Discrete Dynamics in Nature and Society

Volume 2008, Article ID 435787, 14 pages

http://dx.doi.org/10.1155/2008/435787

## A Discrete Monetary Economic Growth Model with the MIU Approach

College of Asia Pacific Management, Ritsumeikan Asia Pacific University, Jumonjibaru, Beppu-Shi, Oita-ken 874-8577, Japan

Received 6 May 2007; Revised 8 January 2008; Accepted 26 February 2008

Academic Editor: Huang Weihong

Copyright © 2008 Wei-Bin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes an alternative approach to economic growth with money. The production side is the same as the Solow model, the Ramsey model, and the Tobin model. But we deal with behavior of consumers differently from the traditional approaches. The model is influenced by the money-in-the-utility (MIU) approach in monetary economics. It provides a mechanism of endogenous saving which the Solow model lacks and avoids the assumption of adding up utility over a period of time upon which the Ramsey approach is based.

#### 1. Introduction

Modern analysis of the long-term interaction of inflation and capital formation begins with Tobin’s seminal contribution [1]. Tobin deals with an isolated economy in which “outside money” (the part of money stock which is issued by the government) competes with real capital in the portfolios of agents within the framework of the Solow growth model. Since then, many models of growth model of monetary economies are built within the OLG framework (see, [2–5]). This paper introduces money into the growth theory proposed by Zhang in the early 1990s (see, [6]).

In nonmonetary growth theory, monetary values, such as wage, rate of interests, prices of goods and services, and land rent, are “fast variables” and are determined by balance conditions of demand and supply of real variables. In frictionless economic systems, issuing money has no effect on economic growth, at least in the long term. Nevertheless, financial assets and paper claims often offer alternatives to hold wealth. In process of exchange and division of labor, money plays an essential role in modern economy [7–11]. In his well-known paper on long-run effects of inflationary policies, Tobin [1] showed that an increase in the level of the inflation rate will increase the capital stock of an economy. Sidrauski [12] constructed an economic model in which no real variable will be affected by the economy’s inflation rate. We will address the issues by Tobin and Sidrauski in the alternative framework. Our approach is strongly influenced by the money-in-the-utility (MIU) approach which was initially proposed by Patinkin [13] and Sidrauski [12]. In this approach, money is held because it yields some services and the way to model it is to enter real balances directly into the utility function. Sidrauski [12] made a benchmark contribution to monetary economics, challenging Tobin’s nonneutrality result. He proposed a framework that explicitly allows for an endogenous treatment of saving behavior. His analytical framework is developed with Patinkin’s idea of ensuring a well-defined demand function for money by assuming that the agent’s utility is directly affected by money. This approach has been widely applied in monetary growth theory (e.g., [14–17]). Rather than following the Ramsey approach, this paper introduces money-in-the-utility function proposed by Zhang to show interactions between money and economic growth. The paper is organized as follows. Section 2 defines the model. Section 3 proves that the dynamic system has a unique unstable equilibrium point and simulates the model. Section 4 examines effects of changes in some parameters on the equilibrium. Section 5 concludes the study. The appendix generalizes the model by treating time distribution between leisure and work as endogenous variables.

#### 2. The Model

We present the model in discrete time, numbered from zero, and indexed by Time being referred to the beginning of period represents the initial situation from which economy starts to grow. The end of period coincides with the beginning of period it can also be called time We assume that transactions are made in each period. The model assumes that each individual lives forever. The production sector in our model is the same as that in the Solow one-sector growth model [18, 19]. The discrete version of the Solow-model is referred to by Diamond [20] and Azariadis [2]. It is assumed that there is only one (durable) good in the economy under consideration. Households own assets of the economy and distribute their incomes to consume and save. Exchanges take place in perfectly competitive markets. Production sectors sell their product to households or to other sectors; and households sell their labor and assets to production sectors. Factor markets work well; the available factors are fully utilized at every moment. Saving is undertaken only by households, which implies that all earnings of firms are distributed in the form of payments to factors of production, labor, managerial skill, and capital ownership.

Let denote the capital existing in period and the flow of labor services used at time for production. In this study, we assume to be fixed. As our model exhibits constant returns to scale, the dynamics will not be affected if we allow the population to change at a constant growth rate over time. We use the conventional production function to describe a relationship between inputs and output. The function defines the flow of production at time The production process is described by some sufficiently smooth function, We assume that is neoclassical. Introduce where The function has the following properties: (i) ; (ii) is increasing, strictly concave on , and is on ; and ; and (iii) and . Let denote the fixed rate of capital depreciation. Markets are competitive, thus labor and capital earn their marginal products, and firms earn zero profits. The real rate of interest, and real wage rate, are determined by markets. Hence, for any individual firm, and are given at each point of time. The production sector chooses the two variables and to maximize its profit. The marginal conditions are given byWe assume that agents have perfect foresight with respect to all future events and capital markets operate frictionless. The government levies no taxes. Money is introduced by assuming that a central bank distributes at no cost to the population a per capita amount of fiat money The scheme according to which the money stock evolves over time is deterministic and known to all agents. With being the constant net growth rate of the money stock, evolves over time according to the following:At the beginning of period the government brings additional units of money per capita into circulation in order to finance all government expenditures via seigniorage. For the seigniorage mechanism to work, injections of the additional units of money take place before the other markets open. Let stand for the real value of money per capita measured in units of the output good, that is, Then, we may rewrite the above equation asThe representative household receives units of paper money from the government through a “helicopter drop,” also considered to be independent of his money holdings. The inflation rate, is given by From and we haveAccording to the definition of per capita physical wealth is equal to Per capita real current income from the interest payment, and the wage payment, is given by

We call the current income in the sense that it comes from consumers’ daily toils (payment for human capital) and consumers’ current earnings from ownership of wealth. As in [6], the disposable income in real terms, is

where We assume that in each point of time the consumer’s utility function for holding money, consuming goods, and saving is represented by the following utility function:

where is called propensity to hold money, propensity to consume, and propensity to own wealth. Here the specified functional form already implies the assumption that real balances and consumption are Edgeworth complements (). If the assumption is replaced with the assumption of Edgeworth substitutability (), then the dynamic properties may be affected. Benhabib et al. [21] show how these two assumptions lead to different dynamic properties of monetary economies in the Ramsey approach.

The real price of holding money is The budget constraint is given byInsert (2.7) into the budget constraintwhere we use Consumers’ problem is to choose money, consumption, and savings in such a way that utility levels are maximized. Maximizing subject to the budget constraint (2.10) yields

where

The expenditure spent on “consuming money,” is proportional to the potential disposable income, and the (relative) propensity to use money. We see that is negatively related to This relation is assumed in the Tobin model and the Levhari and Patinkin’s monetary model [22]. In our approach, this relation results from optimal behavior of households.

According to the definitions of , and as the consumer determines his/her savings in period by maximizing the utility level for that period, the real wealth changes as follows:

We have thus built the model.

#### 3. The Motion, Equilibrium, and Stability

This section examines dynamic properties of the system. We now find dynamics of capital and real money. From the definition of and we have

Inserting (2.5) in the above equation, we solve

Inserting (2.1) in the above equation yields

where

where Hence, we can express as a unique function of and

From (2.13) and (2.3), we have

where we use From the definition of and (2.1), we obtain

Substituting and (3.6) into (3.5) yields

where we also use (3.3). In summary, we have the following lemma.

Lemma 3.1. * The motion of and can be determined by (3.7) and (3.3). For any
positive solution, and of difference equations (3.7) and (3.3), all the
other variables are uniquely determined by the following procedure: and by (2.1) by (3.6) and by (2.11) by (2.5) *

We now examine properties of the dynamic system. From (3.7) and (3.3), an equilibrium point is determined by

where we neglect another possible solution of It is known that when the model is identical to the nonmonetary one-sector model proposed by Zhang [6, Chapter 2]. In the case of the system has a unique stable equilibrium.

From the second equation, in (3.8) we solve as a function of as follows:

where For to be positive, it is necessary to require Denote by the value of such that As tends to be large for small and small for large we see that always exists. As falls in we see that for to be positive, we should require Taking derivatives of (3.9) with respect to yields

We see that the sign of the impact of change in the capital intensity is the same as that of Denote by the value of such that As for we conclude

Inserting (3.9) into the first equation in (3.8) yields

It is straightforward to check the following: and Hence, there is at least one positive solution for For we have

Hence, if or is small in the case of will always be negative. This implies that it is reasonable to consider that any meaningful solution is subject to As

we conclude that there is a unique solution of for The two eigenvalues at the equilibrium point, and are given by

where

We have

We see that the two eigenvalues are positive. From the definitions of and (3.9), we have

where we also use From at the equilibrium point and from the definition of we have From this inequality and (3.17), we have

This implies that at least one of the two eigenvalues is larger than unit. Hence, the system is unstable. In summary, we have the following theorem.

Theorem 3.2. *Let*
or be small in the case of The dynamical system has a unique unstable
equilibrium.

It should be noted that even if is large in the case of the conclusion of Theorem 3.2 may still hold. We now demonstrate Theorem 3.2 with simulation. We specify and We have

where We have the dynamics as follows:

where

We specify the parameters as follows:

It can be shown that (3.14) has a unique solution as shown in Figure 1.

The equilibrium values of the variables are given as follows:

The two eigenvalues, and are given as

Hence, the equilibrium point is a saddle point. We simulate the model with the initial point We simulate the model with periods. We plot the motion in Figure 2. It should be remarked that as the system is unstable, the system does not converge to the equilibrium point with the specified initial condition.

#### 4. Comparative Statics Analysis

This section studies effects of changes in some parameters on the equilibrium. It should be remarked that as the system is unstable, when as the parameters are changed, the system may not move from one steady state to another even when the system is initially located at a steady state. Different from the situation when the system has a unique stable steady state, the comparative statics analysis in the unstable case provides only some insights into the properties of the dynamic systems.

##### 4.1. The Inflation Policy

One of the important issues in monetary growth economics is effects of change of inflation rate, We now examine effects of change in on the economic equilibrium. It should be noted that as the system has a unique equilibrium point, the comparative static analysis examines the shift of the equilibrium point as parameters are changed. Taking derivatives of (3.11) with respect to yields

where as demonstrated in (3.13). We conclude that as the Tobin model predicts, as the inflation rate is increased, the per capita physical capital is increased. From (2.1), we obtain

The output and wage rate are increased and the rate of interest is reduced.

From (3.9), we have

We see that as the inflation rate is increased, the real money is increased. From the total wealth is increased. From (2.11) and at equilibrium, we have

From (4.4), we have

Asa rise in the inflation rate increases the consumption level if the absolute value of is relatively small.

##### 4.2. The Propensity to Use Money

Taking derivatives of (3.11) with respect to yields

Hence, as is increased, is increased. From (2.1), we obtain

The output and wage rate are increased and the rate of interest is reduced.

From (3.9), we have

As the first two terms in the right-hand side are negative and the last term is positive, the impact on the real money is ambiguous. From the impact on the total wealth is ambiguous. From (4.4), we have

##### 4.3. The Propensity to Save

Taking derivatives of (3.11) with respect to yields

As the propensity to save is increased, the per capita physical wealth is increased. From (2.1), we obtain

The output and wage rate are reduced and the rate of interest is increased.

From (3.9), we have

As the propensity to save is changed, the impact on the real money is ambiguous. From (4.4), we have

##### 5. Conclusions

We proposed a one-sector monetary growth model with the MIU approach. The model is much influenced by the Solow-model, the Ramsey model, the Tobin model, and the MIU approach in monetary economics. The main deviation from the traditional approaches is that we proposed an alternative approach to behavior of consumers. It provides a mechanism of endogenous capital and money. The dynamics is two-dimensional. In comparison with the Ramsey approach which would lead to four-dimensional dynamics for a similar problem, the dimension in our approach is reduced. It should be mentioned that the utility function used in this study has been applied to different fields of economics by Zhang (e.g., [6]).

#### Appendix

#### A Monetary Growth Model with Endogenous Labor Supply

Zhang [23] proposed a nonmonetary growth model with endogenous labor supply. The appendix shows that it is straightforward to extend the monetary growth model with fixed time developed in this paper to analyze endogenous labor supply.

Almost all the variables and assumptions are the same as before. Let be the flow of labor services used at time for production. The total labor force is given by where is the work time of a representative household and is the population. Introduce where Equations (2.1)–(2.5) still hold. Let () stand for per capita wealth. According to the definition of and we have Per capita real current income from the interest payment, and the wage payment, is given by

The disposable income in real terms, is

where let denote the leisure time at time We assume that in each point of time the consumer’s utility function for holding money, consuming leisure, consuming goods, and saving is be represented by the following utility function:

where is called propensity to hold money, the propensity to use leisure, propensity to consume, and propensity to own wealth. The real price of holding money is The budget constraint is given by

Denote the (fixed) available time for work and leisure by The time constraint is expressed by

Inserting the time constraint and (2.7) into the budget constraint yields

where we use Consumers’ problem is to choose money, consumption, and savings in such a way that utility levels are maximized. Maximizing subject to the budget constraint (A.6) yields

where

With in this case, we still have (2.13). We thus built the model.

We now find dynamics of capital and real money. From the definition of and we have

Inserting (2.5) in the above equation, we solve

where we use From the definition of and (2.1), we obtain

where Substituting (A.12) into yields

where we use (2.1) and From and (A.13), we solve

We see that the wealth per household can be uniquely expressed as a function of capital intensity in any period of time. It is straightforward to see that the time distribution, the real wage rate, and the real rate of interest are also expressed as functions of Inserting (A.14) and (2.1) into (A.10) yields

Hence, we can express as a unique function of and

Substituting and (A.15) into (A.11) yields

where we also use (A.14) and

Inserting (A.14) into (A.13) yields

Substituting this equation into yields

Inserting this equation in (A.16) yields

The monetary dynamic economy is described by two equations, (A.15) and (A.16). Although it is not difficult to examine properties of the system, we will not further examine them because the tedious expressions make it difficult to interpret the results. It should be noted that the case of is examined by Zhang [23]. It is known that the system has a unique stable equilibrium.

#### Acknowledgment

The author is grateful to Professor Weihong Huang and two anonymous referees for their important comments.

#### References

- J. Tobin, “Money and economic growth,”
*Econometrica*, vol. 33, no. 4, pp. 671–684, 1965. View at Publisher · View at Google Scholar - C. Azariadis,
*Intertemporal Macroeconomics*, Blackwell, Oxford, UK, 1993. View at Zentralblatt MATH - O. Galor and H. E. Ryder, “Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital,”
*Journal of Economic Theory*, vol. 49, no. 2, pp. 360–375, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Tirole, “Asset bubbles and overlapping generations,”
*Econometrica*, vol. 53, no. 5, pp. 1497–1528, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. von Thadden,
*Money, Inflation, and Capital Formation*, Springer, Berlin, Germany, 1999. View at Zentralblatt MATH - W.-B. Zhang,
*Economic Growth Theory*, Ashgate, Hampshire, UK, 2005. View at Zentralblatt MATH - F. H. Hahn, “On money and growth,”
*Journal of Money, Credit and Banking*, vol. 1, no. 2, pp. 172–187, 1969. View at Publisher · View at Google Scholar - D. Gale,
*Money in Disequilibrium*, Cambridge University Press, Cambridge, UK, 1983. View at Zentralblatt MATH - J. M. Grandmont,
*Money and Value*, Cambridge University Press, Cambridge, UK, 1983. View at Zentralblatt MATH - A. Orphanides and R. M. Solow, “Money, inflation and growth,” in
*Handbook of Monetary Economics*, B. M. Friedman and F. H. Hahn, Eds., vol. I, II, North-Holland, Amsterdam, The Netherlands, 1990. View at Google Scholar · View at Zentralblatt MATH - G. W. Stadler, “Business cycle models with endogenous technology,”
*The American Economic Review*, vol. 80, no. 4, pp. 763–776, 1990. View at Google Scholar · View at Zentralblatt MATH - M. Sidrauski, “Rational choice and patterns of growth in a monetary economy,”
*The American Economic Review*, vol. 57, no. 2, pp. 534–544, 1967. View at Google Scholar · View at Zentralblatt MATH - D. Patinkin,
*Money, Interest, and Prices: An Integrating of Monetary and Value Theory*, Harper and Row, New York, NY, USA, 2nd edition, 1965. View at Zentralblatt MATH - W. A. Brock, “Money and growth: the case of long run perfect foresight,”
*International Economic Review*, vol. 15, no. 3, pp. 750–777, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. J. Turnovsky,
*Methods of Macroeconomic Dynamics*, MIT Press, Cambridge, Mass, USA, 2000. View at Zentralblatt MATH - P. Wang and C. K. Yip, “Alternative approaches to money and growth,”
*Journal of Money, Credit, and Banking*, vol. 24, no. 4, pp. 553–562, 1992. View at Publisher · View at Google Scholar - W.-M. Ho, J. L. Zeng, and J. Zhang, “Inflation taxation and welfare with externalities and leisure,”
*Journal of Money, Credit and Banking*, vol. 39, no. 1, pp. 105–131, 2007. View at Google Scholar · View at Zentralblatt MATH - R. M. Solow, “A contribution to the theory of economic growth,”
*The Quarterly Journal of Economics*, vol. 70, no. 1, pp. 65–94, 1956. View at Google Scholar · View at Zentralblatt MATH - E. Burmeister and A. R. Dobell,
*Mathematical Theories of Economic Growth*, Collier Macmillan, London, UK, 1970. View at Zentralblatt MATH - P. A. Diamond, “National debt in a neoclassical growth model,”
*The American Economic Review*, vol. 55, no. 5, pp. 1125–1150, 1965. View at Google Scholar · View at Zentralblatt MATH - J. Benhabib, S. Schmitt-Grohé, and M. Uribe, “Monetary policy and multiple equilibria,”
*The American Economic Review*, vol. 91, no. 1, pp. 167–186, 2001. View at Google Scholar · View at Zentralblatt MATH - D. Levhari and D. Patinkin, “The role of money in a simple growth model,”
*The American Economic Review*, vol. 58, pp. 713–753, 1968. View at Google Scholar · View at Zentralblatt MATH - W.-B. Zhang, “A discrete economic growth model with endogenous labor,”
*Discrete Dynamics in Nature and Society*, vol. 2005, no. 2, pp. 101–109, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet