Economic Theory 21, 767–782 (2003)

Research Articles

Is gold an efficient store of value?

Pradeep Dubey1, John Geanakoplos2, and Martin Shubik2

1 Department of Economics, SUNY at Stony Brook, Stony Brook, NY 11794-4384, USA2 Cowles Foundation for Research in Economics, Yale University, Box 208281,

New Haven, CT 06520-8281, USA

Received: January 2, 2001; revised version: February 22, 2002

Summary. Gold and tobacco have both been used as commodity money. Onedifference between the two is that gold yields utility, on account of its beauty,without diminishing its quantity. Tobacco yields utility when it is consumed. If thiswere the only difference, which would be the better money?

Keywords and Phrases: Gold, Tobacco, Store of value, Liquidity.

JEL Classification Numbers: D4, D5, D61, E40, E41.

1 Introduction

Paper money is a relatively recent innovation. For much of history, various com-modities have played the role of money, serving both as a medium of exchangeand as a store of value. The characteristics that make a commodity suitable to useas money were discussed as far back as Aristotle [1], who emphasized portabilityand intrinsic value. Since that time many distinguished authors have also writtenon this subject ([2,4,5,6,7,9,10,11]), naming other virtues, such as divisibility anddurability.1 In this paper we bring a new aspect of a commodity to light, whichseems curiously to have gone unnoticed, but which has a bearing on its efficiencyas a money.

Correspondence to: J. Geanakoplos1 Modern mathematical analyses have not called attention to new properties of a commodity that

affect its use as money. In both [3] and [8], for example, the importance of the intrinsic value of money(i.e., its desirability vis-a-vis other commodities) is reiterated in a precise manner. It is shown in [3] thatif this value is high uniformly across all agents, and if money is both plentiful and initially distributedin a non-skewed manner, then it will be efficient for the economy, provided that time lasts for just oneperiod.

768 P. Dubey et al.

Our point can be best put across by comparing gold and tobacco, both of whichhave had reputable runs as money. (See, in particular, the historical account in [5].)One difference between the two is that gold yields utility, on account of its beauty,without diminishing its quantity. Tobacco yields utility when it is consumed. If thiswere the only difference, which would be the better money?

Imagine a two-period economy with no uncertainty and with a single durablecommodity. The agents agree on this commodity as money; all trades take placebetween this money and each of the other perishable commodities. Suppose equi-librium is achieved under conditions of perfect foresight and perfect competition.Suppose further that in the first period each agent spends less than his endowmentof money, in fact inventorying a positive amount into the second period, where onceagain he spends only a part of it. Thus the constraint that purchases be financed outof money on hand is not binding on anyone: there is perfect liquidity. Is the resultingallocation necessarily efficient, or does the efficiency of equilibrium depend on anyof the properties of the commodity money?

We show that if money is “gold-like,” i.e., it yields a stream of utility (say asjewelry) without getting dissipated, then equilibrium is inefficient. In particular,gold itself is an inefficient money. This is so even if its utility services are notdiminished when used as money, e.g., even if gold coins are just as beautiful tolook at per ounce as gold necklaces. To the extent that its use as money also re-duces its services, the inefficiency of gold is still worse. On the other hand, if themoney is “tobacco-like,” and yields utility only at the point of consumption (when itdisappears – into smoke!), then equilibrium under the above conditions is efficient.

The question of the relative efficiency of gold and tobacco money cannot beposed in the standard general equilibrium model of exchange because there is nomoney explicitly in that model. Purchases and savings are recorded in imaginaryunits of account. (Implicitly the model allows for fiat money with unlimited credit.)If the general equilibrium model is modified so that payments must be made ina specific commodity, then one necessary property of an efficient money comesimmediately to light: it must be in sufficient supply.

The role of money in transactions at a single point of time is perhaps secondaryto its role as a store of value. To capture this dual role of money we must turn to amultiperiod setting. Then it becomes necessary to distinguish several more featureswhich are not highlighted in the general equilibrium nonmonetary model, or in a oneperiod monetary model of exchange. Commodities can be perishables, or durablesthat are gold-like or tobacco-like. Of course one can imagine durables in betweengold and tobacco, which are partially destroyed depending on the intensity of theirconsumption (see Remark 1). But, for simplicity, we focus on the pure case. Justas we begin by postulating that money is the sole medium of exchange (meaningany purchase is via the commodity money), so we also begin by supposing that thesame money is the only nonperishable good (store of value).

Our first theorem is that under these conditions, gold is (generically) an inef-ficient money. The reason is that it is desired on two counts, to save and to enjoy.Typically those who most desire to save do not also get the most aesthetic enjoymentout of looking at gold. But in the absence of a rental market they cannot decoupletheir savings of gold from their consumption of gold.

Is gold an efficient store of value? 769

Our main result that gold is an inefficient money can be reformulated as theprinciple that in economies with durables, efficiency generally requires that therebe two markets for each durable, a sales market and a rental market. This principleretains its validity even when there are no liquidity constraints, thus implying thatit is generally not true that a rental can be regarded as equivalent to a sale and arepurchase. The equivalence only holds when the economy has available to it analternative store of value, like paper money, which yields no utility services.

The lesson of this paper is not that the best commodity money is the one thatyields the least utility. On the contrary, as long as it is divisible, and as long as its roleas money does not reduce its use value, the more valuable it is the better, since thatensures that agents will tend not to spend all their money and hence the liquidityconstraints will not be binding.2 As we said, durable commodities usually haveboth some consumption value (i.e., utility yielded when they are destroyed), andsome service value (i.e., utility yielded without the commodities’ destruction). Thelesson we draw is that, other things being equal, the higher the ratio of consumptionvalue to service value, the more suitable is the durable commodity to be money.

Our second main result is that if there is enough tobacco-like money at thebeginning, then equilibrium is efficient (in the absence of uncertainty), even if alltransactions and savings must be via the one money. (Since the outstanding stockof the money will constantly be reduced by consumption, its initial supply must beplentiful enough to support efficient trade in subsequent periods.) This conclusionrelies on the assumption that there is diminishing marginal utility to consumingtobacco, and that the marginal utility of consuming tobacco is bounded from below.Our second theorem shows that fixing any economy, then adding more and moretobacco to the endowment of every agent at time 1, eventually creates economieswhich have only efficient equilibria. With enough tobacco at time 1, every agentwill want to defer some tobacco consumption until later (because of the diminshingmarginal utility of tobacco), and this savings of tobacco will be sufficiently valuableto carry out all desired transactions in the future (since the marginal utility of tobaccois bounded from below). The initial distribution of resources among the perishablecommodities does not affect the conclusion.

Section 2 of the paper introduces the formal model, with gold-like or tobacco-like money, and a cash-in-advance constraint. In Section 3, we show the genericinefficiency of equilibrium with gold-like money. In Section 4, we show that withtobacco-like money, once it is in sufficiently large supply in the first period, equi-librium will be efficient.

In Section 5 we give several more thoerems which show that our results carryover with multiple nonperishable commodities. Having many durables, such as goldand silver to use as money, does not change the inefficiency result: in the absenceof credit, the inefficiency of gold-like money(s) persists. Similarly, even with otherdurables, the presence of tobacco-like money in sufficient supply does guarantee

2 A caveat: if the value of money relative to other commodities becomes too high, then the divisibilityassumption becomes implausible. The amount of money needed for small purchases will become soinfinitesimal as to be impossible to use in practice – hence the use of silver and copper, rather than gold,coins for small transactions.

770 P. Dubey et al.

the efficiency of equilibrium.3 Thus the allocation of art or gold coins among privatehomes, under the stylized conditions of this paper, is efficient in equilibrium, evenwhen owners regard the art or coins partly as an investment, if and only if there isa separate storable commodity money in sufficient supply.

In Section 5 we also show that our results have nothing to do with the cash-in-advance constraint. The generic inefficiency of gold-like money is due solely toits role as a store of value, and the absence of credit, i.e. the requirement that allexchange be current value for current value.

2 The formal model

Let H = {1, ..., h} be the set of agents; L = {1, ..., + 1} the set of commodities;and T = {1, 2} the two4 time periods, where h ≥ 2 and ≥ 1. All commoditiesare perishable, except +1, which is perfectly durable and is the stipulated mediumof exchange. We call it “money.”

Any durable commodity can have one of two different characteristics. It maybe “tobacco-like,” i.e., what is consumed yields utility but is destroyed, and whatis not consumed can be stored from period 1 into 2. Or it may be a “gold-like,”i.e., it yields utility in the form of services rendered to the agent who holds it andinventories it from period 1 into 2, without getting destroyed.

As a prototypical example of the first, think of tobacco; and, of the second,think of gold (which can be worn as jewelry at the time of consumption in period1, and carried over into the future, where it is available once more as money fortransactions purposes, and again afterwards as jewelry). Historically, each has hada reputable run as money.

Let us refer to money as “tobacco-like” or “gold-like,” depending on its char-acteristic.

Accordingly, a typical consumption plan is denoted (x, x) ∈ RL+ × R

L+ where

x, x is the consumption in period 1, 2 respectively; and

x�+1 =

money consumed in period 1, if money is tobacco-likemoney “consumed” and inventoried from period1 to 2 if money is gold-like .

The initial endowment of agent is (eα, eα) in RL+ × R

L+ where we assume, for

all α ∈ H:

(i) eαj > 0 for 1 ≤ j ≤ + 1

(ii) eαj > 0 for 1 ≤ j ≤ .

Let � be a hypercube in RL+ × R

L+ with side at least

M > max

{max1≤j≤�

{∑α∈H

eαj ,∑α∈H

eαj

},∑α∈H

(eα�+1 + eα

�+1)

}, i.e.,

3 Historically, tobacco and bricks of tea or salt have been used for extended periods, but only in smalleconomies.

4 This is for ease of notation. Our results easily extend to any finite number of periods.

Is gold an efficient store of value? 771

� = {(x, x) ∈ RL+ × R

L+ : xj ≤ M, xj ≤ M for all j ∈ L}.

Then, for any feasible trade, the consumption of each agent will lie in �.Let uα : R

L+ × R

L+ → R give the utility of consumption of agent α ∈ H . We

assume each uα is strictly concave and strictly monotonic. It will be necessary forus to linearly perturb these utilities. To this end let Uα be an open set in R

L ×R

L. Any vector (v, v) ∈ Uα perturbs uα to uα : � → R, where uα(y, y) =uα(y, y) + (v, v) · (y, y). We naturally assume that 0 ∈ Uα and that Uα is smallenough to ensure that the utility remains monotonic after the perturbation. LetU = U1 × · · · × Uh. We shall keep endowments fixed and view U as the space ofeconomies.

We shall also assume that for any uα ∈ Uα, agent α will do better not to tradethan to accept zero consumption in any good, i.e., introducing the symbol

τ ={

0, if money is tobacco-like1, if money is gold-like

we assume (for all uα ∈ Uα and α ∈ H): if (y, y) ∈ RL+ × R

L+ is zero in some

coordinate, then

(iii) uα(y, y) < max0≤γ≤1 uα(eα1 , ..., eα

� , γeα�+1, e

α1 , ..., eα

� , eα�+1 +(1−γ)eα

�+1 +γτeα

�+1)

The assumption can be dropped, but it makes for a cleaner analysis by keepingequilibrium consumptions in the interior, and enabling us to take derivatives.

Let p = (p1, ..., p�) and p = (p1, ..., p�) be the prices of commodities 1, ...,in periods 1 and 2 (denoted in terms of money) We always take p�+1 = p�+1 =1. We now describe the set of consumption bundles (xα, xα) ∈ R

L+ × R

L+ that

are affordable by agent α at the given prices (p, p). Let c+ ≡ max{0, c} forany real number c. For each of the following constraints (k), denote by ∆(k) thedifference between the RHS and the LSH. Then the budget set Bα(p, p) consistsof all (xα, xα) ∈ R

L+ × R

L+ which satisfy:

�∑j=1

pj(xαj − eα

j )+ ≤ eα�+1 (I)

0 ≤ xα�+1 ≤ ∆(I) +

�∑j=1

pj(eαj − xα

j )+ (II)

�∑j=1

pj(xαj − eα

j )+ ≤ eα�+1 + ∆(II) + τxα

�+1 (III)

xα�+1 = ∆(III) +

�∑j=1

pj(eαj − xα

j )+ (IV)

772 P. Dubey et al.

Define the space of allocations

A =

{(xα, xα)α∈H ∈ (RL

+ × RL+) :

∑α∈H

xαj =

∑α∈H

eαj and

∑α∈H

xαj =

∑α∈H

eαj for 1 ≤ j ≤ ;

∑α∈H

xα�+1 ≤

∑α∈H

eα�+1 and

∑α∈H

xα�+1 =

∑α∈H

eα�+1 +

∑α∈H

eα�+1 −

∑α∈H

(1 − τ)xα�+1

}.

An equilibrium of u ∈ U is a triple (p, p, (xα, xα)α∈H) such that:

(xα, xα) maximizes uα on Bα(p, p) for α ∈ H,

and(xα, xα)α∈H ∈ A.

Recall that an allocation (xα, xα)α∈H is efficient if there does not exist any

(yα, yα)α∈H ∈ A such that

uα(xα, xα) ≤ uα(yα, yα)

for all α, with strict inequality for at least one α.We say that an equilibrium is inefficient if the allocation yielded by it is not

efficient.Finally, we say that a property holds for “generic u ∈ U ,” if there is an open

set U∗ ⊂ U , whose complement has zero Lebesgue measure in U , such that theproperty holds for all u ∈ U∗.

3 Gold

Theorem 1 Let money be gold-like. Then, for generic u ∈ U , any equilibriumof u is inefficient.

3.1 Example

Before giving the proof, we illustrate the theorem with an example. Let there be twoagents α and β, with identical utilities for food (x1, x1), and gold services (x2, x2),given by u(x1, x2, x1, x2) = log(x1x2x1x2). Let the endowment of agent α offood and gold, in periods 1 and 2, be (2, 4, 6, 0) and that of agent β be (6, 4, 2, 0).Evidently there is a unique Walrasian (competitive) equilibrium with consumption(4, 4, 4, 4) for each agent (recall that gold is durable). This allocation is unachievableas a monetary equilibrium, for α would need to give up money to get more x1.

We can compute a monetary equilibrium by solving the following 10 equationsin the 10 unknowns ((p1, p1), (xα

1 , xα2 , xα

1 , xα2 ), (xβ

1 , xβ2 , xβ

1 , xβ2 )):

1xα

1/p1 =

1xα

2+

1xα

2(1)

Is gold an efficient store of value? 773

1xα

1/p1 =

1xα

2(2)

p1(xα1 − eα

1 ) + (xα2 − eα

2 ) = 0 (3)

p1(xα1 − eα

1 ) + (xα2 − eα

2 ) = xα2 (4)

1

xβ1

/p1 =1

xβ2

+1

xβ2

(5)

1

xβ1

/p1 =1

xβ2

(6)

p1(xβ1 − eβ

1 ) + (xβ2 − eβ

2 ) = 0 (7)

p1(xβ1 − eβ

1 ) + (xβ2 − eβ

2 ) = xβ2 (8)

xα1 + xβ

1 = eα1 + eβ

1 (9)

xα1 + xβ

1 = eα1 + eβ

1 . (10)

Equations (1) and (5) say that any agent who consumes food in period 1 shouldnot prefer to sell (a little of) it and buy gold, and then enjoy the pleasure of admiringthe gold in periods 1 and 2; or vice versa. Equations (2) and (6) say that an agentwho consumes food in period 2 should not prefer to sell (a little of) it and buy goldto admire in period 2; or vice versa. Equations (3) and (7) say that households tradevalue for value in period 1, and equations (4) and (8) say that households tradevalue for value in period 2, after inventorying their gold. Equations (9) and (10)say that the market for food clears in both periods.

There are two budget constraints for each agent ((3) and (4) for α, (7) and (8)for β). On account of these budget constraints, the gold market clears automaticallyin each period once the food markets clear.

Any solution (p, p, (xα, xα), (xβ , xβ)) to these equations must be an equilib-rium, provided the variables are all strictly positive.

Solving the equations, we get

((p1, p1), ((xα1 , xα

2 ), (xα1 , xα

2 )), (xβ1 , xβ

2 ), (xβ1 , xβ

2 )))= ((.484, .999), ((3.83, 3.12), (4.56, 4.56)), ((4.17, 4.88), (3.44, 3.44))).

The allocation is inefficient, because agent β is holding more gold than foodin period 1, while α is doing the reverse. The only reason β holds so much gold(which at the margin gives him very little pleasure) is because he wants to save, andholding gold is the only way to do it. The agents could both be made better off ifβ were permitted to rent some of his gold to α for the first period, in exchange forfood, and the absence of this rental market is another way of explaining why themonetary equilibrium is inefficient. A rental is not equivalent to a sale in period 1and a repurchase at the beginning of period 2, because by that time the original foodhas spoiled, and the repurchase must be with period 2 food. In the rental, period2 food is not involved. Note finally that neither agent is liquidity constrained, andboth save, i.e. both inventory positive amounts of money into period 2. ��

774 P. Dubey et al.

Theorem 1 generalizes this example. With three or more commodities the analysisbecomes more difficult because positive consumption of all commodities does notimply the interiority of choices. An agent might sell all the gold on hand to buy food,but still end up with a positive amount of gold after trade, on account of the saleof his labor. Furthermore, the theorem holds only generically; if in the example theinitial endowments corresponded to the unique Walrasian equilibrium, there wouldbe no trade and no inefficiency.

3.2 The proof

To prove Theorem 1 we first establish

Lemma 1 Let money be gold-like. And let (p, p, (xα, xα)α∈H) be an equilibriumof the economy {uα}α∈H ∈ U . Put

∇αj ≡ ∂uα

∂xj(xα, xα)

∇αj ≡ ∂uα

∂xj(xα, xα)

for j ∈ L. Then the equilibrium is efficient only if

(iv)∇α

j

pj= ∇α

�+1 + ∇α�+1

(v)∇α

j

pj= ∇α

�+1

for all α ∈ H and 1 ≤ j ≤ , and

(vi)∇α

l+1 + ∇αl+1

∇αl+1

is invariant of α ∈ H .

Proof of Lemma 1. By condition (iii) on uα, eα, eα we must have (xα, xα) � 0,otherwise α would do better by not trading and inventorying all his gold from period1 to 2. Thus we can take derivatives of agents’ utilities. Note that conditions (iv)and (v) hold by virtue of being in equilibrium (efficient or not) when every agentspends less than the money on hand in both periods, i.e., when the equilibrium isnot liquidity-constrained. When liquidity constraints are binding, (iv) and (v) maynot hold in equilibrium, but the lemma says they must if equilibrium is efficient.

Case I. Suppose xαj − eα

j < 0, i.e., α is a net seller of j. Then since xαj > 0, α must

be selling strictly less than his endowment (or else selling all of eαj and buying back

xαj > 0). If α sells ε/pj more of j (or buys ε/pj less), his increase in utility is

(∇α

�+1 − ∇αj

pj

)ε;

Is gold an efficient store of value? 775

if he sells ε/pj less of j, this is(∇α

j

pj− ∇α

�+1

)ε.

Both terms must be nonpositive at an equilibrium, so (v) holds by virtue of beingin equilibrium.

Case II. Next suppose xαj − eα

j > 0, i.e., α is a net buyer of j. If budget constraint

(III) is not binding on α, i.e., if∑�

i=1 pi[xαi − eα

i ]+ < Mα (where Mα is the moneyon hand for α, prior to trade in period 2), then it is clear that ∇α

j /pj−∇α�+1 = 0. (For

if the term is positive (negative) α should spend a little more (less) on commodityj, improving his utility.) If

∑�i=1 pi[xα

i − eαi ]+ = Mα, then ∇α

j /pj ≥ ∇α�+1 or

else α would spend less on j. Suppose > holds. Then consider β who is a net sellerof j. (Clearly such a β exists since the market for j clears in equilibrium.) We musthave ∇β

j /pj = ∇β�+1 (as shown in Case I). Let β give ε/pj units of j to α and take

ε′ units of gold from α. For ε′ > ε and sufficiently close to ε, and ε′ and ε small,both α and β improve, a contradiction. We conclude again that (v) holds, this timeby virtue of being an efficient equilibrium.

Case III. Finally, suppose xαj − eα

j = 0. Then ∇αj /pj ≥ ∇α

�+1, as just argued(since eα

j > 0 and α can sell j). Suppose

∇αj

pj> ∇α

�+1.

We claim that there exists a β with

∇βj

pj= ∇β

�+1. (11)

If there is a net trade in j at the equilibrium then this is obvious, for (11) willhold for any β who is a net seller of j (by Case I). Suppose there is no net tradein j. If there is a net trade in some other commodity k = j, let β be a net buyerof k, and then ∇β

k/pk = ∇β�+1, as shown in the preceding Case II. But then

∇βj /pj ≤ ∇β

k/pk = ∇β�+1, otherwise if > holds instead of ≤, β would do better

to spend ε less on k and ε more on j. At the same time β could certainly sell j, andthe fact that he does not do so implies ∇β

j /pj ≥ ∇β�+1, again proving (l1). This

leaves the possibility that there is no net trade in any commodity 1 ≤ k ≤ inperiod 2. Then agent α is consuming his initial endowment of commodities 1 till in period 2. Since xα

�+1 > 0, he is also inventorying gold into period 2, or else haspositive endowment of gold in period 2. Hence ∇α

j /pj ≤ ∇α�+1, otherwise α could

buy j with his gold and improve his utility. But this contradicts our assumption that∇α

j /pj > ∇α�+1. We conclude that if ∇α

j /pj > ∇α�+1, then (11) holds.

Now let β give ε/pj units of j to α and take ε′ > ε units of gold from α. Forε′ sufficiently close to ε (and ε′ small), both α and β improve, a contradiction. So(v) holds in Case III as well.

776 P. Dubey et al.

To prove (iv), repeat the above argument, noting that the marginal utility of aunit of gold in period 1 is ∇α

�+1 +∇α�+1, since gold yields utility ∇α

�+1 (as services)in period 1, and can be inventoried into period 2 where it yields ∇α

�+1. To prove(vi), suppose

γα ≡ ∇αl+1 + ∇α

l+1

∇αl+1

>∇β

l+1 + ∇βl+1

∇βl+1

≡ γβ .

Let β give ε units of gold in period 1 to α and get ε(γα + γβ)/2 units of gold fromα in period 2. Both improve their utility. �Proof of Theorem 1. Note that, since (xα, xα) ∈ Bα(p, p), the choice of xα

1 , ..., xα�

and xα1 , ..., xα

� automatically determines xα�+1 and xα

�+1 via the equations:

xα�+1 = eα

�+1 −�∑

j=1

pj(xαj − eα

j )

xα�+1 = xα

�+1 + eα�+1 −

�∑j=1

pj(xαj − eα

j ).

Also, once (xα, xα) is chosen for agents 1, ..., h − 1, then (xh, xh) is alsoautomatically determined at equilibrium by the requirement that (xα, xα)α∈H ∈ A.

Thus the number of independent variables among p, p, (xα, xα)α∈H to solvefor equilibrium is 2(for prices) + (h − 1)2(for allocations) = 2h.

But as we have just seen (recall (iv), (v), (vi)) efficiency implies that the follow-ing 2h + 1 equations must be satisfied (where we have fixed two distinct agents αand β arbitrarily):

∇αj

pj− (∇α

�+1 + ∇α�+1) = 0

∇αj

pj− ∇α

�+1 = 0

for all α ∈ H , 1 ≤ j ≤

∇α�+1

∇α�+1

− ∇β�+1

∇β�+1

= 0.

Theorem 1 will follow, by the transversal density and openness theorems, if wecan show that by linear perturbations of the uα it is possible to arbitrarily vary theseequations one at a time, leaving the others unaffected. For the first 2h equations,note that we can vary ∇α

j (or ∇αj ) by adding to uα(y, y) the linear term δyj (or

δyj). For the last equation add the linear term δ(∑�

j=1 pjyj + y�+1) to agent α’sutility. ��Remark 1. As we mentioned earlier, we could have considered durables betweengold and tobacco. Let τα ∈ [0, 1] denote the quantity of commodity + 1 thatsurvives into period 2 when α consumes 1 unit of it in period 1. The case of goldcorresponds to τα = 1 for all α ∈ H , and that of tobacco to τα = 0 for all

Is gold an efficient store of value? 777

α ∈ H . The generic inefficiency of equilibrium holds so long as +1 is not purelytobacco-like, i.e., τα > 0 for at least one α ∈ H . The proof calls for obviouschanges.

4 Tobacco

Next let us turn to the use of tobacco as money. Here it is possible to get efficientequilibria, provided there is “enough” tobacco in the endowment of each trader andcertain conditions obtain for the utility of tobacco. We will give these conditions intheir simplest (rather than their most refined or tight) form.

Let J ⊂ L, and define

�J =

{(x, x) ∈ R

L+ × R

L+ : xk, xk ≤

∑α∈H

(eαk + eα

k ) for all k ∈ L\J

}.

In other words, �J bounds the consumption of all commodities that are not in J .We will say that “α likes J” if both conditions I and II below hold.

Condition I There exists a B > 0 such that, if j ∈ J ,

∂uα

∂xj(x, x) > B

∂uα

∂xk(x, x)

∂uα

∂xj(x, x) > B

∂uα

∂xk(x, x)

for all k ∈ L\J and (x, x) ∈ �J .

Condition II For any m > 0, there exists M(m) > 0 such that, if j ∈ J ,

(x, x) ∈ �J

xj < mxj > M(m)

⇒ ∂uα

∂xj(x, x) >

∂uα

∂xj(x, x)

(x, x) ∈ �J

xj < mxj > M(m)

⇒ ∂uα

∂xj(x, x) <

∂uα

∂xj(x, x)

Condition I says that (on the margin) j remains desirable for α relative tocommodities in L\J , even when it is consumed in large quantities, provided theconsumption in L\J is bounded.

Condition II says that α prefers to distribute the consumption of j between thetwo periods in a not-too-skewed manner.

When “α likes {j},” we will simply write “α likes j.” (Note that if α likes j,Condition II rules out the possibility that the utility function of α has a separablelinear term in the consumption of xj or xj .)

We shall examine the effect of pouring in money in period 1 while the rest of thedata of the economy is held fixed, i.e., eα

1 , ..., eα� , eα

1 , ..., eα� , eα

�+1, uα are all held

fixed for α ∈ H; while the money endowments {eα�+1}α∈H are varied.

778 P. Dubey et al.

Theorem 2. Suppose the money commodity +1 is tobacco-like, and α likes +1,for all α ∈ H . There exists an M∗ such that, if minα∈H{eα

�+1} > M∗, everyequilibrium is efficient.

Proof. First we establish two claims. Both refer to arbitrary equilibria of the econ-omy, and the first is independent of the levels {eα

�+1}α∈H .

Claim 1. Let m∗ ≡ E/B, where E = max1≤j≤�{max{∑α∈H eαj ,∑

α∈H eαj }}

and B is as in Condition I. Then no agent α spends more than m∗ on purchases ineither period 1 or period 2.

Proof of Claim 1. Suppose α spends m > 0 in (w.l.o.g.) period 1. Then α musthave spent at least m/ on at least one commodity 1 ≤ j ≤ . Consequently

pj ≥ m

E. (12)

Denote the consumption of α by (xα, xα). Since α could buy less of j and doesnot, we must have

pj ≤∂uα

∂xj(xα, xα)

∂uα

∂x�+1(xα, xα)

≤ 1B

(13)

where the second inequality comes from Condition I. Together (12) and (13) imply

m ≤ E

B.

The same inequalities hold in period 2 replacing xj , x�+1 by xj , x�+1 throughout,verifying the claim.

Claim 2. Put M∗ > M(m∗) + 2m∗, where M(m∗) is as in condition II, andsuppose

minα∈H

eα�+1 > M∗.

Then (a) each agent has positive amounts of money on hand for spending in eachperiod; (b) no agent spends all the money at hand in either period.

Proof of Claim 2. Since, by Claim 1, α spends at most m∗ in period 1, there isnothing to check in period 1. Suppose some α ∈ H spends all his money (whichmay be zero) in period 2 on purchases. Let mα = money inventoried by α fromperiod 1 into 2. Then, since he spends all of mα, we see from Claim 1 that

mα ≤ m∗.

Hence the most that α could be consuming of money in period 2 is:

xα�+1 ≤

�∑j=1

pj eαj

≤ E

B≡ m∗

Is gold an efficient store of value? 779

(since eαj ≤ E for all j and pj ≤ 1/B for all j, by the analogue of (13) for period

2).Also the money consumed by α in period 1 is at least

xα�+1 ≥ eα

�+1 − m∗ − mα

> M(m∗) + 2m∗ − m∗ − m∗

= M(m∗).

By Condition II, α would be better off inventorying more money from period 1 intoperiod 2, and consuming it in period 2. This verifies the Claim 2. ��

But then at equilibrium we must have:

∇αj

pj= ∇α

�+1,∇α

j

pj= ∇α

�+1 (14)

for α ∈ H and 1 ≤ j ≤ (otherwise α would either spend more or less on thepurchases of j, both of which choices are available to him in light of the Claim 2).

Further, it must be that∇α

�+1 = ∇α�+1 (15)

for all α ∈ H . For if < (or >), α would do better to consume ε less (or more) inperiod 1, inventory ε more (or less) into period 2, and consuming ε more (or less)in period 2.

But (14) and (15) together imply that all the (∇α, ∇α) point in the same direc-tion, so the equilibrium is efficient proving Theorem 2. ��

5 Generalizations

Suppose we drop the “liquidity constraint” in the definition of the budget set, i.e.,replace (I)–(IV), by

�+1∑j=1

pj(xαj − eα

j ) ≤ 0 (I∗)

�+1∑j=1

pj(xαj − eα

j ) ≤ ∆(I∗) + τxα�+1. (II∗)

This is tantamount to the assumption that α can use the receipts from his sales forpurchases, so that the “cash-in-advance” constraint is no longer binding. (Implicitly,α can borrow money unlimitedly within each period at zero rate of interest but mustpay it back in the same period, after trade and prior to consumption.)

Denote this enhanced budget set by Bα∗ (p, p), and define equilibrium as before,

replacing Bα(p, p) by Bα∗ (p, p) for each α ∈ H .

We shall call this an equilibrium of the liquidity-free model. (Our previousequilibrium was of the liquidity-constrained model.) Theorems 1 and 2 carry overto the liquidity-free model.

780 P. Dubey et al.

So far, we focused attention on the case of a single durable (which we took tobe money) to bring out the results with minimal notation. But our results also carryover to economies with multiple durable commodities of either variety. Further,while money is a commodity, we will no longer insist that it be durable. Equilibria(of the liquidity-free or liquidity-constrained models) is defined as before, with theobvious amendments in the definition of the budget sets.

The analogue of condition (iii) is now: there exist 0 < αj < 1 for each durablej such that uh(eh, eh) > uh(y, y) if (y, y) ∈ � and yi = 0 or yi = 0 for any i ∈ L.Here

ehj =

{ehj if j is perishable or gold-like

αjehj if j is tobacco-like

ehj =

ehj if j is perishable

ehj + eh

j if j is gold-like(1 − αj)eh

j + ehj if j is tobacco-like

This simply says that each agent would do better not trading (and inventorying hisdurables in some manner) than accept zero consumption anywhere.

To generalize our results it will first help to enunciate some lemmas.

Lemma 2 Consider any model (above). Put

τ∗ ={

1, if commodity + 1 is gold-like0, if commodity + 1 is tobacco-like or perishable

.

Then, if the equilibrium is efficient, the following differential conditions musthold for α ∈ H and 1 ≤ j ≤ .

Table 1

Period j is a tobacco-like durable, j is a gold-like durable

or j is perishable

1∇α

j

pj= ∇α

�+1 + τ∗∇α�+1

∇αj + ∇α

j

pj= ∇α

�+1 + τ∗∇α�+1

2∇α

j

pj= ∇α

�+1

∇αj

pj= ∇α

�+1

Proof. Entirely analogous to the proof of Lemma 1. ��Lemma 3 Consider an equilibrium of any model. If the equalities of Table 1 hold,and if there exists at least one tobacco-like durable which is inventoried by everyagent in equilibrium, then the equilibrium is efficient.

Proof. If j∗ is inventoried by α (where j∗ is a tobacco-like durable) then ∇αj∗ = ∇α

j∗ .

But then, using Table 1, (∇α�+1 + τ∗∇α

�+1)/∇α�+1 = (∇α

j∗/pj∗)/(∇αj∗/pj∗) =

Is gold an efficient store of value? 781

pj∗/pj∗ is invariant of α ∈ H . It immediately follows that the gradients (∇α, ∇α)all point in the same direction for α ∈ H , hence the equilibrium is efficient. ��

We are now ready to state the general versions of Theorems 1 and 2.

Theorem 3 Take either the liquidity-constrained or the liquidity-free model. Sup-pose all the durables in the economy are gold-like. Then, generically in utilities,all the equilibria are inefficient.

Proof. In the light of Lemma 2, the proof follows that of Theorem 1 mutatis mu-tandis. ��Theorem 4 Consider the liquidity-free model. Suppose there is at least one tobacco-like durable j∗ in the economy, and that α likes j∗ for all α ∈ H . Fix all thedata of the economy except for {eα

j∗}α∈H . There exists M∗ > 0 such that, ifminα∈H eα

j∗ > M∗, then the equilibria are efficient.

Proof. Reread the proof of Theorem 2, replacing “α spends + 1” by “α sells j∗”(if j∗ = + 1), to obtain an M∗ with the property: eα

j∗ > M∗ ⇒ α does notsell all of eα

j∗ in period 1, nor does he sell all of the j∗ on hand in period 2. But

then, since α can vary the amounts that he inventories, ∇αj∗ = ∇α

j∗ . Also the otherequalities of Table 1 now obtain (as is easily checked, remembering that we are inthe liquidity-free model). Then the result follows from Lemma 3. ��

There are intuitively two reasons for inefficiency of equilibria. The first has to dowith liquidity constraints, since agents must put up cash-in-advance for purchases,and may be stuck with an inefficient supply of money in their endowments. By turn-ing to the liquidity-free model we, of course, wipe out this effect. But inefficiencymay still persist if all durables are gold-like (Theorem 3). This is because agents areunable to decouple consumption (of the durables in period 1) from savings. Hold-ing these durables is the only device whereby an agent can distribute purchasingpower between the two periods, i.e., the durables serve as a “store-of-value.” Thetrouble is that agents are also forced to consume exactly the amount they hold ofthe durables.

Were we to introduce a “rental market” for the durables (where agents could sellthem after trade for others to consume in period 1, and then get them back at the startof trade in period 2), efficiency would be restored (in a liquidity-free model). Forthen consumption is decoupled from savings, and each agent has exactly the sameflexibility as in the Walrasian world. Indeed, in passing, let us note that wheneverour equilibria are efficient, they are also automatically Walrasian.

The case left to be analyzed is that of a liquidity-constrained model with manydurables including at least one that is tobacco-like. Here we need to ensure that notonly the durable, but also money, is in plentiful supply with all agents, so that theliquidity constraints are not binding. More precisely, we have

Theorem 5 Consider the liquidity-constrained model, with commodity +1 beingmoney. Suppose that there is at least one tobacco-like durable j∗ in the economy,and that α likes {j∗, + 1} for each α ∈ H . Then there exists an M∗ such thatany equilibrium is efficient, provided

minα∈H

eαj∗ > M∗

782 P. Dubey et al.

and eitherminα∈H

eα�+1 > M∗ if money is durable

orminα∈H

eα�+1 > M∗

minα∈H

eα�+1 > M∗

if money is perishable

Proof. Repeat the proof of Theorem 2 to conclude that liquidity constraints are not

binding if the endowment of money is above M∗ for each agent. Then the proofproceeds exactly as that of Theorem 4. ��Remark 2. Existence of equilibria (in all our models) follows from standard argu-ments (and hence is omitted).

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