Equilibrium traits of durable commodity money

Journal of Banking and Finance 9 (1985) 5-34. North-Holland

EQUILIBRIUM TRAITS OF DURABLE C O M M O D I T Y MONEY Bart T A U B University of Virginia, Charlottesville, VA 22901, USA Received October 1983 A dynamic model of utility-maximizing agents explains why scarce, durable commodities are typically monetary. The model provides quantitative criteria for distinguishing between monetary and non-monetary durables, and is also used to analyze symmetallic equilibria. The model is then extended to analyze commodity-backed paper money. It is demonstrated that the backing generates trust in the paper money in the dynamic-consistency sense. The model predicts regular devaluations as an equilibrium phenomenon, but finds such behavior to be efficient. Finally, the results are integrated to make a technical point about dynamic models of pure fiat money.

1. Introduction

Recent articles have integrated commodity money with modern analytical models. Whitaker (1979) analyzes the macroeconomic effects of a producible commodity money in a Keynesian model. Barro (1979) analyzes the behavior or metallic standards and backed-paper standards in an adaptive expectations framework. Sargent and Wallace (1982) put commodity money in a rational expectations framework, drawing out implications about the inefficiency of commodity standards, and the mechanics of commodity-backed money. The purpose of this paper is to examine commodity money systems with an emphasis on transaction costs rather than on producibility. With Sargent and Wallace's demonstration that durable commodities are naturally monetary, the models here will assume perfect durability. Assuming this, it must still be explained why certain commodities, particularly gold and silver, are historically favored as monies. The private storage and use costs of these commodities as modeled here makes these choices natural. The simultaneous monetization of gold and silver and the determination of their relative value emerge from the model as well. The model captures the traits of a monetary commodity in parameters related to its quantity and costs. By considering extreme values of these parameters it is possible to examine two broader issues: the distinction between monetary and non-monetary commodities, and the distinction between commodity money and fiat money. 0378--4266/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

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B. Taub, Equilibrium traits of durable commodity money

The distinction between monetary and non-monetary commodities is usually depicted by a set of traits which define a superior commodity money: durability, scarcity, portability, divisibility, and so on. Sargent and Wallace show that durability is not only a desirable trait for a money, but the happy conclusion that such commodities are naturally equilibrium monies. A similar conclusion will be demonstrated here and for some remaining traits: not only will good monetary commodities have them, but they will determine whether or not the commodity is a monetary one. As for the second issue, fiat money is by default the sort of money assumed in most macroeconomic models: few investigators have attempted to model it as an equilibrium standard: Calvo (1978) and Klein (1974) are exceptions. Klein models private profit maximizing firms which produce private money in competition with each other. Calvo posits a seignorage maximizing government. Both investigators found that the central difficulty faced by suppliers was the production of trust: because a currency's value depends on its future value, trust in its future value is essential. Both Klein and Calvo found that, absent some constraint on suppliers, this trust will be lacking. The cost of producing commodity money, as shown in Sargent and Wallace, does provide such a constraint. Commodity money generates trust in another sense as well: its usefulness as a commodity. A well known and unresolved feature of dynamic equilibrium models of fiat money is the multiplicity of equilibria they display, including equilibria in which the (fiat) money is valueless. These valuelessmoney equilibria could be interpreted as ones in which trust is absent: the model which will be developed here shows how the utility of a commodity rules out such equilibria. This result is not completely new: Wallace (1981) eliminates the trustless equilibria by supposing the paper money to be backed by a perishable commodity through the government's commitment to a price level, enforceable by taxation if necessary. Wallace's backing rule is imposed by an exogenous government: this paper generates such a rule as an equilibrium. A motive for the rule must be provided to get such an equilibrium: it is the desire of individuals to economize on the costs of holding commodity money privately. Firms (vaults) which store the commodity issue paper receipts which circulate as money. The firms collect rents through excess issue of receipts. The commodity in the vaults then serves as a form of collateral and constrains the firms from excess issue, and so generates trust. The result vindicates advocates of gold-backed money because of its trustworthiness. The model of commodity vaults issuing receipts then permits the integrated analysis of the costs of each sort of monetary standard. The inefficiencies due to resource waste and the allocative inefficiencies which are their side effects are moderated by the existence of vaults: the vaults ameliorate the mistrust individuals feel toward pure fiat money. The vault

B. Taub, Equilibrium traits of durable commodity money

7

generates new inefficiencies however: it is of a sort similar to the dynamic inconsistency found by Klein and Calvo, but its extent is determined by technology. Debasement of monetary standards is perhaps the strongest empirical regularity in monetary economics. If it is to be realistic, then, a model of backed money must predict its frequent debasement, and the model here accords with the prediction. Surprisingly, the debasement can be viewed as a good thing: it is an efficient seignorage method. Because it is an equilibrium phenomenon, historical debasements are explained. Devaluation is the modern term for debasement: most macroeconomic models have prescriptions about its practice and predict its economic consequences. The model here has no prescriptions, and the predictions cannot be altered by policy. The model's policy conclusions, to the extent that policy itself is not an equilibrium phenomenon, pertain to the sorts of constraints which governments should place on intertemporal exchange: a fuller discussion of this is left to the concluding section of the paper. The analytical sections of the paper follow. The second section presents the basic model of a utilitarian commodity money which is costly to store. The third section expands the model to include two commodities. The fourth section contains the analysis of vaults which use the commodity to back circulating paper. The overlapping generations model as descended from Samuelson (1957) provides the analytical framework for all of the models. The model combines tractability with its provision of a milieu in which money acquires an equilibrium value as a medium of exchange between generations. A superstructure of technical traits can be built on the circulating medium, and their effects can be objectively weighed, because of the equilibrium nature of the model.

2. A single durable commodity The economy consists of a sequence of populations, in which each individual lives two periods, designated youth and old age. A new generation is born each period, so two generations exist contemporaneously in each period. They are asgumed to be able to engage in frictionless exchange: however, time's arrow constrains goods from being sent backwards through time. Young individuals are endowed with a fixed quantity of a completely perishable good, which shall be called ice cream for heuristic purposes. The old have no ice-cream endowment. Although they value ice cream consumption in old age, individuals are unable to store ice cream autarkically: this is the fundamental overlapping generations friction. Unlike the usual overlapping generations model, a durable good (call it gold) exists in the model as well. The aggregate quantity of the gold is fixed,

8

B. Taub, Equilibrium traits of durable commodity money

and production is impossible. Young individuals obtain utility from possessing it, but do not depreciate it or use it up in consumption. To possess it they must expend resources; the resource cost can be interpreted as storage costs or transport costs. The gold serves not only as a substance which produces utility directly, but as a bridge between present and future consumption of ice cream. The young purchase gold from the old with ice cream each period. The gold must have a positive price in terms of ice cream or its use as a bridge for consumption: if granite were the durable commodity it would have no market value, as it is too abundant. The degree of scarcity needed to induce this positive price will emerge as a prediction of the model. Young individuals solve the following maximum problem: (2.1)

max { U ( c , A ) + V(c')}, subject to y = c + pA + s(A),

(2.2)

c' =p'A,

(2.3)

where the following definitions apply:

U, V: y: c, c': A: p, p': s(.):

concave, increasing, differentiable utility functions, endowment of ice cream, current and next-period ice cream consumption, gold purchased and consumed, current and next-period purchasing power of gold in ice cream terms, storage cost function for gold.

The traits of the storage cost function are assumed to be

s(O)=O,

s'(A)>0,

s'(A)>0.

(2.4)

The utility functions U and V are assumed to have traits sufficient to force interior solutions, U'(0)= ~ ,

V'(0)= m.

(2.5)

The necessary condition for a maximum is (p+s'(A))Uc(y-pA-s(A),A)-UA(y-pA-s(A),A)=p'V'(p'A).

(2.6)

This is a demand condition for the nominal quantity of gold. The supply

B. Taub, Equilibrium traits of durable commodity money

9

condition is simply A=<_A,

(2.7)

where A_ is the per capita fixed supply of gold. An inequality rather than an equality condition is imposed because it is possible for individuals to hold less than the total available. The possibilities are most easily analyzed by assuming equality and then observing when the conditions necessary for the existence of an equilibrium are violated. Combining (2.6) and (2.7) under equality yields the equilibrium condition

(p+s'(A))Uc(y-pA_-s(A_),A)-Ua(y-pA-s(A_),A)=p'V'(p'A_).

(2.8)

The separation of utility into current levels, U, and future levels, V, makes the equilibrium condition (2.8) amenable to the use of a diagram to analyze it: the diagram was originated with Brock and Scheinkman (1980) in a similar model of fiat money. Three exhaustive cases are illustrated in fig. 1. The horizontal axis measures the equilibrium purchasing power of gold, p. The left- and right-hand side of (2.8) are plotted as functions of p. In each case, the right-hand side trivially exists and is positive for all positive values of p. The left-hand side is defined for 0

Monetary goods. Fig. 1 (a) represents the situation in which the marginal utility individuals obtain from the durable commodity is large, and hence

LHS

/ RHS

I

P

(a)

(b) Fig. 1

(c)

10

B. Taub, Equilibrium traits of durable commodity money

outweighs the marginal utility 'price' of storing it, s'Uc. Even if the market price of the durable commodity were zero, all of it would be held willingly by individuals. A market price of zero would not, however, be attainable in a stationary equilibrium. Because young individuals put a positive shadow value on the good it cannot be in fixed supply and still be a free good as is implied by a zero market price. There is an interior stationary equilibrium point, where the two schedules cross, at which the durable commodity has a positive market price. This price is the marginal value of the durable commodity in excess of its storage costs. This stationary equilibrium is dynamically unstable in the following sense: an initial level of purchasing power, say for example, /6, which is less than the stationary value,/0, will decline over time until it is zero after finitely many periods. Such dynamically unstable paths in which the purchasing power asymptotically evaporates, plague most sophisticated models of money. I Such paths can be ruled out here, however. Any path for which the purchasing power of money is zero in a finite time must maintain zero as a steady-state equilibrium thereafter. We have already seen, though, that this cannot be the case. The commodity's intrinsic value acts like a transversality condition in capital growth models: it rules out the 'bad paths'. This was noticed by Wallace (1981) who accorded paper money an intrinsic value with a government scheme which backed the money with the perishable good using taxes and transfers.

A non-monetary durable good.

If the fixed supply of the durable good is large relative to the private holding costs, fig. 1 (c) applies. At no price does its net marginal value exceed the marginal cost, and it becomes a free good. The only potential equilibria are of two kinds. The first are dynamic equilibria in which the purchasing power of the durable increases without limit: they can be ruled out because the left-hand side of (2.8) precludes values of p which exceed (y-s(A))/A_. The second kind of equilibria are those in which the market value of the durable is zero, and some quantity A*, A * < A, is held. In that instance the equilibrium is a special case of eq. (2.6),

s'(A*)Uc(Y- s(A*), A*) = U a(y-- s(A*), A*).

(2.9)

It is only in a limited sense that the durable is a free good. Eq. (2.9) is purely a static equilibrium condition: subsequent prices or quantities fail to enter. The benefits and the holding costs are incurred in the same period. If holding cost functions and taste for the commodity were heterogeneously distributed across individuals, then a competitive market would arise in the holding of the durable. The marginal holding cost s'(A*) would then be a 1See Scheinkman (1980) for a discussion.

B. Taub, Equilibrium traits of durable commodity money

11

market price which users of the commodity paid those who had low marginal holding costs. Durables with this property might include steel, aluminum, granite, or the often-lauded bricks. 2 These commodities are in fact non-monetary ones, and it is also true that less than the available known quantities are held, if we count refining as holding costs. The two cases illustrate a clean distinction between monetary and nonmonetary commodities. Monetary commodities must be scarce as well as durable according to the oral tradition: the model here confirms this notion but refines it by quantifying the necessary degree of scarcity. Monetary commodities generate a rent in excess of their immediate net marginal value in consumption, and this makes them a store of value. Non-monetary commodities, being too abundant, do not generate such a rent and carry no residual value from period to period. Any attempt to use them as value stores is defeated because they can be 'counterfeited' by anyone willing to incur the holding costs. Unfortunately, there is a third sort of equilibrium, as illustrated in fig. 1 (b). The left-hand side of (2.8) may intersect the right-hand side at two distinct points, yielding two positive stationary equilibrium values for the purchasing power of the durable commodity. By following the evolution of the diagram from fig. 1 (a) to fig. 1 (b) as A_ rises from an initially small value, we see that the larger stationary value of p is the monetary one, while the smaller rises from an initial value of zero, and is therefore non-monetary. On economic grounds, the monetary equilibrium is preferable, but there are no technical grounds for ruling out the non-monetary equilibrium. Moreover, the non-monetary equilibrium is dynamically stable, so there is a continuum of legitimate paths which are asymptotic to it. An improved model which eliminated the non-monetary stationary equilibrium would also eliminate these asymptotic 'bad paths'. The next section demonstrates a means by which this can be accomplished.

3. Multiple durable commodities Gold and silver have often been used simultaneously as monies, and nonmonetary durables, such as iron, have coexisted with monetary ones. These phenomena will be modeled in this section in order to answer the following questions. First, what determines the relative values of two monetary commodities? Second, what is the intellectual cost of excluding non-monetary durables from models of money? In particular, the omission of crosssubstitution effects between non-monetary and monetary commodities may distort the conclusions of models like that of section 2. Third, to what extent 2See Buchanan (1962) for a discussion of the brick standard. The model here does not address the stabilization qualities which were held up as the main advantage of the brick standard.

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B. Taub, Equilibrium traits of durable commodity money

do non-monetary equilibria like those in fig. 1 (b) remain? Fourth, can fiat money be considered as a special case of commodity money, and hence exist alongside it? The model is a straightforward extension of the model of section 2. Young individuals now have two durable commodities to choose from, denoted A~ and A2. They solve the following maximum problem: max {U(c, A1, A2)-4- V(¢')},

(3.1)

subject to

y=c + plA1 + paA2 + sl(A1) + s2(A2),

(3.2)

c' =P'~A1 + P'2A2.

(3.3)

Substituting, taking the first derivatives, and supposing that the market clearing conditions hold with equality as in section 2 yields the equilibrium conditions

(s'l(AO + pl)U~(y--sl(A1)--s2(A_2)-plA1-P2A_ 2, A1, h2) -- UA1 =pi V'(pia_l + ($2(A2) + P2)Uc -- U.t 2 = P'2 V'.

(3.4.1)

(3.4.2)

By multiplying both sides of each of these equations by the aggregate supply of the corresponding durable, _Ai, the equilibrium conditions can be expressed in real balance form,

(A_ls'I(A_1) + al)Uc- A 1Ua~ =a'l V'(a'1 + a'2),

(3.5.1)

(A-2s2(A-2) + a2)Uc- A-2Ua2 =a'2 V'(a'l + a'2),

(3.5.21

where ai denotes real balances of the ith currency. Summing these equations yields a difference equation in aggregate real balances, (al +a2),

(A_ls'I(A_I)+A_2s'2(A2)+al +a2)Uc-A_IUax-A2UA2=(a'I +a'2)V'. (3.6) This equation can be graphed in a Brock-Scheinkman diagram, illustrating a necessary condition for equilibrium: that aggregate real balances must be in equilibrium. The same three cases illustrated in fig. 1 are possible, including the 'bad paths' of fig. l(b). [-One need only suppose that A_I---A_2 and sl(')=s2(. ) to see that such equilibria are possible.] The various other equilibria will now be examined in turn.

B. Taub, Equilibrium traits of durable commodity money

13

(i) Both commodities are monetary commodities. If both A~ and _,512 are small, then eq. (3.6) will have a single interior stationary solution. The following propositions can now be demonstrated. Proposition 3.1. I f there is a unique stationary solution for aggregate real balances, then the individual stationary real balances al and a 2 are also unique. Proof If al and a 2 are not unique, they are constrained to sum to the constant level of aggregate real balances, a. Eqs. (3.5.1) and (3.5.2) can be solved for al and a2, ai = A_i((U A , - s'(A,)U,)/( U c - V')).

(3.7)

The right-hand side of this equation is fixed by the aggregate conditions in stationary equilibrium. This completes the proof. [] Proposition 3.2. I f there is a unique stationary solution for aggregate real balances, then real balances in the individual commodities must be stationary as well. Proof We can convert (3.5.1) and (3.5.2) into separate linear difference equations at the stationary equilibrium, ai = { V'lU~}a; + {At Val/U¢} - A,s'(A ,)}.

(3.8)

The braced terms are constants at the stationary equilibrium. The coefficient V'/Uc is positive and identical for both equations. As long as this coefficient differs from unity, al and a2 must therefore evolve monotonically in the same direction: this clearly violates the constraint that they sum to a constant. If V'/Uc equals unity, then non-stationary ai must evolve in opposite directions and will be non-convergent. This requires that one or the other becomes negative eventually, which violates the individual optimality conditions. (A negative price requires the seller to subsidize the buyer, which the seller avoids by leaving the market.) The only remaining solutions require real balances in each commodity to be stationary. This completes the proof. [] These two propositions may seem innocuous, but they have an important implication: if multiple equilibria are eliminated from the equilibrium, they are eliminated from the entire economy. Whether this will be so depends then on tastes, technology, and the abundances of the commodities. It might be that a single commodity, by its scarcity, economy of holding, and value in consumption, could drive the aggregate to reflect fig. 1 (a) rather than fig. 1 (b). In other words, the existence of gold eliminates the perverse equilibria

B. Taub, Equilibrium traits of durable commodity money

14

which might arise if, say, copper were the scarcest metal available. The opposite could be argued as well, though: an abundance of iron could spoil this effect and re-introduce the multiple equilibria problem. We will return to this question in a few pages. Having established the uniqueness and stationarity of real balances in the equilibrium is equivalent to establishing the same properties for their purchasing powers, Pi. This equips us to examine the relative values of two commodities which are both monetary. Dividing conditions (3.4) yields the ratio Px/P2=( U A 1- - s i ( A 1)Uc)/(U A2-SI2(A2)Uc) •

Suppose the first commodity is much scarcer than the second, just as gold is scarcer than silver: then _A1 < A 2. If the taste for each is about the same (conditional on equal consumption of both) and the holding cost functions are approximately the same, then UAI>UA2 and s](A0 a 2 .

Substituting the values of these quantities from eq. (3.7), the inequality becomes A : U A 2 > (sl -

sl)Uc.

The left-hand side of the equation can be made arbitrarily small by reducing the marginal value of gold, UA1, and raising that of silver, UA2. The changes in these marginal values can be constrained so as to keep aggregate real balances unchanged in eq. (3.6). A similar variation in the marginal holding cost functions can raise the right-hand side to an arbitrary degree. Therefore the real balances will depend on the elasticities of taste and holding cost, as well as relative scarcity. Debates over the superiority of silver standards versus gold standards, or about the 'correct' symmetallic ratio, seem unproductive in light of this finding: their values will be equated at the margin.

(ii) Both commodities are non-monetary.

Supposing the aggregates of both durables to be large results in a non-monetary equilibrium, like that of fig.

B. Taub, Equilibrium traits of durable commodity money

15

1 (c). Individuals hold less than the available quantity of each, and the net price of each in exchange is zero. The following conditions hold: (3.9)

s'i(A_*)Uc = U A,.

Dividing the two conditions yields the familiar static general equilibrium condition that the marginal rate of substitution in consumption is equal to the marginal rate of technical substitution, or the ratio of marginal costs. (iii) One monetary commodity coexisting with a non-monetary commodity. The world consists of commodities of all three types: monetary, durable non-monetary, and perishable. It will be most instructive, therefore, to examine a model with all three types. In this instance the equilibrium condition (3.6) becomes (AlSI'(A,_

_ ) +A~s'2(A~)+aO U c - A 1 U A 1 - A ~ U A , =alV'(a'l):

(3.10)

The equilibrium quantity of the non-monetary commodity is fixed by the condition t

,

(3.11)

SE(AE)Uc=UA~.

Substituting this into (3.10), the monetary equilibrium condition becomes (A 1si(A 1) + al)U*(Y* -- s,(A a ) - a , , A_1 ) - A, U A1 =a i V'(a'l), where _

:#

t

,

y*=y--A2s2(A2) ,

and U* takes the third argument of U, namely A~, as a fixed parameter. The equation resembles the equilibrium condition of the single commodity model, (2.8). Marginally raising or lowering the supply of the non-monetary commodity, A2, has no effect on the monetary equilibrium in (3.10): hence the influence of non-monetary commodities may be neglected, to a first approximation, in the analysis of monetary commodities. The existence of a single scarce commodity may be enough to drive the equilibrium condition to one like that of fig. 1 (a). The only requirement is that the following inequality be met: r

,

s I ( A 1 ) U c -- UA1 < 0 .

In other words, the monetary durable must have a positive shadow price if its market price is zero. The scarcer the monetary durable, the more the

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B. Taub, Equilibrium traits of durable commodity money

inequality will tend to hold: hence the existence of a commodity such as gold may be sufficient to generate a unique monetary equilibrium and coexist with non-monetary durables. (iv) Fiat money as a special case of commodity money. Fiat money has no intrinsic value in consumption and minimal holding costs. It has coexisted with commodity money, but the extent to which its value is influenced by commodity money has not been made precise. The model shows that, in fact, fiat money has no value if it coexists with a commodity money. Assume the first commodity to be monetary in the sense developed above, and the second to be fiat money. Then both the marginal holding cost of the fiat money and its marginal value in consumption are zero: hence, from eq. (3.7), the equilibrium real balances held in fiat money a r e z e r o . 3 One might have hoped that the unique economy-wide equilibrium generated by a single adequately scarce commodity would rescue models of fiat money from the non-monetary equilibria which plagues them. In fact, the commodity money drives out fiat money entirely: explaining the coexistence of the two sorts of money requires a model with more structure. One approach to adding extra structure, taken up in the next section, supposes paper money to be backed by a monetary commodity.

4. Specie-backed currency The Amsterdam Bank, founded in 1609, was an early example of a bank of deposit which issued receipts for specie, and the receipts then circulating as money. 4 What follows is a model of such a bank. Though such banks or vaults (as they will be called here) have become rare, they seem to have been a key step in the evolution of monetary systems, leading to the decoupling of paper money and the specie which backed it. Such vaults are similar to another, still common, institution: central banks which serve as domestic repositories of commodity reserves or international currencies. The reserves serve to back, at temporarily fixed exchange ratios, the local paper currency. The devaluations of the domestic currencies which regularly occur in such systems are ordinarily analyzed in a normative fashion: models predict the effects of exogenously imposed devaluations on the price level, money supply, and output. The model which is developed below predicts devaluations as an equilibrium phenomenon endogenous to the system. The return function for the central bank is modeled as profit: the assumption seems realistic, especially as a model of developing countries 3This assumes of course that the denominator U c - V' is non-zero. If the denominator is zero, then real balances in the first commodity will be undefined: this violates the assumption of a monetary equilibrium. 4See Vilar (1969, pp. 204-210).

B. Taub, Equilibrium traits of durable commodity money

17

which brazenly use exchange controls and inflation taxes to generate revenue. These taxes could not be as large as they are and be justified on efficiency grounds, and their ubiquitousness belies incompetence as their explanation. Individuals in the model are motivated to hold receipts rather than specie in order to economize on the holding costs of the commodity which were modeled in the previous sections. The vault receipts are assumed to be costless for individuals to hold. The vault itself incurs no costs in storing specie. Assuming the vault to be a monopolist, 5 it will extract a rent from its customers. The vault does this by printing and spending receipts in excess of the value of the specie which backs them. The timing of the issue combined with the cost individuals incur by demanding their specie permits an equilibrium in which the proper rent is collected.

Technology of the vault. The vault operates with a fixed sequence of events. (1) The young sell ice cream to the old in exchange for specie and receipts. The vault also purchases ice cream with the excess receipts it has created. (2) The vault announces the quantity of receipts which it plans to create in the following period and sets the exchange ratio, r', which will apply then. (3) The old [-who were young in step (1)] redeem certificates from the vault or deposit specie there if they so desire. This is done at an exchange rate, r, which was set before the announcement in step (2). (4) The vault creates the new receipts it announced in step (2) and step (1) is repeated. The key aspect of the timing is that the old are allowed to redeem their receipts before the main exchange takes place, at an exchange rate contractually set beforehand. If the vault's intended expansion of receipts as announced in step (2) is too great, then if the old keep their receipts, their original exchange ratio is no longer maintained, and their value becomes diluted. Anticipating this, they will redeem the certificates. The vault perceives the threat, however, and refrains from excessive receipt creation.

Maximum problem of the old. Because the old must decide whether to redeem notes from the vault, they have a non-trivial decision problem, max {c},

(4.1)

M+,A +

subject to c = p M M + + p.4A +,

SCompetition is analyzed in Taub (1981).

(4.2)

18

B. Taub, Equilibrium traits of durable commodity money A +-A_I=r(M_a-M+),

(4.3)

M+>=0,

(4.4)

A+>0,

r, PM, PA, M _ 1, A _ a

(4.5)

given,

where the notation means the following: £:

consumption of the old, M _ l : quantity of receipts carried over from youth, A-a: quantity of specie carried over from youth, quantity of receipts held after redemption, M+: quantity of specie held after redemption, A+: PM, PA: purchasing power of receipts and specie, respectively, r: the redemption ratio of specie to receipts. Since both the objective and the constraints are linear, the problem has a straightforward solution: in terms of final receipt holdings, it is M + = O,

Pa > PM/r,

0 < M + < M _ a + A _ a/r,

PA = PM/r,

M + =M_I+A_I/r,

PA < p ~ / r .

(4.6)

Thus, for example, if individuals can receive more value for specie than for receipts, they will redeem them all. This solution can be substituted into the objective function, yielding optimal consumption as a function of the states, including the carryovers from the previous period, p A ( A - 1 -]- r M _ 1),

PA < PM/r,

pAA-1 +PMM-1,

Pa = pM/r,

t),

PA > pM/r.

p M ( A - 1/r + M -

(4.7)

This value function enters into the maximization problem of the young. M a x i m u m p r o b l e m o f the y o u n g . The young face the usual problem of purchasing some good as a means of storing part of their endowment of the perishable good, but they must balance the effects of holding gold with holdings of receipts. Private gold holdings, A, incur resource costs s(A). The value of gold is, however, not subject to depreciation because it is in fixed supply. There are no resource costs of holding receipts, but the vault's ability to expand their issue makes them lose value.

B. Taub, Equilibrium traits of durable commodity money

19

Young individuals solve the following problem: (4.8)

max {U(c) + V(c')}, M,A

subject to

c = y - PMM - M>_0,

PA A

--

s(A),

A>0,

(4.9) (4.10)

and to the next period version of the value function, (4.7). All variables are as defined previously. Unlike the previous model, no intrinsic value is assumed for the gold. This reduces clutter, and the previous model gives license to focus on the monetary equilibrium. The young have no dealings with the vault, excepting their purchase of its excess issue, for two reasons. First, if the young could store specie in the vault and recover it costlessly, they could escape the holding costs entirely. Second, the withdrawal decision must be anticipated by the young when they make their initial purchases: this adds the dynamic element needed. As will be demonstrated below, convex holding costs and concave utility will generate interior solutions, in the sense that positive quantities of specie and receipts will be held. Assuming this result, the first order conditions are

PMU' =p'uV',

(4.11)

(pA + S'(A))U'=p'A V'.

(4.12)

Maximum problem of the vault. The vault operates under a regime in which it unilaterally sets the exchange ratio between specie and receipts, and expands the issue of the receipts, in the strictly timed fashion outlined above. It is assumed to have no costs of holding specie or of printing receipts. 6 Its profit, therefore, is the discounted purchasing power of the excess receipts it spends. Its maximum problem at time t is max{~~ttt) i=o Pipu(t+i)(M(t+i)-M+(t+i))} '

(4.13)

subject to eqs. (4.3), (4.6), (4.11), (4.12) and to the following conditions:

A(t+i)=A+(t+i),

(4.14)

pA(t +i)=pM(t +i)/r(t +i),

(4.15)

6These elements can be incorporated. See Taub (1981).

B. Taub, Equilibrium traits of durable commodity money

20

r(t + i)= (A_- A(t + i - 1))/M(t + i - 1),

(4.16)

and subject to the optimality of M(t+i), i= 1,2, .... The terms M(t+i) are the total quantity of receipts purchased in period (t+i) by the young. The vault's new issue is the excess of M(t+i) over the receipts the young purchase from the old, M+(t+i). Condition (4.14) is a market clearing condition for specie: it states that the old hold a final quantity of specie, A ÷, which they then sell to the young. This quantity comprises the total supplied to the young, since they do not obtain specie directly from the vault, and thus must equal their demand, A. Condition (4.15) is assumed to hold for all periods. It is equivalent to assuming that an interior equilibrium prevails in all periods. Although held as an assumption a priori, the following exposition will show that it satisfies the equilibrium conditions. Condition (4.16) is also an assumption: the exchange ratio r(t+i) could in principle be a choice variable for the vault. The vault could, in particular, choose to give less than the total specie deposited there, A - A ( t ) , in exchange for the receipts which will be redeemable for it, M(t). The exchange ratio could then be r(t) = O(t)(A - A(t) )/M(t), where O(t) is a positive fraction, chosen by the vault at time ( t - 1 ) . In fact a fractional 0(t) is essentially equivalent to full backing as stated in condition (4.16). That full backing is the vault's natural choice will be demonstrated later; for the time being, (4.16) is assumed to hold. Before attempting to solve the vault's maximum problem, it will be helpful to set out a definition of equilibrium. The nature of the equilibrium conditions permits the simplification of both the vault's maximum problem and the demand system.

Equilibrium. An equilibrium must clear all markets in all periods without violating resource constraints and must leave all arbitrage opportunities exploited.

Definition. An equilibrium is a set of prices {pM(t),pa(t)}, t=O, 1, ... such that: (1) (2) (3) (4)

Eqs. (4.11) and (4.12) are satsfied (intertemporal arbitrage for money). Conditions (4.6) are satisfied (arbitrage between receipts and bullion). Condition (4.14) is met (specie demand equals specie supply). The quantity of receipts M(t) demand equals the sum of those supplied by the old, M+(t), and the vault.

B. Taub, Equilibrium traits of durable commodity money

21

(5) The consumption of the perishable good by the young, the old and the vault, net of real holding costs, sums to the endowment of the young, y (the goods market clears). The analytical construction of an equilibrium is simplified because all terms involving receipts can be eliminated, leaving equations in terms of specie. These equations can then be analyzed like those of the pure specie economy of the first section. The following proposition clarifies the relation between specie and receipts.

Proposition 4.I. If equilibrium is such that conditions (4.15) and (4.16) hold, then the following conditions hold:

Proof.

pM(t)M(t-- 1)=pA(t)(A-- A(t-- 1)),

(4.17)

pM( t)M ( t) = (pA( t) + S'(A( t) ))( A - A(t)),

(4.18)

PM(t)M +(t) = p A(t)( A -- A(t)).

(4.19)

See the appendix.

[]

By substituting the equations of Proposition 4.1 into the first order condition, (4.12), we obtain the equilibrium condition (Pa(t) + s'(A(t)))U'(y-pA(t)A- s(A(t))-s'(A(t))(A- A(t)))

=PA(t + 1)V'(PA(t + 1)_A),

(4.20)

which is a function of specie and its purchasing power. Unlike the model of section 2, it has two unknowns, PA(t) and A(t), because A(t) is not fixed. The behavior of the vault closes the system. Its maximum problem can be converted to cX3

max ~ pis'(A(t+i))(A-A(t +i)), A(t) i =0

subject to O<.A(t)
B. Taub, Equilibrium traits of durable commodity money

22

max A(t)

{s'(A(t))(A-A(t))},

(4.21)

subject to

O
(4.22)

The maximand has an obvious interpretation: it is a price, the marginal holding cost incurred by individuals, multiplied by a quantity, the specie stored in the vault, A - A ( t ) . The vault's rent, then, is the marginal opportunity cost of holding which individuals would incur if they removed their specie and stored it privately. It is instructive to note that individual tastes do not enter into the vault's maximum problem: instead it faces a demand curve fixed by the technology available to individuals. Each individual is like a firm unto himself, supplying at marginal cost. The first order condition of the vault at an interior maximum is

A = s'(A(t))/s"(A(t)) + A(t).

(4.23)

There is no reason to presume the cost function will lend itself to a unique interior maximum: corner solutions are possible, just as in any monopoly analysis. Such solutions are of secondary interest in this context, and so the conditions in Proposition 4.2 will be assumed to hold in order to rule them out:

Proposition 4.2.

Assume the following conditions are met:

(i)

A_
(ii)

s'(O) =0,

(iii)

s(-) convex, increasing, and twice continuously differentiable.

0
Then there is a unique interior maximum for the vault. Proof.

See the appendix.

[]

The next proposition completes the housekeeping needed to proceed to the main discussion.

Proposition 4.3. Eqs. (4.20) and (4.23) define an equilibrium in which condition (4.15) was assumed to hold. Proof. The vault's condition (4.23), since it is based just on holding cost technology, fixes the quantity of specie the young hold, A(t). The demand

B. Taub, Equilibrium traits of durable commodity money

23

condition of the young, (4.20), then becomes a straightforward BrockScheinkman equation in the remaining unknown, PA(t). As in fig. 1, there may be no solution, or two solutions. Choosing the monetary solution, pA(t) is fixed and stationary. If the purchasing power of specie were then in excess of r(t)pM(t), then the old would withdraw it from the vault, violating the market clearing condition for specie, (4.14), because the vault chooses an interior quantity of specie, 0 < A < A by Proposition 4.2. Similarly, if the purchasing power of specie fell belowr(t)pM(t), then the old would deposit all remaining specie in the vault, again violating the interiority of the vault's holdings. This completes the proof. [] Because no intrinsic utility was assumed for the specie, there is still a possibility of multiple equilibria. The bad paths can easily be eliminated by assuming the specie has some intrinsic value if held privately, and that moreover the value can be expressed directly in terms of the perishable good. Call this function w(A): the equilibrium condition of the young is then

(pA(t) + s'(A(t))-- w'(A(t)))U'(y--s(A(t)) + w(A(t)) -(s'( A( t)) + w'( A( t)))( A - A( t)) - pa( t)A) = pA(t + 1) V'(pa(t + 1)A).

(4.24)

The relevant equilibrium concept is then represented by fig. 1 (a), and there is a unique and stationary equilibrium. This model has the added quality that the specie which is held by the young is like gold used in jewelry or in production: it yields utility only if held, while the specie in the vault yields no direct utility. The addition of intrinsic value for specie to the model, then, adds to its realism but does not add any new elements to the behavior of the equilibrium. The intrinsic value can therefore be suppressed as long as attention is focused on the monetary solutions of (4.20). Having established the existence and practical uniqueness of the equilibrium, we can examine some of its attributes. Proposition 4.1 showed that all expressions involving receipts had specie equivalents. The next proposition demonstrates that the economy is entirely equivalent to a pure specie economy.

Proposition 4.4. In equilibrium, the economy is equivalent to a pure specie economy in which vault services are paid for directly. Proof We have already seen that the vault's maximum problem can be stated directly as a function of specie and storage costs. If individuals had to pay for vault services directly, they would solve the following maximum problem:

B. Taub, Equilibrium traits of durable commodity money

24

max {U(y-pA(A + As)-psAs-s(A)) + V(p'a(A+ As))},

(4.25)

A, A s

where As is specie stored in the vault, and Ps is the price the vault charges for storage. The young thus pay to store some of the specie, then reclaim and sell it, along with the quantity they held privately, when old. The equilibrium conditions are

(Pa + ps)U'(Y - PaA_- s'(A)(A- A ) - s(A))= P'a V'(p'aA_), (pA + s'(A)) U' = P'a V'.

Combining these conditions yields

ps=s'(a), and

'A" (pA +S'(A))U'(y-p.4A-s'(A)(A_-A)-s(A))= p A' V " tPA_). The first equation defines the price appearing in the vault's maximum problem. The second is the equilibrium condition for the receipt economy, (4.20). This completes the proof. [] It is instructive at this point to look back at eq. (4.16). If the vault expands its issue of notes each period without changing the amount of specie it stores, A - A ( t ) , then the exchange ratio, r(t), must continually fall. Because the purchasing power of receipts is linked to the (constant) purchasing power of the specie by r(t) in eq. (4.15), it too declines. The vault's strategy, then, is to continually devalue its receipts. What Proposition 4.4 showed is that this is simply the method by which the vault collects seignorage. Devaluations of domestic currencies relative to gold or to international currency are common. The model suggests that the central banks which undertake these devaluations are simply taxing the domestic residents of the country in which the currency circulates. Countries with strict exchange controls on hard currency seem most typical: in such countries the private holding costs, s(.), are made artificially high, and so, consequently, are the potential rents from seignorage. Once devaluation-prone behavior is explained with this positive model, it remains to explain why poor countries seem to prefer seignorage taxes over other forms of taxation. The cost of collection of seignorage taxes is perhaps lower than others, 8 but the quantitative formulation of such a theory remains a topic for future research. 8Keynes (1923, p. 46) said: 'A Government can live by this means [printing paper money] when it can live by no other. It is the form of taxation which the public find hardest to evade and even the weakest government can enforce, when it can enforce nothing else.'

B. Taub, Equilibrium traits of durable commodity money

25

The prevalence of devaluation over direct charges as a method of seignorage suggests that some elements, missing from the model here, make devaluation preferable to direct charges. If the old faced a cost of making trips to the vault to deposit or exchange receipts, devaluation would enable the vault to collect seignorage without forcing the old to incur trip costs, making it the preferred method. Taub (1983) develops a model with trip costs. Welfare. It should be no surprise that the vault, by effecting resource savings, improves welfare. The resource savings are but one saving; a second is the increased efficiency with which the money circulates, as demonstrated in the following propositions: Proposition 4.5. Individual expenditure on combined private and vault storage is less than the expenditure in a pure specie economy, i.e., s(A)+ s ' ( A ) ( A - A ) < s(A). Proof

Obvious from the convexity of s(-).

Proposition 4.6. with the vault. Proof

[]

The purchasing power of the specie is higher in the economy

From the concavity of U(') and Proposition 4.5 we have (PA + s'(A))U'(y-- p_AA-- s'(A)(A -- A ) - s(A)) <(PA + s'(A))U'(y-pAA_-s(A_)).

Here the right-hand side is the left-hand side of the equilibrium condition for an economy with no vault, as in section 2 of this paper. The right-hand sides of the equilibrium conditions are identical whether there is a vault or not. The monetary intersection of the left-hand side with the right-hand side in a Brock-Scheinkman diagram of the vault economy is to the right of that in the vaultless economy. (See fig. 2.) This completes the proof. [] The higher purchasing power combined with the lowered resource costs of maintaining a specie standard effect an improvement in welfare: Proposition 4.7. A vault economy has a Pareto superior allocation of resources over a vaultless economy.

Proof

J.B.F.--B

The initial generation of old individuals, which first deposits the

26

B.

Taub, Equilibrium traits of durable commodity money LHS (no v a u l t ) LHS ( v a u l t )

PI*

PA

Fig. 2

specie in the vault, is better off because the purchasing power of the specie has risen. Subsequent generations' welfare may be examined by the use of a Fisher diagram (see fig. 3). The resource savings shift the budget available to the economy outward, as demonstrated in Proposition 4.5. The marginal rate of substitution between current and future consumption shifts from Cold

s

! y-

s(fi)

Cyo un g

\ y - s(A) - s ' ( A ) ( A - A)

Fig. 3

B. Taub, Equilibrium traits of durable commodity money

27

1/(1 + s'(A_)/p*) to 1/(1 +s'(A)/p**), which is unambiguously larger, as shown in Proposition 4.6. Since the marginal rate of substitution is a fraction in both cases, welfare is improved in the vault economy. This completes the proof. [] Fig. 3 illustrates the proposition. The indifference curve Us intersects the budget in the vaultless equilibrium. Indifference curve Uv intersects the outer budget in the vault equilibrium. Not only is it shifted out, but is higher than the comparable indifference curve had the marginal rate of substitution remained unaffected. The shift in the marginal rate of substitution is a response to the quality of the specie as a store of value. The view of history that paper currency backed by specie represented a social advance seems to be borne out by the model. Fractional reserves. It seemed arbitrary to suppose the vault should be forced to back its paper with all the specie stored in the vault, as in eq. (4.16). The vault, if left to choose the exchange ratio r(t+ 1) at time t, might choose to expropriate some fraction of the specie deposits each period, so that r(t+l)=O(t)((A_-A(t))/M(t)),

0 < 0 ( t ) < 1.

(4.26)

If the vault chose fractional 0(t), the fraction (1-O(t)) of the specie deposited would be at its disposal to spend or hold. If spent, this fraction would have to be held by the young, and the receipts would then be exchangeable for all of the specie remaining in the vault: that is, each receipt would represent a pro-rata share of the vault specie. This is, however, equivalent to the model with full backing already analyzed. By holding the expropriated specie rather than spending it, the vault would be foregoing potential profits. A positive stock of its own specie, held for all time, would represent unexploited profit. By withholding some of the stock of specie from the monetary economy permanently, the vault would raise the purchasing power of specie, and thus the purchasing power of its receipts. Because individuals need to hold less specie, however, their marginal valuation of the stock of specie falls, and so consequently do the potential rents available to the vault. The vault loses, on net, if it expropriates specie. Thus we have: Proposition 4.8. I f O(t+ 1) is a control variable for the vault at time t, it will choose O(t + 1)= 1.

28

Proof

B. Taub, Equilibrium traits of durable commodity money

See the appendix.

[]

Intuitively, the receipts are fractionally backed in a certain sense. If the old choose not to redeem them when the announcement of the new issue is made, then their value falls. By the time the old sell their receipts to the young, they no longer own all the vault's specie: some has in essence been expropriated. Proposition 4.8 is striking because it rules out a particular behavior: the 'destruction' of a deadweight which exists because of the costliness of specie storage. Because the vault's rents are founded on the opportunity cost of that storage, reduction of the extant specie harms it, although efficiency would be served by the reduction. While fractional reserves have been ruled out, the skeptical reader might ask what purpose reserves serve in the first place. Why doesn't the vault simply 'go off gold' and issue a pure fiat currency? The answer is that the timing of the vault's decisions would make such a system untenable. The vault would be able to 'reform' its currency each period, fully capturing the stock value of the new issue of currency each period. Individuals, rationally perceiving that their monetary wealth will be expropriated by the vault in their old age, will refuse to hold the currency: hence it will have no value in equilibrium.

Proposition 4.9. If receipts are completely unbacked, then they will have no value in equilibrium. Proof

See the appendix.

[]

The key to this result is the timing of the vault's policies. The vault is not required here to announce its growth factor for receipts in advance of the creation of the receipts. If they are unbacked, the result is the same as that found by Calvo (1978) and Klein (1974): the vault's incentive is to hyperinflate, and so money has no value. Yet if the receipts are backed, the vault's behavior is constrained by the threat of specie redemption. The vault still need not announce its next period issue: individuals know it will be constrained by fear of specie redemption, and hence trust it. The conventional wisdom, then, that commodity backing makes a currency more trustworthy seems quite true in a technical sense. It is possible to show [Taub (1982)] that a pure unbacked paper currency will be valued if individuals can be assured of the issuer's money growth rate for at least one period ahead of the period in which the currency is acquired. Whether such an announcement interval realistically models the world is open to question. The specie backing scheme achieves a similar quota of trust (although a different equilibrium money growth rate) with a shorter

B. Taub, Equilibrium traits of durable commodity money

29

lead time between the announcement and implementation of the money growth rate. Young individuals hold the paper currency willingly despite the fact that they will not hear the money growth announcement for a period. Their confidence comes from the knowledge that they will have the right to exchange their receipts when old at the prearranged ratio. The vault is therefore easier to implement in reality. A government's role in the system is simply the strict enforcement of the unilaterally contracted exchange ratio r(t), until period (t + 1) begins. 5. Conclusion

What distinguishes money from other goods is the dependence on trust in its future value. The previous section demonstrated that the conditions sufficient to create trust can be described by a set of technical features of the economy, in particular the vault's storage and timing technology. To the extent that the model captures the essence of specie-backed money and, more generally, of fixed-exchange rate regimes, it generates some objective conclusions about such systems. First, of course, is that the system of backing is essential for the paper currency to win acceptance. Advocates of the gold standard who take heart from this conclusion must face the second conclusion, however: that to the extent an economy's monetary instability is caused by the government's generation of inflation through the collection of seignorage taxes, the instability will remain under the gold standard through periodic debasements. A third conclusion is that the backing scheme results in an intimate connection of the paper currency to the monetary base (the commodity) through the exchange ratio between the two. The necessity of this intimacy would seem to cast doubt on the feasibility of schemes like that of Hall (1981), which base paper currency on commodity values but do not connect them to the aggregate physical quantities of the commodities. The connection of commodity to paper is one-to-one and hence the model might not seem to mirror fractional reserve banks. The fact is, however, that fractional reserve banks do indeed back their deposits by other assets 100%. The model characterizes a world with only a single durable real asset: the world has many. The model, of course, does not capture the considerations of risk which fractional reserve banks face, but it does show one important thing: that the monetary base, that is the outside money, constrains the issue of receipts, the inside money. In an extension of the model to competitive vaults, the inside money remains as well-behaved as the commodity base. The instabilities ascribed to free banking do not arise. There is one real-world monetary standard which controverts the findings of this paper: pure fiat money. The results of section 3 seem to rule out its separate existence if monetary commodities also exist. The model must be incomplete. What is the missing element? When fiat currencies are instituted,

B. Taub, Equilibrium traits of durable commodity money

30

they are usually conversions of a reserve-backed currency, retained as the legal tender but floated against the reserve. Private contracts, made before the suspension and denominated in the paper currency, remain in force. 9 Contracts typically offer real goods in exchange for particular quantities of paper currency: in that sense the paper currency is backed by contracts extant at the moment of suspension. This contractual backing is no different on an abstract level from the vault's contractual backing with specie. The extra element needed, then, to model 'pure' fiat money, then, might be its legal tender aspect. This will be taken up in future research.

Appendix P r o o f of Proposition 4.1. Substitution of condition (4.16) into (4.15) yields condition (4.17). To demonstrate (4.18), first divide condition (4.11) by condition (4.12). This yields PM( t)M( t) = (pa( t) + S'( A( t) ) )(p~( t + 1)M( t)/pA( t + 1)).

Now substitute the one step ahead version of (4.17) for the term on the right. To show condition (4.19), begin by eliminating the M+-term using the exchange constraint, (4.3) pM(t)M +(t) = pM(t)(M(t-- 1) + ( A ( t - 1 ) - A +(t))/r(t)).

By the market clearing conditions, (4.14), we have P~t( t ) M +( t) = PM( t)( M ( t - 1) + ( A(t - 1) - A( t) )/r( t) ).

Substituting the pro-rata exchange condition assumed in eq. (4.16) yields PM(t)M +(t) = p~( t ) M ( t - 1)((A -- A(t))/( A - A( t - 1))).

Finally, substituting (4.17) on the right-hand side yields (4.19). This completes the proof. [] P r o o f of Proposition 4.2. As the mathematics is standard, the proof will simply be sketched. Condition (i) guarantees that the second order condition is met. Assumption (ii) pins down the two ends of the maximand, at A = 0 and A = A, to zero. Assumption (iii) rules out perverse behavior between the endpoints. This completes the proof. [] 9Of course it is not always immediately clear cut: the legality of greenbacks was not settled for years after suspension.

B. Taub, Equilibrium traits of durable commodity money

31

Proof of Proposition 4.8. Because the vault can choose fractional redemption ratios, eqs. (4.14) and (4.16) have modified forms,

A(t)=A+(t)+(l_O(t))(A_-A(t-1))-(1-O(t+

1))(A - A(t)),

r(t)=O(t)((A-A(t- 1))/M(t- 1)).

(4.14') (4.16')

Eq. (4.14') takes into account the possibility that the young purchase their specie not only from the old but from the vault's expropriated reserves. These equations, using algebra like that in Proposition 4.1, yield

pM(t)M(t-- 1)=O(t)PA(t)(A--A(t-- 1)),

(4.17')

pM(t)M(t)=O(t + 1)(PA(t) + s'(A(t)))(A_ -- A(t)),

(4.18')

pM(t)M+(t)=O(t + 1)pA(t)(A_-A(t)).

(4.19')

The vault's maximum problem is oo

max ~ p' {PM(t)(M(t)-- M * (t)) + PA(t)[(1 --O(t))(A-- A ( t - 1)) t=O

- ( 1 - 0(t + 1))(A- A(t))-I},

(A.1)

subject to 0 NO(t)< 1, where the extra terms are the profits the vault obtains by spending from its free specie. When converted to specie form, each term takes the form

O's'(A)(A_- A) + Pa[(1 - 0)(_A- A_ ~)-(1 - 0')(A_-- A)]. Suppose the vault does in fact acquire specie. It can do this by purchasing it with receipts, but any receipts so spent cannot be used to purchase the perishable good: thus accumulation of free specie means the vault foregoes current profit, and in the accumulation process, the bracketed term is negative. Meanwhile, the term O's'(A)(A_-A) cannot be larger than the optimal value found in Proposition 4.2, because 0' is a fraction. Thus during the accumulation phase of the vault's strategy, profits are below their potential, and must be compensated by higher future profits. In the stationary state, after the vault has accumulated its desired stock of free specie, the bracketed term vanishes, and the vault's stationary profit thereafter is

O's'(A)(A_-A).

B. Taub, Equilibrium traits of durable commodity money

32

But this cannot exceed the profit generated in Proposition 4.2, and therefore fails to compensate for the losses incurred in accumulating the specie. This completes the proof. []

A second approach. A more formal approach takes into account the demand conditions facing the vault. Using (4.17')-(4.19'), the equilibrium condition for the young is (PA + s'(A))U'(y--s(A)--O'[paA + s'(A)(A--A)]-(1 --O')pAA ) (4.20')

= P'AV'(P'A(A_-(1 - O ' ) ( A - A))).

It is not difficult to demonstrate with a Brock-Scheinkman diagram that decreasing 0', holding A constant, increases the purchasing power of specie, PA.

The vault's formal maximum problem is (A.1) subject to (4.20') as a constraint. The first order condition for 0' is

s'(A)(A_ - A) + pa(A_ - A ) - pp'a(A_ - A ) - 2 + [O's"(A)(A - A) 0A --O' s'( A) + (pA-- pp'a)(1--O')] --~ + [ ( 1 - - 0 ) ( A - A _ , ) - (1 - 0')(_A- A)] -0--07= 0, where 2 is the multiplier for the constraint ( 1 - 0 ' ) ~ 0 , and where the partial derivatives are the implicit derivatives defined by (4.20'),

~A [(PA + s'(A)) 2U" + P'azV"](A - A) < O, ~ 0 ' - [s"U'-(PA+S'(A))2(1-O')U"-O's"(A_-A)U"-p~(1-O')V "] apa aO'

m

[(pA+s'(A))2U" +p'A2V"](A-A) <0. [U'-(pA+S'(A))(O'(A-A)+A)U" ]

We can solve the first order condition for 2, which should be positive if 0 = 1, thus: ~A 2= s'(A)(A_ -- A) + O'[s"(A)(A_ - A ) - s'(A)] - ~ + ( P A -- PP'A)(A - A) -(1

- o')

+

s'(a)Vv" + p V"]

-]

J

× {(I_O,)[(PA+S,(A))2U,,+p,A2V,,]+O,s,,(A__A)U,,_s,,U, } + 1 ~PA

B. Taub, Equilibrium traits of durable commodity money

33

Now consider the long run stationary equilibrium. The last bracketed term is zero, the first and third are positive, and the second term, OA

O'[s"(A)(A_-A)-s'(A)] 00" is ambiguous, if, however, the vault is maximizing, it will maximize s'(A)(A- A) in the stationary state, and hence this term will be zero. This leaves the righthand side positive, and hence 2 > 0, and therefore 0 = 1 in the stationary equilibrium. This completes the proof. []

Proof of Proposition 4.9. Consider the vault's maximum problem (4.13). If the receipts are unbacked, then r(t)=0, and hence the receipts which the old carried into period t, M ( t - 1 ) , cannot be altered by redeeming or depositing specie at the vault: hence

M+(t)=M(t-1).

(A.2)

The vault's new issue of paper at time t, M(t), will be some multiple of the previous issue, M ( t - 1 ) , M ( t ) = M(t-- 1)/R(t),

(h.3)

and hence the vault's problem can be rephrased as oO

max ~ p i ( 1 - R(t + i))M(t + i)pu(t + i),

(A.4)

R(t) i =0

subject to R(t)>0. By using eqs. (A.2) and (A.3) we can convert the demand condition, (4.11), to the equilibrium condition

pu( t)M ( t) U'( y - s( A(t)) - Pa( t)A( t) - P~t(t)M (t) ) = R(t + 1)pM(t + 1)M(t + 1)V'(pa(t + 1)A(t)+ R(t + 1)pM(t + 1)M(t + 1)). (A.5) The key thing to notice about this condition is that current real balances, p~(t)M(t), are entirely demand determined. Eq. (A.5) determines real balances as a function of future real balances and future money growth, over which the vault exercises no control at time t. Thus the vault takes current real balances as given and unaffected by the current growth factor of receipts, R(t). The solution to the vault's maximum problem is then trivial: R(t)=0.

34

B. Taub, Equilibrium traits of durable commodity money

This policy means money growth is infinite, or that there is a currency reform rendering the old receipts worthless. The solution will be the same each period. Since the young get no return from saving money, they will hold none: hence pM(t)=0. This completes the proof. [] References Barro, R., 1979, Money and the price level under the gold standard, Economic Journal 89. Brock, W. and J. Scheinkman, 1980, Some remarks on monetary policy in an overlapping generations model, in: John H. Karaken and Neil Wallace, eds., Models of monetary economies (Federal Reserve Bank of Minneapolis, Minneapolis, MN). Buchanan, J., 1962, Predictability: The criterion of monetary constitutions, in: Leland Yeager, ed., In search of a monetary constitution (Harvard University Press, Cambridge, MA). Calvo, G., 1978, Optimal seignorage from money creation, Journal of Monetary Economics 4, 503-517. Hall, R.E., 1981, Explorations in the gold standard and related policies for stabilizing the dollar, NBER Conference Paper 105. Keynes, J.M., 1923, A tract on monetary reform (Macmillan, London). Klein, B., 1974, The competitive supply of money, Journal of Money, Credit and Banking VI, no. 4. Samuelson, P., 1957, An exact consumption-loan model of interest with or without the contrivance of money, Journal of Political Economy 66, Dec. Sargent, T. and N. Wallace, 1982, A model of commodity money, Mimeo. (Federal Reserve Bank of Minneapolis, Minneapolis, MN). Scheinkman, J., 1980, Comment, in: John H. Karaken and Neil Wallace, eds., Models of monetary economies (Federal Reserve Bank of Minneapolis, Minneapolis, MN). Taub, B., 1981, Specie-backed currency, Mimeo. (University of Virginia, Charlottesville, VA). Taub, B., 1982, Consistency, competition, and the optimal quantity of money, Mimeo. (University of Virginia, Charlottesville, VA). Taub, B., 1983, Notes on specie-backed money with trip costs, Mimeo. (University of Virginia, Charlottesville, VA). Vilar, P., 1969, A history of gold and money 1450-1920 (NLB, London). Wallace, N., 1981, A hybrid fiat-commodity monetary system, Journal of Economic Theory, Dec. Whitaker, J., 1979, An essay on the pure theory of commodity money, Oxford Economic Papers 31.