A Model of Fiat Money - CiteSeerX

A Model of Fiat Money David Andolfatto October 2008

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Fiat Money

The standard dictionary definition of the term fiat is a (government) decree. Fiat money, in this context, might be interpreted (as it has been in the past) to mean a government-issued money that possesses value by virtue of a government decree (for example, as a means by which tax obligations can be discharged). This is not the way in which the term is used in the modern monetary literature. Fiat money is usually defined in the literature as an intrinsically useless unbacked token. At times, it appears that the token is usually restricted further to exist in tangible form—but this seems unduly necessary. The term intrinsically useless means that the token provides no intrinsic value; in particular, in the form of consumption or as input into production. The term unbacked means that no one is under any obligation to redeem the token for anything of intrinsic value. To be honest, I am not sure how to evaluate the profession’s obsession with fiat money. Presumably, it is motivated by the idea that much of what passes for money today appears to consist of intrinsically useless and unbacked notes issued by the government. Whether these notes are truly unbacked is open to interpretation; in particular, one might argue that the government is legally obligated to accept them as tax payments; see also Goldberg (2005). Moreover, it is so patently obvious (to me, at least) that the vast bulk of monetary instruments that have ever existed were clearly backed in some manner. But even if this is true, I think that it is still useful to study models of fiat money. The reason for this is because it is probably the case that most assets have a fiat component embedded in their valuation. One might interpret this fiat component as a liquidity premium (or a bubble component); that is, the market value of the underlying asset beyond its intrinsic worth. To put things another way, an asset may be valued not just for its intrinsic worth, but also for its expected ability to serve as a medium of exchange. In turn, this expectation may not rely entirely on the fact that the object in question is backed in some manner; rather, it may rely on a self-fulfilling expectation that others may believe the same thing.

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If a monetary instrument is completely unbacked, then it is difficult to see how it might circulate in any finite horizon model. In particular, no one is likely to want to work for or hold fiat money going into the final period T. By definition, fiat money is intrinsically useless; an no one is obligated to redeem it for goods. Anticipating this, fiat money will not be valued at T − 1 either. Working backwards in this manner, it follows that fiat money cannot be valued in any period when the horizon is finite. And so, to entertain the prospect of a valued fiat money, the model must have an infinite horizon. This was first accomplished by Samuelson (1958), who developed what was for a long time used as the standard framework for monetary theory—the overlapping generations (OLG) model.

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An Overlapping Generations Model

The OLG framework is in fact very similar to the Wicksellian environment we studied earlier. In the Wicksellian model, we considered a set of agents (t, i) ∈ T × [0, 1]; where T ≡ {1, 2, ..., N } for 3 ≤ N < ∞. A type (t, i) agent was assumed to have preferences, Ut (i) = −yt (i) + u(ct+1 (i));

(1)

where t = 1, 2, ..., N (modulo N ). This structure implies a lack of double coincidence. We can transform this Wicksellian setup into an OLG model by first assuming N = ∞ and dropping the ‘modulo’ qualification. That is, whereas agents are arranged in a circle in the Wicksellian model, they are arranged in a straight line of infinite length in the OLG model. Next, relabel the type N agents in the Wicksellian model as type 0 agents and assume that y0 (i) = 0. With these modifications, we can now redefine the set T ≡ {0, 1, 2, ..., ∞}. A type t agent is now said to belong to a generation ‘born’ at date t. A type t = 0 agent is said to be a member of the ‘initial old’ generation. Type t > 0 agents are said to belong to the set of ‘future’ generations. The initial old are interpreted as ‘living’ for one period only; all future generations are interpreted as ‘living’ for two periods. Hence, the initial old have preferences U0 (i) = u(c1 (i)).

(2)

That is, as with the type N agents in the Wicksellian model, the initial old wish to consume immediately. However, unlike the Wicksellian model, these agents will never be in a position to produce output in the future (they ‘die’ at the end of the period). An important implication of this is that the initial old have zero wealth (unlike the type N agents in the Wicksellian model, they cannot borrow by issuing securities representing claims to future output). 2

As for future generations, they have preferences exactly as in the Wicksellian model. The structure of this economy can be depicted diagrammatically as follows. Type\Date 1 2 3 ··· ∞ 0 u(c1 ) 1 −y1 u(c2 ) 2 −y2 u(c3 ) .. .. . . ∞ As with the Wicksellian model, there is a complete lack of double coincidence of wants. Note that one can assume that these agents live forever, just as in the Wicksellian model. Whether they are interpreted as having finite or infinite lives is inconsequential for the analysis.

2.1

The Golden Rule Allocation

It is useful, as always, to begin by characterizing an efficient (Pareto optimal) allocation. In the present context, the initial old present a bit of a problem; i.e., they are ‘different’ from all other types in that they alone are never in a position to produce output. This difference implies that there are many different allocations that satisfy the Pareto efficiency criterion. Typically, attention is limited to one of these efficient allocations—called the golden rule (GR) allocation. The GR allocation is derived as follows. First, we restrict attention to allocations for which the following is true: (ct+1 (i), yt (i)) = (y, y) for all future generations. That is, apart from the initial old, all generations produce and consume the same level of output. Basically, we are attaching an equal Pareto-weight to all generations t > 0 (so that there is a representative ‘young’ generation). An implicit Pareto-weight is then attached to the initial old by the fact that they are allocated consumption y (the production of the initial young). In this manner, the initial old achieve a utility payoff u(y) and every other type achieves a utility payoff u(y)−y. As it turns out, every allocation y ∈ [y ∗ , y] is Pareto optimal here. The golden rule allocation is given by y ∗ ; i.e., the solution to max {u(y) − y} . That is, in the golden rule allocation, the initial old consume no more than the old do in any other generation; which suggests an element of ‘fairness.’

2.2

Mechanisms

Naturally, if the planner could just force agents to work, then implementing y ∗ is a snap. But what if agents cannot be forced to work (or, equivalently, that they lack commitment)? Then as in the Wicksellian model, the planner will have to put incentives in place that induce agents to work voluntarily; the allocation must be made sequentially rational. As it turns out, the type of 3

incentive mechanisms we described earlier in the Wicksellian model will work here as well. With the exception of the initial old, allocations will have to be made contingent on individual trading histories. As with our earlier discussion, if individual trading histories are private information, then the planner will have to make use of some record-keeping technology. A computerized system of virtual accounts with PINs attached to the account holders will work. If this technology is not available, then physical tokens will work. The main difference between the Wicksellian model and the OLG model is with respect to the type N and type 0 agents, respectively. In the former setup, type N agents were the only ones assumed to have commitment; moreover, they had something to commit to. In the latter setup, type 0 agents have nothing to commit to (hence, we may assume without loss that commitment is completely absent). What this means is that there is no scope for a private liability to circulate in this OLG economy; the monetary instrument will have to be a purely fiat object.

2.3

A Competitive Monetary Equilibrium

As money must take a purely fiat form, the only competitive equilibrium without fiat money is autarky. It is possible, however, to implement the golden rule allocation as a stationary competitive equilibrium with fiat money. Imagine that society creates M units of perfectly divisible, durable, and noncounterfeitable money (tokens). Moreover, assume that this money is endowed to the initial old and that its supply is held fixed forever. At each date t = 1, 2, 3, ..., ∞, agents can buy or sell output at the nominal price 0 < pt < ∞. We can define the value of money by vt ≡ 1/pt . The initial old have a budget constraint, c1 ≤ v1 M. (3) A representative young agent, born at date t, faces the following set of budget constraints, vt mt ct+1

= yt ; ≤ vt+1 mt ;

where mt ≥ 0 denotes a type t individual’s nominal money holdings. Note that the two equations above can be combined to form the following budget constraint, µ ¶ vt+1 ct+1 ≤ (4) yt . vt Here, yt = vt mt corresponds to savings in the form of real money balances; and (vt+1 /vt ) is the (gross) real rate of return on savings (the inverse of the inflation 4

rate). Let me define R ≡ (vt+1 /vt ) (I am implicitly restricting attention to stationary equilibria, so that the inflation rate remains constant over time). The choice problem facing an initial old agent is trivial; i.e., simply set c1 = v1 M. That is, these agents simply dispose of all their (endowed) money balances and use them to purchase whatever output they can get. The choice problem facing a representative young agent is a little more interesting. Such an agent must form a forecast of R (an expected rate of inflation). Assume, for the moment, that R > 0. Then conditional on R, the choice problem is given by, max {u(Ryt ) − yt : yt ≥ 0} . The solution to this problem for real money balances;

yts

(5)

denotes both the supply of labor and the demand

u0 (Ryts ) = R−1 . Notice that since R is constant over time,

yts

(6) s

= y for all t > 0.

Next, impose the market-clearing restrictions, vt M = y s

(7)

for all t ≥ 1. This implies that vt = v = y s /M for all t ≥ 1. In turn, this implies that the equilibrium rate of return on money is given by R∗ = 1. Hence, it follows from (6) that the equilibrium level of output is given by y ∗ . The equilibrium price-level can then be solved by referencing the budget constraint for the initial old; i.e., y∗ v∗ = . (8) M The monetary equilibrium described here implements the golden rule allocation. There is no point in reiterating what the role of money is here (it plays the same role as in the Wicksellian model studied earlier). The only difference here is that there is no explicit or implicit commitment to redeem the fiat token at any date. What this means is that the exchange value of money is driven entirely by speculation. In particular, money will be valued only if it is expected to have value; and this expectation can be a self-fulfilling prophesy. To make this latter point clear, note that there exists another stationary equilibrium here in which money is not valued. Imagine, for example, that the young expect R = 0 (that is, any money they acquire when young will be worthless in the future). Conditional on this expectation, the demand for money is zero; y s = 0. But if the demand for money is zero, then its rate of return to anyone foolish enough to acquire it will also be zero. In short, if everyone believes that money will not be valued, then it will not be—confirming the initial expectation. 5

2.3.1

Fiscal Backing of Government Money

Implicit in all the arguments I have made so far is that society (say, as represented by a government) is limited in its power to tax. If memory is available, the worse punishment that society was assumed to possess is to threaten bad behavior with the prospect of eternal banishment (autarky). Even without memory, such a punishment is possible to deliver here making use of money. That is, if a person shows up to purchase output without money, no one will trade with him (and this is his only chance to purchase consumption in the overlapping generations model). But what if society had the power to physically extract resources from people? If the government had an unlimited ability to tax, then there would be no need for money (at least, in this simple overlapping generations model). In particular, the government could just administer a lump-sum tax equal to y ∗ on each young person and then transfer the output to the current old. Of course, implementing such a policy would require that society be able to observe an agent’s type t at each date. Assuming that there are no distinguishing physical characteristics between different types of agents (remember that “young” and “old” here are just labels that need not be taken literally), then such a policy may be difficult to implement. I will return to this issue later; for now, let me just assume that types are observable. Let me assume, however, that there is an upper bound on how much the government can tax people. In the present context, denote this upper bound by 0 < x < y ∗ . In this case, fiscal policy alone cannot implement the first-best allocation. However, the government can still issue money M along with the promise that each dollar can be redeemed (if the holder so desires) for x units of output. This is an example of government-backed money. It is interesting to note that money that is backed by the government in this manner in no way alters the optimal policy of simply creating M units of money and allowing it to circulate as a medium of exchange (in equilibrium, money would never be redeemed for output). In equilibrium, the exchange value of money is given by v ∗ = y ∗ /M. The intrinsic value of money, in contrast, is given by x/M < y ∗ /M. The difference between exchange value and intrinsic value (y ∗ − x)/M represents the added value that money possesses as an instrument of exchange. The same sort of phenomenon may exist for other assets as well; for example, as when a “thickly traded” stock commands a liquidity premium over a “thinly-traded” stock (with similar fundamental values) in an equity market. Finally, it is of some interest to note that while the policy of backing government money in this manner does not alter the properties of the monetary equilibrium, it does have the effect of eliminating the equilibrium in which money is not valued. Exercise 1 In the case of a pure fiat money instrument, there are two (sta6

tionary) equilibria: one in which money is valued and one in which it is not. Imagine now that money is partially backed by (the threat of ) taxation. Is it still the case that there are two (stationary) equilibria? In particular, is there an equilibrium in which money is valued for its intrinsic worth? Demonstrate mathematically one way or the other and explain.

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Taxation and Inflation

The OLG model can be used to study the implications of monetary and fiscal policies. To do so, let us imagine the existence of an agency called a government that has special powers and its own objective. It’s objective is simply to secure some fraction of the output produced in the economy. For simplicity, assume that the output acquired by the government is utilized in a manner that the general population does not value (e.g., an unpopular war). It’s special power lies in it’s ability to tax. But these powers may be limited, depending on the nature of the environment.

3.1

Lump-Sum Taxation

Assume that the government desires, for its own use, g units of output at every date. The most direct way to achieve this goal, assuming that this is possible, would be to simply take g units of output from those in a position to produce it (the contemporaneous young agents). In this case, the budget constraints are given by, vt mt ct+1

= yt − g); ≤ vt+1 mt .

(9)

ct+1 ≤ R (yt − g) .

(10)

Together, these imply, For a given inflation rate and tax, desired output is characterized by, u0 (R(yts − g)) = R−1 .

(11)

If the money supply is held constant over time, then the equilibrium rate of return on money remains R∗ = 1, so that the equilibrium level of output is characterized by, u0 (ˆ y − g) = 1. (12) Observe that yˆ is an increasing function of g; so that yˆ > y ∗ when g > 0. This is simply a pure wealth effect. That is, the tax policy reduces disposable wealth; and this implies that a young agent reduces his demand for all normal goods (current leisure and future consumption, here). Less leisure means more

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labor, so that production expands. However, while aggregate output expands, private consumption falls; as some output is now allocated to the public sector. Note that this particular policy assumes that the government can distinguish between types of agents. The government could get around this problem by simply applying a uniform lump-sum tax across all agents (the old would pay their taxes in the form of money, which the government could then spend to purchase output). A more serious problem arises if people can somehow evade the tax authority. I consider two examples of how the government may have to modify its tax-collecting activities when some degree of evasion is possible.

3.2

A Sales Tax

Imagine that lump-sum taxes are not feasible (e.g., people can simply hide from the tax man). On the other hand, imagine that the government can observe market sales of output. Thus, while people may have the ability to hide, doing so would mean forgoing market exchange. I assume that in this case, the government can levy a tax 0 < τ < 1 that is proportional to the value of market sales. In this case, the government’s budget constraint is given by g = τ y. In what follows, let me assume that the tax rate τ is exogenous (so that g is then determined residually from the government’s budget constraint). The budget constraint facing a representative young agent is in this case given by, ct+1 ≤ R(1 − τ )yt . (13) The desired level of output is characterized by, (1 − τ )u0 (R(1 − τ )yts ) = R−1 .

(14)

With a constant money supply, R∗ = 1 again. Hence, for a given tax rate, the equilibrium level of output is characterized by, (1 − τ )u0 ((1 − τ )ˆ y ) = 1;

(15)

with the amount of output secured by the government given by gˆ = τ yˆ.

(16)

When lump-sum taxes are feasible, an increase in g leads to an increase in output, as agents try to make up for their lower wealth. Only the wealth effect is at work here, as a lump-sum tax does not distort the relative price of leisure (today) vis-à-vis consumption (tomorrow). The situation is different in the case of a sales tax. This tax makes consumption (tomorrow) more expensive relative to leisure (today). Because of this relative price distortion, there is a substitution effect at work as well (agents are induced to substitute into the cheaper commodity).

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For some commodities, the wealth and substitution effects work in the same direction. For example, a higher sales tax reduces wealth and makes future consumption more expensive; both of these effects serve to reduce desired consumption. On the other hand, while a higher sales tax reduces wealth, it also make leisure relatively cheaper. Here, the two effects work in opposite directions. Less wealth means less leisure, but cheaper leisure means more leisure. Which effect dominates generally depends on parameters. If the substitution effect dominates, then a sales tax will have the effect of reducing labor supply and the level of output. This latter possibility introduces an interesting phenomenon. Unlike the lump-sum tax case, the government budget constraint here is the product of the tax rate τ and the tax base y. A higher tax rate has the effect of increasing government revenue, but it may at the same time reduce the tax base. To the extent that this is true, there may be a limit to how much the government can extract by way of taxation. Indeed, some have gone so far to suggest that it is frequently the case that total tax revenues might be increased by lowering the tax rate. This type of argument forms the foundation of what is commonly labeled supply-side economics.

3.3

An Inflation Tax

Imagine now that the government cannot observe all market exchanges, so that not even a sales tax is feasible. This is, in fact, an accurate description of the constraints facing many governments to a varying degree (e.g., think of the underground economy). This ‘problem’ of government finance appears to be especially acute in lesser-developed economies (where much of the population is dispersed across geographically remote areas). If the government has monopoly control over the money supply, however, it may potentially finance its expenditures by simply printing new money. Revenue generated in this manner is called seigniorage. Imagine that the government expands the money stock at some constant rate μ ≥ 1; so that Mt = μMt−1 , with M0 > 0 given. The government’s budget constraint is now given by, g = vt [Mt − Mt−1 ] ;

(17)

or, using the fact that Mt−1 = μ−1 Mt , g = (1 − μ−1 )vt Mt .

(18)

As an agent faces no explicit taxes, his behavior is characterized by, u0 (Ryts ) = R−1 .

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(19)

The market-clearing conditions at each date require yts = vt Mt . In a stationary state, yts = y for all t, so that market-clearing at each date implies, µ ¶ µ ¶ vt+1 Mt = ; (20) vt Mt+1 or R∗ = 1/μ. With the equilibrium inflation rate determined in this manner, it follows that the equilibrium level of output is determined by, ¢ ¡ μ−1 u0 μ−1 yˆ = 1. (21)

Note the similarity between this expression and condition (3.15). In particular, if μ−1 = 1 − τ , then they are identical. To put things another way, the money growth rate (inflation rate) acts like a distortionary tax. In fact, if we combine the market-clearing restriction with the government budget constraint, we get gˆ = (1 − μ−1 )ˆ y;

(22)

which looks very similar to condition (16). What this analysis suggests is that even if the government cannot employ a sales tax, it can generate revenue in basically the same way by simply printing money (and creating an inflation). While printing money does not sound like taxation, the analysis here demonstrates that inflation is indeed a tax. The inflation reduces the rate of return associated with working for money today and holding it into the next period to purchase output. As with the sales tax, there is a substitution effect at work here that places the same sort of limits on the government’s ability to extract resources from the economy. Exercise 2 Conditions (21) and (22) together determine a seigniorage function gˆ(μ) = (1 − μ−1 )ˆ y (μ). An analytical solution is available if we assume u0 (c) = c−σ , σ > 0. Assume that σ > 1 and demonstrate there exists a revenuemaximizing inflation rate μ∗ . Exercise 3 The analysis above assumes that individuals can only use one money. Assume now that there exists another (foreign) money, say, the U.S. dollar that people may use as well. Moreover, assume that the USD inflation rate is given by some number μ0 < μ∗ . How does the availability of an alternate currency impose further constraints on the ability of a domestic government to extract seigniorage revenue? Why is it, do you think, that many governments impose ‘foreign currency controls’ forbidding, or limiting, the use of foreign money?

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References 1. Goldberg, Dror (2005). “Famous Myths of Fiat Money,” Journal of Money, Credit, and Banking, 37(5): 957-967. 10

2. Samuelson, Paul A. (1958). “An Exact Consumption-Loan Model of Interest With or Without the Social Contrivance of Money,” Journal of Political Economy, 66: 467-482.

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