Thetimeconsistencyoffiscal and monetary policies

Transcription

1 Federal Reserve Bank of Minneapolis Research Department Staff Report November 2001 Thetimeconsistencyoffiscal and monetary policies Fernando Alvarez University of Chicago Patrick J. Kehoe Federal Reserve Bank of Minneapolis anduniversityofminnesota Pablo Neumeyer University Di Tella ABSTRACT We consider the problem of time consistency of the Ramsey monetary and fiscal policies in an economy without capital. Following Lucas and Stokey (1983) we allow the government at any date to leave to its successor a profile of real and nominal debt of all maturities, as a way to influence its decisions. We show that the Ramsey policies are time consistent if and only if the Friedman rule is optimal. We argue that the preferences for which the Friedman rule is optimal, and hence monetary and fiscal policy time consistent, are of applied interest, since they are essentially implied by the restriction that preferences be consistent with balanced growth. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

2 A classic issue in macroeconomics is the time consistency of optimal monetary and fiscal policy. In a monetary economy, Calvo (1978) showed that the incentive for the government to inflate away its nominal liabilities leads to a time inconsistency problem for optimal policy. In a real economy, Lucas and Stokey (1983) showed that the incentive to devalue the real debt typically also leads to a time consistency problem for optimal policy. Lucas and Stokey find that with a carefully chosen maturity structure of real debt, optimal policy can be made time consistent but argue that the analog does not hold for a monetary economy. Here we revisit the problem of time consistency of policies in monetary economies. We find that for the class of economies typically used in applied work, if we carefully choose the maturity structure of both real and nominal debt, optimal policy can be made time consistent. More precisely, we find that optimal policy is time consistent if and only if the Friedman rule is optimal. We consider an infinite horizon model with money in the utility function of the representative consumer. The government has access to nominal and real debt of all maturities and must finance a given stream of government expenditures with a combination of consumption taxes and seignorage. Our approach to the problem of time inconsistency follows that of Lucas and Stokey (1983). We begin by solving for the Ramsey policies in which the government is presumed to have a commitment technology that binds the actions of future government and hence chooses policy once and for all. These Ramsey policies consist of sequences of consumption taxes and money supplies. We then consider an environment without such a commitment technology in which each government inherits a given maturity structure of nominal and real debt. Each such government then decides on the current setting for the consumption tax

3 and the money supply as well as the maturity structure of nominal and real debt that its successor will inherit. We ask whether it is possible to choose the maturity structure of real and nominal debt so that each government carries out the Ramsey policies. If it is we call the Ramsey policies time consistent. In turns out that if the allocations are to be time consistent the structure of the nominal bonds are severely restricted. Not only must they have a present value of zero but, essentially, they must be zero in each period. These restrictions on nominal bonds restrict any government to influence the choices of its successor primarily through real bonds. In general, the government at date t is so restricted in influencing its successor that it cannot induce its successor to carry out the continuation of its plan. In particular, if the Friedman rule does not hold then no matter what the setting of real bonds by the government at t, the successor has an incentive to deviate from the continuation of the date t allocations either by devaluing the real debt or by changing the amount of seignorage raised. When the Friedman rule holds consumers are satiated with money balances and it is as if money disappears so that the economy is equivalent to a real economy. In this economy the government at t can induce its successor to carry out its plan by carefully choosing a maturity structure of the real bonds and by choosing nominal bonds so that the value of nominal liabilities are zero in each period. We argue that the economies for which the Friedman rule is optimal are of applied interest, and given our results, so are the economies for which the Ramsey problem is time consistent. We argue this because the preferences for which the Friedman rule is optimal include the standard ones used in the applied literature. The reason for this connection is that in the applied literature preferences are restricted to be consistent with the growth facts 2

4 and, as we show, for essentially all such preferences the Friedman rule is optimal. We develop a useful analogy between our monetary model and a real economy similar to that in Lucas and Stokey (1983). The real economy has leisure and two types of consumption goods. It is well-known that there is a close analogy between the optimality of uniform commodity taxation in the real economy and the optimality of the Friedman rule in the monetary economy. the Ramsey policies. We show that this close analogy extends to the time consistency of In our monetary economy we showed that the incentive to inflate away nominal debt endogenously restricts the government to basically use only one type of debt, real debt to influence its successors. In the real economy we imitate the endogenous restrictions on debt in the monetary economy by exogenously imposing that the government has access to only one real bond of all maturities. We show that with this restriction the Ramsey policies are time consistent if and only if it is optimal taxes are uniform in the sense that in each period it is optimal to tax the two consumption goods at the same rate. To understand this result for the real economy we consider a T period economy and show that, in general, we cannot choose the T real bonds to ensure that it is optimal not to deviate from the original plan for setting the 2T tax rates, namely two tax rates in each of the T periods. When uniform commodity taxation is optimal there are, in essence, only T such tax rates, one for each period, and by choosing the T bonds appropriately we can ensure that it is not optimal to deviate from the original plan. In the body of the paper we follow the original approach to time inconsistency used by Calvo (1978) and Lucas and Stokey (1983). In the appendix we relate this definition to that of sustainable plans by Chari and Kehoe (1992) and credible policies by Stokey (1991) which explicitly builds the lack of commitment of the government into the environment with 3

5 an equilibrium concept in which governments explicitly think through how their choices of debt influence the choices of their successors. We show that the Ramsey policies are time consistent if and only if they are supportable as a Markov sustainable equilibrium. Our paper is also closely related to that of Persson, Persson and Svensson (1987). They argued that with a sufficiently rich term structure of both nominal and real debt, optimal policy can be made time consistent regardless of whether or not the Friedman rule is satisfied. Unfortunately, their result is not true. Calvo and Obstfeld (1990) sketch a variational argument that suggests that the solution proposed by Persson, Persson and Svensson is not time consistent. Calvo and Obstfeld also conjecture that the mistake made by Persson, Persson and Svensson was that their solution violated the unchecked second order conditions. We formalize the variational argument and we characterize precisely when it applies. We find that the mistake of Persson, Persson and Svensson has more to do with a lack of attention to subtle corners than with the second order conditions. In contrast to Calvo and Obstfeld (1990), our main conclusion is that the Ramsey problem is time consistent for an interesting set of economies. 1. The Ramsey problem and the Friedman Rule Consider a monetary economy with money, nominal debt and real debt. Time is discrete. The resource constraint is given by (1) c t + g t = l t, where c t, g t, l t denote consumption, government spending and labor at t. Throughout the sequence of government spending is exogenously given. 4

6 Consumers have preferences over sequences of consumption c t, real balances M t /p t, and labor l t given by (2) β t U (c t,m t /p t,l t ) with the discount factor 0 < β < 1. Here M t is nominal balances and p t is the nominal price level and we let m t = M t /p t denote real balances. We assume the period utility function U(c,m,l) is concave, twice continuously differentiable, increasing in c, and decreasing in l. We also assume that consumers are satiated at a finitelevelofrealbalances,sothatfor each value of c and l there is a finite level of m such that U m (c, m, l) =0wherehereand throughout the paper we denote partial derivatives by U m,u mm and so on. In terms of assets, we assume there are both nominal and real bonds for each maturity at each date. For the nominal bonds, for each t and s with t s, we let Q t,s denote the price of one dollar at time s in units of dollars at time t and let B t,s denote the number of such nominal claims. Likewise for the real bonds we let q t,s denote the price of one unit of consumption at time s in units of consumption at time t and let b t,s denote the number of such real bonds. We let B t =(B t,t+1,b t,t+2,...) denote the sequence of nominal bonds purchased at t which pay off B t,s at time s for all s t +1. We use similar notation for the real bonds b t, the nominal and real debt prices Q t and q t. For later use it is worth noting that arbitrage among these bonds implies that for all t r s their prices satisfy Q t,s = Q t,r Q r,s, q t,s = q t,r q r,s,q t,s = q t,s p t /p s. By convention, Q t, t =1and q t,t =1. The consumer s sequence budget constraint at date t can be written (3) p t (1+τ t )c t +M t + Q t,s B t,s +p t X X q t,s b t,s = p t l t +M t 1 + Q t,s B t 1,s +p t q t,s b t 1,s s=t+1 s=t+1 s=t s=t 5

7 Thus, at date t, the consumer has nominal balances M t 1, the sequence of nominal bonds B t, and the sequence of real bonds b t and nominal wage income of p t l t. The consumer purchases consumption c t, new money balances M t, and new vectors of nominal bonds B t and real bonds b t. Purchases of consumption are taxed at rate τ t. At date 0 consumers have initial money balances M 1, together with initial sequences of nominal and real debt claims, B 1 and b 1. We assume that in each period the real value of both nominal and real debt purchased is bounded by some arbitrarily large constants. It is convenient to work with the consumer s problem in its date 0 form. To that end, note that the sequence of budget constraints can be collapsed to the date 0 budget constraint (4) q 0,t [(1 + τ t )c t +(1 Q t,t+1 )m t ]= q 0,t l t + M 1 p 0 + Q 0,t B 1,t p 0 + q 0,t b 1,t. The consumer s problem at date 0 is to choose sequences of consumption, real balances and leisure to maximize (2) subject to the (4). The government s sequence budget constraint at date t is (5) Q t,s B t,s +p t X s=t+1 s=t+1 X q t,s b t,s = Q t,s B t 1,s +p t q t,s b t 1,s +p t g t (M t M t 1 ) p t τ t c t s=t s=t At date t, the government inherits the nominal liabilities M t 1 and B t, and real liabilities b t. To finance government spending g t it collects consumption taxes τ t c t and issues new money supply M t, new nominal liabilities B t, and new real liabilities b t. Using the resource constraint and the consumer s date 0 budget constraint we can collapse these sequence of constraints into the government s date 0 budget constraint (6) q 0,t [τ t c t +(1 Q t,t+1 )m t g t ]= M 1 p 0 + Q 0,s B 1,s p 0 + q 0,t b 1,t We find it convenient to use the notation t c =(c t, c t+1,...) for consumption and other variables as well. For given initial conditions M 1,B 1, and b 1, a competitive equilibrium is 6

8 a collection of sequences of consumption, real balances and labor ( 0 c, 0 m, 0 l) together with sequences of prices ( 0 p, Q 0,q 0 ) and taxes 0 τ that satisfy the resource constraint and consumer maximization at date 0. The government budget constraint is then implied. In any equilibrium nominal interest rates are nonnegative so that the one-period bond price Q t,t+1 =1+U mt /U lt 1. Since U lt 0, for interest rates to be nonnegative the marginal utility of money must satisfy the nonnegativity constraint (7) U mt 0. It is easy to show that the allocations in a competitive equilibrium are characterized by three simple conditions: the resource constraint, the nonnegativity constraint (7), and the implementability constraint which is given by (8) β t R(c t,m t,l t )= U l0 p 0 [M 1 + Q 0,t B 1,t ] β t U lt b 1,t where R(c t,m t,l t ) = c t U ct + m t U mt + l t U lt is the surplus of the government τ t c t +(1 Q t,t+1 )m t g t expressed in marginal utility units and Q 0,t = Q t 1 s=0 (1 + U ms/u ls ). This implementability constraint should be thought of as the date 0 budget constraint of either the consumer or the government where the consumer first order conditions have been used to substitute out prices and policies. The following lemma is standard and is included without aproof. Lemma 1. The consumption, real balances and labor allocations together with the price p 0 of a date 0 competitive equilibrium necessarily satisfy the resource constraint, U mt 0, and the implementability constraint. Furthermore, given such allocations and a price p 0 that satisfy these constraints, we can construct nominal money supplies, prices and real and 7

9 nominal debt prices such that these allocations and prices constitute a date 0 competitive equilibrium. Consider now the Ramsey problem at date 0, givenm 1,B 1 and b 1,namelyto choose 0 c, 0 m, 0 l and p 0 to solve (9) max β t U(c t,m t,l t ) subject to the resource constraint, the implementability constraint, and the nonnegativity constraints U mt 0. Our results will depend critically on whether or not the Ramsey allocations satisfy the Friedman rule in that (10) Q t, t+1 =1for all t, so that nominal interest rates are zero in each period. Since Q t,t+1 =1+U mt /U lt and U lt < 0 it follows that the Friedman rule holds if and only if (11) U mt =0for all t. We now discuss the initial conditions for both real and nominal debt that we choose for the Ramsey problem. To make the problem interesting we want initial conditions for which distortionary taxes are necessary. A sufficient condition for this to be true is that (12) g t + b 1,t > 0 for each t. We assume that(12) holds throughout. The solution to the Ramsey problem depends critically on the structure of the value of the initial nominal liabilities as well, namely the initial money supply M 1 and the vector 8

10 of initial nominal debt claims B 1 through the term (13) U l0 p 0 [M 1 + Q 0,t B 1,t ] in the implementability constraint. The term [M 1 + P Q 0,t B 1,t ] is the present value of nominal liabilities of the government in units of dollars at date 0. Dividing by p 0 converts this value into date 0 consumption good units and multiplying by -U l0 converts it into date 0 marginal utility units. We assume that at date 0 nominal liabilities are zero in each period, sothat (14) M 1 + B 1,0 =0and B 1,t =0for all t 1 holds then the present value of nominal liabilities in (13) is identically zero and the Ramsey problem is independent of p 0. In the next section we show that the Ramsey problem is time consistent if and only if the Friedman rule is optimal in each period. Here we establish sufficient conditions for the Friedman rule to be optimal in each period for an economy that has initial nominal claims zero in each period. Letting λ 0, γ t and η t denote the multipliers on the implementability constraint, the resource constraint and the nonnegativity constraint U mt 0, then assuming (14), then for t 1 the first order conditions for c t,m t and l t are (15) (16) (17) U ct + λ 0 [R ct + b 1,t U lct ]+η t U mct + λ 0U l0 p 0 U lt + λ 0 [R lt + b 1,t U llt ]+η t U mlt + λ 0U l0 p 0 U mt + λ 0 [R mt + b 1,t U lmt ]+η t U mmt + λ 0U l0 p 0 Q ct,t+1 s=t+1 Q lt,t+1 s=t+1 Q mt,t+1 s=t+1 Q t+1,s B 1s = γ t, Q t+1,s B 1s = γ t. Q t+1,s B 1s =0, 9

11 where Q t,s = Π s r=t (1 + U mr/u lr ), and Q it,t+1 are the derivatives of 1+U mt /U lt with respect to i = c, m, l. For t =0we add U lc0 λ 0 [M 1 + P Q 0,t B 1t ]/p 0 tothelefthandsideof(15) and we add analogous terms for (16) and (17). We can use these first order conditions to establish circumstances under which the Friedman rule is optimal. Consider an economy with preferences that are separable and homothetic in that (18) U(c, m, l) =u(w(c, m), l) where w is homothetic in c and m and for which initial nominal claims are zero period by period. Preferences that satisfy (18) include commonly used preferences in monetary models like U = w (c, m) 1 σ v (l) /1 σ where w is homogenous of degree one. Such preferences are consistent with some basic facts of economic growth: hours worked per person have been approximately constant, and consumption, real balances and income have grown at approximately the same rate. (See Lucas (2000).) In particular, the utility function must be homothetic in the sense that (19) U m (αc, αm, l) U c (αc, αm, l) = U m(c, m, l) U c (c, m, l) for any α, which is clearly satsified by (18). The following proposition is related to but not covered by the results in Chari, Christiano, and Kehoe (1996). Proposition 1. If preferences are separable and homothetic so that (18) holds and the initial nominal claims are zero in each period so that (14) holds then the Friedman rule is optimal. 10

12 Proof. Assume, by way of contradiction, that U mt > 0. We can arrange the first order conditions for consumption c t and real money balances m t to be (20) (21) Ucct c t + U cmt m t (1 + λ)+λ U ct Ucmt c t + U mmt m t (1 + λ)+λ U mt + λ (l t + b 1t ) U lct U ct = γ t U ct, + λ (l t + b 1t ) U lmt U mt =0. Differentiating (19) with respect to α and evaluating it at α =1gives that (22) c t U cmt + m t U mmt U mt = c tu cct + m t U cmt U ct. By weak separability U lct /U ct = U lmt /U mt. Subtracting (20) from (21) using (22) and weak separability gives that γ t /U ct =0whichisacontradictionsinceγ t and U ct are strictly positive. Q.E.D. The preferences we have assumed in Proposition 1 are consistent with the balanced growth facts. Here we show that, essentially, if preferences are consistent with balanced growth then the Friedman rule is optimal. Define the satiation level of money m (c, l) to be the smallest level of real balances for which its marginal utility is zero, that is m (c, l) =min{m : U m (c, m, l) =0}. Proposition 2. Assume that nominal debt is equal to zero in every period and that U ml 0 whenever U m =0. If preferences are consistent with balanced growth in the sense of (19) then the Friedman rule solves the Ramsey problem. We prove this proposition in the Appendix A. This proposition is related, but not covered by the results in Correia and Teles (1999). In the proposition we show that if U ml 0 and preferences are consistent with balanced growth, then we can construct a 11

13 feasible allocation and non-negative multipliers that satisfies the first order conditions and as well as the implementability condition. It is worth pointing out that preferences such as U = c 1 σ v ³ m c,l / (1 σ) satisfy balanced growth but do not satisfy the Friedman rule if v 12 /v 11 > 0. For such preferences U ml is negative at U m =0. 2. Time consistency and the Friedman rule In this section we give a version of Lucas and Stokey s definition of time consistency and establish that the Ramsey problem is time consistent if and only if the Friedman rule holds. Webeginwithadefinition of time consistency. It is convenient to define the Ramsey problem at period t, given inherited values for money balances M t 1, nominal debt B t 1 and real debt b t 1 to be the problem of choosing allocations from period t onward, namely t c, t l, and t m, and the price level p t (by choosing M t ) to maximize max β s U(c s,m s,l s ) s=t subject to the resource constraint for s t and the implementability constraint at t (23) s=t β s t R s = U lt p t [M t 1 + s=t Q t,s B t 1,s ] s=t β s t U ls b t 1,s ]. where Q t,s = Q t 1 r=t(1 + U mt /U lt ). The Ramsey problem at date t is said to be time consistent for t +1 if there exist values for nominal money balances M t, nominal debt B t, and real debt b t with M t = p t m t such that taking these values as given, the continuation of the Ramsey allocations at period t from t +1 on, namely t+1 c, t+1 l, and t+1 m together with the price level p t+1 solve the Ramsey problem at t +1where the price level p t+1 is a function of the 12

14 allocations and the nominal money supply according to p t+1 = Q t,t+1p t q t,t+1 = β U lt+1 U lt M t /m t (1 + U mt /U lt ). The Ramsey problem at date 0 is time consistent if the Ramsey problem at date t is time consistent for t +1 for all t 0. (In Appendix A we show that if the Ramsey problem is time consistent then the allocations are the outcome of a sustainable Markov equilibrium.) Given this definition, the way to establish that a Ramsey problem at say date 0 is time consistent for date 1 is to show how the initial conditions for the date 1 problem, namely M 0, B 0, and b 0, can be chosen so as give incentives for the government at date 1 to continue with the allocations chosen by the government at date 0. In this section, to keep the proofs simple, we assume that the utility function is weakly increasing in m andhencewecandroptheconstrainttheu mt 0 in the Ramsey problem. Note for later use that under this assumption if U m =0then (24) U mm = U mc = U ml =0. ToseethisnotethatifU m =0at some point m it is also equal to zero at all points m 0 m and thus since U is twice continuously differentiable, U mm =0. To see that U mc =0notice that by concavity U cc U mm Umc 2 0, so U mc =0. A similar argument applies for U ml. We begin with a simple example the illustrates the main ideas behind Propositions 3 and 4. Example. Let the utility function be additively separable in its three arguments. Let g t be zero in all periods, let initial government debt have b 1,0 = b>0 andbezeroinallin other periods and let nominal debt satisfies (14). For the date 0 Ramsey problem consider 13

15 the combined first order conditions for c t and l t (25) (U ct + U cct c t + U lt + U llt l t )+ U ct + U lt λ 0 =0for t 1 and the first order condition for m t (26) U mt + U mmt m t = U mt λ 0 for all t 0. Clearly, c t and l t are constant for t 1 and m t is constant for t 0. It is easy to show that a constant level of positive taxes are levied at each date t 1, so that U ct +U lt > 0 and constant for each t 1. If we are to make this allocation time consistent for date 1, we must be able to choose new debt b 0,t and B 0,t, new nominal money balances M 0 and a new multiplier λ 1 so as to support the continuation of the date 0 allocations. We first show that if the Friedman rule holds, the allocation can be made time consistent. We set M 0 + B 0,t =0and B 0,t =0for all t 1 and construct b 0t and λ 1 in a manner related to that in Lucas and Stokey (1983). The combined first order condition for c t and l t is (27) (U ct + U cct c t + U lt + U llt l t )+U llt b 0t + (U ct + U lt ) λ 1 =0all t 1. Notice that the level of debt b 0t does not vary with t and we denote it by b 0. We use (27) to find b 0 as a function of λ 1, namely b 0 = (U ct + U lt ) U llt 1 λ 1 (U ct + U cct c t + U lt + U llt l t ) U llt We then substitute b 0 into the time t =1implementability constraint to solve for λ 1. Such a λ 1 can be found provided (U ct + U lt ) /U llt 6=0, a condition that we will assume in Proposition 3. Finally, we need to verify that the first order condition with respect to m t, namely, (28) U mt + U mmt m t = U mt λ 1 for all t 1 14

16 holds for the constructed multiplier λ 1. Notice that, if the Friedman rule holds, by (24), this equation is satisfied for any λ 1. In this sense, when agents are satiated with money, it is as if money disappears from the Ramsey problem. Now we show that if the Friedman rule does not hold the date 0 Ramsey problem is not time consistent. In Lemma 3 we will show that if interest rates are always positive then the nominal debt has to be zero in each period. Motivated by this implication of Lemma 3, we suppose that the new nominal debt satsifies M 0 +B 0,t =0and B 0,t =0for all t 1. We argue that with these restrictions on nominal debt, the date 0 allocations cannot be supported. For t 1 taxes are levied and the allocations are constant hence b 0,t is some positive constant b 0 for all t. Comparing the first order conditions for m t at date 0 and date 1 we conclude that λ 1 = λ 0. Using this multiplier we evaluate the first order condition of the date 1 problem (27) at the continuation of the date 0 allocations, which solve (25), to conclude that (U ct + U cct c t + U lt + U llt l t )+U llt b 0 + (U ct + U lt ) λ 0 < 0 all t 1. This inequality means that at the date 0 allocations there is incentive to deviate by increasing taxes. By doing so the date 1 government can devalue the real value of the inherited debt. Hence, the date 0 Ramsey problem is not time consistent. Notice that to arrive our conclusion, it is crucial that U llt 6= 0 for some t which together with concavity implies U llt < 0 (In the nonseparable case consider in Proposition 4 the analogous condition is U clt + U llt < 0.) If the utility function in linear in labor l then the real interest rates are exogenously given and then, with our setting of the nominal debt, the Ramsey problem is time consistent, regardless of whether the Friedman rule holds. Essentially, our setting of the nominal debt eliminates the nominal time inconsistency 15

17 problem identified by Calvo (1978). Making the utility function linear eliminates the ability of the government to manipulate real interest rates and hence eliminates the real time consistency problem identifed by Lucas and Stokey (1983). Finally, notice that in our example, the budget is not balanced at date 0. If in the date 0 Ramsey problem the budget it balanced in every period, then it is possible for the Ramsey policy to be time consistent even if the Friedman rule does not hold in any period. For example, consider an economy where g t is constant and b 1,t = B 1,t =0all t, so that there is no initial government debt. The date 0 Ramsey allocation is constant and prescribes a balanced budget. The Ramsey problem at any future date is the same as the date 0 Ramsey problem, thus its solution is the continuation of the date 0 solution. Therefore, the time 0 Ramsey problem is time consistent. Nevertheless, the solution of the date 0 Ramsey problem may have strictly positive interest rates. Mary put in END OF EXAMPLE space or whatever you do We now use the same logic of the example to show that if the Friedman rule holds, the Ramsey problem is time consistent. To cover the general case we will assume the following regularity conditions U ct + U lt 0, so that taxes are nonnegative, and U lct + U llt < 0 which is essentially normality of consumption. (Lemma 4 in the Appendix gives sufficient conditions for taxes to be nonnegative.) Proposition 3. Assume that the initial nominal claims are zero period by period so that (14) holds and our regularity conditions hold. If the Friedman rule holds for each period the Ramsey problem at date 0 is time consistent. Thelogicoftheproofisverysimilartothatusedintheexampleandisgivenin Appendix A. 16

18 Strictly speaking, in this proposition we show that if the maturity structure of the public debt is adequately managed the continuation of the Ramsey allocation at date 0 satisfies the first order conditions of the Ramsey problems faced by the successor governments. An allocation that satisfies the first order conditions may not solve the Ramsey problem, it could be a local maximum, a minimum or a saddle point. In Appendix B we give conditions under which the first order conditions of the Ramsey problem are sufficient for a maximum. The proof of Proposition 3 makes clear that in order to ensure time consistency there is a unique restructuring of the real debt. However, there are a whole variety of ways to restructure the nominal debt as we now show. Consider nominal debt sequences that satisfy the following (29) M 0 + B 0,t =0 so that at the Friedman rule the value of the initial nominal liabilities of the government of date 1 are zero and the value of nominal liabilities for any date s 1 arenegativeinthat (30) B 0,t 0. t=s We refer to condition (30) as requiring negative tails of the nominal liabilities inherited at date 1. In the following lemma we assume that the multiplier on the implementability constraint ispositive. Lemma5ofAppendixBgivessufficient conditions for this to occur. We have Lemma 2. If the Friedman rule is optimal in an economy in which initial nominal claims are zero period by period so that (14) holds and the multiplier on the implementability constraint is positive then it is also optimal in an economy in which these claims have a present value of zero at the Friedman rule and have negative tails, so that (29) and (30) hold. 17

19 Briefly, if the Friedman rule holds then any deviation from it has to lower some Q t,t+1. Under the negative tails condition such a deviation puts smaller weight on the negative tail and hence raises the present value of initial nominal claims. This deviation does not improve utility and the Friedman rule is optimal. Proof. Consider a relaxed Ramsey problem in which the implementability constraint is written as an inequality and instead of having the bond prices connected to the allocations through Q 1,t = Q t 1 s=1(1 + U ms /U ls ) we simply add the variables Q t,t+1 for all t as new extra choice variables that must only satisfy 0 Q t,t+1 1 and Q 0,t = Q 0,1...Q t 1,t. Given that the multiplier on the implementability constraint is positive, it is clearly optimal to set Q t,t+1 to minimize the present value of nominal claims. Under the negative tails assumption and (29), the lower bound for this present value is zero and is attained at Q t,t+1 =1. Substituting in this value for Q t,t+1 we see that the remaining problem is the same as one for the economy with zero nominal claims in each period, in which the Friedman rule is optimal. Thus, the solution to the relaxed problem is feasible for the original unrelaxed problem and hence is the solution. Q.E.D. We now consider the converse, namely, that when solution to the Ramsey problem violates the Friedman rule then the Ramsey problem is not time consistent. We first present an intermediate result which shows that when the Friedman rule does not hold the nominal debt sequences that can be part of the Ramsey policies are severely restricted. Since the solution of the Ramsey problem at 0 entails an interior solution for p 1 with 0 <p 1 < the first order conditions of the Ramsey problem at date 1 imply that for the Ramsey plan at date 0 to be time consistent it is necessary that M 0 and B 0 satisfy the 18

20 condition (31) M 0 + Q 1,t B 0,t =0. t=1 This condition, however, is not sufficient to eliminate the nominal forces that lead to time inconsistency. We show that the following stronger conditions are necessary. If at some date s, the Friedman rule does not hold so that Q s,s+1 < 1, then the present value of debt from date s +1on must be zero, so that (32) X 0= Q 1,t B 1,t = Q 1s Q s,s+1 Q s+1,t B 1,t. t=s+1 t=s We will assume there is some date, say r, in which consumption taxes are being levied so that U cr /U lr =(1+τ r ) > 1. We will assume that at this date r, the second derivatives satisfies the conditions (33) U mm + U lm < 0 if U m > 0, U ll + U lc 0, U mc + U ml 0. We show the following. Lemma 3. Assume that the solution to the Ramsey problem at date 1 has 0 <p 1 <, thereissomedates at which the Friedman rule does not hold, there is some date r at which consumption taxes are levied and at that date the conditions (33) hold. Then the value of initial nominal debt from s +1on is zero so that (32) holds. Proof. We establish the result by showing that if the assumptions of the lemma hold and (32) does not hold, then we can perturb the allocations and increase utility. If Q s,s+1 < 1 and (31) holds but (32) does not hold then we can make the present value of the government s nominal liabilities negative by a small change in Q s,s+1. This change, which may entail either raising or lowering Q s,s+1, is feasible since the Friedman rule does not 19

21 hold at s. Wedosobychangingc s,m s and l s in a way the satisfies the resource constraint but changes Q s,s+1. Then by lowering the initial price level p 1, we can generate any level of real assets for the government net of the cost of government spending. In the second part of the perturbation we lower the taxes at that date r in which we know positive taxes are being raised in a way that raises utility at that date and holds fixed the Q r,r+1 so that we know the first part of the perturbation still works. To that end note that positive taxes at date r implies that U lr <U cr.sinceu mr 0 we can increase c r and m r and decrease l r in a way that keeps Q r,r+1 constant, satisfies the resource constraint, and increases utility at date r. Clearly, by the implicit function theorem for a fixed Q r,r+1 and g r there exist functions l(c) and m(c) such that c, l(c) and m(c) satisfy (34) U m (c, m(c),l(c)) + (1 Q r,r+1 )U l (c, m(c),l(c)) = 0 and c + g r = l(c). These functions satisfy l 0 (c) =1and if U m > 0 then " # m 0 Umc + U ml +(1 Q r,r+1 )(U ll + U lc ) (35) (c) = U mm +(1 Q r,r+1 )U lm which is nonnegative under our assumptions on second derivatives. (Note that since U mm 0 then even if U cm > 0 the denominator in (39) is nonnegative since U mm + U lm < 0 and 1 Q r,r+1 1.) If U m > 0 then increasing c and thus l 0 (c) and m 0 (c) leads utility at r to change by U c + U l + U m m 0 (c) which is strictly positive since by assumption at r, U l <U c,u m 0 and m 0 (c) 0. If U m =0then by (24), U mc = U mm = U ml =0and thus many values of m solves (34). For the resulting utility U(c, m(c), l(c)) it is irrelevant which value we choose. For concreteness, we choose m to be the smallest value for which U m (c, m, l(c)) = 0. At such a 20

22 point increasing c and thus l 0 (c) leads utility at r to change by U c + U l which is also strictly positive at r. This establishes the contradiction. Q.E.D. Lemma 3 formalizes the variational argument suggested by Calvo and Obstfeld (1990). We use this lemma to establish that if the Ramsey problem is time consistent the Friedman rule must hold. We turn now to the general case. We assume the following regularity condition. At a Ramsey allocation, if the Friedman does not hold for some period t then (36) (R ct + R lt ) U lmt R mt (U lct + U llt ) 6= 0 holds, where R(c, m, l) =cu c + mu m + lu l. (This regularity condition ensures that the first order conditions to the Ramsey problem are not collinear.) Notice that this regularity condition is satisfied for preferences which are additively separable in leisure. For such preferences the term on the left-hand side of (36) reduces to R mt U llt and the first order condition (17) implies R mt = U mt /λ 6= 0. We also assume that at date 0 the budget is not balanced in that (37) τ 0 c 0 +(1 Q 0,1 )m 0 6= g 0 + b 1,0. where Q 0,1 =1+U m0 /U l0 and 1+τ 0 = U c0 /U l0. We then have Proposition 4. Suppose the initial nominal claims are zero in each period so that (14) holds, there is some date r at which consumption taxes are levied and at that date the conditions (33) hold, the regularity condition (36) holds, the normality condition U clt +U llt < 0 holds at all dates athe budget is not balanced at date 0. If the Ramsey problem is time consistent then the Friedman rule holds for each date s 1. 21

23 Proof. We prove this by showing that if the Friedman rule does not hold at some date s 1 the Ramsey problem is not time consistent. By way of contradiction we suppose that thefriedmanruledoesnotholdats 1 but the Ramsey problem is time consistent. We show that this implies at the date 0 Ramsey allocations the first order conditions for the date 1 problem cannot hold. We first show that all the terms involving the nominal debt are zero in the first order conditions for the date 1 problem are zero. Consider taking the first order conditions with respect to c t and l t in the date 1 Ramsey problem. We claim that at any date t 1 the terms of the form (38) Q 0t Q t,t+1 c t v=t+1 Q t+1,v B 1,v =0 Suppose first that t is some date at which the Friedman rule holds. Then U mt =0and from (24) it follows that Q t,t+1 / c t =(U mct U lt U lct U mt )/Ult 2 =0wherewehaveused Q t,t+1 =1+U mt /U lt. A similar argument implies Q t,t+1 / l t =0and hence the corresponding terms are also zero when taking the first order conditions for l t. If t is a date, like date s, at which the Friedman rule does not hold then (32) holds and these terms are zero as well. Moreover, for t =0the first order condition with respect to p 1 implies that terms of the form [M 0 + P Q 1,t B 0,t ]/p 1 =0. We now show that the multipliers on the implementability constraints for the Ramsey problems at date 0 and date 1, λ 0 and λ 1 satisfy λ 0 = λ 1. To see this consider the first order conditions to these problems for date s. From (14) we know that B 1,t is zero for each t 1, the first order condition for m s has the form of (17) which can be written as (39) R ms λ 0 + U lms λ 0 b 1,s = U ms 22

24 Combining the first order conditions for c s and l s gives (40) (R cs + R ls ) λ 0 +(U lcs + U lls ) λ 0 b 1,s = U cs U ls. We can regard (39) and (40) as a system of two linear equations in two unknowns λ 0 and λ 0 b 1,s. For the date 1 Ramsey problem since U ms > 0 for some s 1. Lemma2impliesthat P t=s Q 1,t B 0,t is zero so that the first order conditions for the date 1 problem can be written (41) R ms λ 1 + U lms λ 1 b 0,s = U ms (42) (R cs + R ls ) λ 1 +(U lcs + U lls ) λ 1 b 0,s = U cs U ls which is a system of linear equations in the two unknowns λ 1 and λ 1 b 0,s. By hypothesis the Ramsey problem is time consistent and hence the allocations in the two systems of equations (39) and (40) and (41) and (42) are the same. Our regularity condition (36) implies that there is a unique solution to both and hence that λ 0 = λ 1. NowwewillshowthatthereissomedateT for which b 0,T 6= b 1,T. By way contradiction suppose not. Since the solution to the Ramsey problem at date 1 is interior, the first-order condition for p 1 implies that (43) λ 1 p 1 [M 0 + Q 1,t B 0,t ]=0 t=1 Subtracting β times the date 1 implementability constraint from the date 0 implementability constraint gives (44) U c c 0 + m 0 U m0 + U l0 l 0 = U l0 b 1,0. 23

25 which implies that the budget must be balanced at date 0. This is a contradiction and hence theremustbesomedatet for which b 0,T 6= b 1,T. Using λ 1 = λ 0,b 0,T 6= b 1,T and evaluate the first order condition for the date 1 problem at T at the date 0 allocations gives (45) (R ct + R lt ) λ 1 +(U lct + U llt ) λ 1 b 0,T 6= U ct U lt. wherewehaveusedtheassumptionthat(u lct + U llt ) < 0. Thus, the continuation of the date 0 allocations cannot solve the date 1 problem and hence the Ramsey problem is not time consistent. Q.E.D. It is worth pointing out that if in the date 0 Ramsey problem the budget it balanced in every period then it is possible for the Ramsey policy to be time consistent even if interest rates are strictly positive in all periods. For example, consider an economy where g t is constant and b 1,t = B 1,t =0all t, so that there is no initial government debt. The date 0 Ramsey allocation is constant and prescribes a balanced budget. The Ramsey problem at any future date is the same as the date 0 Ramsey problem, thus its solution is the continuation of the date 0 solution. Therefore, the time 0 Ramsey problem is time consistent. Nevertheless, the solution of the date 0 Ramsey problem may have strictly positive interest rates, see for instance the example in Chari, Christiano and Kehoe (page??) Clearly, our results, especially Proposition 4, are at odds with the results in Persson, Persson and Svensson (1987). In their analysis they constructed a nominal debt sequences to be inherited by the date 1 government and supposed that taking this sequence as an initial condition the date 1 government chose an interior point for p 1, so that 0 <p 1 <. As Lemma 3 shows, unless the Friedman rule is satisfied, the government will choose the lower corner. Thus, there are restrictions on the nominal debt sequence they did not take into 24

26 account which invalidate their construction. 3. An analogy with a real economy In this section we develop the connection between our results for a monetary economy and related results for a real economy. Recall that in the monetary economy the there are two taxes, a tax on consumption goods and a tax on real balances, namely the nominal interest rate. Here we consider a real economy with two consumption goods and two consumption taxes, a common tax on both goods and an extra tax on, say, the second consumption good. The extra tax in the real economy corresponds to the nominal interest rate in the monetary economy. It is well-known that there is a close analogy between the optimality of uniform commodity taxation in the real economy, so that the second consumption tax is zero, and the optimality of the Friedman rule in the monetary economy, so that the nominal interest rate is zero. (See Chari, Christiano and Kehoe 1996.) We show that this close analogy extends to the time consistency of the Ramsey policies. In our monetary economy we showed that the incentive to inflate away nominal debt severely restricted the government at any date from using nominal debt to influence the behavior of its successors. In effect the government was endogenously forced to use only one type of debt, real debt to influence its successors. Here we imitate the endogenous restrictions on debt in the monetary economy by exogenously imposing that the government has access to only one real bond of all maturities. We show that with this restriction the Ramsey policies are time consistent if and only if it is optimal taxes are uniform in the sense that in each period it is optimal to tax the two consumption goods at the same rate. 25

27 The resource constraint for this economy is (46) c 1t + c 2t + g t = l t where c 1t and c 2t are the consumption goods, g t is government consumption and l t is labor. The utility function is (47) β t U(c 1t,c 2t,l t ) and the date 0 budget constraint is (48) q 0t [(1 + τ t )(c 1t +(1+τ 2t )c 2t ) l t ]= q 0t b 1t where b 1 is the sequence of initial real debt of all maturities. The first order conditions for c 1t and c 2t imply that U 2t /U 1t =1+τ 2t and hence, consumption taxes are uniform in the sense that τ 2t =0for all t if and only if U 1t = U 2t for all t. The implementability constraint for this economy is (49) β t R(c 1t,c 2t,l t )= β t U lt b 1t where R(c 1,c 2,l)=c 1 U 1 + c 2 U 2 + lu l. The Ramsey problem at date 0 is to maximize (47) subject to (46) and (49) given a initial real debt b 1. The following well-known result is the analog of Proposition 1. If the utility function is separable and homothetic, so that U(c 1,c 2,l)=u(w(c 1,c 2 ),l) for some homogeneous w, then uniform commodity taxation is optimal. We now establish a proposition that is the analog of Propositions 3 and 4 for this economy. Time consistency is defined in the obvious fashion. We assume the following regularity conditions. As in Proposition 3 we assume that at the Ramsey allocations U ct + 26

28 U lt 0, so that taxes are nonnegative and U lct + U llt < 0 which is essentially normality of consumption. Analogously to Proposition 4 we assume that in any Ramsey allocation if at date t, U 1t 6= U 2t then (50) (R 1t + R lt )(U 2lt + U llt ) (R 2t + R lt )(U 1lt + U llt ) 6= 0. This regularity condition ensures that the first order conditions for the two consumption goods are different from each other when the taxes are different from each other. It can be shown that the regularity condition (50) holds when the utility function is additively separable in leisure. Proposition 5. Assume our regularity condition holds and the budget is not balanced under the Ramsey policies in period 0. If the utility is weakly separable in leisure and uniform commodity taxation is optimal then the Ramsey problem is time consistent. Conversely, if the Ramsey problem is time consistent then uniform commodity taxation is optimal. The proof that if consumption taxes are uniform the Ramsey problem is time consistent uses a construction for the real debt b 0 and multiplier λ 0 identical to the one in Proposition 3. The proof that time consistency implies that consumption taxes are uniform is a simpler version of the one in Proposition 4 because we do not have to deal with nominal debt. To gain some intuition for this proposition consider the first order conditions of the date 1 Ramsey problem for a T period version of this economy. Adding the first order conditions for c 1t and l t we obtain (51) R 1t + R lt +(U 1lt + U llt ) b 0t = (U 1t + U lt ) λ 1 while subtracting the first order conditions for c 1t and c 2t we obtain (52) R 1t R 2t +(U 1lt U 2lt )b 0t = (U 1t U 2t ) λ 1. 27

29 The implementability constraint is (53) TX TX β t R(c 1t,c 2t,l t )= β t U lt b 0t. t=1 t=1 For a given allocation these equations form a linear system of 2T +1 equations in T +1 unknowns, which are the b 0t from t =1,...,T and 1/λ 1. To make the date 0 Ramsey problem time consistent we must be able to choose these T +1 unknowns to solve these 2T +1 equations. In general, this cannot be done and hence the Ramsey problem is not time consistent. In the case in which uniform commodity taxation is optimal, U 1t = U 2t and using weak separability, U 1lt = U 2lt and hence, neither b 0t nor λ 1,enter into (52). Thus, the linear system reduces to T +1equations in T +1unknowns and the Ramsey problem is time consistent. Conversely, if the Ramsey problem is time consistent it must be that T of these equations are redundant. Given our regularity conditions we conclude that U 1t = U 2t,namelythatuniformcommoditytaxationisoptimal. Consider the analogous equations for a T period monetary model in which the utility function is weakly increasing in m. (To make this analogy complete, we include a tax on real balances in period T to imitate the effects of nominal interest rates in a finite horizon monetary economy.) Here we impose that all of the nominal debts are zero, except for the first one for which we set B 0,0 = M 0. This is without loss of generality given our previous analysis: in each period either U mt > 0 andlemma3appliesoru mt =0in which case (24) holds. Either way, the terms in the first order conditions involving nominal debt drop out. For this economy, the first order conditions for c t and l t added together give R ct + R lt +(U lct + U llt )b 0,t = U ct + U lt λ 1 28

30 while the first order condition for money is (54) R mt + U lmt b 0t = U mt λ 1. The implementability constraint is TX TX β t R(c t,m t,l t )= β t U lt b 0t t=1 t=1 As in the real economy this is a linear system in 2T +1equations and T +1 unknowns. In general, this system has too many equations for the unknowns and the Ramsey problem is not time consistent. In the case in which Friedman rule is optimal, U mt =0and using either weak separability or (24), U lmt =0and neither b 0t nor λ 1 enter into (54). Thus, the linear system reduces to T +1 equations in T +1 unknowns and the Ramsey problem is time consistent. Conversely, if the Ramsey problem is time consistent it must be that T equations are redundant. Given our regularity conditions we conclude the T redundant equations are the first order conditions for money which implies that U mt =0,so that the Friedman rule is optimal. 4. Extensions to a cash-credit economy In this section we extend our results to cash-credit goods economy. This economy can be represented as a money-in-the-utility function economy but the assumption that U m 0 everywhere may well be violated. Here we show how to extend Propositions 3 and 4 to cover this economy. Consider an economy with cash and credit goods similar to that in Lucas and Stokey (1983). In particular, we let the period utility function be h(c 1,c 2,l) where c 1 and c 2 are cash and credit goods and l is labor. Assuming that end of period real balances are exclusively used 29

31 to purchase cash goods, we can map the cash-credit economy into our notation by defining U (c, m, l) =h(m, c m, l) where c = c 1 + c 2. Notice that even if h is strictly increasing in c 1 and c 2, the associated utility function U is typically not weakly increasing in m everywhere as we have assumed in Propositions 3 and 4. The monotonicity assumption U m 0 together with the assumptions of concavity and differentiability, imply that U mm =0whenever U m =0. For the cash-credit economy U m =0when h 1 = h 2. It can be shown that our monotonicity and concavity assumptions require that at the point where h 1 = h 2 we have h 11 = h 22 = h 12, which in turns imply an infinite elasticity of substitution between cash and credit goods, a very special condition. Proposition 3 holds for the cash-credit economy if we add an extra regularity condition (55) U mt =0and U mmt < 0 implies (U lct + U llt ) U mmt >U lmt (U mct + U mlt ). This regularity condition is met if U is such that m (c, l) is independent of l and leisure is a normal good. A sufficient condition for this to be true is that U is weakly separable in l, as in U (c, m, l) =u (w (c, m),l). It also holds when U is derived from a shopping time technology, as in U (c, m, l) =u (c, l + s (c, m)), where s is the time an agent that consumes c and has m real balances need to devote to shopping and where the utility is u is defined over c and total labor l + s. We then have Proposition 3. Assume that the initial nominal claims are zero period by period so that (14) holds and our strengthened regularity conditions hold. If the Friedman rule holds for each period the Ramsey problem is time consistent. The proof uses a construction similar to the one in Proposition 3. The only difference is that now we have to construct η t the multiplier on the constraint U mt 0 as well as the b 0 30