Keynesian Multipliers

with Home Production

By

Masatoshi Yoshida

Professor, Graduate School of Systems and Information EngineeringUniversity of Tsukuba

Takeshi Kenmochi

Graduate School of Systems and Information EngineeringUniversity of Tsukuba

The address for editorial correspondence:Masatoshi YoshidaGraduate School of Systems and Information Engineering,University of Tsukuba1-1-1, Tennoudai, Tsukuba, Ibaraki 305-8573, JapanTel: 029-853-5556; Fax: 029-855-3849E-mail: yoshida@sk.tsukuba.ac.jp

Abstract

In a general equilibrium model of monopolistic competition with home production of

a service, this paper explores Keynesian multipliers of expansive government spending on

goods and services �nanced by lump-sum taxation. Without leisure in the utility function,

the short-run national income multiplier for spending on public goods is positive, but that

on public services is negative. These signs are reversed in the long run. With leisure,

the short-run multiplier of public services is positive when price markups in the good and

service sectors are close, and the long-run one of public goods is so when home output is

less than a certain level.

Keywords

Keynesian multipliers, Home production, Monopolistic competition

JEL classi�cation

D13, E62, L13

1

1 Introduction

In many European countries and Japan, since the 1970s public services such as health,

education, old age care, and day care for children have occupied more than half of gov-

ernment consumption, and have taken the place of public goods such as defense, public

order and safety, and justice.1 Hence, it is now an important policy issue to investigate

what impacts a rise in government spending on public services has on the economy. While

public goods are nonrival, public services are rival, so that services should be provided

not only publicly by the government but also privately by �rms and households. This

paper analyzes multiplier e¤ects of public services and public goods with such di¤erent

properties on real national income, employment and so on.

Since the 1980s, Keynesian multipliers of government spending have been studied in

the framework of imperfect competition. Dixon (1987), Mankiw (1988), Startz (1989) and

others have shown that in the short run where the entry and exit of �rms are restricted,

a rise in government spending on public goods �nanced by lump-sum taxation gives rise

to the positive national income multiplier through an increase in pro�t income. On the

other hand, Heijdra and van der Ploeg (1996) and Heijdra and Lighthart (1997) pointed

out that even in the long run where excess pro�ts are driven to zero due to the free entry

and exit of �rms, a positive multiplier arises via a change in the consumer price index,

provided that households have a preference for diversity of goods.

The Keynesian multiplier theory is based on the labour-leisure model, which assumes

that households derive utility from consuming di¤erentiated goods and enjoying leisure.

However, as pointed out by Lindbeck and Weibull (1988) and Lindbeck and Nandaku-

mar (1990), the time allocation between home production and market work seems to be

a more important aspect of the labour supply e¤ects of �scal policies than variations in

1Fiorita and Kollintzas (2004) point out that in 12 European countries except for Italy and Sweden,the share of public services in government consumption has increased, while that of public goods hasdecreased (see Table 2 in their paper). For example, in Germany, the shares of public goods and publicservices were 35.8% and 58.6%, respectively, in the early 1970s, and 26.6% and 69.2% in the early 1990s.On the other hand, following Kaizuka (1990), these shares in Japan changed from 37.5% and 50.3% in1970 to 36.5% and 53.5% in 1987. It seems that this tendency was not as clear as in Europe. However,calculating the shares in the latter half of the 1990s from data in Annual Report on National Accounts,they became about 25% and 66%. Therefore, this trend is evidently observed in Japan recently, too,where the population is rapidly ageing.

2

pure leisure like sleeping, eating and �doing nothing�.2 Therefore, the present paper de-

velops a general equilibrium model of monopolistic competition based on the household�s

choice between market work and home production and investigates features of Keynesian

multipliers both in the short run and in the long run. This model includes di¤erentiated

goods and services. The latter is composed of market-produced services (market services)

and a home-produced service (home service) which is substitutable for them.3

This paper has the following four aims. The �rst is to analyze Keynesian multipliers

of government spending classi�ed into two categories, public services and public goods, on

output of market goods and services, home output, and real national income. The second

is to examine what impacts market production of services has on the multipliers. Since

there is an interaction between the two market sectors, government spending induces not

only intra-sectoral but also inter-sectoral multiplier e¤ects. The latter is due to a labour

movement across the sectors caused by changes in sectoral pro�ts in the short run and

by those in price indices in the long run. The third is to investigate in�uences of home

production on the multipliers.4 Since the household�s choice between market and home

services depends on government�s policies, we are interested in this analysis. The fourth

is to study e¤ects of a policy-induced change in pure leisure on the multipliers.5

At �rst, the following assumptions are set to make a model as simple as possible: (i)

each household�s utility is given by a Cobb-Douglas function of di¤erentiated goods and

services, (ii) a home service is substitutable for market services, (iii) government spending

is �nanced by lump-sum taxation, (iv) both public goods and public services are wasteful,

2The labour supply e¤ects of a policy-induced change in home work have been studied in the economicanalyses on the quantity and quality of children. See Cigno (1986) and Yoshida (1998).

3Following Lindbeck and Nandakumar (1990), Sandmo (1990), Kleven et al. (2000), and Yoshida andYuki (2004), we assume that households produce not goods but services.

4The real business cycle (RBC) theory is an example of the recent work focusing on home production.Benhabib et al. (1991) and Greenwood and Hercowitz (1991) showed that RBC models can successfullyexplain the real business �uctuations by considering home production. In addition, Mcgrattan et al.(1997) investigated e¤ects of tax policies. However, they did not analyze the multiplier e¤ects, since theytreated government spending as a stochastic variable.

5The extended model is also a generalization of the model with monopolistic competition developed byHeijdra and van der Ploeg (1996), where each household allocates his time endowment between marketwork and pure leisure. Such a model, which was originated with Gronau (1977), has been utilized invarious �elds of applied economics. For example, Sandmo (1990), Kleven et al. (2000), Yoshida and Yuki(2004), and Yoshida and Kenmochi (2005) used this model to derive the optimal tax system under thecondition that the government cannot impose any taxes on consumption of home services.

3

and (v) the price mark-up in the good sector is larger than that in the service.6 Under these

assumptions, the two kinds of government spending have the following multiplier e¤ects.

First, in the short run, a rise in spending on public services increases aggregate output

of market services but decreases that of market goods and real national income. In the

long run, it increases not only the aggregate service output but also national income. The

aggregate good output increases if home output is larger than output of public services.

Otherwise, it is ambiguous whether it increases or not. Second, the multiplier e¤ects of

public goods are exactly opposite to those of public services.

Next, the model is extended to include pure leisure. In the case of public services,

although leisure does not a¤ect the signs of the short-run output multipliers of market

goods and services, it does not necessarily decrease real national income. If the price

mark-up in the good sector is su¢ ciently close to that in the service, then real national

income rises. The signs of the long-run multipliers in this case are not entirely changed

by introducing leisure. On the other hand, in the case of public goods, leisure has no

in�uence on the signs of both all multipliers in the short run and the market-service

output multiplier in the long run. However, it alters the signs of the market-good output

and national income multipliers. For example, real national income increases in the long

run if home output is less than a critical level.

The above results on the multiplier e¤ects of public goods are comparable with those

derived by Heijdra and van der Ploeg (1996). In their model, these e¤ects do not occur if

leisure is kept at a constant level. However, this paper shows that the multiplier e¤ects

arise even in such a case, as long as services are produced by �rms and households. In

addition, although they showed that the existence of leisure ensures the short-run and

long-run national income multipliers to be positive, these multipliers are not always pos-

itive in the presence of the time allocation between leisure and home work. For example,

real national income decreases in the long run if home product is su¢ ciently large. This

result suggests that the long-run positive income multiplier of government spending does

not generally hold in an economy with home production.

6This assumption implies that the service sector is more competitive and �rms in this sector face amore elastic demand schedule. This agrees with the notion that the demand for services seems to bemore sensitive to price changes.

4

This paper is organized as follows. Section 2 presents the model and describes a

symmetric monopolistically competitive equilibrium. Section 3 derives the short-run and

long-run multiplier e¤ects of public services and examine properties of these e¤ects. Sec-

tion 4 investigates the multiplier e¤ects of public goods and compare them with those of

public services. Section 5 extends the model to include pure leisure and examine how the

multipliers are modi�ed. Finally, section 6 contains concluding remarks.

2 The Model

This section develops a general equilibrium model of monopolistic competition which in-

cludes not only market production of goods but also market and home production of

services. The closed economy considered here is composed of homogeneous households,

two sectors of market production, and the government. Each household consumes dif-

ferentiated goods and services and derives his utility from not only quantities but also

varieties of them. In addition to purchasing market services, in the home he produces

a service which substitutes for them. Both market sectors consist of monopolistically

competitive �rms. Each �rm produces one variety of the goods or the services by using

labour. The government provides public goods and public services by imposing lump-sum

taxes on households. They are treated as wasteful, entering neither utility nor produc-

tion functions, in order to focus on multiplier e¤ects of government spending on national

income, market output and so on.

2.1 Households

For simplicity, we normalize the number of households to unity. A representative house-

hold derives his utility U from consuming a composite di¤erentiated good Cg and a com-

posite di¤erentiated service Ds. We assume the following Cobb-Douglas utility function:

U = U(Cg; Ds) = (Cg)�(Ds)�; 0 < � < 1; 0 < � < 1; �+ � = 1: (1)

The composite di¤erentiated good is a CES-aggregation of N g varieties of consumption

goods, Cg � (Cg1 ; Cg2 ; � � � ; C

gNg):

5

Cg = (N g)1+�g

"(N g)�1

NgXi=1

(Cgi )(�g�1)=�g

#�g=(�g�1); �g > 1; �g > 0; (2)

where �g is the elasticity of substitution between the di¤erent varieties of goods, and

�g > 0 implies a preference for diversity (PFD) of them.

The composite di¤erentiated service Ds consists of a composite market-produced ser-

vice Cs and a home-produced service Hs:

Ds = Cs +Hs: (3)

The former is a CES-aggregation of N s varieties of market services, Cs � (Cs1 ; � � � ; CsNs):

Cs = (N s)1+�s

"(N s)�1

NsXi=1

(Csi )(�s�1)=�s

#�s=(�s�1); �s > 1; �s > 0; (4)

where �s denotes the elasticity of substitution and �s stands for the degree of his PFD.

The latter is produced by using home labour e through the following production function:

Hs = H(e) = �e1��

1� �; � > 0; 0 < � < 1; (5)

where�� is the elasticity of marginal product of labour. As in Gronau (1977) and Sandmo

(1990), the home production function (5) is subject to decreasing returns to scale.

Assuming that the representative household has one unit of time endowment, his

labour supply to the markets, L, is equal to 1 � e. We also assume that labour is the

numeraire. Then, his budget constraint is given by

1� e+�g +�s � T = PgCg +PsCs; (6)

where �g and �s are aggregate pro�ts in the good and service sectors, respectively, T is

a lump-sum tax, and Pg � fP gi g and Ps � fP sj g are price vectors of the di¤erentiated

goods and services, respectively.

The household�s utility maximization problem is solved through the following two

stages. First, given Ck and Pk, minimize PkCk with respect to Ck for k = g; s subject

6

to (2) or (4). Then, it holds that

minCkPkCk = P kCk; (7a)

Cki = Ck(Nk)�k(�k�1)�1

�P kiP k

���k; i = 1; 2; � � � ; Nk; k = g; s; (7b)

where P g and P s are price indices for the composite good and service, respectively:

P k � (Nk)�(�k+1)

24(Nk)��kNkXi=1

(P ki )1��k

351=(1��k) ; k = g; s: (8)

From (3) and (7a), his budget constraint (6) can be rewritten as

I � 1 + �g +�s +�h � T = P gCg + P sDs; (9)

where I denotes full income and �h stands for home pro�t, that is, a shadow pro�t which

the household receives from the home production. Home pro�t is de�ned as follows:

�h � P sHs � e: (10)

Second, the household maximizes his utility (1) with respect to Cg, Ds and e subject

to (5), (9) and (10). As a result, we obtain the consumption functions of the composite

good and service and the home labour function:

Cg = �I

P g; Cs = �

I

P s�Hs; e = (�P s)

1� : (11)

Substituting the home labour function into (5) and (10) , we obtain the optimal levels of

service and pro�t in home:

Hs = �1�1

1� �(P s)

1��� ; �h = �

1�

�

1� �(P s)

1� :

2.2 The Government

The government provides a composite public good Gg and a composite public service Gs,

which are the following CES functions of all existing varieties of the di¤erentiated goods

7

and services:

Gk = (Nk)1+�k

24(Nk)�1

NkXi=1

(Gki )(�k�1)=�k

35�k=(�k�1) ; k = g; s:

Government spending PkGk is minimized with respect to Gk � (Gk1; � � � ; GkNk), given Pk

and Gk for k = g; s. Then, it holds that

minGk

PkGk = P kGk; (12a)

Gki = Gk(Nk)�k(�k�1)�1

�P kiP k

���k; i = 1; � � � ; Nk; k = g; s: (12b)

Since the government �nances its spending by the lump-sum tax, its budget constraint is

written as P gGg + P sGs = T:

2.3 Monopolistically Competitive Firms

Firm i in sector k produces its output Qki under increasing returns to scale with labour

as the sole production factor. The production technology is given by

Lki = akQki + bk; (13)

where Lki , ak and bk denote the units of labour employed by the �rm, the constant marginal

labour requirement, and the �xed cost in terms of units of labour, respectively.

Since labour is mobile across �rms and sectors, a common wage is paid by all �rms.

Firm i in sector k maximizes its pro�t, �ki � P ki Qki �Lki , subject to (13) and the demand

function which it faces, Cki + Gki , under the Cournot assumption that the other �rms in

sector k do not change their output levels. As a result, marginal revenue should equal

marginal cost, which gives us the following conditions:

P ki = �kak; �k � �k

�k � 1 > 1; i = 1; � � � ; Nk; k = g; s; (14)

8

where �k is price mark-up on variable labour cost of each �rm in sector k. Since private

and public demand for a good have the same price elasticities, �g, the aggregate price

elasticity of demand is not a¤ected by the composition of aggregate demand. Hence, the

price mark-up �g is constant and the same for each �rm in the good sector. The same

property also holds for the price mark-up �s of each �rm producing the service.

Following Kolm (1998), we assume that each �rm producing the service, which can

be easily replaced with a home service by the household, faces a more elastic demand

schedule than each �rm producing the good, i.e.,

�s > �g: (15)

Since the price mark-up is a decreasing function of the demand elasticity, this assumption

can be rewritten as �g > �s. Hence, it implies that the price mark-up in the good sector

is higher than that in the service.

2.4 Symmetric Equilibrium

Since the supply of each �rm i in sector k must equal the demand for its product by the

representative household and the government, it holds that

Qki = Cki +Gki ; i = 1; � � � ; Nk; k = g; s: (16)

Equilibrium in the labour market requires that the demand for labour by all �rms equals

the supply by households:

L =

NgXi=1

Lgi +

NsXj=1

Lsj : (17)

As is conventional in the macroeconomic literature on monopolistic competition, the

attention is restricted to the symmetric equilibrium where the following conditions are

satis�ed: P ki = �P k; Qki = Qk; Lki = Lk; �ki =��k; Cki =

�Ck; Gki =�Gk for k = g; s: In

this equilibrium, it follows from (8) and (14) that price indices are represented as

P k = (Nk)��k �P k = (Nk)��

k

�kak; k = g; s; (18)

and the market clearing condition (17) can be rewritten as

L = N gLg +N sLs: (19)

9

Denote real aggregate output of market goods and services by Y g and Y s, respectively,

that is, Y k �PNk

i=1 Pki Q

ki =P

k for k = g; s: From (18) they can be represented as

Y k = (Nk)1+�k

Qk; k = g; s: (20)

Using the market clearing conditions (16) and the private and public demand func-

tions, (7b) and (12b), we can derive the expressions as follows: NkQk = (Nk)��k(Ck +

Gk) for k = g; s: Eliminating Nk and Qk from these and (20), we obtain the market

equilibrium conditions in aggregate form:

Y k = Ck +Gk; k = g; s:

From (13), (18) and the de�nitions of aggregate market output, aggregate pro�t in sector

k, �k � Nk ��k, is given by �k = (�k)�1P kY k �Nkbk for k = g; s:

We will de�ne real national income in labour units as

Y � P gY g + P sY s:

For convenience, the complete model is summarized in Table 1, although the labour-

market equilibrium condition (19) is omitted from it by Walras�law.

Finally, note that the following identity among real national income, Y , market labour,

L, and aggregate pro�ts in the market sectors, (�g;�s), is satis�ed in the short run:7

Y = L+�g +�s:

Thus, real national income is an increasing function of market labour and aggregate pro�t

in each sector. Even though market labour is �xed, real national income changes as long as

a labour movement between the market sectors arises, so that aggregate pro�ts alter. On

the other hand, in the long run where aggregate pro�ts are driven to zero, real national

income coincides with market labour. Therefore, the only variable factor of income is

market labour because the labour movement does not a¤ect real national income.7This identity can be obtained by combining (19) and the di�nitions of real national income, Y ,

aggregate market output, Y k, aggregate pro�ts, �k, and individual �rm�s pro�ts, ��k. Since labour is thenumeraire, market labour, L, in the identity is equal to total wage income.

10

3 Multipliers of Public Services

This section analyzes the short-run and long-run multipliers of a rise in government spend-

ing on public services (public-service spending) �nanced by lump-sum taxation.

3.1 The Short-run Multipliers

We de�ne the short-run equilibrium as (Y; Y k; Ck; Hs; e; L;�k;�h; I; T; P k; Qk; Lk) for

k = g; s which satisfy (T:1)-(T:14) in Table 1, given the levels of Gg and Gs. Since the

number of �rms in each sector, Nk, is �xed, consumer price indices, P k, are determined

as constant levels by (T:13). It follows from (T:6), (T:7) and (T:8) that home labour, e,

home output, Hs, and home pro�t, �h, are �xed, respectively. Thus, (T:4) indicates that

market labour, L, is also �xed. Note that the price indices, home output and market

labour are not a¤ected by government spending in the short run.

The level of a lump-sum tax, T , is determined by (T:10). (T:1)-(T:3), (T:5) and

(T:9) give us market output, Y k, private consumption of the composite good and service,

Ck, full income, I, and aggregate pro�t, �k. (T:11) and (T:12) determine output and

employment of each �rm, Qk and Lk. Finally, (T:14) provides us real national income, Y .

Di¤erentiating (T:1)-(T:3), (T:5), (T:9)-(T:10) and assuming that dGg = 0, we have

d�k

dGs=P k

�kdY k

dGs; k = g; s; (21a)

dY s

dGs= (1� �) +

�

P sd�s

dGs+

�

P sd�g

dGs; (21b)

P g

P sdY g

dGs= ��+ �

P sd�g

dGs+

�

P sd�s

dGs: (21c)

(21a) implies that in each sector aggregate pro�t increases when aggregate market output

expands. (21b) represents e¤ects of public-service spending on aggregate output of market

services. The �rst term in the RHS is the �rst-round e¤ect in the multiplier process.

Although this is the sum of a positive e¤ect of a rise in government spending and a

negative e¤ect of an increase in the lump-sum tax, it is positive because the former

dominates the latter. The second is an e¤ect of pro�t income from the service sector on

11

private demand for market services. Since this e¤ect elevates aggregate output of market

services and then pro�t income from the service sector again, it induces a multiplier

process within the service sector (an intra-sectoral multiplier process). The third is an

e¤ect of pro�t income from the good sector. (21c) expresses an e¤ect of public-service

spending on output of market goods.

From (21a)-(21c), we obtain the market-output multipliers:

�dY s

dGs

�SR=

(1� �) + (�=�g)M g(��)1� (�=�s)� (�=�g)M g(�=�s)

=1� (�=�g)� �

1� (�=�g)� (�=�s) > 0; (22a)

P g

P s

�dY g

dGs

�SR=

��+ (�=�s)M s(1� �)

1� (�=�g)� (�=�s)M s(�=�g)=

�[(1=�s)� 1]1� (�=�g)� (�=�s) < 0; (22b)

where

M g � 1 +1Xi=1

� ��g

�i=

1

1� (�=�g) > 0; M s � 1 +1Xi=1

��

�s

�i=

1

1� (�=�s) > 0:

The second and third terms in the denominator of the middle expression in (22a) represent

intra-sectoral and inter-sectoral multiplier e¤ects, respectively. The latter is generated

by the following process. A rise in aggregate demand for market services increases pro�t

income from the service sector, so that private demand for goods rises. Since this addi-

tional demand promotes pro�t income from the good sector through the multiplier process

within this sector, it boosts the aggregate service demand again. Since the sum of these

terms is more than �1, the denominator is positive. On the other hand, the �rst term

in the numerator, 1 � �, is the �rst-round e¤ect. The second, (�=�g)M g(��), is a de-

crease in the aggregate service demand due to a fall in pro�t income from the good sector.

This is explained as follows. Public-service spending has the negative �rst-round e¤ect

on aggregate demand for goods, ��, due to lump-sum taxation. This e¤ect depresses

pro�t income from the good sector via the intra-sectoral multiplier process, so that pri-

vate service demand decreases. The numerator is positive, because the positive �rst term

dominates the negative second. Thus, the market-service output multiplier is positive.

We can similarly interpret the middle expression in (22b). While the denominator in

this expression is positive, the numerator is negative because the positive second term

12

representing an increase in aggregate good demand due to a rise in pro�t income from the

service sector, (�=�s)M s(1� �), is dominated by the negative �rst term, ��. Therefore,

the market-good output multiplier is negative.

From (21a), in each sector, the aggregate pro�t and market output multipliers have

the same signs. Di¤erentiating (T:11) and (T:12) and noting that (T:13), (22a) and (22b),

the employment multipliers are given by

1

P s

�d(N sLs)

dGs

�SR=1

�s

�dY s

dGs

�SR> 0;

1

P s

�d(N gLg)

dGs

�SR=1

�gP g

P s

�dY g

dGs

�SR< 0:

These indicate that labour force is moved from the good sector to the service.

Finally, from (T:14), (22a) and (22b), we have the national income multiplier:

1

P s

�dY

dGs

�SR=P g

P s

�dY g

dGs

�SR+

�dY s

dGs

�SR=

�[(1=�s)� (1=�g)]1� (�=�g)� (�=�s) < 0: (23)

This multiplier is certainly negative by the assumption (15). The reason is that under

a constant level of market labour, labour moves from the good sector, where the price

mark-up is relatively high, to the service.

The above results on the short-run multipliers of public-service spending are summa-

rized in Proposition 1.

Proposition 1

In a monopolistically competitive economy where there are market and home produc-

tion of services as well as market production of goods, if each household does not allocate

his time endowment for pure leisure, a rise in government spending on public �services�

�nanced by lump-sum taxation has the following multiplier e¤ects in the �short� run:

(i) Real national income decreases. (ii) Output, pro�t, and employment increase in the

service sector but decrease in the good.

13

3.2 The Long-run Multipliers

The long-run equilibrium is de�ned as (Y; Y k; Ck; Hs; e; L;�k;�h; I; T; P k; Qk; Lk; Nk) for

k = g; s which satisfy (T:1)-(T:15). From (T:15) aggregate pro�t in each sector, �k, is

zero. Combining (T:9), (T:11) and (T:15) yields Qk = (�k � 1)bk=ak, so that the scale of

production for individual �rms is �xed. Thus, (T:12) indicates that employment of each

�rm, Lk, is constant.

(T:1)-(T:3), (T:5)-(T:10), (T:13) and (T:15) determine aggregate market output, Y k,

private consumption of the composite good and service, Ck, full income, I, a lump-sum

tax, T , the number of �rms in each sector, Nk, home labour, e, home output, Hs, home

pro�t, �h, and price indices for goods and services, P k. Market labour, L, is given by

(T:4). Note that the price indices, home output and market labour are in�uenced by

government spending in the long run. Finally, (T:14) gives us real national income, Y .

Di¤erentiating (T:9), (T:11), (T:13) and (T:15) and rearranging, we obtain

dNk

Nk=

1

1 + �kdY k

Y k;

dP k

P k= ��k dN

k

Nk; k = g; s: (24)

Thus, the price index for market services falls because a rise in aggregate service output

increases diversity of the services through the entry of new �rms. By the same reason,

the price index for goods decreases. Di¤erentiating (T:6)-(T:8) and rearranging, we have

de

e=1

�

dP s

P s;

dHs

Hs=1� �

�

dP s

P s;

d�h

�h=1

�

dP s

P s: (25)

(25) implies that home labour, home output and home pro�t are increasing functions of

the service price, P s. From (24) and (25), we �nd that a rise in aggregate service output

reduces home output due to a fall in the service price.

Di¤erentiating (T:10) and assuming that dGg = 0, we get

dT

T= �

dP g

P g+ (1� �)

dP s

P s+ (1� �)

dGs

Gs; (26)

14

where � � P gGg=T is the share of public goods in total spending. The �rst term in

the RHS of (26) represents an e¤ect of a change in the good price on total government

spending. The second represents the similar e¤ect of a change in the service price.

Di¤erentiating (T:1)-(T:3), (T:5) and (T:15) under the assumption that dGg = 0, and

using (24)-(26), we obtain

dY s

Y s= (1� �)(1� !s)

dGs

Gs+

�

1� �AsgdP g

P g+Bs

s

dP s

P s; (27a)

dY g

Y g= �1� �

��(1� !s)

dGs

Gs+1� �

�AgsdP s

P s+Bg

g

dP g

P g; (27b)

where !s � Cs=Y s is the share of private consumption in aggregate service output,

� � P gY g=Y is the share of total value of all existing di¤erentiated goods in real na-

tional income, !g � Cg=Y g is the share of private consumption in aggregate good output,

!h � Hs=Y s is the ratio of home output to aggregate service one, and

Bss � �(!s + !h)� �[(1� !s)� !h]�

�1� �

�

�!h

= �!s � �(1� !s)� (��1 � �)!h < 0;

Asg � ��(1� !g) < 0; Bgg � �!g � �(1� !g) < 0; Ags � ��[(1� !s)� !h]:

The three terms in the RHS of (27a) can be interpreted as follows. The �rst term

represents the �rst-round e¤ect of public-service spending on aggregate service demand.

This e¤ect is positive as in the short run. The second represents a negative indirect e¤ect

of a rise in the good price P g on private service demand Cs through an increase in tax

payment.8 Finally, the third represents a change in the service demand caused by a fall

in the service price P s. This change is decomposed into the following three e¤ects:

(a) A positive direct e¤ect, �(!s+!h)(dP s=P s), under the condition that home output

Hs and full income I are both constant,

8Note that the good price a¤ects private service demand via a change in disposable income, since theconsumption function of the composite service given by (11) has no cross-substitution e¤ects.

15

(b) An indirect e¤ect through a change in full income, ��[(1�!s)�!h](dP s=P s), under

the condition that home output is constant,

(c) A positive indirect e¤ect through a reduction in home output,�[(1��)=�]!h(dP s=P s),

under the condition that total service demand Ds is constant.

The sign of the second e¤ect (b) is ambiguous, since it consists of the positive component

through a decrease in tax payment, ��(1 � !s)(dP s=P s), and the negative through a

decrease in home pro�t, �!h(dP s=P s). It is negative when home output is larger than

output of public services, i.e., !h > 1� !s. However, as the two positive e¤ects, (a) and

(c), dominate the e¤ect (b), the third term is positive.

The RHS of (27b) can be similarly interpreted. The �rst term is the negative �rst-

round e¤ect of a rise in the lump-sum tax on private demand for goods. The second is

an indirect e¤ect of a decrease in the service price on the good demand through a change

in full income. The third is a negative e¤ect of an increase in the good price on the good

demand. This is composed of a direct price e¤ect, ��(1� !g)(dP g=P g), and an indirect

price e¤ect due to an increase in tax payment, �!g(dP g=P g), both of which are negative.

Rearranging (24), (27a) and (27b) gives us the market output multipliers in the long

run:

�dY s

dGs

�LR=(1� �) + Asg�

gRg(��)1�Bs

s�s � Asg�

gRgAgs�s =

1� �

(1 + �g)S> 0; (28a)

P g

P s

�dY g

dGs

�LR=

��+ Ags�sRs(1� �)

1�Bgg�

g � Ags�sRsAsg�

g =(�= )(!h � )

(1 + �s)S: (28b)

where

�k � � �k

1 + �k< 0; Rk � 1 +

1Xi=1

�Bkk�

k�i=

1

1�Bkk�

k; k = g; s;

S � (1�Bss�s)�1�Bg

g�g�� Ags�

sAsg�g; � �

�s(1� �):

The second and third terms in the denominator of the middle expression in (28a) repre-

sent intra- and inter-sectoral multiplier e¤ects, respectively. The former occurs for the

16

following reason: a fall in the service price due to a rise in aggregate service output pro-

motes the private service demand, so that this output increases again. Now, assuming

that this process is stable, it holds that Rs > 0 since the stability condition, Bss�s < 1,

is satis�ed.9 The latter arises in the following process. A change in the private good

demand owing to a fall in the service price a¤ects the good price. Since this price vari-

ation in the good sector has an in�uence on the private service demand, the aggregate

service output changes again. Assuming that the adjustment process consisting of the

intra- and inter-sectoral multiplier e¤ects is stable, the denominator is positive so that

S > 0. On the other hand, the �rst term in the numerator, 1� �, represents the positive

�rst-round e¤ect of public-service spending on the aggregate service demand. The second

term, Asg�gRg(��), represents a decrease in the private service demand due to a rise in

the good price, which is caused by the negative �rst-round e¤ect on the aggregate good

demand, ��. As the negative second term is dominated by the positive �rst, the numer-

ator is positive. Hence, the market-service output multiplier, (28a), is positive. Note that

the sign of this multiplier is not a¤ected by the home service production.

Although (28b) can be similarly interpreted, the sign of the market-good output mul-

tiplier is ambiguous in general. If !h � 1 � !s, this multiplier is negative, because the

second term in the numerator of the middle expression, which represents a change in the

good demand due to a fall in the service price, is non-positive, i.e., Ags�sRs(1 � �) � 0.

Otherwise, its sign is determined by the following condition:

!h

0@ >=<

1A ) P g

P s

�dY g

dGs

�LR0@ >=<

1A 0:Thus, in the case of !h > , the market-good output multiplier is positive, as the positive

second term is large enough to dominate the negative �rst, ��.

From (24), (25) and (28a), we �nd that public-service spending raises aggregate em-

ployment in the service sector but lowers home output and home labour. Since the home

labour multiplier is negative, the market labour multiplier is positive.

9On the other hand, it is easy to show that Bgg�g < 1. Hence, the multiplier process within the good

sector is stable, so that it holds that Rg > 0.

17

Finally, di¤erentiating (T:14) and using (24), (28a) and (28b), we obtain the positive

national income multiplier:

1

P s

�dY

dGs

�LR=

1

1 + �gP g

P s

�dY g

dGs

�LR+

1

1 + �s

�dY s

dGs

�LR=

!h=

(1 + �g)(1 + �s)S> 0: (29)

The sign of the long-run multiplier is opposite to that of the short-run. The reason is that

although home labour is constant in the short run, it decreases in the long run so that

market labour increases. Note that if !h = 0, the long-run national income multiplier

is zero. This is because market labour is constant in the absence of the home service

production.

The above results on the long-run multipliers are summarized in Proposition 2.

Proposition 2

In a monopolistically competitive economy without pure leisure, a rise in government

spending on public services has the following �long�-run multiplier e¤ects: (i) Real national

income increases. (ii) Market output and employment in the service sector increase. (iii)

Market output and employment in the good sector decrease if home output is larger than

output of public services. Otherwise, e¤ects on them are ambiguous.

4 Multipliers of Public Goods

This section analyzes the short-run and long-run multipliers of a rise in government spend-

ing on public goods (public-good spending).

4.1 The Short-run Multipliers

Di¤erentiating (T:1)-(T:3), (T:9) and (T:10) and assuming that dGs = 0, we obtain the

market output multipliers in the short run:

P s

P g

�dY s

dGg

�SR=

�� + (�=�g)M g(1� �)

1� (�=�s)� (�=�g)M g(�=�s)=

�[(1=�g)� 1]1� (�=�g)� (�=�s) < 0; (30a)

18

�dY g

dGg

�SR=

(1� �) + (�=�s)M s(��)1� (�=�g)� (�=�s)M s(�=�g)

=1� �� (�=�s)

1� (�=�g)� (�=�s) > 0; (30b)

which correspond to (22a) and (22b), respectively. The numerator in the middle expression

of (30a) is negative, because the negative �rst term, ��, dominates the positive second,

(�=�g)M g(1 � �), which represents an increase in aggregate service demand through a

rise in pro�t income from the good sector. Hence, the market-service output multiplier is

negative. On the other hand, the numerator in the middle expression of (30b) is positive,

since the negative second term, (�=�s)M s(��), which represents a decrease in aggregate

good demand due to a fall in pro�t income from the service sector, is dominated by the

positive �rst, 1� �. Therefore, the market-good output multiplier is positive.

Since the market output and employment multipliers have the same signs, public-good

spending moves labour force from the service sector to the good. From (T:14), (30a) and

(30b), we obtain the positive national income multiplier:

1

P g

�dY

dGg

�SR=

�[(1=�g)� (1=�s)]1� (�=�g)� (�=�s) > 0: (31)

The above results on the short-run multipliers of public-good spending are summarized

in Proposition 3.

Proposition 3

In a monopolistically competitive economy without pure leisure, a rise in government

spending on public �goods�has the short-run multiplier e¤ects which are exactly �opposite�

to those of public services.

The reason why the short-run multiplier e¤ects of the two kinds of government spend-

ing on aggregate output of market services and goods are opposite is that the signs of the

�rst-round e¤ects of these spending are di¤erent. For example, the �rst-round e¤ect of

public-good spending on aggregate service demand is negative, but that of public-service

spending on it is positive. The reason why the national income multipliers of the two

spending policies have the di¤erent signs is that while public-good spending moves labour

19

force from the service sector to the good, public-service spending moves it in the opposite

direction.

4.2 The Long-run Multipliers

Di¤erentiating (T:1)-(T:3), (T:5), (T:10) and (T:15) under the assumption that dGs = 0

and using (24) and (25), we obtain the market output multipliers in the long run:

P s

P g

�dY s

dGg

�LR=

�� + Asg�gRg(1� �)

1�Bss�s � Asg�

gRgAgs�s =

��(1 + �g)S

< 0; (32a)�dY g

dGg

�LR=

(1� �) + Ags�sRs(��)

1�Bgg�

g � Ags�sRsAsg�

g =�[(1� �)= ](!h � )

(1 + �s)S; (32b)

which correspond to (28a) and (28b), respectively. The numerator in the middle expression

of (32a) is negative, since the negative �rst term, ��, dominates the positive second term,

Asg�gRg(1� �), which represents an increase in aggregate service demand due to a fall in

the good price. Hence, the market-service output multiplier is negative.

On the other hand, the sign of the market-good output multiplier is ambiguous. If

!h � 1� !s, this multiplier is positive, because the second term in the numerator of the

middle expression in (32b), which represents a change in aggregate good demand due to

a rise in the service price, is non-negative, i.e., Ags�sRs(��) � 0. Otherwise, the sign

of this multiplier is determined as follows: public-good spending raises output of market

goods if !h < but reduces it if !h > . These results contrast with those in the case of

public-service spending.

It is evident from (24), (25) and (32a) that public-good spending increases home out-

put but reduces employment in the service sector. From (T:14), (32a) and (32b), the

national income multiplier is calculated as follows:

1

P g

�dY

dGg

�LR=�(1� �)(!h= )

(1 + �g)(1 + �s)S< 0: (33)

This multiplier is certainly negative, because public-good spending raises home labour

and hence decreases market labour.

20

The above results on the long-run multipliers of a rise in public-good spending are

summarized in Proposition 4.

Proposition 4

In a monopolistically competitive economy without pure leisure, the long-run multi-

plier e¤ects of public goods are exactly opposite to those of public services.

5 An Extension: Pure Leisure

This section brie�y analyzes Keynesian multipliers of the two kinds of government spend-

ing in an extended model, where each household allocates some of his time endowment to

pure leisure. By this analysis, we can examine what in�uences a policy-induced change

in leisure has on the multipliers.

The representative household�s utility function (1) is replaced with the following one:

U = U(Cg; Ds; l) = (Cg)�(Ds)�l ; 0 < � < 1; 0 < � < 1; = 1� �� � > 0; (34)

where l is pure leisure. Since his time constraint is rewritten as L = 1� e� l, his budget

constraint (9) is modi�ed to I = P gCg+P sDs+ l. He maximizes the utility function (34)

with respect to Cg, Ds, e and l subject to (5), (10), and the new budget constraint. As a

result, in addition to (11), we have the optimal level of leisure, l = I. Substituting this

into the time constraint gives us the labour supply function: L = 1� e� I. Replacing

(T:4) with this function, Table 1 describes the equilibrium in the extended model.

5.1 Public Services

The output multipliers of market goods and services in the short run are given by (22a)

and (22b), respectively. Thus, the signs of these multipliers are not changed by the in-

troduction of leisure. However, since a variation in leisure a¤ects market labour, the

following multiplier of market labour can be derived:

1

P s

�dL

dGs

�SR=

[1� (1=�s)]1� (�=�g)� (�=�s) > 0:

21

A reduction in leisure due to both a decrease in pro�t income from the good sector and

an increase in the lump-sum tax dominates a rise in it due to an increase in pro�t income

from the service. Therefore, since a net of leisure is declined by expansive spending on

public services, the market labour multiplier is positive.

Leisure also a¤ects the national income multiplier. This multiplier is modi�ed so that

> 0 is added to the numerator in the rightmost expression of (23). Thus, its sign is

ambiguous in general. It is because although national income decreases by the labour

movement from the good sector to the service, it increases by a rise in market labour.

The latter dominates the former if the price mark-up in the good sector is su¢ ciently

close to that in the service. In this case, the national income multiplier is positive.

In the long run, public-service spending has the following multiplier e¤ects. Whereas

the market-good output multiplier is the same as (28b), the market-service output mul-

tiplier is modi�ed so that the numerator in the rightmost expression of (28a) has an

additional term, � g, where � g � �g(1 � !g) > 0. Thus, the signs of these multipliers

are not in�uenced by pure leisure. Although a change in leisure is ambiguous, since a

decrease in home labour dominates it, the market labour multiplier is positive:

1

P s

�dL

dGs

�LR= (1 + � g) + �(!h= )

(1 + �g)(1 + �s)S> 0:

The national income multiplier is positive, because it coincides with market labour mul-

tiplier in the long run.

The above results on the short-run and long-run multipliers of public-service spending

with pure leisure is summarized in Proposition 5.

Proposition 5

In a monopolistically competitive economy with �pure leisure�, a rise in government

spending on public services has the following short run and long run multiplier e¤ects: (i)

In the short run, the signs of the market output multipliers described in Proposition 1 are

not altered by an existence of pure leisure. However, the national income multiplier is not

22

necessarily negative. It is positive if the price mark-up in the good sector is su¢ ciently

close to that in the service. (ii) In the long run, the signs of the market output and

national income multipliers are the same as those in Proposition 2.

5.2 Public goods

In the short-run, the market output multipliers of public-good spending are given by (30a)

and (30b). Hence, their signs are not changed by introducing pure leisure. In addition,

this spending generates the positive market labour multiplier because it decreases pure

leisure. Therefore, it raises real national income by an increase in market labour as well

as the labour movement from the service sector to the good. Thus, the national income

multiplier is positive in the short run.

The long-run multipliers of public-good spending are modi�ed as follows. Since the

market-service output multiplier is the same as (32a), the sign of this multiplier is not

in�uenced. On the other hand, the market-good output multiplier is changed so that

�s[(1�!s)�!h] is added to the numerator in the rightmost expression of (32b). However,

since the �rst equality in (32b) continues to hold in the extended model, too, the sign of

this multiplier is positive if !h � 1 � !s. This implies that the market-good output

multiplier is positive if home output is larger than output of public services. Even though

it is not so, the market-good output multiplier is positive if home output is smaller than

a critical level, i.e., !h < �, where � � [(1� �)=�s + (1� !s)]=[(1� �)=� � �] > 0.

Finally, the market labour and national income multipliers are modi�ed so that the

numerator in the RHS of (33) has an additional term, �s[(1=�s) + (1 � !s) � !h]. As

it is not evident whether public-good spending increases leisure or not, the signs of these

multipliers are ambiguous. However, it is easily shown that they are negative if !h > �,

where � � [(1=�s) + (1 � !s)]=[(1 � �)=� � �] > 0. Hence, as long as home output is

su¢ ciently large, these multipliers are negative.

The above results on the short-run and long-run multipliers of public-good spending

with pure leisure is summarized in Proposition 6.

23

Proposition 6

In a monopolistically competitive economy with pure leisure, a rise in government

spending on public goods has the following short-run and long-run multiplier e¤ects: (i)

In the short run, the signs of the market output and national income multipliers are the

same as those in Proposition 3. (ii) In the long run, whereas the sign of the market-service

output multiplier is not changed, those of the market-good output and national income

multipliers are altered by an existence of leisure. In particular, real national income does

not necessarily decrease. It increases if home output exceeds a certain critical level.

6 Concluding Remarks

This paper has developed a general equilibrium model of monopolistic competition con-

sisting of �rms, which produce di¤erentiated goods and services by using labour as the

unique input, and households facing the two allocation problems: (i) the allocation of dis-

posable income between market goods and services and (ii) the time allocation between

market work and home work. Since there is an interaction between the two sectors of

market production in this model, a rise in government spending induces not only intra-

sectoral but also inter-sectoral multiplier e¤ects. Separating the case with pure leisure

and the case without it, we have investigated the short-run and long-run multipliers of

two kinds of government spending: public services and public goods. The following main

results have been derived (see Table 2).

Without leisure, in the short run, the national income multiplier of public goods is

positive, but that of public services is negative. In the long run, the signs of these

multipliers are reversed. These results suggest that the government should allocate its

spending to public goods in the short run and to public services in the long run. On the

other hand, with leisure, an increase in government spending on public services does not

necessarily decrease real national income in the short run: real national income rises if

the price mark-up in the good sector is su¢ ciently close to that in the service. Similarly,

expansive spending on public goods increases national income in the long run if home

output is less than a critical level. In these cases, the two kinds of government spending

induce the positive national income multipliers both in the short run and the long run.

24

Therefore, since we cannot qualitatively judge which government spending is dominant,

a numerical comparison of the multipliers is required.

For output of market services and goods, the following multiplier e¤ects have been

derived. Without leisure, in the short run, expansive spending on public services increases

aggregate output of market services but decreases that of market goods. The aggregate

service output increases in the long run, too. However, aggregate output of market goods

decreases if home output is larger than output of public services. Otherwise, the sign of the

market-good output multiplier is indeterminate. The short-run and long-run multiplier

e¤ects of public goods are exactly opposite to those of public services. On the other hand,

the market-service output multipliers of public services and public goods with leisure have

the same signs as those without it, although the signs of market-good output multipliers

are ambiguous.

The above results have been obtained under some assumptions. First, government

spending has been �nanced by lump-sum taxation. However, since this taxation is not

actually available for the government, it is necessary to analyze Keynesian multipliers

of government spending �nanced by labour-income taxation. Since a rise in the labour

income tax directly depresses the after-tax real wage, it induces households to substitute

home work for market work. For this, market labour is a¤ected in the short run as well

as in the long run. Thus, it is important to study how such the tax distortion in�uences

the multipliers. Second, both public goods and public services have been assumed to

be wasteful. However, since they are useful in general, it is necessary to study whether

they are pro-cyclical or counter-cyclical by deriving their optimal provision rules. Third,

the increase in government spending has been entirely �nanced by taxes. As in Dixon

(1988), however, it is important to analyze the multipliers when an increase in government

spending on public services is �nanced by a decrease in its spending on public goods. This

analysis would give us a rule for the optimal allocation of government spending. Finally,

the multiplier e¤ects of public services and public goods have been separately examined.

However, in reality, the government changes them simultaneously. Therefore, it is also

important to analyze �scal policy e¤ects when the government increases lump-sum tax

revenue, keeping the shares of public services and goods at constant levels.

25

References

Benhabib, J., Rogerson, R. and Wright, R., �Homework in Macroeconomics : House-

hold Production and Aggregate Fluctuations,�Journal of Political Economy 99 (1991),

pp.1166-1187.

Cigno, A., �Fertility and the Tax-Bene�t System : A Reconsideration of the Theory of

Family Taxation,�Economic Journal 96 (1986) pp.1035-1051.

Dixon, H., �A Simple Model of Imperfect Competition with Walrasian Features,�Oxford

Economic Papers 39 (1987), pp.134-160.

Dixon, H., �Unions, Oligopoly and the natural range of employment,�Economic Journal

98 (1988), pp.1127-1147.

Fioritoa, R. and Kollintzas, T., �Public goods, merit goods, and the relation between

private and government consumption,�European Economic Review, forthcoming.

Greenwood, J. and Hercowitz, Z., �The Allocation of Capital and Time over the Business

Cycle,�Journal of Political Economy 99 (1991), pp.1188-1214.

Gronau, R., �Leisure, Home Production, and Work - the Theory of the Allocation of Time

Revisited,�Journal of Political Economy 85 (1977) pp.1099-1123.

Heijdra, B.J. and van der Ploeg, F., �Keynesian Multipliers and the Cost of Public Fund

under Monopolistic Competition,�Economic Journal 106 (1996), pp.1284-1296.

Heijdra, B.J. and Lighthart, J.E., �Keynesian Multipliers, Direct Crowding Out, and the

Optimal Provision of Public Goods,�Journal of Macroeconomics 19 (1997), pp.803-826.

Kaizuka, K., �Fiscal Expenditure, Budgetary System and Compilation of the Budget,�

Financial Review 17 (1990) pp.1-20 (in Japanese).

Kleven, H.J., Richter, W.F. and Sorensen, P.B., �Optimal Taxation with Household Pro-

duction,�Oxford Economic Papers 52 (2000) pp.584-594.

Kolm, A.S., �Di¤erentiated Payroll Taxes, Unemployment, and Welfare,�Journal of Public

Economics 70 (1998) pp.255-271.

26

Lindbeck, A. and Weibull, J.W., �Welfare E¤ects of Alternative Forms of Public Spending,�

European Economic Review 32 (1988) pp.101-127.

Lindbeck, A. and Nandakumar, P., �Public Spending and Private Services: Macroeconomic

Aspects,�Oxford Economic Papers 42 (1990) pp.620-634.

McGrattan, E.R., Rogerson, R. and Wright, R., �An Equilibrium Model of the Business

Cycle with Household Production and Fiscal Policy,�International Economic Review 38

(1997) pp.267-290.

Mankiw, N.G., �Imperfect Competition and the Keynesian Cross,�Economics Letters 6

(1988), pp.7-13.

Sandmo, A., �Tax Distortions and Household Production,�Oxford Economic Papers 42

(1990) pp.78-90.

Startz, R., �Monopolistic Competition as a Foundation for Keynesian Macroeconomic Mod-

els,�Quarterly Journal of Economics 104 (1989), pp.737-752.

Yoshida, M., �Optimal Taxation with a Trade-o¤ between Income and Children,�Japanese

Economic Review 49 (1998) pp.426-439.

Yoshida, M. and Yuki, K., �Optimal Taxation of Elderly Care Services,�Japanese Economic

Review 55 (2004) pp.86-100.

Yoshida, M. and Kenmochi, T., �Incidence of Consumption Tax and Tax Treatment of

Elderly Care Expenditures,�Public Finance 54 (2005) pp.57-72.

27

Table 1: Summary of the Model

Y k = Ck +Gk; k = g; s (T:1) �k = (�k)�1P kY k �Nkbk; k = g; s (T:9)

Cg = �I

P g(T:2) P gGg + P sGs = T (T:10)

Cs = �I

P s�Hs (T:3) Y k = (Nk)1+�

k

Qk; k = g; s (T:11)

L = 1� e (T:4) Lk = akQk + bk; k = g; s (T:12)

I � 1� T +�g +�s +�h (T:5) P k = (Nk)��k

�kak; k = g; s (T:13)

e = (�P s)1� (T:6) Y � P gY g + P sY s (T:14)

Hs = (�)1�1

1� �(P s)

1��� (T:7) �k = 0; k = g; s (T:15)

�h = (�)1�

�

1� �(P s)

1� (T:8)

NOTE : In the short run, the number of �rms Nk and aggregate pro�t �k in each sector are constant

and non-zero, respectively. However, in the long run, the former is variable and the latter is zero.

Table 2: E¤ects of Two Types of Government Spending

public service public goodvariables short run long run short run long runoutput of market services + (+) + (+) � (�) � (�)output of market goods � (�) ? (?) + (+) ? (?)product of a home service 0 (0) � (�) 0 (0) + (+)home labour 0 (0) � (�) 0 (0) + (+)market labour 0 (+) + (+) 0 (+) � (?)real national income � (?) + (+) + (+) � (?)

NOTE: The signs in parentheses represent e¤ects of government spending in the extended model with

pure leisure.

28

Acknowledgements

This paper was presented at the seminars held at the Doshisha University, the Osaka

University, the Osaka Prefecture University, and the University of Tsukuba. We are grate-

ful to Professors Y. Azuma, K. Eguchi, E. Fujii, K. Futagami, K. Kaneko, S. Matsukawa,

K. Mino, Y.Ono, Y. Tanaka, S. Turnbull, T. Yagi, A. Yakita, M. Yamada, and seminar

participants for constructive comments on an early version of this paper.

29