Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

I

h; hence the name of the ed utility as:

(12.64,

rites the investor's attitude )s out of (12.64) altogether ue of end-of-period wealth. f end-of-period wealth, is

.; utility function, U(2), is , Z) = 0 in this case).

hey receive is certain (has a much higher or lower than are therefore characterized rse investor has an under- r(2) < 0). In order to take The agent must be compen- must receive a risk premium.

-nt is indifferent between

'ratt, 1964):

(12.65)

ple example can be used to f 2 is such that 2 = zo — h

-z q= 0

<0

z

function

Chapter 12: Money

r Z = zo +h with equal probability 2 so that E(Z) = Zo . The risk premium associated fh this distribution is found by applying (12.65). In terms of Figure 12.4, the _ght-hand side of (12.65) is represented by point D which lies halfway along the straight line connecting points A and B. Concavity of the utility function ensures hat the utility of the expected outcome, U(E(Z)) = U(Zo), is higher than expected

utility, E(U(Z)), i.e. point C lies above point D. To find the risk premium we must determine the certain prospect (Zo — 7R) such that (12.65) holds. In Figure 12.4 this

is done by going to point E, which lies directly to the left of point D. The horizontal distance between points D and E represents the risk premium 7R. In order to feel indifferent between, on the one hand, receiving Zo — ITR for sure and, on the other hand, receiving Zo — h or Z0 + h with equal probability, the risk-averse investor must receive a risk premium equal to 7TR.

The third type of agent is described by (12.64) with a negative value for i inserted. Such an agent is called a risk-lover because he prefers an uncertain over a certain outcome when both have equal expected value. He thus enjoys the thrill of a gamble and is willing to pay (rather than receive) a risk premium. In terms of Figure 12.4, a risk-lover has a convex underlying utility function (U'(2) > 0 and U"(2) > 0). In the remainder of this section we focus attention on the portfolio behaviour of

risk-averse investors.

Up to this point, we have described the agent's expected utility in terms of the variable Z which is stochastic only because the return on the risky asset, r, is. Hence, the next step in our exposition of the mean-variance approach consists of postulating a particular probability distribution for r. A particularly simple and convenient

where "—" means "is distributed as," "N" stands for "normal or Gaussian distribution", r is the mean of the distribution, and its variance. The advantage of working with the normal distribution lies in the fact that it is fully characterized by only two parameters, r and All higher-order uneven terms, such as E(r — r)i (for i = 3, 5, 7,•) are equal to zero as the distribution is symmetric around its mean. Furthermore, the higher-order even terms, such as E(r — r)i (for i = 4, 6, 8, ) can be expressed in terms of r and QR (Hirshleifer and Riley, 1992, p. 72). This implies that (12.63) can always be written as in (12.64) even without ignoring the higherorder terms, i.e. preferences are fully described by only two parameters. Another advantage of using the normal distribution is that it enables us to conduct simple comparative static experiments pertaining to r and cR and the optimal portfolio

choice below.

Armed with the distributional assumption in (12.66), the probability distribution of end-of-period wealth can be determined by noting the definition of Z in (12.60). After some manipulation we derive that Z is distributed normally (i.e. Z — N(k , cry))

333

The Foundation of Modern Macroeconomics

with parameters depending on the portfolio fraction of money (0:

By manipulating the portfolio share of money, the investor can influence both the expected value of, and the risk associated with, end-of-period wealth. For example, if only money is held in the portfolio (w = 1), end-of-period wealth equals S(1 + TM) for sure (o-z = 0). This determines point A in Figure 12.5. The top panel of that figure plots combinations of expected return (vertical axis) and risk (horizontal axis), whilst the lower panel plots the relationship between risk and the portfolio share of money. 12 At the other extreme, if no money is held at all ((.0 = 0), expected end-of-period wealth equals S(1 + r) and the standard deviation is az = SaR. In order to have any non-trivial solution at all, the mean return on the risk asset must exceed that on money, otherwise a risk-averse agent would never hold any risky assets. Hence, r > rm must be assumed to hold. This in turn ensures that point B lies north-east of point A in the top panel of Figure 12.5. By connecting points A and B in the top panel we obtain the upward-sloping constraint representing feasible trade-off opportunities between average return and risk. In the lower panel, az and (0 are related by the second definition in (12.67) which can be rewritten as 1 —

The final step in our exposition of the mean-variance model consists of introducing the appropriate indifference curve. According to (12.64), expected utility depends on both i and cri and the indifference curve satisfies dEU(Z) = U'(Z)dZ — 2/7azdaz = 0 from which we derive:

Hence, the typical indifference curve of a risk-averse agent is upward sloping and convex; see for example EU0 in the top panel in Figure 12.5. Since expected return is a "good" and risk is a "bad" for such an agent, expected utility increases if the indifference curve shifts in a north-westerly direction.

It is clear from the slope configuration in Figure 12.5 that a risk-averse investor will typically choose a diversified portfolio. 13 Rather than choosing the safe haven of only money (point A) it is optimal for him to "trade risk for return", i.e. to accept some risk by holding a proportion of his portfolio in the form of the risky asset. In exchange the investor receives a higher expected yield on his portfolio. In Figure 12.5 the optimum occurs at point E0 where the indifference curve is

is convenient to work with the standard deviation of Z (rather than its variance) because it is in the same units as the mean of Z which facilitates the economic interpretation to follow.

13 For a discussion of possible corner solutions, see. Tobin (1958, pp. 77-78).

z

A

CO

(0*

0

Figure 1

tangential to the bu

daz

The left-hand side a1C") whereas the

Although (12.c.) former is merely a

14 The budget line is

1— w = cri/(a„Jj.

noney co:

(12.67

-)r can influence both t!- od wealth. For example, od wealth equals S(1 T M) !.5. The top panel of that ► and risk (horizontal axis), and the portfolio share I at all ((.0 = 0), expected

deviation is az = SCR. In rn on the risk asset must Fuld never hold any risky n turn ensures that point 2.5. By connecting points g constraint representing id risk. In the lower panel, which can be rewritten as

I model consists of intro-

•(12.64), expected utility dies dEU(Z) = U'(Z)dZ —

(12.68)

nt is upward sloping and 2.5. Since expected return ted utility increases if the

it a risk-averse investor choosing the safe haven e risk for return", i.e. to ) in the form of the risky ed yield on his portfolio. the indifference curve is

its variance) because it is in ,tion to follow.

-78).

Chapter 12: Money

Figure 12.5. Portfolio choice

tangential to the budget line. 14 In technical terms we have:

The left-hand side of (12.69) represents the slope of the indifference curve (subscript "IC") whereas the right-hand side is the slope of the budget line (subscript "BL").

Although (12.69) looks different from (12.61), it is not difficult to show that the former is merely a special case of the latter. Since we work with a second-order

The Foundation of Modern Macroeconomics

expansion of utility (see (12.62)), marginal utility can be written as U'(Z) = a — 21)2 , so that (12.61) can be written as:

0 = EU'(2)(rm — = E [(a — 2oZ) — 2o — ZARrm — — —

= — 29Z) (rM — r) 2oE — Z) (i- — r)

= U'(Z) (rM — 27/cov(2, (12.70)

where we have used E(Z) = Z and E(i) = r in going from the first to the second line and where cov(2, r) is the covariance between 2 and F. In view of the definition of Z in (12.61) we find that cov(2, i) = S(1 — (0)ai = azaR • By using this result in (12.70) we find that (12.70) (and thus (12.61)) coincides with (12.69).

Returning now to Figure 12.5, it is clear that a risk-averse agent will hold money even if its return is zero (rM = 0) because it represents a riskless means of investing (at least, under the present set of assumptions). By going to the lower panel of Figure 12.5, the optimal portfolio share of money, w*, can be found which implies that the demand for money equals w*S. Although S is given, w* (and hence money demand) depends on all the parameters of the model such as the yield on money, the mean and variance of the yield on bonds, and the preference parameter(s):

The conventional method of comparative statics can now be used to determine the partial derivatives of the w*(.) function.

First consider the effects of an increase in the yield on money rM (i.e. a reduction in the inflation rate). In terms of Figure 12.5, the budget line shifts up and becomes flatter; see the line A'B in the top panel. We get the result, familiar from conventional microeconomic demand theory, that the ultimate effect on the portfolio share of money (and thus money demand) can be decomposed into income and pure substitution effects. On the one hand, an increase in rM narrows the yield gap between money and the risky asset which induces the investor to substitute towards the safe asset and to hold a higher portfolio share of money. This is the pure substitution effect represented in Figure 12.5 by the move from E0 to E'. On the other hand, an increase in rM also increases expected wealth and the resulting income (or wealth) effect also leads to an upward shift in w. Hence, both income and substitution effects work in the same direction and the new optimum lies at point E1 , where the move from E' to El represents the income effect.

In formal terms, the total effect on w of an increase in rM can be expressed in the form of a conventional Slutsky equation:

where the first term . "compensated" effect an

aw

( ar dEU=0 (i — r

C°)

az — s[c.

The second, much ma effect on the money pa risky asset. Throughout 1 which are downward sic have implicitly assumed result is actually neces , .._

in r causes the budget lin

In contrast to the pre ,. opposite directions and t

where (a /ai_-),dEU=0 = —

In terms of Figure 12.6. the income effect is the or Ei if the

depends positively on t:. Tobin (1958). Under the we have employed t. lio approach does indee postulated by Keynes a:

The third and final demand of the degree ( standard deviation of

happen if aR rises. First, in a clockwise fashion a the investor must be wi,

of az. In the bottom pat to the portfolio share (

fashion around point A

where 0(0/82 is given in

SIR az

Figure 12.6. Portfolio choice and a change in the expected yield on the risky asset

The substitution effect dominates the income effect and money demand rises if the return on the risky asset becomes more volatile.

12.4 The Optimal Quantity of Money

In the previous two sections we have reviewed the main models of money which have been proposed in the postwar literature. We now change course somewhat by

Figure 12.7. Por the risky asset

taking for granted that r process and by posing t money. If fiat money policy maker bring int( _ from Friedman (1969). S of money to be equate tokens) imposes little or up to the point where U

az

B

SCR az

e expected

money demand rises if the

models of money which course somewhat by

Figure 12.7. Portfolio choice and an increase in the volatility of the risky asset

taking for granted that money exists and plays a significant role in the economic process and by posing the question concerning the socially optimal quantity of money. If fiat money is useful to economic agents then how much of it should the policy maker bring into circulation? This question received an unambiguous answer from Friedman (1969). Social optimality requires marginal social benefits and costs of money to be equated. Since the production of fiat money (intrinsically useless tokens) imposes little or no costs on society, the money supply should be expanded up to the point where the marginal benefit of money is (close to) zero and agents

339

The Foundation of Modern Macroeconomics

are flooded with liquidity (money balances). This is the famous full liquidity result proposed by Friedman (1969) and others. 15

Intuitively, people should not economize on resources which are not scarce from a social point of view (like fiat money). Since the opportunity cost of holding money is the nominal rate of interest on bonds, the strong form of the Friedman proposition requires the policy maker to manipulate the rate of money growth (and hence the inflation rate) such as to drive the nominal interest rate to zero (Woodford, 1990, p. 1071). The nominal interest is itself the sum of the real rate of interest (rt , which is largely determined by real factors according to Friedman) and the expected rate of inflation (4), i.e. Rt = rt + 4. Hence, in the steady state (rt = r and 71- = At) and with fulfilled expectations (4 = irt ) the Friedman proposition requires a constant rate of decline in the money supply equal to the (constant) real rate of interest, i.e.

Rt = 0 <#> --itt = -- 7rt = r.

The remainder of this section is dedicated to the following two issues. First, we demonstrate (a version of) the Friedman result with the aid of a simple two-period general equilibrium model. Second, we review the main objections which have been raised against the Friedman argument in the literature.

12.4.1 A basic general equilibrium model

In section 2 above we discussed several justifications for putting real money balances into the felicity function of households. We now postulate that the lifetime utility function of the representative agent can be written as follows:

V = u(ch mo+ ( 1+ p ) m2), (12.78)

where mt denotes real money balances held at the end of period t. 16 Abstracting from bonds, endogenous production, and economic growth, the budget identities in the two periods are given by:

where Mo is given and Pt Tt represents lump-sum cash transfers received from the government. The representative agent takes these transfers as parametrically given in making his optimal plans, but in general equilibrium they are endogenously determined.

15 Other important contributors to the debate are Bailey (1956) and Samuelson (1968b, 1969a). An excellent survey of this vast literature is Woodford (1990).

16 We thus change the timing of the utility-yielding effect of money in comparison to the arguments in section 2. We do so in order to simplify the argument and to retain consistency with Brock's (1975) model of which our model is a special case.

We postulate a simpi, nominal money growth is

AMt

=

Aft-i

where it is a policy instru money supply is disbur , transfers:

PtTt = AMt.

The household chooses C. to (12.79)-(12.80). Assur problem are: I

Uc(CiA rni) = [WC] .n

Pi P

Uc, (C2, m2) = Um(C2,n

I where Uc (.) auoiact 2

marginal utility of spend)- equated to the marginal balances (the right-hand sk reduced transaction cost' money (second term). In u value so only the transactic the expression in (12.84) In the absence of goods c investment, the product equals private consumpu,oi

By multiplying the expres (12.85), the perfect fore ,

[Uc(Y , m1) - U,,,(Y,

Uc(17 , m2) = Um(Y, m21

These two equations recu supply. The trick is to wo, for m2 . Second, by using th an equation deterifuning:

, e famous full liquidity result

s which are not scarce from a inity cost of holding money is of the Friedman proposition toney growth (and hence the ate to zero (Woodford, 1990, real rate of interest (rt , which dman) and the expected rate state (rt = r and n-t = At) and )position requires a constant :#ant) real rate of interest, i.e.

ollowing two issues. First, we aid of a simple two-period a objections which have been

-utting real money balances ulate that the lifetime utility `'l lows:

(12.78)

Id of period t. 16 Abstracting Towth, the budget identities

(12.79)

(12.80)

transfers received from the

,fors as parametrically given 11 they are endogenously

ind Samuelson (1968b, 1969a).

n comparison to the arguments consistency with Brock's (1975)

Chapter 12: Money

We postulate a simple money supply process according to which the rate of nominal money growth is constant:

where is a policy instrument of the government. The increase in the nominal money supply is disbursed to the representative agent in the form of lump-sum transfers:

PtTt = AMt. (12.82)

The household chooses Ct and A (for t = 1, 2) in order to maximize (12.78) subject to (12.79)-(12.80). Assuming an interior solution, the first-order conditions for this

problem are:

marginal utility of spending one dollar on consumption (the left-hand side) must be equated to the marginal utility obtained by holding one dollar in the form of money balances (the right-hand side). The latter is itself equal to the marginal utility due to reduced transaction costs (first term) plus that due to the store-of-value function of money (second term). In the final (second) period, money is not used as a store of value so only the transactions demand for money motive is operative. This is what the expression in (12.84) says.

In the absence of goods consumption by the government, and public and private investment, the product market clearing condition says that endowment income

By multiplying the expression in (12.83) by M1 and using (12.81), (12.84), and (12.85), the perfect foresight equilibrium for the economy can be written as:

These two equations recursively determine the equilibrium values for the real money supply. The trick is to work backwards in time. First, equation (12.87) is solved for m2. Second, by using this optimal value, say m2, in the right-hand side of (12.86), an equation determining m1 is obtained. Since the path of the nominal money

341

The Foundation of Modern Macroeconomics

supply is determined by the policy maker, the nominal price level associated with the solution is given by 11

In our simple two-period model the solution method is quite simple, but the bulk of the literature on the optimal money supply is based on the notion of an infinitely lived representative agent for which a general solution is much harder to obtain. Indeed, in that literature the discussion is often based on simple special cases. In order to facilitate comparison with that literature and to simplify the exposition of our model, we now assume that the felicity function is additively separable:

with u'(Ct) > 0, u"(Ct ) < 0, (mt ) > 0 for 0 < mt < m*, V(mt) = 0 for mt = m*, v'(mt ) < 0 for mt > m* and v'(mt ) < 0. Marginal utility of consumption is positive

throughout but satiation with money balances is possible provided the real money supply is sufficiently high.

In Figure 12.8 these two equilibrium conditions have been drawn. Equation (12.90) is represented by the horizontal line TC, where "TC" stands for "terminal condition". Equation (12.89) is an Euler-like equation and is drawn in the figure as the upward-sloping EE line. 17 The equilibrium is at point Eo. Before going on to the issue of social optimality of the perfect foresight equilibrium at E0, it is instructive to conduct some comparative dynamic experiments. An increase in the money growth rate, for example, leads to an upward shift in the EE line, say to EE 1 in Figure 12.8. The equilibrium shifts to E1 and real money balances in the first period fall, i.e. dmI/dau < 0. Hence, even though only the level of future nominal money balances is affected (M1 stays the same and M2 rises), the rational representative agent endowed with perfect foresight foresees the consequences of higher money growth and as a result ends up bidding up the nominal price level not only in the future but also in the present. A similar effect is obtained if the rate of pure time preference is increased.

12.4.2 The satiation result

We have seen that, in our simple two-period model, the optimal real money balances in the two periods are determined recursively by the expressions in (12.86)-(12.8;)

17 The slope of the EE line is:

dm2 _ + p)(1 + u) [W(') — v'(mi) — rniv"(mi)] > 0. dmi u'(Y)

and can thus be expressed a and the money growth ra' For the separable case of ( derivatives with respect to Since the rate of money g - has the instrument needet in the first period. By sub , • representative agent (12.6

V = u(Y) + v (mi(p, Y,

A utilitarian policy maker c, money growth rate for wh., level. By maximizing (12.91 satiation result: 111

where p,* is the optimal mot supply is such that the chosen by the representat ti balances is zero (and thus e terms of Figure 12.8, the soc level of real money balance

m2

al;

Figure 12.8. foresight 1i,14