Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

ciated with the two budget iget constraints and:

(12.2cor

(12.301

(12.31►

following expression:

(12.32

:e aid of Figure 12.2. The line segment AB with slope has a slope of dC2/dC1 fidget line at point Ec . This e non-negativity constraint lies north-west of point Ec, agent. In economic terms,

!er to attain the consumpindifference curve through he choice set is only AEo C, nents in the two periods.

ci

Chapter 12: Money

In mathematical terms, the slope configuration implies that aL/ami < 0 (lifetime utility rises by supplying money) and complementary slackness results in m 1 = 0.

In the alternative case, for which the income endowment point lies south-east of the consumption point (say at Er) the agent saves in the first period by holding money and the first expression in (12.32) holds with equality so that the Euler equation becomes:

The upshot of the discussion so far is that money will be held under certain circumstances because it provides a means by which intertemporal consumption

smoothing can be achieved.

Of course, the Bewley approach is rather specific and somewhat unrealistic in that interest-bearing financial instrument are widely available in modern market economies. This fact does not, in-and-of-itself invalidate the argument, however, as the following example, inspired by Sargent and Wallace (1982) reveals. Suppose that there are poor agents (with low income endowments) and rich agents (with high income endowments) in the economy, and assume that both types of agents wish to save in the first period. Suppose furthermore that interest-bearing bonds exist but that they come in minimum denominations, say due to legal restrictions or otherwise, and assume there are no savings banks. In this setting the poor agents save too small an amount to be able to purchase even a single bond and they are thus forced to save by holding money. On the other hand, the rich agents will hold all (or part) of their saving in higher-yielding bonds. Aggregating over all agents in the economy, the indivisibility of bonds results in a positive demand for money to be held as a store of value.

12.3.1 Overlapping-generations model of money

In the model of the previous section, money is used as a store of value by an individual agent provided there is some friction which prevents him from using higher-yielding assets for this task. The argument is based on a partial equilibrium investigation, and the first task of this section is to embed the notion of money as a store of value in a general equilibrium economy-wide model. Instead of using the legal restrictions argument of Sargent and Wallace (1982), we introduce an

of the type first emphasized by Samuelson (1958), in order to motivate a meaningful role for money. This allows us to introduce and discuss the socalled overlapping-generations model of money, which has been extremely influential

in modern monetary theory.

At time t the population consists of N / 2 young agents and N/2 old agents and we normalize N to unity to simplify the notation. All agents live for two periods, so the young have two periods to live and the old only one. Agents receive an

endowment, Y, when young, but do not have any endowment income when they

323

The Foundation of Modern Macroeconomics

are old. The output Y is potentially storable and for each unit stored in period t, 1/(1 + 8) units of output will be left over in period t + 1, where 8 > —1. This storage technology nests several special cases. Particularly, if 8 oo, goods spoil immediately and are thus non-storable. If 8 = 0, goods keep indefinitely, and if —1 < 6 < 0 goods reproduce without supervision by the storage process!

The (representative) young agent can either consume his output in youth (Cr, where the superscript denotes "young"), store it (Kt of which Kt /(1 + 8) is available in period t + 1), or trade it for fiat money. Since the money price of output is Pt , the last option yields the agent real money balances at the end of period t (mt Mt /Pr). The budget identity facing a young agent in his youth is thus:

Y = cr +Kt ±MtIPt• (12.34)

Now consider the budget identity of an old agent in period t. This agent stored output in youth (Kt _ i ) as well as nominal money balances (M r_ 1 ) with which he can purchase goods, facing the period-t price level In addition, the agent receives a transfer from the government (Tt), the amount of which he takes as given. The budget identity of an old agent is thus:

• = Kr-1/( 1 + + Tt + (Pt-1/1)t)mt-1, (12.35)

where the superscript denotes "old". But the agent who is young in period t will himself be old in period t + 1, and will thus face a constraint similar to (12.35) but dated one period later in the last period of his life:

dt)+1 = Kt/(1 + 8) + Tt+i + (Pt/PrAmt. (12.36)

The lifetime utility function of the young agent in period t is given by:

and the agent chooses Cr, 0)+1 , Kt, and mt in order to maximize (12.37) subject to (12.34) and (12.36) as well as non-negativity conditions on money holdings and stored output (Mt > 0 and Kt > 0, respectively). The Lagrangean is:

where Au and X2,t are the Lagrangean multipliers of the budget identities in youth and old age respectively. The first-order conditions consist of the budget identities

7 Samuelson gives the examples of rabbits and yeast for this case. In a more serious vein, a negative value for S captures the notion of net productivity in the economy (1958, p. 468).

1

(12.34) and (12.36) ani

acra = u'(q) —

a ( 1

ac() 1 1+p

a

amt = Ai,t + ;-:

where 7tt Pt+11Pt -

the utility function, ..4

;-i,t > 0 and X2,t > 0. young agent will choa

on which one has the 1 then only money will 1 then only goods will

first case the storage tt

.'u which lies below t value (the line AC). Th inflation (irt > 8). 11 The behaviour of tl entered life (in period designated by VI 1 ),

Cct',

Tt+ Y/(1+ it„ C

11

Tt+Y/(1+6) 8

Figure 12.3.

r each unit stored in period xi t + 1, where 6 > -1. Thi<, larly, if 8 oo, goods six, Dods keep indefinitely, and if

he storage process.'

me his output in youth ( Cr, ,f which Kt /(1 + 6) is available , ney price of output is Pt , the

end of period t (mt Aft/Pt)- h is thus:

(12.34)

k period t. This agent stored es (WO with which he can i addition, the agent receives hich he takes as given. The

( 1 2. 35 )

,•ho is young in period t will - raint similar to (12.35) but

(12.36)

Id t is given by:

(12.37)

maximize (12.37) subject nis on money holdings and

7rangean is:

• mt ]

t

(12.38)

e budget identities in youth tsist of the budget identities

a more serious vein, a negative p. 468).

where n- t Pt+ilPt - 1 is the inflation rate. In view of our assumptions regarding the utility function, agents wish to consume in both periods of their lives so that

> 0 and 0. Equations (12.41)-(12.42) imply that, provided 7rt 0 6, the young agent will choose a single type of asset to serve as a store of value, depending on which one has the highest yield. Particularly, if inflation is relatively low (74 < 6)

then only money will be held (Kt = 0 and mt > 0), and if it is relatively high

then only goods will be stored (Kt > 0 and mt = 0). In terms of Figure 12.3, in the first case the storage technology is not productive enough and yields a budget line

AB which lies below the budget line associated with holding money as a store of value (the line AC). The line configuration is switched in the second case with high

inflation (yrt > 8).

The behaviour of the old in period t is quite straightforward. Although they

entered life (in period t - 1) possessing a utility function analogous to (12.37) (and designated by 1/t 1 ), their behaviour in period t -1 (their youth) constitutes "water

Figure 12.3. Choice set with storage and money

325

The Foundation of Modern Macroeconomics

under the bridge" in the sense that it cannot be undone in period t (it is irreversible or "sunk" in economic terms). All that remains for them is to maximize remaining lifetime utility, U(C°) subject to the budget identity (12.35). They simply consume their entire budget.

Following Wallace (1980) we assume that the government pursues a simple money supply rule:

Mt = (1 + /2)Mt-i, (12.43)

with p, > -1 representing the constant rate of nominal money growth. The additional money is used to finance the transfer to the old, i.e. the government budget restriction is Mt — Mt_ i = Pt Tt which implies that the transfer in period t + 1 is:

Equilibrium in the model requires both money and goods markets to be in equilibrium in all periods. By Walras' Law, however, the goods market is in equilibrium provided the money market is, i.e. provided demand and supply are equated in the money market:

m(Tt+1,74) = M t 1 Pt , (12.45)

where m(.) is a function, representing the demand for money by the young in period t, which is implied by the first-order conditions (12.39)-(12.42). For example, if the felicity function in (12.37) is logarithmic, U(x) log x, then this money demand function has the following form:

The model consists of (12.44) and (12.45) and we are looking for a sequence of price

levels (Pt, Pt+i, etc.) such that the equilibrium condition (12.44) holds for all periods given the postulated money supply process. For the logarithmic felicity function the

solution is quite simple and can be obtained by substituting (12.44) into the first line of (12.46) and solving for the equilibrium level of real money balances:

This expression, which is only valid if 74 < 8, shows that real money balances are constant so that the price level is proportional to the nominal money supply and the inflation rate is equal to the rate of growth of the money supply (7r t = ,u). So we reach the conclusion that, provided the money growth rate it is less than the depreciation rate 8, intrinsically useless fiat money will be held by agents in a general equilibrium setting. Intuitively, money is the best available financial instrument to

serve as a store of value course, if the storage teci t4 uilibrium will only ob if there is a constant rat rishable (8 cc) th, represents the only store The existence of a r ii,,nerations model. Index as a store of value and c _ :0 (see the second lir- a:rid is distributed to agLi of value. This implies V.

i.e. 1/Pt = 0 for all t.

12.3.2 Uncertainty al

. • . the basic model dis and bonds are known b ,ids in order to decide

basic model the yi,. former are used as a stoi

:sic model by assum uiat on money) is not kn 4arding consumption this situation as one

to the yield on his invest known with certairr

is known with certain view of the investor.

LAoney as 1 + rit4 1/(1 be expressed in real tern

+ (1 + rn mo + (1 + rr)mi + (1 +

here we have already i cents the present value (

,k-free rate, i.e. Yi* bonds is a stochastic . agent at the end of the yen made (Ci, rni, b,

8 Of course, ro is not sta...1

period.

e in period t (it is irreversible hem is to maximize remaining (12.35). They simply consume 0

nment pursues a simple money

(12.43)

- al money growth. The addi- d, i.e. the government budget transfer in period t + 1 is:

(12.44)

oods markets to be in equilib- )ods market is in equilibrium and supply are equated in the

(12.45)

money by the young in period ))-(12.42). For example, if the then this money demand

7: < 8

(12.46)

7: > 8.

)oking for a sequence of price n (12.44) holds for all periods a rithmic felicity function the `Laing (12.44) into the first

real money balances:

(12.47)

at real money balances are nominal money supply and money supply (7rt = ,u). So A-th rate p is less than the be held by agents in a general hie financial instrument to

Chapter 12: Money

serve as a store of value as it outperforms the storage technology in that case. Of course, if the storage technology yields net productivity (8 < 0) then the monetary equilibrium will only obtain if the money growth rate is negative (p, < 3 < 0), i.e. if there is a constant rate of deflation of the price level. In contrast, if goods are perishable (8 oo) then the monetary equilibrium will always hold since money represents the only store of value in that case.

The existence of a monetary equilibrium is quite tenuous in the overlappinggenerations model. Indeed, if pt > 8 then the storage technology outperforms money as a store of value and consequently the demand for real money balances will be zero (see the second line in (12.46)). Despite the fact that fiat money exists (M t > 0) and is distributed to agents in the economy, it is not used by these agents as a store of value. This implies that money is valueless and the nominal price level is infinite, i.e. 1/Pt = 0 for all t.

12.3.2 Uncertainty and the demand for money

In the basic model discussed in section 2.1 above, the respective yields on money and bonds are known by the agent who consequently only has to compare these yields in order to decide upon the optimal instrument to use as a store of value. In the basic model the yield on bonds is higher than that on money so that only the former are used as a store of value. In this section we introduce a friction into the basic model by assuming that the yield on bonds (though higher on average than that on money) is not known with certainty by the agent when making his decisions regarding consumption and saving in the first period. Sandmo (1970, p. 353) refers to this situation as one in which there exists capital risk; the investor is uncertain as to the yield on his investment. We assume that endowment income in both periods is known with certainty (there is no income risk). Furthermore, the yield on money is known with certainty so that money constitutes a "safe" asset from the point of view of the investor. To simplify the notation somewhat we define the yield on money as 1 + rP4 1/(1 + 7rt). The periodic budget identities (12.1)-(12.2) can then be expressed in real terms as:

Yl + (1 + mo + (1 + ro)bo = + mi + (12.48) (1 + rr)mi + (1 + F1)b1 = (12.49)

where we have already incorporated the fact that m2 = b2 = 0. Note that r represents the present value of present and future endowment income, capitalized at the risk-free rate, i.e. Y1 + Y2/(1 + re). The tilde above r1 denotes that the yield on bonds is a stochastic variable, the realization of which (r 1 ) will only be known to the agent at the end of the first period, i.e. after consumption and savings plans have been made (C1, m1, b1).8 This means (by (12.49)) that consumption in the second

8 Of course, ro is not stochastic as it is a realization of ro which is known at the beginning of the first period.

327

The Foundation of Modern Macroeconomics

period is also a stochastic variable, i.e. C2 appears in (12.49). In the terminology of Dreze and Modigliani (1972, p. 309) the model implies that investing in bonds represents a temporal uncertain prospect, i.e. time must elapse before the uncertainty is removed.

Below it will turn out to be useful to write the budget identities (12.48)-(12.49) in a slightly different manner:

represents the portfolio share of money. In the first period the agent chooses consumption C1 and the portfolio share w l , not knowing how high will be the value of his assets at the beginning of the second period because the yield on the risky investment is uncertain.

Since ri (and thus C2 and A2) is stochastic, the agent must somehow evaluate the utility value of the uncertain prospect C2. The theory of expected utility, which was developed by Von Neumann and Morgenstern (1944) postulates (as indeed its name suggests) that the agent will evaluate the expected utility in order to make his optimal decision, i.e. instead of using V in (12.5) as the welfare indicator the agent uses the expected value of V, denoted by E(V).9 We assume that the agent bases his decisions on a subjective assessment of the probability distribution of the yield on his investment, the density function of which is given by f (TO. We furthermore assume that i1 is restricted to lie in the interval [-1, oo), with the lower bound representing "losing your entire investment principal and all" and the upper bound denoting "striking it lucky by hitting the jackpot". Finally, we assume that the parameters of the model and the stochastic process for r i are such that we can ignore the non-negativity constraint for money holdings. Since there is no sign restriction on bond holdings, this means that we only need to study an internal optimum.

The expected utility of the agent can now be written as follows:

where S1 A 1 + Yl - C1. The agent chooses C 1 (and thus Si) and coi in order to maximize his expected utility, E(V). Straightforward computation yields the following

9 The expected utility theory is discussed in more detail by Hirshleifer and Riley (1992).

0 (1 + p) -1

0 = E (UVC2)(A1

(CO =(1+ 19 ) -1 -1

(CO = (1 +

Technically, (12.53) is the investment portfolio in tt (carrying a stochastic utility per dollar invested pp. 588, 590). Equation ( 1 profile of consumption,

In order to simplify the function, U(Ct ), which

U(Cr) = log Cr

where y < 1 represents The first-order condition

=EV1)2-1 (Al + -

=E[(Ai + Y; - C 1 ,

=E[[(1 + no.)1 +

in going from the first t& ::Jrri (12.51), and in ti., r are non-stochastic vai portfolio share, wi„. plobability distribution (

subjective mean return

(1 + r*) } max E[(1

=E[(1+ ri

12.49). In the terminology riles that investing in bonds apse before the uncertainty

"at identities (12.48)-(12.49)

(12.50)

(12.51)

`al assets inclusive of interId where col ml /(ml + b1 ) -iod the agent chooses con- g how high will be the value -cause the yield on the risky

I

ent must somehow evaluate !ory of expected utility, which 4) postulates (as indeed its led utility in order to make as the welfare indicator the

.° We assume that the agent e probability distribution of which is given by f(i).). We c-val [-1, oo), with the lower ipal and all" and the upper oot". Finally, we assume that s for ri are such that we can

-1gs. Since there is no sign need to study an internal

as follows:

)11f (ii) dr1, (12.52)

is Si ) and col in order to maxutation yields the following

•fer and Riley (1992).

Chapter 12: Money

first-order conditions (for wi and C1, respectively):

00

Technically, (12.53) is the expression determining the optimal composition of the investment portfolio in terms of money (which has a certain yield rr) and bonds (carrying a stochastic yield Intuitively (12.53) says that the expected marginal utility per dollar invested should be equated for the two assets (see Sandmo, 1969, pp. 588, 590). Equation (12.54) is the Euler equation, determining the optimal time profile of consumption, generalized for the existence of capital uncertainty.

In order to simplify the discussion, we now assume that the agent has a felicity function, U(Ct ), which takes the following, iso-elastic form:

where y < 1 represents the degree of risk aversion exhibited by the agent (see below The first-order condition for w i (given in (12.53)) collapses to:

0 = E [&2/-1 (Ai + YiK - Ci)(r14 -

1 (riki

= E [(Ai + Y1 - COY [(1 + r ncui + (1 +1'0(1 -

In going from the first to the second line we have substituted the expression for C 2 from (12.51), and in the final step we have made use of the fact that A1, C1, and III are non-stochastic variables. Equation (12.56) implicitly determines the optimal portfolio share, al, as a function of riti , y, and parameters characterizing the probability distribution of rl . The important thing to note is that ail maximizes the subjective mean return on the portfolio, r*, which is defined as:

The Foundation of Modern Macroeconomics

For the iso-elastic felicity function (12.55), the first-order condition for C1 (given in (12.54)) collapses to:

where c is the marginal propensity to consume out of total wealth:

In going from the second to the third line in (12.58) we have made use of the expression for r* in (12.57). The striking thing to note about (12.58)-(12.59) is that the optimal consumption plan for the first period looks very much like the solution that would be obtained under certainty. Indeed, in the absence of uncertainty about the bond yield, maximization of lifetime utility would give rise to the expression in (12.58)-(12.59) but with r* replaced by max [r1, rr], where rl is the certain return on bonds. Furthermore, in the case of a logarithmic felicity function (y = 0), r* drops out of (12.58)-(12.59) altogether and the capital risk does not affect present consumption at all (see Blanchard and Fischer (1989, p. 285) on this point).

With iso-elastic felicity functions, there thus exists a "separability property" between the savings problem (choosing when to consume) and the portfolio problem (choosing what to use as a savings instrument). 10 Since (as we shall see in subsequent chapters) modern macroeconomics makes almost exclusive use of such felicity functions, it is instructive to turn to a more detailed discussion of the pure portfolio problem. In doing so, we are not only able to characterize more precisely the factors influencing the choice of money versus bonds but it also allows us to introduce the liquidity preference theory of money that was developed by Tobin (1958). This so-called portfolio approach to money played a major role in macroeconomics in the 1960s and 1970s.

The portfolio decision

An important implication of the theory discussed above is that for a certain class of felicity functions, the expected-utility-maximizing household wishes to consume a fraction c of total wealth whilst saving the remaining fraction 1 - c. Designating the amount to be invested by S1 = (1 - c) [A + Yn, the budget equation for the

10 This was first demonstrated by Samuelson (1969b, pp. 243-245) in a multi-period discrete-time setting and generalized to continuous time for a more general class of felicity functions by Merton (1971). See also the discussion by Dreze and Modigliani (1972, pp. 317-323) on the separability property in the context of a two-period model with both capital and income risk.

portfolio problem is S I b1 such as to maximiLc

A2 m Si [(1 rr)wi + (1

Stepping back sonicform of the portfolio pr exactly this form. The ii maximize expected Liu.,

Eu (2), 2 EE S [(1 I

where Z is end-of-perioo free rate (S and rM are bo for this problem is:

E U'(2)(rm - = 0.

Apart from a slight ch condition for wi in the I In order to further Lie now turn to the mean-% case of the model disc,.: utility function, U(2), 1 value (or mean) of Z, dt

U(2) ti U(E(Z)) -7- _

1.1'" ( E(2

I

, .eking expectations for expected utility:

EU(2)=NN=EU(E(2))

= U(E(Z))

In going from the first value of a constant is t., of-period wealth can

1_ I'M on the right-har,- end-of-period wealth (54 The second step in (12.63) so that preferci,

11 See Hirshleifer and Ril.

I

rder condition for C 1 (given in

(12.58)

IT total wealth:

I

(12.59)

i8) we have made use of the about (12.58)—(12.59) is that ks very much like the solution absence of uncertainty about give rise to the expression where ri is the certain return felicity function (y = 0), r* al risk does not affect present

p. 285) on this point).

s a "separability property" ►ume) and the portfolio prob- 1. 10 Since (as we shall see in almost exclusive use of such =ailed discussion of the pure

to characterize more pre- TSUS bonds but it also allows ey that was developed by oney played a major role in

is that for a certain class of usehold wishes to consume a fraction 1 — c. Designating le budget equation for the

'45) in a multi-period discrete-time of felicity functions by Merton -323) on the separability property

- sk.

Chapter 12: Money

portfolio problem is 51 = m1 + bl and the household wishes to choose m 1 and b1 such as to maximize expected utility of end-of-period wealth, E(U(A2)), where

A2 Sl [( 1 + rr )04 + (1 + Fi)( 1 — (01)]•

Stepping back somewhat from the specifics of our two-period model, the general form of the portfolio problem as analysed by Tobin (1958) and Arrow (1965) takes exactly this form. The investor chooses the portfolio share of money w in order to maximize expected utility:

where Z is end-of-period wealth, S is the amount to be invested, and rM is the riskfree rate (S and rM are both exogenously given parameters). The first-order condition for this problem is:

Apart from a slight change of notation, (12.61) coincides with the first-order condition for w i in the two-period model (see (12.53) above).

In order to further develop intuition behind the first-order condition (12.61) we now turn to the mean-variance model, which can be seen as an approximation/special case of the model discussed so far." The first step in the argument is to expand the utility function, U(2), by means of a Taylor approximation around the expected value (or mean) of 2, denoted by E(2):

Taking expectations on both sides of (12.62) yields the (approximate) expression for expected utility:

In going from the first to the second line in (12.63) we use the fact that the expected value of a constant is that constant itself. The expected utility associated with end- of-period wealth can thus be approximated by the utility of expected wealth (first term on the right-hand side in the second line), a term involving the variance of end-of-period wealth (second term), plus higher-order terms subsumed in the dots.

The second step in the argument amounts to ignoring all higher-order terms in (12.63) so that preferences of the investor are (assumed to be) fully described by

11 See Hirshleifer and Riley (1992, pp. 69-73) for a further discussion.

331

The Foundation of Modern Macroeconomics

only the mean and the variance of end-of-period wealth; hence the name of the mean-variance approach. In summary, we write expected utility as:

where ri - 1LI"(E(2)). The sign of 7] fully characterizes the investor's attitude towards risk. Indeed, if 77 = 0, the variance term drops out of (12.64) altogether and the investor is only interested in the expected value of end-of-period wealth.

Such an investor, who totally disregards the variance of end-of-period wealth, is called risk neutral. In terms of Figure 12.4, the underlying utility function, U(2), is simply a straight line from the origin (U'(Z) > 0 and U"(2) = 0 in this case).

In real life, most people do care whether the return they receive is certain (has a zero variance) or is subject to fluctuations and can be much higher or lower than

expected (has a positive variance). Risk-averse investors are therefore characterized by a positive value of r . In terms of Figure 12.4, a risk-averse investor has an underlying utility function which is concave (U'(Z) > 0 and U"(2) < 0). In order to take on additional risk (a "bad" rather than a good) a risk-averse agent must be compen-

sated in the form of a higher expected return, i.e. he must receive a risk premium. In formal terms the risk-premium, n-R , is such that the agent is indifferent between the risky prospect Z and the certain prospect E(Z) (see Pratt, 1964):

In general n-R depends on the distribution of Z but a simple example can be used to illustrate what is going on. Suppose that the distribution of Z is such that 2 Zo -h

Figure 12.4. Attitude towards risk and the felicity function

or 2 = Zo+h with equal with this distribution right-hand side of (12. straight line connectin that the utility of the utility, E(U(Z)), i.e. po determine the certain j is done by going to poi distance between p( indifferent between. o hand, receiving Zo - receive a risk premiu The third type of ae,‘ Such an agent is calle outcome when both and is willing to pay a risk-lover has a a In the remainder of tl risk-averse investors. Up to this point, variable 2 which is sto z.le next step in our ing a particular probd, ,z tribution to choc

r N , ,

here "-" means "is bution", r is the rn,,: working with the nor ,ply two parameters, i = 3, 5, 7, • ) are eqi Jrthermore, the h _ be expressed in tern.: that (12.63) can al•.

...er terms, i.e. pr advantage of using ti comparative static choice below.

Armed with the (! of end-of-period wL.., After some manipula