Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(12.12-'

I terms by:

Bt-1), (12.12 ,

'nt. The sum of spending

t consumption and transfers rn of the labour income tax government debt (right-

Tent price level, and notin lin the government bud . -t

(12.129)

I• Walras' Law. By combinlyernment budget identity

nstraint). But Wt = 1 (by

it follows that Yt Ct +Gt. •:ties are satisfied, so is the

of money growth it is n constraint" (Ljungqvist constraint is obtained by 12.124) into the reg- :ter some manipulation

- rt )Lt

d-xt)Lt] (12.130)

a to the second line. By 2.117)) we derive that period, i.e. A = Uc(xi).

Chapter 12: Money

By substituting this result in (12.130) we obtain the final expression for the adjusted household budget constraint:

t_i

Aouc(xi) = t=i 1 P [Uc(xt) [Ct — Tt] + Um(xt)mt — ui_L(xoLti . (12.131)

The advantage of working with (12.131) instead of with (12.118) is that the former expression no longer contains the distorting tax instruments of the government (namely tt and ,ut ). This facilitates the characterization of the optimal taxation problem because the social planning problem can be conducted directly in quantities (rather than in terms of tax rates). 2°

Optimal money growth revisited

We now have all the ingredients needed to study the optimal tax problem of the gov-

ernment. The social planner chooses sequences for consumption, employment, and real money balances (i.e. (Ct ri° 1 , {Lt }r i , and tnit rt' i ) in order to maximize lifetime utility of the representative household (12.112) subject to the adjusted household

budget constraint (12.131) and the economy-wide resource constraint

We assume that the sequence of government consumption, {G t }c;°,, is exogenously given. The Lagrangian associated with this optimization programme is:

where OG is the Lagrange multiplier for the adjusted household budget constraint and {4}r 1 is the sequence of Lagrange multipliers for the resource constraint.

Let us first assume that the policy maker can freely adjust the level of transfers, Tt, in each period. It is clear that this scenario includes the case of lump-sum financing because negative transfers are allowed. What is the optimal rate of money growth in this economy? The first-order conditions for the sequence of transfers, ITtrt'i, take the following form:

But, since the discounting factor on the right-hand side of (12.133) is strictly positive, and we have ruled out satiation of consumption > 0—see (12.113)),

20 The approach followed here is called the "primal" approach to the Ramsey problem because it uses outputs and the direct utility function. See Atkinson and Stiglitz (1980, pp. 376-382) for a discussion of the primal approach to Ramsey taxation in static models. Jones et al. (1997) and Ljungqvist and Sargent (2000, pp. 319-325) follow Lucas and Stokey (1983) by applying the primal methods in a dynamic context.

353

The Foundation of Modern Macroeconomics

it follows from (12.133) that OG = 0. Intuitively, the availability of the lump-sum instruments means that the adjusted household budget constraint does not represent a constraint on the social optimization programme. The remaining first-order

conditions of the social plan are obtained by setting aLG/aCt = aLG/aLt = aLG/amt = 0 (for t = 1, 2, ...) and noting that OG = 0. After some straightforward

manipulation we find:

Equation (12.134) shows that the marginal rate of substitution between leisure and consumption should be equated to the marginal rate of transformation between labour and goods (which is unity since the production function is linear). Equation (12.135) is the Friedman rule requiring the policy maker to satiate the representative household with money balances. Equations (12.134)-(12.135) characterize the socially optimal allocation in terms of quantities. In the final step we must find out what tax instruments the planner can use to ensure that these conditions hold in the decentralized economy. By comparing (12.134)-(12.135) to the first-order conditions for the household, given in (12.125)-(12.126), we find that they coincide if there is no tax on labour income and the nominal interest rate is zero, i.e. rt = Rt = 0. With a constant level of government consumption (Gt = G for all t) the optimal allocation is constant, i.e. Ct = C, Lt = L, bt = b, mt = m,

Wt = W, and Tt = T for all t. The real interest rate is equal to the rate of pure time preference, rt = p, and, since the nominal rate is zero, it follows that the rate of inflation is constant and equal to Trt = — PI(1+ p). Since m is constant, the rate of money growth equals the rate of inflation, i.e. At = -pi(1+ p).

Ramsey taxation

Matters are not as simple if the policy maker does not have access to a freely adjustable lump-sum instrument like Tt . In the absence of such an instrument the policy maker is forced to raise the required revenue, needed to finance the government's consumption path, in a distortionary fashion, i.e. by means of a tax on labour income and/or by means of money growth (the inflation tax). In the remainder of this subsection we briefly sketch the complications which arise in this setting. As before, the social planner chooses sequences {Ct }ci.' 1 , {Lt }t' 1 , and fmt yl' i which maximize (12.112) subject to (12.131) and the resource constraint Lt = Ct + Gt . We now assume, however, that Tt = 0 for all t.

The first-order conditions for an interior solution for real money balances is given by aLG/amt = 0 for all t. By using (12.132) we derive the following conditions for,

respectively, m1 and Int It

Um(X1) O G [Uni(X1)

I where the term involviht, marginal utility of consurr

glances. In contrast to t: positive. Intuitively, 0 6 r trough distortionary Lonsequence which folk the full liquidity rule is rm

consumption and rt._ separable case (12.136) an

Um(Xt) = — MtUmm(Xt)

A he optimal level of real Friedman result no loft.:

12.5 Punchlines

ney performs three ma serves as a store of value, live functions, the first i _act that every layman N.. to be difficult to come up

..ipter we discuss some c the literature.

The medium of excl-.

.,...)ney reduces the traro. agents. In this view, the e

..ce leisure is valued 13! shopping cost approach

.croeconomic model: function. The ca.u.

for this practice.

The role of money a

in the first model, intrin e - _age in intertempor,: raiAancial assets availa b.,

-ailability of the lump-sum et constraint does not repre-

. The remaining first-order

ng arciaCt = aLG/aLt =

. After some straightforward

(12.134)

(12.135)

ubstitution between leisure nal rate of transformation production function is linthe policy maker to satiate ;quations (12.134)-(12.135) quantities. In the final step an use to ensure that these paring (12.134)-(12.135) to (12.125)-(12.126), we find ad the nominal interest rate ment consumption (Gt = G

Lt = L, bt = nit = pal to the rate of pure time

it follows that the rate of m is constant, the rate of

1 + p).

have access to a freely of such an instrument the

.-eded to finance the gov- t, i.e. by means of a tax on

,9ation tax). In the remain-

.. hich arise in this setting.

{1,}7" 1 , and {mt }"t _i which mstraint Lt = Ct +Gt . We

.11 money balances is given e r !lowing conditions for,

where the term involving Ucm (x1) appearing in (12.136) is due to the fact that the marginal utility of consumption in the first period in general depends on real money balances. In contrast to the lump-sum case, the Lagrange multiplier O G is now strictly positive. Intuitively, O G measures the utility cost of raising government revenue through distortionary taxes (Ljungqvist and Sargent, 2000, p. 323). An immediate consequence which follows from the first-order conditions (12.136)-(12.137) is that the full liquidity rule is no longer optimal even if the felicity function is separable in consumption and real money balances (so that U cm (xt ) = 0). Indeed, in the separable case (12.136) and (12.137) coincide and can be simplified to:

The optimal level of real money balances falls short of its satiation level and the Friedman result no longer obtains in this setting.

12.5 Punchlines

Money performs three major functions in the economy: it is a medium of exchange, serves as a store of value, and performs the role of a medium of account. Of these three functions, the first is the most distinguishing function of money. Despite the fact that every layman knows what money is (and what it can do) it has turned out to be difficult to come up with a convincing model of money. In the first part of this chapter we discuss some of the more influential models that have been proposed in the literature.

The medium of exchange role of money has been modelled by assuming that money reduces the transactions costs associated with the trading process between agents. In this view, the existence of money reduces the time needed for shopping. Since leisure is valued by the agents, the same holds for money. This so-called shopping cost approach is one way to rationalize the conventional practice in macroeconomic modelling of putting money balances directly into the household's utility function. The cash-in-advance approach is another possible rationalization for this practice.

The role of money as a store of value has been modelled in two major ways. In the first model, intrinsically useless money may be held if it allows agents to engage in intertemporal consumption smoothing and either (i) there are no other financial assets available for this purpose at all, or (ii) such assets exist but carry an

355

The Foundation of Modern Macroeconomics

inferior rate of return. The second model of money as a store of value is based on the notion that assets carrying a higher yield than money may also be more risky. In the simplest possible application of this idea, the yield on money is assumed to be certain and equal to zero (no price inflation) whilst the yield on a risky financial asset is stochastic. The risky asset carries a positive expected yield. The actual (realized) rate of return on such an asset is, however, uncertain and may well be negative. In such a setting the risk-averse household typically chooses a diversified portfolio, consisting of both money and the risky asset, which represents the optimal trade-off between risk and return.

In the second part of this chapter we take for granted that money exists and plays a useful role in the economic process and study the socially optimal quantity of money. If fiat money is useful to economic agents then how large should the money supply be? Friedman proposes a simple answer to this question: since fiat money is very cheap to produce, the money supply should be expanded up to the point where the marginal social benefit of money is (close to) zero. This is the famous full liquidity or satiation result. We first demonstrate the validity of the satiation result in a very simple two-period model of an endowment economy with money entering the utility function of the households. Next we extend the model by endogenizing the labour supply decision of households and demonstrate the various reasons why full liquidity may not be socially optimal.

Further Reading

Good textbooks on monetary economics are Niehans (1978), McCallum (1989a), and Walsh (1998). Diamond (1984), Kiyotaki and Wright (1993), and Trejos and Wright (1995) use the search-theoretic approach to model money. The demand for money by firms is studied by Miller and Orr (1966) and Fischer (1974). Romer (1986, 1987) embeds the Baumol-Tobin model in a general equilibrium model. Saving (1971) presents a model of money based on transactions costs. McCallum and Goodfriend (1987) give an overview of money demand theories. Fischer (1979) studies monetary neutrality in a monetary growth model.

On the public finance approach to inflation, see Chamley (1985), Turnovsky and Brock (1980), Mankiw (1987), Gahvari (1988), Chari et al. (1996), Correia and Teles (1996), Batina and Ihori (2000, ch.10), and Ljungqvist and Sargent (2000, ch. 17). On the unpleasant monetarist arithmetic argument, see Drazen and Helpman (1990), Sargent and Wallace (1993), and Liviatan (1984).

Appendix

In this appendix we derive equation (12.116) in the text. As a preparatory step we write (12.115) in short-hand format as follows:

I where zt is the forcing t

zt _=_ Ct + mt — —

a

We wish to solve (Al2.1 1,0 is given. By using

= ( 1

(1 + ro)bo = ( 1 -1 1 +

From the ultimate ex: k substitutions we get:

(1 + ro)bo - 1 +1

By using the definiti, can be written in a cu:

(1 + ro)bo = 4+1 11.-

Next we must simp,.:y t we can write this term a

Eq1+k otzt =E1+k t,

t=1 t=1

The first term on t...

the second term can I]

a store of value is based on may also be more risky. In on money is assumed to be on a risky financial asset yield. The actual (realized) and may well be negative. Doses a diversified portfolio, resents the optimal trade-off

that money exists and plays ocially optimal quantity of low large should the money question: since fiat money expanded up to the point zero. This is the famous full city of the satiation result in lomy with money entering `he model by endogenizing ate the various reasons why

Chapter 12: Money

where zt is the forcing term of the difference equation:

We wish to solve (Al2.1) forwards in time, taking account of the fact that in period bb is given. By using (Al2.1) we find the following expression after two substitutions:

From the ultimate expression it is easy to recognize the pattern and to conclude that after k substitutions we get:

The Foundation of Modern Macroeconomics

qo = qo 1 /(1 + rt_i) for t > 2. By using this result we obtain:

where we have used the fact that the nominal interest rate, Rt , satisfies 1 +Rt = (1 +rt)( 1 +74) in going from the first to the second line. By using (Al2.5) and (Al2.6) in (Al2.4) and rearranging we obtain equation (12.116) in the text.

New Key n

sae purpose of this char*

1.Can we provide m

2.What are the welfai What is the link ID, the marginal cost

3.Does monetary ne,. -

4.What do we mean interact?

13.1 Reconstructi

challenge posed by --,Qmic foundations for as characterized by mono st—h micro-foundations

pace disequilibrium

s.; older literature is &nese models resemble c

_al market coordinati

:.‘es imply the existen, rirestricted market par e question why this 1% Of course some reaso:

but a particularly sinl,

Mt-1

and (Al2.6) in (Al2.4) and

13

New Keynesian Economics

The purpose of this chapter is to discuss the following issues:

1.Can we provide microeconomic foundations behind the "Keynesian" multiplier?

2.What are the welfare-theoretic aspects of the monopolistic competition model? What is the link between the output multiplier of government consumption and the marginal cost of public funds (MCPF)?

3.Does monetary neutrality still hold when there exist costs of adjusting prices?

4.What do we mean by nominal and real rigidity and how do the two types of rigidity interact?

13.1 Reconstructing the "Keynesian" Multiplier

The challenge posed by a number of authors in the 1980s is to provide microeconomic foundations for Keynesian multipliers by assuming that the goods market is characterized by monopolistic competition. This is, of course, not the first time such micro-foundations are proposed, a prominent predecessor being the fixedprice disequilibrium approach of the early 1970s (see Chapter 5). The problem with that older literature is that prices are simply assumed to be fixed, which makes these models resemble Shakespeare's Hamlet without the Prince, in that the essential market coordination mechanism is left out. Specifically, fixed (disequilibrium) prices imply the existence of unexploited gains from trade between restricted and unrestricted market parties. There are f 100 bills lying on the footpath, and this begs the question why this would ever be an equilibrium situation.

Of course some reasons exist for price stickiness, and these will be reviewed here, but a particularly simple way out of the fixity of prices is to assume price-setting

The Foundation of Modern Macroeconomics

behaviour by monopolistically competitive agents. 1 This incidentally also solves Arrow's (1959) famous critical remarks about the absence of an auctioneer in the perfectly competitive framework.

13.1.1 A static model with monopolistic competition

In this subsection we construct a simple model with monopolistic competition in the goods market. There are three types of agents in the economy: households, firms, and the government. The representative household derives utility from consuming goods and leisure and has a Cobb-Douglas utility function:

U Ca(1 - L) 1-" , 0 < < 1, (13.1)

where U is utility, L is labour supply, and C is (composite) consumption. The household has an endowment of one unit of time and all time not spent working is consumed in the form of leisure, 1 - L. The composite consumption good consists of a bundle of closely related product "varieties" which are close but imperfect substitutes for each other (e.g. red, blue, green, and yellow ties). Following the crucial insights of Spence (1976) and Dixit and Stiglitz (1977), a convenient formulation is as follows:

j=1

where N is the number of different varieties that exist, Cj is a consumption good of variety j, and 9 and ri are parameters. This specification, though simple, incorporates two economically meaningful and separate aspects of product differentiation. First, the parameter 9 regulates the ease with which any two varieties (C i and Cj) can be substituted for each other. In formal terms, 9 represents the Allen-Uzawa crosspartial elasticity of substitution (see Chung, 1994, ch. 5). Intuitively, the higher is 9, the better substitutes the varieties are for each other. In the limiting case (as 9 ---> oo), the varieties are perfect substitutes, i.e. they are identical goods from the perspective of the representative household.

The second parameter appearing in (13.2), r1, regulates "preference for diversity" (PFD, or "taste for variety" as it is often called alternatively). Intuitively, diversity preference represents the utility gain that is obtained from spreading a certain amount of production over N varieties rather than concentrating it on a single variety (Benassy, 1996b, p. 42). In formal terms average PFD can be computed by comparing the value of composite consumption (C) obtained if N varieties and X /N units per variety are chosen with the value of C if X units of a single variety

1 See the recent surveys by Benassy (1993a), Silvestre (1993), Matsuyama (1995), and the collection of papers in Dixon and Rankin (1995).

tit •en (N = 1 ):

erage PFD

-ticity of t taste for add

*Ow competition me

for

I

.44.A =q - 1.

S now clear how a v positive any i,ot enjoy diet.::

lie household L

PiCi = WN L

*ml

P, is the price

- later Jectsves from the m

vernF

-fis /or each avaika

(13.1), given ti (13.3A, a: — alit, profit income, a

the cA.

■trisasuaAption,

PC = a L i

!talon

lonopolistic competition in iconomy: households, firms, ives utility from consuming ion:

(13.1)

consumption. The housetime not spent working is consumption good consists are close but imperfect subties). Following the crucial i convenient formulation is

(13.2)

is a consumption good of iugh simple, incorporates oduct differentiation. First, D varieties (Ci and C'i) can

is the Allen-Uzawa cross-

Intuitively, the higher is 0, le limiting case (as 0 -+ oc), goods from the perspective

"preference for diversity" tively). Intuitively, diver- ' t-rom spreading a certain lcentrating it on a single PFD can be computed by

tined if N varieties and 1 units of a single variety

The elasticity of this function with respect to the number of varieties represents the marginal taste for additional variety2 which plays an important role in the monopolistic competition model. By using (13.3) we obtain the expression for the marginal

preference for diversity (MPFD):

It is now clear how and to what extent 77 regulates MPFD: if 77 exceeds unity MPFD is strictly positive and the representative agent exhibits a love of variety. The agent does not enjoy diversity if Ti = 1 and MPFD = 0 in that case.

j=i

where Pi is the price of variety j, WN is the nominal wage rate (labour is used as the numeraire later on in this section), n is the total profit income that the household

receives from the monopolistically competitive firms, and T is a lump-sum tax paid to the government. The household chooses its labour supply and consumption levels for each available product variety (L and Cj , j = 1, , N) in order to maximize utility (13.1), given the definition of composite consumption in (13.2), the budget constraint (13.5), and taking as given all prices (Pi, j = 1, . ,N ), the nominal wage

rate, profit income, and the lump-sum tax.

By using the convenient trick of two-stage budgeting, the solutions for composite

where P is the so-called true price index of the composite consumption good C. Intuitively, P represents the price of one unit of C given that the quantities of all varieties are chosen in an optimal (utility-maximizing) fashion by the household.

It is defined as follows:

The Foundation of Modern Macroeconomics

Intermezzo

Two-stage budgeting. As indeed its name strongly suggests, the technique of two-stage budgeting (or more generally, multi-stage budgeting) solves a relatively complex maximization problem by breaking it up into two (or more) much less complex sub-problems (or "stages"). An exhaustive treatment of two-stage budgeting is far beyond the scope of this book. Interested readers are referred to Deaton and Muellbauer (1980, pp. 123-137) which contains a more advanced discussion plus references to key publications in the area.

We illustrate the technique of two-stage budgeting with the aid of the maximization problem discussed in the text. Since C and 1 - L appear in the utility function (13.1) and only ci (j-_-_-_- 1, ... N) appear in the definition of C in (13.2) it is natural to subdivide the problem into two stages. In stage 1 the choice is made (at the "top level" of the problem) between composite consumption and leisure, and in stage 2 (at the "bottom" level) the different varieties are chosen optimally, conditional upon the level of C chosen in the first stage.

We postulate the existence of a price index for composite consumption and denote it by P. By definition total spending on differentiated goods is then equal to Ei p; c; = PC so that (13.5) can be re-written as:

PC + WN (1 -- L) = WN +11 - IF, (a)

which says that spending on consumption goods plus leisure (the left-hand side) must equal full income (IF on the right-hand side). The top-level maximization problem is now to maximize (13.1) subject to (a) by choice of C and 1 - L. The first-order conditions for this problem are the budget constraint

(a) and:

The marginal rate of substitution between leisure and composite consumption must be equated to the real wage rate which is computed by deflating the nominal wage rate with the price index of composite consumption (and not just the price of an individual product variety!). By substituting the right-hand expression of (b) into the budget identity (a), we obtain the optimal choices of C and 1 L in terms of full income:

Finally, by substituting these expressions into the (direct) utility function (13.1) we obtain the indirect utility function expressing utility in terms of full income

:id a cost-of-livi _

Tv,

here Pv is the true of utility (a "ut..

Stage 2. In the sec cr to "construct" 4 nion. The formal

Max .NP/[N-1 1:

ic-,} .

shich the first- ,

ac lac; _

C/aCk — Pk

c marginal rate u:

.ated to the rela'

.1A-order cond e following expr.

N-

Ci =

[Ek=1N N- -

)stituting (h) .

uex P is obtained:

/si

P,C1 =

r=1 1

t.---

P --- N [ '

Sy using this price mption g. x1

=

is the expit.4).

411111.11.111101111111•1111110111111111111.0