THE CAMBRIDGE EQUATION WITH GOVERNMENT ACTIVITY REVISITED *

This paper offers an analysis of the steady-state distributional features found in a Kaldor-Pasinetti process, in which the government sector is allowed to run persistent defi cits that may be fi nanced through different instruments. Productive capital and bonds generate single rates of return, while workers’ saving propensity remains uniform. This paper seeks to establish a generalization of Cambridge Eauqtion, considering the specifi c contributions of Steedman (1972), Pasinetti (1989), Dalziel (1991), and Faria (2000).


INTRODUCTION
The Kaldor -Pasinetti process (K-P) could be characterized as having three groups of savers: the government plus two socioeconomic classes with differentiated propensities to save.As a socioeconomic class, capitalists save from profi t income, while also exhibiting a higher propensity to save than workers, the second socioeconomic class, who are characterized as saving both from profi ts and wages.The government's savings behaviour is modelled symmetrically to other saving groups through fi xing a constant ratio between budget defi cits and disposable income, measured as net taxes of interest payments.
One of the features of this type of model is the existence of a steady-state in which the profi t rate as well as the capitalist's savings behaviour, and possibly the levels of government savings and tax rates, depend only on the growth, rate of output.An alternative equilibrium could arise, however, if the system's given parameters prove to be inconsistent with the existence of capitalists.The existence of this alternative steady state has been explored by Meade (1966), Samuelson andModigliani (1966), andSteedman (1972), showing that the alternative equilibrium could also emerge as a function of the profi t rate, and that this rate is not necessarily equal to the ratio of the growth rate to the workers' propensity to save, but it could coexist with the Pasinetti equilibrium under the same savings and tax regime, along with a generalized production technology.Samuelson and Modigliani (1966) argue that Pasinetti's model could give rise to an alternative balanced growth path if "pure" capitalists would cease to exist, violating the Cambridge Equation.Their anti-Pasinetti Theorem can be seen, however, as a simple attempt to resurrect the marginal productivity theory, the condition required to attain the dual result being far from real world magnitudes [see Pasinetti (1974) and Kaldor (1966)].
If the capitalist's propensity to save (and hence their existence) is inconsistent with the level of taxation, technology and government defi cits, the model could also fi nd a solution only in the absence of that class of savers.The possibility of the Pasinetti equilibrium's inconsistency with exogenous elements of the system warrants the analysis of the necessary restrictions for a Pasinetti's steady-state, which is one of the interesting features in the literature, combined with a simple formulation of a "Cambridge Equation." An additional and equally important discussion involving the theorem behind the Cambridge Equation concerns the effects of government activity.Fleck andDomenghino (1987, 1990) have argued that the Cambridge Theorem fails to hold if there exists a steady-state budget defi cit.Pasinetti (1989aPasinetti ( , 1989b) ) and Dalziel (1991) 1 demonstrate errors in Fleck and Domenghino's approach, while also demonstrating that Steedman's (1972) analysis could be extended to consider the budget defi cit case.
This paper considers the case of a private economy in which government does not own capital goods.Section 2 provides an overview of the literature.Section 3 considers the assumption that government fi nances its budget defi cits by selling bonds.In section 4, the government monetizes public defi cits.We seek to establish that versions of Pasinetti's Result, advanced by Steedman (1972), Pasinetti (1989aPasinetti ( , 1989b) ) and Dalziel (1991), serve as particular cases of the K-P model presented in this paper.The corollary we advance is that fi nancing a budget defi cit by debt creation or by issuing money fails to alter the essence of the Cambridge Theorem.Pasinetti's model (1962) stems from Kaldor's (1955) macroeconomic theory of growth and distribution.According to the former, when workers save, they gradually become owners of fi nancial assets.As a result, they receive both profi ts and wages as income.Pasinetti then shows that the equilibrium rate of profi ts, r, consistent with full employment, does not depend on workers' propensity to save, but is equal to the natural growth rate, g n , divided by capitalists' propensity to save, s c .This relationship between and among variables is known as the Cambridge Equation:

AN OVERVIEW OF THE DEBATE
This result, also termed the "Pasinetti Paradox" by Samuelson and Modigliani (1966), was derived through considering the long-runsteady-state equilibrium for a closed economy without government, and it assumes that capitalists' marginal propensity to consume is greater than that of workers.Subsequently: where I stands for investment and Y for production.Rewriting this inequality in the form of s w = I/Y, Meade (1966), as well as Samuelson and Modigliani (1966), concluded that the "Cambridge Result" provided a possible steady-state solution, and not the most general solution.According to these contributors, the dual result, that is, the anti-Pasinetti equilibrium, also appears and stands fully symmetric to the Cambridge Equation.
However, Meade's attack has limited signifi cance, since the anti-Pasinetti equilibrium holds only in the case where the steady-state solution implies the capitalists' euthanasia.Following Pasinetti (1974, p. 130), "[i]f capitalists were not to exist anymore, their propensity to save obviously could not determine the rate of profi t.There is therefore a way of preventing the Cambridge Equation from operating, and that is by eliminating capitalists from the system." After this initial debate, the focus on the Cambridge Equation shifted toward an analysis of its validity when considering government taxation and spending activities.Steedman (1972, p. 138) has shown that in this case a Pasinettian equilibrium emerges, in which the rate of profi t is not dependent upon production methods.By considering a perfectly balanced government budget, Steedman arrived at the following version of the Cambridge Theorem: (1) where 0 < t p < 1 is the (average and marginal) tax rate on profi ts.
Steedman's extension involves direct taxation and a balanced budget, holding on to the general thrust and refl ecting the essence of the original Cambridge Theorem.Namely, the propensity for savings on the part of capitalists proves to be the key for determining the rate of profi t.Fleck andDomenghino (1990, 1987) challenged Steedman's view, arguing that his contribution remains a limited case of a perfectly balanced government budget.They also argue that, in cases of a steady-state budget defi cit, the workers' propensity to save is what matters when determining rates of profi t.Pasinetti (1989aPasinetti ( , 1989b)), however, counter-attacked their challenge, showing that, even in the cases of budget defi cits or superavits, a more correct version of the expression should be presented as: where s g , is the government's savings rate.Naturally, if s g < 0 or s g > 0, the equation above is still valid, with the proviso that the second term in the denominator is negative or positive, respectively. 2Here we will not be concerned with the case in which s g > 0, the case of government permanent superavit.
This debate seems to have been settled.However, a minor disagreement arose between those who defended the Cambridge Equation.On one side of the debate, Denicolò and Matteuzzi (1990), Dalziel (1991) and Araujo (1992) have argued that equation (1) would also hold independently of government budget defi cits or surpluses.In short, the government's propensity to save does not matter when determining the rate of profi t, as argued by Pasinetti in expression (2).In section 3, below, we shall deconstruct and expose shortcomings that have not been addressed nor resolved, so far.This constitutes the main contribution of our paper.

PUBLIC DEBT CREATION
The economy consists of households, fi rms and the government.Households are divided into two classes: capitalists, whose main source of income is earnings from capital, and workers, who are mixed-income receivers.In this economy, ownerships are private.Capitalists and workers decide their own savings ratios.But the government may affect the overall savings ratio through the redistribution of income.For the sake of clarity of exposition, we shall consider an elementary version of the Kaldor-Pasinetti process in which there is direct taxation 3 only.Suppose, for example, that a unique net direct tax, t p , applies to all kinds of profi t income, irrespective of it being earned by capitalists or workers.It is also reasonable to assume that the government will not pay taxes to itself.According to Dalziel (1991), the government's savings, S g , is given by: , (3) where T stands for the total taxation, G for the government's expenditures, and A for the public debt in period t.Let us fi rst consider a case in which the government fi nances its budget defi cits by issuing bonds.By making A the stock of government bonds, then it varies according to: We consider that i is the nominal interest rate that in the absence of an expected and/or actual infl ation is equal to the real interest rate r.In this case, expression (4) may be written as: Let us keep the assumption that capitalists fi nance a constant fraction λ, 0 ≤ λ ≤ 1 , of the public debt, which is the same as assuming that they finance the same proportion of government defi cits each period of time.Hence, Consequently, capitalists receive an interest payment r λ A in every period.Assuming that interest receipts are taxed at the same rate as profi ts, capitalists' income may be written as: In this case, their savings is: As long as a fraction λ of the budget defi cits is fi nanced by capitalists in each period of time, the capitalists' investment I c , is given by: Since capitalists invest in capital as well as government bonds, therefore: . This identity is also limited by their budget constraint, namely: Dividing both sides of ( 9) by K c , we obtain: (10) In a steady-state, the growth rate of capital and government bonds remains constant, namely .Noticing that A c = λA, equa tion (10) can be rewritten as: where and .After some algebraic manipulation, we could derive an extended version of the Cambridge Equation as: (12) Note that if λ = 0, that is, when only workers own government bonds, we obtain the Steedman's (1972) version of the Cambridge Equation with government.If a particular fraction of the budget defi cits or surpluses are financed or invested by capitalists, then we obtain Pasinetti's (1989aPasinetti's ( , 1989b) ) version of the Cambridge Result.The value of λ that gives rise to Pasinetti's version is formulated in the following expression: (13) In conclusion, both Stedman's (1972) and Pasinetti's (1989aPasinetti's ( , 1989b) ) versions may well be more correctly viewed as particular cases of our approach.In our formulation, workers' savings behaviour does not play a role in the determination of the rate of profi t.That is, by fi nancing budget defi cits through issuing bonds, the government fails to alter the essence of the Cambridge Equation.

MONETIZING PUBLIC DEFICITS
Recently a number of authors, such as Commendatore (2002), Palley (1996Palley ( , 2002)), Park (2002Park ( , 2004Park ( , 2006) ) and Seccareccia (1996), have dealt with the case in which money is introduced in the K-P framework.In this model, the existence of money may well be explained by purely transactional reasons.The general result found is that the validity of the Cambridge Equation holds independently of the way in which money is introduced in the economy.Here, let us assume that money is incorporated into the K-P process in order to monetize budget defi cits.In this vein, the government does not issue government bonds, instead it monetizes budget defi cits according to the following equation: where M is the stock of money, p is the price level, and real money balances are defi ned as m ≡ M / p.We assume that capitalists hold a constant fraction θ , 0 ≤ θ < 1 of the stock of money, thus: Capitalists hold their wealth in physical capital and in real money balances, therefore: where is the rate of infl ation.In a steady-state, we have , consequently, .Dividing both sides of ( 16) by K c yields: (17) Rewriting ( 17) and considering that m • c = g n m c and , we obtain: From (18) we could then derive the Cambridge Theorem for the case in which the government monetizes the public defi cits as: (19) Expression ( 19) keeps the essence of the Cambridge Equation despite the inclusion of monetary variables: the rate of profi t is determined independently of the workers' propensity to save.If θ = 0, that is, if capitalists do not hold money in the steady-state equilibrium, we arrive at Faria's (2000) version of the Cambridge Result with money and infl ation.However, if capitalists hold money in equilibrium, we arrive at Dalziel's (1991) version of the Cambridge Equation, where θ is given by the following expression: (20) Therefore, equation ( 19) can be seen as a more general version of the Cambridge Equation when the government monetizes public defi cits.An interesting property that arises from this analysis is that money is not superneutral, since the rate of monetary expansion has effects over the long run capital stock of the economy and functional distribution of income.It is worth to remember that, by performing his analysis in a Neoclassical model, Sidrauski (1967) has found that money is super-neutral.In addition, this new version keeps the essence of the Cambridge Theorem since workers' propensity to save does not matter when determining the rate of profi t.

CONCLUDING REMARKS
In this paper, through modelling an economy in which government budget defi cits -or superavits -is either fi nanced or invested by capitalists and/ or workers, we have derived a new version of the Cambridge Equation.When public defi cits are totally fi nanced by workers, the model generates the Steedman's (1972) version of the Cambridge Result.When capitalists hold part of the government bonds, we obtain Pasinetti's (1989aPasinetti's ( , 1989b) ) version.If the public defi cits are monetized, we arrive at Faria's (2000) ver-sion of the Cambridge Equation, with money and the rate of infl ation derived for the case in which capitalists fail to hold money in a steady-state.Dalziel's (1991) case is obtained if and when capitalists hold money in equilibrium.Finally, it should be stressed that the government's decisions regarding the fi nancing of a budget defi cit through debt creation or issuing money fails to alter the essence of the Cambridge Equation.This result reinforces previous positions by Dalziel (1989), Pasinetti (1989aPasinetti ( ,1989b)), Denicolò & Matteuzi (1990), Araujo (1992) and Teixeira (1999), who have shown that the validity of a version of the Cambridge Theorem is not limited to the case of balanced budget.NOTES 1. Bortis (1993, p. 115) concludes that "the debate provoked by Fleck and Domenghino (1990) has been largely settled by a 'generalization and a simplifi cation of the Cambridge Theorem with budget defi cits' (Dalziel 1991)."The expressions Cambridge Theorem, Cambridge Result, Pasinetti's and Cambridge Equation are interchangeable.
2. As a consequence, expression (1) appears to be a particular case of a balanced budget, in which s g = 0.
3. Pasinetti (1989) and O'Connel (1995) deal with a more complex arrangement, since their approaches consider both direct and indirect taxes.