Debt Maturity Management, Monetary and Fiscal Policy Interactions


 Jacob Porter
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1 Debt Maturity Management, Monetary and Fiscal Policy Interactions Hao Jin April 22, 23 Abstract This paper examines the interactions of debt maturity management, monetary and fiscal policy in a DSGE model. When the economy reaches either the zero lower bound of nominal interest rates or a fiscal limit in which fiscal policy becomes exogenous, inflation must be used to finance government liabilities which include both budget deficits and capital gains/losses on government debt. Fiscal policy determines the budget deficits, monetary policy and debt maturities jointly control the capital gains/losses, and then monetary policy allocates inflation across time. More aggressive monetary policy with longterm debt buffers more inflation into the future at higher rollover cost. Longer average debt maturities mitigates this cost because a smaller proportion of debt is rolled over each period. On the other hand, longterm debt amplifies the impacts of monetary policy shocks on bond prices so that a contractionary monetary policy could generate either a current inflation or deflation depending on the outstanding debt maturity structure. Lengthening the average maturities has real effects which raises current inflation and output. Keywords: Debt Maturity Management, Monetary and Fiscal Policy Interactions, Longterm Bond Prices JEL Classification Numbers: E62, E63, E43, H6 Preliminary Department of Economics, Indiana University, Bloomington.
2 Introduction Recently, much attention is drawn to the public debt maturity structure as governments intervene the economy by altering the compositions of public debt. Figure documents that outstanding government debt maturities vary significantly both across countries and over time. Despite its empirical importance, debt maturity structure has been largely neglected in the monetary and fiscal policy analysis framework, which makes most of the existing literature silent on the policy implications of debt maturity management and its interactions with monetary and fiscal policy. 6 4 years UK US Japan Italy Germany France Canada Figure : Average Term to Maturity of Outstanding Government Debt in G7 countries An important question then is how does the maturity structure of government debt affect the conduct of monetary and fiscal policy? Moreover, does debt maturity management has real effect on the economy? Earlier works on monetary and fiscal policy interactions abstract from the maturity structure by assuming the only debt instrument government can issue is oneperiod bond, such as Leeper (99) and Woodford (996) among others. They show that monetary and fiscal policy interacts with each other in a very generic way. While the monetary policy targets inflation only if fiscal policy adjusts to stabilize debt, which is called Active Monetary/Passive Fiscal or AM/PF regime in Leeper s (99) term, a different policy mix can achieve the same goal, called Passive Monetary/Active Fiscal or PM/AF regime. In the latter regime, fiscal policy controls inflation and monetary policy stabilizes debt. Cochrane (2) extends the framework incorporating longterm debt and studies the policy 2
3 implications of maturity structure in PM/AF regime. Eusepi and Preston (22) examine effects of altering debt level and maturity in AM/PF regime under learning environment. This paper takes a comprehensive view of debt maturity structure, monetary and fiscal policy interactions and shed some new light on the consequences of debt maturity management. The main findings of this paper are the following: First, the debt maturities do not matter for determinacy as long as there is no real risk premium on longterm debt. In other words, the determinacy conditions with longterm debt collapses to the ones with oneperiod debt. More specifically, for the monetary policy to control inflation, the fiscal authority must adjust surplus at a speed greater than the oneperiod real interest rate no matter what debt maturities are. Otherwise, the monetary authority must allow inflation rises to prevent debt from exploding. Second, when fiscal instruments are unwilling or unable to stabilize debt, inflation must rise to make the debt level sustainable. This scenario is plausible particularly after the recent financial crisis when many advanced countries face the peak of the laffer curve or for political and economic reasons not willing to enforce austerity policy. Under such a condition, fiscal policy determines budget deficits that need to be financed, meanwhile the monetary policy and average debt maturities pin down the capital gains/losses. The sum of deficits and capital gains/losses is the total government liabilities. Then monetary policy allocates inflation across time to bring the total liabilities back to its steady state level. More aggressive monetary policy spreads more inflation into the future at the cost of higher present value of rollover cost. Longer average debt maturities reduces this cost because a smaller proportion of debt is rolled over each period, hence realized rollover cost is lower. Sticky price allows more inflation to be postponed into the future. Third, the impacts of monetary policy shocks on bond prices are amplified by longer average maturities since longer term debt is more sensitive to monetary policy shocks. A contractionary monetary policy has two channels to affect the current inflation in a sticky price model. One channel is that it increases the real interest rates and discounted future fiscal surpluses more heavily, therefore reduces the fiscal backing of government debt. The other channel is that it also lowers the bond prices, hence devalues the government outstanding debt. The total effect on the current price level depends on the average 3
4 maturities. When the average maturities is sufficiently long, the bond prices channel dominates and leads to a current deflation. This is because when taxes are exogenous, Ricardian equivalence breaks down and households view debt holding as their net wealth so that a reduction in debt value generates a negative wealth effect driving down aggregate demand and price level. Finally, a lengthening in debt maturities has no real impact in AM/PF regime since Ricardian equivalence holds and debt management is irrelevant for agent s decision making. On the other hand, lengthening in debt maturities raises current inflation in PM/AF regime since longer term debt portfolio generates higher coupon payments to the bond holder as net wealth. Households feel wealthier and want to convert these bond holdings into consumption until this wealth effect disappears. 2 Simple Analytic: Constant Endowment Economy 2. A constant endowment Economy Suppose there is a simple constant endowment cashless economy. A representative household which is endowed each period with a constant quantity of nonstorable goods, y, maximize lifetime utility subject to a budget constraint. s.to : c t + P M,t B M,t max : E β t u(c t ), (2.) t= + τ t = y + ( + ρp M,t ) B M,t. (2.2) The agent consumes c t and invests in riskfree government bonds, plus he has to pay a lumpsum tax τ t every period. is the current price level. There is a general portfolio of government bond, B M,t, in nonzero net supply with price P M,t. The household clears the bond holding each period. Following Woodford (2), the general bond portfolio is defined as a consol with a coupon decay factor ρ. The debt issued at time t is assumed to pay ρ j dollars j + periods in the future. When ρ is high, it means the coupon payments decay slowly and the cash flows are more spread out into the far future. On the other hand, when ρ is low, the coupon payments decay fast and the cash flows are concentrated in the near future. I use this specification to mimic the cash flow streams of bonds with 4
5 different maturities. Bonds with higher ρ represents longer maturities. Hence, by altering the value of ρ, we can model the maturity structure of outstanding government debt using only one parameter. In extreme case ρ =, the debt portfolio collapses to oneperiod debt and if ρ =, it becomes a consol. In later analysis, we are going to generalize the parameter ρ to be timevarying and examine the policy implications of debt maturity management. The firstordercondition with respect to B M,t is: β P M,t = E t ( + ρp M,t+ ), (2.3) π t+ And the Fisher equation that connects oneperiod nominal and real interest rates and expected inflation is: β = E t, (2.4) R t π t+ Combining the above two equations yields the noarbitrage condition: P M,t = E t + ρp M,t+ R t. (2.5) The government collects a lump sum tax τ t and government spending is assumed to be zero each period, so the government budget constraint is the following: P M,t B M,t + τ t = ( + ρp M,t ) B M,t. (2.6) 2.2 Policy Rules The monetary authority is assumed to set a Taylor type feedback rule in which nominal interest responds to the contemporaneous inflation: R t = ϕ π π t + u R t, (2.7) 5
6 Where u R t is an AR() process with stochastic disturbance ε R t. u R t = ρ R u R t + εr t. The fiscal authority is assumed to collect lumpsum tax in response to the outstanding real government debt level: τ t = ϕ b ( + ρp M,t ) B M,t + u τ t, (2.8) where u τ t is an AR() process with stochastic disturbance ε τ t. u τ t = ρ τ u τ t + ετ t. We loglinearize the model around steady state reduce the model into a dynamic system in (π t, b M,t, P M,t ). Write in matrix form as the following: Γ EX t+ = Γ X t + Φ Z t+ + Φ Z t, (2.9) where the vector of endogenous variables X t (ˆπ t, ˆb M,t, ˆP M,t ), the vector of exogenous variables Z t (û R t, û τ t, û ρ t ). ˆx denotes deviation from the deterministic steady state. Proposition. The following conditions are necessary and sufficient for a unique bounded equilibrium: AM/PF regime: ϕ π, ϕ b > β, PM/AF regime: ϕ π <, ϕ b β. Detailed Proof can be found in the Appendix A. These results are identical to the Leeper(99) conditions in the sense that the determinacy of the solution depends on the policy parameters ϕ π and ϕ b and the deep behavior parameter β. Although we relax the assumption of government issuing only oneperiod debt, maturity structure plays no role in the determinant conditions. The speed of fiscal responsiveness to stabilize debt depends only on oneperiod real interest rate. And because there is no uncertainty on the real interest rate, the real interest rates of short and longterm bonds are the same. 6
7 Nevertheless, the total amount of taxes needed to stabilize debt are different under various maturity structures since different coupon payment schedules affect the total value of outstanding debt. Leeper(99) provides an intuitive interpretation for these two regimes. In the AM/PF regime, central banks fight inflation aggressively by setting nominal interest rate response to inflation more than onetoone. This anchors household s inflation expectation and brings inflation back to its target level. The fiscal policy then needs to adjust lumpsum tax in order to at least pay back real interest rate, preventing explosive debt path. Introducing longterm debt does not change the speed of fiscal adjustment, but needs a higher level of total amount of taxes to be collected. In the PM/AF regime, fiscal policy is unable to raise sufficient taxes to service debt interest, so that the monetary policy must allow inflation above target level to accommodate fiscal policy. Higher inflation revalues the outstanding debt, which acts as an inflation tax to help pay back debt interests. In the AM/PF regime, inflation is uniquely determined by the monetary policy, altering current debt maturities does not change the expected future policies hence does not affect the equilibria. This is analog to the Ricardian Equivalence which states that a tax change has no effect on household s consumption streams since household would expect future taxes to adjust so that the present value of taxes is the same. Altering debt maturities would incur expected future taxes adjustment so that it has no real impact. However, if the monetary policy cannot anchor the inflation on its target because of the crisis, then Ricardian Equivalence breaks down and debt maturity management has real impacts Fixed Maturity Structure We have the following Fisher relation by taking FOC with respect to B S,t, P S,t = βe t ( ), (2.) π t+ And the FOC with respect to B M,t implies an arbitrage condition given by P M,t = E t P S,t ( + ρp M,t+ ), (2.) 7
8 Iterate on (2.) and impose transversality condition to obtain P M,t = β (βρ) j E t j= i= j ( ), (2.2) π t+i+ The price of multiperiod bond equals the present value of all the future coupon payments discounted by the nominal interest rates. Combining Fisher equation (2.) with this bond pricing equation yields P M,t = E t j= ρ j j ( ), (2.3) R t+j Iterate on the GBC and impose the no arbitrage condition and transversality condition to yield the intertemporal equilibrium condition ( + ρp M,t ) B M,t = i= β j E t τ t+j. (2.4) Consider the case that monetary policy has full control over inflation, both current and future, by setting the shortterm nominal interest rate path. In this case, fiscal policy has to adjust government surpluses to accommodate interest payment changes, so that government budget constraint is separated from the dynamic system and the economy exhibits Ricardian Equivalence or AM/PF. Suppose there is a surprise shock to the current nominal interest rate that affect the current inflation level. Compare to the economy with only oneperiod debt, longterm debt enables the monetary policy to absorb part of the current inflation shift and spread it to the future. Hence, for countries with longer debt maturity, it is less likely to see large variations in inflations. If an economy hits a fiscal limit or the zerolowerbound of interest rate,i.e. operates in the PM/AF regime, then monetary policy loses control of the current inflation, and fiscal shocks have impacts on the current price level through the intertemporal equilibrium condition (2.4). Ricardian Equivalence breaks down and a change of the present value of fiscal backing of government debt imposes restrictions on the inflation path. However, central banks still face the tradeoff between current inflation and future inflation in response to a surplus shock as j= in the AM/PF regime. As a negative tax shock hits the economy, the intertemporal equilibrium condition (2.4) requires a jump in the current price to equate, but the longterm debt could act as a cushion by decreasing its nominal market value. The decline in market value represents future inflation. 8
9 Another policy experiment is that the central bank announces a shift in inflation target today. For example, the Japanese central bank increases its inflation target in February 23. This announcement devalues the outstanding debt portfolio since households will discount future coupon payments more heavily. Without any fiscal policy adjustment, households feel less wealthier and consume less, this in turn reduces aggregate demand and drives down the current price level until this wealth effect disappears. To prevent current deflation, fiscal policy must accommodate by committing to cut taxes in the future so that monetary policy controls current inflation as well. Longer average maturity amplifies this effect since it is more sensitive to the changes in future expected inflations. Until now, we have fixed the average maturity of government debt portfolio to understand the mechanisms of maturity structure to affect the conduct of monetary and fiscal policy. Next, we are going to allow the average maturity to be timevarying, then we can to analyze the impacts of debt maturity management as a policy tool Timevarying Debt Maturity The arbitrage condition is now P M,t = E t P S,t ( + ρ t P M,t+ ), (2.5) Iterate forward and impose transversality condition to obtain the bond pricing equation β P M,t = E t ( + π t+ = E t R t ( + i= β i i i= j= i j= ρ t+j π t+j+ ) (2.6) ρ t+j R t+j ), (2.7) And we can obtain the intertemporal equilibrium condition by iterating the GBC and imposing transversality condition. ( + ρ t P M,t ) B M,t = E t β i τ t+i. (2.8) i= Holding fixed the taxes and shortterm interest rate streams, under an exogenous debt management rule, an extension of current or future average maturities, ρ t+j, will raise the bond price. A higher bond 9
10 price would then raise the current inflation in order to equate the intertemporal equilibrium condition (2.8). This model prediction undermines the effectiveness of Federal Reserve s recent quantitative easing operation that sells shortterm Treasuries to purchase longterm Treasuries. Such an operation shortens the average maturities hold by the households and generates a negative wealth effect which is contractionary to the economy. Utilizing the debt management as an additional policy tool gives the monetary authority more flexibility in controlling the inflation path. While always facing a current and future inflation tradeoff in the presence of longterm debt, the central bank can manage a sequence of lower expected interest rate, which will in turn result a higher inflation today holding other exogenous process fixed. However, even when the shortterm nominal interest rate reaches its zero lower bound and can not be further reduced, active debt management can still transfer future inflation into current by extending the average debt maturities. This can be seen clearly from equation (2.6) and (2.8). Although a pegged shortterm nominal interest rate sequence holds constant the expected inflation, an extension in average maturities ρ t+j could lead to a raise in the current inflation. 3 Debt Management in New Keynesian Model With the basic transforming mechanism understood from a constant endowment economy model, we now move to a basic New Keynesian sticky price model. Under the New Keyesian sticky price model, monetary policy has real effect. Under sticky price setting, inflation rises with lags so that future inflation will play a more significant role in terms of deficit financing. 3. The Model 3.. Households An infinitely lived representative household derives utility from consumption, c t, and real money balance, M t. The household derives disutility from hours worked. Specifically, the household chooses sequences of consumption, real money balance, hours worked and government debt
11 to maximizes E β t [ c t σ t= σ N t +ν + ν + (M t/ ) θ θ ], (3.) where c t is a consumption index given by ( c t c t (i) ϵ di ) ϵ ϵ. with c t (i) representing the quantity of good i consumed by the household in period t. < β < is the discount factor, σ > is the household s risk aversion, ν > is the inverse of the Frisch labor elasticity, and θ > determines the interest elasticity of real money demand. The general portfolio of government debt, B M,t, in nonzero net supply with price P M,t is the same as in the constant endowment model The household clears the bond holding each period. The price of shortterm nominal bonds satisfies P S,t = Rt, where R t is the gross nominal interest rate. Following Woodford (2), longterm debt issued at time t is assumed to pay ρ j t dollars j + periods in the future, for j and ρ t. The average duration of this longterm bond is ( βρ t ). The household s flow budget constraint is given by c t + M t B S,t B M,t + P S,t + P M,t + τ t = y t + M t + B S,t + ( + ρ t P M,t ) B M,t. (3.2) Taking prices and B S, >, B M, >, and M as given. The household pays lumpsum taxes, τ t, each period Firms A continuum of monopolistically competitive firms produce goods using labor. Production of good j is given by the following production function. Y t (j) = N t (j), (3.3)
12 Following Calvo (983), each firm may reset its price only with probability ω in any given period Fiscal Policy The government issues nominal bonds, one period bond B S,t, and multi periods bond B M,t, prints money M t and collects lumpsum taxes τ t. Government spending is assumed to be zero in all periods. The government s flow budget constraint is M t B M,t + P M,t + τ t = M t + ( + ρ t P M,t ) B M,t, (3.4) The fiscal authority is assumed to collect lumpsum tax in response to the outstanding real government liability level: log τ t = log τ + ϕ b log b M,t P M,t + log u τ t, (3.5) where τ is the target tax level, and u τ t is an AR() process with stochastic disturbance ε τ t. u τ t = δ τ u τ t + ετ t Monetary Policy The monetary authority is assumed to set the nominal interest in response to the contemporaneous inflation: log R t = log R + ϕ π log π t + log u R t, (3.6) Where log R is the target interest rate level, and u R t is an AR() process with stochastic disturbance ε R t. u R t = δ R u R t + εr t debt management policy To ensure that the coupon payment parameter ρ t always be bounded between zero and one, we assume ρ t is a logistic function of a latent policy variable h t. The 2
13 functional form is: ρ t = + e h t, (3.7) Notice that ρ t is a monotone transformation of the latent policy variable h t and always lies between zero and one, so we can specify rules on h t and shock on it to examine the effects of altering debt compositions have on the economy. Debt Maturity management follows an exogenous meanreverting rule:. log h t = log h + log ε h t, (3.8) (3.9) where h is the target maturity structure of government debt, and ε h t is a stochastic disturbance. The linearized system of equations is derived in the Appendix. 3.2 Calibration and impulse responses The model is calibrated to a annually frequency to examine the impacts of policy shocks over a horizon of 4 years. All the parameter values are in the range of standard values in the literature. The real interest rate is set to percent(β =.99). The preference over consumption is assumed to be logarithmic, so σ =. We set the inverse of the Frisch labor supply elasticity ν =. The parameter θ, which determines the interest elasticity of real money balance, is set to 2.6. Two thirds of firms cannot reset their price each period, so ω =.66. The monetary,fiscal and maturity shocks are AR() process with the persistence parameter set to.8. For simplicity, the steady state inflation is set to zero. The steady state debttogdp ratio(s b ) is set to.4 and we consider a cashless limit version of the model. The lumpsum tax rate(s τ ) is set to.2 in steady state. The implied steady state nominal interest rate R =.. 3
14 To study the policy implications when the economy reaches zero lower bound and fiscal limit, we only allow weak response of shortterm nominal interest rate to inflation and exogenous lumpsum taxation. Particularly, we set the policy parameters ϕ π =,.3 and.8 respectively and ϕ b =. These policy specifications characterize PM/AF equilibrium. Then we vary the outstanding debt average maturities to year, 5 years and years. The model is solved using Sims (22) gensys algorithm A tax cut Two central equations in this system are the bond pricing equation (3.) and Intertemporal Equilibrium Condition (3.). Detailed derivation is given in the Appendix. ˆP M,t = E t (βρ) i [ ˆR t+i + βρˆρ t+i ], (3.) i= ( β ) β i+ E tˆτ t+i β i+ ( ρ)e t ˆPM,t+i + βρ β i+ E t ˆρ t+i i= } {{ } PV of Taxes ( ˆb M,t βρˆρ t ) } {{ } = i= } {{ } PV of RollOver Cost + ˆπ }{{} t + Surprise Revaluation i= i= } {{ } PV of Coupon Payments β i+ˆπ t+i+ } {{ } PV of Future Inflation, = (3.) The IEC (3.) must hold in equilibrium. In the case of exogenous taxes and fixed average maturities, a tax cut today and the capital gains or losses induced by the tax cut must be financed by either current inflation or future inflation. How the deficit financing is allocated across time depends on both the monetary policy and the maturity structure. The following Table shows the impacts of a tax cut on inflation under different policy combinations. The left panel illustrates the deficits financing decompositions under flexible prices as benchmark and the right panel is for sticky price case. The mechanism for longterm debt to buffer fiscal shocks is to use longterm bond prices as a cushion. When fiscal shock hits, the bondprices will absorb part 4
15 Table : Deficit Impacts Decomposition Flexible Price Sticky Price ˆπ t PV( ˆP M ) PV(ˆπ future ) P V (ˆπ future) ˆπ t ˆπ t PV( ˆP M ) PV(ˆπ future ) P V (ˆπ future) ˆπ t ϕ π = year % 6.8% 39.2% year % 6.8% 39.2%.645 year % 6.8% 39.2%.645 ϕ π =.3 year % 4.6% 4.6% % 4.6% 74.8%.2 5 year 76.4% 8.3% 32.9% % .8% 58.5%.2 year 73.% 3.7% 3.6% % 5.4% 55.7%.2 ϕ π =.8 year % % 362.9% % % 378.% year 36.5% 69.% 32.5% % 78.6% 45.9% 4.46 year 28.6% 32.7% 4.% % 37.9% 2.6% 4.46 of the shock and release it in the future by changing the prices. The part is not absorbed becomes current inflation and the absorbed part becomes future inflation. Hence, the ability to adjust bond price is necessary for the cushion effect to work. More aggressive monetary policy with longterm debt enhances the cushion effect, but leads to higher rollover cost. Longer average debt maturities reduces this cost since a smaller proportion of debt is rolled over each period. When the central bank pegs nominal interest rate, as ϕ π = in the top of Table, then to satisfy the IEC (3.) with flexible price, the current price has to adjust upward to accommodate this negative tax shock. Intuitively, a tax cut today produces a positive wealth effect, which stimulates aggregate demand and pushes up the current price level. When price is sticky, future inflation will increase as well to reduce the future real interest rate and revalue the government s present value of deficits. Debt maturity structure does not play a role here since the monetary policy fixes the bond prices by pegging interest rates. The middle part of the table shows the decomposition when allowing nominal interest rate to weakly respond to the inflation level. Since a tax cut causes deficit which then raises inflation, a Taylor type monetary policy to raise nominal interest rate imposes higher rollover cost on the government. Longer maturity mitigates this problem by rolling over debt less frequently. For example, with average debt maturities of year, the government will suffer 4.6% of additional rollover cost at present value sense, but with average maturities of 5 year, the rollover cost decreases to.8%. At the same time, longterm bond cushions the impacts of deficit shocks on current inflation as reduced from 5
16 % to 76.4% as average maturities goes from year to 5 year. The total inflation is determined by the fiscal shock and the magnitude of rollover cost which depends both on the monetary policy and the debt maturity structure. The allocation of total inflation between current and future inflation is controlled by the monetary policy. As shown in the fourth column, the ratio of P V (ˆπ future) π t is constant under such monetary policy configuration. This is because monetary policy controls the evolution path of all future inflation. We see a similar pattern of inflation tradeoffs when monetary policy is more aggressive as ϕ π =.8. The sticky price setting enables even more cushion than flexible price environment as a further reduction in current inflation happened. The reason is that when inflation is sluggish, full financing through current inflation is no longer available. There has to be some part of the inflation to be postponed into the future. Therefore, all the decomposed percentages in P V (ˆπ future ) under sticky prices contains two effects. The longterm debt cushion effect and the sticky price delaying effect. In sum, how the tax cut is financed depends on the interactions of debt maturity, monetary and fiscal policy. Holding fixed the tax stream, more aggressive monetary policy generates larger capital losses which needs to be financed by current and future inflation. Longterm debt reduces the capital losses by rolling over debt less frequently. Therefore the magnitude of the rollover cost is determined by both monetary policy and debt maturity structure. A more aggressive monetary policy spreads out more current inflation into the future at higher rollover cost, while longer maturities mitigate this problem by rolling over debt less frequently. Sticky price setting builds up another channel that current inflation can be postponed into the future An extension of average debt maturities Figure 2 shows the effects of an operation of the government to extend the average maturities of outstanding government debt. For example, a lengthening of average maturities could be that the government sells shortterm and to purchase longterm bond. This action raises the bond average maturity parameter ρ at time t. Higher ρ also represents a higher debt coupon payment, so the government s liabilities rise. Initially, the real government debt is not backed up by enough expected tax revenues, so households will convert their liability holdings 6
17 2 Inflation 2 4 Government Real Liabilities Output Nominal Interest Rate 2 4 Bond Price Labor Hours Real interest rate 2 4 Lump sum Tax 2 4 Weighted Average Maturities.5 year debt year debt 2 4 Figure 2: responses to a shock that lengthens the average outstanding debt maturities into consumption, which increases the aggregate demand of goods and then pushes up the current price level, which dilutes the government liabilities and restores the equilibrium. Monetary authority holds the nominal interest rate fixed which in turn leads real interest rate to fall. The bond pricing relation (3.) states that a longer average debt maturities with constant shortterm interest rate, without expected future maturities changes, increases the bond prices. Lower real interest rates distort the saving incentives for households, the wealth effect in this case dominates the substitution effect, so that households work more to consume. This result opposes the recent quantitative easing operations by many central banks that sell shortterm and purchase longterm government securities. This operation is likely to generate negative wealth effect when the fiscal policy is exogenous because households will take holdings of government securities as net wealth. If the central bank reduces the average maturities of public holdings, negative wealth effect will cause a recession in contrast of an expansion A contractionary monetary policy Next, the effects of a contractionary monetary policy are summarized in Figure 3. The most striking result is that there could be either a current inflation 7
18 or a current deflation depending on the average debt maturities in the economy. This is because a rise in nominal interest rate has two effects on the intertemporal equilibrium condition (3.). It increases the real interest rate due to sticky price which means the government tax revenues are discounted more heavily. This will require a current inflation to devalue the outstanding government debt if there is only oneperiod debt. With multiperiod debt, a higher nominal interest rate also leads to decrease in the bond prices, and the longer the maturities, the more sensitive is the price to the nominal interest rate. When the average maturities are sufficiently long, the decrease in bond price may exceeds the decrease in present value of government tax revenues and price level has to go down to reestablish the equilibrium. Empirically, one will expect countries with different maturity structures to have different inflation response to monetary policy under PM/AF regime. Assuming only oneperiod debt, the results are similar to Kim (23) study of monetary policy shock under PM/AF regime. Output and labor supply decline since real interest rate decreases. Inflation Nominal Interest Rate 2 Real interest rate Government Real Liabilities Output Bond Price Labor Hours Lump sum Tax Weighted Average Maturities year debt year debt Figure 3: responses to a positive shock to the nominal interest rate 8
19 4 Conclusion This paper introduces multiperiod debt into a standard New Keyesian sticky price model to study the interactions among debt maturity management, monetary and fiscal policy. We find that first, the debt maturities do not matter for determinacy as long as there is no real risk premium associated with longterm debt. In other words, the determinacy conditions with longterm debt collapses to the ones with oneperiod debt. Second, Fiscal policy determines the total deficits, monetary policy and debt maturities jointly influence the capital gains/losses, then monetary policy allocates inflation across time. More aggressive monetary policy with longterm debt cushions more inflation into the future at higher rollover cost. Longer average debt maturities mitigates this cost because a smaller proportion of debt is rolled over each period. Third, longer average debt maturities amplifies the impacts of monetary policy shocks on bond prices, which would generate either a current inflation or deflation depending on the debt maturity structure. Finally, Lengthening the average maturities has real effects which raises current inflation and output. 9
20 A Proof of Proposition The equilibrium is completely characterized by three equations. The first one is the noarbitrage condition (2.5). The second equation comes from combining fisher equation (2.4) and the monetary policy (2.7) and the last equation comes from imposing fiscal policy (2.8) on GBC (2.6). The equilibrium dynamic system is summarized below: P M,t = E t + ρp M,t+ R t, (A.) E t π t+ = ϕ π βπ t + βu R t, P M,t b M,t + ϕ b ( + ρp M,t )b M,t + u τ t = ϕ b ( + ρp M,t )b M,t π t. (A.2) (A.3) where b M,t B M,t denotes real debt. Linearize the above system and write in matrix form: } {{ λ } Γ E t ˆπ t+ ˆbM,t+ ˆP M,t+ + } {{ } Φ ûr t+ û τ t+ ϕ π ˆπ t = λ 2 λ 2 ˆbM,t ˆP M,t, } {{ } Γ + ûr t û τ t } {{ } Φ (A.4) where λ = βρ π, λ 2 = ϕ b β. All the variables without time index refer to the steady state values. 2
21 The above system has two forecast errors, η π t+ and ηp M t+, which are defined as: η π t+ = π t+ E t π t+, (A.5) η P M t+ = P M,t+ E t P M,t+. (A.6) so that two of eigenvalues of the transition matrix Γ Γ should be greater than one in absolute value in order for a unique equilibrium to exist. The transition matrix is given below: Γ Γ = ϕ π ϕ π λ λ λ 2 λ 2 = ϕπ λ λ 2 ϕ π λ λ, λ λ (A.7) The eigenvalues of the transition matrix can be directly read off from the diagonal elements. The three eigenvalues are: (ϕ π, λ 2, λ ). Notice that the discount factor β and the average maturity parameter ρ are bounded between and, and the steady state inflation is greater than, so that the eigenvalue λ > always. Then for a unique equilibrium to exist, we need one of the remaining roots to be outside unit circle and the other one to be inside. Plug in the value of λ and λ 2, we can write the determinacy conditions as: ϕ π, ϕ b β < or ϕ π <, ϕ b β. Ignoring the cases of negative response coefficients, we can rewrite the conditions as in Proposition. For exogenous timevarying debt management rule ρ t = ϕ ρ π t + u ρ t, t, plug this rule into the dynamic equilibrium system, the coefficient matrix Γ is unchanged, the eigenvalues of this system are unchanged, therefore, the determinacy conditions are also unchanged. 2
22 B LogLinearized System of Equations for the New Keynesian Model Let ˆx t log x t log x represents the log deviation of a variable from its steady state value x. For simplicity, we assume the steady state inflation is zero. ARC: ĉ t ŷ t =, (B.) Production: ŷ t ˆn t =, (B.2) No arbitrage: βρe t ˆρ t + βρe t ˆPM,t+ = ˆP M,t + ˆR t, (B.3) Money demand: β ˆR t σ ( β)ĉ t + θ( β) ˆm t =, (B.4) NK Phillips Curve: βe tˆπ t+ = ˆπ t + ( ωβ)( ω) ( νˆn t ω σ ĉt), (B.5) GBC: ˆbM,t + βρˆρ t = sm β P M b M ˆµ t + ( β )ˆτ t + βˆb M,t + ( ρ)β ˆP M,t + ˆπ t, (B.6) where ˆµ t = ˆm t ˆm t + ˆπ t is the seigniorage revenue. 22
23 Fisher Equation: σ E tĉ t+ + E tˆπ t+ = ˆR t + σ ĉt, (B.7) Fiscal Policy: ˆτ t = ϕ b (ˆb M,t + ˆP M,t ) + û τ t, (B.8) Fiscal Policy shock: û τ t = ρ τ û τ t + ε τ t, (B.9) Monetary Policy: ˆR t = ϕ π ˆπ t + û R t, (B.) Monetary Policy shock: û R t = ρ R û R t + ε R t, (B.) latent policy variable: ( + e h )ˆρ t = he h ĥ t, (B.2) exogenous debt maturity management rule: ĥ t = û h t, (B.3) 23
24 C Derivation of Bond Pricing Equation and IEC for the New Keynesian Model The noarbitrage condition comes from combining the FOCs of c t, B s,t and B M,t : P M,t = E t ( + ρ t+ P M,t+ ) R t, (C.) Rewrite this noarbitrage condition as: P M,t = R t + E t ρ t+ R t P M,t+, (C.2) Iterating on P M,t and imposing transversality condition yields the bond pricing equation: P M,t = E t R t ( + i i= j= ρ t+j R t+j ). (C.3) Impose equilibrium noarbitrage condition and rewrite the GBC (3.4) as: M t + ( + ρ t P M,t )B M,t = M t + E t ( + ρ t+ P M,t+ ) R t B M,t + τ t, (C.4) adding and subtracting β( c t+ c t ) σ Mt + relation R t = βe t ( c t+ c t ) σ π t+ yields: and using the shortterm nominal interest rate noarbitrage M t + ( + ρ t P M,t )B M,t = M t + ( + ρ t+ P M,t+ )B M,t + β( c t+ c t ) σ + ( R t ) M t + τ t, (C.5) Iterate and impose transversality condition to get the IEC: M t + ( + ρ t P M,t )B M,t = i= [ β i E t η i τ t+i + ( ) M ] t+i. (C.6) R t+i +i where η i = ( c t+i+ c t ) σ for i and η = 24
25 References [] Cochrane, J.H. (2): Long Term Debt and Optimal Policy in the Fiscal Theory of the Price Level, Econometrica. [2] Cochrane, J.H. (2): Understanding Policy in the Great Recession: Some Unpleasant Fiscal Arithmetic, European Economic Review. [3] Davig, T., Leeper, E. M. and Walker, T. B. (2): Inflation and the Fiscal Limit, European Economic Review. [4] Eggertsson, G., and Woodford, M. (23): The Zero Bound on Interest Rates and Optimal Monetary Policy, Brookings Papers on Economic Activity. [5] Eusepi, S., and Preston, B. (2): Learning the Fiscal Theory of the Price Level: Some Consequences of Debt Management Policy, Journal of the Japanese and International Economies [6] Eusepi, S., and Preston, B. (22): Fiscal Foundations of Inflation: Imperfect Knowledge, Working Paper, Monash University [7] Galí, J. (28): Monetary Policy, Inflation, and the Business Cycle. Princeton University Press, Princeton. [8] Kim, S. (23): Structural Shocks and the Fiscal Theory of the Price Level in the Sticky Peice Model, Macroeconomic Dynamics. [9] Leeper, E.M. (99): Equilibria under active and passive monetary and fiscal policies, Journal of Monetary Economics. [] Leeper, E. M. and Walker, T.B. (2): Perceptions and Misperceptions of Fiscal Inflation, Manuscript, Indiana University. [] Sims, C.A. (22): Solving Linear Rational Expectations Models, Computational Economics. [2] Woodford, M. (996): Control of Public Debt: A Requirement for Price Stability?, NBER working paper
26 [3] Woodford, M. (2): Fiscal Requirement for Price Stability, Journal of Money, Credit, and Banking. 26
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