Price Level Determination when Requiring Tax Payments in Paper Money

Transcription

1 Price Level Determination when Requiring Tax Payments in Paper Money Hannes Malmberg and Erik Öberg Work in progress, September 25, 2013 Abstract We explore the consequences of requiring consumers to pay taxes in paper money in a general equilibrium model without frictions. We show that a government policy that determines 1) the price by which the government exchanges money for goods, 2) the nominal tax level and 3) the interest rate at which the government trade money intertemporally together form a sufficient condition for having a unique positive price sequence for money. We also show this set of nominal policy tools uniquely determines the level of real government consumption. 1 Introduction In this paper we develop general equilibrium model which pins down the price level as a function of fiscal policy. We show that Hahn s problem, We have benefited greatly from insightful comments and help by Phillipe Aghion, David Domeij, Tore Ellingsen, John Hassler, Per Krusell, Lars Ljungqvist, Dirk Niepelt, Lars EO Svensson and seminar participants at the SUDSWEC conference in Stockholm. All mistakes are our own. hannes.malmberg@iies.su.se; erik.oberg@iies.su.se 1

2 that of excluding equilibria were money has zero value, can be solved by requiring tax payments in government printed paper money. Given that we can exclude non-monetary equilibria, we also show that by emitting money via purchases of real goods, the government can uniquely pin down the price level with fiscal policy. The intuition of the model can be explained in a simple story. Suppose we live in a one-period world, where the state pays $10,000 to anyone working in the military and requires every citizen to pay $1,000 in a lump sum tax. Assume that taxes can be paid in dollars only. Given sufficiently hard punishment for not paying taxes, it is clear that a 1/10 th of the population will work in the military in order to achieve the necessary money for tax payments. Also, if there is an option is to produce y of a private good, labor market equilibrium requires that yp = 10, 000 p = 10, 000/y, where p is the money price of the private good. Government ensures a unique value for government consumption and price level given nominal government wages and taxes. Thus, the model ensures that paper money has positive value in equilibrium. In this paper, we will formalize and extend this story in a general equilibrium setting. We will show how the theory sheds light on the fiscal determination of monetary variables, including the price level, and the interconnection of bond issues, government consumption, government wage levels, tax levels and nominal interest rates. Our theory manages to determine the price level without resorting to the tools used by other fiscal explanations of monetary phenomena such as the fiscal theory of the price level (FTPL) and seigniorage explanations 2

3 of the price level. In the FTPL, the price level is determined by equating the real value of outstanding nominal debt with the real value of all future primary surpluses (?). Our model can explain the price level even when the government runs a balanced budget and there is no outstanding government debt. In seigniorage theories of the price level, a money demand function is posited and the monetary effects of fiscal policy is analyzed by finding the price level equilibrating money demand with supply given the government s seigniorage generating strategy, for example see (?). Our model, on the other hand, also works in a cashless economy where there is no transaction demand for money. It will turn out that our model has important implications for the interpretation of monetary phenomena. To illustrate its workings, we introduce progressively more complex models, allowing us to sequentially illustrate the different mechanisms in operation. We begin in section 2 with a one-period endowment model where we show the basic workings of price level determination in our theory. In section 3, we add a transaction induced demand for money and show that the proposed mechanism of price level determination still hold and also that the model provide a novel interpretation of the quantity equation of money. In section 4, we use a multi-period model to illustrate how bond policy, interest rates, government consumption, taxes, and the price level are jointly determined in our economy. In section 5, we perform a number of monetary experiments and discuss policy implications. We conclude the paper by discussing to model s ability to meet with requirements needed for any monetary theory. Before closing the introduction we review the history of legal restriction theories of money. The idea is old, notably put forth by? and?. The first attempt to formalize it is due to?. In a static Walrasian economy with 3

4 exchanges constrained to goods for money -barters (exchanging a real good for another good thus requires at least two exchanges) and where money were given as endowments to agents, he showed that requiring taxes to be paid in money can indeed ensure a positive value of money in equilibrium. However, as Starr also showed, this is not true for all parametrizations of the tax system. If the tax system is designed such that after taxes has been collected, the money endowment is not exhausted, money is no longer a scarce resource and therefore cannot maintain a positive value in equilibrium. This non-robustness was raised by? as a critique against the legal restrictions theory of money. However, as will be clear after having introduced our model this non-robustness arose in Starr s economy solely because money was introduced as an endowment and not treated endogenously.? also raised the issue that even if requiring taxes to be paid in money can support a positive price of money, it might be exceeded by the price implied by the transaction induced demand for money. If money is more highly valued as a medium of exchange than as a specie for tax payments, we cannot interpret the legal restriction equilibrium as the true monetary equilibrium. This question on price level determination is the principal motivation for the model we present in section 3. As we will show there, this critique does not take into account the equilibrium effects when the stock of money is endogenous - the price level will still be determined through the same channel as in the economy without any transactions motive for holding money. We therefore argue that legal restrictions can indeed provide a foundation of monetary analysis. There has also been an attempt to integrate legal restrictions in a search model of money (?). Agents are randomly matched with other agents, government purchasers and government tax collectors. Taxes are required to 4

5 be paid in government printed money. If not having enough money when meeting a collector, a large punishment is dealt. Tax collectors are only encountered directly after trade with other agents. Agents are thus willing to give up goods to the government purchasers in order to avoid the punishment after trading with other agents, which provides government money a positive value. Government money is made unique medium of exchange if the punishment is large enough. The model thus have the virtue of explicitly accounting for how government policy can promote the use of government money as medium of exchange in competition with other currencies, but the result hinges on the assumption that tax collectors are met either directly or never after a trade has been made. We treat the legal restrictions in the same manner as Starr (1972), by requiring taxes to paid at the end of every period with certainty instead. 2 One-period model A representative consumer seeks to maximise his utility U which is increasing in consumption C. His problem is given below. The consumer has an endowment of W real goods. There exist two markets in which trades can be done. A private market where the consumer can exchange goods for privately held money. The relative price of private money M p in terms of real goods is denoted p m. The inverse of p m will be interpreted as the price level. In the other market, the consumer can sell real goods to the government in exchange for government money M g. The government sets the price of government money in terms of real goods, which will be denoted p g. Ex ante we need to separate between government and private money because the government will be able to control the price p g but not the 5

6 market price of money p m. We capture this scheme in the first constraint of the consumer problem. At the end of the period, the consumer is required to pay a lump sum tax T in money. This is captured in the second constraint. The government will be a monopolist in the production of money and is thus indifferent whether the consumer pay his taxes in money received from a private source, M p, or via direct trade with the government M g. M p is government money that has been bought by the consumer second hand in the private market. The government only trades one way, and will not relinquish goods in return for money, so even if the consumer had a wish to go short in trade with the government, this is not possible. There is, however, no constraint on trading in privately held money. Finally, we also impose non-negativity on consumption. max U(C) (1) C,M p,m g s.t. C + p m M p + M g p g W (2) T M p + M g (3) M g 0 (4) C 0 (5) In this model, the quantity M g p g represents the real goods in government hands at the end of the period, and thus, government consumption is determined by the consumer s choice of M g p g. Thus, we will write G = M g p g, while recognizing that G is an endogenous variable. The government chooses the tax level T and how many goods it is willing to buy for one unit of money p g. We will assume that only the government 6

7 can print money and thus there is no net supply of money in the private market, i.e. M p = 0. To exclude a situation where people cannot possibly pay their taxes, we forbid the government to choose parameters such that money does not suffice to pay taxes even though people sell all their endowment to the government. That is, we require T p g W. Otherwise, the government is free to let its nominal parameters take any value. However, as soon as the government seeks to achieve a consumption target, it is constrained by having to set its parameters to make its consumption target a market outcome. Thus, if the government aims at consumption level Ḡ, it has the constraint Ḡ = M g (T, p g )p g (6) The reader may already see the direction of this exercise. As mentioned above, we must have M p = 0 in equilibrium. Hence, when the government sets T > 0 the second constraint gives us M G = T > 0. Arbitrage conditions then give us that p m = p g ; if p m > p g, the consumer would gain unbounded utility by selling goods to government and exchanging money for goods in the private market. If p m < p g the consumer would choose M p > 0, thereby violating the condition of zero net supply of money in the private market. As long as the government set p g > 0, government policy has, using nominal instruments only, ensured government consumption, given money value, and pinned down the price level. We provide a formal argument following a standard equilibrium definition: Definition 1 A competitive equilibrium as an allocation {C, G, M P, M g }, a 7

8 government policy {T, p g } and a price system {p m } s.t. Given prices and government policy, the allocation solves the consumer s problem The real resource constraint is satisfied: C + G W The money resource constraint is satisfied: M p + M g M Only the government can print money: M p 0 The equilibrium calculations are then simple: Proposition 1 Given that the government policy satisfies T p g < W and p g > 0, there exist a unique equilibrium where p m > 0. Moreover, in equilibrium government consumption G is determined by government policy {T, p g } Proof. Money market constraints require M p = 0. The consumer s problem gives p g p m with equality if M g > 0 As T > 0 and M p = 0, we have M g = τ > 0. Thus, we get and allocations are then p m = p g G = T p g C = W T p g We see from the resource constraint that this equilibrium exist if and only if T p g W. 8

9 Given that all money the is sent into circulation via government expenditure is taken out of circulation via tax payments, the price level is uniquely determined by the nominal price offered by the government is exchange for real goods. Note that the fact that quantity of money M = M g is endogenous make the equilibrium robust. Starr s analysis was on the presumption of money being a fixed endowment. Hence, taxation could support a positive price of money if and only if the tax system is designed such that for all money prices p m < b, the sum of taxes collected will exceed the money endowment. This non-robustness has been used as critique against the legal tender theory of money (Calvo, 2012). However, in the light of our model, this critique seems invalid. If the equilibrium quantity of money is endogenous, the consumers will never acquire more money than what is required to pay their taxes. As such, the only requirement on government tax policy is that the consumer knows what her nominal tax level when she engages in the market. Moreover, proposition 1 shows that any level of government consumption G [0, W ] can be implemented and is uniquely determined by the policy decision {T, p g }. It may thus be reasonable to understand the existence of money as a solution to a fiscal problem. In an unpublished manuscript under construction, we raise the question if requiring taxes to be paid in government printed money is an distortion-minimizing solution for implementing a government tax system given any government consumption target Ḡ when there is imperfect information on equilibrium prices. The general problem is that a real goods tax system will distort production if the relative taxation value of real goods differs from market relative prices. If requiring taxes to be paid in government printed money, no information on market prices is needed to implement a non-distorting tax system.. 9

10 3 One period model with a money-in-the-utility function The model presented in section II cannot in a strict sense be said to be a monetary model, since money did not carry any transaction-facilitating function. We will now incorporate such a function and study the implications of a legal restriction constraint. This can be done in many ways, we will proceed with the ordinary money-in-the-utility function for no other reason than its mathematical simplicity. The exercise is important for three questions. First, as Calvo (2012) conjectured, although a legal restriction might exclude equilibria were money is not valued, the transaction-facilitating function of money may raise its price above the price implied by a legal restriction. As such, we cannot treat the legal tender equilibrium as the true monetary equilibrium. However, we will show that the price level determination mechanism will not change when adding transaction purposes to our model. Rather, the equilibrating mechanism is adjusting the money stock to equate supply and demand at any given price level. Secondly, we will see that our model flips the interpretation of the quantity equation. In standard environments where money carries a transactionfacilitating function, the money supply is exogenous. Here it is endogenous, and as such, the interpretation of the quantity equation will go from prices to the money stock rather than the other way around. We proceed with setting up the model. We will consider two environments, differing in the timing of tax payments. If taxes are collected after trade in the private market has been conducted, the money holdings entering utility is M p + M g. If taxes are collected pre-trade, the argument is 10

11 M p + M g T. This timing difference will have implications for the money stock in equilibrium, and as such, also the amount of government consumption in equilibrium. Hence, we will study both environments. 3.1 Post-trade tax payments The model will be exactly the same as in section II, apart from the incorporation of real money balances in the utility function. We will assume that utility U is additively separable in consumption and real money holdings, U = u(c) + v((m p + M g )p m ), that both u and v are increasing and concave and that u satisfies the Inada conditions. It is common to assume that also v satisfies Inada conditions, since this excludes any equilibrium where money is not held and valued. However, this assumption seems empirically implausible since it is observable that trade can be conducted without money. We do not need it since non-monetary equilibria can be excluded with the legal tender constraint. The consumer problem is max u(c) +v((m p + M g )p m ) (7) C,M p,m g s.t. C + p m M p + M g p g W (8) T M p + M g (9) M g 0 (10) C 0 (11) The equilibrium definition is identical to the model in section II. As in section II, uniqueness follows: Proposition 2 Given that the government policy satisfies T p g < W and p g > 0, there exist a unique equilibrium where p m = p g. Moreover, in 11

12 equilibrium government consumption G is determined by government policy {T, p g } The proof is established in much the same fashion as in section II. It is left in the appendix. Notice that even though we have added a utility increasing function of money, the price level in equilibrium does not change. No arbitrage stills implies that given a government policy for p g we have that p m = p g. Only the quantity of money in equilibrium can be affected. The latter effect can be better understood by looking at the first order conditions: u (C) λ 1 + λ 3 = 0 (12) v ((M p + M g )p m )p m λ 1 p m + λ 2 = 0 (13) v ((M p + M g )p m )p m λ 1 p g + λ 2 + λ 4 = 0 (14) where λ 1,..., λ 4 are the Lagrange multipliers. Now, λ 3 = λ 4 = 0 holds in optimum. p m = p g, M p = 0 and C = W M g p m holds in equilibrium. We thus get the equilibrium condition u (W M g p m )p m + λ 2 = v (M g p m )p m (15) By the proof of 2:, we know that there exist a T such that for all T T, λ 2 = 0 and for for all T > T, λ 2 > 0 in any solution to the consumer s problem. Moreover, we also know that λ 2 is strictly increasing in the parameter value T whenever T > T. With this characterization at hand, equation 15 is easy to interpret. We know that u is increasing and v decreasing in real money holdings M g p m. Assuming that taxes are such that the tax constraint is not binding, the quantity of money is determined by the intratemporal tradeoff shown in graph 1. This means that the consumer will buy more 12

13 money from the government than the amount she is required to pay back in taxes. The equilibrium quantity of money is then equal to a model with only consumption in the utility function. Now consider the case where the tax level is increased such that the tax constraint becomes binding, say from T to T. This means that the required real tax payments T p m are higher than the amount of real money holdings that would be optimal without any legal restriction constraint. Then λ 2 > 0 which will bump up the left hand side of 15 such that it intersects the right hand side where the M g = T. This is illustrated in graph 2. 13

14 3.2 Pre-trade tax payments With pre-trade tax payments, the real money balances entering utility is (M p + M g T )p m. Everything else will be the same as in the model above. The equilibrium tradeoff between real money balances and consumption becomes u (W M g p m )p m + λ 2 = v ((M g T )p m )p m (16) The only thing that has changed is that the argument on the left hand side has been reduced by a constant. As such, 1 and 2 still applies. However, the tax constraint will start binding at a higher level. If taxes are high enough, the Inada conditions on u will shift all spending towards consump- 14

15 tion. 3.3 Comments We have thus seen that adding money in the utility function can increase the equilibrium quantity of money compared to the model with only consumption in utility. The requirement is that taxes are low enough. If we were to make the marginal utility of holding real money balances high enough, it is clear that the consumer will demand a greater money balances than the required tax payments for any given price level. A peculiar feature of the model is that it produces a reversed interpretation of the quantity equation. We cannot perform the classical monetarist experiment of doubling the money supply since the stock of money is endogenous. We can, however, pursue the experiment of doubling the price level, that is decreasing p g by 50 per cent. To hold government real consumption constant, this must be done in conjunction with doubling T. Equilibrium condition 16 then requires that M g is doubled. Our model thus implies a causal relationship from the price level to the stock of money rather than the other way around. We urge the reader to take this as a preliminary result. As we will see, we need to be more careful when analysing the stock of money in a multiperiod framework, as will be discussed in section 5. An well-known implication is that transaction induced demand for money creates the opportunity for seinorage taxing. When the quantity of money increases, so must government consumption G for any given price policy p g. Consumers will for the purpose of getting enough real money balances sell more real goods to the government than what is required to pay back in taxes and as such G = M g p g > T p g. 15

16 We stress that the equilibrating mechanism is by adjusting the quantity of money M g since the price level 1 p m is indirectly determined by the government via equilibrium condition p m = p g. Giving money an extra utility increasing function (such as facilitating trade) does not change the fact that government purchasing policy determines the price level. Hence, The critique put forth by Calvo (2012) that a legal tender constraint will not affect the price level when money carries such a function does seem invalid. 4 Infinite horizon model We extend our theory to an infinite horizon setting. To this aim, we make only one non-trivial modification to the one period-model, namely the introduction of a one-period risk-free money bond. We view money borrowing as running a deficit on one s tax account, or equivalently as getting a resource which is perfectly substitutable with M pt and M gt from the one-period model as means of paying taxes. This market structure allows us to understand the intertemporal constraints on government s behavior in our model, as well as laying foundations for more sophisticated models with realistic policy experiments. Also, as there is no uncertainty, government bonds will provide complete markets provided has a positive value in every period. Thus, the restriction of the set of intertemporal assets to government bonds is non-essential for the economic interpretation of the model. Our aim in this section is to establish the criteria for government money being valued, and express the value of money as a function of the government s choice variables. Furthermore, we want to show that for every feasible sequence of {g t } t=0, there exists a sequene of nominal government 16

17 prices {p gt } t=0, nominal taxes {T t} t=0 and nominal interest rates {i t+1} t=0 implementing {g t } as a unique equilibrium outcome. There is one important technical aspect of the model: the form of the no Ponzi-condition. As always with per-period trading, we have to rule out Ponzi schemes to ensure that households optimization problems are wellbehaved. Usually, this is done by noting that the market would not advance any loans to an agent planningan unsustainable sequence of debt holdings, as they will not be paid back. However, the government is not profit-making, so we cannot rely directly on this argument to conclude that a Ponzi game in money is unsustainable. Furthermore, the constraint when money has zero value in every period is not easy to interpret. In this case, people can run up debts growing as fast as the rate of interest, while the value of the debt in terms of individual goods endowments would always be 0. We need to exclude this case, as our price level determination mechanism relies on non-arbitrage conditions between government and private prices, and this mechanism requires an interior solution. If people can run Ponzi schemes, they can pay taxes by taking money loans, and pay interest rates and refinancing by lifting new loans. If the value of money is zero, the real value of debt is also zero. Thus, no one needs to work for the government, and in this case, we cannot exclude the non-valued money equilibrium. Thus, it is clear that we need a nominal no Ponzi-constraint. Analyzing the problem further, we note that if someone is allowed to run a Ponzi scheme, the government will advance loans, the repayments of which are inconsistent with the price sequence {p gt } that the government seeks to implement. Assuming that the government has a belief in its ability to implement a particular price sequence, we will constrain the agents to be consistent with 17

18 repayment of their loans if the state is successful in implementing its desired price sequence. This is also a canonical borrowing constraint in that it agrees with the natural borrowing limit whenever money is valued. 1 Below are the equations governing our model. Noe the similarity to the first period model. Equation (19) formalizes the tax account interpretation of money loans, but can equivalently be seen as a consolidation of a tax constraint with a flow account balance at the central bank. Equations (22)- (23) implements the nominal borrowing constraint discussed above. max {C t,m pt,m gt} t=0 U(C t ) (17) s.t. C t + p mt M pt + p gt M gt W t (18) where I t+1,s = s j=t+1 sup t 0 T t M p + M g (B t+1 (1 + i t )B t ) (19) C t 0 (20) M gt 0 (21) B t+1 W t+s p g,t+s = A t I t+1,t+s (22) (1 + i j ) s=1 A t < (23) Notice that the bonds B t+1 do not show up in the real budget constraint (18). However, B t+1 can be traded for money M pt as shown in the tax constraint (19). Assuming p mt = p gt 0 (as we will show below), (18) and (19) can then be consolidated: 1 The fact that this agrees with the natural borrowing limit means that this borrowing constraint on money loans leads to markets being incomplete once money is valued, a private market will yet again be redundant 18

19 C t + T t p mt + B t+1 p mt W t + (1 + i t )B t p mt (24) which the reader will recognize as a standard consumer per-period budget constraint. 4.1 Equilibrium in the multi-period model As in the single-period model, we are interested in how the government can use the tools at its disposal to implement a preferred sequence of government consumption. There are different choices for what the government takes as exogenous or endogenous: the government can for example choose to target bond issues or interest rates. However, throughout this section, we will assume that the government chooses sequences {p gt, T t, i t+1 } t=0. We will show that subject to very mild restrictions, every policy choice by the government implies a unique equilibrium. Secondly, we will show that any feasible sequence of government consumption can be implemented by an appropriate tax/pricing policy. Definition 2 A competitive equilibrium is a set of government policies {p gt, T t, i t+1 } t=0, a set of allocations {c t, g t, M gt, M pt, B t+1 } t=0 prices {p mt } t=0 such that: and a set of private money Consumers maximize (17) given prices, government policy, and constraints (18)-(22). Resource constraints for goods C t + p g,t M t W t, and private money M pt = 0, hold in every period. 19

20 Proposition 3 Suppose the government chooses a policy {p gt, T t, i t+1 } t=0 satisfying T t > 0 and i t > 0 in every period, the sequence {p gt } t=0 is bounded, taxes satisfy lim s s t=0 T t and t=0 T t I t < t=0 p gt W t I t, where I 0 = 1 and I t = t s=1 (1 + i s). Then, there exists Ī0,t s.t. for all I 0,t > Ī0,t there exist a unique competitive equilibrium, and it will have p mt = p gt. The proof is included in the technical appendix. The idea behind it is this. We first establish that p m,t 0 in every period. This is true as consumers otherwise will try to settle all their tax liabilities for all time by trading private goods for money at an infinite exchange rate in the period where the price of money is zero. Some technical issues arise because binding borrowing constraints on the optimal path may hinder the consumers to settle all tax liabilities in this manner. However, we show that such a tax repayment schedule is possible, and this establishes that M g,t = 0 for all time periods i.e. no one will ever work for the government. The second step of the proof is to show that an equilibrium where no one works for the government is not possible if taxes are positive. In particular, we show that not working for the government leads to an exponentially increasing nominal debt burden which will eventually break the borrwing constraint (22). Proposition 3 shows that any feasible government policy uniquely pins the down the price level of the economy and the level of government consumption. The imposed requirements are technical. The government bonds 20

21 serve the purpose of tax smoothing and hence consumption smoothing. Also equivalent to the one-period model, we can show that the government can use the set of policy tools to implement any feasible government consumption target {G t } t=0 : Proposition 4 Assume all conditions in Proposition 3. Given a government policy target Ḡ = {Ḡt} t=0 that satisfies G T < W t for all t, there exist a policy sequence {p gt, T t, i t+1 } t=0 s.t. in equilibrium. {M gtp gt } t=0 = {Ḡt} t=0 Proof. By the proof of proposition 1, we know that there exist a interest rate paths I 0,t such that p mt = p gt for all t. Assume the government set I 0,t such that this holds. In equilibrium, the Euler equation then become u (W t M gt p gt ) = β(1 + i t+1 ) p gt+1 p gt u (W t+1 M gt+1 p gt+1 ) (25) Together with the consolidated budget constraint, this equation uniquely pins down the the path of {M gt } t=0 = {Ḡt} t=0 for any choice of {p gt} t=0 = {Ḡt} t=0. 5 Policy implications in the infinite-horizon model 6 Conclusion This paper presents a skeleton for a novel monetary theory. There are four important features that we believe a good monetary theory should possess, which are the requirements we want to measure our theory against. 1. The theory should explain why money is valued, and exclude a zerovalue equilibrium. 21

22 2. The theory should uniquely pin down the price level as a function of the behaviour of the monetary authority. 3. The theory should be possible to integrate into a wide range of economic environments. 4. The theory should allow us to analyze monetary policy within the setting of the theory. In this paper, we have showed that our theory satisfies the first three requirements. We have demonstrated that non-zero taxes and the pricing policy of the government uniquely determines the sequence of prices in the economy. Furthermore, this has been done both in a single-period and a multi-period model, both with and without a transactions motive for holding money. All analysis has also been done in a standard neo-classical framework, which makes extensions to a wide range of canonical models straightforward. What we have not yet achieved within the model is to analyze monetary policy. For an interesting model of monetary policy, we neede a multi-period model with a transactions motive for money, and this is beyond the scope of this paper, but is the topic of a sister paper to this one (Malmberg & Öberg, 2013). It is also important to point out that this is not a model of fiat money as characterized by?. Wallace stressed that fiat money should be inconvertible and intrinsically useless. In our main model, money is intrinsically useless, but they are not in a strict sense inconvertible. They can be interpreted as being convertible into get out of jail free -cards, in the sense that the transfer of money to the government relieves you of the punishment of not paying taxes (as the requirement is strict, we can view this as an infinite punishment for paying to little taxes). Indeed, one purpose of our model is 22

23 to show that even after formal convertibility to specie has been abolished, government issued paper money in modern economies is not fiat money in the sense suggested by Wallace. Technical appendix Proof of Proposition 2. Optimality conditions of the consumer s problem are u (C) λ 1 + λ 3 = 0 (26) v ((M p + M g )p m )p m λ 1 p m + λ 2 = 0 (27) v ((M p + M g )p m )p m λ 1 p g + λ 2 + λ 4 = 0 (28) where λ 1,..., λ 4 are the Lagrange multipliers. Now, λ 3 = 0 by Inada and λ 4 = 0 holds in equilibrium since M p = 0. If p m > p g the consumer would sell all his endowment W to the government in return for W/p g money units. Since W/p g > T, the consumer can then buy (W/p g T )p m > 0 in the private market which violates the resource constraint. If p m < p g the consumer would buy all money needed for tax payments in the private market which would violate equilibrium condition M p > 0. We must thus have p g = p m in equilibrium. Optimality conditions then reduces to u (W M g p m )p m + λ 2 = v (M g p m )p m (29) We provide to intermediate results: Lemma 1 There exist a T such that for all T T, λ 2 = 0 and for all T > T, λ 2 > 0 in any solution to the consumer s problem. 23

24 Proof: We know that the solution is unique and interior. If M g = T we have that u (W T p m )p m + λ 2 = v (T p m )p m (30) Now, this can be true if and only if v (T p m )p m u (W T p m ) > 0. Since both u and v are concave there exist a T such that v (T p m )p m + u (W T p m ) 0 for all T T and v (T p m )p m + u (W T p m ) > 0 for all T > T. Lemma 2 There exist a T such that for all T > T, λ 2 is strictly increasing in the parameter value T in any solution to the consumer s problem. Proof: By 1, we know that there exist a T such that for all T > T, λ 2 > 0. Whenever λ 2 > 0, we have that λ 2 = v (M g p m )p m u (W M g p m )p m (31) by 29, which by strict concavity of u and v is strictly increasing in T. With these results we know that M g is strictly increasing in T in equilibrium. Uniqueness follows. Proof of Proposition 3. We first establish p mt > 0 for all t. Let us proceed by contradiction, and assume that t is the first period for which p mt = 0. We seek to show that the condition M ps 0 will be violated for some time period s, that is, we will have an excess demand for money in the private market at some point if p mt = 0 for any time period. To do so, we will first show that we have M gs = 0 in all periods. 24

25 If p mt = 0, M gs = 0 for all s t. That is, after the time period where money is free, no one will ever again sell goods to the government. Indeed, any plan involving M gs > 0 can be improved upon by reducing M gs to 0, increasing consumption by p gs M gs, and replacing the money shortfall by increasing M pt by M gs /( s j=t+1 (1 + i j)) and invest it in bonds. As p mt = 0, this can be done without forfeiting any time-t consumption. Furthermore, if the borrowing constraint is not binding at any time s < t, we also have M gs = 0. Indeed, if M gs > 0, we could increase consumption at time s by reducing M gs by ɛ and borrow money to make up the shortfall. We repay ɛ(1 + r s+1 ) (1 + r t ) in time-t at zero cost in consumption terms. Thus, if there does not exist any binding borrowing constraint at any time s < t, we have M gs = 0 for all s. Let us now assume that s is the last time period for which the borrowing constraint is binding, that is B s+1 = A s. However, as the borrowing constraint is not binding at any time between s+1 and t, we get that M g,s+1 = 0, and equilibrium gives us that M p,s+1 0. Therefore, using T t > 0, the tax constraint gives us B s+2 M p, s+1 + M g, s+1 (1 + i s+1 )B s+1 T t < (1 + i s+1 )B s+1 = (1 + i s+1 )A s = (1 + i s+1 )W s+1 p g, s+1 A s+1 < A s+1. Thus, we violate the borrowing constraint, which is not allowed. Hence, we 25

26 conclude that the borrowing constraint does not hold in any time period, and therefore, M g,s = 0 for all s. Now, we show that an allocation with M g,s an equilibrium. M p,s = 0, we get = 0 for all s cannot be Using the tax constraint and the equilibrium constraint B t+1 t I s,t T s (1 + i) t T 0. s=0 As sup t A t <, we eventually will break the borrowing constraint. Hence, we have shown that p m,t > 0 in all time periods, and thus fiat money is valued in our model. We seek to show that given a price sequence {p gt, i t+1, T t }, there exists a unique competitive equilibrium. First, we use sup t A t < + to consolidate the consumer s tax constraint (using that it will always hold with equality, as p mt > 0 for all t), to get t=0 M gt + M pt I 0,t = t=0 T t I 0,t. Thus, we reduce the consumer s choice set to private and government money holdings in every period. After substituting in constraint (18) into the consumer s objective function, and using M gt 0, the consumer s Lagrangian becomes ( β t u (W t M gt p gt M pt p mt )+µ t M gt +λ max M gt,m pt t=0 The f.o.c.:s become t=0 M gt + M pt I 0,t T t I t=0 0,t β t p gt u (W t M gt p gt M pt p mt ) = λ I 0,t + µ t (32) β t p mt u (W t M gt p gt M pt p mt ) = λ (33) I 0,t t=0 M gt + M pt I 0,t = 26 t=0 T t I 0,t. (34) ).

27 An equilibrium is a solution to the conditions above with M pt = 0 in every period, reflecting the zero net supply of private money. We note first that p gt = p mt for all t where M gt > 0. Furthermore, we have that M gt = 0 if and only if ( ) β t p gt u λ (W t ) λ/i 0,t W t φ β t I 0,t p g,t (35) where φ( ) = u 1 ( ). Using this result, and re-arranging (32), we get 0 if (35) holds M gt = ( 1 p gt (W t φ )) λ β t p mti 0,t otherwise Substituting this result into (34) with M pt = 0, we get ( )) 1 λ p gt (W t φ ( ( )) β t p mti 0,t λ I W t > φ I 0,t β t = I 0,t p g,t t=0 t=0 T t I 0,t. where I is the indicator function. As φ is strictly decreasing, the left hand ( ) λ side is increasing in λ. Furthermore, as λ 0, we have that φ β t I 0,t p g,t by Inada conditions. Thus, the indicator function will go uniformly towards 0 (as W t is bounded), which means that LHS goes to 0. Thus LHS ( ) λ starts below RHS. Furthermore, as λ we get that φ β t I 0,t p g,t 0 by the Inada condition. Thus, we get LHS t=0 W t p gt I 0,t > T t I t=0 0,t by the assumption in theorem. Thus, there exists a unique λ = λ compatible with equilibrium. We now go from the unique λ to fully characterize the equilibrium. For ( λ all t with W t φ β t I 0,t p g,t ), we know that W t = c t, and use (33) to get p mt = λ β t u (W t ) p gt. 27

28 And since by no-arbitrage p mt For t such that ( λ ) W t > φ β t I 0,t p g,t, we have M gt > 0 and hence µ t = 0. We can then use (32) to get ( ) λ c t = φ β t < W t. p gt I 0,t The quantities c t in periods with M g,t = 0 are already trivially determined, and so are prices p m,t when M g,t > 0. Thus, we have fully characterized the equilibrium. 28