1 – Introduction
1The neoKaleckian approach to growth and distribution, which follows the heritage of Michael Kalecki, John Maynard Keynes, Josef Steindl and Alfred Eichner, was originated during the 70’s and has been developed and systematised during the 80’s and the 90’s. [2] The standard model describes an economy in which industrial production is undertaken by an oligopolistic sector and income distribution is the final outcome of wage bargaining and firms price setting behaviour. Firms are endowed with a large stock of capital build as a precaution to unexpected rises in demand and as a deterrent for potential entrants. The downside is that scarce demand accompanied by extensive plant underutilisation could trigger a vicious circle of demand reductions and low capacity utilisation. An increase in one of the components of aggregate demand does not have only a temporary effect, as theorised by mainstream economists, but it can also permanently modify the growth path of the economy.
2In this paper we deal with the component of aggregate demand that is managed by the public sector with a particular stress on the effects on longrun growth of the composition of public expenditure. As noticed by Pressman (1994), Keynes was wellaware that certain kinds of public expenditures are preferred to others for economic, social or political reasons. Kaldor (1958; 1966; 1967; 1971) specified the relation between longrun growth and the composition of government expenditure, mentioning how a large government consumption could be detrimental to growth and international competitiveness given the higher rates of variation in productivity enjoyed by the investment goods sector compared to the consumption goods sector. The neoKaleckian literature on growth and distribution, when dealing with the role of the government sector, has been mainly concerned with public consumption expenditure (see You and Dutt, 1996; Lavoie, 2000). More recent contributions, however, explore the impact of different types of public expenditure when economic growth is driven by effective demand (see Dutt, 2010; and Commendatore et al., 2010; 2011). Commendatore et al. (2010; 2011) consider the effects on longrun growth of a government expenditure that may affect labour productivity. The impact on growth of this type of government expenditure is compared with that of public consumption expenditure, that does not affect the input coefficients. Dutt (2010) explores the effects on growth of changes in the composition of government expenditure. In his analysis, given the government size, the replacement of public consumption with public investment has a positive impact on growth due to a “crowding in” effect on private investment.
3Building upon the previous literature, we put forward a neoKaleckian model of growth and distribution where the presence of two types of government expenditure is considered: public consumption expenditure and public provision of capital in the form of infrastructures and other facilities to production. Moreover, we also study the consequences of changes in the composition of public expenditure. Like in Dutt (2010), we assume that some types of government expenditure may affect positively private investment decisions; we take also into account that the capital input coefficient can be affected. In this sense, the present paper represents a follow up of Commendatore et al. (2010; 2011). Our study can also be considered as an heterodox contribution to the broader literature investigating these issues. [3] Unlike the mainstream approach, as developed by Barro (1990) and others, in our analysis the economy is characterised by the underutilisation of the capital stock Consequently, a greater provision of public capital to the private sector does not involve faster growth, if it is not accompanied by a sufficient increase in demand.
4The structure of the paper is the following. In section 2 we put forward the economic framework of our model; in section 3, we study the effects of variations of the two types of public expenditure when they are allowed to vary in the same direction; in section 4, we fix the size of the government sector and we explore the role of changes in the public expenditure composition; in section 5 we provide a brief comparison between our results and those put forward by the mainstream endogenous growth approach; in this section some final remarks are also presented.
2 – The economic framework
5We assume the simplest possible economic framework: a singlegood closed economy, where labour and capital are the only factors of production. The labour supply is abundant, there is no capital depreciation and technological progress is excluded. Production involves a Leontief technology with a and b representing the reciprocals of the capital and labour input coefficients, respectively. Demand determines the current level of output, Y, which is typically below its potential level, Y^{P}, determined by the current stock of capital, K. The implication is that capacity utilization is only partially utilised, ? = Y/Y^{P} denoting the current degree of capacity utilisation. Labour employment, L, accommodates the production decisions of firms. We may concisely write:
7With the price of the unique good as the obvious numéraire, in absolute terms income distribution between wages and profits corresponds to Y = wL + rK, where w is the wage rate and r is the rate of profit; whereas, in relative terms, income distribution corresponds to 1 = ? + ?, where ? is the wage share in national income and ? is the profit share in national income. Moreover, with labour wages as the only variable cost, markup pricing determines the profit share; more specifically, the profit share depends only on the average productivity of labour, i.e. it depends on the ratio w/b. Furthermore, it is possible to express the positive relationship between the rate of profit and the degree of capacity utilisation which characterises neoKaleckian growth and distribution models:
9Expression (2) shows that, as pointed out by Rowthorn (1981, p. 16), for a given degree of capacity utilization, profitability increases when costs fall. In this expression, a reduction in costs may take the form of an increase in the profit share (induced by a reduction in the wage rate and in the wage share), or in a decrease in the capital input requirement (represented by the reciprocal of the capital input coefficient, a).
10We introduce a government sector for which we assume a balanced budget:
12where ? represents taxation in terms of the capital stock, G_{1} represents government consumption expenditures in terms of the capital stock and G_{2} government capital expenditures, in the form of infrastructures and other facilities to production which are privately produced and publicly provided to the private sector, also expressed in terms of the capital stock. ? also represents the overall government size compared to the existing stock of capital in the economy. [4]
13We assume further that government spending in infrastructure, G_{2}, may enhance input productivity. This implies a reduction in the capital input requirement:
15where a’(G_{2}) > 0 and a’’ (G_{2}) ? 0. For the function a(G_{2}) we choose a linear form:
17with a_{0} > 0 and a_{1} > 0.
18In a similar manner, the labour input requirement b is reduced as well. However, a change in b does not modify the profit share, and therefore has no consequences for the results, if it is accompanied by a similar change in the wage rate. We make this assumption in what follows. [5]
19Concerning the saving and investment decisions of the private sector, we put forward the following equations:
23where s is the ratio between saving and capital, S_{?} is the propensity to save out of profits, t ? ?/au is the income ta1 rate, g is the of rate capital accumulation. According to equation (5) the only source of saving is net profit; according to equation (6), investment decisions depend on the degree of capacity utilization u and on public investment; with ?, ?, ? > 0. That is, we maintain the standard neoKaleckian assumption that investment is driven by the current degree of capacity utilisation. Moreover, we assume that government expenditure may also positively affect private investment. Specifically, a higher G_{2} may strengthen firms’ incentive to invest determining a “crowding in” effect (see Dutt 2010; and for empirical analyses Aschauer, 1989; and de Haan and Romp, 2007). Finally, equation (7) represents the equilibrium condition saving equal to investment.
24In the following sections, we study the effects of variations of the two types of public expenditure G_{1} and G_{2} considering two cases. In the first, presented in section 3, the government size is allowed to vary. For this case, we explore the implications for equilibrium capacity utilisation and growth of a change of one type of public expenditure keeping the other fixed or of a variation of both in the same direction. In the second, presented in section 4, the government size is fixed. For this case, we verify the implications of changes in the composition of public expenditure.
3 – Variable government size
25When the size of the government sector is allowed to vary, the equilibrium solutions for the degree of capacity utilisation and the rate of growth are:
27In order to have definite and positive solutions, we need to assume s_{?}? (a_{0} + a_{1}G_{2}) > ? for any G_{2} ? 0.
28The impact of a change of G_{1} on the equilibrium solutions is described by the following derivatives:
31That is, the impact of G_{1} both on equilibrium capacity utilisation and equilibrium growth is positive (see Figure 1).
32These results confirm to the socalled “paradox of thrift”. In the context of our analysis, a reduction in the propensity to save of the overall economy, induced by an increase in the size of the government sector (characterised by a zero propensity to save), has a positive effect on capacity utilisation and growth.
33The impact of a change of G_{2} on the equilibrium degree of capacity utilisation is:
35where for or, alternatively, for .
36That is, the impact of G_{2} on u^{*} depends on the parameters configuration and, in particular, on the parameters which are directly linked with this type of public expenditure. Specifically, the impact is positive provided that the effect of this type of public expenditure on the technological output/capital ratio is not too strong or its effect on private investment is sufficiendy large [see Figure 2(a)]; otherwise, it is negative [see Figure 2(b)]. This result can be explained considering that when a_{1} is sufficiently small or ? is sufficiently large, the positive effect on the equilibrium capacity utilization of the reduction in savings (due to the increase in the government size) and of the increase in private investment demand (due to the crowding in effect) overcomes the negative effect on u^{*} induced by the increase in average capital productivity favoured by the public provision of capital. The latter effect (originated by the change in the capital input requirement, a) is analogous to the socalled “paradox of costs”. On the other hand, when a_{1} is sufficiently large or ? is small, the opposite holds: the negative effect induced by the increase in average capital productivity is stronger than the positive effect induced by the increase in the government size or by the crowding in of private investment.
37The impact of a change of G_{2} on the equilibrium rate of growth is:
38 for
39where and
41As previously seen for the degree of capacity utilisation, the impact of G_{2} on equilibrium growth also depends crucially on the coefficients a_{1} and ? as the following proposition shows:
42Proposition 1. The effect of G_{2} on the rate of capital accumulation, when the government budget is balanced and the government size is variable, depends on the coefficients a_{1} and ? as follows:
 when or for any G_{2} ? 0 (see Figure 3(a)) ;
 when a_{1} > ã_{1} or for and for , where solves AG^{2} + BG + C = 0 and minimises g^{*} with respect to G_{2} [See Figure 3(b)].
43Therefore for a_{1} ? ã_{1} or , AG^{2} + BG + C > 0 and ?g^{*} / ?G_{2} > 0.
44Instead, for a_{1} > ã_{1} or we have that C < 0. The equation admits a positive solution and a negative solution, which are both real. Denoting by the positive solution and taking into account that
46for or , we deduce that corresponds to a minimum for g^{*} and therefore for and for
47Part a) and part b) of Proposition 1 are represented in Figure 3(a) and 3(b), respectively.
48The results concerning the effects of G_{2} on the equilibrium capacity utilization and growth are summarized by the following Table 1:
Table I
Table I
49According to these results, following a change in G_{2}, the degree of capacity utilisation and the rate of growth could move in the same direction or in opposite directions depending on the influence of this type of government expenditure on average capital productivity and investment decisions. In particular, when a_{1} is sufficiently small or ? is sufficiently large, the positive effect of the reduction in savings (determined by the increase in the government size) and of the rise in private investment (crowded in by the public provision of capital) overcomes the negative effect induced by the increase in average capital productivity both on the degree of capacity utilisation and on the rate of capital accumulation, u^{*} and g^{*} both increase; when a_{1} and ? take intermediate values, the effect of public investment expenditure on the degree of capacity utilisation is negative, u^{*} decreases; however, the crowding in effect on private investment is sufficiently strong to determine a positive impact of G_{2} on capital accumulation, g^{*} rises. Finally, when a_{1} is sufficiently large or ? is sufficiently small, for an initial range of values of G_{2}, i.e. for , the negative effect on the degree of capacity utilisation overcomes the positive effects on private investment of a rise in G_{2}, u^{*} and g^{*} both decrease. However, when G_{2} is above , the crowding in effect is so large to prevail over the negative effect, g^{*} increases whereas u^{*} decreases. [6]
50The parameters a_{1} and ? are also crucial to determine when G_{2} is more effective than G_{1} to intensify capital utilisation or to enhance growth (see the Appendix 1). For instance, the condition
52gives and, consequently, it is sufficient in order to have .
53Finally, note that the overall effect of G_{2} on u^{*} and g^{*} crucially depends on the value of G_{1}, as and ã_{1} () are both negative (positive) functions of G_{1}. The level of public consumption expenditure is also crucial in determining when G_{2} is more effective than G_{1} for enhancing capacity utilisation and growth. The effects of a simultaneous change of G_{1} and G_{2} on the equilibrium degree of capacity utilisation and on the equilibrium rate of growth are presented in Figure 4. In Figure 5, we plot the corresponding contour lines whose properties are the following: points belonging to the same line represents all the combinations of G_{1} and G_{2} characterised by the same values of capacity utilisation and growth; [7] moreover to lines more distant from the origin correspond higher degrees of capacity utilisation and rates of growth. Figure 4 and 5 are plotted for different values of the ratio ? / a_{1}. In panels 4(a) and 4(b), which are plotted for ? / a_{1} = 5, [8] a simultaneous increase of G_{1} and G_{2} determines a rise in both u^{*} and g^{*}. As can be inferred looking at Panels 5(a) and 5(b), where the corresponding contour lines for u^{*} and g^{*} are plotted, the effect of both types of public expenditure goes in the same direction, with G_{2} more effective than G_{1} (the slope of the contour lines is flatter than the 45° line). In Panel 4(c) and 4(d), which are plotted for ? / a_{1} = 0.25, a simultaneous increase of G_{1} and G_{2} also determines an increase in u^{*} and g^{*}. The corresponding contour lines, plotted in Panels 5(c) and 5(d), show that with a considerably smaller ratio ? / a_{1} effect of both types of public expenditure still goes in the same direction, however now G_{1} is more effective than G_{2} (the slope of the contour lines is steeper than the 45° line). Finally, in Panels 4(e) and 4(f), which are plotted for ? / a_{1} = 0.05, the effects of G_{1} and G_{2} do not always go in the same direction, i.e. the effect of G_{1} is always positive whereas the effect of G_{2} can also be negative. This can be verified looking to the corresponding contour lines of u^{*} and g^{*} presented in Panels 5(e) and 5(f). Concerning u^{*}, Panel 5(e) shows that when G_{1} is sufficiently large, to higher values of G_{2} correspond contour lines closer to the origin (i.e. smaller values of the degree of capacity utilisation). Concerning g^{*}, Panel 5(f) shows that when G_{1} is sufficiently large, to higher values of G_{2} initially correspond contour lines closer to the origin and after contour lines more distant from the origin (where the latter phenomenon is not always visible for the values of G_{1} and G_{2} chosen to plot the Figures). In this case, since along their corresponding contour lines the degree of capacity utilisation and the rate of growth are constant, in order to maintain the same values of u^{*} and g^{*} the negative effect of an increase in G_{2} has to be compensated by the positive effect of a simultaneous increase in G.
4 – The government size is fixed
54When the size of the government sector cannot exceed a certain proportion of the overall economy (in our formulation the stock of capital), the government has to choose among alternative combinations of the two types of public expenditure. Taking into account that the government size is fixed, ? = ? _{f}, it follows that G_{1} = ? _{f} – G_{2} or that G_{2} = ? _{f} – G_{1}. The equilibrium solutions for the degree of capacity utilisation and the rate of growth become:
56and
58To ascertain what are the effects of alternative combinations of G_{1} and G_{2} on the equilibrium solutions, we introduce the following functions
59 and
61for the degree of capacity utilisation and
62 and
64for the rate of growth.
65u1 and u2 and gl and g2 are symmetric with respect to ? _{f} / 2.
66By differentiating u1 and u2, we obtain the following expressions:
67 for
69That is, both functions are monotonic. Moreover, from ? / a_{1} ? (<)h it follows u2(0) ? (>)u1(0) and, by symmetry, u_{2} (? _{f} ) ? (<)u_{1} (? _{f}).
70That is, as shown in the Figure 6 below, when a) ? /a_{1} > h, G_{2} has a positive effect and G_{1} a negative effect on instead, when ? /a_{1} < h the opposite holds, i.e. G_{2} has a negative effect and G_{1} a positive effect on .
71With a fixed government size, there is no change in the propensity to save of the economy. The impact of a change in the public expenditure composition on the degree of capacity utilisation depends only on the counterbalancing of two effects. [9] The first is the change in the average productivity of capital, induced by an increase in G_{2}, which has a negative impact. The other is the positive effect, induced by an increase in G_{2}, on the rate of capital accumulation. When ? / a_{1} relatively large, the second effect exceeds the first: the degree of capacity utilisation increases when the public expenditure composition changes in favour of G_{2}; whereas, when ? / a_{1} relatively small, the first effect exceeds the second: the degree of capacity utilisation increases when the public expenditure composition changes in favour of G_{1}.
72Finally, we note also that for ? = ? _{ f}, it is possible to determine the maximum degree of capacity utilization, which is
74In order to assess the impact of changes in the composition of public expenditure on the equilibrium rate of growth, we state the following proposition:
75Proposition 2. The effect of a change in the composition of public expenditure (i.e. of a change in G_{1} or G_{2} when the government budget is balanced and the government size is fixed) on the rate of capital accumulation crucially depends on the ratio ? / a_{1} as follows: fir a sufficiently large value of ? / a_{1} a shift in the composition of public expenditure in favour of G_{2} enhances growth; whereas, for smaller values of ? / a_{1} various possibilities may occur: 1) a shift in favour of G_{2} can slow down growth initially and increase growth subsequently. Moreover, the highest rate of growth is obtained at G_{2} = ? _{f} ; 2) a shift in favour of G_{2} still slows down growth initially and increases growth subsequently. However, in this case, the highest rate of growth is obtained at G_{2} = 0; 3) an increase in G_{2} slows down growth for any possible public expenditure composition (See Figure 7).
76Proof. See the Appendix 2
77As suggested by Proposition 2, concerning the effect of a change in the composition of public expenditure on the equilibrium rate of growth, various possibilities may emerge. The occurrence of a specific case depends on the relative strength of the two opposite effects, induced by a change in G_{2}, on the average productivity of capital and on the investment decisions. The various possibilities are shown in figure 7, where on the horizontal axis G_{1} and G_{2} are represented, both varying from 0 to ? _{f} (G_{2} is increasing from left to right and G_{1} from right to left); and on the vertical axis the equilibrium rate of growth is plotted. Moreover, proceedings from Figure 7(a) to Figure 7(h) we have progressively reduced the ratio ? / a_{1}. In Figures 7(a) and 7(b), public capital expenditure crowds in a sufficiently large amount of private investment to overcome the negative effect of a reduction in the capital input requirement. G_{2} is always more effective than G_{1}. In Figures 7(c)7(f) and 7(h), the positive or the negative effect prevails depending on the composition of public expenditure: the negative effect dominates the positive one when G_{2} is relatively small (G_{1} is relatively large) and the positive effect dominates the negative one when G_{2} is relatively large (G_{1} is relatively small). Finally, in Figure 7(h) the negative effect prevails on the positive effect for any possible public expenditure composition.
78From this analysis, we can deduce that, in general, more unbalanced public expenditure compositions correspond to higher growth rates. Moreover, for relatively high values of ? / a_{1}, the composition of public expenditure that gives the highest possible rate of growth corresponds to G_{2} ^{=}? _{f} and G_{1} = 0. Instead, for relatively low values of ? / a_{1} composition of public expenditure that gives the highest possible rate of growth corresponds to G_{2} = 0 and G_{l} = ? _{f}.
5 – Final remarks
79The study presented above is concerned with the effects on capacity utilisation and growth of different combinations of two types of public expenditure, for consumption and investment, in the context of a neoKaleckian model of growth and distribution. Our conclusions do not always agree with those drawn by the mainstream theory of endogenous growth, the socalled new economic growth (NEG) theory. In particular, under the assumptions of full capital utilisation, a production function that exhibits decreasing returns in private capital and investment decisions that depend on saving (i.e. it is excluded an autonomous investment function) – and further imposing a balanced budget constraint –, Barro (1990) identifies two possible effects of public investment expenditure on growth. Such effects are both operating through the saving rate of the economy: a positive effect induced by a rise in the aftertax marginal product of capital, in the aftertax rate of profit and in the saving rate; and a negative effect induced by the corresponding rise in taxation and reduction of saving. Public consumption, instead, only impacts negatively on growth through the increase in the size of the government sector and therefore through the increase in taxation and the reduction of saving. Barro’s conclusions on the effects of public expenditure on equilibrium capacity utilisation and growth are summarised as follows:
 considering public investment expenditure, at low values of G_{2} the positive effect on productivity dominates the negative effect of taxation: the rate of growth of the economy icreases. As G_{2} increases the effect of taxation becomes stronger, until the rate of growth reaches a peak. After further increasing G_{2}, the rate of growth decreases as the negative effect dominates the positive one (i.e. the relationship between G_{2} and g^{*} is bellshaped);
 considering public consumption expenditure, a rise in G_{1} always determines a reduction of the equilibrium rate of growth;
 it follows that when the effect of the size of the government sector is neutralised (when to a rise in G_{2} corresponds a simultaneous reduction of G_{1}); a rise in G_{2} always determines an increase in the rate of growth;
 Finally, public expenditure of any kind has no effect on the equilibrium degree of capacity utilisation which is fixed at the value of one by assumption. [10]
80We must acknowledge that our analysis is quite simplified. By focussing only on public consumption and on government expenditure in infrastructure (having in mind ’basic’ services such as, for example, those provided by road, electricity and water supply networks), we did not take into account other types of public capital expenditure (for example, those on ICT infrastructures or on R&D investment) or government incentives aiming to stimulate specific private investments (for instance those in the green economy). These types of public expenditure have, conceivably, a strong impact on the decisions concerning complementary private investment. Moreover, they could have a further positive effect on investment by accelerating the rate of obsolescence of the existing capital stock. Finally, we did not consider the effects of government policies contributing to human capital formation.
81In order to verify that the parameters a_{1} and ? are crucial to determine when G_{2} is more effective than G_{1} in order to enhance growth, we proceed as follows:
82Consider the derivatives of g^{*} with respect to G_{1} and G_{2}
83Therefore:
85 is a sufficient condition in order to have .
86More generally, for
87where and where A, B, C are defined as in the main text in section 3.
88We have that:
90If then . The equation has two real roots, one negative and one positive. We indicate the positive root with . We have that for . It follows that for .
91If then . If it is also true that ? > ? _{B}, then . It follows that the equation has either two complex conjugate roots or two real negative roots. Therefore, and for any G_{2} > 0.
92Instead when ? < ?_{B}, we need to distinguish between two subcases:
93For ?_{?} < ? < ?_{B}, and for G_{2} > 0, where ?_{?} is the positive value of ? that corresponds to a zero discriminant, ? = 0, for the equation , where gives and where
95b) For ? < ?_{?}, and for and , where and are the two real positive roots of the equation .
96In order to prove proposition 2, we proceed as follows:
97By differentiating g2, we obtain
99where and
101We have that g2? ? (<)0 for .
102We know that and (since s_{?}?a_{0} > ?). Moreover
104When , it follows that . Therefore, g2? > 0 always.
105Instead, from it follows that . In this case, the equation admits two real solutions, and , with . Moreover, since, , the positive solution minimizes growth.
106In summary, when for any G_{2} ? 0; when , g2? < 0 for and g2? > 0 for , where corresponds to a minimum for g^{*} since g2? > 0 for .
107In order to verify if , we need to study the shape of g1. We note that, since g2? = – g1?, the sign of g1? depends on , where , and .
108We know that and that . We need to determine the sign of .
109We proceed as follows:
110When and therefore . The term is always negative and g1? < 0 or the equation , admits two real positive solutions and . However, given that g2? > 0 for 0 ? G_{2} ? ? _{f}, by symmetry g1? < 0 for 0 ? G_{1} ? ? _{f}.
111Consequently, it must be and .
112When things are more complicated given that when can take any sign (i.e.,). The sign of is crucial to determine the sign of the two real solutions of .
113Moreover,
 when ; considering that and that and , it necessarily follows and ;
 when ; it follows that ,
114After defining and
115, we disentangle the various cases
116as follows:
117If always;
118If for and for or , where and are values of a_{1} which solve and enjoy the following properties:
119 decreases (increases) when ? is increased, with
120 at ;
123Moreover, given the shape of the relationship between and ? and since at , it follows that always.
124Finally, after labelling and , we have that g2(0) ? (<)g1(0) and, by symmetry, g2(? _{f} ) ? (>)g1(? _{f}) for
125 and , with
127and g2(0) < g1(0) and by symmetry, g2(? _{f} ) ? (>)g1(? _{f}) for
129Therefore, for 0 ? G_{1} ? ? _{f} and 0 ? G_{2} ? ? _{f}, we have that:
 For any ? and is increasing and g1 is decreasing; g2(0) ? g1(0) [see Figures 7(a) and 7(b)];
 For and , g2 is decreasing for and increasing for and g1 is decreasing for and increasing for [see Figure 7(c)];
 For and , g2 is decreasing for and increasing for and g1 is decreasing for and increasing for [see Figure 7(d)];
 For and , g2 is decreasing for and increasing for and g1 is decreasing for and increasing for [see Figure 7(e)];
 For and g2 is decreasing for and increasing for and g1 is decreasing for and increasing for [see Figure 7(f)];
 For and g2 is decreasing and g1 is increasing; g2(0) > g1(0) [see Figure 7(g)];
 For and g2 is decreasing for and increasing for and g1 is decreasing for and increasing for [see Figure 7(h)].
Notes

[1]
(Corresponding author) Pasquale Commendatore, Department of Economics, University of Naples ’Federico II’, Via Mezzocannone 16, I80134, Naples, Italy. Email commenda@unina.it

[2]
See among the others the contributions by Del Monte (1975), Rowthorn (1981), Dutt (1984, 1990), Amadeo (1986); and, for reviews, Blecker (2002) and Commendatore et al. (2003).

[3]
See Barro (1990), Devarajan et al. (1996), Rivas (2003), Agénor (2006) and Irmen and Kuehnel (2009).

[4]
Both types of public expenditure are expressed in terms of capital in order to facilitate the comparison. Moreover, the analysis is considerably simplified.

[5]
But see Commendatore et al. (2010 and 2011), where government expenditure is allowed to affect income distribution.

[6]
In order to better understand this final case, we note that and that . For initial values of is large in absolute value, therefore and . As G_{2} is increased, decreases in absolute value until when and .

[7]
So we could label these lines isoutilisation and isogrowth curves.

[8]
In order to plot figure 4, for the other parameters we have chosen the following values: ? = 0.02, ? = 0.15, s_{?} = 0.8, ? = 0.5, a_{0} = 0.5; moreover, the two types of public expenditure in terms of the capital stock have been varied within the intervals: 0 ? G_{1} ? 0.25 and 0 ? G_{2} ? 0.25.

[9]
This point can be further illustrated considering that the derivative u2? can be expressed as the difference between the derivatives (10) and (8), which holds for the case of a variable government size, evaluated at ? = ? _{f} :The positive effect on the equilibrium capacity utilisation of the reduction of savings induced by the increase in G_{2} is exactly offset by the reduction in G_{1}.

[10]
Devarajan et al. (1996) and Rivas (2003) study also the effects of the composition of public expenditure on growth in the context of mainstream endogenous growth models. Deverajan et al. (1996) consider the case of two different types of public expenditure that both affect productivity. Rivas (2003), instead, explores the case of different ta1 rates attached to wages and profits.