Research Article A Discrete Monetary Economic Growth Model with the MIU Approach

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.


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.

The model
We present the model in discrete time, numbered from zero, and indexed by t 0, 1, 2, . . . .Time 0, being referred to the beginning of period 0, represents the initial situation from which economy starts to grow.The end of period t − 1 coincides with the beginning of period t ; it can also be called time t.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 K t denote the capital existing in period t and N the flow of labor services used at time t for production.In this study, we assume N 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 F t defines the flow of production at time t.The production process is described by some sufficiently smooth function, F t F K t , N .We assume that F is neoclassical.Introduce f k t ≡ F k t , 1 , where k t ≡ K t /N.The function f has the following properties: i f 0 0; ii f is increasing, strictly concave on R , and C 2 is on R ; f k > 0 and f" k < 0; and iii lim k→0 f k ∞ and lim k→ ∞ f k 0. Let δ k 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, r t , and real wage rate, w t , are determined by markets.Hence, for any individual firm, r t and w t are given at each point of time.The production sector chooses the two variables K t and N t to maximize its profit.The marginal conditions are given by We 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 M t > 0. The scheme according to which the money stock evolves over time is deterministic and known to all agents.With μ being the constant net growth rate of the money stock, M t evolves over time according to the following: At the beginning of period t, the government brings M t − M t − 1 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 m t stand for the real value of money per capita measured in units of the output good, that is, m t M t /P t .Then, we may rewrite the above equation as The representative household receives μm/ 1 μ units of paper money from the government through a "helicopter drop," also considered to be independent of his money holdings.The inflation rate, π t , is given by π t P t 1 − P t P t .

2.4
From m t M t /P t and M t According to the definition of k t , per capita physical wealth is equal to k t .Per capita real current income from the interest payment, r t k t , and the wage payment, w t , is given by y t r t k t w t .

2.6
We call y t 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, y, is where a t ≡ k t m t .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 ε 0 is called propensity to hold money, ξ 0 propensity to consume, and λ 0 propensity to own wealth.Here the specified functional form already implies the assumption that real balances and consumption are Edgeworth complements u cm > 0 .If the assumption is replaced with the assumption of Edgeworth substitutability u cm < 0 , then the dynamic properties may be affected.where The expenditure spent on "consuming money," r π m, is proportional to the potential disposable income, y a t , and the relative propensity to use money.We see that m t is negatively related to r π.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 a t , s t , and τ t 1 , as the consumer determines his/her savings in period t by maximizing the utility level for that period, the real wealth changes as follows: a t 1 s t τ t 1 .

2.13
We have thus built the model.

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 y a t and r t π t m t εy a t , we have 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 m 0. It is known that when m 0, the model is identical to the nonmonetary one-sector model proposed by Zhang 6, Chapter 2 .In the case of m 0, the system has a unique stable equilibrium.
From the second equation, in 3.8 we solve m as a function of k as follows: where λ 1 ≡ 1/λ − δ > 0. For m to be positive, it is necessary to require f/k > λ 1 .Denote by k * the value of k such that f/k λ 1 .As f/k tends to be large for small k and small for large k, we see that k * always exists.As f/k falls in k, we see that for m to be positive, we should require 0 < k < k * .Taking derivatives of 3.9 with respect to k yields

3.10
We see that the sign of the impact of change in the capital intensity is the same as that of f − λ 1 .
Denote by k * * the value of k such that f λ 1 .As f/k > f for k > 0, we conclude 0 < k * * < k * .Inserting 3.9 into the first equation in 3.8 yields It is straightforward to check the following: H 0 < 0 and H k * * > 0. Hence, there is at least one positive solution for 0 < k < k * * .For k ≥ k * * , we have Hence, if μ ≥ δ k or μ − δ k is small in the case of μ < δ k , H k will always be negative.This implies that it is reasonable to consider that any meaningful solution is subject to 0 < k < k * * .As we conclude that there is a unique solution of H k 0 for 0 < k < k * * .The two eigenvalues at the equilibrium point, φ 1 and φ 2 , are given by where We have We see that the two eigenvalues are positive.From the definitions of Ω j and 3.9 , we have 3.17 where we also use m λ 1 μ f k − λ 1 k .From f − λ 1 > 0 at the equilibrium point and from the definition of λ 0 , we have f δ λ > 1.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 μ ≥ δ k or μ − δ k be small in the case of μ < δ k .The dynamical system has a unique unstable equilibrium.
It should be noted that even if μ − δ k is large in the case of μ < δ k , the conclusion of Theorem 3.2 may still hold.We now demonstrate Theorem 3.2 with simulation.We specify N 1 and F AK α N β .We have where f Ak α .We have the dynamics as follows: where

3.22
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: k 0.790, m 0.576, a 1.362, r 0.307, w 0.539, f 0.829, c 0.138, s 1.381.

3.24
Hence, the equilibrium point is a saddle point.We simulate the model with the initial point k 0 , m 0 0.7, 0.3 .We simulate the model with 9 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.

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.

The inflation policy
One of the important issues in monetary growth economics is effects of change of inflation rate, μ.We now examine effects of change in μ 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 μ yields Wei-Bin Zhang 9 where ∂H/∂k > 0 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 a k m, the total wealth is increased.From 2.11 and π μ at equilibrium, we have From 4.4 , we have a rise in the inflation rate increases the consumption level if the absolute value of f"/ r μ is relatively small.

The propensity to use money
Taking derivatives of 3.11 with respect to ε 0 yields Hence, as ε 0 is increased, k 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 a k m, the impact on the total wealth is ambiguous.From 4.4 , we have

The propensity to save
Taking derivatives of 3.11 with respect to λ 0 yields ∂H ∂k 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

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 .
Almost all the variables and assumptions are the same as before.Let N t be the flow of labor services used at time t for production.The total labor force N t is given by N t T t N 0 , where T t is the work time of a representative household and N 0 is the population.With a k m in this case, we still have 2.13 .We thus built the model.

Figure 1 : 2 Figure 2 :
Figure 1: The existence of a unique equilibrium point.
Benhabib et al.21show how these two assumptions lead to different dynamic properties of monetary economies in the Ramsey approach.The real price of holding money is 1 r t π t .The budget constraint is given by The motion of k t and m t can be determined by 3.7 and 3.3 .For any positive solution, k t and m t , of difference equations 3.7 and 3.3 , all the other variables are uniquely determined by the following procedure: a t k t m t →r t and w t by 2.1 →y a t by 3.6 →c t and s t by 2.11 →π t by 2.5 →f k t →F t Nf k t .
where δ ≡ 1 − δ k .Hence, we can express m t 1 as a unique function of k t and m t .wherewe use a t ≡ k t m t .From the definition of y a t and 2.1 , we obtain Introduce f k t ≡ F k t , 1 , where k t ≡ K t /N t .Equations 2.1 -2.5 still hold.Let k t ≡ K t /N 0 stand for per capita wealth.According to the definition of k t and k t , we have k t k t T t .Per capita real current income from the interest payment, r t k t , and the wage payment, w t T t , is given by .Let T h t denote the leisure time at time t.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:U t T σ 0 h t m ε 0 t c ξ 0 t s λ 0 t , σ 0 , ε 0 , ξ 0 , λ 0 > 0, A.3where ε 0 is called propensity to hold money, σ 0 the propensity to use leisure, ξ 0 propensity to consume, and λ 0 propensity to own wealth.The real price of holding money is 1 r t .The budget constraint is given by Denote the fixed available time for work and leisure by T 0 .The time constraint is expressed by Consumers' problem is to choose money, consumption, and savings in such a way that utility levels are maximized.Maximizing U t subject to the budget constraint A.6 yields