Budget deficit in a growing economy and impossibility of fiscal collapse: A continuous time analysis

Tanaka, Yasuhito Article Budget deficit in a growing economy and impossibility of fiscal collapse: A continuous time analysis Central European Economic Journal (CEEJ) Provided in Cooperation with: Faculty of Economic Sciences, University of Warsaw Suggested Citation: Tanaka, Yasuhito (2024) : Budget deficit in a growing economy and impossibility of fiscal collapse: A continuous time analysis, Central European Economic Journal (CEEJ), ISSN 2543-6821, Sciendo, Warsaw, Vol. 11, Iss. 58, pp. 305-319, https://doi.org/10.2478/ceej-2024-0020 This Version is available at: https://hdl.handle.net/10419/324617 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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Central European Economic Journal, 11(58), 305-319. DOI: 10.2478/ceej-2024-0020 To link to this article: https://doi.org/10.2478/ceej-2024-0020 Budget Deficit in a Growing Economy and Impossibility of Fiscal Collapse: A Continuous Time Analysis Yasuhito Tanaka Open Access. © 2024 Yasuhito Tanaka, published by Sciendo. This work is licensed under the Creative Commons Attribution 4.0 International License. Yasuhito Tanaka Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, Japan corresponding author: [email protected] Budget Deficit in a Growing Economy and Impossibility of Fiscal Collapse: A Continuous Time Analysis Abstract Using a continuous time dynamic model of growing economy we will show the following results. 1) When people derive utility from their money holding (or government bond holding) along with their consumption, a budget deficit is essential to achieve and maintain full employment under stable prices or inflation in a growing economy. 2) If we take into account that government spending due to budget deficits increases financial assets held by the private sector, and then consumption will occur from assets in addition to consumption from income, even when the interest rate on government bonds is higher than the real economic growth rate, the ratio of government debt to GDP can not diverge and the divergence is naturally prevented by mild inflation. The required inflation rate is such that the interest rate of the government bonds is smaller than the weighted average of the rate of return on capital and the nominal growth rate. Since the interest rate of the government bonds is usually considered smaller than the rate of return on capital, this is not a very demanding requirement. Thus, we need not worry at all about the accumulation of government debt or about the divergence of the debt to GDP ratio, which is often taken as an indicator of fiscal collapse. Keywords budget deficit | growing economy | infinitely living consumers | continuous time model | impossibility of fiscal collapse JEL Codes E12, E24 1. Introduction The purpose of this paper is to prove the following results using a relatively simple mathematical model. 1. When people derive utility from their money holding (or government bond holding) along with their consumption, a budget deficit is essential to achieve and maintain full employment under stable prices or inflation in a growing economy. 2. With the following items (a) and (b) in mind, even when the interest rate on government bonds is higher than the real economic growth rate, the ratio of government debt to GDP does not diverge and the divergence is naturally prevented by mild inflation. The required inflation rate is such that the interest rate of the government bonds is smaller than the weighted average of the rate of return on capital and the nominal growth rate. Since the interest rate of government bonds is usually considered smaller than the rate of return on capital, this is not a very demanding requirement. (a) Government spending due to budget deficits increases government bonds or base money held by banks and increases financial assets held by the private sector. (b) As financial assets increase, consumption will occur from assets in addition to consumption from income. If the GDP ratio of government debt diverges and becomes infinitely large, then the GDP ratio of private financial assets also becomes infinitely large, and the GDP ratio of consumption from assets also becomes infinitely large. However, since consumption is a part of GDP, such a situation cannot occur and a contradiction arises. In such a case, an increase in consumption demand would cause a rise in prices, which would prevent the debt-to-GDP ratio from diverging. This is a naturally occurring phenomenon, not caused by some policy. Thus, we need not worry at all about the accumulation of government debt or about the divergence of the debt-to-GDP ratio, which is often taken as an indicator of fiscal collapse. CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 307 If current account deficits continue and net external debt accumulates, it will eventually have to be repaid, which will be a major obstacle to the country’s economy. However, as long as the government debt remains domestic, there is no problem at all. In Section 2, we explain the methodology of this paper and present a literature review. Section 3 presents this paper’s model and analyses of the behaviour of consumers, firms, and market equilibrium. We will prove the necessity of a budget deficit for full employment under constant prices. We show also that larger budget deficits cause inflation, or we need a larger budget deficit to realise full employment under inflation. The policy of increasing money through budget deficits in line with the rate of economic growth as indicated by the conclusions of this section is consistent with the Monetarist k% rule. Section 4 presents the explicit values of the savings and that of money holding in the steady state. In Section 5, a case with interest-producing government bonds instead of money will be considered. It will be shown that the debt-to-GDP ratio should be constant in a steady-state growth path, and the large propensity to consume leads to a small debt-to-GDP ratio. In Section 6, we will prove that divergence of the debt-to-GDP ratio to infinity cannot occur even if the interest rate of the government bond is larger than the real growth rate, that is, fiscal collapse is impossible. In that case, inflation raises the nominal growth rate and prevents the debt-to-GDP ratio from diverging to infinity. The required inflation rate is such that the interest rate of the government bonds is smaller than the weighted average of the rate of return on capital and the nominal growth rate. Hyper-inflation will not occur. It is the interest on government bonds that causes divergence. Without interest, divergence does not occur. Section 7 is the concluding section. 2. Methodology and literature review A macroeconomic model that includes microeconomic foundations for consumers’ behaviour and firms’ behaviour will be used. Consumers are assumed to live infinitely, and utility maximisation over an infinite time is considered. People’s utility depends on their holding of money or government bonds as well as their consumption. A continuous time dynamic model will be used. Economic growth is not based on the assumption that new generations will be born one after another, but rather on the assumption that the population of the same generation will increase. This can be interpreted as economic growth due to technological progress that increases labour productivity. As a result, the size of the economy continuously increases, and the accumulated financial assets must be discounted by the growth rate. In the above discussion, the permanence of the state or mankind is assumed. Under this assumption, both government debt and private financial assets will continue to accumulate forever. If the destruction of the nation or the extinction of the human race are foreseen, people will try to use up all of their assets by then, so consumption will increase further, full employment can be maintained even with budget surpluses, and both private financial assets and government debt will gradually decline and disappear on the day of destruction. In Sections 3 and 4, we consider budget deficit due to money issuance, not government bonds. In Sections 5 and 6, we examine the case of budget deficit by government bonds. Please see Oguri (2011) for the relationship between government bonds and money. There is little literature written from the viewpoint of not considering the accumulation of government debt to be a problem, except for those who belong to MMT (Modern Money Theory, for example, Wray (2015), Mitchell, Wray and Watts (2019), Kelton (2020) or its periphery. Since MMT members do not like to use mathematical models, such papers are even scarcer. In this section, we would like to briefly mention some of the literature that we consulted in writing this paper. Most of the discussions about the debt-to-GDP ratio use a simple calculation about primary budget balances, the interest rate, and the growth rate. In his 2022 and 2023 articles, Blanchard introduces the following problem: “When does the level of debt become unsafe? To answer this question, we need a definition of ‘unsafe’. I propose the following: Debt becomes unsafe when there is a non-negligible risk that, under existing and likely future policies, the ratio of debt to GDP will steadily increase, leading to default at some point. The natural way to proceed is then straightforward. The dynamics of the debt ratio CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 308 depend on the evolution of three variables: primary budget balances (that is, spending net of interest payments minus revenues); the real interest rate (the nominal rate minus the rate of inflation); and the real rate of economic growth.” (Blanchard, 2002/3) However, in a steady state under full employment with or without inflation, the necessary budget deficit for full employment is determined by several parameters of the economy. The larger budget deficit raises the inflation rate, and the large (or small) propensity to consume leads to the small (or large) budget deficit required for full employment under constant price or inflation. Therefore, the large propensity to consume leads to a small debt-to-GDP ratio. While this paper uses a continuous time model that assumes people live infinitely, we have also analysed problems related to budget deficits using overlapping generation models. In doing so, we referred to Diamond (1965), J. Tanaka (2010, 2011a, 2011b, 2013) and Otaki (2007, 2009, 2015). As for economic growth, this paper uses an exogenous growth model, but we have also used an endogenous growth model due to investment by firms with reference to Grossman and Yanagawa (1993) and Maebayashi and J. Tanaka (2021). There are papers, for example, Weil (1987, 1989), that use continuous time models which assume people live infinitely. However, in this research, the budget constraint in the discrete time model is considered implicitly in our mind with reference to Tachibana (2006), and the budget constraint in the continuous time model is derived by making the time interval of the discrete time model infinitesimal. We follow Lerner’s functional finance theory (Lerner, 1944). He did not consider whether the government should run surpluses or deficits to be meaningful in and of itself. He believes that fiscal policy should be used to realise full employment avoiding inflation. For more on Lerner’s functional finance theory, see Forstater (1999). Our model is a kind of neoclassical model, but its spirit may be post-Keynesian in that it does not abandon the goal of full employment out of a dislike of budget deficits or the accumulation of government debt. Lopez-Gallardo (2000) is a study of budget deficits and full employment from a post-Keynesian standpoint. It is inspired by Minsky (1986) and related to Mosler (1997-1998), Wray (1998) and Kregel (1998). Mosler, Wray and Kregel (M-W-K), as a policy for full employment, proposed the following, as described in Lopez-Gallardo ((2000), p. 550): “Let the government assume the role of employer of last resort at a given wage rate, so that anybody willing to work at that rate will get a job from the government. Government expenditure will thus expand, but will not entail any complication because “the purchasing ability of the government is limited only by what is available for sale in exchange for dollars” (Mosler, 1997-1998, p.169), while this availability, we are told, is elastic below full employment. Now, government expenditure will grow probably over and above tax receipts, and a budget deficit will ensue. However, M-W-K demonstrate, with explanations rich in theoretical, historical, and institutional details, that the government can simply create enough new money, or otherwise sell securities, to finance the deficit with an unchanging rate of interest.” Lopez-Gallardo (2000) discusses various problems with this proposal, which are not of interest to this paper. Another description from a post-Keynesian standpoint, according to Sawyer (2020), is that “Kalecki (1944) argued that there would be the need for permanent budget deficits in the face of intentions to save exceeding intentions to invest.” This paper is also an example of an analysis, using a simple mathematical model, of the following statement by J. M. Keynes: “Unemployment develops, that is to say, because people want the moon; — men cannot be employed when the object of desire (i.e. money) is something which cannot be produced and the demand for which cannot be readily choked off. There is no remedy but to persuade the public that green cheese is practically the same thing and to have a green cheese factory (i.e. a central bank) under public control.” (Keynes (1936), Chap. 17) The goal of macroeconomic policy is, motto-wise, “full employment without inflation”. It is not right to be concerned about fiscal surpluses or deficits, since they are merely the means to that goal and the result of that goal. Whether or not to repay government debt with taxes should be determined based on how it will affect prices and employment. Repayment is not a natural assumption, and whether it is repaid or not has no value in itself. CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 309 3. Holding of money and budget deficit in a growing economy 3.1. Consumers’ behaviour Utility function We consider an exogenous growth model in which consumers infinitely live and hold money for the reason of liquidity and so on. The consumer’s utility over an infinite time is ∫∞ 𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑. (1) (1) The utility function is 𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�=𝛼𝛼𝛼𝛼ln 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡+ (1 −𝛼𝛼𝛼𝛼)ln 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡. (2) (2) ct is the real value of the consumption by a consumer, and pt is the price of the good. mt is the nominal value of the money holding of the consumer. Therefore, mt/pt is the real money holding. The consumer’s utility depends on the consumption and the real money holding. δ >0 is the discount rate. α is the propensity to consume of the consumers. 0<α<1. Budget constraint The budget constraint for the consumer is  t=(1τ )wt lt-pt ct-rt mt+(rt-n)bt.(3) bt is the per capita savings of the consumer, and  t is the time derivative of bt. Generally, the time derivative of a variable x is denoted by  . wt is the wage rate, lt is an indicator of whether the consumer is employed or not. 1 if employed, 0 if not. The meanings of this equation are as follows. 1.  t is the change in the per capita value of the savings. 2. (1τ )wt lt-pt ct is the difference between the per capita disposable labour income and consumption. 3. bt-mt is the portion of the per capita savings that is invested in productive capital, which generates interest (return) rt. 4. We assume that people live infinitely and no new generation will be born, however, the current generation’s population will grow. Hence the savings must be discounted by the growth rate as expressed by -nbt. Utility maximisation The present value Hamiltonian is written as follows. 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�+𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡−𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡−𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]. (4) (4) 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢�𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡,𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�+𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡−𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡−𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]. (4) λ t is the Lagrange multiplier. The first order conditions are 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑡𝑡𝑡𝑡 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒 −𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡−𝜆𝜆𝜆𝜆 𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡 = 0, (5) (5) and 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑡𝑡𝑡𝑡 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑒𝑒𝑒𝑒 −𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 1−𝛼𝛼𝛼𝛼 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡−𝜆𝜆𝜆𝜆 𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡 = 0. (6) (6) The costate equation is 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑡𝑡𝑡𝑡 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=(𝑟𝑟𝑟𝑟 𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛)𝜆𝜆𝜆𝜆 𝑡𝑡𝑡𝑡 =−𝜆𝜆𝜆𝜆󰇗 𝑡𝑡𝑡𝑡 . (7) (7) By (5) and (6), we get 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏 󰇗𝑡𝑡𝑡𝑡�, and 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏 󰇗𝑡𝑡𝑡𝑡�. (8) They mean 𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 𝜆𝜆𝜆𝜆𝛿𝛿𝛿𝛿= (1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝛿𝛿𝛿𝛿𝑙𝑙𝑙𝑙𝛿𝛿𝛿𝛿+(𝑟𝑟𝑟𝑟𝛿𝛿𝛿𝛿−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝛿𝛿𝛿𝛿−𝑏𝑏𝑏𝑏 󰇗𝛿𝛿𝛿𝛿. Thus, 𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 =1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡. Differentiating this with respect to t, CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 310 𝜆𝜆𝜆𝜆󰇗𝑡𝑡𝑡𝑡=−𝛿𝛿𝛿𝛿1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒−𝛿𝛿𝛿𝛿𝑡𝑡𝑡𝑡 =−𝛿𝛿𝛿𝛿𝜆𝜆𝜆𝜆𝑡𝑡𝑡𝑡. Then, from (7) we find rt=n+δ. (9) This is the equilibrium interest rate (rate of return). Steady state Let us consider a steady state. Under constant prices, wt ct, mt and bt are constant in the steady state. They are the wage rate, and the per capita values of real consumption, nominal money holding and nominal savings. Then,  t=0. On the other hand, under inflation at a constant rate π , ct is constant, but wt, mt and bt increases at the rate of π . Then,  t=bt π . Denote the labour supply or the employment under full employment by Lt f. Also, we denote Bt=bt Lt f,Ct=ct Lt f,Mt=mt Lt f. The real value of the capital is 𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡. Denote the real capital per labour under full employment by 𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡=𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡 𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓. In the steady state under full employment, k t=0. 3.2. Firms’ behaviour Let yt be the output, Kt be the capital, and Lt be the employment of a firm. Then, the production function is written as follows. yt=F(Kt, Lt)=Lt f(kt)=Lt F(kt,1). We assume the constant returns to scale property for the production function. We normalise so that the number of firms is one. Each firm maximises its profit. The profit of a firm is pt yt-pt rt Kt-wt Lt=pt Lt f(kt)-pt rt Kt-wt Lt. The first order conditions for profit maximisation are 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓′(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡), and 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡[𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡)−𝑓𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡)𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡]. 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡 and 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 are the marginal productivity of capital and that of labour. From them, we have wt Lt=pt [f(kt)-f‘(kt)kt] Lt, and pt rt Kt=pt f‘ (kt) Kt = pt f‘(kt) kt Lt Then, we obtain py yt=wt Lt+pt rt Kt. This is the total nominal supply of the good. The real value of the capital increases at the rate of n. Then, 𝐾𝐾𝐾𝐾󰇗𝑡𝑡𝑡𝑡=𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡. 𝐾𝐾𝐾𝐾󰇗𝑡𝑡𝑡𝑡 is the time derivative of Kt. This increase in capital is the investment. We assume full employment. Therefore, lt=1 for all people. 3.3. Market equilibrium We consider two cases, with and without inflation. Without inflation Denote the constant price by p . In the steady state under full employment and constant price, the total consumption demand is CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 311 p Ct=Lt f p ct=α[(1τ )wt+(rt-n)bt] Lt f=α[(1τ )wt Lt f+(rt-n)Bt]. The total money holding is 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�. (10) (10) 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡[(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡]𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�. (10) Let Gt be the nominal value of the fiscal expenditure. The total nominal demand is Gt+α[(1τ )wt Lt f+(rt-n)Bt]+ p nKt. p nKt is the nominal investment. The market clearing condition is Gt+α[(1τ )wt Lt f+(rt-n)Bt]+ p n Kt=wt Lt f+ p r t Kt.(11) From this we obtain (see Appendix 1) Gtτ wt Lt f=nMt. (12) So long as 0<α<1 and n>0, this is positive. It is the budget deficit. Therefore, we have shown the following result. Proposition 1 If consumers’ utility depends on holding of money, a budget deficit is necessary for economic growth under full employment and constant prices. Suppose that Mt is given. If, prior to that time, full employment was achieved, the budget deficit shown in (12) is necessary and sufficient for continuous full employment without inflation. The policy of increasing money through budget deficits in line with the rate of economic growth as indicated by (12) is consistent with the Monetarist (Friedman’s) k% rule (Halton, 2023). Under inflation at a constant rate of π In this case,  t= π bt. Therefore, (10), (11), (A-1) in Appendix 1 and (12) are rewritten as 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�, 𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡+𝛼𝛼𝛼𝛼�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+ (𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡, 𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡+𝛼𝛼𝛼𝛼�(1 −𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+ (𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡=𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾𝑡𝑡𝑡𝑡, 𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡−𝜏𝜏𝜏𝜏𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝛼𝛼𝛼𝛼�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡+𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓, 𝐺𝐺𝐺𝐺𝑡𝑡𝑡𝑡−𝜏𝜏𝜏𝜏𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝛼𝛼𝛼𝛼�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�+(𝑛𝑛𝑛𝑛−𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡+𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓, and Gtτ wt Lt f=nMt+ π Bt. This is greater than the value in (12). It means the following results. Proposition 2 1. A budget deficit larger than its value under full employment and constant prices in (12) leads to inflation. Or, 2. To achieve full employment under inflation, a budget deficit larger than that under constant prices in (12) is required. 4. Explicit values of the savings and money holding in the steady state Without inflation From (9) in the steady state, the value of the interest rate is rt=n+ δ . Denote this value with r . Then, the capital-labour ratio, kt, which is constant in the steady state, satisfies f‘(kt)=r . (13) Denote this value of kt with k. Then, the steady-state value of the capital is 𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡=𝑘𝑘𝑘𝑘 �𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓, CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 312 also, we have 𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡=𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑝 . (14) (14) The steady-state value of the nominal wage rate is 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝�𝑓𝑓𝑓𝑓(𝑘𝑘𝑘𝑘 �)−𝑓𝑓𝑓𝑓′(𝑘𝑘𝑘𝑘 �)𝑘𝑘𝑘𝑘 ��. The steady-state value of the money holding is 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡� . It is rewritten as 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟 𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡. (15) (15) From (14), 𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=𝑝𝑝𝑝𝑝𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 (16) (16) By (15) and (16), we obtain 𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟 �−𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 � . (17) (17) This is the explicit solution for the value of the savings. By similar calculations, we get 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟 �−𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟 �−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 (18) (18) = (1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟 =(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟. =(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟. This is the explicit solution for the value of the money holding. Under inflation at a constant rate of π Under inflation at a constant rate of π , 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟𝑟 �(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟)𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡−𝜋𝜋𝜋𝜋𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡�. Therefore, we get 𝐵𝐵𝐵𝐵𝑡𝑡𝑡𝑡= (1−𝛼𝛼𝛼𝛼)1 𝑟𝑟𝑟𝑟𝑟 (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 1−(1−𝛼𝛼𝛼𝛼)𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟 (19) and 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟) (20)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟). 𝑀𝑀𝑀𝑀𝑡𝑡𝑡𝑡=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟− 𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟 −(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟) (20)=(1−𝛼𝛼𝛼𝛼)(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓+(𝑟𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟)𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 �𝑡𝑡𝑡𝑡 𝛼𝛼𝛼𝛼𝑟𝑟𝑟𝑟𝑟 +(1−𝛼𝛼𝛼𝛼)(𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟). 5. Government bond holding in the steady state In this section, we consider the case where financial assets are held not in money but in interest-producing government bonds that have almost the same liquidity as money. Budget constraint and utility maximisation Let i<r be the interest rate of the government bonds. It is usually smaller than the rate of return on capital for the risk premium. The budget constraint for the consumer is1 b t=(1τ ) wt lt-pt ct-(rt-i) mt+(rt-n) bt. imt is the interest income from the government bond holding. Except for that, it is the same as (3). The consumption and the government bond holding are 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡=𝛼𝛼𝛼𝛼 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏 󰇗𝑡𝑡𝑡𝑡�, and 𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡=�1−𝛼𝛼𝛼𝛼 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑖𝑖𝑖𝑖��(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡+(𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏 󰇗𝑡𝑡𝑡𝑡�. (21) (21) 1 We do not consider taxation of interest on government bonds, but it could be included. An interest tax would reduce the likelihood of a divergence of the debt-to-GDP ratio. CEEJ • 11(58) • 2024 • pp. 305-319 • ISSN 2543-6821 • DOI: 10.2478/ceej-2024-0020 319 Since, from (26) 𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=(𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡)𝜋𝜋𝜋𝜋 , we obtain 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=�𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋. This means 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋. Therefore, 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=(1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋. Again by (21), (1−𝜏𝜏𝜏𝜏)𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡. From this, 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−�𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋 =��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡+𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋. By 𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=(𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡)𝜋𝜋𝜋𝜋, 𝑏𝑏𝑏𝑏󰇗𝑡𝑡𝑡𝑡−𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋=𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋. Thus, 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡= ��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡+𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋, Therefore, 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋= ��𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡−(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡� 𝜋𝜋𝜋𝜋+(𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛)𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−�𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡=�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�(𝑚𝑚𝑚𝑚󰇗𝑡𝑡𝑡𝑡−𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝜋𝜋𝜋𝜋) =�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡. Then, by (28) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡��=�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟−𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼�𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�− 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�𝜋𝜋𝜋𝜋 =�𝑟𝑟𝑟𝑟 −𝑛𝑛𝑛𝑛− 𝑟𝑟𝑟𝑟 −𝑖𝑖𝑖𝑖 1−𝛼𝛼𝛼𝛼−𝜋𝜋𝜋𝜋� 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�. Since r - n= δ , 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡��=�𝑖𝑖𝑖𝑖−𝑛𝑛𝑛𝑛−𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼−(1 −𝛼𝛼𝛼𝛼)𝜋𝜋𝜋𝜋 1−𝛼𝛼𝛼𝛼 �𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕�𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑡𝑡�.