On 'rusting' money: Silvio Gesell's Schwundgeld reconsidered. Part II: The long run

Rehme, Günther Article On 'rusting' money: Silvio Gesell's Schwundgeld reconsidered. Part II: The long run Review of Economic Analysis (REA) Provided in Cooperation with: International Centre for Economic Analysis (ICEA), Waterloo, Ontario Suggested Citation: Rehme, Günther (2024) : On 'rusting' money: Silvio Gesell's Schwundgeld reconsidered. Part II: The long run, Review of Economic Analysis (REA), ISSN 1973-3909, International Centre for Economic Analysis (ICEA), Waterloo (Ontario), Vol. 16, Iss. 2, pp. 133-174, https://doi.org/10.15353/rea.v16i2.4945 This Version is available at: https://hdl.handle.net/10419/328163 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Part II: The Long Run GÜNTHER REHME† Technische Universität Darmstadt  Silvio Gesell argued that 'rusting' money is economically and socially beneficial; that claim has often been contended. In Part II of the paper, I concentrate on the long-run implications of his ideas. I show that introducing money depreciation in isolation may be economically non-beneficial in a typical long-run equilibrium. But money depreciation, when coupled with expansionary monetary policy, is a necessary condition for a positive Mundell-Tobin effect on long-run real variables and so creates wealth in the model. It is found that this also holds in the transition to the long-run equilibrium. Hence, the spirit of Gesell's hypotheses can be verified for a plausible, long-run environment as well, and may, thus, be relevant for long-run economic policy problems. Keywords: Economic Performance, Depreciating Money, Zero Lower Bound, Demonetization, Love of Wealth JEL classification: E1, E5, O4 † Professor Rehme passed away last year. The paper is published as it was originally submitted, with the exception of the quotes in part D, which are the same as the first part of the work, pages 91-131 in this issue.  I am indebted to Ingo Barens and Thomas Fischer for valuable help and insightful comments. I have also benefitted from discussions with Parantab Basu, Volker Caspari, Christiane Clemens, Alex Cukierman, Soumya Datta, Hartmut Egger, Sabine Eschenhof-Kammer, Christian Gelleri, Rafael Gerke, Chetan Ghate, Charles Goodhart, Marcus Miller, Michael Neugart, Uwe Sunde, Werner Onken, and from feedback at Bayreuth, LMU Munich, the 5th International Conference on South Asian Economic Development (SAED), South Asian University (SAU), New Delhi, the 5th HenU/INFER Workshop on Applied Macroeconomics, Kaifeng, Henan, the 50th Anniversary "Money, Macro and Finance" (MMF) conference at the London School of Economics (LSE), London, the 10th RCEA "MoneyMacro-Finance" Conference in Waterloo, Ontario, the 15th Annual Conference on Economic Growth and Development, New Delhi, in 2019, and the 65th Münden Talks "Proudhon, Gesell, Keynes and Negative Interest Rates", Wuppertal, in 2021. The usual disclaimer shields them all. © 2024 Günther Rehme. Licensed under the Creative Commons Attribution-Noncommercial 4.0 Licence (http://creativecommons.org/licenses/by-nc/4.0/). Available at http://rofea.org. Review of Economic Analysis 16 (2024) 133-173 134 www.RofEA.org 1 Introduction "Money is the football of economic life." Silvio Gesell (1920) The Natural Economic Order. In his main piece of work, "The Natural Economic Order" Silvio Gesell developed his idea of Schwundgeld (demurrage) and its consequences on economic performance. In part I of the paper it is shown that Gesell's claim can be justified in a short-run IS-LM-AS-AD environment. In Part II I now analyze whether his key conjectures can be justified in a parsimonious, modern theoretical framework for the long run. Gesell (1920), p. 78, acknowledges that money is "the football of economic life", but to him placing money and commodities on equal 'physical' footing as commodities is necessary and requires that money depreciate so that it performs its prime task, namely that of being the medium of exchange. For him, the face value of (paper) fiat money should be irredeemable and depreciate at a certain percentage over a particular period of time. In order to regain the previous face value of the money (note) used, people would have to buy stamps to make up for the depreciation the monetary authority would decree for the money note. As pointed out in Part I Gesell formulated four hypotheses about such a monetary arrangement. 1 Gesell Conjecture 1 (GC1) The introduction of, and, when present, an increase in, the money depreciation rate leads to a higher velocity of money in circulation. Gesell Conjecture 2 (GC2) Money depreciation coupled with expansionary monetary policy stimulates aggregate demand and through that output and employment. Gesell Conjecture 3 (GC3) A money depreciation rate is welfare enhancing. Gesell Conjecture 4 (GC4) A money depreciation rate benefits workers relatively more than capital owners. The present paper complements research that investigates whether the Gesell hypotheses can be replicated in modern standard model frameworks for the long run. One finds that the results of previous research are mixed. 2 For example, Rösl (2006) finds that only the first hypothesis can be derived from Sidrauski (1967), that is, in a money-in-the-utility set-up. He concludes that Gesell neglected an analysis 1 More verbal justifications for Gesell's claims and his ideas can be found in the working paper version of this paper; see Rehme (2018), especially appendix F, and at the end of this paper. 2 Gesell's ideas have been important in recent discussions about overcoming the problems after the Great Recession. For good surveys on the relevance of Gesell's ideas see, for example, Darity (1995), Ilgmann and Menner (2011), and Svensson and Westermark (2016). REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 135 www.RofEA.org of the long run and any possible effects on capital accumulation so the other three hypotheses turn out to be non-valid in his model. In turn, Menner (2011), for example, uses an elaborate and involved New Monetarist DSGE model to find that "inflation and 'Gesell taxes' maximize steady-state capital stock, output, consumption, investment and welfare at moderate levels. ... In a recession scenario, a Gesell tax speeds up the recovery in a similar way as a large fiscal stimulus but avoids 'crowding out' of private consumption and investment." Thus, he finds support for the Gesell hypotheses at moderate levels in his business cycle model of the third-generation monetary search models. The present paper uses an alternative micro-founded and simple dynamic general equilibrium model to analyze whether the depreciation of money is socially beneficial. For that, we abstract from fiscal policy, as Gesell did not consider the interaction of fiscal and monetary policy in detail. Following him we assume that the state issues a homogenous money and by legal coercion that money is legal tender. That is explained in some detail in Part I of this contribution. In the present paper, the basic Sidrauski framework is coupled with the additional motive of an agent to derive utility from (real) wealth. 3 People are taken to be rational and are not fooled by money illusion. Thus, the agents only consider real, physical capital as wealth. In that way, I relate to this as a 'love of wealth' as in Rehme (2011). In Part II I use a standard Ramsey-Cass-Koopmans framework where markets are assumed to clear at each point in time, and demand equals supply. Importantly, and as is standard in the literature, the marginal productivity theory of distribution is (now) assumed to hold in this model framework. It turns out that this yields interesting insights about Gesell's advocated monetary system where fiat (paper) money is irredeemable and, thus, directly related to a basket of real goods in an economy. 4 These insights refer, in particular, to the idea of money depreciation and its consequences for the steady state of an economy and its transitional dynamics. In such an optimal growth framework these results then emerge. 3 This has been done, for example, by Weber (1930) and Pigou (1941) who argue that individuals derive utility from the mere possession of wealth and not simply its expenditure. Later Kurz (1968) provided a thorough analysis of an optimal growth model where wealth features in utility. Furthermore, Zou (1994), Bakshi and Chen (1996) and Carroll (2000) relate to Max Weber and argue that the dependence of utility on wealth captures the "spirit of capitalism" in competitive market economies. More generally, it captures the 'love of wealth' in more general set-ups, including competitive market economies, as argued in Rehme (2017). 4 Irredeemability implies that you cannot exchange a banknote back into another banknote or any collateral that might possibly back the face or any other (real) value of the banknote. For instance, in the Euro and the Fed system you can in principle redeem your banknote, but only to get another banknote with an equally denoted face value. This is not possible under the irredeemability of the banknote and plays a role when there is a depreciation of the face or other value of the banknote. On the issue of irredeemability and fiat money see, for example, Buiter (2003). Review of Economic Analysis 16 (2024) 133-173 136 www.RofEA.org In the steady state, inflation depends on the sum of the money growth and depreciation rate. It turns out that the model dichotomizes into a monetary and real sector if there is no money depreciation. If the latter is present, the model features non-superneutrality. Thus, in the model money depreciation is a necessary condition for particular forms of a Mundell-Tobin effect. That effect is present if inflation leads people to hold less money and more real capital, implying a lower real interest rate. More precisely, it is found that the introduction of or an increase in money depreciation in isolation reduces the steady state capital stock (wealth), consumption, income and welfare. It also implies a higher return to capital, but a lower steady-state wage rate. Thus, more money depreciation seems to destroy wealth and implies lower wages. The only hypothesis that is validated is that higher money depreciation implies a higher velocity of money, [GC1]. Some authors have stopped here to argue that money depreciation is generally a bad idea, because it just destroys long-run wealth, instead of fostering it. However, in light of the quotes above, that view does not do justice to Gesell's thinking. He was not arguing solely about money depreciation. Of course, he knew that the monetary authority was also issuing new and withdrawing old money. Here it turns out that, for a given positive money depreciation rate, an increase in the money growth rate produces a Mundell-Tobin effect. Thus, higher money growth increases steadystate inflation, but also the steady-state capital stock, output, and consumption. It implies a higher long-run wage rate and a lower return to capital. The consequences for the holdings of real money balances and so for total welfare are not unambiguously clear. But the velocity of money increases. However, the partial welfare channels through consumption and wealth work clearly in a positive direction. Hence, the conjectures GC1 and GC2 can be validated for the long run. But given the necessary nature of money depreciation for these results one may argue that GC3 and GC4 are also not too far off their marks. In terms of the economic effects the conjectures ultimately wish to capture they are not wrong because of the possibility of a positive Mundell-Tobin effect which would indeed support GC3 and GC4. The analysis of the transitional dynamics reveals that the speed of convergence increases if money depreciation increases, and decreases if the money growth rate is raised. That complements Fischer (1979) who finds that more money growth speeds up convergence when utility is non-logarithmic and the steady state features asymptotic superneutrality. Here the steady state generally features non-superneutrality, utility is logarithmic, and convergence is slower when the money growth rate increases. A simulation exercise based on some standard calibration values reveals that the response of the key variables to permanent changes in the monetary policy variables is the same in the transition as in the steady-state. That also holds for the jump variables, namely, initial money holdings and consumption. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 137 www.RofEA.org Furthermore, for temporary changes in the policy variables, one obtains the temporary responses that, again, qualitatively equal those for the steady state. Summarizing these findings yields that the present model framework is indeed capable of verifying most of Gesell's claims, also in the long run. In Part II and, thus, for a long-run equilibrium two claims of Gesell's follow directly, and the other two indirectly, because money depreciation is a necessary condition for a positive Mundell-Tobin effect. This may justify why Gesell's ideas may have significance for a description of long-run macroeconomic phenomena and realistically relate to the current economic situation in many countries. The paper is organized as follows. Section 2 presents the model and its set up. Section 3 derives and analyzes the long-run equilibrium and section 4 the transitional dynamics. Section 5 concludes. 2 The Model The set up of the model is explained in detail in Part I of the paper. For the purposes of Part II, I only restate the main ingredients of the model. I use a continuous time framework and for all variables that are continuous functions of time the subscript 𝑡 is used to denote their dependence on time. Thus, ℎ𝑡≡ℎ(𝑡) for some variable ℎ depending on time. Furthermore, the change of a variable ℎ over time, i.e. 𝑑ℎ𝑡 𝑑𝑡, is denoted by ℎ˙𝑡. The economy has many, price-taking households. The aggregate resource constraint of the households is 𝐶𝑡+𝐾˙𝑡+𝑀˙𝑡 𝑃𝑡+𝜎⋅𝑀𝑡 𝑃𝑡=𝑤𝑡𝑁𝑡+𝑟𝑡𝐾𝑡+𝑋𝑡(1) where 𝐶𝑡 and 𝐾𝑡 denote aggregate real consumption and the aggregate real capital stock, respectively. 𝑀𝑡 represents the aggregate nominal money holdings and 𝑃𝑡 is the price level. 𝑁𝑡 denotes population and 𝑤𝑡 is the real wage rate. 𝑟𝑡 denotes the real rate of return on capital, net of depreciation of physical capital 𝐾𝑡. The lump-sum (real) transfers of the government are denoted 𝑋𝑡. Thus, the right-hand side of the budget constraint captures aggregate income, consisting of total wage (𝑤𝑡𝑁𝑡) and capital income (𝑟𝑡𝐾𝑡) as well as government transfers (𝑋𝑡) and the lefthand side, captures aggregate spending. Thus, income is spent on consumption (𝐶𝑡), investment in new capital (𝐾˙𝑡) and acquisitions of new, real money holdings (𝑀˙𝑡 𝑃𝑡). The aggregate budget constraint in equation (1) corresponds to the conventional money-in- the-utility-function model. The novel feature here is the term 𝜎⋅𝑀𝑡 𝑃𝑡. It captures the Gesell tax and so the idea of "rusting money". That can be interpreted as a depreciation on the circulating real money holdings of the households and is tantamount to a tax on them. Review of Economic Analysis 16 (2024) 133-173 138 www.RofEA.org Now consider a representative agent economy, and define per capita consumption 𝑐𝑡, real money balances 𝑚𝑡, as well as the per capita capital stock 𝑘𝑡 and transfers 𝑥𝑡 as follows 𝑐𝑡≡𝐶𝑡 𝑁𝑡, 𝑚𝑡≡𝑀𝑡 𝑃𝑡𝑁𝑡, 𝑘𝑡≡𝐾𝑡 𝑁𝑡, and 𝑥𝑡≡𝑋𝑡 𝑁𝑡 One verifies that the budget constraint of the representative household is then given by 𝑐𝑡+𝑘˙𝑡+𝑛𝑡𝑘𝑡+𝑚˙𝑡+𝜋𝑡𝑚𝑡+𝑛𝑡𝑚𝑡+𝜎𝑚𝑡=𝑤𝑡+𝑟𝑡𝑘𝑡+𝑥𝑡. Again, the right-hand side corresponds to the household's income and the left-hand side captures the household's expenditure. Notice 𝜎𝑚𝑡 is the outlay for the household. The longer the household holds real money balances 𝑚𝑡, the more is foregone (a form of expenditure) in terms of real income. 5 For simplicity let 𝑎𝑡≡𝑘𝑡+𝑚𝑡. Thus, the household has real resources in the form of physical capital and real money balances. Then 𝑎˙𝑡=𝑘˙𝑡+𝑚˙𝑡. After collecting terms and rearrangement one then obtains 𝑎˙𝑡=[(𝑟𝑡−𝑛𝑡)𝑎𝑡+𝑤𝑡+𝑥𝑡]−[𝑐𝑡+(𝑟𝑡+𝜋𝑡+𝜎)𝑚𝑡]. (2) Thus, the change in real per capita resources 𝑎˙𝑡 depends on the household's income from capital and real money balances (𝑟𝑡−𝑛𝑡)𝑎𝑡, labor income 𝑤𝑡 and transfers 𝑥𝑡. Consumption then consists of the consumption of goods 𝑐𝑡 and the expenses for using money services. The latter depends on the user cost of money (𝑟𝑡+𝜋𝑡+𝜎)𝑚𝑡. Here we employ the Fisher relation that nominal interest rates 𝑖𝑡 equal the real interest rate 𝑟𝑡 plus the inflation rate 𝜋𝑡. The user cost of holding money, thus, depends on the nominal interest rate 𝑖𝑡 and the depreciation of money 𝜎. To simplify the analysis assume a stationary population 𝑛𝑡=0 and set its size to 𝑁𝑡=1 for all 𝑡. As an important departing point from a standard Sidrauski model the representative household also "loves wealth". The household is not fooled by money illusion and only physical capital is considered to be "wealth" that directly bears on welfare. However, the household also values real money balances as they facilitate exchange and transactions. Thus, (real) money balances are also taken to bear on welfare as in Sidrauski (1967). Although both money and capital feature directly in utility, they do so for different reasons. Money is valued because it facilitates exchange, whereas physical capital is valued as an expression of wealth. 5 To capture the Gesell tax in this way see, for example, Rösl (2006), and the explanations in Part I. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 139 www.RofEA.org The household's problem is then taken to be to maximize the functional 𝑊=∫ ∞ 0 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡)𝑒−𝜌𝑡𝑑𝑡 (3) where 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) is period utility depending on consumption, real money balances and physical capital. Welfare is discounted at the (positive) rate of time preference 𝜌, capturing how patient households are, and the convergence of the utility function. In order to derive clear predictions that also allow for an analysis of transitional dynamics, and building on previous own work, cf. Rehme (2011), we now make the following assumptions about the period utility function 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡). 1. 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) is taken to be separable in 𝑐𝑡,𝑚𝑡 and 𝑘𝑡. In particular, assume that of the project. ∂2𝜑(⋅)/∂𝑖∂𝑗=0 for all 𝑖,𝑗=𝑐𝑡,𝑚𝑡,𝑘𝑡 and 𝑖≠𝑗. 2. 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) is increasing and concave in each (own) argument, that is, ∂𝜑(⋅)/∂𝑖>0 and ∂2𝜑(⋅)/∂𝑖2<0 for all 𝑖=𝑐𝑡,𝑚𝑡,𝑘𝑡. 3. 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) satisfies the Inada conditions for each (own) argument, that is, lim 𝑖→0 𝜑(⋅)/∂𝑖→∞ and lim 𝑖→∞ 𝜑(⋅)/∂𝑖→0 where 𝑖=𝑐𝑡,𝑚𝑡,𝑘𝑡. A simple and convenient period utility function that satisfies all these requirements is the logarithmic one. So we invoke Assumption 1 Period utility 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) is separable and logarithmic in each argument and given by 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡)=ln 𝑐𝑡+𝛿ln 𝑚𝑡+𝛽ln 𝑘𝑡 where 𝛿,𝛽>0 (4) The parameter 𝛿 measures how people value the transaction services real money balanced render, and 𝛽 captures "love of wealth". The assumption that 𝛿 and 𝛽 are positive means that the model is structurally different from the more conventional setups of "money-in-the-utility- function"-models without "love of wealth". 6 6 From the logarithmic utility set-up it is immediate that relative wealth, for instance, the logarithm of the ratio of individual to total (aggregate) wealth would be separable in the two concepts. If the representative individual takes total wealth as given, then both approaches, that is, working with relative Review of Economic Analysis 16 (2024) 133-173 140 www.RofEA.org Let [ℎ𝑡]𝑡=0 +∞ denote the continuous time path of variable ℎ𝑡 and use the following definitions: 𝑘𝑡≡(1−𝑧𝑡)𝑎𝑡 and 𝑚𝑡≡𝑧𝑡𝑎𝑡 where 𝑎𝑡 is an indicator of the total real resources of the household, and 𝑧𝑡 denotes the share of the real resources held in terms of real money balances. These definitions serve to facilitate the analysis, and i.a. imply 𝜑(𝑐𝑡,𝑚𝑡,𝑘𝑡) =ln 𝑐𝑡+𝛿ln [𝑧𝑡⋅𝑎𝑡]+𝛽ln [(1−𝑧𝑡)⋅𝑎𝑡] ln 𝑐𝑡+(𝛿+𝛽)ln 𝑎𝑡+𝛿ln 𝑧𝑡+𝛽ln (1−𝑧𝑡) (5) We can then formulate the representative household's problem as the maximization of intertemporal welfare based on equation (5) subject to the flow budget constraint in equation (2). Thus, the household's problem is max 𝑐𝑡,𝑧𝑡 ∫ ∞ 0 [ln 𝑐𝑡+(𝛿+𝛽)ln 𝑎𝑡+𝛿ln 𝑧𝑡+𝛽ln (1−𝑧𝑡)]𝑒−𝜌𝑡𝑑𝑡 s.t. 𝑎˙𝑡=[𝑟𝑡𝑎𝑡+𝑤𝑡+𝑥𝑡]−[𝑐𝑡+(𝑟𝑡+𝜋𝑡+𝜎)𝑧𝑡𝑎𝑡]. Here consumption 𝑐𝑡 and real money balances 𝑚𝑡 in terms of per capita resources 𝑎𝑡, that is, 𝑧𝑡 are the control variables, and 𝑎𝑡 is the state variable. The household takes the paths of the real interest rate, the wage rate, the inflation rate and government transfers [𝑟𝑡,𝑤𝑡,𝜋𝑡,𝑥𝑡]𝑡=0 +∞ and the (constant) policy parameter 𝜎 as given. Recall that 𝑛𝑡=0,∀𝑡, (no population growth) has been assumed. Furthermore, the household takes as given his initial level of real resources, 𝑎0. Setting up the current-value Hamiltonian for this problem and denoting 𝜇𝑡 as the currentvalue costate variable 7 the necessary first-order conditions for this maximization problem is 1 𝑐𝑡−𝜇𝑡 =0 (6) 𝛿 𝑧𝑡−𝛽 1−𝑧𝑡−𝜇𝑡⋅𝑎𝑡(𝑟𝑡+𝜋𝑡+𝜎) =0 (7) −[𝛿+𝛽 𝑎𝑡+𝑟𝑡𝜇𝑡−𝜇𝑡(𝑟𝑡+𝜋𝑡+𝜎)⋅𝑧𝑡] =−𝜌𝜇𝑡+𝜇˙𝑡(8) where we also require that equation (2) holds (with 𝑛𝑡=0 ) and the transversality condition is satisfied, i.e. or absolute wealth would not make a difference in the individual’s decision and would yield similar results. As argued above I follow Plutarch here. 7 For what is to follow we now use subscripts, except subscript t, to denote partial derivatives. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 147 www.RofEA.org For convenience rearrange the last expression to obtain Δ=𝛽 where Δ≡(1+ 𝛿𝜎 𝜌+𝜋∗+𝜎)⋅𝜌−𝑟∗ 𝑟∗⋅𝛼 and 𝜋∗=𝜃+𝜎 (20) which implicitly defines the capital stock in steady state, that is, 𝑘∗, as a function of the model's parameters, that is, 𝑘∗=𝑘∗(𝜎,𝛽,𝛿,𝜌,𝜃,𝛼). From that, we obtain an important result. If 𝜎=0, then 𝑘∗ would be independent of monetary variables and the model would dichotomize into a monetary and real sector. To see this consider equation (20) to find that 𝑘∗ would then be independent of 𝜃 and 𝜎. Furthermore, given that, 𝑐∗ and 𝑦∗ would also be independent of 𝜎 and 𝜃. 8 In contrast, if 𝜎 is non-zero, then one easily verifies that the steady state capital stock depends on the money growth rate 𝜃 and the money depreciation rate 𝜎. Thus, the model is then not super-neutral. 9 Proposition 1 Without a Gesell tax, that is, when 𝜎=0, the model's steady state dichotomizes into a monetary and real sector. Monetary variables would then be neutral and superneutral in a long-run equilibrium. In contrast, if 𝜎≠0, the model implies non-superneutrality. For the rest of the paper assume that 𝜎 is non-zero. The economy does not dichotomize in that case and has, in general, a non-superneutral long-run equilibrium. As a consequence, the model features some form of a Mundell-Tobin effect. Recall that Tobin (1965) and Mundell (1963) argued that monetary variables, in particular, realized or expected inflation, may have an effect on the real variables, especially on the (longrun) real interest rate of an economy. The effect is usually taken to be positive because it is argued that higher inflation causes people to hold less money and more real capital. That would then imply a lower real interest rate. 10 In this model 𝜋∗=𝜃+𝜎 in the steady state which, according to equation (20), bears on 𝑘∗ and so the long-run real interest rate 𝑟∗. Thus, it is through 𝜃 and 𝜎 that the model features 8 As argued above the model also dichotomizes when β=0. This is the world that Rösl (2006) analyzed. Clearly, and rather unsurprisingly, neutrality and super neutrality are then a feature of such a model. For this reason, amongst others, a positive β is one constitutional feature of the present model. 9 Recall that non-neutrality implies that money supply variables bear on long-run real variables like the steady-state capital stock. Non-super neutrality means that the rate of money supply growth has an effect on real variables. See, for example, Ahmed and Rogers (1996) for a clarifying study of this issue. 10 Fischer (1988), p. 296/7 explains where the differences in the respective contributions of Tobin and Mundell lie. See also Temple (2000) for a more recent literature survey on the interaction of inflation and economic growth. Review of Economic Analysis 16 (2024) 133-173 148 www.RofEA.org Mundell-Tobin effects. However, the effects of 𝜃 and 𝜎 will be shown to be different. When any (positive) change in the variables leads to a higher real interest rate, I call that a reverse Mundell-Tobin effect. We now analyze the comparative static properties of the steady state values of 𝑘,𝑚, and 𝑐, and other variables of interest. I analyze the effects on 𝑘 in more detail in the main text and present the derivation for the other variables in the appendix. For a change in 𝜎 on 𝑘 note that Δ𝑟=−(1+ 𝛿𝜎 𝜌+𝜋∗+𝜎)⋅ 𝛼𝜌 (𝑟∗)2<0 (21) As 𝑟=𝛼𝑘𝛼−1 we have 𝑟𝑘<0. But then Δ𝑘=Δ𝑟⋅𝑟𝑘>0 by equation (20), where again subscripts denote partial derivatives. Furthermore, it turns out that, if 𝛿>0, Δ𝜎=(𝛿(𝜌+𝜃+2𝜎)−2𝛿𝜎 (𝜌+𝜃+2𝜎)2)⋅(𝜌−𝑟∗ 𝑟∗)⋅𝛼>0 Then we have that Δ𝑘⋅𝑑𝑘+Δ𝜎⋅𝑑𝜎=0 has to hold from equation (20). But consequently, we get 𝑑𝑘/𝑑𝜎=−Δ𝜎/Δ𝑘<0, that is, a higher money depreciation rate implies a lower steadystate capital stock. Thus, households choose to hold less physical capital which implies some form of a reverse Mundell-Tobin effect. Higher 𝜎 may require more outlays for money holdings. These more "expensive" money holdings also make it more costly to hold physical capital. Holding less capital, in turn, entails a higher long-run real interest rate 𝑟∗, that is, it makes physical capital more "expensive". Hence, raising 𝜎 appears to 'destroy' long-run wealth, that is, it implies a smaller, long-run physical capital stock. For the effect of "love of wealth" 𝛽 one easily verifies that 𝑑𝑘/𝑑𝛽=1/Δ𝑘>0 so that an increase in the 'love of wealth' raises the long-run capital stock. Valuing monetary transactions more (larger 𝛿 ) implies Δ𝛿=( 𝜎 𝜌+𝜃+2𝜎)⋅𝜌−𝑟∗ 𝑟∗⋅𝛼>0 so that 𝑑𝑘/𝑑𝛿=−Δ𝛿/Δ𝑘<0. Clearly, if people derived more utility from money transactions (higher 𝛿 ) they might wish to hold more money, but in the model they definitely want to have less physical capital, implying a higher real interest rate 𝑟∗. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 149 www.RofEA.org The effect of more impatience (larger 𝜌 ) depends on Δ𝜌 =𝛼 𝑟∗[1+ 𝛿𝜎 𝜌+𝜃+2𝜎]−𝛼(𝜌−𝑟∗ 𝑟∗)[𝛿𝜎 (𝜌+𝜃+2𝜎)2] =𝛼 𝑟∗[1+ 𝛿𝜎 𝜌+𝜃+2𝜎−(𝜌−𝑟∗)𝛿𝜎 (𝜌+𝜃+2𝜎)2] =𝛼 𝑟∗[1+𝛿𝜎(𝜌+𝜃+2𝜎)−(𝜌−𝑟∗)𝛿𝜎 (𝜌+𝜃+2𝜎)2]=𝛼 𝑟∗[1+𝛿𝜎(𝜃+2𝜎)+𝑟∗𝛿𝜎 (𝜌+𝜃+2𝜎)2]>0 so that 𝑑𝑘/𝑑𝜌=−Δ𝜌/Δ𝑘<0. Thus, when the representative household is more impatient, there will be less physical capital in steady state. For the impact of the money growth rate 𝜃 I find Δ𝜃=−( 𝛿𝜎 (𝜌+𝜃+2𝜎)2)(𝜌−𝑟∗ 𝑟∗)⋅𝛼<0 from which it follows that 𝑑𝑘/𝑑𝜃=−Δ𝜃/Δ𝑘>0 so that 𝑘∗ would be larger. Thus, with a positive money depreciation rate a higher money growth rate implies a positive Mundell-Tobin effect. This is because for a given positive 𝜎 an increase in 𝜃 entails a higher steady state inflation rate 𝜋∗. But a higher 𝜃 has just been found to raise the long-run capital stock, coupled with a lower real interest rate. Therefore, this captures the main point of a positive Mundell-Tobin effect. The parameter 𝛼 represents the elasticity of output with respect to capital, but also the capital share, since it is assumed that firms are profit maximizers under conditions of perfect competition. As 𝑟=𝛼𝑘𝛼−1 we can express Δ=𝛽 in equation (20) as Δ=(1+ 𝛿𝜎 (𝜌+𝜃+2𝜎))(𝜌−𝑟∗)⋅(𝑘∗)1−𝛼=(1+ 𝛿𝜎 (𝜌+𝜃+2𝜎))(𝜌⋅(𝑘∗)1−𝛼−𝛼)=𝛽 Then it follows that Δ𝛼=−(1+ 𝛿𝜎 (𝜌+𝜃+2𝜎))(𝜌⋅ln 𝑘∗⋅(𝑘∗)1−𝛼) which is negative as long as ln 𝑘∗ is larger than zero. 11 I assume this to be true, because it only depends on mild theoretical assumptions and very plausible values for the capital-labor ratio, 11 Note that 𝑘1−𝛼=𝑒(1−𝛼)ln 𝑘 and 𝑑𝑘1−𝛼/𝑑𝛼=−ln 𝑘⋅𝑒(1−𝛼)ln 𝑘=−ln 𝑘⋅𝑘1−𝛼. Review of Economic Analysis 16 (2024) 133-173 150 www.RofEA.org often shown in the empirical literature. 12 As a consequence, 𝑑𝑘/𝑑𝛼=−Δ𝛼/Δ𝑘>0 so that a higher capital share implies a higher steady state capital stock. Summarizing these findings, the model features the following properties of the steady-state capital stock 𝑘∗=𝑘∗𝜎 (−),𝛽 (+),𝛿 (−),𝜌 (−),𝜃 (+),𝛼 (+)). (22) Clearly as 𝑦=𝑓(𝑘) is monotonically increasing in 𝑘, the properties of 𝑘∗(⋅) carry over to steady state output 𝑦∗=𝑓(𝑘∗(⋅)), and - in our Cobb-Douglas world - also to the wage rate 𝑤∗=𝑓(𝑘∗)−𝑓′(𝑘∗)⋅𝑘∗=(1−𝛼)𝑓(𝑘∗) and the real interest rate 𝑟∗=𝑓′(𝑘∗). The latter immediately follows from assumption 2. From equation (12) consumption in a steady state is given by 𝑐∗=(𝜌−𝑟∗ 𝛽)⋅𝑘∗ (23) In Appendix A. 1 the reaction of steady-state consumption is analyzed and found to be characterized by 𝑐∗=𝑐∗(𝜎 (−),𝛽 (+),𝛿 (−),𝜌 (−),𝜃 (+),𝛼 (+)) 𝑐∗=𝑐∗(𝜎 (−),𝛽 (+),𝛿 (−),𝜌 (−),𝜃 (+),𝛼 (+))) (24) Two results are noteworthy here. The monetary policy variables 𝜃 and 𝜎 have opposite effects on steady-state consumption. A higher money depreciation rate lowers it, whereas a higher money growth rate raises it. This is probably less surprising if one notes that higher 𝜃 raises income and capital, but 𝜎 does not. Actually, more money depreciation is 'bad' for capital as well as income, and it competes through money depreciation outlays with consumption. The second interesting finding is that more 'love of wealth' makes more consumption possible in a steady state. Even though higher 𝛽 may seem to be only conducive to more investment, it leads to more steady-state capital and income, making a higher level of steady-state consumption possible. A related finding is presented in Rehme (2017) and analyzed there in more detail. From equation (19) the demand for real balances in a steady state is given by 12 Clearly the model requires ρ>r* and so 𝜌>𝛼(𝑘∗)𝛼−1 and 𝑘∗>(𝛼/𝜌)1/(1−𝛼) . Thus, as long as α is larger than ρ then the condition k^*>1 is met. It is conventionally assumed that α is around 1/3 and ρ<<0.33. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 151 www.RofEA.org 𝑚∗=𝛿𝑐∗ 𝜌+𝜃+2𝜎 As 𝜈≡𝑐/𝑚 it follows that in steady state 𝜈∗ is increasing in 𝜎 and 𝜃. In that sense, the shortrun and long-run effects of monetary policy on the velocity of money are very similar. Next, in Appendix A. 2 the reaction of steady-state real money balances is analyzed. The findings there can be summarized by 𝑚∗=𝑚∗(𝜎 (−),𝛽 (+),𝛿 (?),𝜌, (−),𝜃 (?),𝛼 (+)) (25) Interestingly, households hold less money in a steady state when money depreciation is increased. This is because a higher 𝜎 implies a higher velocity of money so that households need to hold less money in a long-run equilibrium to conduct their monetary transactions. In turn, the effect of 𝜃 is not unambiguously clear and depends on the parameter values of the model. If 𝛿 and/or 𝜎 are sufficiently small, then a higher money growth rate is coupled with less money holdings, but a higher velocity of money. From equations (22), (24), and (25) and the expression of the welfare function in equation (4) the reactions of the steady state variables and welfare to changes in the variables of interest here yield the following. Proposition 2 Given everything else, the introduction of a positive, previously nonexistent Gesell tax, which is kept in place forever, implies a higher velocity of money 𝜈∗, a lower capital stock 𝑘∗, lower consumption 𝑐∗, and less holdings of real money balances 𝑚∗ and so lower welfare 𝜑∗(𝑐∗,𝑚∗,𝑘∗) in steady state. The steady-state return on capital 𝑟∗ rises so that some form of a reverse Mundell-Tobin effect is present. Thus, when looking at the effect of money depreciation on long-run outcomes in isolation, it seems that it would be a 'bad' policy option to introduce a depreciation rate on money holdings. Only [GC1] is validated. However, the introduction of a Gesell tax may not be too 'bad' an option because of the following. Proposition 3 Given everything else and conditional on a positive (possibly very small) Gesell tax, a higher rate of money growth 𝜃 that is kept in place forever, implies a Mundell-Tobin effect. The capital stock 𝑘∗, output 𝑦∗, consumption 𝑐∗, and the velocity of money 𝜈∗ would be higher,the long-run real interest rate 𝑟∗ lower and the wage rate 𝑤∗ higher. Steady-state real money balances 𝑚∗ may be higher or lower, depending on the parameter values of the model. The effect on long-run welfare is in general not unambiguously clear. For sufficiently high values of 𝜎 and/or 𝛿, an increase in 𝜃 may raise 𝑘∗,𝑐∗ and 𝑚∗ and long-run welfare 𝜑∗(𝑐∗,𝑚∗,𝑘∗). Review of Economic Analysis 16 (2024) 133-173 152 www.RofEA.org Those findings would lend clear support to the Gesell Conjectures 1, and 2, [GC1], and [GC2]. Note that the proposition requires a positive Gesell tax. The latter is, thus, a necessary condition for any Mundell-Tobin effect to work. In order to see this more clearly consider the effects of joint variations in 𝜎 and 𝜃 on steady state 𝑘∗. They can be determined from the differential Δ𝑘⋅ 𝑑𝑘+Δ𝜎⋅𝑑𝜎+Δ𝜃⋅𝑑𝜃=0 using equation (20). We know that Δ𝑘>0. Thus, the reaction of 𝑘 is, for example, positive, if Δ𝑘⋅𝑑𝑘>0. But that requires that −Δ𝜎⋅𝑑𝜎−Δ𝜃⋅𝑑𝜃>0, that is: −(𝛿(𝜌+𝜃+2𝜎)−2𝛿𝜎 (𝜌+𝜃+2𝜎)2)𝑄⋅𝑑𝜎+( 𝛿𝜎 (𝜌+𝜃+2𝜎)2)𝑄⋅𝑑𝜃>0 where 𝑄=𝛼(𝜌−𝑟∗ 𝑟∗) and the expressions for Δ𝑖,𝑖=𝜎,𝜃 follow from above. This holds if both 𝜎 and 𝜃 are changed. Again we see that, if 𝜎 is zero, 𝜃 does not affect 𝑘∗. For a non-zero 𝜎, and simultaneous changes in both policy variables simplification yields that a positive effect on steady-state 𝑘 is present if 𝜎⋅𝑑𝜃>(𝜌+𝜃)⋅𝑑𝜎 As a higher 𝜎 lowers 𝑘∗ whereas a higher 𝜃 raises it, the change in 𝜃 must be sufficiently strong, that is, it must obey 𝑑𝜃/𝑑𝜎>(𝜌+𝜃)/𝜎 to have an overall positive effect on 𝑘∗. Result 1 In general monetary policy conducted through changes in 𝜎 and/or 𝜃 has ambiguous effects on the steady-state capital stock 𝑘∗. If the relative changes in the two monetary policy variables satisfy 𝑑𝜃/𝑑𝜎>(𝜌+𝜃)/𝜎, that is, if the change in 𝜃 is sufficiently strong and positive, given that money depreciation is present or its change is positive and given, then the long-run capital stock 𝑘∗ can be increased and the long-run real interest rate 𝑟∗ decreased. Thus, by a right combination of 𝜎 and 𝜃 the monetary authority can generate a Mundell-Tobin effect with a higher long-run physical capital stock and lower real interest rate. This appears to be in line with Gesell's idea that expansionary monetary policy increases real activity. Only here it is found that simply focusing on money depreciation alone may not be enough for generating a positive effect on real variables. Although a necessary condition in this model, money depreciation has to be coupled with (new) money creation, that is, it must be accompanied by the injection of "new money" into the economy to have any positive effect on real variables in the long run. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 153 www.RofEA.org 4 Transitional Dynamics The dynamic system of the equations in (15) can be log-linearized in a standard way to yield insights into the transitional dynamics and convergence properties of the system. The technical details for that are presented in Appendix C. As the dynamics of the system are essentially governed by the same variables as in the standard Sidrauski model, one can employ the same arguments as in, for example, Blanchard and Fischer (1989), Appendix B of chapter 4, and Fischer (1979). Thus, note that the capital stock is given, but the money stock and consumption can jump at any point in time. As a consequence, if the system is to have a (locally) unique stable path, it must have two positive roots (or a pair of complex roots with positive real part) and one negative root. If that is the case, the jump variables take on (initial) values that make the system converge. The analysis of the roots that govern the speed of convergence of the system is presented in the appendix. For the present model that implies the following result. Proposition 4 Given a positive money depreciation rate, an increase in 𝜎 speeds up, and an increase in 𝜃 lowers the speed of convergence to the steady state. Interestingly, that is a complement of the result in Fischer (1979), who shows that more money growth would lead to faster convergence when the utility function is non-logarithmic and the steady state features asymptotic superneutrality. In turn, in this paper, the presence of (positive) money depreciation entails that the steady state is non-superneutral, but convergence is slower if the money growth rate 𝜃 is increased and we have a logarithmic utility function. 13 ⬚ 4.1 Numerical simulation The model is calibrated along some commonly observed magnitudes. The resulting system is then solved for those values. As a starting value assume that the initial capital stock, which is a given (state) magnitude, is taking a value of 5, that is, by assumption 𝑘0=5. For the other parameters of the model consider the following values. Table 1: Simulation 𝛼 𝛽 𝛿 𝜌 𝜃 𝜎 0.33 0.1 0.02 0.1 0.01 0.01 13 Recall that if σ=0, then r^* is independent of σ and θ. So the latter variables would not impinge on convergence in that case. A similar result for logarithmic utility can be found in Fischer (1979). Review of Economic Analysis 16 (2024) 133-173 154 www.RofEA.org The major reason for working with these values is that they command wide support in the literature. For example, the value for 𝛼 is pretty standard and that for 𝛿 is almost the same as in Walsh (2010), p. 72. The money growth rate implied by Walsh is roughly equivalent to 𝜃= 0.01 for quarterly U.S. data on money supply M1, but in the model here I take the sum of 𝜃 and 𝜎 to equal the long-run inflation rate, which many people consider to be around two percent. An exception may be the value of 𝜌 which is taken to be a lot higher than what is conventionally used in empirical work. However, when one reminds oneself that the time preference rate is an important and somehow pervasive, but, nevertheless, ultimately quite unobservable concept, I assume a value of 10 percent, because it will make the other calibrated values correspond to ranges one finds in the literature. Furthermore, note that 𝛽 is also very difficult to measure. Even some data of the World Value Service are not clearly established to be good measures of the "love of wealth", although the latter has clearly been identified by hermeneutic thinking (e.g. in philosophy, psychology, history and sociology among others) to be an important deep fundamental for social and, particularly, economic relationships. Here I calibrate 𝛽 so the long-run interest rate assumes a reasonable value. With that in mind, the parameter values generate the following steady state magnitudes of the variables of interest. Table 2: Simulated Steady State Values 𝑘∗ 𝑦∗ 𝑘∗/𝑦∗ 𝑟∗ 𝑚∗ 𝑘∗/𝑚∗ 𝑐∗ 𝑣=𝑐∗/𝑚∗ 𝜋∗ 9.016 2.081 4.332 0.077 0.320 28.200 2.078 6.500 0.020 These numbers imply a steady-state inflation rate 𝜋∗=𝜎+𝜃 of two percent. The steady-state capital stock and output are then calculated as 𝑘∗=9.016 and 𝑦∗= 2.081. 14 That implies a capital-output ratio of about four which seems realistic for many countries. Then the steady state (i.e. long-run) return on capital is about 7.7 percent, which is broadly in line with many findings in the literature. See, for example, Jordá, Knoll, Kuvshinov, Schularick, and Taylor (2017), Table 11, for recent evidence. Furthermore, the implied velocity of money in circulation is around 6.5 for measures such as 𝑣=𝑐∗/𝑚∗ or 𝑣1=𝑦∗/𝑚∗ which one approximately finds as a period average, for example, for the United States for the period 1960-2015. 14 The simulation and the numerical convergence analysis were carried out in MATHEMATICA. The code used for the results and graphs below is available upon request. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 155 www.RofEA.org From equation (35) in the appendix, we get the following numerical representation of the calibrated, log-linearized system (dln 𝑘/𝑑𝑡 (𝑑ln 𝑐)/𝑑𝑡 (𝑑ln 𝑚)/𝑑𝑡)=(0.0769 −0.2305 −0.0004 −0.0743 0.0230 0.0000 −0.0744 −0.1069 0.1300)×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚)+(−0.035𝑑𝜎 0 2𝑑𝜎+1𝑑𝜃) where 𝑑𝜎 and 𝑑𝜃, our variables of interest here, denote the differentials of 𝜎 and 𝜃 which are constants. From that one obtains (𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚)=𝜉1(−0.713 −0.469 −0.496)𝑒𝜆1⋅𝑡+(−0.045𝑑𝜎+0.0041𝑑𝜃 −0.144𝑑𝜎+0.0131𝑑𝜃 −15.529𝑑𝜎−7.6792𝑑𝜃) (26) as the solution to the system. The derivation can be found in Appendix C.1. Here 𝜆1=−0.084 is the only negative root of the system for the given parameter values. Its associated eigenvector is (−0.713,−0.469,0.496) and 𝜉1 is a constant that needs to be definitized. As 𝑐 and 𝑚 are jump variables we concentrate on the first component of this system, i.e., the equation for the capital stock 𝑘 to determine the constant 𝜉1 from initial conditions. Thus, we solve for 𝜉1 when 𝑡=0, that is, we solve (𝑑ln 𝑘)𝑡=0=ln 𝑘0−ln 𝑘∗=𝜉1⋅(−0.713)⋅𝑒𝜆1⋅0−(0.045)⋅𝑑𝜎+(0.0041)⋅𝑑𝜃 for 𝜉1 with 𝑒𝜆1⋅0=1 This yields the definitized constant 𝜉1∗=(ln 𝑘0−ln 𝑘∗)+0.045⋅𝑑𝜎−0.0041⋅𝑑𝜃 −0.713 (27) where 𝑘0 and 𝑘∗ are predetermined (non-jump) variables which are constant like the chosen values of 𝑑𝜎 and 𝑑𝜃. Hence, 𝜉1∗ is the constant sought after. Clearly, 𝜉1∗ is also important for the paths of the jump variables 𝑐 and 𝑚 and it depends on 𝑑𝜎 and 𝑑𝜃. The paths of 𝑘𝑡,𝑐𝑡 and 𝑚𝑡 in natural logarithms are presented in the next figure, and those for the levels are presented in the appendix. Review of Economic Analysis 16 (2024) 133-173 156 www.RofEA.org Figure 1: The paths of 𝑘𝑡,𝑐𝑡 and 𝑚𝑡 in natural logarithms We now conduct the following experiment for 𝜉 when each policy variable 𝜃 and 𝜎 has a value of one percent so that the steady-state inflation rate is two percent, i.e. around 𝜎=𝜃=0.01. The experiment is to increase the variables by one percentage point. For instance, we look at the system if 𝜎 is raised from one to two percentage points, given 𝜃. The same is done for 𝜃. A final experiment is to consider a joint increase of one percentage point each, given that they were one percent. Table 3: Changes in 𝜎 and 𝜃 and the resulting 𝜉1∗ Case 𝑑𝜎 𝑑𝜃 𝜉1∣ Case ∗ 0 0.00 0.00 0.827303 1 0.01 0.00 0.826675 2 0.00 0.01 0.827360 3 0.01 0.01 0.826732 The changes are taken around 𝜎=𝜃=0.01. From the table the differences are small. But it can be verified that 𝜉1∣2 ∗>𝜉1∣0 ∗>𝜉1∣3 ∗>𝜉1∣1 ∗ which one may have expected from the theoretical predictions. First, consider the equation for the capital stock when 𝜆1=−0.084. Given that 𝑑ln 𝑘= ln 𝑘𝑡−ln 𝑘∗, we obtain from equation (26) that at any point in time 𝑡 REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 163 www.RofEA.org The sign of 𝑑𝑚∗/𝑑𝜎. 𝑑𝑚∗ 𝑑𝜎 =𝛿(𝑑𝑐∗/𝑑𝜎)(𝜌+𝜃+2𝜎)−2𝛿𝑐∗ (𝜌+𝜃+2𝜎)2<0 because 𝑑𝑐∗ 𝑑𝜎<0 The sign of 𝑑𝑚∗/𝑑𝛽 𝑑𝑚∗ 𝑑𝛽 =𝛿(𝑑𝑐∗/𝑑𝛽) 𝜌+𝜃+2𝜎>0 because 𝑑𝑐∗ 𝑑𝛽>0 The sign of 𝑑𝑚∗/𝑑𝛿. I want to show that 𝑑𝑚∗ 𝑑𝛿 =𝑐∗+𝛿(𝑑𝑐∗/𝑑𝛿) 𝜌+𝜃+2𝜎 ⋛0 To this end recall that 𝑐∗=(𝜌−𝑟∗ 𝛽)𝑘∗ and 𝑑𝑐∗ 𝑑𝛿=1 𝛽(−𝑟𝑘∗⋅𝑘∗⋅𝑑𝑘∗ 𝑑𝛿+(𝜌−𝑟∗)⋅𝑑𝑘∗ 𝑑𝛿) and −𝑟𝑘∗⋅𝑘∗=(1−𝛼)𝑟∗. So we get 𝑑𝑚∗ 𝑑𝛿 =𝑐∗+𝛿(𝑑𝑐∗/𝑑𝛿) 𝜌+𝜃+2𝜎 =[(𝜌−𝑟∗)𝑘∗+𝛿[(1−𝛼)𝑟∗+(𝜌−𝑟∗)]𝑑𝑘∗ 𝑑𝛿] 𝛽(𝜌+𝜃+2𝜎) =[(𝜌−𝑟∗)𝑘∗+𝛿[(𝜌−𝛼𝑟∗)]𝑑𝑘∗ 𝑑𝛿] 𝛽(𝜌+𝜃+2𝜎) where we know that 𝑑𝑘∗ 𝑑𝛿=−Δ𝛿 Δ𝑘=− (𝜎 𝜌+𝜃+2𝜎)⋅𝜌−𝑟∗ 𝑟∗⋅𝛼 −(1+ 𝛿𝜎 𝜌+𝜋+𝜎)⋅𝛼⋅𝜌 (𝑟∗)2⋅𝑟𝑘∗=𝜎(𝜌−𝑟∗)⋅(𝑟∗/𝜌) (𝜌+𝜃+(2+𝛿)𝜎)⋅𝑟𝑘∗ Making the substitution above yields [(𝜌−𝑟∗)𝑘∗+𝛿[(𝜌−𝛼𝑟∗)]{𝜎(𝜌−𝑟∗)⋅(𝑟∗/𝜌)) (𝜌+𝜃+(2+𝛿)𝜎)⋅𝑟𝑘∗}]⋅𝐵 (30) Review of Economic Analysis 16 (2024) 133-173 164 www.RofEA.org where 𝐵≡[𝛽⋅(𝜌+𝜃+2𝜎)]−1>0. Pulling out (𝜌−𝑟∗) the expression in square brackets is positive, zero or negative if 𝑘∗⋛−𝛿[(𝜌−𝛼𝑟∗)]{𝜎⋅(𝑟∗/𝜌) (𝜌+𝜃+(2+𝛿)𝜎)⋅𝑟𝑘∗}. (31) As 𝑟𝑘∗=(𝛼−1)⋅𝑟∗⋅(𝑘∗)−1 the inequality boils down to (1−𝛼)𝑟∗𝑘∗ ⋛𝛿[(𝜌−𝛼𝑟∗)]⋅𝜎⋅(𝑟∗/𝜌)⋅𝑘∗ (𝜌+𝜃+(2+𝛿)𝜎) (1−𝛼)𝑟∗𝑘∗⋅(𝜌+𝜃+(2+𝛿)𝜎) ⋛𝛿⋅[(𝜌−𝛼𝑟)]⋅𝜎⋅(𝑟∗/𝜌)⋅𝑘∗ No clear relationship can be established for this inequality. For example, if 𝛿 or 𝜎 are very low (𝛿,𝜎→0), then the inequality is positive and 𝑑𝑚/𝑑𝛿>0 would follow. In turn, if, for example, 𝜎 is very large (e.g. 𝜎→∞ ) then 𝑑𝑚/𝑑𝛿>0 would be implied. Hence, the sign of 𝑑𝑚/𝑑𝛿 is generally not unambiguously clear. The sign of 𝑑𝑚∗/𝑑𝜌. 𝑑𝑚∗ 𝑑𝜌 =𝛿(𝑑𝑐∗/𝑑𝜌)(𝜌+𝜃+2𝜎)−𝛿𝑐∗ (𝜌+𝜃+2𝜎)2<0 because 𝑑𝑐∗ 𝑑𝜌<0 The sign of 𝑑𝑚∗/𝑑𝜃. We have 𝑑𝑚∗ 𝑑𝜃 =𝛿(𝑑𝑐∗/𝑑𝜃)(𝜌+𝜃+2𝜎)−𝛿𝑐∗ (𝜌+𝜃+2𝜎)2 where the sign of that expression depends on the sign of (𝑑𝑐∗/𝑑𝜃)(𝜌+𝜃+2𝜎)−𝑐∗ for nonzero 𝛿, and 𝑐∗=(𝜌−𝑟∗ 𝛽)𝑘∗,𝑑𝑐∗ 𝑑𝜃=1 𝛽(−𝑟𝑘∗⋅𝑘∗⋅𝑑𝑘∗ 𝑑𝜃+(𝜌−𝑟∗)⋅𝑑𝑘∗ 𝑑𝜃)=(𝜌−𝛼𝑟∗ 𝛽)𝑑𝑘∗ 𝑑𝜃 and 𝑑𝑘∗ 𝑑𝜃=−Δ𝜃 Δ𝑘=(𝛿𝜎 (𝜌+𝜃+2𝜎)2)⋅𝜌−𝑟∗ 𝑟∗⋅𝛼 −(1+ 𝛿𝜎 𝜌+𝜃+2𝜎)⋅𝛼⋅𝜌 (𝑟∗)2⋅𝑟𝑘∗=𝛿𝜎(𝜌−𝑟∗)⋅𝑟∗𝑘∗ (𝜌+𝜃+(2+𝛿)𝜎)⋅𝜌⋅(1−𝛼)𝑟∗ where again I have used that −𝑟𝑘∗⋅𝑘∗=(1−𝛼)𝑟∗. Making the appropriate substitutions yields after simplification that the sign of (𝑑𝑐∗/𝑑𝜃)(𝜌+𝜃+2𝜎)−𝑐∗ depends on whether ((𝜌−𝑟∗)𝑘∗ 𝛽)[(𝜌−𝛼𝑟∗)𝛿𝜎(𝜌+𝜃+2𝜎) (𝜌+𝜃+(2+𝛿)𝜎)𝜌(1−𝛼)−1]⋛0. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 165 www.RofEA.org The sign of the expression in square bracket depends on the model's parameters. For example, if 𝜎 or 𝛿 are sufficiently small, the expression in square brackets is negative, if they are sufficiently large, it is positive. Hence, the sign of 𝑑𝑚∗/𝑑𝜃 is not unambiguously clear. B Long-run welfare effects Long-run period welfare is given by 𝜑∗ and by equations (19) and (23) amounts to 𝜑∗=ln 𝑐∗+𝛿ln 𝑚∗+𝛽ln 𝑘∗ =(ln (𝜌−𝑟∗ 𝛽)+ln 𝑘∗) +𝛿(ln ( 𝛿 𝜌+𝜃+2𝜎)+ln (𝜌−𝑟∗ 𝛽)+ln 𝑘∗)+𝛽ln 𝑘∗. Collecting terms then reveals that long-run period welfare is 𝜑∗=(1+𝛿+𝛽)ln 𝑘∗+(1+𝛿)ln (𝜌−𝑟∗ 𝛽)−𝛿ln (𝜌+𝜃+2𝜎 𝛿) where 𝑣=𝜌+𝜃+2𝜎 𝛿 equals the velocity of money. Notice it has a negative effect on long-run welfare in this model. B.1 The effect of 𝝈 and 𝜽 As 𝑐∗,𝑚∗ and 𝑘∗ all depend negatively 𝜎 if follows that 𝑑𝜑∗/𝑑𝜎<0. For the money growth rate 𝜃 one calculates 𝑑𝜑∗ 𝑑𝜃=(1+𝛿+𝛽)⋅𝑑𝑘∗ 𝑑𝜃⋅1 𝑘∗+(1+𝛿)(−𝑟𝑘⋅𝛽 𝜌−𝑟∗)⋅𝑑𝑘∗ 𝑑𝜃−𝛿 𝜌+𝜃−2𝜎 From the main text, we know that 𝑑𝑘∗ 𝑑𝜃=−−( 𝛿𝜎 (𝜌+𝜃+2𝜎)2)(𝜌−𝑟∗ 𝑟∗)⋅𝛼 −(1+ 𝛿𝜎 𝜌+𝜃+2𝜎)(𝛼𝜌 (𝑟∗)2)⋅𝑟𝑘 I want to check whether 𝑑𝜑∗ 𝑑𝜃 >0. This boils down to analyze whether Review of Economic Analysis 16 (2024) 133-173 166 www.RofEA.org (𝛿𝜎 (𝜌+𝜃+2𝜎)2)(𝜌−𝑟∗ 𝑟∗)⋅𝛼⋅[(1+𝛿+𝛽)⋅1 𝑘∗+(1+𝛿)(−𝑟𝑘⋅( 𝛽 𝜌−𝑟∗))]> (𝛿 𝜌+𝜃−2𝜎)⋅(−1)⋅(1+ 𝛿𝜎 𝜌+𝜃+2𝜎)(𝛼𝜌 (𝑟∗)2)⋅𝑟𝑘. Cancellation by common terms then yields 𝜎⋅(𝜌−𝑟∗)⋅[(1+𝛿+𝛽)⋅1 𝑘∗+(1+𝛿)(−𝑟𝑘⋅( 𝛽 𝜌−𝑟∗))]> (𝜌+𝜃+(2+𝛿)𝜎)(𝜌 𝑟∗)⋅(−𝑟𝑘). Note that −𝑟𝑘=𝛼(𝛼−1)𝑘𝛼−2=(𝛼−1)⋅𝑟∗⋅(𝑘∗)−1. Substituting this in and rearrangement yields 𝜎⋅(𝜌−𝑟∗)⋅[(1+𝛿+𝛽)+(1+𝛿)((1−𝛼)⋅𝑟∗⋅( 𝛽 𝜌−𝑟∗))]> (𝜌+𝜃+(2+𝛿)𝜎)⋅𝜌⋅(1−𝛼). It is not difficult to see that the last inequality does not always hold and depends in an important way on the parameters of the model. For example, if 𝜎 is very low, it does not hold. It may hold for sufficiently large values of it, though. It may also hold if 𝛽 is sufficiently large. But in general, no clear overall relationship between 𝑤∗ and 𝜃 holds. But it is definitely so that the partial effects of 𝜃 on welfare through the consumption and capital channel raise welfare derived from them, that is, they raise welfare conditionally. In the model, the impact of the velocity of money and its reaction to changes in 𝜃 are so large that the other partial effects are outweighed. C Analysis of the Transitional Dynamics The dynamic system of the equations in (15) can be formulated in (natural) logs as 𝑑ln 𝑘 𝑑𝑡 =𝑒(𝛼−1)ln 𝑘−𝑒ln (𝑐/𝑘)−𝜎𝑒ln (𝑚/𝑘) (32𝑎) 𝑑ln 𝑐 𝑑𝑡 =𝛽𝑒ln (𝑐/𝑘)+𝛼𝑒(𝛼−1)ln 𝑘−𝜌 (32𝑏) 𝑑ln 𝑚 𝑑𝑡 =𝜃+2𝜎−𝛿𝑒ln (𝑐/𝑚)+𝛽𝑒ln (𝑐/𝑘)+𝛼𝑒(𝛼−1)ln 𝑘 (32𝑐) REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 167 www.RofEA.org In steady state 𝑑ln 𝑘 𝑑𝑡 =𝑑ln 𝑐 𝑑𝑡 =𝑑ln 𝑚 𝑑𝑡 =0 so that 𝑒(𝛼−1)ln 𝑘∗ =𝑒ln (𝑐∗/𝑘∗)+𝜎𝑒ln (𝑚∗/𝑘∗)(33𝑎) 𝛽𝑒ln (𝑐∗/𝑘∗)+𝛼𝑒(𝛼−1)ln 𝑘∗ =𝜌, (33𝑏) 𝛿𝑒ln (𝑐∗/𝑚∗) =𝜃+2𝜎+𝛽𝑒ln (𝑐∗/𝑘∗)+𝛼𝑒(𝛼−1)ln 𝑘∗. (33𝑐) From these equations, it then follows that in a steady state 𝑒ln (𝑐∗/𝑚∗)=𝜌+𝜃+2𝜎 𝛿 and 𝜎𝑒ln (𝑚∗/𝑘∗)=𝑒(𝛼−1)ln 𝑘∗−𝑒ln (𝑐∗/𝑘∗)=𝑟∗ 𝛼−𝜌−𝑟∗ 𝛽 where 𝑓(𝑘∗)/𝑘∗=(𝑘∗)𝛼−1=𝑟∗/𝛼 and 𝑟∗=𝛼(𝑘∗)𝛼−1=𝛼𝑒(𝛼−1)ln 𝑘∗(34) Now we linearize the system in (32) to get ( 𝑑ln 𝑘 𝑑𝑡 𝑑ln 𝑐 𝑑𝑡 𝑑ln 𝑚 𝑑𝑡 ) =Δ×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚) where 𝑑ln 𝑗=ln 𝑗−ln 𝑗∗=ln (𝑗/𝑗∗) for 𝑗=𝑘,𝑐,𝑚, and the star * denotes variables that are in their steady state. Δ is defined as 𝚫≡(Δ1𝑘 Δ1𝑐 Δ1𝑚 Δ2𝑘 Δ2𝑐 Δ2𝑚 Δ3𝑘 Δ3𝑐 Δ3𝑚)𝑘∗,𝑐∗,𝑚∗ and represents the Jacobian of the system, evaluated in steady state equilibrium. Its elements are given by Using the information about the steady state values yields the following: Δ1𝑘=(𝛼−1)𝑒(𝛼−1)ln⁡𝑘∗+𝑒ln⁡(𝑐∗/𝑘∗)+𝜎𝑒ln⁡(𝑚∗/𝑘∗),Δ1𝑐=−𝑒ln⁡(𝑐∗/𝑘∗),Δ1𝑚=−𝜎𝑒ln⁡(𝑚∗/𝑘∗), Δ2𝑘=−𝛽𝑒ln⁡(𝑐∗/𝑘∗)+𝛼(𝛼−1)𝑒(𝛼−1)ln⁡𝑘∗,Δ2𝑐=𝛽𝑒ln⁡(𝑐∗/𝑘∗),Δ2𝑚=0, Δ3𝑘=−𝛽𝑒ln⁡(𝑐∗/𝑘∗)++𝛼(𝛼−1)𝑒(𝛼−1)ln⁡𝑘∗,Δ3𝑐=−𝛿𝑒ln⁡(𝑐∗/𝑚∗)+𝛽𝑒ln⁡(𝑐∗/𝑘∗),Δ3𝑚=𝛿𝑒ln⁡(𝑐∗/𝑚∗). Review of Economic Analysis 16 (2024) 133-173 168 www.RofEA.org Δ1𝑘 =(𝛼−1)𝑒(𝛼−1)ln 𝑘∗+𝑒ln (𝑐∗/𝑘∗)+𝜎𝑒ln (𝑚∗/𝑘∗) =𝛼𝑒(𝛼−1)ln 𝑘∗−𝑒(𝛼−1)ln 𝑘∗+𝑒ln (𝑐∗/𝑘∗)+𝜎𝑒ln (𝑚∗/𝑘∗)=𝑟∗ because 𝑒(𝛼−1)ln 𝑘∗=𝑒ln (𝑐∗/𝑘∗)+𝜎𝑒ln (𝑚∗/𝑘∗) in steady state and 𝛼𝑒(𝛼−1)ln 𝑘∗=𝑟∗. Δ1𝑐=−𝑒ln (𝑐∗/𝑘∗)=𝑟∗−𝜌 𝛽<0 On account of equations (33b) and (34). Furthermore, Δ1𝑚=−𝜎𝑒ln (𝑚∗/𝑘∗)=−𝑒(𝛼−1)ln 𝑘∗+𝑒ln (𝑐∗/𝑘∗)=−𝑟∗ 𝛼+𝜌−𝑟∗ 𝛽=𝜌−𝑟∗(1+𝛽 𝛼) 𝛽<0, i.e. for a positive money depreciation rate Δ1𝑚 is negative. 15 Next, we have Δ2𝑘=−𝛽𝑒ln (𝑐∗/𝑘∗)+𝛼(𝛼−1)𝑒(𝛼−1)ln 𝑘∗=−(𝜌−𝑟∗)+(𝛼−1)𝑟∗=𝛼𝑟∗−𝜌<0 Δ2𝑐=𝛽𝑒ln (𝑐∗/𝑘∗)=𝜌−𝑟∗>0 and Δ2𝑚=0. For the effect on money growth, we get Δ3𝑘 =−𝛽𝑒ln (𝑐∗/𝑘∗)++𝛼(𝛼−1)𝑒(𝛼−1)ln 𝑘∗=Δ2𝑘=𝛼𝑟∗−𝜌<0 Δ3𝑐 =−𝛿𝑒ln (𝑐∗/𝑚∗)+𝛽𝑒ln (𝑐∗/𝑘∗)=−𝛿[𝜌+𝜃+2𝜎 𝛿]+𝜌−𝑟∗=−(𝑟∗+𝜃+2𝜎) Δ3𝑚 =𝛿𝑒ln (𝑐∗/𝑚∗)=𝜌+𝜃+2𝜎. All of this and the definition Δ imply that the log-linearized system is given by ((𝑑ln 𝑘)/𝑑𝑡 (𝑑ln 𝑐)/𝑑𝑡 (𝑑ln 𝑚)/𝑑𝑡) =(Δ1𝑘 Δ1𝑐 Δ1𝑚 Δ2𝑘 Δ2𝑐 Δ2𝑚 Δ3𝑘 Δ3𝑐 Δ3𝑚)𝑘∗,𝑐∗,𝑚∗×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚) = ( 𝑟∗𝑟∗−𝜌 𝛽𝜌−𝑟∗(1+𝛽 𝛼) 𝛽 𝛼𝑟∗−𝜌 𝜌−𝑟∗0 𝛼𝑟∗−𝜌 −(𝑟∗+𝜃+2𝜎)𝜌+𝜃+2𝜎 ) ×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚). 15 Note that for 𝜎=0, that is, when the model dichotomizes we would, of course, have Δ1𝑚=0. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 169 www.RofEA.org In order to analyze the question of how the log-linearized system reacts to a change in monetary policy, that is, to changes in the Gesell Tax and the money growth rate one verifies that the complete linearized system is really given by 16 ((𝑑ln 𝑘)/𝑑𝑡 (𝑑ln 𝑐)/𝑑𝑡 (𝑑ln 𝑚)/𝑑𝑡)=Δ×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚)+ ( 𝜌−𝑟∗(𝛽+𝛼 𝛼) 𝜎𝛽 02 ) ×𝑑𝜎+(001)×𝑑𝜃 (35) where 𝑑𝜎 and 𝑑𝜃 are scalars that represent the differential of 𝜎 and 𝜃, respectively, and the entries of the column vector 𝑣 represent the response of the (log-linearized) differential system to a change in 𝜎 when the transpose of 𝑣 is given by 𝑣′≡(𝜌−𝑟∗(𝛽+𝛼 𝛼) 𝜎𝛽 ,0,2)′ and in steady state Δ1𝜎=𝑒ln (𝑚∗/𝑘∗) and 𝜎𝑒ln (𝑚∗/𝑘∗)=𝑟∗ 𝛼−𝜌−𝑟∗ 𝛽. 17 In turn, the entries of the column vector 𝑤 represent the response of the (log-linearized) differential system to a change in 𝜃 when the transpose of 𝑤 is given by 𝑤′≡(0,0,1)′ We can then express the system in (35) in compact form as 𝐉′ =Δ𝑱+𝐠 where 𝐉′= ( 𝑑ln 𝑘 𝑑𝑡 𝑑ln 𝑐 𝑑𝑡 𝑑ln 𝑚 𝑑𝑡 ) ,𝐉=(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚), and 𝐠 = ( 𝜌−𝑟∗(𝛽+𝛼 𝛼) 𝜎𝛽 02 ) ×𝑑𝜎+(001)×𝑑𝜃= ( 𝜌−𝑟∗(𝛽+𝛼 𝛼) 𝜎𝛽 𝑑𝜎 0 2𝑑𝜎+𝑑𝜃 ) . This is a nonhomogeneous differential equation system. The homogeneous part is 𝐉′= 𝚫𝑱 and depends in an important way on the Jacobian 𝚫. In turn, the term g makes the system nonhomogeneous. 16 Here the assumption is, of course, that the initial values are close to the steady state. Although loglinear approximations are widely used in macroeconomics, the requirement that they apply only as approximations in the neighborhood of the steady state can be regarded as disadvantage. See, for example, Barro and Sala-i-Martin (2004), p. 111. 17 Again note that this only holds for a non-zero 𝜎. If 𝜎=0, then in view of equation (19) we would have 𝑣′≡(𝛿 𝛽[𝜌−𝑟∗ 𝜌+𝜃+2⋅0],0,2)′ and then there is no effect of a change in 𝜃 on 𝑘 in the transition. Review of Economic Analysis 16 (2024) 133-173 170 www.RofEA.org First, we solve the homogeneous part 𝐉′=𝚫𝑱, that is 𝐉′−𝚫𝑱=𝟎, by employing the guess 𝐉=𝒙𝑒𝜆𝑡. From this, we get 18 𝐉′=𝜆𝒙𝑒𝜆𝑡=𝚫𝒙𝑒𝜆𝑡, hence 𝜆𝒙=𝚫𝒙. For a nontrivial solution, we need the eigenvalues (roots) and the eigenvectors of this threedimensional system. The general solution of the homogeneous system is given by 𝑱𝒉=𝜉1𝒙(𝟏)𝑒𝜆1⋅𝑡+𝜉2𝒙(𝟐)𝑒𝜆2⋅𝑡+𝜉3𝒙(𝟑)𝑒𝜆3⋅𝑡. For 𝑖=1,2,3 the roots of the system are given by 𝜆𝑖, the eigenvectors by 𝒙(𝑖), and the arbitrary constants by 𝜉𝑖. As the dynamics of the system are essentially governed by the same variables as in the standard Sidrauski model, we can employ the same arguments as in, for example, Blanchard and Fischer (1989), Appendix B of chapter 4, and Fischer (1979). Hence, we note that the capital stock is given, but the money stock and consumption can jump at any point in time. As a consequence, if the system is to have a (locally) unique stable path, it must have two positive roots (or a pair of complex roots with positive real part) and one negative root. If this is the case the jump variables will take on (initial) values that make the system converge. 19 In fact, we can determine this more rigorously for the present model by following arguments. It is well known that the product of the roots is equal to the determinant of 𝚫. Calculating the determinant then yields |Δ|=𝜆1⋅𝜆2⋅𝜆3 =𝜌2𝑟∗−𝜌2𝑟∗ 𝛼+2𝜌𝑟∗𝜎−2𝜌2𝑟∗𝜎 𝛼+𝜌𝑟∗𝜃−𝜌2𝑟∗𝜃 𝛼 =−(1−𝛼)𝜌𝑟∗(𝜌+2𝜎+𝜃) 𝛼<0. Thus, either all three roots are negative, or there are two positive and one negative root. If the system features saddle path stability, the latter is true and we additionally should have that the trace of Δ which equals the sum of the eigenvalues be non-negative. The latter is easily calculated as tr (𝚫)=𝜆1+𝜆2+𝜆3=2𝜌+2𝜎+𝜃 18 In this section I follow the solution method presented in Kreyszig (2006), ch. 4. 19 If the system had, for example, three negative roots, then starting from any value of c and m. the system would - locally - converge. There would be nothing to tie down the money stock or the level of consumption c. See Blanchard and Fischer (1989), p. 204. REHME Silvio Gesell's Schwundgeld Reconsidered, Part II 171 www.RofEA.org which is indeed positive. Hence, at least one root is positive. With |𝚫|<0 and tr (𝚫)>0 the system is, therefore, saddle-path stable, because for our 3×3 system at least one root is positive so that there can only be one negative root. In summary, there will be two positive and one negative eigenvalue in the system. One can also calculate the eigenvalues of the system. They are given by 𝜆1,𝜆2,𝜆3={−−𝛼𝜌±√𝛼𝜌⋅√𝛼𝜌+4(1−𝛼)𝑟∗ 2𝛼 ,𝜌+2𝜎+𝜃}. Let 𝜆1 denote the negative root. Given the parameters it satisfies 𝜆1=−−𝛼𝜌+√𝛼𝜌⋅√𝛼𝜌+4(1−𝛼)𝑟∗ 2𝛼 <0. It is important to note that the negative root governsthe speed of convergence of the system. The more negative the negative eigenvalue 𝜆1 is, the faster the speed at which the system converges to its steady state. In this context, it is not difficult to verify that 𝑑𝜆1/𝑑𝜎<0 and 𝑑𝜆1/𝑑𝜃>0. That means that as you increase 𝜎, the root 𝜆1 will be more negative and so the convergence to the steady state will be faster, whereas an increase in 𝜃 is associated with a less negative root, implying that convergence will be slower. From that Proposition 4 in the main text follows in a straightforward manner. Notice that we cannot have a convergent system when any of the roots is positive and the associated eigenvector 𝑥(𝑖) is non-zero. One way to rule out explosive paths is to set the arbitrary constant associated with a positive root equal to zero. In our context, 𝜆1<0, and 𝜆2,𝜆3>0, and 𝜉1≠0, but then we need 𝜉2=𝜉3=0 to rule out explosive behavior. As a consequence, the solution to the homogenous system boils down to 𝑱𝒉=𝜉1𝒙(1)𝑒𝜆1⋅𝑡. For a particular solution of the nonhomogeneous system above and since the vector 𝐠 is constant, we try a constant column vector 𝑱𝒑=𝒂 with components 𝛼1,𝑎2 and 𝑎3 20 As a consequence, 𝐽𝑝′=𝟎 and substitution in the system 𝐉′=Δ𝑱+g yields Δ𝒂+g=𝟎. Solving for the components of a, we get the following system under the assumptions made so far 𝑱=𝑱𝒉+𝑱𝒑=𝜉1𝒙(𝟏)𝑒𝜆1⋅𝑡+𝐚. The last step then is to use the initial conditions to definitize the constant 𝜉1. Let 𝜉˜1 denote the definitzed constant and let 𝜉˜1⋅𝒙(1)≡𝒙(1). Then the solution of our system is given by 𝑱=𝑱𝒉+𝑱𝒑=𝒙(𝟏)𝑒𝜆1⋅𝑡+𝐚. 20 In this paragraph I closely follow Kreyszig (2006), p. 133. Review of Economic Analysis 16 (2024) 133-173 172 www.RofEA.org The numerical simulation below clarifies the procedure in more detail. C.1 Numerical simulation From the values in Tables 1 and 2, one gets the following numerical representation of the system in (35), ( 𝑑ln 𝑘 𝑑𝑡 𝑑ln 𝑐 𝑑𝑡 𝑑ln 𝑚 𝑑𝑡 ) =(0.0769 −0.2305 −0.0004 −0.0743 0.0230 0.0000 −0.0744 −0.1069 0.1300)×(𝑑ln 𝑘 𝑑ln 𝑐 𝑑ln 𝑚)+(−0.035𝑑𝜎 0 2𝑑𝜎+1𝑑𝜃) where 𝑑𝜎 and 𝑑𝜃, our variables of interest in this section, denote the differentials of 𝜎 and 𝜃 which are constants. The numerical convergence analysis was carried out in MATHEMATICA. The code used for the results is available upon request. The 3×3 matrix represents the Jacobian 𝚫. The roots 𝜆 of the homogenous part 𝑱𝒉 are given by (−0.0837,0.1300,0.1838). As outlined above I concentrate on the negative root and call it 𝜆1. Thus, 𝜆1=−0.084. Associated with 𝜆1 is the eigenvector (−0.713,−0.469,0.496). Hence, the general solution for our system is given by 21 𝑱𝒉=𝜉1𝒙(1)𝑒𝜆1⋅𝑡=𝜉1(−0.713 −0.469 −0.496)𝑒𝜆1⋅𝑡 For the particular solution 𝑱𝒑 one obtains 𝑱𝑝=(−0.045𝑑𝜎+0.0041𝑑𝜃 −0.144𝑑𝜎+0.0131𝑑𝜃 −15.529𝑑𝜎−7.6792𝑑𝜃) that solves Δ𝑎+g=0 when looking at changes in 𝜎 and 𝜃. From that we get (𝒅𝐥𝐧 𝒌 𝒅𝐥𝐧 𝒄 𝒅𝐥𝐧 𝒎)=𝑱=𝑱𝒉+𝑱𝒑=𝝃𝟏(−𝟎.𝟕𝟏𝟑 −𝟎.𝟒𝟔𝟗 −𝟎.𝟒𝟗𝟔)𝒆𝝀𝟏⋅𝒕+(−𝟎.𝟎𝟒𝟓𝒅𝝈+𝟎.𝟎𝟎𝟒𝟏𝒅𝜽 −𝟎.𝟏𝟒𝟒𝒅𝝈+𝟎.𝟎𝟏𝟑𝟏𝒅𝜽 −𝟏𝟓.𝟓𝟐𝟗𝒅𝝈−𝟕.𝟔𝟕𝟗𝟐𝒅𝜽) as the solution to the system which features in the main text as equation (26). With the definitized constant 𝜉1∗=0.827 the graphs of the variables of interest in levels are presented in the following figure. 21 Notice that with these values we get |Δ|=-0.002 and tr⁡(Δ)=0.23.