Optimal Money and DebtManagement:

Friedman and Barro Revisited

Matthew Canzoneri∗ Robert Cumby† Behzad Diba‡

April 10, 2013

Abstract

How do the fundamental insights of Milton Friedman[16] and RobertBarro[6] fare when we recognize the fact that government bonds pro-vide liquidity services? To answer this question, we extend a standardcash and credit good model by assuming that bonds may be used ascollateral in the purchase of some kinds of consumption goods. In theextended model, the government cannot use its liabilities to financedeficits without affecting the amount of liquidity available to supportconsumption. The insights of Friedman and Barro survive largelyintact if the government has an additional debt instrument that is Ri-cardian – in the sense that its outstanding stock has no direct effecton consumption decisions. In particular, an extended Friedman Ruleis optimal when prices are flexible, and when prices are sticky, theRamsey planner will choose to smooth short run tax distortions atthe expense of longer run consumption (when adjusting to, say, an in-crease in government spending). Absent a Ricardian debt instrument,the planner’s options are much more limited. It would not appear thatgovernments currently have available to them such a debt instrument.

∗Georgetown University, canzonem@georgetown.edu†Georgetown University, cumbyr@georgetown.edu‡Georgetown University, dibab@georgetown.edu

1

1 Introduction

Milton Friedman[16] and Robert Barro[6] made seminal contributions toour u nderstanding of the optimal management of money and public debt.Friedman held that the cost of producing fiat money is negligible, and there-fore real money balances should be driven to satiation levels. This couldbe accomplished using the celebrated Friedman Rule: drive the interest ratethat measures the opportunity cost of holding money to zero. In a largelyforgotten part of his essay, Friedman noted that bonds also provide liquid-ity. However, Friedman had private debt in mind; so he missed a potentiallyimportant link between monetary and fiscal policy.

Barro held that short run tax distortions could be alleviated at the ex-pense of some long run consumption. In response to, say, a temporaryincrease in government spending, new liabilities should be issued to smooththe path of the tax hikes that would ultimately be needed; long run con-sumption would of course have to fall a little to service the higher level ofdebt. In stochastic versions of Barro’s world, debt and taxes follow unit rootprocesses as they move to efficiently absorb fiscal shocks.1 However, Barrodid not contemplate the possibility that government bonds might provide liq-uidity to the private sector, and that the paths of both money and debt mayaffect the path of equilibrium allocations. So he too missed a potentiallyimportant link between monetary and fiscal policy.

In this paper, we ask how the insights of Friedman and Barro fare whenwe assume that money and government bonds provide differing degrees ofliquidity to the private sector. While the Friedman Rule is optimal in manymodels, it is not optimal in all models.2 So, we begin with a commonlyused cash and credit good model in which the Friedman Rule is optimalunder flexible prices; adding price rigidities, public liabilities should be usedto smooth tax distortions, ala Barro. Then, we expand the model in anatural way to allow for the liquidity of government bonds.3 We do this by

1Barro’s original contribution did not include uncertainty and focused on real debt.The unit root implication of his tax smoothing result is highlighted in discussions ofstochastic general equilibrium models with price rigidities. See for example Albanesi[3].

2See Woodford.[28] and DeFiore and Teles.[14]3We are of course not the first to study the liquidity services of bonds. In addition to

Friedman, contributions to the literature include: Patinkin[24], , Bansal and Coleman[8],Holmstrom and Tirole[18], Calvo and Vegh[10], Hu and Kam[19], and Linnemann andSchabert[23]. Woodford[29], Aiyagari and McGratten[2] and Angeletos et al[4] study therole of bonds as collateral in real models with heterogenous agents.

2

assuming that bonds may be used as collateral in the purchase of some kindsof consumption goods.

In what follows, we will refer to a government debt instrument as being“Ricardian” if its outstanding (real) stock has no direct effect on consump-tion decisions. Money is clearly not a Ricardian debt instrument since itprovides liquidity that is necessary to purchase cash goods. In the stan-dard cash and credit good model, government bonds are Ricardian in thissense.4 But when we extend the model to let government bonds provide liq-uidity, then public debt is no longer Ricardian. Absent a Ricardian debtinstrument, the government cannot choose to finance a short run deficit (alaBarro) independently of its decision of how much liquidity to provide forconsumption. This basic fact will play a prominent role in our results.

If the government has a Ricardian debt instrument, then the Ramseyproblem for our expanded model is similar to that of the standard cash andcredit good model.5 An Extended Friedman Rule is optimal when prices areflexible; that is, optimality requires satiation in both money and governmentbonds, and this can be achieved by driving the interest rates that measure theopportunity costs of holding money and bonds to zero. Absent a Ricardiandebt instrument, the Extended Friedman Rule is optimal in the steady state,but the cash and collateral constraints may be binding in the transition toit.

When we introduce staggered price setting (thereby entering an environ-ment considered by Barro), the need for a Ricardian debt instrument becomeseven more apparent. Without such an instrument, the government wouldnot want to smooth short run tax distortions by running deficits; issuing newliabilities would distort consumption patterns. In this case, the Ramsey so-lution is stationary; the unit root characterization of optimal taxes and debtis lost.

In this paper, we characterize the Ricardian debt instrument that isneeded as a nominal government asset that can be used to finance a tem-porary deficit. We model these assets as government loans to the privatesector. We will discuss the possible interpretations of this in the conclusion.

A brief review of the literature will put our work in context. Chari, Chris-tiano, and Kehoe[11][12] – using a cash and credit goods framework – extend

4Note however that the standard model does not obey the “Ricardian Principle” unlessthe government is able to finance its spending with lump sum taxes.

5See Chari, Christiano and Kehoe[11] for a discussion of the standard model.

3

Friedman’s discussion to a stochastic general equilibrium model with nomi-nal government bonds and a distortionary tax on labor. In their model, theFriedman Rule characterizes monetary policy in the Ramsey solution. Theirbasic insight is that unanticipated inflation provides a non-distortionary taxon nominal assets, and these tax revenues can be used to limit fluctuationsin the distortionary wage tax. Optimal inflation is very volatile in theirmodel,6 but that does not matter since they assume prices are flexible. Morerecently, Benigno and Woodford[7] and Schmitt-Grohe and Uribe[26][27] addstaggered price setting to the mix. The inefficient price dispersion createdby staggered price setting leads to a fundamental inflation tradeoff for theRamsey planner: the price dispersion can be eliminated by holding the ag-gregate price level constant, but this conflicts with the use of unanticipatedinflation as a fiscal shock absorber and with implementation of the FriedmanRule. In numerical exercises, the tradeoff is generally resolved in favor ofprice stability, and the Friedman Rule is lost.

Finally, Correia, Nicolini, and Teles [13] reach a striking conclusion: witha sufficiently rich menu of taxes, nominal rigidities are irrelevant to the op-timal conduct of monetary policy. They consider a stochastic economywith cash goods and credit goods, monopolistically competitive firms, andnominal debt. Optimal tax rates are set on labor income, dividends, andconsumption. The authors show that the Ramsey solution for an economywith nominal rigidities is identical to that for an economy with flexible prices;the Friedman rule is optimal despite the nominal rigidities.

We reviewed this literature extensively in Canzoneri, Cumby and Diba[9].The model we used there was essentially the Correia-Nicolini-Teles model,minus the consumption tax;7 here, we call that model the Standard Model.We extend the Standard Model in a natural way to allow for the liquidityof government bonds; that is, we add a third good – a bond good – forwhich consumers hold bonds as collateral. We call the extended model theLiquid Bonds (or LB) Model.8 The LB Model comes in two forms: in the

6In a calibrated exercise, the authors show that the labor tax is quite stable, while thestandard deviation of annual inflation is 20 percent.

7Eliminating this tax breaks up Correia et al’s basic result, and restores the fundamen-tal inflation tradeoff described above. Our reasons for eliminating this tax were discussedat length in our handbook chapter.

8Unfortunately, neither the Standard Model nor the LB Model offers a deep theoryof liquidity demand. Such theories do exist. For example, Holmstrom and Tirole[18]show private provision of liquidity will be inefficiently low and that public debt can fill

4

Benchmark LB Model, the government is able to buy private sector bonds(which have no liquidity value); that is, it has available the Ricardian debtinstrument discussed above. In the Constrained LB Model, the governmentis not able to lend to the private sector.

Before we proceed to the analysis, we should perhaps discuss our assertionthat government bonds provide liquidity services. The basic premise shouldnot be controversial. U.S. Treasuries facilitate transactions in a number ofways: they serve as collateral in many financial markets, banks hold themto manage the liquidity of their portfolios, individuals hold them in moneymarket accounts that offer checking services, and importers and exporters usethem as transactions balances (since so much trade is invoiced in dollars).The empirical literature finds a liquidity premium on government debt, andmoreover that the premium depends upon the quantity of debt.

Empirical contributions to the this literature include: Friedman andKuttner[15], who study the imperfect substitutability of commercial paperand U.S. Treasuries; Greenwood and Vayanos[17], who find that the supplyof long-term relative to short-term bonds is positively related to – and pre-dicts – the term spread. Krishnamurthy and Vissing-Jorgensen[20], who findthat the spread between liquid treasury securities and less liquid AAA debtmoves systematically with the quantity of government debt; and Pfluegerand Viceira[25], who document a liquidity premium in the spread betweeninflation-indexed and nominal government bonds.

The paper proceeds as follows: In section 2, we present both versions ofthe LB Model, and show that it encompasses the Standard Model. In the LBModel the government has two seigniorage taxes at its disposal: the usualone on cash holdings, and a new one on government bond holdings. Wealso discuss how the various taxes distort household decisions. In section3, we compare the Ramsey solutions to these models. And in section 4, webriefly summarize our conclusions and discuss the availability of Ricardiandebt instruments. In an appendix, we provide a full derivation of the Ramseysolution to the LB Model in the case of flexible prices.

the gap. And, other theories of money abound. Here, we use the financial frictions thathave generally been used in discussions of optimal money and debt management.

5

2 The Liquid Bonds (LB) Model

The basic structure of our LB Model is easily explained. There arethree consumption goods: a cash good, cm, a bond good, cb, and a creditgood, cc. Households face a cash in advance constraint for the cash goodand a collateral constraint for the bond good; government bonds serve ascollateral. Households pay for their credit and bond goods, receive theirwages and dividends, and pay their taxes at the beginning of the period thatfollows. A competitive bundler produces a perishable final product, y, whichcan be sold as a cash good, a bond good, or a credit good. Governmentpurchases, g, are assumed to be credit goods. Consolidating the treasuryand the central bank, the government has four taxes at its disposal: a labortax, τw, a profits tax, τΥ, and two seigniorage taxes, one on money and theother on bonds. All of the taxes are distortionary except for the profits tax.In the Benchmark LB Model, the government is able to lend to the privatesector; that is, it can hold bonds issued by the private sector. These privatesector bonds cannot be used as collateral in purchasing the bond good. Inthe Constrained LB Model, the government is not allowed to hold privatesector bonds, even though (as we shall see) it would be welfare improving forit to do so.

The LB Model encompasses the Standard Model, which is obtained bydropping the bond good (and therefore the seigniorage tax on bonds). Gov-ernment bonds do not provide liquidity in the Standard Model, and there isnot an optimum quantity of debt in the steady state. In our numerical exer-cises, the steady state value of the debt is pinned down by initial conditions;to facilitate comparisons, we set the debt to GDP ratio equal to the optimaldebt to GDP ratio in the LB Model. Apart from such issues, the StandardModel is identical to the model we analyzed in some detail in Canzoneri,Cumby and Diba[9]. In the next subsection, we present the LB Model, anddescribe the way in which it reduces to the Standard Model.

2.1 Structure of the Liquid Bonds (LB) Model

In each period t, one of a finite number of events, st, occurs. st denotesthe history of events, (s0, s1, ..., st), up until period t. The initial realization,s0, is given. The probability of the occurrence of state st is ρ(st).

A continuum of monopolistically competitive firms produce intermediategoods using a common technology, which is linear in labor and has an aggre-

6

gate productivity shock, z(st). Competitive retailers buy the intermediategoods and bundle them into the final good, using a CES aggregator withelasticity σ. The final goods are then sold to households (as cash goods,bond goods, or credit goods), and to the government (as credit goods). Thefeasibility constraint is then

y(st) = cm(st) + cb(st) + cc(s

t) + g(st) (1)

Government purchases are assumed to follow an exogenous AR(1) processesin our numerical exercises. There is no cb(s

t) in the Standard Model.Labor markets are competitive; there is no wage rigidity. However,

intermediate goods producers engage in Calvo price setting. In every period,each producer gets to reset its price with probability 1−α; otherwise, its priceremains unchanged from the previous period. There is no price indexationto lagged inflation, or to steady state inflation.

The utility of the representative household is

U =∞∑t=0

∑st

βtρ(st)u[cm(st), cb(st), cc(st), nt(st)]

(2)

= E∞∑t=0

βtu [cm,t, cb,t, cc,t, nt]

where nt is hours worked. As these equations suggest, we will suppress thenotation for the state, st, when we think that it will cause no confusion.

In period t (or more precisely, state st), households may purchase statecontingent securities, x(st+1), that pay one dollar in state st+1 and costQ(st+1|st). Households can construct a riskless bond, Bc, by purchasinga portfolio of claims that costs one dollar and pays a gross rate of returnIc(st) in every future state st+1:

1 = Ic(st)∑st+1|st

Q(st+1|st) (3)

In our LB Model, there are actually two risk free bonds, Bc and B. Bc isissued by the private sector and cannot be used as collateral to purchase bondgoods. We will refer to Ic as the CCAPM rate (for reasons that will becomeevident in the next subsection). Since households are identical in our models,there will be no private sector demand for Bc in equilibrium. However, the

7

government can hold private sector bonds in the Benchmark LB Model; so,the net demand for these bonds will be Bc

g(st), the government’s holding of

Bc. In the Constrained LB Model, Bcg = 0 is a constraint on government

behavior. B is the bond issued by the government, and it pays a gross rateof return I. This is the liquid bond that can be used as collateral in thepurchase of bond goods. In the Standard Model, government bonds do notprovide liquidity, and B and Bc are perfect substitutes and I = Ic.

Each period is divided into two exchanges. In the financial exchange,public and private agents do all of their transacting except for the actualbuying and selling of the final product; purchases of the final good occur inthe goods exchange that follows, subject to the cash and collateral constraint:Mt ≥ Ptcm,t and Bt ≥ Ptcb,t, where Pt is the price of the final product. Onceagain, there is of course no bond good or collateral constraint in the StandardModel. Households come into the financial exchange in period t with nominalwealth A(st). Their budget constraint in the financial exchange is

M(st) +B(st)−Bc(st) +∑st+1|st

Q(st+1|st)x(st+1) ≤ A(st) (4)

where x(st+1) is the number of dollar claims purchased for state st+1 . Theevolution of wealth is governed by

A(st+1) = I(st)B(st)− Ic(st)Bc(st) + x(st+1) + [M(st)− P (st)cm(st)] (5)

− P (st)[cb(st) + cc(st)] + [1− τw(st)]W (st)n(st) + [1− τΥ(st)]Υ(st)

where Υ is profits.First, we discuss the pricing of state contingent claims, Q(st+1|st), and

conditions required for the cash and collateral constraints to be binding.This will be useful when we derive the Ramsey solution. Then, we turn tothe first order conditions that illustrate the tax distortions.

8

2.2 Pricing of State Contingent Claims

Households maximize

V [A(st)] = max{u[cm(st), cb(st), cc(st), nt(st)]

(6)

+ λm(st)

[M(st)− P

(st)cm(st)]

+ λb(st)

[B(st)− P

(st)cb(st)]

+ λA(st)

[A(st)−M(st)−B(st) +Bc(st)−∑st+1|st

Q(st+1|st)z(st+1)]

+ β∑st+1|st

ρ(st+1|st)V [A(st+1)]}

First order conditions from this maximization give the pricing equation forstate contingent claims

Q(st+1|st) = βρ(st+1|st)[um(st+1)

um(st)

P (st)

P (st+1)

](7)

The first order conditions also imply that It ≤ Ict ; this reflects the factthat government bonds pay a non-pecuniary return (or yield transactionsservices). Moreover, the cash in advance constraint is binding (λm (st) > 0)if Ict > 1 and the collateral constraint is binding (λb (st) > 0) if Ict > It.

2.3 Tax Distortions and Euler Equations

The household’s first order conditions imply

um,t = Ict uc,t (8)

If the household gives up one dollar’s worth of the cash good, it can spendIct dollars on the credit good, because credit goods avoid the cash in advanceconstraint. Since the final good can be sold as either a cash good or a creditgood, the marginal rate of transformation is one for one. So if Ict > 1,the household buys too few cash goods. Ict − 1 is the standard seignioragetax on cash goods, and it is distortionary. This tax is also available to thegovernment in the Standard Model.

Similarly, the first order conditions imply

ub,t = (1 + Ict − It)uc,t (9)

9

If the household gives up one dollar’s worth of the bond good, it can spend1+Ict −It dollars on the credit good, because credit goods avoid the collateralconstraint. So if Ict −It > 0, the household buys too few bond goods. Ict −Itis a new seigniorage tax on bond goods, and again this tax is distortionary.This tax is not available in the Standard Model.

The bond good - cash good margin can also be distorted by the seignioragetaxes. From (8) and (9),

ub,t =

(1 + Ict − It

Ict

)um,t (10)

The wage tax, τw,t, is also distortionary. First order conditions imply

−un,t = (1− τw,t)(Wt

Pt

)uc,t (11)

−un,t =

(1− τw,tIct

)(Wt

Pt

)um,t (12)

−un,t =

(1− τw,t

1 + (Ict − It)

)(Wt

Pt

)ub,t (13)

Note that τw,t, Ict − 1 and Ict − It distort the consumption - leisure margins;

they make the household want to work less than is efficient.Now, we can derive the Euler Equations for the three goods. Recalling

that the gross nominal CCAPM rate is defined by

1 = Ic(st)∑st+1|st

Q(st+1|st) (14)

and using the pricing equation for contingent claims, (7), we arrive at thestandard Euler equation

1 = Ict βEt

[um,t+1

um,t

PtPt+1

](15)

Then, (8) implies the Euler equation for credit goods

1 = βEt

[Ict+1

uc,t+1

uc,t

PtPt+1

](16)

10

Note that intertemporal smoothing of the credit good depends on next pe-riod’s interest rate; that is, Ict+1appears in this Euler equation instead of Ict ;The reason is that the credit good is not paid for until period t+ 1. Finally,(9) implies the Euler Equation for bond goods

1 = βEt

[Ict+1

(1 + Ict − It

1 + Ict+1 − It+1

)(ub,t+1

ub,t

PtPt+1

)](17)

The difference in the timing of these seigniorage taxes will play a role in thenext section.

2.4 The Government’s Present Value Budget Constraint

So, where does the second seigniorage tax – the one on bonds – show upin the government’s fiscal accounting? In the LB Model, the consolidatedgovernment flow budget constraint can be written as

Mt +Bt −Bcg,t + St = Mt−1 + It−1Bt−1 − Ict−1B

cg,t−1 (18)

where St is the primary surplus, and where it will be recalled that Bcg,t is the

government’s stock of private sector bonds. Let Lt (≡ Mt−1 + It−1Bt−1 −Ict−1B

cg,t−1) be total government liabilities at the beginning of period t. Then,

iterating the flow budget constraint forward, and applying the householdtransversality condition, yields the present value budget constraint.

LtPt

= Et

∞∑j=0

αct+j

[St+j +

(Ict+j − 1

Ict+j

)Mt+j

Pt+j+

(Ict+j − It+j

Ict+j

)Bt+j

Pt+j

](19)

where αct+j is the stochastic discount factor. The next to last term in theequation is the standard seigniorage tax. The last term is the seignioragetax on bonds; it reflects the fact that the government borrows at a discount,due to the non-pecuniary return on its bonds. Of course, this last tax is notavailable in the Standard Model.

3 Optimal Money and Debt Management

We can solve the Ramsey problem analytically for the Benchmark LBModel with flexible prices, and we will begin with that case. Then, we will

11

discuss the complications that arise in the Constrained LB Model, wherethe government does not have access to a Ricardian debt instrument. Andfinally, we will relate our results to those found (by others) in the StandardModel.

With staggered price setting, we can no longer solve the Ramsey problemanalytically. The reason is that, as noted by Schmitt-Grohe and Uribe[26]and others, price rigidity adds a new state variable, one cannot reduce thesequence of flow implementability constraints to a single present value con-straint. However, we can illustrate the optimal solution using numericalmethods. Here, the results are quite different for the Benchmark LB Modeland the Constrained LB Model.

In what follows, we will work with the utility function

u(cm,t, cb,t, cc,t, nt) = φm log(cm,t) + φb log(cb,t) + φc log cc,t −1

2n2t , (20)

although our results readily extend to more general preferences. In particu-lar, as explained in the Appendix, our derivation of the Ramsey solution forthe Benchmark LB Model with flexible prices extends to preferences that arehomothetic over the three consumption goods and weakly separable acrossemployment and consumption. In our numerical exercises, we will also letφm = φb = φc; this symmetry will make some of the impulse response func-tions easier to interpret.

One final note before we begin the derivations: Λ(st) ≡ m(st) + b(st)−bcg(s

t) represents net real government liabilities. We will let the Ramseyplanner set the initial values of m(s0), b(s0), and bcg(s

0) subject to a giveninitial value of Λ(s0) (≥ 0).9 This allows the planner to choose arbitrarilylarge values of initial gross liabilities, m(s0) + b(s0), in the Benchmark LBModel (the only model for which bcg(s

t) is available). The choice of initialgross liabilities will be important in what follows.

3.1 The Case of Flexible Prices

We begin with the Benchmark LB Model. We present the full derivationof the Ramsey Solution in an appendix. In the main text, we present a sketch

9There are important questions about how we should treat the choice of the initialconditions (or the price level, given some initial nominal liabilities). However, theseissues are well known, and we have nothing new to contribute.

12

of the derivation, give intuition for how the solution works, and illustrate theintuition with impulse response functions from a numerical example.10

3.1.1 The Benchmark LB Model with Flexible Prices

We follow the usual procedure in solving for the Ramsey problem: wederive flow implementability conditions by using household first order condi-tions to eliminate relative prices and tax rates in the household’s flow budgetconstraints; then, we choose the quantities of consumption and labor thatoptimize household utility subject to the relevant constraints. However, theRamsey planner will also choose m(st) and b(st), because we have inequalityconstraints on the consumption of cash and bond goods. The Ramsey plan-ner will also choose its holdings of private sector debt, bcg(s

t). As we shallsee, the flow implementability conditions reduce to a single present valuecondition if this Ricardian debt instrument is available to the planner.

The first step is to derive the flow implementability conditions, and in-corporate the complimentary slackness conditions into them. As shown inthe appendix, the household’s flow budget constraint can be written as

A(st) =∑st+1|st

[Q(st+1|st)A(st+1)] +M(st)

[1− 1

Ic(st)

]+B(st)

[1− I(st)

Ic(st)

](21)

+P (st)

Ic(st)[cm(st) + cb(st) + cc(st)]−

[1− τw(st)

Ic(st)

]W (st)n(st)

The cash in advance constraint is binding for Ic(st) > 1, and we have Ic(st) =1 if this constraint is not binding. So for all Ic(st) ≥ 1, we have

M(st)

[1− 1

Ic(st)

]= P (st)cm(st)

[1− 1

Ic(st)

](22)

or, rearranging terms,

M(st)

[1− 1

Ic(st)

]+P (st)cm(st)

Ic(st)= P (st)cm(st) (23)

10For our numerical examples, the Calvo parameter, α, is set at 0.75 for the case ofstaggered price setting. Government spending follws an AR(1) process, with the auto-regressive parameter set at 0.9. σ, the CES aggregator for the final good is set equal to7. The steady state debt to GDP ratio in the Standard Model is set equal to the optimalsteady state debt to GDP ratio in the Benchmark LB Model.

13

Similarly, the collateral constraint is binding if Ic(st) > I(st), and we haveIc(st) = I(st) if this constraint is not binding. So, for all values of I(st)satisfying 1 ≤ I(st) ≤ Ic(st), we have

B(st)

[1− I(st)

Ic(st)

]= P (st)cb(st)

[1− I(st)

Ic(st)

]= P (st)cb(st)

[Ic(st)− I(st)

Ic(st)

](24)

or, rearranging terms,

B(st)

[1− I(st)

Ic(st)

]+P (st)cb(st)

Ic(st)= P (st)cb(st)

[1 + Ic(st)− I(st)

Ic(st)

](25)

Substituting the complimentary slackness conditions – (23) and (25) – into(21), and using household first order conditions to eliminate relative pricesand Q(st+1|st), we arrive at the flow implementability conditions

φmA(st)

P (st)cm(st)= β

∑st+1|st

ρ(st+1|st)[

φmA(st+1)

P (st+1)cm(st+1)

]+ 1−

[n(st)

]2(26)

The Ramsey planner maximizes household utility subject to (26), thefeasibility constraint

cm(st) + cb(st) + cc(s

t) + g(st) = z(st)n(st) , (27)

and the inequality constraints for cash and bond goods. We have put thecomplementary slackness conditions into the implementability conditions,they are valid whether or not the constraints are binding. So, all we haveleft are the inequality constraints themselves.

As shown in the appendix, two of the first order conditions for this max-imization problem are

λ(st+1)− λ(st) = γm(st+1)cm(st+1)

φm(28)

andγm(st) = γb(st) ( ≡ γ(st)) (29)

where γm(st) and γb(st) are the Lagrange multipliers for the inequality con-straints on the cash and bond goods, and λ(st) is the Lagrange multiplier onthe implementability conditions (26).

14

The cash and collateral constraints are both binding, or they are bothslack. If they are binding (γ(st) > 0), then (28) says that the Ramsey plannerissues more liabilities, easing the constraints but increasing the tax burdenand λ(st+1). If the inequality constraints are never binding (γ(st) = 0),then λ(st) = λ, a constant over time and across states. In this case, wecan iterate the flow implementability conditions to obtain a single presentvalue condition. If however the cash and collateral constraints are binding,then λ(st) increases over time, and the flow constraints do not collapse intoa present value constraint.

So, are the cash and collateral constraints binding in the Benchmark LBModel, or are they not? We have made no mention of the Ramsey planner’suse of its Ricardian debt instrument, bcg(s

t). It turns out that the derivativewith respect to this instrument implies

γm(st) = 0 (30)

and therefor λ(st) = λ. We will give a fuller explanation of the role playedby the Ricardian debt instrument later in the section.

As shown in the appendix, the Ramsey planner’s first order conditionsfor consumption imply

φmcm(st)

=φb

cb(st)=

φccc(st)

(31)

when γ(st) = 0. And since the implementation of the optimal allocation issubject to household first order conditions (8) and (9), we must have

Ic(st) = I(st) = 1 (32)

So, the Ramsey planner follows an Extended Friedman Rule: the costof holding money is eliminated (Ic(st) − 1 = 0), and the cost of holdinggovernment bonds is eliminated (Ic(st) − I(st) = 0). We have satiation inboth money and government debt. And of course, Ic(st) = 1 implies thatprices are expected to deflate at the real rate of interest.

The wage tax is the third distortionary tax. As shown in the appendix,the optimal tax rate is positive.

τw(st) = 1−(

1

1 + 2λ(st)

)(σ

σ − 1

)> 0 (33)

15

And since λ(st) = λ, the optimal wage tax rate is constant. The wage taxdistortion is perfectly smoothed over time and across states. Why is the wagetax rate positive while the the two seigniorage taxes are set to zero? As seenin section 2.3, any of these taxes would distort the labor - leisure margin.However, using either of the seigniorage taxes would also distort the relativeconsumption margins. The wage tax does not distort the consumptionmargins; so it is efficient to use it to tax work.11

But if the wage and seigniorage tax rates are held constant, how does theRamsey planner accommodate shocks that have fiscal consequences? Figure1 illustrates the Ramsey planner’s response to an auto-corellated increase ingovernment spending.12 The planner engineers a surprise (shock dependent)inflation; this is a non-distortionary tax that lowers the real value of govern-ment liabilities, or equivalently households’ real assets. Households respondby decreasing consumption and increasing their work effort to rebuild theirwealth. The three goods are perfect substitutes in production, and theyenter utility in an identical way. So, their consumption paths are identical,declining by equal amounts and then returning to their steady state values atequal rates. A rising path of consumption requires a real interest rate thatis above its steady state value. The nominal interest rate enters the threeEuler equations – (15), (16) and (17) – in different ways, and the ExtendedFriedman Rule (Ic(st) = I(st) = 1) nullifies these differences. The requiredmovements in the real interest rate are accomplished by expected inflation.

Since all of the tax rates are unchanged, the persistent increase in govern-ment purchases results in a deficit. The Ramsey planner finances the deficitby issuing new liabilities, and this is of course what allows the households torebuild their wealth.

But what if the initial inflation tax on money and bonds makes the cashand collateral constraints binding? This would distort consumption patternsand raise interest rates, none of which happens. Here is where bcg(s

t) comesin. Recall that we have let the Ramsey planner set the initial values of itsliabilities, m(s0) and b(s0), subject to a given (possibly zero) initial valueof net liabilities, Λ(s0) = m(s0) + b(s0) − bcg(s0). By lending to the privatesector, the planner can build up m(s0) and b(s0) to a level where the cashand collateral constraint will be never binding. This debt instrument, unlike

11Chari et al[11] relate this result to Atkinson and Stiglitz[5].12We use numerical methods for illustration only. We provide an analytical derivation

of the Ramsey solution in the Appendix.

16

government bonds, is Ricardian; its supply has no direct effect on householdconsumption decisions.

3.1.2 The Constrained LB Model with Flexible Prices

In the Constrained LB Model, the government is not allowed to holdprivate sector debt; that is, bcg(s

t) ≡ 0. The government does not have aRicardian debt instrument. The steady state solution and all of the firstorder conditions for this problem are the same as in the Benchmark caseexcept of course for the condition with respect to bcg(s

t), which made λt(st)

constant and kept the cash and collateral constraints from binding. If initialliabilities are small, then the cash and collateral constraints will bind forlarge shocks. In this case, (28) implies that the government will issue moreliabilities going into states with binding inequality constraints (recall that arising shadow price λ corresponds to higher public-sector liabilities). We willassume that initial liabilities are large enough for the inequality constraintsnot to bind. In this case the Ramsey solution coincides with the solutionwhen we let the government buy private bonds.

3.1.3 The Standard Model with Flexible Prices

In the Standard Model, there is no bond good and no collateral con-straint. As Chari, Christiano and Kehoe[11] have shown, the original Fried-man Rule is optimal in the Standard Model with flexible prices.13 There isno need for government lending, bcg(s

t), here. Public debt, b(st), is a Ricar-dian instrument. It can be issued to finance temporary deficits. And if,say, the response to an increase in government spending requires an initialinflation tax that would make the cash constraint binding, then the Ramseyplanner can use open market operations to change the composition of totalliabilities and avoid the problem.

Surprise inflation is quite volatile in Chari, Christiano and Kehoe’s nu-merical simulations. In their benchmark calibration, inflation has a standarddeviation of 20 percent per annum. This price volatility can also be seen inFigure 1 for the Benchmark LB Model.

13Canzoneri, Cumby and Diba[9] provide a fuller discussion the earlier literature.

17

3.2 The Case of Staggered Price Setting

With flexible prices, we saw that the Friedman Rule was optimal and theexpected rate of deflation was equal to the real rate of interest; moreover, theRamsey Planner made substantial use of unanticipated inflation to absorb thefiscal consequences of macroeconomic shocks. This is costless when pricesare flexible. But, as is well known, staggered price setting can create adispersion of intermediate goods’ prices that distorts household consumptiondecisions.14 And if (as we assume) prices are not indexed to inflation, therecan also be a dispersion of prices in the steady state. This relative pricedistortion can be eliminated if the Ramsey Planner fixes the price level; inthis case, price setters will not want to change their prices even when giventhe opportunity.

So, staggered price setting creates a fundamental inflation tradeoff forthe Ramsey Planner. Fixing the price level to eliminate the price dispersionclashes with the implementation of the Friedman Rule, and with the use ofunanticipated inflation as a fiscal shock absorber. The question becomes:how important is the distortion created by the price dispersion relative tothe distortions created by the seigniorage taxes and the wage tax? Benignoand Wooford[7] and Schmitt-Grohe and Uribe[26], using calibrated modelssimilar to our Standard Model, found that the inflation tradeoff was basicallyresolved in favor of price stability.

As noted above, we can not solve the Ramsey problem analytically withstaggered price setting. So, we have to resort to numerical methods.15 Wewill begin with a discussion of optimal taxation and inflation in the variousmodels’ steady states. Then, we will show how the Ramsey planner respondsto an unexpected increase in government spending. We will begin withthe Standard Model, since the planner’s response reflects the conventionalwisdom that grew out of the work of Barro. Then, we will proceed to theBenchmark LB Model, which retains the basic insights of Barro. Finally,we will discus the Constrained LB Model, in which the government does nothave available a Ricardian debt instrument and the insights of Barro are lost.

14Woodford[30] is the classic reference.15More precisely, we use the Get Ramsey program developed by Levin and Lopez-

Salido[21] and Levin, Otnatski, Williams, and Williams[22].

18

3.2.1 Optimal Tax Rates and Inflation in the Steady State

Table 1 reports optimal steady state tax rates and inflation for our nu-merical calibrations of the various models. With flexible prices, the seignior-age taxes are set to zero and the wage and profit taxes service the publicdebt; deflation is equal to the real rate of interest. With staggered price set-ting, prices are virtually fixed, and the seigniorage taxes are positive. Calvotrumps Friedman: the inflation tradeoff is resolved decidedly in favor of pricestability.

Table 1 shows that Ic > I > 1. Why is this? If I were set equal to 1,(10) implies that the bond good - cash good margin would not be distorted,but (9) implies that the bond good - credit good margin would be distorted.Similarly setting Ic = I would leave the bond good - credit good marginundistorted, but the bond good - cash good margin would be distorted.Optimal policy spreads the consumption distortions by setting Ic > I > 1.Moreover, the cash seigniorage tax is larger than the bond seigniorage tax.This must be the case, since I > 1 implies that Ic − 1 > Ic − I.16

3.2.2 Optimal Responses in the Standard Model

Figure 2 shows the Ramsey planner’s response to an auto-corellatedincrease in government spending, and it reflects the conventional wisdom.As before, households respond to the increased tax burden by decreasingconsumption and working more. Here however, there is virtually no unan-ticipated inflation; it is just too costly in terms of the price dispersion itcreates. The spending increase has to be financed another way.

Raising the seigniorage tax, I, or the wage tax, τw, creates a distortion,and the Ramsey planner minimizes these distortions by issuing liabilities tofinance part of the increase in government spending. This leaves a largerpublic debt to be rolled over in the new steady state, and the wage tax israised permanently to finance that. In this way, the Ramsey planner tradesoff short run tax smoothing against lower long run consumption. Thistradeoff, and the unit root it engenders, captures the fundamental insight ofBarro.

The consumption paths of the cash good and the credit good are verysimilar after the first few periods, and they resemble the consumption paths

16With the symmetry in our utility function, I is set exactly half way between zero andIc.

19

for the flexible price case. As before, the optimal allocation requires thatconsumption falls initially and then rises over time, requiring that the realinterest rate be above its steady state value. Now, however, the nominalinterest rate needs to change because inflation is too costly to move. And asa result, the paths of cash and credit good consumption are similar, but notidentical.

The Ramsey planner takes some artful steps in the first few periods tosmooth initial distortions. The need for these steps arises because of thedifference in the timing of the interest rates in Euler equations (15) and(16). From (16), a drop in period 1 credit good consumption requires anincrease in Ic2, the nominal interest rate in period 2. But this will distortthe relative consumption of cash and credit goods in period 2; that is, cashgood consumption will be too low. To mitigate the cost of this, the plannermakes cash good consumption fall by less than credit good consumption inperiod 1, which requires a decrease in Ic1.

In order to achieve the optimal paths of consumption, the Ramsey plannerhas to alter the composition of government liabilities so that household moneybalances are consistent with cash good consumption. Since public debt isRicardian, open market sales of bonds allow the cash in advance constraintto be satisfied with no further consequences.

The details of the artful steps taken by the Ramsey planner in the first fewperiods depend on the special timing in the cash and credit good apparatus.They may not be robust to changes in the way of modeling liquidity. Theimportant result here is that optimal policy involves short run tax smoothingand permanent effects on debt, consumption, and the wage tax rate.

3.2.3 Optimal Responses in the Benchmark LB Model

Figure 3 shows the Ramsey planner’s response to a persistent increasein government spending, and it is similar to that in Figure 2. In particular,Barro’s fundamental insight survives: short run tax distortions are smoothedat the expense of longer run consumption. There is, however, a twist inthe Ramsey planner’s implementation of the policy due to the fact thatgovernment bonds are now used as collateral; they are no longer Ricardian.Government assets, bcg, are Ricardian, and they play an important role inwhat follows; we return to this issue at the end of the section.

As before, the increase in government spending crowds out consumptionand increases work effort in the short run, and once again the three consump-

20

tion goods’ paths are very similar after the first few periods. Inflation (notpictured) is too costly to move. So, the real interest rate increases, associ-ated with the rising consumption profiles, require changes in the seignioragetaxes, Ic − 1 and Ic − I. Consequently, the consumption paths are notidentical.

As with the standard model, the Ramsey planner takes some compli-cated steps in the first few periods. The increase in Ic2 again implies thatperiod 2 cash good consumption will fall too much relative to credit goodconsumption, and this loss is mitigated by a decrease in Ic1. But the diver-gence in period 1 cash and credit good consumption has a utility cost. TheRamsey planner must choose bond good consumption in a way that miti-gates that utility cost while achieving a drop in overall period 1 consumptionlarge enough to accommodate the increase in government purchases. Thisis achieved by balancing the smaller drop in cash good consumption with alarger drop in bond good consumption; that is, the decline in credit goodconsumption is between these two.

The difference in bond and credit good consumption must be reflected inthe interest rate spread, Ic − I. From (17), the interest rate entering theEuler equation for the bond good is Ict+1

spreadtspreadt+1

, which differs from that inthe Euler equation for the credit good by the spread terms. If the spreadwere to remain constant at its steady state value, the response of bond goodconsumption would be identical to that of credit good consumption. Butsince bond good consumption falls by more than credit good consumptionand is therefore expected to rise by more, the real interest rate for bondgoods must be higher and the spread must be expected to fall. And since thespread returns to its steady state value, a declining spread implies an initialincrease in the spread.

Now, we return to the role played by bcg. The Ramsey planner cannotsimply alter the composition of government liabilities to achieve the optimalallocation: satisfying the cash in advance constraint requires selling gov-ernment bonds. But in the LB Model, this would loosen the collateralconstraint, and too many bond goods would be consumed. The governmentcan finance its short run deficit in the normal way (by issuing more total lia-bilities), but then the government sells bcg for m and b. In this way, the cashand collateral constraints can be made consistent with optimal allocation ofcash and bond good consumption.

21

3.2.4 Optimal Responses in the Constrained LB Model

Figure 4 shows the Ramsey planner’s response to a sustained increase ingovernment spending, and it clearly violates the conventional wisdom. TheRamsey solution is stationary; the government does not smooth short runtax distortions at the expense of longer run consumption.

The reason is that both money and government bonds provide liquidity inthe LB Model, and the path of consumption is tied to their supplies. With-out a Ricardian debt instrument (like b in the Standard Model or bcg in theBenchmark LB Model), the planner cannot finance temporary surpluses ordeficits without affecting the supply of liquidity to households. Stated differ-ently, without the Ricardian instrument, fiscal policy and liquidity provisioncannot be determined independently.

Consumption is again crowded out by the increase in government pur-chases. The decline in cash and bond good consumption requires a corre-sponding decline in bothm and b. (This means of course that the governmenthas to run a surplus.) Comparing Figures 3 and 4, the Ramsey planner hasto raise the wage tax more by a full order of magnitude. And this meansthat consumption has to fall by about twice as much and the additional workeffort is curtailed.

The Ramsey planner’s steps in the first few periods are even more artfulthan in the previous cases. Private sector holdings of nominal governmentliabilities are inherited from the previous period and real liabilities will ini-tially change only as a result of inflation. Since inflation is small, the sum ofthe change in m and b – and therefore the sum of the changes in cash andbond good consumptions – is close to zero. But since interest rates mustchange to reflect the optimal consumption paths, the ratio of consumption ofthe cash and bond goods must change. If the sum of the changes is nearlyzero and the ratio changes, one must rise and one must fall. From above,we know that it is cash good consumption that will rise because Ic1 can beused to mitigate the distortion resulting from an increase in Ic2. In orderto mitigate the utility cost of differing consumption of the 3 goods, bondgood consumption will fall less than credit good consumption. (If it were tofall by more, cash good consumption would need to rise by even more.) Inaddition, to mitigate the effect of this distortion, inflation while negligible ismuch greater here than in the previous case.

Because the decline of credit good consumption exceeds the decline ofbond good consumption, expected growth of credit good consumption ex-

22

ceeds expected growth of bond good consumption. Therefore the interestrate in the credit good Euler equation exceeds that in the bond good Eu-ler equation. The spread is therefore expected to grow and must thereforedecline initially.

3.2.5 The Benefit of having a Government Lending Instrument

Figure 5 compares the Ramsey planner’s responses to an increase ingovernment spending in three models: the Benchmark LB Model with flex-ible prices, the Benchmark LB Model with staggered price setting, and theConstrained LB Model with staggered price setting. The figure illustratesthe potential benefit of the government’s having a Ricardian debt instrumentlike bcg.

Staggered price setting adds a nominal distortion to the model. Butwhen bcg is available, the consumption paths are quite similar to those withflexible prices. When bcg is not available, consumption of all three goods fallsmuch further. Of course, long run consumption is a little lower when bcg isused.

In contrast, the response of government liabilities to a spending shock ismuch smaller when the Ramsey planner does not have access to a Ricardiandebt instrument. This stands in sharp contrast to the much greater responseseen for consumption and the wage tax rate. The reason is, once again,that tax policy and liquidity provision cannot be determined independentlywithout a Ricardian debt instrument. The liquidity value of governmentliabilities introduces a motive for the Ramsey planner to smooth debt as wellas the labor tax rate. Alternatively, the Ramsey planner needs to trade offsmoothing the wage tax rate and smoothing the bond seigniorage tax rates.That trade-off is resolved in this model by a much greater wage tax ratevolatility.

4 Conclusion

In this paper, we asked how the fundamental insights of Friedman andBarro fare when we recognize the fact that bonds provide liquidity services.To investigate this question, we augmented a standard cash and credit goodmodel by allowing government bonds to be used as collateral in purchasingcertain types of consumption goods. In this environment, the government

23

cannot use its bonds to finance a short run deficit (ala Barro) independentlyof its decision of how much liquidity to provide for consumption. We foundthat the insights of Friedman and Barro survive largely intact if the govern-ment has at its disposal an additional debt instrument that is “Ricardian”– in the sense that it does not directly affect economic activity. Absent aRicardian debt instrument, the Friedman Rule may not be optimal in thetransition to a steady state, even with flexible prices. And with stickyprices, the Ramsey planner would no longer choose to smooth short run taxdistortions at the expense of some long run consumption (ala Barro).

We model the Ricardian debt instrument as government holdings of pri-vate sector bonds, which by our assumptions did not provide liquidity ser-vices. But, it would not seem that governments currently utilize a Ricardiandebt instrument. They certainly do not use private sector debt in the wayour Ramsey planner would. They do of course have non-liquid assets suchas parks, oil rights, grazing rights, spectrum rights, and state owned enter-prises. However these assets would be hard to use to finance short rundeficits without creating disruptions in the real economy.

Finally, future research might usefully focus on relaxing some of the sharpdistinctions that we have made in our modeling of money and bond liquidity.For example, some kinds of private debt may well compete with governmentbonds in the provision of liquidity services. Such debt would not qualify asa Ricardian debt instrument for use in the manner described here. A moreelaborate modeling of the use of various financial assets would seem to bewarranted. In a similar vein, long term government debt may not providethe liquidity services envisioned in our model. If this is the case, then longterm government bonds may qualify as the Ricardian debt instrument thatis needed here. Modeling a term structure of debt may help differentiate thefinancial assets that do and do not provide liquidity. More generally yet,the substitutability between money and bonds in the provision of liquiditycould be explored further. This could be done by say increasing the sub-stitutability of the cash and bond goods in utility. As the cash and bondgoods became perfect substitutes, money would presumably be crowed out.Or alternatively, holding the elasticity of substitution constant, one couldexplore the implications of the central bank’s paying interest on reserves.This could lead to a discussion of the optimal approach to a winding downof the “unconventional policies” pursued by central banks in recent years.

24

5 Appendix: Ramsey Solution to the LB Model

with Flexible Prices

We can solve the Ramsey problem analytically for the LB Model withflexible prices. In the Benchmark LB Model, the government is able to lendto the private sector; that is, bcg is a policy instrument. In the ConstrainedLB Model bcg is not available.

We follow the usual procedure in solving for the Ramsey solution: wederive flow implementability conditions by using household first order con-ditions to eliminate relative prices and tax rates from the household’s flowbudget constraints; then, we choose the quantities of consumption and laborthat optimize household utility subject to the relevant constraints. How-ever, the Ramsey Planner will also choose m(st) and b(st), because we haveinequality constraints on the consumption of cash and bond goods. We haveto insert complementary slackness conditions into the implementability con-ditions to allow for the possibility that these constraints are not binding inthe optimal solution. Finally, the Ramsey planner will also choose its hold-ings of private sector debt, bcg(s

t). As we shall see, the flow implementabilityconditions reduce to a single present value condition if this Ricardian debtinstrument is available to the planner.

One final note before we begin the derivations: there are important ques-tions about how we should treat the choice of the initial government liabil-ities (or the price level, given some initial nominal liabilities). However,these issues are well known, and we have nothing new to contribute. Inthe benchmark model, we will let the Ramsey planner set the initial valuesm(s0), b(s0), and bcg(s

0), subject to a given (possibly zero) value of net lia-bilities, m(s0) + b(s0) − bcg(s0). We will return to the question of where theplanner would set these initial values later in the appendix.

We begin by deriving the flow implementability constraint. Assumingnon-satiation in some state, the household budget constraint, (4), holds withequality; using (5) to eliminate x(st+1), we get

A(st) = M(st) +B(st) +Bc(st) +∑st+1|st

Q(st+1|st) {A(st+1)

−M(st)− I(st)B(st)− Ic(st)Bc(st)

+ P (st)[cm(st) + cb(st) + cc(st)]− [1− τw(st)]W (st)n(st)} (34)

We have set after tax profits to zero because profits are pure monopoly rents

25

in our model, and we know that the Ramsey planner will tax them away.Using

∑st+1|st Q(st+1|st) = 1

Ic(st), the budget constraint becomes

A(st) =∑st+1|st

[Q(st+1|st)A(st+1)] +M(st)

[1− 1

Ic(st)

]+B(st)

[1− I(st)

Ic(st)

](35)

+P (st)

Ic(st)[cm(st) + cb(st) + cc(st)]−

[1− τw(st)

Ic(st)

]W (st)n(st)

Next, we incorporate the complementary slackness conditions for the cashand collateral constraints. The cash in advance constraint is binding forIc(st) > 1, and we have Ic(st) = 1 if this constraint is not binding. So forall Ic(st) ≥ 1, we have

M(st)

[1− 1

Ic(st)

]= P (st)cm(st)

[1− 1

Ic(st)

](36)

or

M(st)

[1− 1

Ic(st)

]+P (st)cm(st)

Ic(st)= P (st)cm(st) (37)

Similarly, the collateral constraint is binding if Ic(st) > I(st), and we haveIc(st) = I(st) if this constraint is not binding. So, for all values of I(st)satisfying 1 ≤ I(st) ≤ Ic(st), we have

B(st)

[1− I(st)

Ic(st)

]= P (st)cb(st)

[1− I(st)

Ic(st)

]= P (st)cb(st)

[Ic(st)− I(st)

Ic(st)

](38)

or

B(st)

[1− I(st)

Ic(st)

]+P (st)cb(st)

Ic(st)= P (st)cb(st)

[1 + Ic(st)− I(st)

Ic(st)

](39)

Substituting (37) and (39) into (35), the budget constraint becomes

A(st) =∑st+1|st

[Q(st+1|st)A(st+1)] + P (st)

{cm(st) +

cb(st) + cc(st)

Ic(st)(40)

+

[Ic(st)− I(st)

Ic(st)

]cb(st)

}−[

1− τw(st)

Ic(st)

]W (st)n(st)

26

Next, we use the household’s first order conditions (8), (9), and (12) toeliminate relative prices and tax rates,

A(st) =∑st+1|st

[Q(st+1|st)A(st+1)] + P (st)

{cm(st) +

ub(st)cb(st)

um(st)(41)

+uc(st)cc(st)

um(st)+un(st)n(st)

um(st)

}And using (7), we get

A(st)um(st)

P (st)= β

∑st+1|st

ρ(st+1|st)[A(st+1)um(st+1)

P (st+1)

]+ um(st)cm(st) (42)

+ ub(st)cb(st) + uc(st)cc(st) + un(st)n(st)

Equation (42) is a series of implementibility constraints, one for eachperiod and state, st . With flexible prices, in the Standard Model the flowconstraints can be reduced to a single present value constraint beginning ins0 . See Canzoneri, Cumby and Diba ([9]) This is also true in the LB Modelif bcg is available. However, if bcg is not available, the cash and collateralconstraints can be binding and we have to work with the flow implementibilityconstraints, even with flexible prices.

We should also note that there are implementability constraints impliedby 1 ≤ I(st) ≤ Ic(st) (see equations (9) and (10)),

uc(st) ≤ ub(st) ≤ um(st) (43)

We will tentatively ignore these constraints and leave them to be verifiedlater.

To lighten our exposition, we will work with the utility function

φm log(cm,t) + φb log(cb,t) + φc log cc,t −1

2n2t , (44)

although our results readily extend to preferences that are homothetic overthe three consumption goods and weakly separable across employment andconsumption, as in Chari, et. al.[11][12].17 With this specification of prefer-ences, the implementability constraints (42) simplify to

φmA(st)

P (st)cm(st)= β

∑st+1|st

ρ(st+1|st)[

φmA(st+1)

P (st+1)cm(st+1)

]+ 1−

[n(st)

]2(45)

17We don’t present these derivations because they mimic very closely the derivations inChariet.al.[12]

27

The Ramsey planner maximizes household utility subject to (45), thefeasibility constraint

cm(st) + cb(st) + cc(s

t) + g(st) = z(st)n(st) , (46)

and the inequality constraints for cash and bond goods. We have alreadyput the complementary slackness conditions into the implementability con-straints; they are valid whether or not these constraints are binding. So, allwe have left are the inequality constraints themselves,

cm(st) ≤ m(st) (47)

andcb(s

t) ≤ b(st) (48)

We can now solve the planner’s problem. Let ω(st) denote the left handside of the implementability constraint,

ω(st) ≡ φmA(st)

P (st)cm(st)=φm[m(st) + b(st)− bcg(st)

]cm(st)

(49)

Let λ(st) denote the Lagrange multipliers for the implementability con-straints, and incorporate these constraints in the Lagrange function as

βtρ(st)λ(st)

β ∑st+1|st

ρ(st+1|st)ω(st+1) + 1−[n(st)

]2 − ω(st)

(50)

Let µ(st) denote the multipliers for the feasibility condition, (46), and in-corporate these constraints in the Lagrange function as

βtρ(st)µ(st)[z(st)n(st)− cm(st)− cb(st)− cc(st)− g(st)

](51)

Finally, let γm(st) and γb(st) be the multipliers for the inequality constraints,and (since m = ωcm/φm− b− bcg) incorporate them in the Lagrange functionas

βtρ(st)

{γm(st)

[ω(st)cm(st )

φm− b(st)− bcg(st)− cm(st)

]+γb(st) [b(st)− cb(st)]

}(52)

28

We begin with the first order conditions for consumption and work effort:The The FOC for cc(st) gives

φccc(st)

= µ(st) (53)

The FOC for cb(st) gives

φbcb(st)

= µ(st) + γb(st) (54)

The FOC for cm(st) gives

φmcm(st)

= µ(st) + γm(st)

[1− ω(st)

φm

](55)

orφm

cm(st)= µ(st) +

γm(st)

cm(st)[cm(st)− A(st)] (56)

And, the FOC for n(st) gives[1 + 2λ(st)

]n(st) = µ(st)z(st) (57)

Next, we have the first order conditions for the financial variables: TheThe FOC for ω(st+1) gives

βtρ(st)λ(st)βρ(st+1|st)−βt+1ρ(st+1)λ(st+1)+βt+1ρ(st+1)γm(st+1)cm(st+1)

φm= 0

(58)We have ρ(st)ρ(st+1|st) = ρ(st+1), or else ρ(st+1|st) = 0, because st+1 is justthe history st followed by the realization st+1 of the random variables at datet+ 1. So, this FOC reduces to

λ(st+1)− λ(st) = γm(st+1)cm(st+1)

φm(59)

This equation says that the shadow price of the implementability constraintis increasing over time if the inequality constraint for cash goods is binding.The The FOC for b(st) gives

γm(st) = γb(st) ≡ γ(st) (60)

29

We also have the non-negativity conditions with the complementary slacknessconditions

γm(st)[m(st)− cm(st)] = γb(st)[b(st)− cb(st)

]= 0 (61)

In the Benchmark LB Model, the government is allowed to buy householddebt; that is, it has another policy instrument, bcg(s

t). In this case, the Ram-sey solution simplifies quite dramatically: TheThe FOC for bcg(s

t) givesγm(st) = 0

Then, (59) implies that λ(s0) = λ(st) ≡ λ for all states and dates; the seriesof implementibility constraints can be iterated forward to obtain a singlepresent value implementability constraint.

More importantly, the inequality constraints on cash and bond holdingswill never bind if the government can buy private debt. And this is whythe Lagrange multipliers, λ(st), are constant. The Ramsey planner does nothave to use b(st) to finance fiscal imbalances, which could be distortionary.The Constrained LB Model does not share these features.

When γ(st) = 0, the FOC’s for consumption imply

φmcm(st)

=φb

cb(st)=

φccc(st)

= µ(st) (62)

Since the optimal allocation satisfies (62), and its implementation is subject

to (8) and (9), we haveIc(st) = I(st) = 1

The extended Friedman Rule is optimal. This also confirms that (43) issatisfied.

Using (62) and (57), the Ramsey allocation satisfies

(1 + 2λ)n(st)cm(st) = φmz(st) (63)

Since the implementation of optimal policy is subject to (12), with the realwage equal to (σ − 1)z(st)/σ, (63) and Ic(st) = 1 imply

(1 + 2λ)[1− τw(st) ] =σ

σ − 1

30

and a constant optimal tax rate

τw(st) = 1−(

1

1 + 2λ

)(σ

σ − 1

)(64)

To calculate optimal allocations, we need to pin down the value of λ.Using (46), (62) and (63), we get

(1 + 2λ)nt (ztnt − gt) = zt

which implies

nt =1

2zt

gt +

√(gt)

2 +(zt)

2

1 + 2λ

(65)

Iterating (45), we get

φma0

cm0

= E0

+∞∑t=0

βt{

1− [nt]2} ,

which, together with (55) and (57), implies

a0(1 + 2λ)n0

z0

= E0

+∞∑t=0

βt{

1− [nt]2}

Substituting (65) and taking expectations will pin down λ.

References

[1] Aiyagari, S. Rao, Albert Marcet, Thomas Sargent, and Juha Seppala,2002, Optimal Taxation without State-Contingent Debt, Journal of Po-litical Economy, University of Chicago Press, vol. 110(6), December,1220-1254.

[2] Aiyagari, S. Rao, and Ellen McGrattan, 1998, The optimum quantity ofdebt, Journal of Monetary Economics, 42, 447-469.

[3] Albanesi, Stefania, 2003, “Comments on Optimal Monetary and Fis-cal Policy: A Linear-Quadratic Approach by Pierpaolo Benigno andMichael Woodford, in Mark Gertler and Kenneth Rogoff (eds.), NBERMacroeconomics Annual 2003.

31

[4] Angeletos, George-Marios, Fabrice Collard, Harris Dellas, and BehzadDiba, 2012, Public Debt as Optimal Liquidity Provision, mimeo.

[5] Atkinson, Anthony, and Joseph Stiglitz, 1976, The design of tax struc-ture: Direct versus indirect taxation, Journal of Public Economics, vol.6(1-2), 55-75.

[6] Barro, Robert, 1979, On the Determination of the Public Debt, TheJournal of Political Economy, Vol. 87, No. 5, Part 1, pp. 940-971.

[7] Benigno, Pierpaolo and Michael Woodford, 2003,“Optimal Monetaryand Fiscal Policy: A Linear-Quadratic Approach,” in Mark Gertler andKenneth Rogoff (eds.), NBER Macroeconomics Annual.

[8] Bensal, Ravi, John Coleman, 1996, “A Monetary Explanation of theEquity Premium, Term Premium, and Risk-Free Rate Puzzles,” Journalof Political Economy, No 6, 1135-1171.

[9] Canzoneri, Matthew, Robert Cumby and Behzad Diba, 2011, “The In-teraction Between Monetary and Fiscal Policy” in Handbook of Mon-etary Economics, Vol. 3, Benjamin Friedman and Michael Woodfordeditors, Elsevier.

[10] Calvo, Guillermo and Carlos Vegh, 1995, “Fighting inflation with highinterest rates: the small open economy case under flexible prices,” Jour-nal of Money Credit and Banking, v. 27 no. 1, 49-66.

[11] Chari, V., Lawrence Christiano and Patrick Kehoe, 1991, Optimal Fiscaland Monetary Policy: Some Recent Results, Journal of Money, Creditand Banking, Ohio State University Press, vol. 23(3), pages 519-39, Au-gust.

[12] , 1996, Optimality of the Friedman Rule in Economies withDistorting Taxes, Journal of Monetary Economics, Vol. 37, 202-223.

[13] Correia, Isabel, Juan Nicolini and Pedro Teles, 2008, “Optimal Fiscaland Monetary Policy: Equivalence Results,” Journal of Political Econ-omy, 168, 1, pp. 141-170.

[14] Defiore, Fiorella, and Pedro Teles, 2003, The Optimal Mix of Taxes onMoney, Consumption and Income, Journal of Monetary Economics, Vol.50, 871-887.

32

[15] Friedman, Benjamin, and Kenneth N. Kuttner, 1998, Indicator Proper-ties Of The Paper-Bill Spread: Lessons From Recent Experience, TheReview of Economics and Statistics, MIT Press, vol. 80(1), pages 34-44,February.

[16] Friedman, Milton, 1969, The Optimum Quantity of Money and OtherEssays, Aldine Publishing Company, Chicago.

[17] Greenwood, Robin, and Dimitri Vayanos, 2010, “Bond Supply and Ex-cess Bond Returns,” NBER Working Paper Series No. 13806.

[18] Holmstrom, Bengt and Jean Tirole, 1998, Private and Public Supplyof Liquidity, 1998, The Journal of Political Economy, Vol. 106, No. 1(February 1998), pp. 1-40.

[19] Hu, Yifan, and Timothy Kam, 2009, Bonds with transactions serviceand optimal Ramsey policy, Journal of Macroeconomics, Volume 31,Issue 4, December, Pages 633-653.

[20] Krishnamurthy, Arvind, and Annette Vissing-Jorgensen, 2012, “The Ag-gregate Demand for Treasury Debt”, Journal of Political Economy, Vol.12, No. 2, pp 233-267.

[21] Levin, Andrew, and David Lopez-Salido, 2004, “Optimal Monetary Pol-icy with Endogenous Capital Accumulation”, unpublished manuscript,Federal Reserve Board.

[22] Levin, Andrew, Alexei Onatski, John C. Williams, and Noah Williams,2005, “Monetary Policy Under Uncertainty in Micro-Founded Macroe-conometric Models,” in Mark Gertler and Kenneth Rogoff (eds.), NBERMacroeconomics Annual 2005, Cambridge, MIT Press, pp. 229-287.

[23] Linnemann, Ludger and Andreas Schabert, 2010, “Debt Nonneutrality,Policy Interactions, And Macroeconomic Stability, International Eco-nomic Review, vol. 51(2), pages 461-474.

[24] Patinkin, Don, 1965, Money, Interest, and Prices: An Integration ofMonetary and Value Theory, 2nd edition, (Harper & Row, New York).

[25] Pflueger, Carolin and Luis Viceira, 2011, “An Empirical Decompositionof Risk and Liquidity in Nominal and Inflation Indexed GovernmentBonds”, NBER Working Paper #16892, March.

33

[26] Schmitt-Grohe, Stephanie, and Martin Uribe, 2004, “Optimal Fiscal andMonetary Policy under Sticky Prices,” Journal of Economic Theory, 114,February, pg. 198-230.

[27] Schmitt-Grohe, Stephanie, and Martin Uribe, 2005, “Optimal Fiscaland Monetary Policy in a Medium-Scale Macroeconomic Model,” inMark Gertler and Kenneth Rogoff (eds.), NBER Macroeconomics An-nual, Cambridge, MIT Press, pp. 383-425.

[28] Woodford, Michael, 1990a, “The Optimum Quantity of Money,” in B.M.Friedman and F.H. Hahn eds., Handbook of Monetary Economics, Vol-ume II, Amsterdam, Elsevier Science, pp. 1067-1152.

[29] Woodford, Michael, 1990b, Public Debt as Private Liquidity, AmericanEconomic Review, Vol. 80, No. 2, Papers and Proceedings, pp. 382 -388.

[30] Woodford, Michael, 2003, Interest and Prices: Foundations of a Theoryof Monetary Policy, Princeton University Press, Princeton.

34