {Υ,W}:=FindBetterParams(N(Λˆ(s t ,·,I,M),W),Υ∗,{Υ,W}) 7: if Υ=Υ∗ then 8: Q:=false 9: end if 10: end while 11: end procedure 1: procedure FindBetterParams(N(Λˆ(s t ,·,I,M),·),Υ∗,{Υ 0 ,,W 0 }) 2: ν ∈(0,1) 3: γ :=1 4: Q:=true 5: while Q do 6: W∗ :=N(Λˆ(s t ,η(γ,Υ 0 ,Υ ∗ ),I,M),W) 7: if W∗ =∅ then 8: γ :=ν×γ; 9: else 10: Q:=false 11: end if 12: end while 13: return({η(γ,Υ 0 ,Υ ∗ ),W∗})) 14: end procedure 23

Algorithm 2 Range Homotopy Procedures 1: procedure WidenRangeToFull(N(Λˆ(s t ,Υ,I(·),M),W 0 )) 2: Q:=true 3: W =W 0 4: while Q do 5: {γ,W}:=FindWiderRange(N(Λˆ(s t ,·,I,M),W),α∗,W) 6: if γ :=1 then 7: Q:=false 8: end if 9: end while 10: end procedure 1: procedure FindWiderRange(N(Λˆ(s t ,Υ,I(·),M),W 0 )) 2: ν ∈(0,1) 3: γ :=1 4: Q:=true 5: while Q do 6: W∗ :=N(Λˆ(s t ,Υ,I(γ),M),W 0 )) 7: if W∗ =∅ then 8: γ :=ν×γ 9: else 10: Q:=false 11: end if 12: end while 13: return(γ,W∗})) 14: end procedure 24