Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

gists, the technique of budgeting) solves a rela- _ it up into two (or more) n exhaustive treatment of ;)ook. Interested readers are which contains a more

ms in the area.

w'h the aid of the maxi- _ 1 — L appear in the utility he definition of C in (13.2) s. In stage 1 the choice is nposite consumption and

- ent varieties are chosen the first stage.

for composite consump- m differentiated goods is ritten as:

(a)

plus leisure (the left-hand de). The top-level maxi-

(a)by choice of C and

-e the budget constraint

(b)

composite consumption )mputed by deflating the to consumption (and not `- stituting the right-hand n the optimal choices of

(c)

► utility function (13.1) tv in terms of full income

Chapter 13: New Keynesian Economics

and a cost-of-living index:

where PV is the true price index for utility, i.e. it is the cost of purchasing one unit of utility (a "util"):

Stage 2. In the second stage the agent chooses varieties, ci = 1,2, ..., N), in order to "construct" composite consumption in an optimal, cost-minimizing,

fashion. The formal problem is:

- 0/(0-1)

for which the first-order conditions are the constraint in (f) and:

The marginal rate of substitution between any two product varieties must be equated to the relative price of these two varieties. By repeatedly substituting the first-order condition (g) into the definition of C (given in (13.2)), we obtain the following expression for

[ENk _ i N__i p1-0] - 0/(1-0

By substituting (h) into the constraint given in (f) the expression for the price index P is obtained:

[Ej=i Pi

(i)

By using this price index we can re-express the demand for variety j of the consumption good (given in (h)) in a more compact form as:

which is the expression used in the text (namely equation (13.7)).

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The Foundation of Modern Macroeconomics

It must be pointed out that we could have solved the choice problem facing the consumer in one single (and rather large) maximization problem, instead of by means of two-stage budgeting, and we would, of course, have obtained the same solutions. The advantages of two-stage budgeting are twofold: (i) it makes the computations more straightforward and mistakes easier to avoid, and (ii) it automatically yields useful definitions for true price indexes as by-products.

Finally, although we did not explicitly use the terminology, the observant reader will have noted that we have already used the method of two-stage budgeting before in Chapter 10. There we discussed the Armington approach to modelling international trade flows and assumed that a domestic composite good consists of a domestically produced good and a good produced abroad.

The firm sector is characterized by monopolistic competition, i.e. there are very many small firms each producing a variety of the differentiated good and each enjoying market power in its own output market. The individual firm j uses labour to produce variety j and faces the following production function:

which incorporates the assumption that labour is perfectly mobile across firms, so that all firms are forced to pay a common wage (147 N does not feature an index j). The firm chooses output in order to maximize its profits (13.11) subject to its priceelastic demand curve. We assume that it acts as a Cournot competitor in that firm j takes other firms' output levels as given, i.e. there is no strategic interaction between producers of different product varieties.

In formal terms, the choice problem takes the following form:

where the notation Pi(Yi) is used to indicate that the choice of output affects the price which firm j will fetch (downward-sloping demand implies aPdaY; < 0).

first-order conditioi texts:

= +

dYi al-,

pi = WNk,

re Ili is the markup we (absolute value ui

E.

Pi = E — 1 Ei -

1

higher is the

the solution to the peril _ sensible if ii is p The government du,..; ;, given below), it Levi( employs civil serval.. analogously to C in (13

I

G N' [N-1 E

where Gi is the govern m efficient in the sense 1 minimizing, fashion, to

oven. This implies

where the similarity feature the same funL in (13.9).

Total demand facii.t and (13.16) shows tha the markup is const,.. composition of deman firms face the same pro

he choice problem facing ization problem, instead of course, have obtained the are twofold: (i) it makes easier to avoid, and (ii) it

ndexes as by-products. minology, the observant the method of two-stage -1 *.he Armington approach at a domestic composite

(2- ood produced abroad.

•

mpetition, i.e. there are very lifferentiated good and each individual firm j uses labour , n function:

(13.10)

zr used by the firm, F is fixed marginal labour requirement. t expend a minimum amount ny output at all (see Mankiw, c.- ale at firm level as average

!nue minus total costs:

L- tly mobile across firms, so does not feature an index j). Its (13.11) subject to its price- - - ot competitor in that firm j strategic interaction between

.ng form:

(13.12)

choice of output affects the mand implies aPilaYi < 0).

Chapter 13: New Keynesian Economics

The first-order condition yields the pricing rule familiar from first-year microeconomic texts:

(13.13)

here ui is the markup of price over marginal cost (i.e. variable labour cost) and ci

• the (absolute value of the) price elasticity of demand facing firm j:

The higher is the elasticity of demand, the smaller is the markup and the closer is the solution to the perfectly competitive one. Clearly, the pricing rule in (13.13) is only sensible if p 1 is positive, i.e. demand must be elastic and ci must exceed unity.

The government does three things in this model: it consumes a composite good (G, given below), it levies lump-sum taxes on the representative household (T), and it employs civil servants (LG ). To keep things simple we assume that G is defined analogously to C in (13.2):

N e/0-1)

G -...=-MI[N-1 E G,(0_1),0

where Gi is the government's demand for variety j. It is assumed that the government is efficient in the sense that it chooses varieties Gi (j = 1, ...,N) in an optimal, costminimizing, fashion, taking a certain level of composite public consumption (G) as given. This implies that the government's demand for variety j is:

G =N-(9+0+0 (Pi)_c

P , 1

where the similarity to (13.7) should be apparent to all and sundry. Since C and G feature the same functional form, the price index for the public good is given by P

in (13.9).

Total demand facing each firm j equals Yi which in view of (13.7) and (13.16) shows that the demand elasticity facing firm j equals ci = 0 so that the markup is constant and equal to ,a = ,u = 1). In this simplest case, the composition of demand does not matter. The model is completely symmetric: all firms face the same production costs and use the same pricing rule and thus set the

The Foundation of Modern Macroeconomics

Table 13.1. A simple macro model with monopolistic competition

so that the aggregate goods market equilibrium condition can be written as in (T1.1) in Table 13.1.

For convenience, we summarize the model in aggregate terms in Table 13.1. Equation (T1.1) is the aggregate goods market clearing condition and (T1.2) is household demand for the composite consumption good (see (13.6)). Equation (T1.3) relates aggregate profit income (11) to aggregate spending (PY) and firms' outlays on overhead labour This expression is obtained by using the symmetric pricing rule, P1 = i) = ,u,WN k, in the definition of firm profit in (13.11) and aggregating over all active firms. The government budget restriction (T1.4) says that government spending on goods (PG) plus wage payments to civil servants (WN LG) must equal the lump-sum tax By using the symmetric pricing rule in the definition of the price index (13.9) expression (T1.5) is obtained. Labour supply is given by (T1.6). Finally, (T1.7) contains some welfare indicators to be used and explained below in section 1.4.

Equilibrium in the labour market implies that the supply of labour (L) must equal the number of civil servants employed by the government (LG) plus the number of

Walras' Law ensures tha ,.'ther imply that (13.1 There is no money in t - is convenient to use I , ,) measured in wage ur first case, the number of

version of the model is short-run multipliers (N. s i able and exit/entry of uilowing Startz (1989) t

13.1.2 The short-run

In the (very) short run, = No) and the model ii ois can be demonstrate the aggregate consumpt 4.id constants:

C = co + (a / 0)Y —

where co

1.5) that the real wa . tion looks rather KeynL and Additional spends a fraction of leisure). The consumpt

el of government spe is obtained. The initial production and equilib Now consider what h G0 to G1, and finances

tax. Such a balanced-' negative effect on the a holds have to pay high in Figure 13.1. Seco: .. for-one because the g( propensity to consur effect dominates the pi

a) dG), as is illy

3 The number of prod labour requirement (k).

'^rn petition

(T1.1)

(T1.2)

(T1 .3)

(T1 .4)

(T1.5)

(T1 .6)

(T1.7)

(13.17)

n can be written as in (T1.1)

e terms in Table 13.1. Equaion and (T1.2) is household ). Equation (T1.3) relates and firms' outlays on over4ng the symmetric pricing 3.11) and aggregating over

1.4) says that government rvants (WNLG ) must equal

in the definition of the supply is given by (T1.6). xl and explained below in

of labour (L) must equal it (LG ) plus the number of

I

(13.18)

Chapter 13: New Keynesian Economics

Walras' Law ensures that the labour market is in equilibrium, i.e. (T1.1)-(T1.6)

;ether imply that (13.18) holds.

There is no money in the model so nominal prices and wages are indeterminate. It is convenient to use leisure as the numeraire, i.e. WN is fixed and everything

is measured in wage units. The model can be analysed for two polar cases. In the first case, the number of firms is constant and fluctuations in profits emerge. This version of the model is deemed to be relevant for the short run and gives rise to short-run multipliers (Mankiw, 1988). In the second case, the number of firms is variable and exit/entry of firms ensures that profits return to zero following a shock. Following Startz (1989) this can be seen as the long-run version of the model.

13.1.2 The short-run balanced-budget multiplier

In the (very) short run, Mankiw (1988) argued, the number of firms is fixed (say N = No) and the model in Table 13.1 exhibits a positive balanced-budget multiplier. This can be demonstrated as follows. By substituting (T1.3) and (T1.4) into (T1.2), the aggregate consumption function can be written in terms of aggregate output and constants:

where co a [1 - NoF - LG] W and W WN/P is the real wage. It follows from (T1.5) that the real wage rate is constant in the short run. 3 The consumption func-

tion looks rather Keynesian and has a slope between zero and unity since 0 < a < 1 and 6 > 1. Additional output boosts real profit income to the household which spends a fraction of the extra income on consumption goods (and the rest on leisure). The consumption function has been drawn in Figure 13.1 for an initial level of government spending, G o . By vertically adding Go to C, aggregate demand is obtained. The initial equilibrium is at point Eo where aggregate demand equals production and equilibrium consumption and output are, respectively, Co and Yo.

Now consider what happens if the government boosts its consumption, say from Go to G1, and finances this additional spending by an increase in the lump-sum tax. Such a balanced-budget policy has two effects in the short run. First, it exerts a negative effect on the aggregate consumption function (see (13.19)) because households have to pay higher taxes, i.e. the consumption function shifts down by a dG in Figure 13.1. Second, the spending shock also boosts aggregate demand one- for-one because the government purchases additional goods. Since the marginal propensity to consume out of full income, a, is less than unity, this direct spending effect dominates the private consumption decline and aggregate demand increases

as is illustrated in Figure 13.1. The equilibrium shifts from E 0 to E1 ,

3 The number of product varieties (N) is fixed as are (by assumption) the markup (A) and the marginal labour requirement (k).

367

The Foundation of Modern Macroeconomics

Figure 13.1. Government spending multipliers

output increases from (Y0 to Y1), but consumption falls (Co to C1). Formally, the short-run income and profit multipliers are:

An increase in government spending increases aggregate demand on impact by (1 - a) dG and causes additional real profits to the tune of 0 -1 (1 - a) dG. Although aggregate household consumption declines at impact by adG, the rise in profit income mitigates this reduction somewhat. This furnishes a second round in the multiplier process, which ultimately converges to the expression given in (13.20). Under perfect competition, there is no profit effect and hence the ultimate effect of a change in government consumption coincides with the impact effect, 1 - a.

Although (13.20) looks like a Keynesian multiplier (and certainly was sold as one by the initial authors), 4 some features are distinctly un-Keynesian. For one, household consumption falls as a result of the increase in government consumption:

which is at odds with the usual Haavelmo balanced-budget multiplier (see Chapter 1). Furthermore, it turns out that the same reason that makes households cut back consumption (i.e. the higher tax burden, which lowers full income) also

4 With the notable exception of Dixon (1987) who argued that the multiplier was more Walrasian than Keynesian.

-s them cut back on

3)) and increase labo.

dLyR

0 < W (— (- dG =

.ute, the Keynesian r more labour bet:, with the new I

The short-run

_,Aw (1988) uses an

, ultic model (like the vtion is not finar

isicad is paid for by I.. sentative householc

consumption

C= a[1-NF]W -t-

-ure the real tax bi (1

(dy) sR (odn

dG LG PdG

output multiplier ( tithe representative 1..

:mption rises and

,wets of labour that d: lie public sector. The

_:te sector) doming tan expand.

X 3.1.4 The "long-i,

Siam (1989) sugge -,, A-o subsections are ittiALs lying around

s And with disconnec...

r 6.

Y=C+G,

Y=C+Go

-C = co+ (x/0) Y--aGo C = co+ (a/O) Y—orGi

Chapter 13: New Keynesian Economics

makes them cut back on leisure consumption (since leisure is a normal good, see (13.8)) and increase labour supply. In aggregate terms we have:

a dG, the rise in profit tes a second round in the pression given in (13.20). !nce the ultimate effect of impact effect, 1 - a.

J certainly was sold as un-Keynesian. For one, ernment consumption:

(13.21)

:idget multiplier (see that makes households lowers full income) also

(13.24)

(13.25)

The Foundation of Modern Macroeconomics

in Chapter 5, not all trading opportunities are exhausted in the short-run equilibrium that emerges following a public spending shock. Indeed, as both (13.20) and (13.24) demonstrate, additional profits emerge as a result of the increase in government spending. In the absence of barriers to entry, one would expect new firms to commence operations as long as super-normal profits persist. Following Heijdra and van der Ploeg (1996, p. 1291) we capture this idea with the following simple specification:

where 1S/- dN I dt is the rate of change in the number of firms over time and yN is finite so that exit/entry occurs gradually over time.

To keep the discussion as simple as possible, it is assumed in the remainder of this section that the government employs no civil servants (i.e. LG = 0). The goods market equilibrium (GME) condition is obtained by substituting (T1.2)-(T1.4) into (T1.1):

where we have solved for output and used the pricing rule (given in (T1.5) above) to relate the real wage to the number of firms in the second line of (13.27). For future reference we rewrite this pricing rule as follows:

Finally, the zero-profit condition, ZP, which is obtained by setting 11 = 0 in (T1.3), collapses to Y = OWNF which can be re-expressed with the aid of the pricing rule (13.28) in terms of the number of firms:

The intuition behind the short-run, transitional, and long-run effects of a taxfinanced increase in public consumption can now be explained with the aid of Figure 13.2. In the top panel ZP represents combinations of output and the number of firms for which profits are zero. In view of (13.29) the ZP line goes through the origin and is upward sloping:

Furthermore, (13.26) shows that profits are positive (negative) for points to the left (right) of the ZP line so that the entry dynamics is as indicated by horizontal arrows. Still in the top panel, GME0 represents the initial goods market equilibrium locus

Yo

W

wo

Figure 13.;

as defined in equation differentiate (13.2 -

dY

[1 - a 10] y- = —

=[1

I

=

where we have used t second line and

uatial output shares c oc C IY and coG E- 1

ed in the short-run equilib-

,Indeed, as both (13.20) and suit of the increase in gov-

r, one would expect new firms i ts persist. Following Heijdra a with the following simple

(13.26)

0

r of firms over time and yN is

assumed in the remainder of servants (i.e. LG = 0). The `)y substituting (T1.2)-(T1.4)

(13.27)

rule (given in (T1.5) above) to )nd line of (13.27). For future

(13.28)

I

ed by setting 11 = 0 in (T1.3), the aid of the pricing rule

(13.29)

I

long-run effects of a taxexplained with the aid of s of output and the number

the ZP line goes through the

p

(13.30)

dative) for points to the left r4 icated by horizontal arrows. xis market equilibrium locus

Chapter 13: New Keynesian Economics

Figure 13.2. Multipliers and firm entry

as defined in equation (13.27). In order to study the properties of the GME-locus we differentiate (13.27) around an initial zero-profit equilibrium:

where we have used the zero-profit condition (in levels) in going from the first to the second line and the pricing rule in going from the second to the third line. The initial output shares of private and public consumption are given, respectively, by (oc C/Y and wG 1 - 0_)c G/ Y.

371

The Foundation of Modern Macroeconomics

Equation (13.31) shows that, for a given number of firms, an increase in government consumption leads to an upward shift of the GME-locus. Note furthermore that the output-related profit effect appears on the left-hand side of (13.31). There are two distinct mechanisms by which a change in the number of firms affects the GME-locus, namely the diversity effect and the fixed-cost effect. The first term on the right-hand side of (13.31) represents the positive effect on aggregate demand of an increase in the real wage which occurs as a result of an increase in the number of firms provided the agents exhibit love of variety 1). This is the diversity effect. The second term is potentially offsetting and represents the negative effect on aggregate demand of fixed costs: as the number of firms increases, total overhead costs rise and profits fall. This is the fixed-cost effect.

The overall effect of N on Y along the GME-locus is thus theoretically ambiguous because the diversity and fixed-cost effects work in opposite directions. Our usual ploy to be used in the face of ambiguity, the Samuelsonian correspondence principle (see Chapter 2), does not help to resolve this issue because the model is stable for all parameter values. Indeed, in view of (13.26) the stability condition (aN/aN < 0) amounts to requiring that the ZP line is steeper than the GME line.

where the latter inequality holds as both terms on the left-hand side are strictly between zero and unity. 6

Two often-used approaches lead to a resolution of the ambiguity regarding the slope of the GME-locus. In the first approach the ambiguity is resolved by ignoring the conceptual distinction between the price elasticity of demand (9) and the preference for diversity (r7) and imposing a single utility parameter to regulate these two effects. Technically, the standard Dixit and Stiglitz (1977, p. 298) formulation is used for composite consumption by setting ri = 9/(9 — 1) in (13.2). Since 0 > 1 is required to guarantee a meaningful monopolistically competitive equilibrium (i.e. to ensure that x > 1), diversity preference is operative

to render the slope of the GME-locus positive:

aY ria (1— a)wc

aN GME (9 -1)[1— a/9J] >0.

This is the case drawn in Figure 13.2. An increase in government consumption shifts the GME locus from GME0 to GME1 . At impact the number of firms is predetermined

6 For a more general utility function than (13.1), the stability condition does furnish additional information that is useful for comparative static purposes. See Heijdra and van der Ploeg (1996, p. 1291), Heijdra and Ligthart (1997, p. 817), and Heijdra et al. (1998, p. 86) for different examples.

= No) and output 7 san -,tiplier given ►., ad and entry of new fin

ri E1 to E2 and both

,Autiorium- values. Furth( rate also increases di

—oe Keynesian in

sr e the real wage and areas in the first a;

.-ft multiplier, this conLI

-- 411 implicitly resole,

__...hating preference t1/4.- uNIE locus is downward sl le economy (such

.cur):

iwthermore, the pricing :- am like Figure 13 dad the wage curve is hL,i

as in (13.20) but during ti in aggregate o.

mast round of the multipim

prompts Startz (1989.

..dlier is eliminated In the most general ‘c

iplier can be solve b:

dY \ LR dL kciG =

re the inequality f( pro and unity if ri > 1

,_plain the multiplicati, diversity effect whi,:1 1

Although Startz (1989, - ce by appealing to col:

all as the discussion ab rater than unity there a