Growth, Income Distribution and Public Debt

Transcription

1 Growth, Inome Distribution and Publi Debt A Post Keynesian Approah João Basilio Pereima Neto José Luis Oreiro Abstrat: The objetive of this paper is to evaluate the long-run impat of hanges in fisal poliy and inome distribution over the degree of apaity utilization, within the ontext of a high publi debt so that risk premium, paid by the government to the apitalists/inome earners, is an inreasing funtion of the level of publi debt. In order to do that, a maro dynami model of Post Keynesian inspiration is presented, whereby (i) apitalists obtain inome from either the profits ahieved from the produtive use of existing apital stok or from interest derived from the aquisition of government bonds; and (ii) the rate of return (required) in government bonds is an inreasing funtion of the degree of indebtedness of publi setor. The model shows that the eonomy does not follow a linear response to the fisal shoks in the sense that an expansionary fisal poliy ould have positive or negative effets over the level of apaity utilization. It is also shows that the effet of hanges in distribution of inome over the level apaity utilization is dependent on the level of publi debt. If the eonomy is operating with a high level of publi debt, an inrease in profit share will result in an inrease in the degree of apaity utilization, thus defining a profit-led aumulation regime in the long run. Key-words: Eonomi Growth, Inome Distribution, Fisal Poliy, Publi Debt, Risk Premium. JEL Classifiation: E12 July, 2008 Assistant Professor of Eonomis at Federal University of Parana (UFPR), Curitiba, Brazil. joaobasilio@ufpr.br. Assoiate Professor of Eonomi at University of Brasilia (UNB), Brazil. jloreiro@terra.om.br. Web-page: 1

2 1 Introdution This paper presents a dynami model of eonomi growth, in whih the degree of indebtedness of the publi setor plays an important role in determining the growth dynamis, as it an modify the nature of aumulation regime. The eonomi theory has traditionally emphasized the pro-ylial role of fisal defiits, established through mehanisms suh as the Keynesian multiplier, even in the presene of some rowding out effets that may attenuate the impats of an expansionary fisal poliy. The debate surrounding the rowding out effets between monetarists and Keynesians, whih ourred during the 1970s, led to divergent onlusions regarding the effets of an expansionary fisal poliy that is finaned by the emission of publi debt and the effets it has on the level of employment in the short and in the long-run. Sine it is widely understood, that the monetarist position, defended espeially by Friedman (1972), suggested that a fisal expansion finaned by the emission of publi bonds would generate a so strong wealth effet in demand of money, inreasing interest rates, so that aggregate demand and, onsequently, the level of real output, would remain more or less onstant, resulting in the thesis of an ineffiay in the fisal poliy in the long-term. The Keynesian position - defended by Blinder and Solow (1973) - was that wealth effets are present not only in the LM urve, but also in the IS urve and that the latter are stronger than the in suh a way that the final result of a fisal expansion would produe an inrease in the aggregate demand and hene the level of real output. The issues regarding the long-run effets of publi debt over the real eonomy was reently revisited by You and Dutt (1996) in the ontext of a Post Keynesian model of growth and inome distribution. The authors onluded that an inrease in the level of publi debt (as a ratio to apital stok) in the long-term will ause an inrease in the rate of growth of real output and, at the same time, a derease in the wage share. However, the model ignores the sensibility of investment to interest rate, sine this variable is exogenously determined, so that an important rowding out mehanism is not taken into aount in this model. The hypothesis regarding exogeneity in the interest rate implies that the money supply is ompletely horizontal whih suggests an aeptane of the horizontalist theory of demand for money, based on Kaldor and Moore (1988). However, the exogeneity of the interest rate does not seem to be a reasonable hypothesis, espeially in the ontext of monetary eonomies suh as the modern apitalist eonomies, sine they possesses more than two assets for whih private wealth an be aumulated. In fat, the issuing of bonds that provide interest payments by the private and publi setor, used to finane their onsumption and investment expenditures, rules out the equality between interest rate and 2

3 profit rate, an assumption usually made in ontext of models of balaned growth 2. Given the existene of finanial assets, it is very likely that the behavior of the interest rate within the eonomy beomes influened not only by the money demand and supply, as it is traditionally supposed, but also by the subjetive evaluations about the risk of apital or inome losses of these assets. Kaleki (1971) had already drawn attention to a theory of interest rate that would take into aount the so alled priniple of inreased finanial risk, whih is nowadays fully onsidered by the literature on orporate finane (Brealey and Myers, 1996). This priniple establishes that the investors and reditors demand a higher interest rate as the degrees of leverage inreases. In the model developed below, the government will be the only agent that issues debt bonds, making it possible to analyze not only the maroeonomi dynamis that results from publi debt, but also the effetiveness of fisal poliies. The inlusion of these elements in a model of growth and distribution of inome shows that the eonomy an present differentiated regimes of aumulation depending on the degree of leverage or indebtedness in whih it is operating. This introdues the possibility, whih has not been onsidered by the Post Keynesian models of growth, suh that the long-run effets on the degree of indebtedness depend on the level of publi debt. In fat, You and Dutt (1996) reahes the onlusion that the degree of apaity utilization in short-term and the rate of growth in the longterm are always positively orrelated with the level of publi debt (as a ratio to apital stok). However, reent studies 3 have demonstrated that the maroeonomi dynamis does not respond in a linear way to expansionary and ontrationary fisal poliy. Many of the works have found empirial evidene of nonlinear responses in the eonomy to the fisal measures that affet the relation between debt and GNP. They also found that, amongst the various hannels, the effet it had on the expeted interest rates and the effet of inome and wealth on onsumption, together with the investment and savings made by the agents were extremely important. In the following, we will present a model of growth and inome distribution that inludes these nonlinear mehanisms. In setion 2, the basi struture for the theoretial model is presented. Setion 3 is devoted to the attainment of the short-run equilibrium of the model. Setion 4 presents the long-run equilibrium and the analysis of stability. Setion 5 presents the long-run effets of hange on the fisal poliy, while setion 6 analyses the long-run effets of hanges in the funtional distribution of inome based over the degree of apaity utilization. Finally, setion 7 summarizes the paper s onlusions. 2 This priniple is found, for example, in the redued version of the Cambridge equation attributed to Pasinetti (1974), a result obtained by the hypothesis that there is only one asset in the eonomy, whih is physial apital. 3 Regarding this, see Giavazzi and Pagano (1990), Alesina and Perotti (1995,1997), IMF (1996), MDemott and Wesott (1996), Alesina and Ardagna (1998), Alesina et al. (1999) and Blanhard and Perotti (1999). 3

4 2. The model struture Let us onsider a losed eonomy, whih has government and is a mono-produer. Without an external setor, the eonomy produes only one produt, whih is alloated for both onsumption and investment. There are only two fators of prodution, apital (K) and labour (L), whih are ombined in fixed ratios, in a sense that the prodution funtion, in the absene of tehnologial progress, an be expressed by a Leontieff type in the form: X = min [K,L/a] (1) Where: X is the prodution and inome level and a is the unitary requirement of labour. In this ase, the amount of labor employed is a diret funtion of the prodution level and an be expressed by the equation: L = ax (2). The total inome (X) generated during the produtive proess is distributed between wages and profits suh as speified in the equation (3): X W = L + rk P (3) Where: W/P represents the real wage and r is the rate of return on apital. Following the tradition of a lassial politial eonomy and Marx, the interest rate is not onsidered as part of the inome generated during the produtive proess. In the model presented here, the hypothesis is easily justified by the fat that the interest rate is a simple transferene of tax resoures (urrent and future) from government to the owners of publi bonds (the apitalists), not being part of the "added value" generated by the eonomy in a ertain period of time. Dividing (3) for K and defining the real wage as V = W/P 4, the degree of produtive apaity utilization as u = X/K, the profit share as m = rk/x 5, we an then express (3) as follows: 4 And by the definition (2), L = a.x 5 As K/X is the inverse of the degree of use of the produtive apaity, it is possible either to express the partiipation of the profits in the inome as m = r/u or to define de profit rate as r = mu. 4

5 u = Va.u + mu (3a) Where: isolating m (partiipation of the profits in the inome) and Va (partiipation of the wages in the inome), we obtain by inome side a result as follows: m = 1-Va Va = 1 m (3b) (3). Furthermore, we an assume that the real wage V is determined by the subsistene level of the work fore, so that it an be onsidered an exogenous variable of the model and represented by V. Sine the eonomy under srutiny is deprived of tehnologial progress, the parameter a an also be onsidered as a onstant. Thus, the partiipation of profits (and of the wages) in the inome is determined by the following equation: m = 1 Va (3d). With regards to the expenditures, the inome of the eonomy is distributed between onsumption (C), investment (I) and publi expenses (G): X = C + I + G (4). The total onsumption is determined by the onsumption of workers and apitalists. Following Kaleki (1954), Kaldor (1956) and Robinson (1956, 1962), we assume that the workers spend all their inome on onsumption; while the apitalists save a onstant fration of their inome, whih is obtained either in the form of profits over the stok of existing apital or interests on bonds that they own. Finally, the government harges a tax τ only on the inome that is obtained through profits and interests. The wages are exempt from taxation. Thus, the onsumption funtion is given by: C = VaX + (1 s )(1 )( rk i D τ + ) P (5) 6 6 Despite the fat that the interests are a simple transferene of urrent and future tax revenues from the government to the apitalists, the apitalists onsider the inome obtained from the interests as part of their available inome for finane their onsumption expenses. This is the ase beause we assume that the government is a typial Ponzi agent; so part of the government expenses with the payment of interest is funding with the emission of a new debt by the government. Thus, part of the interest paid to the apitalists today is, indeed, funded by the taxes that will be harged from the apitalists on the future. Assuming that the urrent generation of apitalists does not are about the well-being of the future generations and/or is not able to foresee the moment at whih the government will have to inrease the taxes for the payment of its debt; it follows that the inome that they obtain in the form of interests will be onsidered as 5

6 where: τ is the tax rate, s is the propensity to save by the apitalists, r is the rate of profit on apital, i is the nominal interest rate and D is the nominal stok of debt and and P is the level of pries. The speifiation for the investment funtion follows the onlusions by Steindl (1952), Spene (1977) and Cowling (1982), and hene we assume that the deision regarding investment by the firms depends, amongst other things, on the rate of produtive apaity utilization, due to the strategy of reating barriers for potential new entrants. Thus, in an oligopoly market, the firms keep a surplus apaity level as a means of reating rapidly to demand hanges, whih prevents stimulating the entrane of new firms. Besides that, investment in fixed apital is also negatively orrelated to the real interest rate, suh as in Keynes (1936), and therefore the higher the interest rate, the lower the amount of investment in the eonomy. Thus, the investment funtion assumes the following form: I = I α + βx - φ(i π).k (6). Dividing it by K, we obtain the equation for the growth rate of apital: g = Ι/Κ = α + βu φ(i π) (6a) Where: g is the growth rate of apital, α is the autonomous investment as a ratio of the apital stok, β is a parameter that measures the sensitivity of investment to the produtive apaity utilization, φ measures the sensitivity of investment to the real interest rate, i is the nominal interest rate and π is the inflation rate. Differing from previous works in the Post Keynesian tradition 7, we hoose to resue the role played by the interest rate as an expliative variable of investment in our model. In fat, many models in the Post Keynesian tradition have assumed an exogenous and onstant interest rate. This has allowed them to introdue a profit rate as an expliative variable based on the behavior determined by the deisions of firm s investments. A reent example of this approah is given by part of their available inome, affeting then their expenses of onsumption. As a orollary of this argument, it follows that the form of finaning the government expenses has an effet on the deisions about expenses of the eonomi agents, thus the Riardian equivalene is not valid in the model under onsideration. 7 On that, we differ from other ways to represent the investment funtion. Aording to Robinson (1956, 1962), Kaleki (1971), Rowthorn (1981) and Dutt (1984, 1990), the investment positively depends on the profit rate. For Bhaduri and Marglin (1990), the investment monotonially depends on profit share in inome. And, more reently, Lima (1998) makes the investment to depend not linearly, but in a quadrati form, on the wage partiipation in the inome. 6

7 You and Dutt (1996). In their model, they assume an exogenous and onstant interest rate and take the profit rate as an expliative variable of the investment funtion. The hypothesis of an exogenous and onstant interest rate is endorsed by the so alled horizontalist view of the monetary endogeneity, as developed by Kaldor (1982) and Moore (1988). Aording to this approah, the ommerial banks meet all the demands for redit by means of a onstant interest rate, whih is determined through a fixed mark-up over the ost of the resoures obtained in the inter-banking market (Rousseas, 1992, p.85). However, the horizontalist approah to money and redit has been ritiized by Post Keynesians authors. The main ritiism is that it ignores the ommerial banks liquidity preferene (Carvalho, 2005, p ). Indeed, if the banks were to provide all the redit demanded by a onstant interest rate, then, in as far as money and redit supply would inrease, the banks themselves would have less liquidity, sine the relationship between reserves and demand deposits would be redued, whih then inreases the risk of illiquidity for banks. If they had a preferene for liquidity, as any other agent, then they would only aept an inrease in the illiquidity risk, if they were ompensated through higher profitability. Hene, an inrease in the interest rate that they harged on loans would be reommended. Another boundary within the Kaldor-Moore approah is that it does not onsider the issue regarding the limits of indebtedness. With regards to these limits, Kaleki (1954) argues that the ompanies with a higher degree of leverage have greater osts on apital following extreme inreases in their liabilities, onsequently, a greater ompromise towards a short run solveny. In an extreme ase, the ompanies may be unable to get new loans. A simple way to formalize this argument, suh as in Bresser and Nakano (2002) and Oreiro (2002, 2004), is to assume that the interest rate paid on the debt is positively influened by the level of indebtedness of the ompany, whih makes it an endogenous variable. Thus, we an determine the interest rate on bonds aording to the following equation: i = ρ ρ > 0 (7) Where ρ is a fixed and positive parameter and is the degree of publi indebtedness that an be defined as: = D P. K (8) 8. 8 Atually, the degree of indebtedness should be expressed by a relation between the real stok of debt and gross domesti produt (D/PX). For the purpose of modeling, we are using as proxy the definition of the indebtedness degree as the relation between real debt and apital stok, given that the rate of eonomi growth is gotten from the relation investment (I) and apital supply (K). 7

8 In the eonomy under onsideration, we assume that the firms determine the pries of their produts based on a fixed mark-up over the diret unitary osts of prodution. The mark-up effetively pratied by the firms may however be smaller than the mark-up that the ompanies would wish for. The desired mark-up is determined by the long-run strategi deisions of the ompanies (f. Kaleki, 1954, p.17). The effetive mark-up must be seen as a ommitment solution between the desired mark-up and the existing ompetition onditions within the eonomy (f. Possas and Dwek, 2005, p. 12); that is to say, that the ompanies an fix, in the short-run, a markup that is smaller than the desired one with an aim to, for example, attain a greater market-share. In this ontext, the inflation is derived from the attempt of firms to line up the effetive mark-up with the desired mark-up. Thus, if the effetive mark-up is lower than the desired mark-up; then the firms must inrease the pries of their produts in the long-run as a strategy to reah the desired mark-up. As the profit share is given by z, where z is the effetive rate of the mark-up; 1 + z then the firms will inrease the pries harged for their produts whenever the desired partiipation of the profits in the inome (suh as determined by the desired rate of mark-up) is greater than the effetive partiipation of the profits in the inome (suh as determined by the effetive rate of markup). That is: P f π = = ε ( m m) P (9) Where: m f is the partiipation of the profits in the inome that is desired by the apitalists (by the firms), and m the effetive partiipation. 3. The behaviour of the eonomy in the short-run In the short-run, the publi debt is onsidered to be a onstant ratio of the apital stok. As the real wage is onstant and exogenous, it follows that the profit share in inome is also onstant, suggesting a fixed mark-up rate. Thus, the prodution level is determined by the effetive demand, whih is given by the equations (5) to (9). Replaing these equations in (4), dividing the resultant expression by K, and defining u = X/K and γ = G/K, we get the following equation: f [((1 s )(1 τ ) ρ φρ ) + α + φε( m m) + γ ] 1 2 u = u = ) λ( m) (10) 8

9 Where: u is the degree of the produtive apaity utilization of short run equilibrium and λ( m) = { 1 ( 1 s )( 1 τ ) m β} is assumed to be bigger than zero 9. [ ] Substituting (7) and (10) in (6a), we obtain the expression for growth rate of apital supply within the short-run equilibrium of the eonomy under onsideration. This is given by: g f = α + φε( m m) + βu φρ (11). Based on the equations (10) e (11), we an appraise the effets of exogenous hanges in the funtional distribution of inome, in government expenses and publi setor level of indebtedness over the degree of apaity utilization and on the rate of growth in the short-run equilibrium. Differentiating (10) e (11) to m, we get the following expressions: u g [ = 1 { φε + 1 (1 s )(1 ) τ u ]} < 0 (12a) λ( m) u = φε + β < 0 (12b) The expression (12a) shows that an inrease in the profit share (m) will ause a redution in the degree of apaity utilization in the short-run equilibrium. This is beause an inrease in the partiipation of profits in inome will redue the effetive demand for two mehanisms. The first mehanism is the traditional Kalekian one, that is: a redistribution of inome in favour of apitalists whih will redue the aggregate onsumption expenses sine workers propensity to onsume is bigger than the apitalists. The seond mehanism is kind of a Mundell-Tobin effet in the ontext of a model of growth and distribution. An inrease in the atual profit share will redue the differene between this variable and the desired partiipation of profits on inome by apitalists, whih leads to a redution in the inflation rate. For a given nominal interest rate, there will be an inrease in the real interest rate, whih will lead the apitalists to invest less, thus reduing the effetive demand and a degree of use in produtive apaity. 9 This hypothesis is neessary to guarantee the stability of the short-run equilibrium position. In eonomi terms, this hypothesis establishes that the sensitivity of the apitalists saving to a variation of the degree of use of the produtive apaity is bigger than the sensitivity of the investment to hanges in the utilization degree. It is worth emphasizing that this hypothesis is usually adopted in the ontext of Post Keynesian models of growth and distribution. 9

10 The expression (12b) shows that in an eonomy, suh as that being onsidered, a regime of aumulation of wage-led type prevails, sine a redution in the partiipation of profits in inome (that is, an inrease in wage share) will result in an inrease in the rate of growth of apital stok. The short-run effets of a fisal expansion, that is, an inrease of the government expenses as a ratio of apital stok, an be appraised by the following expressions: u = γ 1 λ( m) > 0 (13a) g β = γ λ( m) > 0 (13b) The equations above show that a fisal expansion will produe an inrease of the degree of apaity utilization and of the rate of growth of the apital stok of the short-run equilibrium in the eonomy under onsideration, like in the traditional Keynesian models. Finally, the effets of an inrease of the publi setor indebtedness as a ratio of the apital stok an be appraised by the following expressions: u = 2(1 s )(1 τ ) ρ φρ λ( m) (14a) g β = λ( m ) { 2( 1 s )( 1 τ ) ρ ρφ} ρ (14b) The signs of expressions (14a) e (14b) are ambiguous, depending on the value of the publi setor indebtedness as a ratio of the apital stok. Based on (14a), we an onlude that the sign of this partial derivative will be positive if the following ondition is satisfied: φ > = ; while it will be negative, on the ontrary. In this ontext, the relation 2(1 )(1 τ ) s between the degree of use of the produtive apaity and the indebtedness of the publi setor as a ratio of the apital supply is nonlinear, being able to be visualized by the Figure 1 below: 10

11 u Figure 1 Regimens of Indebtedness Regime of Low Indebtedness b i Regime of High Indebtedness In figure 1, we observe that for low levels of indebtedness of publi setor as a ratio of apital stok, an inrease of will redue the short-run equilibrium level of apaity utilization; while for high levels of indebtedness, the inverse effet ours. This results from the fat that variations of generate effets of positive and negative signs over aggregate demand. On the one hand, an inrease of disourages the aggregate demand in as far as it generates an inrease on the interest rate paid on the publi bonds; hene, inreasing the opportunity of ost on the investment in fixed apital. On the other hand, the inrease of has a wealth effet and a positive inome effet on the onsumption for the apitalists, sine the inome interests are an important part of the available inome for the apitalists. In this ontext, the figure 1 shows that the first effet tends to be stronger than the seond one for low values of the level of indebtedness of the publi setor; while for higher variable values, the seond effet tends to be stronger than the first one. Finally, we observe in the expression (14b) that the sign of the partial derivative will be ( β + λ) φ positive if the following ondition is satisfied: > = β 2(1 )(1 τ ) s, while on the ontrary it will show negative. 4. The behaviour of the eonomy in the long-run. In the long-run, the indebtedness of the publi setor as a ratio to apital stok is an endogenous variable; being affeted by the primary defiit of the government, by the rate of growth of apital stok and by the inflation rate. Differentiating to time from (8), we obtain the following expression: d D = & dt PK ( π + g) (15). 11

12 equation: The debt of the publi setor hanges with time, based on the following differential ( G T ) + i D D & = P. (16). The first part of the equation (16) represents the primary defiit of the government, therefore emphasizing differenes between the expenses and tax revenues of the government, apart from the payment of interests over the existing debt. The seond part of represents the finanial harges (interests) on the total debt of the publi setor. The real value of the taxes harged by the government is determined by the following equation: ( mu i ) K T = τ + (17). Replaing (17) with (16) and the resultant of (15) and then exeuting some mathematial transformations, we have the following dynami equation to : d = γ + ( 1 τ ) i τ mu ( π + g) (18). dt Substituting (9), (10) and (11) in (18), we get the following expression: β = t λ( m ) 1 λ( m ) τm γ λ( m ) 1 λ( m ) 3 [( 1 s )( 1 τ ) ρ] [ τ m( 1 s )( 1 τ ) ρ + φρβ] [( 1 τ ) + φρ] f [ β( α + γ + φε( m m) τmρφ) ] + α + ( 1+ φ ) ε( m m ) + f [ α + γ + φε( m m )] ( 19) f 2 The equation (19) is, indeed, a third degree polynomial differential equation, whih an be rewritten as follows: d dt 2 3 = A + B + C + D (20) 12

13 Where: β A λ( m ) 1 B λ( m ) C 1 λ( m ) τm D γ λ( m ) [( 1 s )( 1 τ ) ρ] [ τ m( 1 s )( 1 τ ) ρ + φρβ ] [( 1 τ ) + φρ] f [ β ( α + γ + φε( m m) τmρφ) ] f [ α + γ + φε( m m )] + α + ( 1+ φ ) ε( m f m ) (21a) (21b) (21) (21d). In the long-run equilibrium, the publi debt as a ratio of the apital stok will be onstant though time, that is: d dt = 0. Thus, the equation (20) is redued to a polynomial of the third degree type: 3 2 A + B + C + D = 0 (22) The roots of this polynomial equation are the values of long-term equilibrium of publi debt as a ratio of the apital stok. As it is a polynomial of the third degree, we know that three roots satisfy the ited equation. However, we are only interested in the real positive roots, given that a negative root would denote a situation in whih the government was a net reditor of the private setor. Therefore, the possibility exists for only three distint situations regarding the hypothesis of three different real roots: the equation presents one, two or three positive roots. If two equal roots our, the graph will be interepted on the horizontal axis in only two points. Graphially, we would have the following representations 10, with the hypothesis that the equation would be suh that there are three distintive real roots, as shown in figure 2: 10 Perhaps it is important to emphasize that the sine format of the urve depends on the existene of opposing signs between the parameter A and B, with A < 0 and B > 0. The bigger the value of B, the greater is the amplitude of the undulation.. In eonomi terms, more aented undulations favor the ourrene of a bigger interval between the roots as well as inrease the domain of the debt/apital relation in whih stable equilibria are observed (points where the urve uts the horizontal axis). 13

14 Figure 2 Distint Real Roots of the Equation of the Degree of Indebtedness (a) One positive root (b) Two positive roots () Three positive roots In example (a), the positive root is given by 3 ; although it is stable, it is the only possibility of equilibrium with regards to the debt/apital relationship being greater than zero. The example (b) shows two points of equilibrium, the smaller one ( 2 ) being unstable, while the bigger one ( 3 ) being stable. Furthermore, the stable level of indebtedness ( 3 ) will be situated in a higher level in relation to example (a). Finally the situation () enables the eonomy to find three equilibrium points, with the stable points ourring in a low level and another high one for indebtedness. Furthermore, we an inlude some additional onditions to the parameters regarding the attainment of a onfiguration that allows us to reprodue the situation (). Based on the Theorem of the Deomposition and Girard Relationships, we find that the roots of a third degree polynomial obey the following properties in the parameters: B + = A C = A (23a) (23b) D = A (23). Moreover, it is possible to establish the following onditions with regard to the parameters, ourring in situations (a), (b) or () represented above, knowing beforehand that the parameter A is definitely negative: 14

15 Table 1 Neessary Conditions Parameter 1 Positive Root 2 Positive Roots 3 Positive Roots (a) (b) () A ( - ) ( - ) ( - ) ( - ) ( - ) ( - ) B ( + ) ( - ) ( - ) ( - ) ( + ) ( + ) C ( + ) ( + ) ( - ) ( + ) ( - ) ( - ) D ( + ) ( + ) ( + ) ( - ) ( - ) ( + ) In the expression (20) above, only the sign of the oeffiient A is known for ertain, sine we know that in the equation (10) that λ (m) > 0 and therefore A < 0. All the other oeffiients have ambiguous signs. To solve the ambiguity we must impose additional restritions to the values of the parameters. In this ontext, the oeffiient B is positive if the following ondition is satisfied: [ m( 1 s )( 1 τ ) + φβ ] [( 1 τ ) + φρ] ρ τ B > 0 λ ( m ) > = λ (24) 11. The oeffiient C requires a more areful analysis, though it has only a positive term, whih ould lead us to onlude that C is possibly negative. As the expression of the parameter is the same for all the domain of the funtion, we an use some of the equilibrium points in whih m f = m. Thus, the expression an be rewritten as follows: 1 C < 0 [ β ( α + γ τmρφ) ] + α > 0 λ( m ) (21 ). Solving it for γ, we obtain the ondition for that C < 0: τ mρφ λ( m ) C > 0 γ > α 1 + β β (21 ). Finally, adopting the same proedure for the oeffiient D, suh that m f = m in equilibrium, we obtain the following inequality: 11 It is possible to demonstrate easily that this ondition an be satisfied if the sensitivity of the investment to hanges of the degree of the use of the produtive apaity is low, or if the partiipation of the profits in the inome is high. 15

16 τm D > 0 γ γ > λ ( m ) [ α + ] 0 (21d ). Solving for γ, we obtain the ondition for that D > 0: τ mα D > 0 γ > λ( m ) τ m (21d ). Moreover, to have three positive real roots it is neessary that their produt > 0. In order to obey Girard s third relation, whih establishes that the produt > -D/A, it is possible to find that the imposed onditions on the parameters D/A indeed suggests that the result of the relationship is greater than zero, for A < 0 and D > 0. Satisfying all these requirements, we an represent the dynami equation for the degree of indebtedness in the long-run as follows, representing the situation () previously shown. So we have the following A < 0, B > 0, C < 0 and D > 0, in whih the polynomial has 3 positive roots. Thus, we Figure 3 Phase Diagram For Level of Indebtness d/dt 1 L 2 M 3 H an visualize the fixed points of (20) through figure 3 represented below. It is observed that the eonomy has three values of long-run equilibrium level of publi debt as a ratio of apital stok, that is: 1 L (equilibrium with low debt), 2 M (equilibrium with medium debt) and 3 H (equilibrium with high debt). It is also observed that the equilibrium with medium debt is unstable, while the equilibrium points with low and high debt are stable. From that, it follows that the initial value of debt as a ratio of the apital supply is greater than 2 M; the eonomy will present a transient dynamis haraterized by the rise of publi debt as a ratio of the apital supply and an inrease in the nominal and real interest rate, thus defining a viious irle of inrease in the debt/rise of interests/inrease of the debt. Finally, it is important to explain why the situation () with three positive real roots was hosen as representative of the eonomy under study. The situations with one root (a) and two roots (b) were disarded beause of the following reasons. From a numerial simulation, with different values for the parameters, it is possible to find that the parameters A < 0, B < 0, C > 0 e D > 0, the 16

17 ase (a), is only reprodued for eonomially unrealisti values for the parameters. Moreover, the eonomy would be suh that only one equilibrium with a positive level indebtedness would be obtained, whih does not seem to be a realisti situation and therefore we have disarded the ase (a). However the differene between the situation (b), with two positive roots, and the situation (), with three positive roots is limited to the position of the stable equilibrium point with low indebtedness. If the parameters are suh that the ase (b) prevails, then one of the stable equilibrium will our with a negative value for debt-apital relation, that is, a reditor government, instead of a resoures borrower, a situation that is not empirially verifiable. The other stable equilibrium would only be possible with a very high degree of indebtedness. But if the parameters values are suh that three real positive roots are obtained, so the maroeonomi dynamis will be able to reflet, in fat, a more realisti situation in whih the eonomy an operate, with two positive and stable degrees of indebtedness, one low and one high. The (b) and () ases and the neessary onditions imposed on the parameters, suh as in table 2, are fully feasible 12. In the following, we will fous on ase () to analyze the long-run dynamis and the impliations of eonomi poliies in the ontext of different regimens of indebtedness. The qualitative onlusions are valid for both ases. 5. Long-Run Effets of Changes in Fisal Poliy The next step in our analysis onsists of determining the long-run effet of variation in the government expanses as a ratio of apital stok and in the profit share over the level of apaity utilization and over the rate of growth of apital stok. The differene regarding the analysis made in setion 3 is that now we will take into aount the impat of these variations over the level of indebtedness of the publi setor and, therefore, the indiret effets of these hanges on the variables under onsideration. For that, we will initially appraise the impat of hanges in the fisal poliy and in the distribution of inome on long-run equilibrium values of the degree of indebtedness of the publi setor. One way to analyze that, without having to appeal to the numerial alulation of the roots of the expression (22), is to appraise the impat of hanges in the variables under onsideration for the position of lous d dt, in order to appraise graphially the effet over the fixed points of the lous under onsideration. Variations in the fisal poliy and in the share of profits will make that the equilibrium points of the lous d dt move from the right to the left, as well as alter the distane between them, aording to the type of the variation onsidered. 12 The proof of this statement is made by means of numerial simulations, whih an be obtained with authors by request. 17

18 Going bak to equation (18), we an rewrite it emphasizing the degree apaity utilization (u) and also as a funtion of γ. Doing this we have: d = & = γ + ( 1 τ ) i τ mu( γ ) dt ( π + g) (18b) Differentiating (18b) to γ, we obtain the following expression: & λ( m) τ m β = γ λ( m) (25a). The result is ambiguous. Based on expression (25a) we an onlude that & γ > 0 if the following ondition is satisfied: = = s (1 τ ) m 1 > 0 β (25b). That is, the effet of a hange in the fisal poliy over the position of lous d dt will depend on whether the indebtedness of the publi setor as a ratio of the apital supply is smaller or bigger than a ertain ritial value. For degrees of indebtedness lower than this value the derivative (25a) is positive, then a fisal expansion will disloate the lous d dt to above. On the other hand, for levels of indebtedness bigger than this ritial value, a fisal expansion will disloate this lous to below. Given that the equation of the movement of the long-run degree of indebtedness has three roots, then the ritial value will be able to be loated in four different points, as follows: Table 2 Positions of the Critial Value Case I Case II Case III Case IV > 3 H 2 M < < 3 H 1 L < < 2 M < 1 L Case I - > 3 H In the ase where the ritial value of the publi indebtedness level is bigger than the value of this variable in the initial equilibrium with high indebtedness ( 3 H), then & γ > 0, thus a displaement to above of the entire lous d dt, as represented by Figure 4 below: 18

19 Figure 4 Case I > 3 H d/dt 1' L 2' M 3 H 3' H In figure 4 we observe that a fisal expansion generated an inrease of the publi indebtedness as a ratio of apital stok in the long-run equilibrium (from 3 H to 3 H). A redution of the distane also ours between the points of low 1 L and medium 2 M indebtedness, whih in eonomi term means an inrease of the expenses, in this situation, redues the region of stability with low indebtedness and inreases the possibility of the eonomy to enter the high equilibrium earlier. Remember, as shown in figure 3, the 2 M is instable. The room for the adoption of an expansionary fisal poliy diminishes. Case II - 2 M < < 3 H In the ase where the ritial value of the indebtedness level is between the medium and high values of equilibrium, the effet of an inrease of government expenses has different effets as the degree of initial indebtedness is below (to the left) or above (to the right) this ritial value. If it is below & γ > 0, then the urve will disloate to above. If it is above & γ < 0, then the urve is disloated below. These two movements ause a twist in the lous d dt around the point A, whih is the limit between the two regions, as demonstrated in figure 5 below: In this ase, a fisal expansion inreases the degree of indebtedness of low equilibrium 1 H and redues the medium equilibrium to 2 M and, as before, redues the interval of low stability in the eonomy. On the other hand, it auses a redution in the degree of indebtedness of high equilibrium of 3 H to 3 H. 19

20 Figure 5 Case II 2 M < < 3 H d/dt A 1' L 2' M 3 ' H 3 H Case III - 1 L < < 2 M In the ase that the ritial value level of indebtedness lies between the low and medium equilibrium, an inrease in the expenses has different effets aording to the initial degree of indebtedness shown above (to the left) or below (to the right) this is the ritial value, suh as in ase 2. If it is below & γ > 0, then the urve moves up. If it is above & γ < 0, then the urve moves down. These two movements ause a twist of the lous d dt around point A, whih is the Figure 6 Case III 1 L < < 2 M d/dt 1' L 2' M 3 ' H 3 H A limit between two regions, as it is shown in figure 6. The differene of this ase from ase II is related to the point of unstable equilibrium 2 M. This point is now plaed in a higher level. However, the point of low equilibrium of the eonomy also moved up. In eonomi terms, ase III is preferable to ase II due to bigger distane between 1 L e 2 M, whih means that there is more room for the exeution of expansionary fisal poliies with inreases of government expenses, before the equilibrium point 2 M is exeeded and the eonomy starts a transient trajetory to the diretion of a higher equilibrium. The eonomi ost of this is that the low equilibrium also inreases to 1 L. 20

21 Case IV - < 1 L Finally we have ase IV, where the ritial value of the indebtedness level is below of the low equilibrium. In the ase where the ritial value of the level of publi indebtedness is lower than the value of this variable in the initial equilibrium with low indebtedness ( 1 L), then & γ < 0. So there is a moving down of the entire lous d dt, suh as represented by figure 7 below. This is the best one of the worlds. The low stable equilibrium point diminishes for 1 L, at the same time that the medium stable equilibrium point inreases for 2 M, whih makes the differene between them bigger than all the other ases. Additionally, the high stable equilibrium point also diminishes for 3 H. How is it possible that a fisal expansion generates a redution in the publi debt as a ratio of the long-run apital supply? This ounter-intuitive result an be explained by the fat that in the ase IV above, a fisal expansion generates a very strong expansion of the aumulation of apital and in the level of apaity utilization (and, therefore, of the tax revenue of the government) so that debt as a ratio apital stok is redued. Figure 7 Case IV < 1 L d/dt 1' L 2' M 3 ' H 3 H Although this result is a logial possibility of the model presented here, we have to keep in mind that it is not very probable that the same would be observed in the real world. This is beause for minimal and realisti values of the parameters s, m, τ eβ, the ritial value of must be very high, so that ase VI would be disarded as a simple theoretial uriosity. As for ase III, though a fisal expansion inreases the long-run degree of indebtedness as expeted, it is not feasible either, 21

22 sine it requires a lower than that whih an be effetively obtained with plausible values for the parameters 13. So therefore, ase I and ase II are situations that are muh loser to reality. 5.1 SHORT AND LONG RUN FISCAL MULTIPLIER We now need to analyze the effets of a fisal expansion over the level of apaity utilization in the long-run equilibrium, so that it is possible to alulate the long-term fisal multiplier. For that, we must differentiate the equation (10) to γ, taking into onsideration, however, the effets of γ on. We have, then, that: u γ CP 1 = > 0 λ( m ) (13a) u γ LP 1 = λ( m ) ( s )( 1 τ ) ρ( ) γ (26). In the expression (26) we observe that if >, that is, if the eonomy is operating in a regime of high publi indebtedness, the long-run fisal multiplier will, ertainly be positive. It must also be observed that the long-run fisal multiplier - given by the equation (26) - is greater than the short-run fisal multiplier - represented by the equation (13a). From that, it follows that a fisal expansion will have a bigger impat on the aggregate demand and the level of eonomi ativity in the long-run than in the short-run in eonomies that operate in a regime of high publi indebtedness. 6 Long-run effets of a hange in the inome distribution We will now analyze the long-run effets of a hange in the funtional distribution of the inome, more speifially, the effet of an inrease of the partiipation of profits in inome. For 13 Indeed, taking s = 0.75; τ = 0.20; m = 0.40 and β = 0. 10, we obtain a ritial value of equal to 2,40. Assuming a degree of utilisation of produtive apaity equal to 0,85 and a apital-produt relation equal to 2,5; this ritial value of implies a ritial value for the publi debt as a ratio of the GDP of 5.10 or 510%. To get this value suffies to ( D / P) Y Y d u remember that: = =. Where: d is the publi debt as a ratio of the GDP, u is the degree of the use Y Y K σ of the produtive apaity, σ is the apital/produt relation and Y is the potential produt. There is no ase in the real world of a sovereign government that has a publi debt as a ratio of the GDP greater than 200%, so the effetive value of must be a lot smaller than the ritial value of this variable, taking then ase III and this as a simple theoretial uriosity. 22

23 that, we must initially appraise the impat of a variation in m over the values of long-run equilibrium of publi debt as a ratio of apital stok. From equation (18), substituting π, and realling that u and g also depend on m, we have: f { ξ ( m m) + ( m } & d = = γ + ( 1 τ ) i τ m u( m) g dt ) (27) Differentiating (27) to m, we get: & u = τ u τ m g + ξ (28). Remembering that the partial derivative g was already alulated in (12), and then substituting, we have: & = u u ( 1+ φ) ε β τ m + u (29) Defining η u, m m = u u as the elastiity of the degree of apaity utilization with respet to the partiipation of the profits in the inome (f. Oreiro, 2004, p.46), the expression (29) an be rewritten as follows: & u = ( 1+ φ) ε β τ [ 1 η ] u. m (30). Assuming that η 1, and given that u < 0 as shown in (12a), in order the partial u. m < derivative & > 0, it is neessary that is above a ertain ritial value given by. If is below the ritial value, the derivative will be negative. The ritial value, whih is obtained equalizing (30) to zero and solving it to, is given by: = ( 1 η ) u ( ) 1+ φ ε β τ u,m (31). of lous That is, the effet of an inrease on the partiipation of profits on the inome of the position d dt will depend on whether the publi setor debt as a ratio of the apital stok is smaller or bigger than a ertain ritial value. For degrees of indebtedness bigger than this 23

24 ritial value, an inrease of the partiipation of the profits in the inome will move the lous d dt up. Otherwise, to degrees of indebtedness smaller than this ritial value, an inrease of m will move the lous down. In ontrast to the ase analyzed in the previous setion, whih referred to a fisal expansion, the ritial value of for a hange in the partiipation of the profits in the inome must be very low 14. Thus, the eonomially relevant ase is the one that orresponds with a situation in whih: 1 < L above 3 H. or even 1 L < <, disarding both possibilities in whih is either above 2 M or 2 M 1 Case I - < L In this ontext, as we an see in figure 8 below, the three equilibrium levels of indebtedness Figure 8 Case I < 1 L d/dt 1' L 2' M 3 H 3' H are above the ritial value, whih means, taking the results obtained in (30) and (31), that the partial derivative will be & > 0, therefore, in this situation an inrease of the partiipation of the profits in the inome auses the lous d dt to move up in all the extension above. The onsequene of this is that the low stable equilibrium point inreases to 1 L, the medium unstable point diminishes to 2 M, so that the distane between them is small, whih diminishes, therefore, the range of the stability of the eonomy. 1 2 Case II - < < L M In this ase, the values below, the partial derivative will be negative, as before, and for values above it, it will be positive, as represented in the following figure 9: 14 Assuming s = 0. 75; τ = 0. 20;m = e β = as we did before, and η = 0. 5 e u = 0. 5, we get = u,m

25 Figure 9 Case II 1 L < < 2 M d/dt 1' L 2' M A 3 H 3' H The urve suffers then a ounter-lokwise rotation around point A, whih delimits the two regions. A onsequene of this is the low stable equilibrium point redues to 1 L, and the medium stable point diminishes to 2 M, so that the distane between them depend on how muh eah one of the points withdrew. In the graph above, as is loser to 2 M, this means that the relative displaement to left from the stability point 2 M will be smaller than 1 L, inreasing then the interval of stability of the debt-apital relation. The inverse would our if were loser to 1 L. Anyway, given that 2 M moves to the left, there will be less room for the exeution of expansionary fisal poliies without breahing the barrier of sustainable debt. 6.1 LONG-RUN EFFECTS OF AN INCREASE IN m ON THE DEGREE OF THE CAPACITY UTILIZATION (u) In the short-run, as we have seen before, in equation (12a), the effet of an inrease in profit share has a negative effet over the degree of produtive apaity utilization, thus onfiguring a regime of aumulation of waged-led type, as reprodued below. u CP = λ( m) { φε + [ 1 (1 s )(1 τ ) u ]} 1 < 0 (12a) In the long-run, the effet of an inrease in the partiipation of the profits an be appraised differentiating the equation (10) to m, taking into aount the effets of hanges of the partiipation of the profits in the inome on. This proedure results in: 25

26 u LP 1 = λ( m) 1 [ φε + ( 1 (1 s )(1 τ )) u ] + 2( 1 s )( 1 τ ) ρ ( ) λ( m) (32). As it an be observed, the first term of the expression (32) is nothing but the short-run effet of a variation of the partiipation of the profits in the inome over the degree apaity utilization, whih - on the basis of the equation (12a) - is negative. The seond term presents the indiret effet of hanges in the funtional distribution of inome over apaity utilization. The sign of this indiret effet depends, however, on the regime of indebtedness in whih the eonomy is. If the eonomy is operating in a regimen of low indebtedness, then the indiret effet will be negative, thus strengthening the diret or short-run effet of hanges on the inome distribution. If the eonomy is operating in a regime of high indebtedness, then the indiret effet would be positive, whih in turn ould ause that an inrease in profit share ould produe a rise in the level of apaity utilization in the long-run. This result will be more likely as higher is the level of indebtedness of the publi setor. As summary of this result, it follows that if the eonomy is operating in a regime of high indebtedness; then the regime of aumulation will be of profit-led type. 7 Final Comments As demonstrated, the short and long-run eonomi dynamis differ from the traditional Keynesian models when the interest rate is make an endogenous variable that responds to the degree of indebtedness, sine it introdues a region in whih the ourrene of persistent publi debt, in as muh as it provokes disequilibrium in the debt stok and is able to alter the regime of aumulation. In the short-term, the effetiveness of publi expenses depends on the initial onditions and the degree of indebtedness in whih the eonomy finds itself. If, on the one hand, expenses finaned with emission of bonds an provoke an inrease in the aggregate demand, either via onsumption or via publi investment, then on the other hand, the existene of a risk premium on the publi debt has a negative effet on private investments, so that there is a point where the latter effet is greater than the former and throws the eonomy into a region where the indebtedness degree produes a viious yle for fisal poliies. This behavior, as demonstrated, ould be an extension on the traditional Keynesian models, for whih the positive effets of fisal poliies are always expansionary. Here we demonstrated that there an be a differene with inverse dynamis. 26