1 – Introduction

1The neo-Kaleckian 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 long-run growth of the composition of public expenditure. As noticed by Pressman (1994), Keynes was well-aware 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 long-run 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 neo-Kaleckian 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 long-run 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 neo-Kaleckian 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 single-good 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:

6

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, mark-up 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 neo-Kaleckian growth and distribution models:

8

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:

11

12where ? represents taxation in terms of the capital stock, G₁ represents government consumption expenditures in terms of the capital stock and G₂ 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, G₂, may enhance input productivity. This implies a reduction in the capital input requirement:

14

15where a’(G₂) > 0 and a’’ (G₂) ? 0. For the function a(G₂) we choose a linear form:

16

17with a₀ > 0 and a₁ > 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:

20

21

22

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 neo-Kaleckian 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 G₂ 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 G₁ and G₂ 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:

26

27In order to have definite and positive solutions, we need to assume s_?? (a₀ + a₁G₂) > ? for any G₂ ? 0.

28The impact of a change of G₁ on the equilibrium solutions is described by the following derivatives:

29

30

31That is, the impact of G₁ both on equilibrium capacity utilisation and equilibrium growth is positive (see Figure 1).

Figure 1

32These results confirm to the so-called “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 G₂ on the equilibrium degree of capacity utilisation is:

34

35where for or, alternatively, for .

36That is, the impact of G₂ 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 a₁ 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 so-called “paradox of costs”. On the other hand, when a₁ 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.

Figure 2

37The impact of a change of G₂ on the equilibrium rate of growth is:

38 for

39where and

40

41As previously seen for the degree of capacity utilisation, the impact of G₂ on equilibrium growth also depends crucially on the coefficients a₁ and ? as the following proposition shows:

42Proposition 1. The effect of G₂ on the rate of capital accumulation, when the government budget is balanced and the government size is variable, depends on the coefficients a₁ and ? as follows:

1. when or for any G₂ ? 0 (see Figure 3(a)) ;
2. when a₁ > ã₁ or for and for , where solves AG² + BG + C = 0 and minimises g^* with respect to G₂ [See Figure 3(b)].

43Therefore for a₁ ? ã₁ or , AG² + BG + C > 0 and ?g^* / ?G₂ > 0.

44Instead, for a₁ > ã₁ 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

45

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.

Figure 3

48The results concerning the effects of G₂ 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 G₂, 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 a₁ 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 a₁ 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 G₂ on capital accumulation, g^* rises. Finally, when a₁ is sufficiently large or ? is sufficiently small, for an initial range of values of G₂, i.e. for , the negative effect on the degree of capacity utilisation overcomes the positive effects on private investment of a rise in G₂, u^* and g^* both decrease. However, when G₂ is above , the crowding in effect is so large to prevail over the negative effect, g^* increases whereas u^* decreases. [6]

50The parameters a₁ and ? are also crucial to determine when G₂ is more effective than G₁ to intensify capital utilisation or to enhance growth (see the Appendix 1). For instance, the condition

51

52gives and, consequently, it is sufficient in order to have .

53Finally, note that the overall effect of G₂ on u^* and g^* crucially depends on the value of G₁, as and ã₁ () are both negative (positive) functions of G₁. The level of public consumption expenditure is also crucial in determining when G₂ is more effective than G₁ for enhancing capacity utilisation and growth. The effects of a simultaneous change of G₁ and G₂ 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 G₁ and G₂ 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 ? / a₁. In panels 4(a) and 4(b), which are plotted for ? / a₁ = 5, [8] a simultaneous increase of G₁ and G₂ 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 G₂ more effective than G₁ (the slope of the contour lines is flatter than the 45° line). In Panel 4(c) and 4(d), which are plotted for ? / a₁ = 0.25, a simultaneous increase of G₁ and G₂ 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 ? / a₁ effect of both types of public expenditure still goes in the same direction, however now G₁ is more effective than G₂ (the slope of the contour lines is steeper than the 45° line). Finally, in Panels 4(e) and 4(f), which are plotted for ? / a₁ = 0.05, the effects of G₁ and G₂ do not always go in the same direction, i.e. the effect of G₁ is always positive whereas the effect of G₂ 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 G₁ is sufficiently large, to higher values of G₂ 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 G₁ is sufficiently large, to higher values of G₂ 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 G₁ and G₂ 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 G₂ has to be compensated by the positive effect of a simultaneous increase in G.

Figure 4

Figure 5

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 G₁ = ? _f – G₂ or that G₂ = ? _f – G₁. The equilibrium solutions for the degree of capacity utilisation and the rate of growth become:

55

56and

57

58To ascertain what are the effects of alternative combinations of G₁ and G₂ on the equilibrium solutions, we introduce the following functions

59 and

60

61for the degree of capacity utilisation and

62 and

63

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

68

69That is, both functions are monotonic. Moreover, from ? / a₁ ? (<)h it follows u2(0) ? (>)u1(0) and, by symmetry, u₂ (? _f ) ? (<)u₁ (? _f).

70That is, as shown in the Figure 6 below, when a) ? /a₁ > h, G₂ has a positive effect and G₁ a negative effect on instead, when ? /a₁ < h the opposite holds, i.e. G₂ has a negative effect and G₁ a positive effect on .

Figure 6

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 G₂, which has a negative impact. The other is the positive effect, induced by an increase in G₂, on the rate of capital accumulation. When ? / a₁ relatively large, the second effect exceeds the first: the degree of capacity utilisation increases when the public expenditure composition changes in favour of G₂; whereas, when ? / a₁ relatively small, the first effect exceeds the second: the degree of capacity utilisation increases when the public expenditure composition changes in favour of G₁.

72Finally, we note also that for ? = ?  _f, it is possible to determine the maximum degree of capacity utilization, which is

73

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 G₁ or G₂ when the government budget is balanced and the government size is fixed) on the rate of capital accumulation crucially depends on the ratio ? / a₁ as follows: fir a sufficiently large value of ? / a₁ a shift in the composition of public expenditure in favour of G₂ enhances growth; whereas, for smaller values of ? / a₁ various possibilities may occur: 1) a shift in favour of G₂ can slow down growth initially and increase growth subsequently. Moreover, the highest rate of growth is obtained at G₂ = ? _f ; 2) a shift in favour of G₂ still slows down growth initially and increases growth subsequently. However, in this case, the highest rate of growth is obtained at G₂ = 0; 3) an increase in G₂ 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 G₂, on the average productivity of capital and on the investment decisions. The various possibilities are shown in figure 7, where on the horizontal axis G₁ and G₂ are represented, both varying from 0 to ? _f (G₂ is increasing from left to right and G₁ 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 ? / a₁. 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. G₂ is always more effective than G₁. 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 G₂ is relatively small (G₁ is relatively large) and the positive effect dominates the negative one when G₂ is relatively large (G₁ 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 ? / a₁, the composition of public expenditure that gives the highest possible rate of growth corresponds to G₂ ⁼? _f and G₁ = 0. Instead, for relatively low values of ? / a₁ composition of public expenditure that gives the highest possible rate of growth corresponds to G₂ = 0 and G_l = ? _f.

Figure 7

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 neo-Kaleckian model of growth and distribution. Our conclusions do not always agree with those drawn by the mainstream theory of endogenous growth, the so-called 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 after-tax marginal product of capital, in the after-tax 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:

1. considering public investment expenditure, at low values of G₂ the positive effect on productivity dominates the negative effect of taxation: the rate of growth of the economy icreases. As G₂ increases the effect of taxation becomes stronger, until the rate of growth reaches a peak. After further increasing G₂, the rate of growth decreases as the negative effect dominates the positive one (i.e. the relationship between G₂ and g^* is bell-shaped);
2. considering public consumption expenditure, a rise in G₁ always determines a reduction of the equilibrium rate of growth;
3. it follows that when the effect of the size of the government sector is neutralised (when to a rise in G₂ corresponds a simultaneous reduction of G₁); a rise in G₂ always determines an increase in the rate of growth;
4. 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.