Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

la' price level associated w:".

d is quite simple, but the btyli r the notion of an infinit, in is much harder to obtain cl on simple special cases. ' - to simplify the exposition of

is additively separable:

(12.P('

ins, v'(mt ) = 0 for mt = m", y of consumption is positive -',1e provided the real money

(12.891

(12.90')

seen drawn. Equation (12.90)

'stands for "terminal condi-

-drawn in the figure as the t Eo . Before going on to the librium at E0 , it is instrucits. An increase in the money re EE line, say to EE 1 in Figure

'Ices in the first period fall,

I future nominal money bal- - -.tional representative agent ices of higher money growth not only in the future but of pure time preference is

optimal real money balances -ressions in (12.86)-(12.87)

Chapter 12: Money

and can thus be expressed as implicit functions of taste and endowment parameters and the money growth rate, i.e. we can write mI = ml (p, Y, ,u) and rti2 = frq(p, Y, p). For the separable case of (12.88) these implicit functions feature the following partial derivatives with respect to the money growth rate: amI/am, < 0 and amt/aµ = 0. Since the rate of money growth is a policy variable it follows that the policy maker has the instrument needed to influence the equilibrium of money balances, at least in the first period. By substituting rn,*() and (12.85) into the utility function of the representative agent (12.88) we obtain:

A utilitarian policy maker can pursue an optimal monetary policy by choosing the money growth rate for which the welfare of the representative agent is at its highest level. By maximizing (12.91) by choice of it we obtain (a variant of) the Friedman satiation result:

where is the optimal money growth rate. This optimal growth rate of the money supply is such that the corresponding demand for current real money balances chosen by the representative household is such that the marginal utility of these balances is zero (and thus equal to the social cost of producing these balances). In terms of Figure 12.8, the social optimum is at point E s° and corresponds to a higher level of real money balances and a lower money growth rate than at point Eo .

EE

Figure 12.8. Monetary equilibrium in a perfect foresight model

343

The Foundation of Modern Macroeconomics

The satiation result does not hold in the final period, of course, as the terminal condition pins down a positive marginal utility of money balances needed for transaction purposes (see (12.87)). It is straightforward to generalize the Friedman result to a setting with an infinitely lived representative agent. 18 In that case terms like (1 +p)t-1 U(Ct, mt) are added to the utility function in (12.78) and budget equations like A are added to (12.79) (both for t = 3, 4, 5, , oo). Equation (12.86) is then generalized to:

The thing to note about (12.93) as compared to (12.89)-(12.90) is that the terminal condition is no longer relevant. Brock (1975, pp. 138-141) shows that the equilibrium solution to (12.93) will in fact be the steady-state solution for which

mt = mt+i = m*:

Since both the endowment and real money balances are constant over time, lifetime utility of the infinite lived representative agent is equal to:

p

Maximizing (12.96) by choice of p, yields the result that the optimal money supply is such as to ensure that v'(m*) = 0 for all periods. In view of (12.94), this is achieved if the money supply is shrunk at the rate at which the representative household discounts future utility:

Although there are no interest-bearing assets in our model, equation (12.97) can nevertheless be interpreted as a zero-interest rate result (see Turnovsky and Brock, 1980). Indeed, the pure rate of time preference represents the psychological costs associated with waiting and can be interpreted as the real rate of interest. Furthermore, since real money balances are constant, the money growth rate also represents the rate of price inflation. The nominal rate of interest in the optimum is thus R p I (1 + p) + = p I (1 + p) + ,u* =0.

18 Much of modern macroeconomic theory makes use of such a fictional agent. See e.g. section 4.4 below and Chapters 14-17.

12.4.3 Critiques of the f

The Friedman satiation rule money growth instrument i ances to zero, has come un demonstrate the two most is invalidated. In order to do basic model of section 4.1 a economy and by introduL...

I

Introducing endogenous prc

We assume that the repress sumption of goods and real (12.78) is replaced by:

I

V = u(Ci , 1 - Li, mi)

where the time endowmen period t (= 1, 2). The hou-

Wr (1 - TOL' + Mo + 14/(1 - t2)L2 + M, + P,

where A) is given, Pt T r rep ernment, 147%1 is the nomii things simple we assume that the production functio price equal to marginal c( does not consume any goo consumption by househole have that:

Ct = Yr = Lt, W =1

Rather than analysing the separately, it is legitima have the household-prof money balances directly. .1 characterizing the opting

ui_L(xt) = (1- 11)

[Uc (x - Um (x )1 m =

Uc(x2) = Um(X2),

of course, as the termin i! iey balances needed for trans-

• ..-ralize the Friedman result ■ent. 18 In that case terms like (12.78) and budget equations both for t = 3, 4, 5, . , cc).

, ^c). (12.93)

2.89)412.90) is that the terpp. 138-141) shows that the iy-state solution for which

(12.94)

(12.95)

constant over time, lifetime to:

(12.96)

t the optimal money supply is of (12.94), this is achieved he representative household

(12.97)

^ ,del, equation (12.97) can t (see Turnovsky and Brock, ants the psychological costs

7preted as the real rate of )nstant, the money growth i nal rate of interest in the

-)nal agent. See e.g. section 4.4

Chapter 12: Money

12.4.3 Critiques of the full liquidity rule

The Friedman satiation rule, according to which the policy maker should use its money growth instrument in order to drive the marginal utility of real money balances to zero, has come under severe criticism in the literature. We now wish to demonstrate the two most important mechanisms by which the full liquidity result is invalidated. In order to do so we return to the two-period setting but we enrich the basic model of section 4.1 above by moving from an endowment to a production economy and by introducing (potentially) distorting taxes.

Introducing endogenous production

We assume that the representative household derives utility not only from consumption of goods and real money balances but also from leisure. Hence, equation (12.78) is replaced by:

where the time endowment is unity, Lt is labour supply, and 1 - Lt is leisure in period t (= 1, 2). The household budget identities in the two periods are:

where Mo is given, Pt Tt represents lump-sum cash transfers received from the government, Wt is the nominal wage rate, and rt is the tax rate on labour. To keep things simple we assume that production is subject to constant returns to scale and that the production function is given by Yt = Lt . Perfectly competitive producers set price equal to marginal cost which implies that Pt = WN. As before, the government does not consume any goods so that the goods market clearing condition requires consumption by households to equal production in both periods. In summary, we have that:

Rather than analysing the behaviour of the representative household and firm separately, it is legitimate to incorporate (12.101) into (12.99)-(12.100) and to have the household-producer choose consumption (and thus production) and real money balances directly. Assuming an interior solution, the first-order conditions characterizing the optimum are given by:

The Foundation of Modern Macroeconomics

where xt [Ct, 1 - Ct, mt] and we have used the definition of the money growth rate (given in (12.81)) to simplify (12.103). Equation (12.102) shows that the household equates the marginal rate of substitution between leisure and consumption to the after-tax wage in both periods. Equations (12.103)-(12.104) generalize (12.86)- (12.87) by accounting for an endogenous labour supply (and thus production) choice. Armed with this minor modification to our original model, the robustness of the full liquidity result can be examined.

Non-separability

The model is solved recursively by working backwards in time, just as in section 4.1 above. We assume that both tax rates are constant. Equations (12.102) (for t = 2) and (12.104) then pin down optimal levels of consumption (and labour supply) and money balances for the final period (q and m2, respectively) which are constant and independent of the rate of money growth IL Given these values for C2 and rrG, equations (12.102) (for period t = 1) and (12.103) together constitute a system of implicit equations expressing CI and m1 in terms of the rate of money growth it (as well as p, r1 , and r2, but these are held constant). Denoting these implicit functions by C1(µ) and m1 (µ), we obtain the following derivatives by means of standard techniques:

where I AI is the (negative) Jacobian of the system and where the partial derivatives Ucc, Ucm, UCJ-L, U1-L,1-L, and U1-L,m are all evaluated in the optimum point (CI,

1-

The expression in (12.106) shows that the sign of diri/d,u is ambiguous in the

generalized model. The existence of diminishing marginal utility of leisure and consumption ensures that Ui _LJ _L and Ucc are both negative, but the cross-term,

===- (kJ-La can have either sign. Turnovsky and Brock (1980, p. 197) argue that it is reasonable to assume on economic grounds that Uc,1_L is positive, i.e. the marginal utility of consumption rises with leisure. With that additional assumption it is clear that optimal money holdings in the current period fall as the money growth rate is increased, i.e. dmI/d,u < 0. This conclusion generalizes our earlier result obtained for the basic model of section 4.1 above (see (12.95)).

As the expression in (12.105) shows, the sign of dq/dp, is also ambiguous in general as it depends on the cross-partial derivatives U1_L,m and UCm which can have either sign and about which economic theory does not suggest strong priors. In economic terms the ambiguity arises because it is not a priori clear how (or even whether) the rate of money growth affects the marginal rate of substitution between consumption and leisure, i.e. how ,u influences the consumption-leisure trade-off.

The issue can be invest stitution between leisure an form as g(Ci,m1):

Ui_L (CI , 1 g(Ci, mi) = Uc(Ci, 1 -

By partially differentiating

gm (Ci, mi) = Uc

th-L,rn -

.■■■■11

I

1

where we have used (12.1 intimate link which exists 1 the marginal rate of subs: real money balances, g,„(C rate leads to an increase

The upshot of the

rate of money growth and plugging C1(µ) and m*, (pc for household utility in ter

I

V U (cl(u), 1 - c; (m

The policy maker selects tl V, a problem which yields

where we have used equati sion we can re-examine th to which should be set to zero. Equation (12.11( ues to hold in our exten,., first period (ri = 0) then change in the money b

it does affect consumptior that case and the sign (or the right-hand side of (1_ money growth rate entails

ition of the money growth

.102) shows that the houseleisure and consumption to 12.104) generalize (12.86)- v (and thus production ) :inal model, the robustness

I time, just as in section 4.1 tions (12.102) (for t = 2) :1 (and labour supply) and lively) which are constant these values for C2 and )gether constitute a system the rate of money growth ). Denoting these implicit g derivatives by means of

here the partial derivatives in the optimum point (CI,

is ambiguous in the pal utility of leisure and f:itive, but the cross-term,

.ck (1980, p. 197) argue t Uc,i_L is positive, i.e. the t- it additional assumption period fall as the money ion generalizes our earlier

e (12.95)).

.1g is also ambiguous in

1-4„m and Ucm which can not suggest strong priors. a priori clear how (or even of substitution between imption-leisure trade-off.

Chapter 12: Money

The issue can be investigated more formally by writing the marginal rate of substitution between leisure and consumption (for period t = 1) in a general functional

form as g(C1, m1):

By partially differentiating g(.) with respect to m1 we obtain the following result:

Ul-L,m g(C1,M1)UCm

Uc

Ul-L,m — (1 — TOUCm (12.108)

Uc

where we have used (12.102) in the final step). The expression in (12.108) shows the intimate link which exists between gm (Ci, m1) and the sign of dq/dit in (12.105): if the marginal rate of substitution between leisure and consumption rises (falls) with real money balances, gm (Ci , m1 ) > 0 (< 0), then an increase in the money growth rate leads to an increase (decrease) in goods consumption, i.e. dCl/d,u > 0 (< 0).

The upshot of the discussion so far is that Cl; and rrq do not depend on the rate of money growth and that CI and m1 do so but in an ambiguous fashion. By plugging CI (4) and mi (A) into the utility function (12.98) we obtain an expression for household utility in terms of the policy variable it:

The policy maker selects the optimal money growth rate it* in order to maximize V, a problem which yields the following first-order condition:

where we have used equation (12.102) to simplify (12.110). Armed with this expression we can re-examine the validity of the Friedman full-liquidity result according to which i_t* should be set such as to drive the marginal utility of money balances to zero. Equation (12.110) shows the various cases under which this result continues to hold in our extended model. First, if there is no initial tax on labour in the first period (t1 = 0) then the leisure-consumption choice is undistorted so that a change in the money growth rate does not create a first-order welfare effect even if it does affect consumption in the first period. In terms of (12.110), Uc = Ui_L in that case and the sign (or magnitude) of deildit does not matter. The first term on the right-hand side of (12.110) drops out and, provided dmI/dp, 0 0, the optimal money growth rate entails driving Um to zero.

347

The Foundation of Modern Macroeconomics

The second case for which the satiation result obtains is one for which the tax is strictly positive (ti > 0) but consumption is independent of the money growth rate (dCl/d,u, = 0). This case was emphasized by Turnovsky and Brock (1980). In terms of (12.110) and (12.108) this holds if the marginal rate of substitution between leisure and consumption does not depend on A. If that result obtains, the felicity function U(.) is said to be weakly separable in (Ct , 1 —Li ) on the one hand and mt on the other. It can then be written as:

u(ct , 1 — Lt, mt) = U [Z(ct, 1 — Li), mt] (12.111)

where Z(.) is some sub-felicity function. Note that (12.111) implies that the marginal rate of substitution between leisure and consumption only depends on the properties of Z(.), as th-L/Uc = = Z1_L/Zc and thus does not depend on mt.

In summary, the Friedman satiation result holds in our model if (i) there is no initial tax on labour income (ti = 0), and (ii) if ti is positive but preferences display the weak separability property. In general, however, (12.109) implicitly defines the optimal money growth rate and Um will not be driven to zero. Turnovsky and Brock refer to (12.110) as a "distorted" Friedman liquidity rule (1980, p. 197).

The government budget restriction

The second major argument against the validity of the Friedman result is based on the notion that steady-state inflation (caused by nominal money growth) can be seen as a tax on money balances and thus has repercussions for the government budget constraint especially in a "second-best" world in which lump-sum taxes are not available to the policy maker. In such a world, Phelps (1973) argues, government revenue must be raised by means of various distorting taxes, of which the "inflation tax" is only one. The literature initiated by Phelps is often called the "public finance" approach to inflation and optimal money growth. Briefly put, the Phelps approach is an application to monetary economics of the optimal taxation literature in the tradition of Ramsey (1927). 19 We return to the insights of Phelps (1973) below.

12.4.4 An infinite horizon model

Up to this point we have employed simple two-period models in order to demonstrate (some of) the key issues in monetary macroeconomics. Although such two-period models are convenient for some purposes, they also have some undesirable features. For example, as the model only distinguishes two periods ("today" and the "future"), there is no third period and the model economy "closes down" at the end of period 2. The aim of this subsection is to get rid of this rather unattractive feature of the model. To that effect, we develop a general equilibrium model of a

19 We briefly discussed Ramsey taxation in the context of Chapter 10 above.

monetary economy in w time in the future but use this multi-period MO( different assumptions I

Households

We assume that the bet with the fictional reprt lifetime utility function:

V = ( 1

At=i—•

where the felicity funci... diminishing marginal fed satiation level for real mo diminishing. Since timiR the notation by defini: felicity function as follui,

au(xt) uc(xi) = act >

au(xi

ui_L(xi) =

=au(xt) um(xt) aam[ 1

To keep the model as capital, and assume th.. time by means of govern!

in period t (= 1, 2, ....) i

I

where Bt _i is the stock oi the nominal interest or- _ stock of money balance tt is the proportional tax government. Equation fers, recognizing endoc income), and by distin,

ins is one for which the tax i -:t of the money growth rate and Brock (1980). In terms )f substitution between leisure

obtains, the felicity functior `ne hand and mt on the

(12.111►

it (12.111) implies that tl IF l mption only depends on = Z i _L/Zc and thus does n , •

n our model if (i) there is no nitive but preferences display ' 2.109) implicitly defines the to zero. Turnovsky and Brock 'e (1980, p. 197).

e Friedman result is based on i nal money growth) can be cussions for the government in which lump-sum taxes are Rs (1973) argues, government taxes, of which the "inflation 'i called the "public finance"

\- put, the Phelps approach nal taxation literature in the s of Phelps (1973) below.

models in order to demoneconomics. Although such they also have some undeuishes two periods ("today" 1 economy "closes down" at

' of this rather unattractive oral equilibrium model of a

!O above.

Chapter 12: Money

monetary economy in which the economy does not come to a full stop at some time in the future but instead runs on indefinitely through time. We subsequently use this multi-period model to demonstrate the validity of the Friedman rule under different assumptions regarding government financing.

Households

We assume that the behaviour of households in the economy can be captured with the fictional representative agent who is infinitely lived and has the following lifetime utility function:

where the felicity function, U(.), has the usual properties: (i) there is positive but diminishing marginal felicity for both consumption and leisure, (ii) there exists a satiation level for real money balances, Tit, and (iii) marginal felicity of real money is diminishing. Since timing issues will prove extremely important below, we simplify the notation by defining xt E-- [Ct, 1 - and by writing the properties of the felicity function as follows:

(12.113)

To keep the model as simple as possible we continue to abstract from physical capital, and assume that the representative household can shift resources through time by means of government bonds and/or money. The periodic budget constraint in period t (= 1, 2, ....) is given in nominal terms by:

where Bt_1 is the stock of government bonds held at the end of period t - 1, Rt _ i is

the nominal interest on government bonds paid at the end of period t -1, Mt is the stock of money balances at the beginning of period t, WPT is the nominal wage rate, rt is the proportional tax on labour income, and PtTt is transfers received from the government. Equation (12.114) generalizes (12.1)-(12.2) by adding taxes and trans-

fers, recognizing endogenous labour income (rather than exogenous endowment income), and by distinguishing multiple periods.

349

The Foundation of Modern Macroeconomics

By dividing both sides of (12.114) by the current price level, Pt, we obtain the household's budget identity in real terms:

7rt Pr-FilPt--1 is the inflation rate, and rt -.=_Pt(l+Rt)11)t+i-1 is the real interest rate.

The household's budget identity is a difference equation in bond holdings, bt, which can be solved forwards in time by repeated substitutions. After some tedious but straightforward manipulations we find the following general expression (see the appendix to this chapter):

where Ao (1 + ro)bo + mo/(1 + 70) and q(t) is a rather complicated discount factor involving the real interest rates in future periods:

provided the following so-called transversality conditions hold:

lim 4+i bk± i = 0, k—,00

0

qk+1 Mk+ 1 lim= k--+. 1 ± Rt+k

We postpone a more extended discussion of transversality conditions to Chapter 14. Intuitively, equation (12.119) means that the household's assets (bt positive) cannot grow faster than the rate of interest. In the case of household debt (bt negative) the household would, of course, be perfectly happy to let the expression in (12.119) be negative but there will be no lenders in the market allowing this to happen. Technically, (12.119) is a terminal condition on the household's debt as time goes to infinity. It does the same thing in the infinite-horizon model as the assumption of B2 = 0 does in the two-period model of section 12.2.1, namely to ensure that the household is solvent. The intuition behind (12.120) is similar.

Equation (12.118) is the present-value budget constraint of the household. It shows that the excess of spending on consumption and money balances over the

after-tax wage income plus in present-value terms to the Equation (12.118) is the in:, The household chooses se money balances (i.e. {Ct} utility (12.112) subject to to expression associated with U

where A. is the Lagrangian optimum are the constr.-at

a = 0: act

a = 0: aLt

amaLt =o:

By eliminating the Lagrange conditions equating ma:

In each period, the marginal should be equated to the al tution between real mono opportunity cost of holding

Firms, government, and Wal

The firm sector is very sin., produced with labour only_ representative firm maxin nology. The perfectly comps

I

t price level, Pt, we obtain t!-•

Wtiv IPt is the real wage rate, Pt-,-1-1 is the real interest repquation in bond holdings, br , bstitutions. After some tedious -ing general expression (see t! -

(12.116►

T complicated discount factor

(12.117)

(12.118)

hold:

(12.119)

(12.120)

I

v conditions to Chapter 14.

s assets (bt positive) cannot hold debt (bt negative) the t the expression in (12.119) allowing this to happen. xisehold's debt as time goes model as the assumption 1. namely to ensure that the

4-lint of the household. nd money balances over the

The Foundation of Modern Macroeconomics

where Gt is the consumption of goods by the government. The sum of spending on interest payments on outstanding debt plus government consumption and transfers to households (left-hand side) must be equal to the sum of the labour income tax revenue, newly issued money balances, and newly issued government debt (righthand side). By dividing both sides of (12.128) by the current price level, and noting (12.4) and the definition of the real interest rate, we obtain the government budget identity in real terms:

Before discussing the key features of the model we check Walras' Law. By combining the household budget identity (12.115) with the government budget identity (12.129) we obtain is the resource constraint). But Wt = 1 (by (12.127)) and the production function implies Yt = Lt so it follows that Yt = Ct +Gt . So, provided the household and government budget identities are satisfied, so is the economy-wide resource constraint.

The adjusted household budget constraint

In order to prepare for our discussion of the optimal rate of money growth it is useful to derive the so-called "adjusted household budget constraint" (Ljungqvist and Sargent, 2000, pp. 319-325). This adjusted budget constraint is obtained by substituting the household's first-order conditions (12.122)-(12.124) into the regular, unadjusted, household budget constraint (12.118). After some manipulation we obtain the following expression:

where we have used (12.122)-(12.124) to get from the first to the second line. By applying (12.122) for t = 1 and noting that q7 = 1 (from (12.117)) we derive that A equals the marginal utility of consumption in the first period, i.e. A = Uc(x1)-

By substituting this result household budget constrd;;

Aouc(xi) =-- 2_,( 1+ t=1

The advantage of working 10 expression no longer conta (namely rt and At ). This fa, lem because the social (rather than in terms of tax

a

Optimal money growth re.

We now have all the ingrec ernment. The social planner real money balances (i.e. (Ci utility of the representative budget constraint (12.131) a We assume that the seque:.. given. The Lagrangian assoc

O G (UC(Xt) [Ct —

OGAOUC(X1),

where 9G is the Lagrange m and {4} ' 1 is the sequence Let us first assume that the in each period. It is clear ft.. because negative transfers a in this economy? The firs! take the following form: 1

But, since the discounting f itive, and we have ruled oL,

20 The approach followed here outputs and the direct utility fur of the primal approach to Rar - Sargent (2000, pp. 319-325) foil , dynamic context.