Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

p

omy

:(1) money as a medium of

3) money as a store of value

h the aid of Figure 12.1. Supeconomy who each produce a n product but also all other nts formulate their supply of at a central market place i ch the equilibrium relative

it example, there are in total s place without the use of for good 2, etc. Aside from xis etc., a centralized market itively, without some kind

r, z to have.

Nace in a centralized full- s more complicated. Assume hat the agent does not know e at any particular time. Supirket in each period. Then

need for a double coincimired with agent 2 who may ce of money, an exchange Flo wants to have his good

Denoting as the relative price of P14, P23, P24, and p34. Obviously,

Chapter 12: Money

and who himself has a good which agent 1 is looking for. Hence, in such a setting it may take a lot of effort and a long time before agent 1 can actually trade.

Even if agents are perfectly informed about the location of trading partners, the problem may still persist. Cass and Yaari (1966) present a case in which the double coincidence of wants always fails. Assume that agents only wish to consume their own good and the good produced by the agent located closest (in a clockwise direction), i.e. agent 1 would like to consume the bundle (1,2), agent 2 (2,3), agent 3 (3,4), and agent 4 (4,1). Assume that the goods are non-storable and that each agent can at most travel halfway towards his adjacent neighbours. This means that agent 1, for example, can attempt to trade with agents 4 and 2, agent 2 with 1 and 3, etc. It is easy to see, however, that no trading will actually take place. Agent 1, for example, cannot trade with 2 because the latter is not interested in good 1 at any price. Similarly, agent 1 will not trade with agent 4 for the same reason. The double coincidence of wants fails, all agents consume only their own good, and a situation of autarky persists.

Now assume there is a durable "thing" which is storable and can be transferred across agents at zero cost, and call this thing money. Then agents will actually be able to trade with each other by using this money rather than bartering. Agent 1, for example, sells his good to agent 4, and receives money for it with which he purchases good 2 from agent 2. Since the other agents do the same with their neighbours, an equilibrium can be attained in which all agents are better off (in welfare terms) as a result of the existence of a medium of exchange called money.

Of course, the circle model is a highly stylized account of the trading process but it is nevertheless useful because it motivates the following medium-of-exchange "test". Something serves the role of medium of exchange if its existence ensures that agents can attain a higher level of welfare. 3 In the "random-encounters" model and in the "circle" model money serves as a medium of exchange in the sense of this proposed definition. Indeed, in the former model the trading friction is reduced (but not totally eliminated) 4 by the existence of a medium of exchange, whereas in the latter the friction is completely eliminated.

There is nothing in the theory which suggests that the medium of exchange must be an intrinsically valuable commodity such as gold or silver (or rare shells) which enhance people's utility or can be put to productive uses. Indeed, an intrinsically low-valued good (such as paper) can also serve as a medium of exchange provided it is generally accepted in exchange. To the extent that gold and silver are better

3 This test is similar to (but more general than) the one suggested by McCallum (1983b). His requirement is more strict in that it requires the medium of exchange to expand production possibilities.

Indeed, he call this the "traditional presumption" (1983, p. 24).

4 Agent 1 may meet an agent from whom he does not want to buy anything but who does want to buy good 1. The transaction takes place against money, which agent 1 can use at some later encounter. If agent 1 instead meets an agent who does not want good 1 and whose good agent 1 does not want, then no trade takes place. Hence, some frictions remain in the random-encounters model.

313

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The Foundation of Modern Macroeconomics

used for productive purposes, it is actually preferable for society to use intrinsically low-valued material as a medium of exchange (McCallum, 1989a, p. 17).

The second major function of money is that of medium of account. As was explained above, an economy with four distinct goods exhibits six distinct relative prices. For an economy with N different goods the number of distinct relative prices amounts to N(N-1)/2, which is a rather large number even for a modestly large N. If all goods are expressed in terms of money, and money is thus the medium of account, then only N different (absolute) prices for the different goods need be recorded. Denoting these absolute prices by the relative prices are then implied,

e.g. Ai

The third function of money is that of store of value. In a monetary economy money can be used to buy goods and vice versa, not only today but (more than likely) also tomorrow. Hence, a stock of money represents "future purchasing power". In the future the money can be exchanged for goods which can be consumed or used in the production process. Money is thus capable of being used as a store of value, but there are other assets (bonds, company shares, real estate, etc.) which typically outperform it in this role because they yield a positive rate of return whereas money (typically) does not.

Of the three major roles played by money, only the medium-of-exchange role is the distinguishing feature of money. Any commodity can serve as a medium of account (without at the same time serving as a medium of exchange) and there are various non-money assets which can serve as a store of value.

12.2 Modelling Money as a Medium of Exchange

In Chapter 1 we discussed the Baumol (1952)—Tobin (1956) inventory-theoretic model of money demand in an intermezzo. The basic idea behind that model is that money is held through the period between income receipts, despite the fact that it does not yield any interest, because it is needed to make purchases. The baker will sell you a loaf of bread in exchange for money but not for bonds. At a more general level the model suggests that money facilitates transactions. Of course, the Baumol—Tobin model is rather restrictive in its scope and partial equilibrium in nature, and the task of this section is to study how money as a medium of exchange can be cast in a general equilibrium framework. In what follows the Baumol—Tobin model is shown to be a special case of a more general framework in which money helps to "grease the wheels" of the economy by minimizing liquidity costs.

12.2.1 Setting the stage

Suppose an individual agent lives for two periods, "now" (period 1) and "in the future" (period 2), and possesses stocks of bonds (Bo) and money (Mo) that were

accumulated in the pi periods (Y1 and respectively). The pri4 respectively. The per.._

P1 Y1 + MO + ( 1

P2 172 + M1 +

I

where Ri is the nom ii these expressions rei the right-hand side re, Since the agent N, (see Chapter 6), he %%,., (i.e. M2 < 0 and B2 < (B2 > 0) and the ag, requirements yields 30, following consolidaI _

where mt Mt /Pt is n is negative), and rt is 1

rt = Pt(1 + Rt)

Pt+i

If the price level is st,, exceeds) the nominal The agent has the u in the two periods in

v = u( ci) O( TH

where p > 0 is the pui Chapter 6). The how and its desired mor and the non-negati .4 predetermined stocks with this problem is:

for society to use intrinsically Ilum, 1989a, p. 17).

As was explained six distinct relative prices. For stinct relative prices amounts I modestly large N. If all goods the medium of account, then ads need be recorded. Denotttive prices are then implied,

lue. In a monetary economy t only today but (more than 7resents "future purchasing for goods which can be con- is capable of being used as a pany shares, real estate, etc.) yield a positive rate of return

le medium-of-exchange role v can serve as a medium of n of exchange) and there are I value.

Exchange

(1956) inventory-theoretic c idea behind that model is e receipts, despite the fact to make purchases. The ney but not for bonds. At a -fes transactions. Of course, e and partial equilibrium in Eiy as a medium of exchange t follows the Baumol-Tobin ramework in which money

n g liquidity costs.

--" (period 1) and "in the arid money (M0 ) that were

Chapter 12: Money

accumulated in the past. The agent has fixed real endowment income in the two

periods (Y1 and Y2, respectively) and consumes in the two periods (C1 and C2, respectively). The price of the good in the two periods is denoted by Pi and P2,

respectively. The periodic budget identities are then given by:

P1 Y1 + Mo + + Ro)Bo = P1 C1 +M1+ Bli

P2Y2 ± M1 + ( 1 + R1)131 = P2C2 + M2 + 132,

where R, is the nominal interest rate on bonds in period i. The left-hand side in these expressions represents the total resources available to the household whereas

the right-hand side represents what these resources can be spent on.

Since the agent will not be around in period 3 and there is no bequest motive (see Chapter 6), he will not wish to die with positive stocks of money and/or bonds

(i.e. M2 < 0 and B2 < 0). The financial sector will not allow him to die indebted (B2 > 0) and the agent cannot create money (M2 > 0). Hence, combining all these

requirements yields M2 = B2 = 0, so that (12.1)-(12.2) can be combined into the following consolidated budget constraint:

where mt Mt/Pt is real money balances, bt Bt/Pt is real bonds (or real debt if bt is negative), and rt is the real rate of interest which is defined as:

rt-Fi

If the price level is stable (rising, falling), the real interest rate equals (falls short of,

exceeds) the nominal interest rate.

The agent has the usual lifetime utility function which depends on consumption in the two periods in a time-separable manner:

where p > 0 is the pure rate of time preference and U(.) has the usual properties (see Chapter 6). The household chooses consumption in the two periods (C1 and C2)

and its desired money holding (m 1 ) in order to maximize (12.5) subject to (12.3) and the non-negativity condition on money holdings (m1 > 0), and given the predetermined stocks of money and bonds (mo and bo). The Lagrangean associated

with this problem is:

The Foundation of Modern Macroeconomics

where A is the Lagrangean multiplier. The first-order conditions are:

Equations (12.7)—(12.8) are exactly the same as in a model without money and in combination yield the usual Euler equation relating the optimal time profile of consumption to the divergence between the real interest rate and the rate of time preference. The existence of money does not affect this aspect of the intertemporal model. Equation (12.9) is new and warrants some further discussion. First consider the normal case with a strictly positive rate of interest (R1 > 0) so that the term in round brackets in (12.9) is strictly negative and the complementary slackness condition suggests that no money is held by the agent:

m1 = 0 if R1 >0. (12.10)

The intuition behind this result is that the opportunity cost of holding money consists of foregone interest, which is positive. Since money is not "doing" anything useful in the model developed thus far, the rational agent refrains from using money altogether.

The second, at first view rather pathological, case describes the situation in which the nominal interest rate is negative (R 1 < 0), so that the term in round brackets in (12.9) is positive. Now the agent wishes to hold as many money balances as possible. By simply holding these money balances they appreciate in value (relative to goods). To put it differently, money has a positive yield if the interest rate is negative.

oo if R1 < 0. (12.11)

Of course, negative nominal interest rates do not represent a particularly realistic phenomenon. We shall nevertheless have a need to return to this case in section 4.2 below where we discuss the optimal quantity of money argument. In the remainder of this section, however, we restrict attention to the normal case, i.e. we assume that the nominal interest rate is strictly positive. The challenge is then to modify the basic model in such a way that money will play a non-trivial role for the agent (and thus for the economy as a whole).

12.2.2 Shopping costs

In section 1 it was argued that money as a medium of exchange reduces the transactions costs associated with the trading process between agents. A particularly

simple and elegant way to (1983b, 1989a). He assuil time endowment is spent sense that it makes shot- . time otherwise spent on st shopping costs.

Suppose that the househ of time units, Si, and sper (1 — N —Se) units of leisu account that the agent h.,

v = u(ch i — — S.

The intertemporal budgel come now representi: nominal wage rate in pt..,

following form:

1 — = (nit-i,C

I

'sere the *(.) function i! is.ven level of goods consu a finite reduction of ti T-

..c. > 0. Second, oney balances decreases words, the shopping 1_

.-Loney balances. Third, in a diminishing rate, i.e.

,..,uunded, i.e. 0 < Vi(oc) <

The household chooses order to maximize (1_1 constraint on money balm

U(Ci, — N — S i ,

+A [A —

conditions are:

(12.7)

(12.8)

(12.9)

t model without money and the optimal time profile of rest rate and the rate of time is aspect of the intertemporal her discussion. First consider ct (R1 > 0) so that the term he complementary slackness

t

I tv cost of holding money coney is not "doing" anything

nt refrains from using money p

-gibes the situation in which the term in round brackets LS many money balances as appreciate in value (relative yield if the interest rate is

esent a particularly realistic to this case in section 4.2 argument. In the remainder tormal case, i.e. we assume '- a llenge is then to modify on-trivial role for the agent

‘- 7fiange reduces the transeen agents. A particularly

Chapter 12: Money

simple and elegant way to capture this aspect of money was suggested by McCallum (1983b, 1989a). He assumes that households value leisure time and that part of their time endowment is spent on "shopping around" for goods. Money is useful in the sense that it makes shopping easier, i.e. by using money the agent can save leisure time otherwise spent on shopping. We now modify our basic model to incorporate

shopping costs.

Suppose that the household has a time endowment of unity, works a fixed amount

of time units, N, and spends Sr units of time on shopping. Then the agent enjoys (1 — N — Sr) units of leisure in period t. The utility function is modified to take into

account that the agent likes leisure time:

v = un, 1 -+ 1 )u(c2,1- N — S2), p > 0. (12.12)

1+ p

The intertemporal budget constraint is still given by (12.3), with endowment

income now representing real labour income, Yt (Wt /Pt )/■/, where Wt is the nominal wage rate in period t. The shopping technology is assumed to take the

following form:

1 — N — St = Cr), (12.13)

where the *(.) function is assumed to have the following properties. First, for a given level of goods consumption, raising the level of real money balances results in a finite reduction of time spent shopping and thus an increase in available leisure, i.e. *,„ > 0. Second, the reduction in shopping cost due to a given increase in money balances decreases as more money balances are used, i.e. < 0 or, in words, the shopping technology features diminishing marginal productivity of money balances. Third, increasing consumption requires more shopping costs but at a diminishing rate, i.e. *c < 0 and Ikcc (.) > 0. Finally, the shopping costs are

bounded, i.e. 0 < Igoe) < *(0) < 1 — N.

The household chooses Cr , Sr (for t = 1, 2), and m 1 (mo being predetermined) in order to maximize (12.12) subject to (12.3), (12.13), and the non-negativity constraint on money balances (m 1 > 0). The Lagrangean expression is:

- ST - SO+ 1 \ u(c2,1 -- s2)

1+ p

317

The Foundation of Modern Macroeconomics

where At are the Lagrangean multipliers associated with the shopping technology in the two periods. The first-order conditions are:

where Lk(.) and UL(.) respectively.

The first thing to note about these expressions concerns equation (12.19), which is the first-order condition for optimal money balances. Comparing this expression to its counterpart in the basic model (i.e. equation (12.9)) reveals that the existence of shopping costs indeed gives rise to an additional positive term in the first expression of (12.19), UL (C2, 1 — N — S2 )irm (mi , C2) / (1 + p) (we have used (12.18) to eliminate A.2). This term represents the marginal utility of money balances. It must be stressed, however, that this does not in-and-of-itself ensure that the agent will choose to hold positive money balances. Indeed, given the assumptions made so far, it is quite possible that m1 = 0 is the best available option for the household. Specifically if the marginal utility of leisure and/or the marginal productivity of money balances are low, the first expression in (12.19) will be strictly negative so that the complementary slackness condition ensures that m1 = 0 is optimal, as in the basic model. Intuitively, no money is held in that case because the agent does not really mind shopping (UL low) and/or because money does not reduce shopping costs by much (V'm low).

In the remainder of this section we assume that Ifrin and/or UL are high enough to ensure that a strictly positive amount of money is held by the agent. The first expression in (12.19) holds with equality and the Lagrange multipliers (A1 and A 2 ) can be eliminated by combining (12.15)—(12.18) after which the following optimality conditions are obtained:

=LIc(C 1, 1 — N — Si) + UL(Ci, 1 — N — si)*c(mo, C i)

=( 1 + ) [ Lk (C2, 1 — N — 52) + UL(C2, 1 — N — S2)*c(ini, C2)] p

I

where A represents the optimal consumption le net marginal utility of c of consumption (Lk() ii caused by the additie:.. terms). For consumptior net discounting factor that the marginal opportunity costs associ,

12.2.3 Money in the

Inspection of equations I approach in effect amp by substituting (12.13 indirect felicity functit consumption and mone to rationalize the cony( money directly into tLL In a recent paper, Feer tice by demonstrating t one hand, models with and, on the other han affects "liquidity costs" Baumol—Tobin model gi strated that, in a gent money in the utility fur In a classic paper on I

complained that (at is not allowed to play the budget identities

in exactly the same wa any item (be it goods. r item, i.e. goods for b, . that: "... an economy t Classical economist 1% fact that fiat money is ii irrelevant; the role of m ii- om that of any othL . economy, Clower argu4 transactions, and "mu,.. goods" (1967, p. 86).

h the shopping technology

consumption and leisure,

Tns equation (12.19), which Comparing this expression 12.9)) reveals that the exisonal positive term in the first p) (we have used (12.18) to 1- of money balances. It must

elf ensure that the agent will n the assumptions made so option for the household. the marginal productivity of ° will be strictly negative so that m1 = 0 is optimal, as in t case because the agent does v does not reduce shopping

Nor UL are high enough to by the agent. The first expres-

!multipliers (A 1 and A2 ) can h the following optimality

C1 )

)ific (m C2)]

(12.20)

Chapter 12: Money

where A. represents the marginal utility of wealth (see Chapter 6). In planning his optimal consumption levels, the agent equates the marginal utility of wealth to the net marginal utility of consumption, which consists of the direct marginal utility of consumption (L/c(.) in the first and second lines of (12.20)) minus the disutility caused by the additional shopping costs which must be incurred (the UL (.)1frc(.) terms). For consumption taking place in the future the expression is augmented by a net discounting factor (see the second line of (12.20)). The third line in (12.20) shows that the marginal utility of money balances (UL 01/4,,(.)) must be equated to the opportunity costs associated with holding these balances expressed in utility terms.

12.2.3 Money in the utility function

Inspection of equations (12.12)-(12.13) of the shopping-cost model reveals that this approach in effect amounts to putting money directly into the utility function, i.e. by substituting (12.13) into the felicity function U(Ct , 1 - N - St ) we obtain an

indirect felicity function, CI (Ct , mt_i) U (Ct, (mt-i Ct)), which only depends on consumption and money balances. Hence, the shopping cost approach can be used

to rationalize the conventional practice in macroeconomic modelling of putting money directly into the utility function.

In a recent paper, Feenstra (1986) has provided further justifications for this practice by demonstrating that there exists a functional equivalence between, on the one hand, models with money entered as an argument into the utility function and, on the other hand, models in which money does not enter utility but instead affects "liquidity costs" which in turn show up in the budget restriction. Since the Baumol-Tobin model gives rise to such liquidity costs, Feenstra (1986) has demonstrated that, in a general equilibrium setting, it too is equivalent to a model with money in the utility function.

In a classic paper on the micro-foundations of monetary theory, Clower (1967) complained that (at least in models such as developed up to this point) money is not allowed to play a distinctive role in the economy. Indeed, by looking at the budget identities (12.1)-(12.2), it is clear that money enters these expressions in exactly the same way that goods and bonds do. Implicitly, this suggests that any item (be it goods, money, or bonds) can be directly exchanged for any other item, i.e. goods for bonds, bonds for money, etc. This makes Clower complain that: "... an economy that admits of this possibility clearly constitutes what any Classical economist would regard as a barter rather than a money economy. The fact that fiat money is included among the set of tradeable commodities is utterly irrelevant; the role of money in economic activity is analytically indistinguishable from that of any other commodity" (Clower, 1967, p. 83). In a pure monetary economy, Clower argues, there is a single good, "money", which is used in all transactions, and "money buys goods and goods buy money; but goods do not buy goods" (1967, p. 86).

319

The Foundation of Modern Macroeconomics

In the context of our basic model of section 2.1, Clower's idea can be formalized by requiring that spending on consumption goods cannot exceed cash balances carried over from the previous period.5 The so-called Clower or cash-in-advance constraint thus amounts to:

PtCt <Mt-i <#. Ct < (Pt-i/Pt)mt-i• (12.21)

The basic model, augmented with the Clower constraint (12.21), can be solved as follows. To keep things simple, we assume that the Clower constraint holds with equality in the first period. Since mo is predetermined, the same then holds for consumption in the first period, i.e. Ci = Pomo/Pi . The household chooses C2 and mi in order to maximize (12.5), subject to (12.3) and (12.21). The Lagrangean is:

U((Po/Pi)mo) ( 1 +1 p U(C2) + A2 [(P1/132)M1 — C2]

where A2 is the Lagrangean multiplier associated with the Clower constraint. The first-order conditions consist of the budget constraint (12.3) and:

The marginal utility of wealth is strictly positive, i.e. > 0, so that (by (12.23)) the

+

marginal utility of consumption is bounded. Since U'(Ct) = 0 by assump-

tion, this implies that the consumer chooses a strictly positive consumption level in period 2, i.e. C2 > 0 and (by the first inequality in (12.25)) m i > 0. Hence, the cash- in-advance constraint does indeed deliver the "goods" desired by Clower. Money is essential, not because it is valued intrinsically, but rather because households wish to consume in the second period. It can also be shown that the household will not hold excess cash balances. Since m1 > 0, the first expression in (12.24) holds with equality, which ensures that the shadow price of cash balances is strictly positive:

This implies that the first expression in (12.25) holds with an equality, i.e. the household will hold just enough cash to be able to finance their optimal consumption plan in the future. This result is not specific to our simple two-period model and easily generalizes to a multi-period setting.

5 For simplicity we assume that the cash-in-advance constraint does not affect purchases of bonds.

As is the case for ti cash-in-advance appro: money approach. Inde

(Ct = (Pt-i/Pt)mt-i), th U(Ct, mt_i) min[Ct , Feenstra, 1986, p. 285).

le substitution elastic .1 aspect the cash-in-advar model and the Baumol-

I

I

12.3 Money as a

In the basic model of se

.dividual agent to trai.: serving as a store of valu

:ter as it yields a hi, the basic model. It u.

is technically capable i,cwley (1980) press:..

approach can be illustral

•jives is that money is 4,;,- the budget equations

I

—re in Pt+i/Pt – 1 0 two periods and money

2.5) subject to (12.2: j .

C= U(Cl ) + ( 1

A.2 [Y2 +

If m i is strictly positive. tt,

shows that the mimpli

-1 + 74)

I er's idea can be formalized by Dt exceed cash balances carried rr or cash-in-advance constraint 1

(12.21)

I nt (12.21), can be solved as Clower constraint holds with the same then holds for he household chooses C2 and

12.21). The Lagrangean is:

1 1 — C21

(12.22)

the Clower constraint. The 12.3) and:

> 0, so that (by (12.23)) the LP(Ct ) = 0 by assump- )ositive consumption level in ►) rn l > 0. Hence, the cashdesired by Clower. Money is r because households wish I that the household will not ssion in (12.24) holds with

balances is strictly positive:

(12.26)

th an equality, i.e. the housetheir optimal consumption rnple two-period model and

11/-

oes not affect purchases of bonds. 111

Chapter 12: Money

As is the case for the shopping model and the Baumol-Tobin model, the cash-in-advance approach can also be shown to be equivalent to a utility-of- money approach. Indeed, as the Clower constraint always holds with equality

(Ct = (Pt_1/Pt)int-i), the same results are obtained if the indirect felicity function 0(Ct , mt_i) min[Ct , mt_ i ] is maximized subject to the budget constraint only (see

Feenstra, 1986, p. 285). An important aspect of this indirect felicity function is that the substitution elasticity between consumption and money balances is zero. In this aspect the cash-in-advance formulation differentiates itself from both the shopping model and the Baumol-Tobin model.

12.3 Money as a Store of Value

In the basic model of section 2.1 above, both bonds and money can be used by the individual agent to transfer resources across time and both assets are thus capable of serving as a store of value, although the former does so in a superior fashion to the latter as it yields a higher rate of return. For that reason, money is not generally held in the basic model. It thus does not actually serve as a store of value even though it is technically capable of doing so.

Bewley (1980) presents a model in which money is used as a store of value. His approach can be illustrated with the aid of our basic model. The key assumption he makes is that money is the only asset available to the agent, i.e. Bo = B1 = B2 = 0 in the budget equations (12.1)-(12.2). These can then be expressed in real terms as:

where Trt Pt+111)t- — 1 is the inflation rate.6 The agent chooses consumption in the two periods and money holdings (C1, C2, m1) in order to maximize lifetime utility

(12.5) subject to (12.27). The Lagrangean for this problem is given by:

6 If m1 is strictly positive, the first two expressions in (12.27) can be consolidated:

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The Foundation of Modern Macroeconomics

where Xi and A2 are the Lagrangean multipliers associated with the two budget restrictions. The first-order conditions are the two budget constraints and:

Equations (12.29)-(12.31) can be combined to yield the following expression:

The intuition behind (12.32) can be illustrated with the aid of Figure 12.2. The consolidated budget equation is drawn as the straight line segment AB with slope dC2/dCi = -1/(1 + Tr1). The indifference curve, Vo, has a slope of dC2/dCi = -(1 + p)L1/(C1)11F(C2) and has a tangency with the budget line at point Ec . This is the privately optimal consumption point ignoring the non-negativity constraint on money holdings. If the income endowment point lies north-west of point Ec, say at money is of no use as a store of value to the agent. In economic terms, the agent would like to be a net supplier of money in order to attain the consumption point Ec but this is impossible. Graphically, the indifference curve through q7 (the dashed curve) is steeper than the budget line, the choice set is only AE,317 C, and the best the agent can do is to consume his endowments in the two periods.

Figure 12.2. Money as a store of value

In mathematical tern utility rises by supp .

In the alternative ( of the consumption I money and the fir- equation becomes:

Er(c2 ) = (1 + k

(C

The upshot of the di cumstances because

smoothing can be d1/4., Of course, the Bev interest-bearii:0 economies. This fact

• the following ex.. ,„It there are poor a h income endow(

tio..,h to save in the exist but that they cc

therwise, and a , e too small an am

.s forced to save kor part) of their s,

the economy, the inc 'e held as a store of

12.3.1 Overlappin

model of the 4o.idual agent pro

r-yielding as•

..›agation, and -,.. &Sore of value in a g

31 restrictions a: 0

*narrational friction, ate a meanl• overlapping-s,

is 'modem moneta - :le t the poi

we normalize N to u young have t Anent, Y,