Keynesian Multipliers with Home Production By Masatoshi Yoshida Professor, Graduate School of Systems and Information Engineering University of Tsukuba Takeshi Kenmochi Graduate School of Systems and Information Engineering University of Tsukuba The address for editorial correspondence: Masatoshi Yoshida Graduate School of Systems and Information Engineering, University of Tsukuba 1-1-1, Tennoudai, Tsukuba, Ibaraki 305-8573, Japan Tel: 029-853-5556; Fax: 029-855-3849 E-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 government 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 Nandakumar (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 1 Fiorita 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 has decreased (see Table 2 in their paper). For example, in Germany, the shares of public goods and public services 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% in 1970 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 develops 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, 2 The labour supply e ects of a policy-induced change in home work have been studied in the economic analyses on the quantity and quality of children. See Cigno (1986) and Yoshida (1998). 3 Following Lindbeck and Nandakumar (1990), Sandmo (1990), Kleven et al. (2000), and Yoshida and Yuki (2004), we assume that households produce not goods but services. 4 The 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 successfully explain 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 they treated government spending as a stochastic variable. 5 The extended model is also a generalization of the model with monopolistic competition developed by Heijdra and van der Ploeg (1996), where each household allocates his time endowment between market work and pure leisure. Such a model, which was originated with Gronau (1977), has been utilized in various 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 the condition 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 positive 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. 6 This assumption implies that the service sector is more competitive and rms in this sector face a more elastic demand schedule. This agrees with the notion that the demand for services seems to be more 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. Section 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 includes 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 differentiated 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 production 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 household derives his utility U from consuming a composite di erentiated good C g and a composite di erentiated service D s. We assume the following Cobb-Douglas utility function: U = U(C g ; D s ) = (C g ) (D s ) ; 0 < < 1; 0 < < 1; + = 1: (1) The composite di erentiated good is a CES-aggregation of N g varieties of consumption goods, C g (C g 1; C g 2; ; C g N g): 5
" N g # X g=(g 1) C g = (N g ) 1+g (N g ) 1 (C g 1)= g i )(g ; g > 1; g > 0; (2) i=1 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 D s consists of a composite market-produced service C s and a home-produced service H s : D s = C s + H s : (3) The former is a CES-aggregation of N s varieties of market services, C s (C1; s ; CN s s): " N s # X s=(s 1) C s = (N s ) 1+s (N s ) 1 (Ci s ) (s 1)= s ; s > 1; s > 0; (4) i=1 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: H s = H(e) = e1 ; > 0; 0 < < 1; (5) 1 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 = P g C g + P s C s ; (6) where g and s are aggregate pro ts in the good and service sectors, respectively, T is a lump-sum tax, and P g fp g i g and Ps fp s j 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 C k and P k, minimize P k C k with respect to C k for k = g; s subject 6
to (2) or (4). Then, it holds that C k i = C k (N k ) k ( k 1) 1 min P k C k = P k C k ; (7a) C k P k k i ; i = 1; 2; ; N k ; k = g; s; (7b) P k where P g and P s are price indices for the composite good and service, respectively: 2 XN k P k (N k ) (k +1) 4(N k ) k (Pi k ) 1 i=1 k 3 5 1=(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 g C g + P s D s ; (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 s H s e: (10) Second, the household maximizes his utility (1) with respect to C g, D s 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: C g = I P g ; Cs = I P s H s ; 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: H s = 1 2.2 The Government 1 1 (P s ) 1 ; h = 1 1 (P s ) 1 : The government provides a composite public good G g and a composite public service G s, which are the following CES functions of all existing varieties of the di erentiated goods 7
and services: 2 3 X G k = (N k ) 1+k 4(N k ) 1 N k (G k i ) (k 1)= k 5 i=1 k =( k 1) ; k = g; s: Government spending P k G k is minimized with respect to G k (G k 1; ; G k N k ), given P k and G k for k = g; s. Then, it holds that min P k G k = P k G k ; (12a) G k G k i = G k (N k ) k ( k 1) 1 P k k i ; i = 1; ; N k ; k = g; s: (12b) P k Since the government nances its spending by the lump-sum tax, its budget constraint is written as P g G g + P s G s = T: 2.3 Monopolistically Competitive Firms Firm i in sector k produces its output Q k i under increasing returns to scale with labour as the sole production factor. The production technology is given by L k i = a k Q k i + b k ; (13) where L k i, a k and b k 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, k i P k i Q k i L k i, subject to (13) and the demand function which it faces, C k i + G k i, 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 k i = k a k ; k k k 1 > 1; i = 1; ; N k ; 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 Q k i = Ci k + G k i ; i = 1; ; N k ; k = g; s: (16) Equilibrium in the labour market requires that the demand for labour by all rms equals the supply by households: XN g N s L = L g i + X L s j: (17) i=1 j=1 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 k i = P k ; Q k i = Q k ; L k i = L k ; k i = k ; C k i = C k ; G k i = G k for k = g; s: In this equilibrium, it follows from (8) and (14) that price indices are represented as P k = (N k ) k P k = (N k ) k k a k ; k = g; s; (18) and the market clearing condition (17) can be rewritten as L = N g L g + N s L s : (19) 9
Denote real aggregate output of market goods and services by Y g and Y s, respectively, that is, Y k P N k i=1 P k i Q k i =P k for k = g; s: From (18) they can be represented as Y k = (N k ) 1+k Q k ; k = g; s: (20) Using the market clearing conditions (16) and the private and public demand functions, (7b) and (12b), we can derive the expressions as follows: N k Q k = (N k ) k (C k + G k ) for k = g; s: Eliminating N k and Q k from these and (20), we obtain the market equilibrium conditions in aggregate form: Y k = C k + G k ; k = g; s: From (13), (18) and the de nitions of aggregate market output, aggregate pro t in sector k, k N k k, is given by k = ( k ) 1 P k Y k N k b k for k = g; s: We will de ne real national income in labour units as Y P g Y g + P s Y s : For convenience, the complete model is summarized in Table 1, although the labourmarket 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. 7 This 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 the numeraire, 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 spending 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 ; C k ; H s ; e; L; k ; h ; I; T; P k ; Q k ; L k ) for k = g; s which satisfy (T:1)-(T:14) in Table 1, given the levels of G g and G s. Since the number of rms in each sector, N k, 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, H s, 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, C k, full income, I, and aggregate pro t, k. (T:11) and (T:12) determine output and employment of each rm, Q k and L k. Finally, (T:14) provides us real national income, Y. Di erentiating (T:1)-(T:3), (T:5), (T:9)-(T:10) and assuming that dg g = 0, we have d k dg = P k dy k ; s k dgs k = g; s; (21a) dy s dg = (1 s ) + d s P s dg + d g s P s dg ; s (21b) P g dy g P s dg = s + d g P s dg + d s s P s dg : s (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: where dy s SR = (1 ) + (=g )M g ( ) dg s 1 (= s ) (= g )M g (= s ) = 1 (=g ) > 0; (22a) 1 (= g ) (= s ) P g dy g SR + (= s )M s (1 ) = P s dg s 1 (= g ) (= s )M s (= g ) = [(1= s ) 1] < 0; (22b) 1 (= g ) (= s ) M g 1 + 1X i=1 g i = 1 1 (= g ) > 0; M s 1 + 1X i = i=1 s 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 additional 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 decrease 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 private 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 d(n s L s SR ) = 1 dy s SR > 0; P s dg s s dg s 1 d(n g L g SR ) = 1 P g dy g SR < 0: P s dg s g P s dg s 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: SR 1 dy = P g P s dg s P s dy g SR dy s + dg s dg s 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 summarized in Proposition 1. Proposition 1 In a monopolistically competitive economy where there are market and home production 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 ; C k ; H s ; e; L; k ; h ; I; T; P k ; Q k ; L k ; N k ) 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 Q k = ( k 1)b k =a k, so that the scale of production for individual rms is xed. Thus, (T:12) indicates that employment of each rm, L k, 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, C k, full income, I, a lump-sum tax, T, the number of rms in each sector, N k, home labour, e, home output, H s, 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 dn k N k = 1 1 + k dy k Y k ; dp k k dn = k ; k = g; s: (24) P k N k 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 dh s = 1 dp s H 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 dg g = 0, we get dt T = dp g P g + (1 )dp s + (1 )dgs P s G ; (26) s 14
where P g G g =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 dg g = 0, and using (24)-(26), we obtain dy s = (1 )(1! s ) dgs Y s G + dp g s 1 As g P g dy g Y g = 1 (1!s ) dgs G + 1 s + dp s Bs s P ; s dp s Ag s P + dp g s Bg g P ; g (27a) (27b) where! s C s =Y s is the share of private consumption in aggregate service output, P g Y g =Y is the share of total value of all existing di erentiated goods in real national income,! g C g =Y g is the share of private consumption in aggregate good output,! h H s =Y s is the ratio of home output to aggregate service one, and B s s (! s +! h ) [(1! s )! h ] 1 =! s (1! s ) ( 1 )! h < 0;! h A s g (1! g ) < 0; B g g! g (1! g ) < 0; A g s [(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 C s 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, H s and full income I are both constant, (! s +! h )(dp s =P s ), under the condition that home output 8 Note that the good price a ects private service demand via a change in disposable income, since the consumption 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 D s 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 rstround 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: where k dy s LR = (1 ) + As g g R g ( ) dg s 1 Bs s s A s g g R g A g s s = 1 > 0; (28a) (1 + g )S P g dy g LR + A g = s s R s (1 ) P s dg s 1 Bg g g A g s s R s A s g g = (= )(!h ) : (28b) (1 + s )S k 1 + k < 0; Rk 1 + 1X i=1 S (1 B s s s ) 1 B g g g A g s s A s g g ; Bk k k i 1 = ; k = g; s; 1 Bk k k s (1 ) : The second and third terms in the denominator of the middle expression in (28a) represent 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 promotes the private service demand, so that this output increases again. Now, assuming that this process is stable, it holds that R s > 0 since the stability condition, B s s 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 variation 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, A s g g R g ( ), 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 numerator 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 multiplier 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., A g s s R s (1 ) 0. Otherwise, its sign is determined by the following condition:! h 0 @ > = < 1 0 A ) P g dy g LR @ P s dg s > = < 1 A 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 employment in the service sector but lowers home output and home labour. Since the home labour multiplier is negative, the market labour multiplier is positive. 9 On the other hand, it is easy to show that B g g g < 1. Hence, the multiplier process within the good sector is stable, so that it holds that R g > 0. 17
Finally, di erentiating (T:14) and using (24), (28a) and (28b), we obtain the positive national income multiplier: LR 1 dy = P s dg s = 1 1 + g P g P s dy g dg s! h = (1 + g )(1 + s )S LR + 1 1 + s dy s dg s LR > 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 spending 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 dg s = 0, we obtain the market output multipliers in the short run: P s P g dy s SR = dg g + (= g )M g (1 ) 1 (= s ) (= g )M g (= s ) = [(1= g ) 1] 1 (= g ) (= s ) 18 < 0; (30a)
dy g SR = (1 ) + (=s )M s ( ) dg g 1 (= g ) (= s )M s (= g ) = 1 (=s ) > 0; (30b) 1 (= g ) (= s ) 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: SR 1 dy = [(1=g ) (1= s )] P g dg g 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 spending 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 dg s = 0 and using (24) and (25), we obtain the market output multipliers in the long run: P s dy s LR + A s g g R g (1 ) = P g dg g 1 Bs s s A s g g R g A g s s = < 0; (32a) (1 + g )S dy g LR = (1 ) + Ag s s R s ( ) dg g 1 Bg g g A g s s R s A s g g = [(1 )= ](!h ) ; (32b) (1 + s )S 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, A s g g R g (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., A g s s R s ( ) 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 output but reduces employment in the service sector. From (T:14), (32a) and (32b), the national income multiplier is calculated as follows: LR 1 dy = (1 )(!h = ) P g dg g (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 multiplier 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 spending 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(C g ; D s ; l) = (C g ) (D s ) 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 g C g + P s D s + l. He maximizes the utility function (34) with respect to C g, D s, 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 introduction of leisure. However, since a variation in leisure a ects market labour, the following multiplier of market labour can be derived: 1 SR dl = P s dg s [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 multiplier 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: LR 1 dl = (1 + g ) + (! h = ) P s dg s (1 + g )(1 + s )S > 0: The national income multiplier is positive, because it coincides with market labour multiplier 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 consisting 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 disposable 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 intrasectoral 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
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