On the stability of a certain Keynes-Metzler-Goodwin monetary growth model

Damian Sołtysiak

doi:10.18559/ebr.2023.1.2

Sołtysiak, Damian

2023-04-01 00:00:00

Economics and Business Review Vol. 9 (1), 2023: 26-64 https://www.ebr.edu.pl https://doi.org/10.18559/ebr.2023.1.2 On the stability of a certain Keynes-Metzler- -Goodwin monetary growth model Damian Sołtysiak Abstract Keywords The article has three aims. The first aim is to develop an im - • Keynesian proved version of the Keynes-Metzler-Goodwin (the KMG) macroeconomics monetary growth model originally presented and analysed • disequilibrium in a series of publications by Carl Chiarella, Peter Flaschel macroeconomics and Willi Semler. The improvement of the model is obtained • monetary growth by modifying some of its equations in a way which ensures models that they reflect real macroeconomic dependencies more • nonlinear economic properly. The equations that have been modified describe dynamics final demand expectations, determinants of production de - • stability cisions, fixed capital accumulation, tax revenues, govern - ment budget deficit and money demand. The second aim is to transform the model into an intensive form described by seven non-linear differential equations and determine its unique steady state which shows proportions between vari - ables on the balanced growth path. The third ultimate aim is to present a mathematical proof that the new improved version of the KMG model is locally asymptotically stable. JEL codes: C62, E12, E40 Article received 21 August 2022, accepted 14 December 2022. Suggested citation: Sołtysiak, D. (2023). On the stability of a certain Keyens-Metzler-Goodwin monetary growth model. Economics and Business Review, 9(1), 26–64. https: //doi.org/10.18559/ ebr.2023.1.2 This work is licensed under a Creative Commons Attribution 4.0 International License https: //creativecommons.org/licenses/by/4.0 Department of Applied Mathematics, Poznań University of Economics and Business, al. Niepodległości 10, 61-875 Poznań, Poland, damian.soltysiak@ue.poznan.pl, https://orcid. org/0000-0002-1110-7554. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 27 Introduction The mainstream economic theory has been increasingly questioned since the global financial crisis of 2007–2008. This raises interest in monetary mo - dels of economic growth which are related to Keynesian economics. Especially important in this regard are the works of three well known Keynesian econo- mists: Carl Chiarella, Peter Flaschel and Willi Semmler who have been deve- loping Keynesian economics over the last two decades. They have published jointly (sometimes with other co-authors) a series of books on Keynesian eco- nomics emphasing interrelations between real and financial spheres of the economy. Among others, one should mention the comprehensive monograph The dynamics of Keynesian monetary growth (Chiarella & Flaschel, 2000) and the fundamental trilogy Reconstructing Keynesian macroeconomics (Chiarella et al., 2012, 2013, 2014), in which they make an aemp tt t at completely rein - terpreting and reconstructing the whole Keynesian macroeconomics. Other important works by these authors include Chiarella et al. (2000), Asada et al. (2003), Chiarella et al. (2005), and Charpe et al. (2011). The most important model developed and analyzed (in various variants) by Chiarella, Flaschel and Semmler was named by them “the Keynes-Metzler– Goodwin model” (abbreviated as the KMG model) to emphasize its relation - ship to the concepts developed earlier by these economists. The KMG model is a disequilibrium monetary growth model which refers to ideas expressed in Keynes’s General theory (1936) and in Goodwin’s (1967) work on the interaction of growth and income distribution; these are the K and G components of the model. The KMG models take into account a gradual adjustment of inventories to its desired level. The dynamics of inventories is related also to the concept of expected sales, which is formulated in Metzler (1941) and constitutes the - refore the M-component of the KMG approach. To the same extent, the KMG model refers also to the Keynes-Wicksell type models presented, among others, in Stein (1966), Rose (1966) and Fischer (1972). Besides, it is worth mentio - ning its similarity to the Keynesian model presented in Sargent (1987, Ch. 5). All versions of the KMG model describe the functioning of the economy with the use of sixth or seventh dimensional systems of non-linear dier ff ential equations that reflect adaptive decision-making processes. A characteristic feature of any KMG model is that it can be transformed into so called intensi- ve form model, in which all original variables are replaced with new variables describing proportions between them. The intensive form model enables in turn the derivation of the steady state of the economy, which describes pro - portions between variables of the original model maintained in the process of balanced growth at a certain constant rate. Another alternative and important current in economics referring to Keynes is post- Keynesian economics (Lavoie, 2014). 28 Economics and Business Review, Vol. 9 (1), 2023 The main theoretical results which are obtained with the use of the inten - sive form of KMG models are stability theorems showing conditions under which the economy converges toward the steady state, which is equivalent to approaching the balanced growth path. Since the intensive form model is a system of nonlinear differential equ - ations to prove its (local) asymptotic stability, one has to show that all of the eigenvalues of the Jacobian matrix of this system are either negative num - bers or complex numbers with negative real parts. Investigating the eigenva - lues is a standard way of proving stability, which has been applied hundreds of times to different dynamic models. Since eigenvalues are also roots of the corresponding polynomial, their analysis is quite easy if one deals with a sys- tem of two differential equations; however, it becomes very difficult and so - phisticated in the case of high dimensional systems such as the KMG model. In the laer tt case, probably the only way to show that the eigenvalues of a gi- ven Jacobian matrix guarantee stability is a subsequent zeroing of appropria- te matrix parameters, which enables a multiple Laplace expansion of the de - terminant of the Jacobian matrix in order to obtain a sequence of matrices of an increasingly lower order whose eigenvalues can be more easily analysed. First, it is shown that the matrix of the lowest order has appropriate eigenva- lues. Next, through a subsequent restoration of small positive values of pre - viously zeroed parameters, it is demonstrated that also the original Jacobian matrix has eigenvalues which are either negative numbers or complex num - bers with negative real parts. The first rigorous proof of stability of the seven-dimensional KMG model based on the idea outlined above was presented twenty years ago in Chiarella et al. (2002). Other versions of the proof of stability (concerning modified KMG models) can be found in Asada et al. (2003) and Chiarella et al. (2006). In the laer tt publication the authors use the term cascade of stable matrices approach to name the general idea lying behind the proof. Work on the extension and modification of the KMG model is continuous. One example is Ogawa (2019a, 2019b, 2020), who extended the KMG mo- del to a two-sector model. Another is Flaschel (2020), in which an extensive KMG model is used to analyze how taxes, transfer payments and government spending improve the social protection of the employee household sector. It is worth mentioning the most recent work by Chiarella et al. (2021), which is the culmination of the development of the Bielefeld school of macroecono - mic thought. The book is authored by major representatives of this approach, namely Flaschel, Franke and the late Chiarella. Although the term the cascade of stable matrices approach comes from Chiarella et al. (2006), the idea of proving the stability hidden behind it was used independently of them by other authors in their proofs of stability of high dimensional dynamical systems completely different from the KMG model, e.g. Duménil and Lévy (1991), Kiedrowski (2018). D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 29 An interesting variant of the KMG model with private corporate debt, aimed at modeling the eects ff of fiscal and monetary policies, has been presented recently by Asada et al. (2018, 2019). The authors of these publications also deal with the question concerning the existence of limit cycles in the KMG mo - del. Keynesian monetary growth models similar to the KMG model in which limit cycles or periodic orbits may exist have also been presented recently in Murakami (2016, 2018, 2020) and Araujo et al. (2020). This article has three aims. The first aim is to demonstrate a new, improved KMG model which differs from the versions analyzed by the aforementioned authors through a number of modifications. The second aim is to transform the model into its intensive form and determine its unique steady state. The most important and challenging is the third aim, which is to prove that the new version of the KMG model is locally asymptotically stable, which means that the economy described by this model has an intrinsic ability to converge toward the balanced growth path. The modifications are aimed at improving the model by eliminating from its equations some especially simplistic, questionable elements which seem to have been introduced not for their economic relevance but primarily for their simplicity, which essentially eases a mathematical analysis of the mo - del. The r fi st two modic fi ao ti ns concern equao ti ns describing determinants of the growth rate of fixed capital K/K and the growth rate of expected demand e e Y /Y . In all the KMG models presented by Chiarella et al. above equations contain parameter n added simply to other components of these equations. Depending on the particular version of the model, n expresses either the con- stant growth rate of labour supply or the growth rate of labuor productivity or the sum of these two. Adding n to these equations is very convenient sin - ce, in the steady state, the remining components of the equations become zero, which implies immediately that, in the steady state, both fixed capital and expected demand grow at rate n. In this article, the dynamics of fixed capital and expected demand are much more thoroughly elaborated. In the equation describing K/K, the constant parameter n has been replaced by a e e variable expressing the growth rate of expected demand Y /Y . At the same e e time, parameter n has been removed also from the equation for Y /Y and substituted with the growth rate of the real wage as one of the two factors e e influencing Y /Y , which is economically much more justifiable. The growth rate of the real wage is an endogenous variable which depends on other mo- del variables: primarily on the growth rate of the nominal wage and the in- flation rate. Similar in character is the third modification concerning the equ - ation showing three factors which determine output level Y. In this case, one d d doubtful factor (component) nN (N – desired level of inventories) has been replaced with Y , i.e. the change in expected demand. Other important modifications concern the assumption about taxes and real interest on bonds. In all versions of the KMG model presented by Chiarella 30 Economics and Business Review, Vol. 9 (1), 2023 et al., in order to ease the derivation of the intensive form of the model, it is assumed that lump-sum real taxes net of interest are collected in such a way that their ratio to the capital stock remains constant. As a consequence, the dependence of tax revenues on tax rates imposed on labuor or capital inco- mes is not visible. The dependency above is explicitly taken into account only in the present model, which allows for a more comprehensive analysis of sc fi al policy regarding taxes and the government budget deficit. In particular and contrary to the previous models, the ratio of the government budget deficit to fixed capital ceases to be constant in time, despite the assumption about a constant ratio of government spending to capital. The last modic fi ation introduced into the model concerns the money market and the interest rate. In the earlier versions of KMG models, a linear money de- mand function is considered, whose values depend on expected demand and the deviation of the actual interest rate from the steady state interest rate, which is known in advance. As a consequence, money demand in the steady state is determined exclusively by the expected demand being independent of the value of the exogenously given steady state interest rate. This also raises some doubt. Therefore, in the present article, a nonlinear money demand function is considered which depends on expected demand and the actual, endogenously determined interest rate. The steady state interest rate is not known in advance. The aforementioned improvements make the KMG model closer to reality. Despite the increased mathematical complexity of the modified model it is still possible to transform the model into its intensive form and determine its unique steady state which is the second aim of the article. Determination of the intensive form model and its steady state presented in the article opens the way to prove the stability of the model, i.e. to the realization of the third and the main aim of the article. The proof of stability presented in the article exploits the aforementioned idea cascade of stable matrices approach, as named and originally applied by Chiarella et al. Despite this, due to at least two reasons, it is not just a re-nar- ration of the proofs of stability of the earlier versions of the KMG model. The first reason is that the mathematical structure of the KMG model developed in the article is much more sophisticated than its earlier versions analyzed by Chiarella et al. This is especially visible in the intensive form of the model (described by seven non-linear differential equations with two additional con - ditions) for which the Jacobian matrix in the steady state had to be determi - ned, and whose eigenvalues had to be examined. The second reason is that the proof of stability is obtained under a different set of assumptions about model parameters. Some of these assumptions concern tax rates, which are not present at all in the KMG models analysed by Chiarella et al. Chiarella et. al. admit that the above assumption was introduced in order to ease the analysis of the model (Chiarella et al., 2000, p. 279). D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 31 To prove stability two fundamental difficulties had to be overcome. The first one was the derivation of the values of partial derivatives at the steady state of all functions defining the dynamics of the intensive form model in order to examine the Jacobian matrix. The second difficulty concerned the question as to which parameters in the Jacobian matrix should be zeroed and in which sequence, to show finally that all eigenvalues of the Jacobian matrix are either negative or complex numbers with negative real parts. There was no indication as to how to deal with this crucial question. The proper way of proceeding was found in a laborious heuristic process by undertaking a series of aemp tt ts that only finally led to success. The article consists of three parts. Section 1 is devoted to the presentation of the new version of the KMG model. In Section 2, the model is transformed into its intensive form, and the steady state is determined. Finally, in Section 3, the proof of the local, asymptotic stability of the steady state is presented. 1. The model This section provides an overview of the author’s version of the KMG mo - del, which is a modification of the KMG model considered in Chiarella and Flaschel (2000) as well as in Chiarella et al. (2013). The model is presented in the following seven sub-sections. 1.1. Consumption, wages and prices Total final demand Y (in real terms) is the sum of private sector consump- tion C, private-sector gross investment I and government (public) sector de- mand G: Y = C + I + G (1) Consumer demand is described by the following equation: C = (1 – τ )ωL (2) where ω – real wage, L – labor demand (by assumption equal to employ - ment), τ – tax rate on income from work. Despite the simplicity of Equation (2), the consumption dynamics is the result of a complex processes on the one hand being formed by production dynamics, which determines employment, and on the other hand being de- 32 Economics and Business Review, Vol. 9 (1), 2023 termined by real wage ω= , (the ratio of nominal wage to price level the ratio of nominal wage w to price level p). The growth rate of real wage ω= ω /ω equals the growth rate of nominal wage w minus inflation rate π = p (the growth rate of the price level): ˆ ˆ ω= w− p (3) Wage and price dynamics are determined by two separate equations of two Phillips curves. The rate of growth in the nominal wage is calculated according to the fol- lowing equation of the wage Phillips curve: w= β V −V +κ p+ 1−κ π +n (4) ( ) ( ) w w w where β > 0 is a parameter of sensitivity of the nominal wage growth rate to the deviation of the employment rate 0 < V < 1 (employment to labor sup- ply ratio) from natural employment rate V . It is also influenced by the labor productivity growth rate ( n > 0) and the linear combination of the current in - flation π = p and the expected inflation π (0 < κ < 1). Inflation π = p, in turn, is described by the equation of the price Phillips curve p= β u−u +κ w−n + 1−κ π (5) ( ) ( ) ( ) p p p according to which inflation depends on the deviation of the capacity utili - zation rate 0 ≤ u ≤ 1 from its normal level 0 < u < 1 (β > 0 is the response parameter) and on the linear combination of the surplus of nominal wage growth rate w over labour productivity growth rate and the expected infla - tion (0 < κ < 1). Coefficient β > 0 is the reaction parameter of the price level p p to deviation u – u. The change in expected inflation π depends on the difference between the linear combination of current inflation p and its normal value π (equal to inflation at steady state π = μ – n) and expected inflation π : e e π = β αp+ (1− α)π− π (6) e( ) where β > 0 is the adjustment parameter. The capacity utilization rate 0 < u < 1 and the employment rate 0 < V < 1 are model vari- ables defined in section 1.5 (Equations (22) and (25)). The basic idea of Equation (6) is borrowed from Groth (1988, p. 254). Here, the revisions of π are a combination of two rules with weighting factor α, where the adjustments take place at a general speed of adjustment β . > The 0 polar case α = 1 represents adaptive expectations. The other extreme case α = 0 is a regressive mechanism. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 33 1.2. Capital dynamics and investment demand Private sector investment demand I equals the sum of net investments, increasing the fixed capital stock K of firms, and restitution investments in - curred to replace depreciated capital. Net investments are described by the derivative of capital with respect to time , while restitution investments are given as δK, where 0 < δ < 1 is the capital depreciation index. Thus: I = K + δK (7) The increase in fixed capital K is described by the following behavioural equation: e e e  ˆ K = i ρ − (r− π ) K + i u− u +Y K (8) ( ) ( ) 1 2 where: ρ – the expected rate of return on fixed capital, r – the nominal in- terest rate on government bonds r – π – the expected real interest rate, e e e Y =Y /Y – the expected growth rate of final demand, i > 0 and i > 0 – re- 1 2 action parameters. The expected rate of return on capital is defined as the ratio of expected pro - fit to capital, where the expected profit is the difference between the expected amount of final demand Y and the costs of labour and capital depreciation: e d Y −ωL −δK (9) ρ = e e e The growth rate of expected final demand is assumed to be Y =Y /Y shaped according to equation: d e Y −Y Y = ω+ β (10) d e Y −Y ˆ  where ω= ω /ω is the real wage growth rate, (resulting from (3)–(5)), β > 0 is the relative error in final demand expectations. Symbol denotes a re- action parameter. The view that an increase in the real wage increases demand expectations is quite obvious and expressed by many economists (e.g. Napoletano et al., 2012). Introducing such an assumption to the KMG model is a novelty pro- posed by the author. d e Y −Y 7 e In all earlier versions of the KMG model it is assumed that Y = n+ β , where n y e is an exogenously given constant labour growth rate or the sum of labour growth rate and the growth rate of labour productivity (e.g. Chiarella et al., 2013, p. 247). Such an assumption is quite doubtful from the economic point of view. 34 Economics and Business Review, Vol. 9 (1), 2023 According to (Equation 8), the first two factors contributing to high fixed capital dynamics are: – an excess of the real expected rate of return on fixed capital ρ over the expected real interest rate on bonds r – π , – an excess of the capacity utilization rate over its natural level ( u > u). The third term Y K in equation (8) is another novelty introduced by the author. According to it, the growth rate of fixed capital /K depends also e e on the expected growth rate of final demand Y /Y . Such an assumption is consistent with the Keynesian theory, which emphasizes the key role of final demand in the economy. “The feature that is uniquely Keynesian in growth models, and is found in all such models, however, is the role of aggregate de- mand as a determinant of growth” (Dutt, 2012, p. 42). It is worth emphasizing that component Y K in Equation (8) introduces a new large loop into the model since, on the one hand, investments I = + δK are a component of final demand Y = C + I + G and, on the other, they are dependent on final demand. The last dependence is realized directly through Equations (8) and (10), and indirectly by the fact that, according to (19)–(20), final demand influences production decisions, which in turn, through (22), (24)–(25) and (3)–(5), have an eect ff on growth in the real wage and expec - ted demand, and ultimately on capital growth. 1.3. Government budget deficit, issuance of bonds and money creation Following Sargent (1987), Asada (2011), Asada et al. (2012) and Chiarella et al. (2000, 2005), for the sake of simplicity, the government sector’s demand is assumed to be proportional to fixed capital, i.e.: G = gK (11) where g is a constant ratio. Government expenditures comprise also interest payments on bonds, paid to the private sector. Government expenditures are covered mainly by taxes. The government’s total tax income (in real terms) is the following sum: In other KMG models, fixed capital dynamics is described by a simpler equation of the e e form: K = i ρ − r− π K + i u− u + nK (e.g. Chiarella et al., 2013, p. 246). Since in the ( ) ( ( )) 1 2 e e steady state ρ = (r – π ) and u = u, this implies immediately that, in the steady state, the growth rate of fixed capital K /K equals labour growth rate n. Such an equation makes it easier not only to derive the steady state of the model but also to prove its stability. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 35  rB T = τ ωL + τ ρK + (12) w c     rB d d τ ρK + where τ ωL (0 < τ < 1) are taxes on wages ωL and are taxes c  w w   on capital gains imposed on profits from fixed capital ρK and profits (in real rB terms) from government bonds (both capital gains are taxed by the same 9 p capital tax rate 0 < τ < 1). Similarly, as in other KMG models (Asada, 2012; Asada et al., 2011; Chiarella et al., 2000) it is also assumed that the government budget deficit is financed either by the government selling new bonds to the public sector or through open market operations by the central bank, which buys short-term bonds from asset holders when issuing new money. The open market operations by the central bank are a unique channel through which money enters the eco- nomy. Hence the government budget deficit equation has the form:   B+ M = pG+rB− pT (13)   where B denotes government fixed-price bonds in the hands of the public, B + M = pG+rB− pT   describes changes in bonds and B+ M = rep flects G+chang rB− pes T in the amount of money in the hands of the private sector. The central bank’s monetary policy rule is to keep a constant growth rate of money supply μ > 0, so: M = = μ (14)   B+ M = pG+rB− pT In view of (14), changes in bonds supplied by the government which ap- pear in Equation (13) are given residually. Since the money market cannot be in disequilibrium, the constant growth rate of money supply necessitates an identical growth in money demand, so that in every moment of time: M = M (15) where M is money demand. Assumptions about taxes in the presented KMG model dier ff essentially from those made in its earlier original versions. Usually, in the KMG models, real taxes, net of interest, remain T−rB / p in a constant proportion to the capital stock, e.g. = t = const. Such an assumption, together with G = gK, imply that the ratio of government budget deficit G + rB – T to fixed capital K remains constant in time. Moreover, it eliminates both tax rates and bonds from the stage which makes easier derivation of intensive form model and calculation of its steady state (Chiarella et al., 2013, p. 247). A similar simplifying assumption can be found also in Sargent (1987, p. 16) and Rødseth (2000, p. 122). 36 Economics and Business Review, Vol. 9 (1), 2023 Unlike in the earlier versions of the KMG model a non-linear money de- mand function is considered: pY M = h (16) in which pY is the nominal value of expected demand in the goods market, r is the interest rate on bonds and h > 0 is a reaction parameter. In view of (14)–(16), to enable a constant growth in money supply, the inte- rest rate on bonds in every moment of time must satisfy the following equation: pY r = h (17) where M = μ = . = μ 1.4. Determinants of production decisions To counteract the difficulty with maintaining the continuity of sales caused by too low levels of stocks, or the reduction in revenues resulting from too high levels of stocks, producers strive to maintain a desired ratio of stocks to expected demand. Hence the desired level of stocks N satisfies equation: d e N = β Y (18) where β is the desired ratio of inventory to the expected demand. Change in actual inventories N equals the difference between output Y and demand (equivalent to sales) Y : N = Y – Y (19) The decision about output level is based on three factors: currently expect- e e e d ed demand Y , change in expected Y = demand Y + β Y (r+ esulting β (N − frN om ) Equation (10)) Z n and the deviation of actual inventories from their desired level N – N. These assumptions are reflected in the following behavioural equation: Money demand function in the earlier KMG models is usually a linear function of the d e form M = h pY + h pK(r – r), where r is the steady state interest rate and r is the actual 1 2 o o interest rate (Chiarella et al., 2013, p. 247). Such a formula assumes that the steady state in- terest rate is known in advance. In the present KMG model money demand depends not on deviations r – r but on the actual interest rate r. Consequently, the steady state interest rate (denoted in the article by r ) is an endogenous variable whose value depends on the model parameters. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 37 e e d Y =Y + β Y + β (N − N) (20) Z n where β > 0, β > 0 are reaction parameters. n Z 1.5. Constraints of output by capital and labor resources Production requires inputs of fixed capital K and labor L. The dynamics of K are described by Equation (8). Labour supply is constant in time and equal to L . Capital and labour are complementary, so both production factors are necessary in specific amounts to generate a given volume of production (no substitution possible). As a consequence, the production technology is de - scribed by two coefficients: the potential capital efficiency coefficient y and p p p y = the labour productivity coefficient x. Coefficient y is ratio , where Y denotes potential output defined as the maximum production that can be ob- tained with the use of fixed capital K (and sufficient labour supply). It is assumed that decisions on production made in line with Equation (20) are always feasible with respect to capital, which means that in every mo- ment of time: p p Y ≤ Y = y K (21) The utilization degree of the existing fixed capital is measured by the capa - city utilization rate 0 ≤ u ≤ 1 representing the ratio of output Y (determined p p by (20)) to potential output Y = y K: u= (22) Deviations of the capacity utilization rate from its natural level 0 < u < 1 influence price dynamics, as shown in Equation (5). (In particular, according to (5), if u exceeds u̅ firms have limited possibilities of increasing output in a short time and then are more likely to raise prices.) By assumption, labo - ur demand L never exceeds labour supply L , so labour demand is identical The equivalent of Equation (20) in the earlier versions of the KMG models does not con - e e d tain component β Y and takes a simpler form: Y = Y + nN + β (N – N) (e.g. Chiarella et al., n d 2013, p. 247). The above equation is interpreted in such a way that output Y is designed to meet e d expected demand Y and adjust inventories to the desired level given by Z = nN + β (N – N). n d Factor nN however has no clear economic meaning and is probably added to ensure balanced growth in the steady-state. 38 Economics and Business Review, Vol. 9 (1), 2023 to employment. Hence, labour productivity coefficient x, defined as output Y 1 per worker, is expressed as ratio x = , and labuor demand equals L = Y . L x Labour productivity x grows exogenously at a constant rate n: n= (23) Thanks to this, despite constant labour supply L , the constraint: (24) L = Y ≤ L is satisfied at any moment of time, even when output grows infinitely (later a balanced growth path, at which all quantitative variables, such as output, fixed capital, consumption and investments, grow at rate n will be considered). A counterpart to capacity utilization rate in the case of labour market is the employment rate defined as a ratio of employment to labour supply: V = (25) Deviations of V from its natural level 0<V < 1 influence wage dynamics according to Equation (4). 1.6. Investments and savings So far, only sources of the financing of private consumption C and govern- e e e  ˆ ment spending G have been presented. As far as investments I = K + δK = i ρ ar− e (r− π ) K + i u− u +Y K ( ) ( ) 1 2 concerned the focus has been on the behavioural Equation (8): e e e  ˆ K = i ρ − (r− π ) K + i u− u +Y K ( ) ( ) 1 2 e e e which presents factors determining the decision on net investments K with = i -ρ − (r− π ) K + i u− u +Y K ( ) ( ) 1 2 out indicating sources of their financing. To fill this gap the standard identi - e e e e e e ˆ ˆ   ty K = S = i will ρ −be (r− deriv π ) ed K + belo i u w − which u +Yimplies K that ex post net investments K = i ρ − (r− π ) K + i u− u +Y K ( ) ( ) ( ) ( ) 1 2 1 2 equal total saving S composed of private savings S and government savings S : p g S = S + S (26) p g D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 39 Private savings S equal taxed capital income (on fixed capital and bonds) and are expressed by the Equation below:  rB S = (1− τ ) ρK + (27) p c     Equation (27) results from the simplifying assumption (reflected by Equation (2)) that, rather than saving, workers spend their whole taxed labo- ur income entirely on consumption while capital owners do not consume at all, devoting their whole taxed capital income (from fixed capital and bonds) to savings (an assumption often found in post-Keynesian literature). Government savings S (if negative—government deficit) are the differen - ce between tax revenues and the sum of government spending on goods and services G and the real value of paid interest on bonds B/p:  rB S =T− G+ (28) g     Equations (27)–(28) imply that:     rB rB S= 1−τ ρK + +T− G+ ( ) (29) c     p p     d d According to (9) the realized profit on fixed capital equals ρK = Y – ωL – δK.   rB Taxes in view of (12) satisfy Equation T = τ ωL + τ ρK + . Substituting w c    both equations into (29) results in: d d S = Y – (1 – τ )ωL – δK – G (30) e e e  ˆ In view of C = (1 – τ )ωL and I = K + δK = i ρ (E− qua (r− tions π )(2) Kand + i (7)) u− u the +final Y K ( ) ( ) 1 2 demand Y = C + I + G equals: e e e d d  ˆ Y = (1 – τ )ωL + K + δK + G = i ρ − ( r− π ) K + i u− u(31)+Y K ( ) ( ) 1 2 Equations (30)–(31) imply directly that: e e e  ˆ K ̇= S = i ρ − (r− π ) K + i u− u +Y K (32) ( ) ( ) 1 2 e e e  ˆ which means that ex post net investments K ar= ie financed ρ − (r−bπ y t)otK al +sa i vings u− u S+ . Y K ( ) ( ) 1 2 40 Economics and Business Review, Vol. 9 (1), 2023 2. The steady state stability of the intensive form KMG In this section, the KMG model is reduced to an intensive form and its steady state is derived. Next, a theorem about local asymptotic stability is formulated. 2.1. Derivation of the intensive form KMG model As in the case of many other models of economic growth, an interesting question is whether the economy described by this version of KMG model can evolve sustainably along the balanced growth path. Balanced growth is defined as a growth in which all quantitative variables of the model grow at the same growth rate. This implies that, in the process of balanced growth, proportions between model variables remain constant. For this reason, to verify if balanced growth is a possible outcome in the presented model, one needs to find such proportions between variables of the model that allow for such growth. To do this, the original model must be transformed into the intensive form model, whose variables describe proportions between varia - bles of the original model. The variables of the intensive form KMG model are as follows: d d real labour income ωL per unit of output: U = ωL /Y, eectiv ff e labour supply xL per unit of capital: l = xL/K, real money supply M/p per unit of capital: m = M/pK, e e expected final demand per unit of capital: y = Y /K, inventory stock per unit of capital: ν = N/K, real value of bonds B/p per unit of capital: b = B/pK, expected inflation: π , d d final demand per unit of capital: y = Y /K, output per unit of capital: y = Y/K. To derive the intensive form KMG model the start is from the real labour d d d income ωL per unit of output: U = ωL /Y. Ratio Y/L is labour productivity, denoted by x. Hence U = ω/x, which implies that the growth rate of U equals: ˆ ˆ U = ω− x (33) ˆ ˆ ˆ The formula for the real wage ω = w/p implies that ω= w− p. According to ˆ ˆ (23), labour productivity grows exponentially at a constant ratU e = n, ω so − x = x. In view of these: ˆ ˆ U = w− p−n (34) D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 41 ˆ ˆ ˆ ω= w− p According to (4) and (5), the growth rate of nominal wage and the infla - ˆ ˆ ˆ ω= w− p tion rate are given by equations: w= β (V −V )+κ p+ (1−κ )π +n (35) w w w p= β u−u +κ w−n + 1−κ π (36) ( ) ( ) ( ) p p p Solving the above system of equations for and yields: w= κ β V −V +κ β (u−u) + π +n (37) ( ( ) ) w w p (38) p= κ κ β V −V + β (u−u) + π ( ( ) ) p w p where κ= , κ κ ≠ 1. w p 1−κ κ w p Substituting (37)–(38) into (34) aer ft simplifications results in the first equ - ation of the intensive form model:     y y   U = κ β 1−κ −V − β (1−κ ) −u (39)  ( )  w p   p w     l y       y y where V = , u= . l y The formula for eectiv ff e labor supply per unit of capital, l = xL/K, implies that: ˆ ˆ ˆ l = x+ L− K (40) According to (8), the growth rate of fixed capital is given by: e e e ˆ ˆ K = i ρ − r− π + i u− u +Y (41) ( ) ( ) ( ) 1 2 where: e d e d Y −ωL −δK Y ωL δK e e ρ = = − − = y −Uy−δ (42) K K K K d e d Y −Y  y  ˆ ˆ Y = ω+ β = ω+ β −1 (43) e e   y e y e Y y   e e pY py (44) r = h = h M m 42 Economics and Business Review, Vol. 9 (1), 2023 By assumption labor supply is constant, so L= 0. Moreover x = n. Hence in view of (41)–(44) the second equation of the intensive form model is obtained: d e         y  hy  y e e l = n− U+ n+ β −1 −i y −Uy− δ− − π −i − u (45)  e        e 1   2 p   y m y           where u = y/y and U+n= ω (see Equation (34)). Now proceed to variable m = M/pK which is the real money supply M/p per unit of fixed capital. The growth rate of m equals: ˆ ˆ ˆ ˆ (46) m= M− p−K By assumption, the nominal money supply grows at a constant rate μ, so ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ M = μ. The inflation mra=te M− p − is K described by (38). The growth ratm e = of M capit − pal −K , ˆ ˆ according to (40) and Equations x = n, L= 0 equals K = n− l . Taking all of the- se into account the third equation is obtained:   y  y    m= μ−π −n−κ κ β −V + β −u + l (47)   p w  p    l y       The fourth equation of the intensive form model describes changes in in - flation expectations. It is the same as in Section 1.1, i.e.: e e π = β αp+(1− α)π− π (48) e( ) ˆ ˆ mwher = M e − p sa−K tisfies (38). The formula for expected demand per unit of capital implies that: e e ˆ ˆ y =Y − K (49) Hence, by substituting (43) and K = n− l to (48) the fifth equation is ob - tained:    y  e e y = y U+ n+ β −1 −n+ l (50)  e    y e       ˆ ˆ where satisfies (45). K = n− l The sixth equation describes the dynamics of inventory stocks per unit of capital . To derive its intensive form start is from Equation: v=        NK − NK N N K N N K N v= = − = − = −vK (51) K K K K K K K K D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 43  ˆ According to (19), N =Y −Y . Hence, by using K = n− l , y = Y/K, and d d y = Y /K, the sixth Equation is obtained: ν= y− y − ν(n− l ) (52) The seventh equation of the intensive form model reflects changes in the real value of bonds per unit of capital = . The definition of b implies that: pK   B B B  ˆ ˆ ˆ ˆ b= − K + p = −b K + p (53) ( ) ( ) Kp Kp Kp   According to (13) and (11), B= pG+rB− pT− M and G = gK. Hence: T M  ˆ b= g +rb− − −b K + p (54) ( ) K Kp ˆ ˆ   In the next step EquaB tions = p (12), G+r(5) B−and pT− M = μM and to (54) are K = n− l substituted which gives the final form of the seventh equation of the inten - sive form model: b= g + rb−τ y −Uy−δ+ rb −τ Uy− μm− ( ) c w      y  y   −b n−l +κ β −u +κ β −V + π (55)     p  p w      y l         where m=M/pK. By collecting together the seven equations derived above the following system of seven nonlinear differential equations is obtained which describe the dynamics of proportions between variables of the original KMG model from Section 1.     y y   U =Uκ β 1−κ −V − β (1−κ ) −u (56)  ( )  w p   p w     l y       d e        y   hy   y  e e  ˆ l = l n− U+n+ β −1 −+ i y −Uy− δ− − π −i − u    e      1   2  e p     y m y            d e          y  hy  y e e  ˆ l = l n− U+n + β −1 − –+ i y −Uy− δ− − π −i − u (57)  e        1   2  y e p     y m y                 y  y    m = m μ−π −n−κ κ β −V + β −u + l (58)     p w  p      l y        44 Economics and Business Review, Vol. 9 (1), 2023        y y   e e e π = β α κ κ β −V + β −u + π + 1−α π−π  (59) e   ( )    p w  p π p       l y               e e ˆ ˆ  (60) y = y U+ β −1 + l  e          ν= y− y − ν(n− l ) (61) b= g + rb−τ y −Uy−δ+ rb −τ Uy− μm− ( ) c w      y  y   − b n−l +κ β −u +κ β −V +π (62)     p  p w      y l         e e It should be emphasized that besides variables U, l, m, π , y , v, b there are also two additional variables which appear in Equations (56)–(62). These are the final demand per unit of capital y = and the production per unit of capital y = . To identify interrelations between the above variables it sho - uld be noted that, according to (1) and (20), variables Y and Y appear in the following equations of the original model from Section 1: Y = C + I + G (63) e e d Y =Y + β Y + β (N − N ) (64) Z n By dividing both equations by the result is: C I G y = + + (65) K K K e d   Y N N y = y + β + β − (66) Z n  K K K   e where y = and v = N/K. Substituting Equations (2), (7) and (11) into (65) and Equations (43), (18) into (66) results in the following system of two linear equations (variables Y and Y appear on both sides of these equations): D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 45    hy  d e e y = 1−τ yU+i y − yU−δ− −π +i u−u + δ+ g + ( ) ( )   w 1   2           y y y   (67) + κ β 1−κ −V − β (1−κ ) −u +n+ β −1  ( )  e w p   p w     p e   l y y         e e y = y + β β y − β ν+ n n        y y y   + y β κ β 1−κ −V − β 1−κ −u +n+ β −1 (68)  ( ) ( ) e       Z w p   p w p y e     l y y           Since variables y and y appear also in differential Equations (56)–(62), Equations (67)–(68) must be taken into account in any analysis of the above system of differential equations. This means that the complete KMG mod- el in intensive form is composed not only of Equations (56)–(62) but also of Equations (67)–(68). 2.2. The steady state The economy described by the KMG model presented in section 1 rema - ins in the steady state if the proportions between its variables allow for a ba- lanced growth of the economy at a constant growth rate equal to the growth rate of labour productivity n. e e d Formally, the steady state is described by a vector U, l , m, π , y , υ, b , y, y ( ) for which the right hand sides of all Equations (56)–(62) are equal to zero and additionally Equations (67)–(68) are satisfied. By solving Equations (56)–(62) with left hand sides zeroed out and consi - dering (67)–(68) it is easy to obtain analytically that the formulas describing the steady state values of the intensive form KMG model are as follows: p p p γuy − (n+ δ+ g ) uy huy U = , l = , m= γ 1−τ uy V r ( )  1−γ d e p p π = μ−n, y = y = γuy , v = uy n    c c g −τ γuy − −δ −τ − μm    c w   1−τ 1−τ ( ) ( ) w w    b =   c c p p n γuy δ τ γuy δ π (69) − + + + − − +     1−τ 1−τ ( ) ( )  w w    46 Economics and Business Review, Vol. 9 (1), 2023 where: g + μ−τ μ−n−δ+ γuy n+ β ( ) n p γ = , r = , c = γuy −(n+ δ+ g ) n+ β n + β β n+ β 1−τ ( ) Z n n w g + μ−τ μ−n−δ+ γuy ( ) n+ β w n 12 γ = , r = , c = γuy − n+ δ+ g ( ) n+ β n + β β n+ β (1−τ ) Z n n w To show that proportions described by the above formulas indeed allow for balanced growth of the economy at the rate n the first focus is on the eec ff - tive labor supply per unit of capital l = xL/K. In the steady state, the growth ˆ ˆ ˆ rate of l is zero, so l = x+ L− K = 0. By assumption, labour supply L is con- stant while labour productivity x grows at rate n. Hence, in the steady state, the growth rate of fixed capital equals the growth rate of labour productiv - p p p ˆ ˆ ˆ l = x+ ity L− , K = = n 0 . According: to (21) and (22), Y ≤ Y = y K and u=Y/Y . In view of p p p this, uy =Y / K . Since uy = is Y cons / K tant in time, output uy =Y /in Kthe steady state must also grow at rate n. When examining other formulas, one can easily no- tice that in the steady state also other quantitative variables, such as invest - ments, consumption and government purchases, grow at rate n. Additionally, taking into account that in the steady state the expected inflation equals the actual inflation, it is easy to find that in the steady state also the growth rate ˆ ˆ of the real wage ω= w− p̂ is equal to n. What seems to be interesting is that, according to equation c = γuy −(n+ δ+ g), consumption per unit of capi - tal (and thus also per unit of output) decreases in the steady state when the growth rate of labour productivity n (equal to the balanced growth rate) in- e e d creases. Moreover, an increase in n also lowers the ratio U ,of l , labor m, π ,inc y ome , υ, b , y, y ( ) to output. Finally, it is worth stressing that the only variable which depends in the steady state on the growth of money supply μ is inflation rate π = μ−n. Thus money neutrality is confirmed. 2.3. Stability of the steady state The main topic of the paper is the local asymptotic stability of the discussed KMG model starting from the formal definition of stability of the steady state of the KMG model in intensive form from the previous subsection. e e Definition. The steady state (U , l , m, π , y , υ, b ) of model (56)–(62), (67)– (68) is locally asymptotically stable if there is such neighborhood of this steady Due to the limited number of words imposed by the publisher, the article omits the derivation of Equation (69). However, this derivation as well as all other derivations and other details may be made available to readers by the author on request at his e-mail address. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 47 state that every solution of the model starting from this neighborhood, con - verges toward it when time tends to infinity, i.e.: e e e e (U , l , m , π , y , v , b ) → (U , l , m, π , y , v , b ) t t t t t t t t → ∞ To prove the stability defined above the following assumptions will be used. Assumption 1. Parameters β , β , α, β , β , β > 0 occurring in equations: z n p w e e d – Y =Y + β Y + β (N − N ) z n e e – π = β αp+ (1−α)(μ−n)−π e( ) ˆ ˆ p= β (u−u)+κ (w−n)+ (1−κ )π p p p – w= β (V −V )+κ p+ (1−κ ) π +n w w w d e Y −Y – Y = ω+ β are sufficiently small. According to Assumption 1 output is determined mainly on the basis of de - e e mand expectations Y , whose growth rate Y depends mainly on the growth ˆ ˆ rate of the real wage ω= w− p. Changes in inflation expectations π are stabilized strongly by a constant (μ – n) factor reflecting inflation in the steady state. Moreover, deviations of the capacity utilization rate from its normal level ˆ ˆ u−u weakly influence the inflation rω ate = w− p, and deviations of the employment rate from its natural level V −V weakly influence the growth rate of nomi - ˆ ˆ nal wag ω e = w.− p Assumption 2. The growth rate of the nominal money supply μ cannot ex- ceed the sum of labour productivity growth rate n and the capital deprecia- tion rate δ: Assumption 3. Parameter occurring in the investment equation: e e e I = i ρ − (r− π ) K + i (u− u)K +Y K + δK ( ) 1 2 is sufficiently large. According to Assumption 3, investment demand is assumed to be highly sensitive to the dier ff ence between the expected profit from fixed capital and the expected real interest rate. Assumption 4. Parameter κ , (0 < κ < 1) occurring in the price dynamics p p equation: p= β (u−u)+κ (w−n)+ (1−κ )π p p p is sufficiently close to 1. 48 Economics and Business Review, Vol. 9 (1), 2023 According to Assumption 4, the inflation rate is more sensitive to the dif - ˆ ˆ ference between nominal wage growth rω ate = w − and p the rate of growth in la- bour productivity n= than to the expected inflation rate π . Assumption 5. The growth rate of money supply satisfies the inequality: (1− τ )r < μ Assumption 6. The nominal interest rate in the steady state is positive, e.g.: g + μ− τ (μ−n− δ+ γuy ) r = > 0 (1− τ ) It is worth noting that Assumption 5 is satisfied for a sufficiently high ca - pital tax rate τ , while Assumption 6 is met for a sufficiently low labour inco - me tax rate τ . Assumptions 1–6 allow for the proof of the main result of the paper, which is the following stability theorem. Theorem 1. If Assumptions 1–6 are satisfied, then the steady state of model (56)–(62), (67)–(68) is locally asymptotically stable. 3. The proof of stability of the KMG model 3.1. General remarks about the proof To prove Theorem 1, it must be shown that all of the eigenvalues (charac- teristic roots) of the 7 × 7 Jacobian matrix J of model (56)–(62), (67)–(68) in e e the steady state x = U , l , m, π , y , v , b are either negative numbers or com - ( ) plex numbers with negative real parts (see Gandolfo, 1996, pp. 360–362). As already mentioned in the introduction, the examination of the eige - nvalues of the Jacobian matrix J is based on the idea of the cascade of sta- ble matrices approach applied originally by Chiarella, Franke, Flaschel and Semmler in the stability proof of their version of the KMG model (Chiarella Examination of the eigenvalues of Jacobian matrix is a standard way of proving local as- ymptotic stability. In most cases it is applied however to two dimensional dynamical systems (e.g. Filipowicz et al., 2016). The difficulties in examining eigenvalues grow very rapidly with the dimension of the system becoming an extremely complex maer tt in a case of high dimen- sional systems like the KMG model. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 49 et al., 2006). The implementation of this idea in the stability proof presented below is realized in four steps. In the first step, the Jacobian matrix J is determined. In the second step, the characteristic polynomial of Jacobian matrix J (considered to be determi- nant W (λ) = det(J – λI)) is reduced to the second degree polynomial W 7 7 2 by resetting the values of some parameters to zero. This is realized in three stages. The first vector of parameters β = (β , β , α) is reset to zero. The next n z parameter β and finally parameter β are also reset to zero. The third step w p consists in showing that both roots of polynomial W have negative real parts. In the last (fourth) step, by gradually restoring the positive values of the parameters previously reset to zero, it is demonstrated that all eigenva- lues of Jacobian matrix J have roots with negative real parts. To examine the Jacobian matrix subsequent functions which appear on the right hand sides of Equations (56)–(62), (67)–(68) are denoted by: F = F (X) (i = 1, 2, …, 7) i i where: e e X = (x = U, x = l, x = m, x = π , x = y , x = v, x = b) 1 2 3 4 5 6 7 Consequently, the Jacobian matrix J in the steady state can be expressed in the following way: F F F F F F F  e e  1U 1l 1m 1v 1b 1π 1y   F F F F F F F e e  2U 2l 2m 2v 2b 2π 2 y   F F F F F F F e e 3U 3l 3m 3v 3b 3π 3 y     F F F F F F F J = e e (70) 4U 4l 4m 4v 4b 4π 4 y    F F F F F F F  e e 5U 4l 5m 5v 5b 5π 5 y   F F F F F F F e e  6U 6l 6m 6v 6b 6π 6 y     F F F F F F F e e 7U 7l 7m 7v 7b 7π 7 y   where elements of matrix J are derivatives of functions F with respect to 7 i e e model variables calculated in the steady state , which x = U, l ,m,π , y ,v ,b ( ) means that: ∂F (X) F = , (i, j= 1, 2,…, 7) ij ∂x x=x It is worth emphasizing that due to many modifications introduced into the KMG model the proof of its stability presented in the article although based on the idea of the cascade of stable matrices approach differs essentially in many details from the proofs of stability of other versions of KMG models. 50 Economics and Business Review, Vol. 9 (1), 2023 The eigenvalues λ , …, λ of matrix J are the roots of the following charac- 1 7 7 teristic equation of this matrix: W (λ) = det(J – λI) = 0 (71) 7 7 As stated at the beginning of this section at the first stage the vector of three reaction parameters is zeroed: β = (β , β , α) = 0. This reduces derivati - n z ∂F (X) ves F = (e.g. elements of Jacobian matrix J ) to derivatives: ij ∂x x=x ∂F (X) , (i, j = 1, …, 7) F = (β= 0) ij ∂x x=x Similarly symbol J (β = 0) or simply J is used to denote the 7 × 7 matrix of 7 7 such derivatives. Let F (X, 0), (i = 1, 2, …, 7), be functions obtained from F (X) by setting i i β = (β , β , α) = 0. n z On the basis of (56)–(62), it can be easily shown that: ∂F (X, 0) ∂F (X) 0 i F = (β= 0)= , (i, j = 1, …, 7) (72) ij ∂x ∂x j j x=x x=x Equation (72) simplifies the derivation of Jacobian matrix J because, in- ∂F (X) F = stead of determining derivatives and then reducing them to ij ∂x x=x ∂F (X) ∂F (X) 0 i 0 i F = (β= 0), one can obtain F = more easily (bβy =nu 0)llifying the first ij ij ∂x ∂x j j x=x x=x parameters β = (β , β , α) in functions F (X) and then determining derivati - n z i ∂F (X,0) ves . ∂x x=x In particular, the derivation of Jacobian matrix J in the method described above reveals that some of its elements F are zeros: ij 0 0  0 F 0 0 F 0 0  1l 1y   0 0 0 0 0  F F F F F 0 0  e e 2u 2l 2m 2π 2 y   0 0 0 0 0 F F F F F 0 0  e e  3u 3l 3m 3π 3 y   0 0 0 0 0 F F 0 0 (73) J = e e   4π 4 y   0 0 0 0 F 0 F F F 0 0 e e 5u 5m   5π 5 y   0 0 0 0 0 0 F F F F F F 0 e e 6u 6l 6m 6v  6π 6 y   0 0 0 0 0 0 0    F F F F F F F e e 7u 7l 7m 7v 7b 7π 7 y   D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 51 Because of the limited length of this paper, it is not possible to present all the formulas describing elements of J . At the next stages of the proof only some of them will be presented where this is especially needed. 3.2. Reducing polynomials degrees 0 0 λ =−β < 0, λ =−n< 0 Lemma 1. Matrix J has three negative eigenvalues 1 2 and λ = (1− τ )r − μ< 0. 3 c 0 0 Proof. In view of (73), polynomial W (λ)= det( J − λI) can be expanded to 7 7 the following form: 0 0 0 0 0 W (λ)= (F − λ)(F − λ)(F − λ)⋅W (λ) (74) 7 7b 6v 4 4π where: 0 0 −λ F 0 F  1l 1y   0 0 0 0  F F − λ F F  2u 2l 2m 2 y 0 0 W (λ)= det(J − λI)= det  (75) 4 4 0 0 0 0 F F F − λ F  e  3u 3l 3m 3 y   0 0 0   F 0 F F − λ 5u 5m 5 y   ∂F 0 7 F = The first step in obtaining formula for is to note that in view 7b ∂b x=x e e of (62) function F = F (U , l, m, π , y , v, b) has the following form: 7 7 e e hy  hy  F = g + b−τ y −Uy−δ+ b −τ Uy− μm− 7 c  w m m         y y   − bn−l +κ β −u +κ β −V + π      p p w      y l         0 0 In general, according to (72), F is a derivative of function F obtained from ij i function F by nullifying the vector of parameters β = (β , β , α) = 0. Function i n z F , however, does not contain the above parameters, so F = F . Hence: 7 7 7 52 Economics and Business Review, Vol. 9 (1), 2023 0 e e         ∂F ∂F ∂ hy  hy  y y   e e 7 7 ˆ = =  g + b−τ y −Uy−δ+ b −τ Uy− μm−b n−l ++κ β −u ++κ β −V + π =       c  w p p w        ∂b ∂b ∂b m m y l             0 0 0 e e e e e e        ∂F∂F∂F ∂F∂F∂F ∂∂∂ hyhyhy  hyhyhy yy y yy y   e e e e e e 7 7 7 7 7 7 ˆ ˆ ˆ === === g g+g++ bb−b−τ−τ τyy−y−U− U yU y−y−δ−δ+δ++ bb−b−τ−τUτU yU y−y−μ−m μm μ−m−b−−b nbn−n−l−+l +l++κ+κ+βκββ −−u−uu++++κ+κ+βκββ −−V− VV++π+ππ===       ccc  w w w p pp  p pwpw w  p p p       ∂b∂b∂b ∂b∂b∂b∂b∂b∂b mmm mmm yy y l l l              e e hy  ∂y hy  = −τ −U + − c  m ∂b m       ∂y  y  y   − − − + − + − + − τ U n l κ β u κ β V π    w p  p w      ∂b y l          ∂y ∂y        ∂b ∂b – b κ β +κ β     p p w y l           It follows form Equations (68) and (20) that y does not depend on b, hence: ∂y = 0 ∂b This in turn implies that: 0 0 e e e e     ∂F∂F ∂F∂F hyhy hyhy yy  yy   e e 7 7 7 7 ˆ ˆ == == −−ττ −−nn−−l + l +κκ ββ −−uu++++ κκββ −−VV ++ππ    (76)   c c p p  p pw w  p p     ∂b∂b ∂b∂b mm mm yy l l         Substituting values of the variables U, l, m, π, y, v in the steady state into ∂F above Equation yields the formula for the derivative in the steady state: ∂b ∂F F = = r − τ r − μ= (1− τ )r − μ 7b c c ∂b x=x which in view of Assumption 5 implies that F < 0. 7b 0 0 Similarly it can be derived that: F =−n and F =−β . Hence, in view of e e 6v 4π π 0 0 W (λ) (74) the polynomial has three negative roots λ =−β < 0, λ =−n< 0 7 e 1 2 and λ = (1− τ )r − μ< 0, which are the eigenvalues of matrix J . 3 c  D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 53 0 0 0 0 0 The remaining roots λ , λ , λ , λ of polynomial W (λ) are also roots of 4 5 6 7 7 the fourth degree polynomial W (λ) defined by (75). 00 00 Let be J the matrix obtained from J under assumption that not only 4 4 β= (β , β , α)= 0 but also β = 0. The characteristic polynomial of matrix J n z 4 is given by: −λ 0 0 F  1y   00 00 00  F −λ F F  2u 2m 2 y 00 00   W (λ)= det(J − λI)= det (77) 4 4 00 00 00 F 0 F − λ F  e  3u 3m 3 y   00 00 00   F 0 F F − λ 5u 5m 5 y   Expanding W (λ) yields:   −λ 0 F 1y   00 00 00 00 00 00   (78) W (λ)= λ⋅W (λ)= λ⋅det(J − λI)= λ⋅det F F − λ F 4 3 3 3u 3m 3 y   00 00 00   F F F − λ 5u 5m 5 y   00 00 00 00 λ , λ , λ , λ which implies that one of the roots of polynomial W (λ) is 4 5 6 7 zero. Suppose that: λ = 0 (79) Finally add to Assumptions β= (β , β , α)= 0, β = 0 that also β = 0. Matrix n z w p 000 000 reduces then to . The polynomial: J J 3 3   −λ 0 0   000 000 000 000 000   (80) W (λ)= det(J − λI)= det F F − λ F 3 3 3u 3m 3 y   000 000 000   F F F − λ 5u 5m 5 y   aer e ft xpansion takes the form: 000 000   F − λ F 3m 3 y 000 000 000   W (λ)= λ⋅W (λ)= λ⋅det(J − λI)= λ⋅det (81) 3 2 2 000 000   F F − λ 5m 5 y   000 000   F − λ F 3m 3 y 000 000 000 000 000 000   W (λ)= λ⋅W (λ)= λ⋅det(J − λI)= λ⋅det This implies that one of the roots of is zero. Suppose λ , λ , λ 3 2 2 5 6 7 000 000   F F − λ 5m that: 5 y   000 000 000 = 0 (82) λ , λ , λ 5 6 7 54 Economics and Business Review, Vol. 9 (1), 2023 000 000  F F  3m 3 y 000 00 00 00 000000 000 000 Lemma 2. Both eigenvalues and of matrix   are ei- λ ,λλ ,,λ λ , λ J = 5 56 6 7 7 2 000 000   F F 5m 5 y   ther negative or complex numbers with negative real parts. 000 000   F F 3m 3 y 000 000   Proof. The characteristic equation det(J − λI)= 0 of matrix J c=an be ex- 000 000   F F 5m pressed as: 5 y   λ + a λ+ a = 0 (83) 1 2 000 000 000 where: a =−tr J =−(F + F ) (84) 1 2 3m 5 y 000 000 000 000 000 a = det J = F F − F F (85) e e 2 2 3m 5m 5 y 3 y 000 000  F F  3m 3 y It can be verified that elements of matrix ar e described by the follo- J = 000 000   F F wing formulas: 5m 5 y     uy F =−i r  (86) 3m 1   uy − β      y −1 κ κβ   huy κβ r i w p y w 2   F =− V + + β  +i 1−U − + (87) e e   3 y p p y p p p   r  uy y uy uy y        F = β −i (δ− μ+ n)− ui (88) e e 1 2 5 y y 000 2 F =− r (89) 5m F < 0 F < 0 Assumptions 2, 3 and 6 imply that and . Hence: 3m 5 y (90) a =−tr J > 0 1 2 Inserting (86)–(89) into (85) aer simplific ft ations implies that: p p −β − uy + (n+ δ+ g)+ 2i (δ− μ+ n)− β uy  e e y y a = i rβ   (91) 2 1 y p   uy − β   Assumption 2 implies that δ – μ + n > 0. Hence, i > 0 if is sufficiently large (Assumption 3) and parameter is sufficiently small (Assumption 1), then: β > 0 (92) a = det J > 0 2 2 In view of Routh-Hurwitz theorem (Gandolfo, 2005, pp. 221–222), condi- 000 000 tions (90) and (92) imply that both eigenvalues λ and of matrix J have 6 7 negative real parts. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 55 3.3. Restoring positive values of reaction parameters In the further part of the proof of Theorem 1 the positive values of tem - porarily nullified parameters , β , β will be gradually restored. β= (β , β , α) n z w p First by restoring positive value of parameter β and exploiting Lemma 2 00 000 the following Lemma 3 on the eigenvalues of matrix J → (which J appear when s in β → 0 3 3 p Equation (78)) will be proved. Lemma 3. Suppose that β= (β , β , α) = 0 and β = 0. Then, for sufficiently n z 00 00 00 00 000 small values of parameters β > 0 and β > 0, all λ , λ , λ of matrix J → areJ when β → 0 5 6 7 3 3 p p y either negative or complex numbers with negative real parts. Proof. The starting point are matrices:     00 00 F F ee 00 00 00 11yy     0000 0000 0000 0000 000000 000000 000000 000000     JJ == F F F F F F a and an nd d JJ == F F F F F F ee ee 33 33u u 33m m 33 33u u 33m m 33yy 33yy     0000 0000 0000 000000 000000 000000     F F F F F F F F F F F F ee ee 55u u 55m m 55u u 55m m 55yy 55yy     00 000 00 000 Ma J trix → J is w ob hteained n β fr→ om 0 J → by Jnullifying when parame β → ter 0 β > 0 in all ele- 3 3 p 3 3 p 00 000 ments of J .→ J when β → 0 3 3 p 0000 000000 Moreover: JJ →→ JJ when w wh heen n β β →→ 00 (93) 33 33 pp According to (80) and (81): 000 000 000 W (λ)= det(J − λI)= λ⋅det(J − λI) (94) 3 3 2 000 000  F F  3m 3 y which implies that the two eigenvalues of matrix ar e also eigen values of J = 000 000 00 000 00 000   F F ma J trix → J . Hence whenin vie β w → of 0 Lemma 2 and (94) it is conclud5ed m that ma J trix → J when β → 0 5 y 3 3 p  3 3 p 000 000 000000 000 has one zero eigenvalue λ = 0 and two eigenvalues λ ⋅and λλ ⋅> λ 0 which > 0 are 5 6 67 7 either negative or complex numbers with negative real parts. In both cases: 000 000 λ ⋅ λ > 0 (95) 6 7 00 000 Due to the continuity of matrix J → with J respect when to β β ≥ → 0 0 (condition (93)) 3 3 p 00 000 and the continuity of the eigenvalues of matrix J wi→th J respw ect hetn o its βelemen → 0 ts 3 3 p 000000 000000 00 000 λλ ⋅⋅λλ >>00 for a sufficiently small value of β > 0, the two eigenvalues , of matrix J → J when β → 0 6 6 7 7 3 3 p 000 000000 000 λ ⋅λλ ⋅ λ> 0 > 0 (corresponding to and ) are also either negative or complex numbers 6 67 7 with negative real parts, satisfying inequality 000000 000000 λλ ⋅⋅λλ >>00 ∙ > 0 (96) 66 77 56 Economics and Business Review, Vol. 9 (1), 2023 000 000 To complete the proof the third eigenvalue λ corresponding to λ = 0 5 5 must be examined. For this purpose the determinant will be considered:   0 0 F 1y 00 00     F F 3u 3m 00 00 00 00 00   detJ = det F F F = F ⋅det   (97) e e 3 3u 3m 3 y 1y 00 00     F F 5u 5m   00 00 00   F F F 5u 5m 5 y   The determinant of any matrix equals the product of its eigenvalues. Hence: 00 00 00 00 det J = λ ⋅ λ ⋅ λ (98) 3 5 6 7 The elements of matrix are described by the following formulas:  β (1−κ ) p w F =−Uκ (99)   1y p     β − τ β −i uy e e h w 1 y y 00 p 2   (100) F =− (uy ) 3u   r uy − β      uy − (n+ δ+ g uy 00 p F =−i uy − − δ+ μ−n  (101) 3m 1    (1−τ ) uy − β    00 p 2 F = (uy ) i (102) 5u 1   i uy −(n+ δ+ g) 00 1 p F =− uy − −δ+ μ−n (103) 5m   h (1−τ )  w  Inserting (99)–(103) into (97) yields aer simplific ft ations:   (1− τ )β β (1−κ ) e   w 00 p w p 2 y   det J =−i rUκ (uy ) (104) 3 1   p p   y uy − β       In view of inequalities 0 < κ < 1 and 0 < τ < 1, it follows from (60) that w w for a sufficiently small β > 0: 00 000 detJ → < 0J when β → 0 3 3 p Hence, in view of (96) and (98) it is obvious that that for a sufficiently small 00 000 000 β > 0 and βy > 0, the third eigenvalue of matrix J is → a Jnegativw e h numbe en βr → λ 0 < 0. 3 3 p 5  D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 57 Below, by restoring the positive value of parameter β and using Lemma 3 it will be proven Lemma 4 on the eigenvalues of the following matrix: 0 0  0 F 0 F  1l 1y   0 0 0 0  F F F F  2u 2l 2m 2 y (105) J =  0 0 0 0 F F F F  e  3u 3l 3m 3 y   0 0 0   F 0 F F 5u 5m 5 y   appearing in (73)–(74), in which β= (β , β , α)= 0 while β > 0 and β > 0. n z p w 0 0  0 F 0 F  1l 1y Lemma 4. Suppose that β = (β , β , α) = 0 and β > 0, β > 0. Then, under n z p w   0 0 0 0 Assumptions 1–6 for sufficiently small values of parameters β > 0 and β > 0,  F F F F  pe w 2u 2l 2m 2 y 0 0 0 0 0   all eigenvalues λ , λ , λ , λ of matrix J are = either negative or complex num - 4 5 6 7 4 0 0 0 0 F F F F  e  3u 3l 3m bers with negative real parts. 3 y   0 0 0   F 0 F F 5u 5m Proof. The proof of Lemma 4 is similar to that of Lemma 3. The start will be 5 y   by considering the following matrix:   0 0 0 F 1y   00 00 00  F 0 F F  2u 2m 2 y J =  00 00 00 F 0 F F  e  3u 3m 3 y   00 00 00 00 0 0      F 00 F0 F 0 F 0 F 0 F e e e 1l 5u 5m 5 y 1y 1y        0 0 0 F 00 00 00 0 0 0 0 1y  F 0 F F  F F F F  e e  2u 2m 2u 2l 2m 2 y 2 y 00 0 00 00 00 which appears in (46). Matrix J is = obtained from J b=y nullifying parameter   F 4 0 F F  4 00 0 e 00 0 00 00 0 0 0 0 2u 2m 2 y 00     0 F F 0 0 00F F0 F F F F F F e e  1l e  e  3u 3m 3u 3l 3m β in all elements of J . Besid = e this:  1y 1y 3 y 3 y w 00 00 00        F 0 F F 0 0 00 0 000 00  00 e  00 00 0 0 0 3u 3m 3 y  F  FF F0 FF F   F 0 F F F 0 F F e e 2u 2l2u 2m 2m e e 5u 5m 5u 5m 2 y 5 y 2 y 5 y  0 00     00 00 00 J →= J when = β → 0   (106)   F4 0 4 F F 0 0 00 w 0 000 00 5u 5m 5 y  F FF F0 FF F     e e 3u 3l3u 3m 3m 3 y 3 y     0 00 0 000 00 According to (75):     F 0F F0 FF F e e 5u 5u 5m 5m 5 y 5 y       −λ 0 F 1y   00 00 00 00 00 00   (107) W (λ)= det(J − λI)= λ⋅det(J − λI)= λ⋅det F F − λ F 4 4 3 3u 3m 3 y   00 00 00   0 0 0 F  e  F F F λ e 1y 5u 5m 5 y     00 00 00  F 0 F F  2u 2m 2 y 00 0000 0000 0000 0000 It follows from Equation (107) that matrix J has = three eigen ddee vttalues JJ ==λλ , ⋅⋅λλ , ⋅⋅ λ λ 4 33 55 66 77 00 00 00 00 00 00 00 00 00 000   F 0 F F 0 0 0 F   e 3u 3m det J = λ ⋅ λ ⋅ λ which are identical to the eigenvalues of matrix J . → As Jshown wh in en the pr β oof → 0 1y 3 y 3 5 6 7 3 3 p   00 0000 0 00 0 0 00 0 00   00 00 00 00 00 00 of Lemma 3, the two eig deten Jde vtalues =J λ=⋅ λ λ ⋅⋅and λ λ ⋅ λ are either negative or complex   3 3 5 5 6 7 6 7 F 0 F F F 0 F F  e 2u 2m 5u 5m 5 y 2 y 00 00 00 00 00   numbers with negative real parts. The third eigen dvealue t J = λ ⋅of λ J⋅ λ is = a negati -  3 5 6 4 7 00 00 00 F 0 F F   3u 3m ve number: 3 y   00 00 00   F 0 F F 5u 5m 5 y   58 Economics and Business Review, Vol. 9 (1), 2023 00 00 00 00 det J = λ ⋅< 0 λ ⋅ λ (108) 3 5 6 7 Equation (108) also implies that the fourth eigenvalue of equals zero: λ = 0 (109) Hence, in view of (106) and the continuity of the eigenvalues of matrix J with respect to its elements, for a sufficiently small β > 0 the three eigenva- 0 0 0 00 00 00 lues λ , λ , λ of matrix (corresponding to λ , λ , λ ) are also either negative 5 6 7 5 6 7 or complex numbers with negative real parts. What remains to be investiga - 0 0 0 0 0 00 00 000 ted is the fourth eigenvalue λ of , λ J, λ corr , λ esponding to λ = 0 of J .→ J when β → 0 4 5 6 7 4 3 3 p For this purpose the sign of the determinant will be determined: 0 0 0 0 0 det J = λ ⋅ λ ⋅ λ ⋅ λ (110) 4 4 5 6 3 and use the inequalities: 00 00 00 (111) λ < 0, λ ⋅ λ > 0 5 6 7 (the second inequality is identical to (96)). It follows from (105) that: 0 0 0 detJ =−F ⋅ A − F ⋅ A (112) 4 1l 1 2 1y β (1−κ ) β (1−κ ) V   0 0 w p p w where F =−Uκβ (1−κ ) < 0, F =Uκ V − 1l w p   p 1y p p uy uy y   0 0 0 0 0 0   F F F e  F F F  2u 2m 2 y 2u 2l 2m     0 0 0 0 0 0   A = det F F F , A = det F F F e   1 3u 3m 2 3u 3l 3m 3 y    0 0  0 0 0 F 0 F   F F F 5u 5m e   5u 5m 5 y   Elements of determinants and have the form:   uy − 2β   F = κβ (1−κ )V (113) 2l w p   uy − β    β (1−κ ) β (1−κ ) w p p w F =Uκ V − (114)   p p 1y uy y   p p     i uy (uy − (n+ δ+ g) 0 p   F =− uy − −δ+ μ−n (115)   2m p   Vh uy − β (1−τ )  w    D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 59 p 2   (1− τ )β −i uy (uy ) w 1 0 y F =−   (116) 2u   V uy − β   p p     (uy − (n+ δ+ g) uy 0 p   F =−i uy − −δ+ μ−n (117)   2m 1   (1−τ ) uy − β  w      β − τ β −i uy e e h w 1 0 p 2 y y   F =− (uy ) (118) 3u   r uy − β    uy − 2β + κ β  e e h p 0 2 y y   F = κβ V (119) 3l w   r uy − β     i (uy −(n+ δ+ g) 0 p F =− uy − −δ+ μ−n (120) 5m   h (1−τ )  w  0 p 2 F = (uy ) i (121) 5u 1 F =−Uκβ (1−κ ) (122) 1l w p uy It follows from the above formulas that the first component of determinant 0 0 0 0 0 det J = λ ⋅ λ ⋅ λ ⋅(λ−F ⋅ A ) , is equal to and converges to zero when 0 < κ < 1 converges 4 4 5 6 3 1l 1 to 1 (Assumption 4). After being expanded and simplified, determinant takes the form: A = F (−F F )+ F (F F ) (123) 2 5u 3l 2m 5m 2u 3l Inserting (113)–(122) into (123) yields, aer simplific ft ation:    uy − 2β + κ β (1−τ )β e e e p w y y y p 2 A = (uy ) i rκβ V   (124) 2 1 w p p    uy − β uy − β e e y y    It follows from (124) that β > 0 if is sufficiently small (Assumption 1), then: A > 0 At the same time, when 0 < κ < 1 converges to 1 (Assumption 4), then: β (1−κ ) β (1−κ )   w p p w F =Uκ V − < 0   p p 1y uy y   60 Economics and Business Review, Vol. 9 (1), 2023 Consequently, if 0 < κ < 1 is sufficiently close to 1 (Assumption 4) and pa - rameters β > 0, β > 0, β > 0 and are sufficiently small (Assumption 1), then: p w 0 0 0 0 0 (125) detJ = λ ⋅ λ ⋅ λ ⋅ λ > 0 4 4 5 6 7 00 0 0 Since λ < 0, the corresponding eigenvalue λ of J for a sufficiently small 5 5 4 β > 0 can be either a negative number or a complex number with a negati - ve real part. 0 0 If λ < 0, then, in view of the continuity of the eigenvalues of matrix J with 5 4 respect to its elements, taking into account (106) and (111), for a sufficien - tly small β > 0: 0 0 0 λ ⋅ λ ⋅ λ < 0 (126) 5 6 7 is obtained, which in view of (125) implies directly that λ < 0. 0 0 0 The second possibility should be considered that when eigenvalue λ ⋅is λ ⋅ λ < 0 5 6 7 a complex number with a negative real part. Then, since complex eigenvalues (roots of a polynomial) always appear as conjugate numbers, the fourth eige- 0 0 0 0 0 0 0 nvalue λ must be the conjugate of λ ha⋅ λv⋅ing λ < the 0 same negative real part as λ ⋅ λ . ⋅ λ < 0 4 5 6 7 5 6 7 00 000 000 0 It has been demonstrated earlier that λ λ , ⋅⋅λ λλ ⋅⋅and ⋅ λ λλ <<⋅ λ0 0 ar< 0 e either negative or 55 665 776 7 complex numbers with negative real parts. Thus, the two remarks above on λ complete the proof of Lemma 4. To complete the whole proof of stability Theorem 1 it should be noted 0 0 0 0 0 0 0 that in view of lemmas 1 and 4 all eigenvalues λ , λ , λ , λ , λ , λ , λ of ma- 1 2 3 4 5 6 7 0 0 0 W (λ)= d trix et(J de− λ fined I)= b(( y 1(73) − τ )ar r e − either μ− λ)(neg −n− ativλe )(or −βcomple − λ)⋅W x number (λ) s with negative 7 7 c 4 π real parts, since: 0 0 0 W (λ)= det(J − λI)=((1− τ )r − μ− λ)(−n− λ)(−β − λ)⋅W (λ) (127) 7 7 c 4 π 0 0 0 W (λ)= det(J − λI)= (1− τ )r − μ− λ (−n− λ)(−β − λ)⋅W (λ) Matrix is obtained ( from Jacobian ) matrix J by nullifying the vector of pa- 7 7 c 4 π rameters β = (β , β , α) > 0. It can also be verified that: n z 0 0 0 W (λ)= det(J →− λ JI) when = (1−βτ → )r 0− μ− λ (−n− λ)(−β − λ)⋅W (λ) ( ) e 7 7 c 4 7 π In view of this and the continuity of the eigenvalues of matrix with respect to its elements, it may be concluded that all eigenvalues λ , λ , λ , λ , λ , λ , λ 1 2 3 4 5 6 7 of Jacobian matrix J are either negative or complex numbers with negative real parts. This proves theorem 1 stating that the steady state of model (56)– (62), (67)–(68) is locally asymptotically stable. 15 0 λ = (a−ib) The product of conjugate complex numbers λ = a+ ib and different from zero 4 5 0 0 2 2 is always positive since (a + ib)(a – ib) = a + b > 0. Hence, in view of (65), there is λ ⋅ λ > 0 4 5 0 0 0 0 0 0 0 and λ ⋅ λ > 0, which implies that detJ = λ ⋅ λ ⋅ λ ⋅ λ > 0. 6 7 4 4 5 6 7 D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 61 Conclusions In their (sometimes co-authored) books, Flaschel, Franke and Chiarella have contributed significantly to the development of Keynesian monetary macro - economics. One of their main achievements is the development of the KMG model and a stability analysis of its various variants. Proving the stability of complex, high dimensional systems like the KMG model is always a complex, difficult task. For this reason, to make the stability analysis easier, they sim - plified some equations of the model at the expense of its adequacy to reality. In the present article, an aemp tt t has been made to improve the way the KMG model describes the functioning of the economy by modifying some of its equations. The modifications introduced have resulted in the appearance of new loops in the model, thereby increasing its complexity. This is particu - larly evident in the intensive form model (56)–(62) with additional Equations (67)–(68). What is worth emphasizing is that all of these modifications have been introduced in a way which retains the possibility of transforming the KMG model into its intensive form and deriving its steady state, which is ne- cessary for proving the stability of the model. The modifications of the KMG model have influenced all the results pre - sented in the article. Firstly, they are reflected in the formulas describing va - lues of the variables in the steady state presented in section 2.2. In particu - lar, this is visible how tax rates (which are not present in other versions of the KMG model) influence the values of some variables in the steady state, such as the interest rate, the real labour income per unit of output, the ra- tio of the real value of bonds to fixed capital. (On the other hand, an inte - resting finding is that, in the steady state, the ratio of consumption to fixed capital does not depend on tax rates.) Secondly, the Jacobian matrix of the intensive form of the new KMG model differs from the Jacobian matrices of KMG models analysed by Chiarella et al. This meant that the proof of stabili- ty presented in section 3, although based on the general idea of the cascade of stable matrices approach, differs essentially from the proofs of stability of other versions of KMG models. Thirdly, due to the equation’s modifications, the set of Assumptions 1–6 exploited in the proof also differs from that used by Chiarella et al. In particular, Assumptions 5 and 6, which feature tax rates on labour and capital incomes are completely new, as these tax rates are not considered at all in other versions of the KMG model. According to the proven stability theorem, when Assumptions 1–6 are sa - tisfied, the economy described by the modified KMG model approaches the balanced growth path when time goes to infinity. Since the proven stability theorem concerns only local stability, the convergence to the balanced growth path is guaranteed only if the initial structure of the economy does not dif - fer too much from that on the balanced growth path, described by the ste- 62 Economics and Business Review, Vol. 9 (1), 2023 ady state of the intensive form model. How much the initial structure of the economy may depart from the steady state without losing the stability of the model may be verified only through computer simulations. Also, the speed at which the economy converges toward the balanced growth path can only be tested by computer experiments. Despite these limitations, mathematical proofs of the stability of econo - mic systems like that presented in the article are important for the develop - ment of economic growth theory since they reveal an intrinsic ability of the analyzed economy to achieve a structure which allows for balanced growth. Lack of stability is a serious deficiency of the economy because it is equiva - lent to the existence of a self-deepening disequilibrium mechanism leading to economic collapse. References Araujo, R. A., Flaschel, P., & Moreira, H. N. (2020). Limit cycles in a model of supply-side liquidity/prot— fi rate in the presence of a Phillips curve. Economia, 21(2), 145–159. Asada, T. (2012). Modeling financial instability. Intervention: European Journal of Economics and Economic Policies, 9, 215–232. Asada, T., Chiarella, C., Flaschel, P., & Franke, F. (2003). Open economy macrodynam- ics. Springer. Asada, T., Chiarella, C., Flaschel, P., Mouakil, T., Proaño, C., & Semmler, W. (2011). Stock flow interactions, disequilibrium macroeconomics and the role of economic policy. Journal of Economic Surveys, 25(3), 569–599. Asada, T., Demetrian, M., & Zimka, R. (2018). On dynamics in a Keynesian model of monetary and fiscal policy with debt ee ff ct. Communications in Non-linear Science and Numerical Simulations , 58, 131–146. Asada, T., Demetrian, M., & Zimka, R. (2019). On dynamics in a Keynesian mod- el of monetary and fiscal stabilization policy mix with twin debt accumulation. Metroeconomica, 70(3), 365–383. Charpe, M., Chiarella, C., Flaschel, P., & Semmler, W. (2011). Financial assets, debt and liquidity crises: A Keynesian approach. Cambridge University Press. Chiarella, C., & Flaschel, P. (2000). The dynamics of Keynesian monetary growth. Cambridge University Press. Chiarella, C., Flaschel, P., Franke, R. (2005). Foundations for a disequilibrium theory of the business cycle. Cambridge University Press. Chiarella, C., Flaschel, P., Franke, R., Araujo, R., Charpe, M., Proaño, Ch., & Szczutkowski, A. (2021). Unbalanced growth from a balanced perspective . Edward Elgar. Chiarella, C., Flaschel, P., Franke, R., & Semmler, W. (2002). Stability analysis of a high- dimensional macrodynamic model of real-financial interaction: A cascade of matri - ces approach. Working Paper, 123. School of Finance and Economics, UTS Business School, University of Technology, Sydney. D. Sołtysiak, On the stability of a certain Keynes-Metzler-Goodwin monetary growth 63 Chiarella, C., Flaschel, P., Franke, R., & Semmler, W. (2006). A high-dimensional model of real-financial market interaction: The cascade of stable matrices approach. In C. Chiarella, R. Franke, P. Flaschel & W. Semmler (Eds.), Quantitative and empirical analysis of nonlinear dynamic macromodels, contributions to economic analysis (vol. 277, s. 359–384). Elsevier. Chiarella, C., Flaschel, P., Groh, G., & Semmler, W. (2000). Disequilibrium, growth and labor market dynamics. Springer. Chiarella, C., Flaschel, P., & Semmler, W. (2012). (2013). (2014). Reconstructing Keynesian macroeconomics, vol. 1–3: Integrated approaches. Routledge. Duménil, G., & Lévy, D. (1991). Micro adjustment toward long-term equilibrium. Journal of Economic Theory, 52(2), 369–395. Dutt, A. (2012). Distributional dynamics in post Keynesian growth models. Journal of Post Keynesian Economics, 34(3), 431–451. Filipowicz, K., Misiak, T., & Tokarski, T. (2016). Bipolar growth model with investment flows. Economics and Business Review, 2(3), 32–56 https://doi.org/10.18559/ ebr.2016.3.4 Fischer, S. (1972). Keynes-Wicksell and neoclassical models of money and growth. American Economic Review, 62, 880–890. Flaschel, P. (2020). A baseline model of ’Social Protection’ in open economies of the KMG variety. Working Paper: Bielefeld University. Gandolfo, G. (2005). Economic dynamics. Springer Verlag. Goodwin, R. M. (1967). A growth cycle. In C. H. Feinstein (Ed.), Socialism, capitalism and economic growth (pp. 54–58). Cambridge University Press. Groth, C. (1988). IS-LM dynamics and the hypothesis of adaptive-forward-looking expectations. In P. Flaschel & M. Krüger (Eds.), Recent approaches to economic dynamics (s. 251–266). Verlag Peter Lang. Keynes, J .M. (1936). The general theory of employment, interest and money . Macmillan. Kiedrowski, R. (2018). Profit rates equalization and balanced growth in a multi-sector model of classical competition. Journal of Mathematical Economics , 77, 39–53. Lavoie, M. (2014). Post-Keynesian economics: New foundations . Edward Elgar. Metzler, L. A. (1941). The nature and stability of inventory cycles. Review of Economic Statistics , 23, 113–129. Murakami, H. (2016, December). Alternative monetary policies and economic stabil - ity in a medium-term Keynesian model. Evolutionary and Institutional Economics Review, 13(2), 323–362. Murakami, H. (2018). Existence and uniqueness of growth cycles in post Keynesian systems. Economic Modelling, 75, 293–304. Murakami, H. (2020). Monetary policy in the unique growth cycle of post Keynesian systems. Structural Change and Economic Dynamics, 52(C), 39–49. Napoletano, M., Dosi, G., Fagiolo, G., & Roventini, A. (2012). Wage formation, invest - ment behavior and growth regimes: An agent-based analysis. Revue de L’OFCE, 5(124), 235–261. https://www.cairn.info/revue-de-l-ofce-2012-5-page-235.htm Ogawa, S. (2019a). Dynamic analysis of a disequilibrium macroeconomic model with dual labor markets. Metroeconomica, 70(3), 525–550. 64 Economics and Business Review, Vol. 9 (1), 2023 Ogawa, S. (2019b). Eective ff demand and quantity constrained growth: A simple two - -sector disequilibrium approach. MPRA Paper 93336. University Library of Munich. Ogawa, S. (2020). Monetary growth with disequilibrium: A non-Walrasian baseline model. MPRA Paper 101236. University Library of Munich. Rødseth, A. (2000). Open economy macroeconomics. Cambridge University Press. Rose, H. (1966). Unemployment in a theory of growth. International Economic Review , 7, 260–282. Sargent, T. (1987). Macroeconomic theory. Academic Press. Stein, J. (1966). Money and capacity growth. Journal of Political Economy , 74, 451–465.

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Economics and Business Review de Gruyter

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Publisher: de Gruyter
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DOI: 10.18559/ebr.2023.1.2
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Economics and Business Review – de Gruyter

Published: Apr 1, 2023

Keywords: Keynesian macroeconomics; disequilibrium macroeconomics; monetary growth models; nonlinear economic; dynamics stability; C62; E12; E40

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On the stability of a certain Keynes-Metzler-Goodwin monetary growth model

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On the stability of a certain Keynes-Metzler-Goodwin monetary growth model

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