Download A Model of Fiat Money

A Model of Fiat Money
David Andolfatto
October 2008
1
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.
2
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).
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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
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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
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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.
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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.
3
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 (ŷ − g) = 1.
(12)
Observe that ŷ is an increasing function of g; so that ŷ > 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 )ŷ) = 1;
(15)
with the amount of output secured by the government given by
ĝ = τ ŷ.
(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 ŷ = 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
ĝ = (1 − μ−1 )ŷ;
(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 ĝ(μ) = (1 − μ−1 )ŷ(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.
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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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