Essays in Monetary Economics

Liu, Ding (2015) Essays in monetary economics PhD thesis http://theses.gla.ac.uk/6692/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ theses@gla.ac.uk Essays in Monetary Economics by Ding Liu Submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy Adam Smith Business School College of Social Sciences University of Glasgow August 2015 c Ding Liu Abstract This dissertation can be thematically grouped into two categories: monetary theory in the so called New Monetarist search models where money and credit are essential in terms of improving social welfare, and optimal time-consistent monetary and fiscal policy in New Keynesian dynamic stochastic general equilibrium (DSGE) models when the government cannot commit Arguably, the methodology and conceptual frameworks adopted in these two lines of work are quite different However, they share a common goal in helping us understand how and why monetary factors can affect the real economy, and how monetary and fiscal policy should respond to developments in the economy to improve social welfare There are two chapters in each part In the first chapter, recent advances based on the pre-eminent Lagos-Wright (LW) monetary search model are reviewed Against this background, chapter two introduces collateralized credit inspired by a communal responsibility system into the creditless LW model, in order to study the role of money and credit as alternative means of payment In contrast, the third chapter revisits the classic inflation bias problem associated with optimal time-consistent monetary policy in the cashless New Keynesian framework In this chapter, fiscal policy is trivial, due to the assumption of lump-sum tax As a follow-up work, chapter four studies optimal time-consistent monetary and fiscal policy mix as well as debt maturity choice in an environment with only distortionary taxes, endogenous government spending and government debt of various maturities Chapter introduces the tractable and influential Lagos-Wright (LW) search-theoretic framework and reviews the latest developments in extending it to study issues concerning the role of money, credit, asset pricing, monetary policy and economic growth In addition, potential research topics are discussed Our main message from this review is that the LW monetary model is flexible enough to deal with numerous issues where fiat money plays an essential role as a medium of exchange Chapter 2, based on the LW framework, develops a search model of money and credit motivated by a historical medieval institution - the community responsibility system The aim is to examine the role of credit collateralized by the community responsibility system as a supplementary medium of exchange in long-distance trade, assuming that iii entry cost and the cost of using credit are proportional to distance, due to factors like direct verification and settlement cost and indirect transportation cost We find that both money and credit are useful in the sense of improving welfare In addition, the Friedman rule can be sub-optimal in this economy, due to the interaction between the extensive margin (that is, the range of outside villages which the representative household has trade with) and the intensive margin (that is, the scope of villages where credit is used as a supplementary medium of exchange) Finally, higher entry cost narrows down the extensive margin, and similarly, higher cost of using credit, ceteris paribus, reduces the usage of credit and hence lowers social welfare Chapter reconsiders the inflation bias problem associated with the renowned rules versus discretion debate in a fully nonlinear version of the benchmark New Keynesian DSGE model We ask whether the inflation bias problem related to discretionary monetary policy differs quantitatively under two dominant forms of nominal rigidities Calvo pricing and Rotemberg pricing, if the inherent nonlinearities are taken seriously We find that the inflation bias problem under Calvo contracts is significantly greater than under Rotemberg pricing, despite the fact that the former typically exhibits far greater welfare costs of inflation In addition, the rates of inflation observed under the discretionary policy are non-trivial and suggest that the model can comfortably generate the rates of inflation at which the problematic issues highlighted in the trend inflation literature Finally, we consider the response to cost push shocks across both models and find these can also be significantly different Thus, we conclude that the nonlinearities inherent in the New Keynesian DSGE model are empirically relevant and the form of nominal inertia adopted is not innocuous Chapter studies the optimal time-consistent monetary and fiscal policy when surprise inflation (or deflation) is costly, taxation is distortionary, and non-state-contingent nominal debt of various maturities exists In particular, we study whether and how the change in nominal government debt maturity affects optimal policy mix and equilibrium outcomes, in the presence of distortionary taxes and sticky prices We solve the fully nonlinear model using global solution techniques, and find that debt maturity has drastic effects on optimal time-consistent policies in New Keynesian models In particular, some interesting nonlinear effects are uncovered Firstly, the equilibrium value for debt is negative and close to zero, which implies a slight undershooting of the inflation target in steady state Secondly, starting from high level of debt-GDP ratio, the optimal policy will gradually reduce the level of debt, but with radical changes in the policy mix along the transition path At high debt levels, there is a reliance on a relaxation of monetary policy to reduce debt through an expansion in the tax base and reduced debt service costs, while tax rates are used to moderate the increases in inflaiv tion However, as debt levels fall, the use of monetary policy in this way is diminished and the policy maker turns to fiscal policy to continue the reduction in debt This is akin to a switch from an active to passive fiscal policy in rule based descriptions of policy, which occurs endogenously under the optimal policy as debt levels fall It can also be accompanied by a switch from passive to active monetary policy This switch in the policy mix occurs at higher debt levels, the longer the average maturity of government debt This is largely because high debt levels induce an inflationary bias problem, as policy makers face the temptation to use surprise inflation to erode the real value of that debt This temptation is then more acute when debt is of shorter maturity, since the inflationary effects of raising taxes to reduce debt become increasingly costly as debt levels rise Finally, in contrast to the Ramsey literature with real bonds, in the current setting we find no extreme portfolios of short and long-term debt In addition, optimal debt maturity, implicitly, lengthens with the level of debt v Table of Contents Abstract iii List of Tables xiii List of Figures xv Acknowledgements xvii Declaration xxi Preface xxiii References xxix Search Models of Money: Recent Advances 1.1 Introduction 1.2 The Underlying Model 1.2.1 The Basic Environment 1.2.2 Decisions and Equilibrium Equilibrium with Money Being Essential 1.3.1 Essentiality of Money 1.3.2 Properties of Monetary Equilibrium 10 1.3.3 Discussion 19 Models with Competing Media of Exchange 19 1.4.1 Money and Real Assets 20 1.4.2 Money and Nominal Assets 21 1.3 1.4 vii 1.4.3 1.5 1.6 1.7 1.8 1.9 Discussion 23 Models with Money and Credit 24 1.5.1 Money and Credit as Means of Payment 24 1.5.2 Credit, Banking and Liquidity Reallocation 26 1.5.3 Discussion 30 Liquidity, Asset Prices and Monetary Policy 31 1.6.1 Asset Prices with Liquidity Premia 32 1.6.2 Monetary Policy and Asset Prices 33 1.6.3 Discussion 35 Monetary Propagation and Business Cycles 36 1.7.1 Monetary Transmission Mechanism 36 1.7.2 Optimal Monetary Policy 40 1.7.3 Welfare Costs of Inflation 45 1.7.4 Discussion 47 Money in Economic Growth Models 48 1.8.1 Discussion 51 Concluding Remarks 51 Appendices 53 1.A Shi (1997) in Detail 53 References 57 A Spatial Search Model with Money and Credit 2.1 2.2 71 Introduction 71 2.1.1 Related Literature 73 The Environment 75 2.2.1 Agents 75 2.2.2 Preferences and technology 76 2.2.3 Money Supply 77 2.2.4 Market Structure 78 viii 2.2.5 Communal Responsibility System 80 2.2.6 First Best 81 The Representative Household’s Problems 82 2.3.1 The Problem in Day Markets 82 2.3.2 The Problem in Night Markets 84 2.3.3 The Envelope Conditions 85 2.3.4 Market Clearing Conditions 86 2.3.5 The Value of Defection 86 Symmetric Stationary Equilibrium 86 2.4.1 Steady State Welfare 91 2.4.2 Welfare Without Credit 91 2.5 Numerical Analysis 92 2.6 Discussion 97 2.7 Conclusion 98 2.3 2.4 Appendices 99 2.A Technical Appendix 99 2.A.1 The Problem in Day Markets 99 References 101 The Inflation Bias under Calvo and Rotemberg Pricing 103 3.1 Introduction 103 3.2 The Model 108 3.3 3.4 3.2.1 Households 108 3.2.2 Firms 110 3.2.3 Aggregate Conditions 112 Optimal Policy Problem Under Discretion 114 3.3.1 Rotemberg Pricing 114 3.3.2 Calvo Pricing 115 Numerical Analysis 118 ix Technical Appendix −1 C = (1 − τ ) Y G C = − = − (1 − τ ) Y Y PMb = Y ϕ+σ G 1− Y Y− −1 −1 β τ 1−β Note that, 1/σ σ = (1 − τ ) − ϕ+σ σ 1/σ Y− ϕ+σ σ G Y Y −1 (4.39) which will be used to contrast with the allocation that would be chosen by a social planner 4.A.3 Numerical Algorithm Let st = (bt−1 , at ) denote the state vector at time t, where real stock of debt bt−1 is endogenous and technology At = exp(at ) is exogenous and respectively, with the following law of motion: PtM bt = (1 + ρPtM ) bt−1 − wt Nt τt + Gt Πt at = ρa at−1 + eat where ≤ ρa < and technology innovation eat is an i.i.d normal random variable, which has a zero mean and a finite standard deviation σa There are endogenous variables and Lagrangian multipliers Correspondingly, there are 10 functional equations associated with the 10 varaibles Ct ,Yt ,Πt ,bt ,τt ,PtM ,Gt ,λ1t ,λ2t ,λ3t Let’s define a new function X : R2 → R10 , in order to collect the policy functions of endogenous variables as follows: X(st ) = Ct (st ), Yt (st ), Πt (st ), bt (st ), τt (st ), PtM (st ), Gt (st ), λ1t (st ), λ2t (st ), λ3t (st ) Given the specification of the function X, the equilibrium conditions can be written more compactly as, Γ (st , X(st ), Et [Z (X(st+1 ))] , Et [Zb (X(st+1 ))]) = 207 Technical Appendix where Γ : R2+10+3+3 → R10 summarizes the full set of dynamic equilibrium relationship, and     Z1 (X(st+1 )) M (bt , At+1 )     Z (X(st+1 )) =  Z2 (X(st+1 ))  ≡  L(bt , At+1 )  −1 M Z3 (X(st+1 )) (Πt+1 ) + ρPt+1 λ3t+1 with M (bt , At+1 ) = (Ct+1 )−σ Yt+1 Πt+1 Π∗ Πt+1 −1 Π∗ M L(bt , At+1 ) = (Ct+1 )−σ (Πt+1 )−1 (1 + ρPt+1 ) and   Zb (X(st+1 )) =  ∂Z1 (X(st+1 )) ∂bt ∂Z2 (X(st+1 )) ∂bt ∂Z3 (X(st+1 )) ∂bt     ≡  ∂M (bt ,At+1 ) ∂bt ∂L(bt ,At+1 ) ∂bt M ∂ [(Πt+1 )−1 (1+ρPt+1 )λ3t+1 ] ∂bt     More specifically, L1 (bt , At+1 ) = M ∂ (Ct+1 )−σ (Πt+1 )−1 (1 + ρPt+1 ) ∂bt M = −σ(Ct+1 )−σ−1 (Πt+1 )−1 (1 + ρPt+1 ) M − (Ct+1 )−σ (Πt+1 )−2 (1 + ρPt+1 ) ∂Ct+1 ∂bt ∂P M ∂Πt+1 + ρ(Ct+1 )−σ (Πt+1 )−1 t+1 ∂bt ∂bt and M1 (bt , At+1 ) = = −σ(Ct+1 )−σ−1 Yt+1 + (Ct+1 )−σ Yt+1 Π∗ Yt+1 Π∗ Πt+1 Π∗ −1 ∂bt Πt+1 ∂Ct+1 Πt+1 −σ Πt+1 + (C ) − −1 t+1 Π∗ ∂bt Π∗ Π∗ ∂Πt+1 Yt+1 Πt+1 ∂Πt+1 −1 + (Ct+1 )−σ ∗ ∂bt Π Π∗ ∂bt Πt+1 Π∗ Πt+1 Π∗ Πt+1 Πt+1 −1 Π∗ Π∗ 2Πt+1 ∂Πt+1 −1 ∗ Π ∂bt = −σ(Ct+1 )−σ−1 Yt+1 + (Ct+1 )−σ ∂ (Ct+1 )−σ Yt+1 ΠΠt+1 ∗ ∂Ct+1 Πt+1 + (Ct+1 )−σ ∗ ∂bt Π Πt+1 −1 Π∗ ∂Yt+1 ∂bt ∂Yt+1 ∂bt Note we are assuming Et [Zb (X(st+1 ))] = ∂Et [Z (X(st+1 ))] /bt , which is normally valid using the Interchange of Integration and Differentiation Theorem Then the problem 208 Technical Appendix is to find a vector-valued function X that Γ maps to the zero function Projection methods, hence, can be used Following the notation convention in the literature, we simply use s = (b, a) to denote the current state of the economy st = (bt−1 , at ), and s to represent next period state that evolves according to the law of motion specified above The Chebyshev collocation method with time iteration which we use to solve this nonlinear system can be described as follows: Define the collocation nodes and the space of the approximating functions: • Choose an order of approximation (i.e., the polynomial degrees) nb and na for each dimension of the state space s = (b, a), then there are Ns = (nb + 1) × (na + 1) nodes in the state space Let S = (S1 , S2 , , SNs ) denote the set of collocation nodes • Compute the nb + and na + roots of the Chebychev polynomial of order nb + and na + as zbi = cos zai = cos (2i − 1)π , for i = 1, 2, , nb + 2(nb + 1) (2i − 1)π , for i = 1, 2, , na + 2(na + 1) • Compute collocation points as = a+a a−a i a−a i + za = za + + a 2 for i = 1, 2, , na + Note that the number of collocation nodes is na + Similarly, compute collocation points bi as bi = b+b b−b i b−b i + z = zb + + b 2 b for i = 1, 2, , nb + 1, which map [−1, 1] into [b, b] Note that S = {(bi , aj ) | i = 1, 2, , nb + 1, j = 1, 2, , na + 1} that is, the tensor grids, with S1 = (b1 , a1 ), S2 = (b1 , a2 ), , SNs = (bnb +1 , ana +1 ) • The space of the approximating functions, denoted as Ω, is a matrix of 209 Technical Appendix two-dimensional Chebyshev polynomials More specifically,         Ω (S) =         Ω (S1 )         =  Ω (Sna +1 )       Ω (S2 ) Ω (SNs )  T0 (ξ(b1 )T1 (ξ (a1 )) T0 (ξ(b1 )T2 (ξ (a1 )) ··· Tnb (ξ(b1 )Tna (ξ (a1 ))         =        T0 (ξ(b1 )T1 (ξ (a2 )) T0 (ξ(b1 )T2 (ξ (a2 )) ··· Tnb (ξ(b1 )Tna (ξ (a2 )) ···                T0 (ξ(b1 )T1 (ξ (ana +1 )) T0 (ξ(b1 )T2 (ξ (ana +1 )) ··· T0 (ξ(b1 )Tna (ξ (ana +1 )) ··· T0 (ξ(bnb +1 )T1 (ξ (ana +1 )) T0 (ξ(bnb +1 )T2 (ξ (ana +1 )) ··· T0 (ξ(bnb +1 )Tna (ξ (ana +1 )) Ns ×Ns where ξ(x) = (x − x) / (x − x) − maps the domain of x ∈ [x, x] into [−1, 1] • Then, at each node s ∈ S, policy functions X(s) are approximated by X(s) = Ω(s)ΘX , where ΘX = θc , θy , θπ , θb , θτ , θp , θg , θλ1 , θλ2 , θλ3 is a Ns × 10 matrix of the approximating coefficients Formulate an initial guess for the approximating coefficients, Θ0X , and specify the stopping rule tol , say, 10−6 At each iteration j, we can get an updated ΘjX by implement the following time iteration step: • At each collocation node s ∈ S, compute the possible values of future policy functions X(s ) for k = 1, , q That is, X(s ) = Ω(s )Θj−1 X where q is the number of Gauss-Hermite quadrature nodes Note that Ω(s ) = Tjb (ξ(b ))Tja (ξ(a )) is a q × Ns matrix, with b = b(s; θb ), a = ρa a + zk 2σa2 , jb = 0, , nb , and ja = 0, , na The hat symbol indicates the corresponding approximate policy functions, so b is the approximate policy for real debt, for example 210 Technical Appendix Similarly, the two auxilliary functions can be calculated as follows: M (s ) ≈ C(s ; θc ) −σ Y (s ; θy ) Π(s ; θπ ) Π∗ Π(s ; θπ ) −1 Π∗ and, L(s ) ≈ C(s ; θc ) −σ Π(s ; θπ ) −1 1+ ρP M s ; θ p Π∗ − ρβ Note that we use PtM = (Π∗ − ρβ) PtM rather than PtM in numerical analysis, since the former is far less sensitive to maturity structure variations • Now calculate the expectation terms E [Z (X(s ))] at each node s Let ωk denote the weights for the quadrature, then E [M (s )] ≈ √ π E [L(s )] ≈ √ π q −σ ωk C(s ; θc ) Y (s ; θy ) k=1 q −σ ωk C(s ; θc ) Π(s ; θπ ) Π∗ Π(s ; θπ ) −1 1+ k=1 Π(s ; θπ ) −1 Π∗ ρP M s ; θ p Π∗ − ρβ ≡ M (s , q) ≡ L (s , q) and Et M + ρPt+1 Πt+1 λ3t+1 ≈ √ π q   ρP M (s ;θp )  Π∗ −ρβ 1 + ωk  Π(s ; θπ ) k=1 λ  λ3 (s ; θ ) ≡ Λ (s , q) Hence,   M (s , q)   E [Z (X(s ))] ≈ E Z (X(s )) =  L (s , q)  Λ (s , q) • Next calculate the partial derivatives under expectation E [Zb (X(s ))] M • Note that we only need to compute ∂Ct+1 /∂bt , ∂Yt+1 /∂bt , ∂Πt+1 /∂bt and ∂Pt+1 /∂bt , which are given as follows: ∂Ct+1 ≈ ∂b j ∂Yt+1 ≈ ∂bt j nb na b =0 ja =0 nb na b =0 ja =0 2θjcb ja b−b 2θjyb ja b−b Tjb (ξ(b ))Tja (ξ(a )) ≡ Cb (s ) Tjb (ξ(b ))Tja (ξ(a )) ≡ Yb (s ) 211 Technical Appendix ∂Πt+1 ≈ ∂bt j M ∂Pt+1 ≈ ∂bt j nb nb na 2θjπb ja b =0 ja =0 na b =0 ja =0 b−b Tjb (ξ(b ))Tja (ξ(a )) ≡ Πb (s ) 2θjpb ja b−b (Π∗ − ρβ) Tjb (ξ(bi ))Tja (ξ(aj )) ≡ PbM (s ) Hence, we can approximate the two partial derivatives under expectation ∂E [M (s )] ∂b  ≈√ π c −σ−1 π ; θy ) Π(sΠ;θ∗ ) Y (s  −σ C(s ; θ )  −σ Π(s ;θπ ) ωk  + C(s ; θc )  Π∗  −σ k=1 Π(s ;θπ ) + C(s ; θc ) Π∗ q Π(s ;θπ ) Π∗  − Cb (s )   Π(s  − Y (s ) b ∗  Π  π 2Π(s ;θ ) − Π (s ) b Π∗ ;θπ ) ≡ Mb (s , q) , ∂E [L(s )] ∂b −σ−1 −1 ρP M (s ;θp ) c π −σ C(s ; θ ) Π(s ; θ ) (1 + )Cb (s ) Π∗ −ρβ  q  −σ −2 ρP M (s ;θp ) ωk  ≈√ − C(s ; θc ) Π(s ; θπ ) (1 + Π∗ −ρβ )Πb (s )  π k=1  −σ −1 +ρ C(s ; θc ) Π(s ; θπ ) PbM (s )        ≡ Lb (s , q) That is, E [Zb (X(s ))] ≈ E Zb (X(s )) = Mb (s , q) Lb (s , q) At each collocation node s, solve for X(s) such that Γ s, X(s), E Z (X(s )) , E Zb (X(s )) =0 The equation solver csolve written by Christopher A Sims is employed to solve the resulted system of nonlinear equations With X(s) at hand, we can get the corresponding coeffcient ΘjX = Ω (S)T Ω (S) −1 Ω (S)T X(s) Update the approximating coefficients, ΘjX = η ΘjX + (1 − η) Θj−1 X , where ≤ η ≤ is some dampening parameter used for improving convergence 212 Technical Appendix Check the stopping rules If ΘjX − Θj−1 < tol , then stop, else update the X approximation coefficients and go back to step When implementing the above algorithm, we start from lower order Chebyshev polynomials and some reasonable initial guess Then, we increase the order of approximation and take as starting value the solution from the previous lower order approximation This informal homotopy continuation idea ensures us to find a solution Remark Given the fact that the price PtM fluctuates significantly for larger ρ, in numerical analysis, we scale rule for PtM by (Π∗ − ρβ), that is, PtM = (Π∗ − ρβ)PtM In this way, the steady state of PtM is very close to β, and PtM does not differ hugely as we change the maturity structure 4.A.4 Optimal Policy Under Discretion With Endogenous ShortTerm Debt In this case, the government is allowed to issue new bonds of a different maturity and swap these for existing bonds, in a way which does not affect the wealth of the bond holders at the time of the swap, such that the exchange is voluntary The policy under discretion in this case can be described as a set of decision rules for {Ct , Yt , Πt , bt , τt , Gt , bSt } which maximise, 1−σ V (bt−1 , At , bSt−1 ) = max (Yt /At )1+ϕ G g Ct1−σ − +χ t + βEt V (bt , At+1 , bSt ) 1−σ − σg 1+ϕ subject to the following constraints: Yt φ 1− Πt −1 Π∗ + φβEt Ct Ct+1 σ = Ct + Gt Πt −1 Π∗ Yt+1 Πt+1 Πt+1 −1 Yt Π∗ Π∗ = (1 − ) + mct − φ Πt Π∗ 213 Technical Appendix σ Ct Ct+1 βEt = + ρβEt − τt − τt Pt Pt+1 Ct Ct+1 Yt At σ M + ρPt+1 σ Pt Pt+1 Ct Pt Ct+1 Pt+1 S bt−1 bt−1 + Πt Πt bt + βEt M + ρPt+1 bSt 1+ϕ Ctσ + Gt where bSt is the level of real short-term debt Defining auxilliary functions, M (bt , At+1 , bSt ) = (Ct+1 )−σ Yt+1 Πt+1 Π∗ Πt+1 −1 Π∗ M L(bt , At+1 , bSt ) = (Ct+1 )−σ (Πt+1 )−1 (1 + ρPt+1 ) −σ −1 K bt , At+1 , bSt = Ct+1 Πt+1 we can rewrite the NKPC and government budget constraints as, respectively, (1 − ) + (1 − τt )−1 Ytϕ Ctσ A−1−ϕ −φ t Πt − + φβCtσ Yt−1 Et M (bt , At+1 , bSt ) = Π∗ Πt Π∗ = βbt Ctσ Et L(bt , At+1 , bSt ) + βbSt Ctσ Et K bt , At+1 , bSt − bt−1 + ρβCtσ Et L(bt , At+1 , bSt ) Πt − bSt−1 + Πt τt − τt Yt At 1+ϕ Ctσ − Gt The Lagrangian for the policy problem can be written as, 1−σ L= Gt g Ct1−σ (Yt /At )1+ϕ +χ − + βEt [V (bt , At+1 , bSt )] 1−σ − σg 1+ϕ + λ1t Yt φ 1− Πt −1 Π∗ − Ct − Gt (1 − ) + (1 − τt )−1 Ytϕ Ctσ A−1−ϕ − φ ΠΠ∗t t +φβCtσ Yt−1 Et M (bt , At+1 , bSt ) + λ2t Πt Π∗ −1  βbt Ctσ Et L(bt , At+1 , bSt ) + βbSt Ctσ Et K bt , At+1 , bSt  t−1 + ρβCtσ Et L(bt , At+1 , bSt ) − bΠ + λ3t  t  1+ϕ bS τt Yt t−1 −Π + Ctσ − Gt 1−τ A t t t We can write the first order conditions for the policy problem as follows: 214     Technical Appendix Consumption, Ct−σ − λ1t + λ2t σ (1 − τt )−1 Ytϕ Ctσ−1 A−1−ϕ + σφβCtσ−1 Yt−1 Et M (bt , At+1 , bSt ) t  +λ3t  σβbt Ctσ−1 Et L(bt , At+1 , bSt ) + σβbSt Ctσ−1 Et K bt , At+1 , bSt t−1 Ctσ−1 Et −ρσβ bΠ t L(bt , At+1 , bSt ) τt 1−τt +σ Yt At 1+ϕ Ctσ−1  =0 Government spending, −σg χGt − λ1t − λ3t = Output, −Ytϕ A−1−ϕ + λ1t − t +λ2t Πt −1 Π∗ φ 2 ϕ(1 − τt )−1 Ytϕ−1 Ctσ A−1−ϕ − φβCtσ Yt−2 Et M (bt , At+1 , bSt ) t +λ3t (1 + ϕ)Ytϕ Ctσ τt − τt A−1−ϕ =0 t Taxation, λ2t + λ3t Yt = Inflation, −λ1t Yt +λ3t φ Π∗ Πt −1 Π∗ − λ2t φ Π∗ bt−1 + ρβCtσ Et L(bt , At+1 , bSt ) Πt 2Πt −1 Π∗ + bSt−1 =0 Π2t Government debt, bt , βEt [V1 (bt , At+1 , bSt )] + λ2t φβCtσ Yt−1 Et M1 (bt , At+1 , bSt ) +βCtσ λ3t Et L(bt , At+1 , bSt ) + bt Et L1 (bt , At+1 , bSt ) + bSt Et K1 bt , At+1 , bSt t−1 −ρ bΠ Et L1 (bt , At+1 , bSt ) t where V1 (bt , At+1 , bSt ) ≡ ∂V (bt , At+1 , bSt )/∂bt L1 (bt , At+1 , bSt ) ≡ ∂L(bt , At+1 , bSt )/∂bt M1 (bt , At+1 , bSt ) ≡ ∂M (bt , At+1 , bSt )/∂bt K1 (bt , At+1 , bSt ) ≡ ∂K(bt , At+1 , bSt )/∂bt 215 =0 Technical Appendix Short-term government debt, bSt , βEt [V3 (bt , At+1 , bSt )] + λ2t φβCtσ Yt−1 Et M3 (bt , At+1 , bSt ) +βCtσ λ3t bt Et L3 (bt , At+1 , bSt ) + Et K bt , At+1 , bSt + bSt Et K3 bt , At+1 , bSt t−1 Et L3 (bt , At+1 , bSt ) −ρ bΠ t =0 where V3 (bt , At+1 , bSt ) ≡ ∂V (bt , At+1 , bSt )/∂bSt L3 (bt , At+1 , bSt ) ≡ ∂L(bt , At+1 , bSt )/∂bSt M3 (bt , At+1 , bSt ) ≡ ∂M (bt , At+1 , bSt )/∂bSt K3 (bt , At+1 , bSt ) ≡ ∂K(bt , At+1 , bSt )/∂bSt Note that by the envelope theorem, V1 (bt−1 , At , bSt−1 ) = − λ3t + ρβCtσ Et L(bt , At+1 , bSt ) Πt =− λ3t + ρPtM Πt V3 (bt−1 , At , bSt ) = − λ3t Πt hence, V1 (bt , At+1 , bSt ) = λ3t+1 M + ρPt+1 Πt+1 V3 (bt , At+1 , bSt ) = − λ3t+1 Πt+1 and the FOCs for government debt bt and bSt can be rewritten as, respectively, −βEt +βCtσ λ3t λ3t+1 M (1 + ρPt+1 ) + λ2t φβCtσ Yt−1 Et M1 (bt , At+1 , bSt ) Πt+1 Et L(bt , At+1 , bSt ) + bt Et L1 (bt , At+1 , bSt ) + bSt Et K1 bt , At+1 , bSt t−1 −ρ bΠ Et L1 (bt , At+1 , bSt ) t and −βEt [ λ3t+1 ] + λ2t φβCtσ Yt−1 Et M3 (bt , At+1 , bSt ) Πt+1 216 =0 Technical Appendix +βCtσ λ3t bt Et L3 (bt , At+1 , bSt ) + Et K bt , At+1 , bSt + bSt Et K3 bt , At+1 , bSt t−1 Et L3 (bt , At+1 , bSt ) −ρ bΠ t =0 A Note on Numerical Analysis Given the solution from the case with exogenously given short-term debt, we can use a penalty term to gradually get the solution for the case with endogenous short-term debt Specifically, we modify the period utility function into the following form, 1−σ Ct1−σ G g (Yt /At )1+ϕ S +χ t − − ζ bSt − b 1−σ − σg 1+ϕ where ζ is a scale parameter of penalty Note that the modified period utility function is reduced into the original form, when ζ = This fact motivates a homotopy continS uation idea Initially, we can set ζ large enough to make sure bSt close to b Then, we incrementally decrease ζ down to zero Now, the FOC for bSt becomes, −2ζ bSt − b +βCtσ λ3t S − βEt [ λ3t+1 ] + λ2t φβCtσ Yt−1 Et M3 (bt , At+1 , bSt ) Πt+1 bt Et L3 (bt , At+1 , bSt ) + Et K bt , At+1 , bSt + bSt Et K3 bt , At+1 , bSt t−1 Et L3 (bt , At+1 , bSt ) −ρ bΠ t 217 =0 REFERENCES References S Rao Aiyagari, Albert Marcet, Thomas J Sargent, and Juha Seppala Optimal Taxation without State-Contingent Debt Journal of Political Economy, 110(6):1220– 1254, 2002 Gary S Anderson, Jinill Kim, and Tack Yun Using a projection method to analyze inflation bias in a micro-founded model Journal of Economic Dynamics and Control, 34(9):1572–1581, 2010 doi: http://dx.doi.org/10.1016/j.jedc.2010.06.024 URL http://www.sciencedirect.com/science/article/pii/S0165188910001545 George-Marios Angeletos, Fabrice Collard, Harris Dellas, and Behzad Diba Optimal public debt management and liquidity provision NBER Working Paper 18800, 2013 Robert J Barro On the determination of the public debt The Journal of Political Economy, pages 940–971, 1979 Pierpaolo Benigno and Michael Woodford Optimal monetary and fiscal policy: A linear-quadratic approach In NBER Macroeconomics Annual 2003, Volume 18, pages 271–364 The MIT Press, 2003 Saroj Bhattarai, Gauti B Eggertsson, and Bulat Gafarov Time Consistency and the Duration of Government Debt: A Signalling Theory of Quantitative Easing Mimeo, 2014 Huixin Bi, Eric M Leeper, and Campbell Leith Uncertain fiscal consolidations The Economic Journal, 123(566):F31–F63, 2013 Qingqing Cao Optimal Fiscal and Monetary Policy with Collateral Constraints Mimeo, 2014 Han Chen, Vasco Curdia, and Andrea Ferrero The Macroeconomic Effects of Largescale Asset Purchase Programmes The Economic Journal, 122(564):F289–F315, 2012 doi: 10.1111/j.1468-0297.2012.02549.x URL http://dx.doi.org/10.1111/ j.1468-0297.2012.02549.x John H Cochrane Monetary policy with interest on reserves Journal of Economic Dynamics and Control, 49(0):74 – 108, 2014 ISSN 0165-1889 doi: http://dx.doi org/10.1016/j.jedc.2014.09.003 Alexandre Corhay, Howard Kung, and Gonzalo Morales Government maturity structure twists Mimeo, 2014 Davide Debortoli and Ricardo Nunes Lack of Commitment and the Level of Debt Journal of the European Economic Association, 11(5):1053–1078, 2013 Davide Debortoli, Ricardo Nunes, and Pierre Yared Optimal Time-Consistent Government Debt Maturity Mimeo, 2015 Stefano Eusepi and Bruce Preston The maturity structure of debt, monetary policy and expectations stabilization mimeo, Columbia University, 2011 218 REFERENCES Stefano Eusepi and Bruce Preston Debt, policy uncertainty, and expectations stabilization Journal of the European Economic Association, 10(4):860–886, 2012 Stefano Eusepi and Bruce J Preston Fiscal Foundations of Inflation: Imperfect Knowledge FRB of New York Staff Report No 649, (649), 2013 Elisa Faraglia, Albert Marcet, Rigas Oikonomou, and Andrew Scott The impact of debt levels and debt maturity on inflation The Economic Journal, 123(566):F164– F192, 2013 Elisa Faraglia, Albert Marcet, and Andrew Scott Modelling Long Bonds-The Case of Optimal Fiscal Policy Mimeo, 2014 Daryna Grechyna Debt and Deficit Fluctuations in a Time-Consistent Setup Available at SSRN 1997563, 2013 Robin Greenwood, Samuel G Hanson, Joshua S Rudolph, and Lawrence H Summers Government Debt Management at the Zero Lower Bound Mimeo, 2014 George J Hall and Thomas J Sargent Interest Rate Risk and Other Determinants of Post-WWII US Government Debt/GDP Dynamics American Economic Journal: Macroeconomics, 3(3):192–214, 2011 International Monetary Fund Balancing Fiscal Risks Fiscal Monitor, April 2011, 2012 Kenneth L Judd Numerical Methods in Economics MIT press, 1998 Kenneth L Judd Existence, uniqueness, and computational theory for time consistent equilibria: A hyperbolic discounting example mimeo, Stanford University, 2004 Paul Klein and Jose-Victor Rios-Rull Time-consistent optimal fiscal policy* International Economic Review, 44(4):1217–1245, 2003 Paul Klein, Per Krusell, and Jose-Victor Rios-Rull Time-consistent public policy The Review of Economic Studies, 75(3):789–808, 2008 Per Krusell, Fernando Martin, and José-Víctor Ríos-Rull Mimeo, 2006 Time-consistent debt Ricardo Lagos and Randall Wright A Unified Framework for Monetary Theory and Policy Analysis Journal of Political Economy, 113(3):pp 463–484, 2005 Eric Leeper and Campbell Leith Inflation Through the Lens of the Fiscal Theory Forthcoming in the Handbook of Macroeconomics, volume 2, 2017 Eric Leeper, Campbell Leith, and Ding Liu Optimal Time-Consistent Monetary, Fiscal and Debt Maturity Policy Mimeo, 2015 Eric M Leeper and Xuan Zhou Inflation’s Role in Optimal Monetary-Fiscal Policy NBER Working Paper No 19686, 2013 219 REFERENCES Campbell Leith and Ding Liu The Inflation Bias Under Calvo and Rotemberg Pricing University of Glasgow Discussion Paper No 2014-6, 2014 Campbell Leith and Simon Wren-Lewis Fiscal Sustainability in a New Keynesian Model Journal of Money, Credit and Banking, 45(8):1477–1516, 2013 ISSN 15384616 Robert E Jr Lucas and Nancy L Stokey Optimal fiscal and monetary policy in an economy without capital Journal of monetary Economics, 12(1):55–93, 1983 Fernando M Martin A positive theory of government debt Review of economic Dynamics, 12(4):608–631, 2009 Fernando M Martin On the joint determination of fiscal and monetary policy Journal of Monetary Economics, 58(2):132–145, 2011 Fernando M Martin Government Policy in Monetary Economies International Economic Review, 54(1):185–217, 2013 Fernando M Martin The Value of Constraints on Discretionary Government Policy Mimeo, 2014 Dmitry Matveev Time-Consistent Management of a Liquidity Trap: Monetary and Fiscal policy with Debt Mimeo, 2014 Taisuke Nakata Optimal fiscal and monetary policy with occasionally binding zero bound constraints Federal Reserve Board Finance and Economics Discussion Series, 2013-40, 2013 Phuong V Ngo Optimal discretionary monetary policy in a micro-founded model with a zero lower bound on nominal interest rate Journal of Economic Dynamics and Control, 45(0):44 – 65, 2014 Stefan Niemann and Paul Pichler Optimal fiscal and monetary policies in the face of rare disasters European Economic Review, 55(1):75–92, 2011 Stefan Niemann, Paul Pichler, and Gerhard Sorger Public debt, discretionary policy, and inflation persistence Journal of Economic Dynamics and Control, 37(6):1097– 1109, 2013 Salvador Ortigueira Markov-perfect optimal taxation Review of Economic Dynamics, 9(1):153–178, 2006 Salvador Ortigueira, Joana Pereira, and Paul Pichler Markov-perfect optimal fiscal policy: the case of unbalanced budgets Mimeo, 2012 Julio J Rotemberg Sticky Prices in the United States Journal of Political Economy, 90 (6):1187–1211, 1982 doi: 10.2307/1830944 URL http://www.jstor.org/stable/ 1830944 Stephanie Schmitt-Grohe and Martin Uribe Optimal fiscal and monetary policy under sticky prices Journal of Economic Theory, 114(2):198–230, 2004a 220 REFERENCES Stephanie Schmitt-Grohe and Martin Uribe Optimal fiscal and monetary policy under imperfect competition Journal of Macroeconomics, 26(2):183–209, 2004b Christopher A Sims Paper money The American Economic Review, 103(2):563–584, 2013 Henry E Siu Optimal fiscal and monetary policy with sticky prices Journal of Monetary Economics, 51(3):575–607, 2004 Willem Van Zandweghe and Alexander Wolman Discretionary monetary policy in the Calvo model Federal Reserve Bank of Richmond Working Paper Series, WP11-03, 2011 Michael Woodford Fiscal Requirements for Price Stability Journal of Money, Credit and Banking, 33(3):669–728, 2001 221