Multiple equilibria and indeterminacy in money-in-the-production-function model

Transcription

1 Multiple equilibria and indeterminacy in money-in-the-production-function model Akihiko Kaneko a,, Daisuke Matsuzaki b a School of Political Science and Economics, Waseda University, Nishiwaseda, Shinjuku-ku, Tokyo , Japan. b Department of Economics, Okinawa International University, 2-6-1, Ginowan, Ginowan-City, Okinawa , Japan Abstract In this study, we assume the situation that government resort to tax on asset and monetary expansion. Government expenditure finance by tax on asset and monetary expansion may generate multiple equilibria and indeterminacy in a money-in-the-production-function model. We show that both the growth rate and the welfare are always higher at the determinate balanced growth path equilibrium than the ones at the indeterminate balanced growth equilibrium. Keywords: money-in-production function, multiple equilibria, indeterminacy JEL classification: E60, O42 We thank seminar participants for their helpful comments and suggestions. Corresponding author. phone & fax:+81 (0) addresses: akaneko@waseda.jp (Akihiko Kaneko), daisuke@okiu.ac.jp (Daisuke Matsuzaki) 1

2 1. Introduction Since the pioneering works on the endogenous growth theory, such as Romer (1986) and Lucas (1988), the effect of taxation on economic growth has been one of the most carefully examined topics (for example Rebelo (1991) and Jones et al. (1997) ). Recently, as a related topic, the effect of different government expenditure financing on economic growth or welfare has been examined in many literatures, such as Grinols and Turnovsky (1993), Turnovsky (1993), Palivos and Yip (1995), Pecorino (1997), and Gokan (2002). These papers considers the mixture of several tax financing, bond financing and money financing to meet a given government expenditure. Our paper considers the effect of government policy when the government can resort to an asset tax and money financing. Another feature of our paper is that we use the money-in-the-productionfunction (MIP) model. Grinols and Turnovsky (1993), Turnovsky (1993), Palivos and Yip (1995), and Gokan (2002) adopted the money-in-the-utility-function (MIU) model. Palivos and Yip (1995) used the cash-in-advance (CIA) model and Pecorino (1997) developed his model based on the transaction cost model. But, up till now, to the best of our knowledge, no study has been dealt with the effect of government finance by tax on asset and monetary expansion in the MIP model. There are some studies which considers the MIP model theoretically and empirically. The MIP approach, originally developed by Levhari and Patinkin 2

3 (1968) and Fischer (1974), assumes that economic agents, especially firms, have to divert part of the employed factor of production to costly activities. Even in recent studies, such as Benhabib et al. (2001), Meng and Yip (2004), and Suen and Yip (2005), the implications of monetary policy or the dynamic property of a balanced growth path (BGP) equilibrium under the situation where money is introduced as a factor of production is examined. 1 In the literature on instability-indeterminacy property of BGPs equilibrium, Schmitt-Grohé and Uribe (1997) investigated the relation between an indeterminate BGP and a balanced-budget fiscal policy rule where exogenous government spending is financed by taxing labor. Guo and Harrison (2004) and Giannitsarou (2007) examined the possibility of arising indeterminacy BGP under the balanced-budget rule with using various endogenous policy variables. However, their studies did not consider a monetary growth model. In this paper, we introduce money with using the MIP model. We show that there is an indeterminate BGP equilibrium under balanced-budget policy where asset subsidy is financed by monetary expansion policy. The main result of our paper is as follows: Suppose that the government can resort to an asset tax and money financing. Multiple equilibria and global indeterminacy occur when the government subsidizes (imposes a negative tax) 1 For empirical evidences that money is an important factor in the production function, see Sinai and Stokes (1972), Hasan and Mahmud (1993), and Saygili (2009). 3

4 on asset. The reason for our result is as follows: In the MIP model, money is one of factors of production unlike other kind of monetary model, such as the MIU model and the CIA model. The asset tax is levied on the level of money holdings and capital stock. The consolidated government issues new money at a constant rate. Thus, in the MIP model with asset tax and money financing, both factors of production are a kind of tax base. This is the main source of multiple equilibria (or self-fulfilling equilibria). The remainder of this paper proceeds as follows. Section 2 develops a MIP endogenous growth model. Section 3 discusses the economy s BGP equilibrium, and then examines the possibility of multiple equilibria and indeterminacy. In Section 4, we compare the growth rate and welfare of both equilibria. Finally, concluding remarks are provided in Section Model 2.1. Firms Prior to conducting any analysis, we first normalize the number of households and firms as equal to 1. We then specify the technology as having the following Cobb-Douglas form: y(t) = Λk(t) α m(t) 1 α, 0 < α < 1. (1) where y(t) is output, k(t) is physical capital, and m(t) is real money at time t. This production function reflects the role of money that money is useful in reducing costs from goods market transactions, such as bargaining, setting prices, 4

5 and conducting wholesale operations. A Cobb-Douglas specification of MIP was specified in Pecorino (1995) as well. Shaw et al. (2005) illustrated how we can derive the Cobb-Douglas form of a MIP from more general specification. A firm s profit is defined as the after-tax revenue minus the cost of hiring physical capital and borrowing money from households: y(t) r(t)k(t) R(t)m(t), where r(t) is the rental rate of physical capital, R(t) is the rental rate of money or the cost of borrowing money from households. The profit maximization of the representative firm are written as follows: αλ(m(t)/k(t)) 1 α r(t) = 0, (2) (1 α)λ(m(t)/k(t)) α R(t) = 0. (3) The firm equates the after tax marginal product of each input to its rental rate, r(t) or R(t) Households A representative household gains utility from its consumption. This representative household can earn revenue in two ways: from its savings and through lending money to firms. The household s nominal budget constraint is as follows: p(t) k(t)+ṁ(t) = p(t)r(t)k(t)+p(t)r(t)m(t) p(t)c(t) p(t)τa(t), where τ is an asset tax rate, M(t) is nominal money, p(t) is the commodity price, and a(t) is the total real asset, that is, the sum of k(t) and M(t)/p(t)( m(t)). 5

6 A dot ( ) means a time derivative. Dividing both sides by p(t), we obtain the household s real budget constraint as follows: ȧ(t) = (r(t) τ)a(t)+(r(t) π(t) r(t))m(t) c(t), (4) where π(t) is the inflation rate of the commodity price p(t). The lifetime utility of the household is as follows: 0 c(t) exp( ρt)dt, (5) where ρ is the rate of time preference and 1/ is the elasticity of substitution for the utility function. Throughout this paper, is assumed to be greater than 1. 2 The household maximizes its lifetime utility (5) subject to its budget constraint (4). The Hamiltonian function associated with this problem is J(t) = c(t) λ(t)((r(t) τ)a(t)+(r(t) π(t) r(t))m(t) c(t)), where λ(t) is the costate variable. Then, the necessary conditions for optimality are J(t) c(t) = 0;c(t) λ(t) = 0, (6) λ(t) = ρλ(t) J(t) a(t) ; λ(t) = (ρ r(t)+τ)λ(t), (7) J(t) = 0;λ(t)(R(t) π(t) r(t)) = 0. (8) m(t) From the above necessary conditions, we develop the following Euler equation 2 Ogaki and Reinhart (1998) reported that was larger than 1 in the US. 6

7 and no-arbitrage condition: c(t)/c(t) = (r(t) τ ρ)/, (9) r(t) = R(t) π(t). (10) Though the firms pay R(t) to the household for hiring one unit of real money, the value of real money decreases at the rate of inflation. Therefore, the return on real money to the household is R(t) π(t). The above equation indicates that the return on k(t) must be equal to that on m(t) under the equilibrium in which both physical capital and real money are held by the household. The transversality condition must be also satisfied as the following necessary condition: lim λ(t)a(t)exp( ρt) = 0. (11) t In Appendix A, we illustrate that the transversality condition is satisfied at the BGP when is greater than Government The government imposes an asset tax or issues new money, for nominal wasteful government consumption of p(t)g(t). The government s nominal budget constraint then becomes p(t)g(t) = τp(t)a(t)+µm(t), where µ is the growth rate of nominal money. In real terms G(t) = τa(t)+µm(t). (12) 7

8 2.4. Dynamics For ease of exposition, we suppress the time index in what follows. As the economy will exhibit long-run growth, we assume the the ratio of government expenditure to output is constant, g such that g = G(t)/y(t). Using the production function in (1), the government budget condition (12) becomes µ = Λω α g τ ω τ. (13) Given a constant monetary growth rate µ, the money market equilibrium requires that ṁ/m = µ π. Using Equations (2), (3), and (10) to eliminate π, we have ṁ/m = µ+λ { α(m/k) 1 α (1 α)(m/k) α}. (14) From Equations (2), (3), (4), (12), (14), and the total asset constraint, the market equilibrium condition for a commodity is given by k/k = y/k (G/y)(y/k) c/k,= (1 g)λ(m/k) 1 α c/k. (15) From Equations (2) and (9), the growth rate of consumption can be rewritten as follows: ċ/c = { Λα(m/k) 1 α τ ρ } /. (16) Defining c/k and m/k as χ (the consumption-to-capital ratio) and ω (the real money holdings-to-capital ratio), we have the following autonomous dynamic 8

9 system from Equations (13), (14), (15), and (16): [ 1 χ = χ ω = ω [ Λαω 1 α τ ρ ] ] (1 g)λω 1 α +χ, (17) [ Λω α g τ ω τ +Λ{ αω 1 α (1 α)ω α} (1 g)λω 1 α +χ 3. Financing methods and growth ]. (18) In this section, we compare the effect of each financing method on the BGP growth rate of the economy. At a BGP, χ/χ = ω/ω = 0. Imposing Equation (13) and χ/χ = ω/ω = 0 in Equation (17) and (18) results in ( 1 1 ) αλω 2 α +Λ(1 α g)ω 1 α = (1 )τ +ρ ω τ. (19) When we call the left-hand side (LHS) of Equation (19) F(ω) and the right-hand side (RHS) G(ω), we can depict the graphs of each function as in Figure 1 when τ < 0 (see Appendix B for detail). As can be seen from the figure, there must be multiple equilibria. The intuition behind our result is as follows: Since the government must finance a given level of expenditures, the asset tax rate or the monetary expansion rate endogenous. The government has to set its policy variable relatively high if the tax base is significantly reduced by a change in the policy variable, and vice versa. Since the tax base (real money balance) affects the interest rate and the growth rate in the MIP, it generates self-fulfilling equilibria. If the economy s agents expect a relatively worse economy with a high monetary expansion 9

10 rate, then an equilibrium emerges. In contrast, if they expect a relatively better economy with a low money expansion rate, the low monetary expansion rate is actually enough to finance the target level of the expenditure. Moreover, we find that the one is determinate and the other is indeterminate Property of the BGPs Here, we examine the dynamic property of the BGPs and derive the condition for the equilibrium to be determinate or indeterminate. Linearizing Equations (17) and (18), we obtain the following: where ( ) χ = ω ( )( χ ζ 12 χ χ ω ζ 22 ω ω ), (20) χ 1 ( Λαω 1 α τ ρ ) +(1 g)λω 1 α, (21) ζ 12 χ { α (1 g) } Λ(1 α)ω α, (22) ζ 22 ω [ τ ω 2 +Λα{(1 α) g}ω α 1 +Λ(1 α){α (1 g)}ω α]. (23) Using Equations (21), (22), and (23), the trace, T, and determinant, D, of the coefficient matrix are given by T = α ( 1 ) +(1 g)+(1 α) Λω 1 α + 1 (τ +ρ)+λα(1 α g)ω α + τ ω, D = αλω α χ [( 1 1 )(1 α)ω (1 α g) ] τ Λαω 1 α. Suppose that D > 0. In this case, the following condition holds (24) ( ) 1 1 (1 α)ω τ (1 α g) < 0. (25) Λαω 1 α 10

11 Using Equations (24) and (25), the following inequality holds: (T =)ω α Λα [{(1 1 }ω )+(1 α g) 1 + (τ +ρ)ω α > ω α Λα {(1 α g)ω + 1 (τ +ρ)ω α Λα Λα +(1 α g)+ Thus, if g < 1 α, then (25) holds and T is positive. 3 When T > 0 and D > 0, }. ] τ Λαω 1 α the coefficient matrix from Equation (20) has two positive eigenvalues. Since both ω and χ are jumpable, the economy immediately jumps to its BGP and is on a determinate equilibrium. If D < 0, there are two real eigenvalues, of which one is negative and the other is positive whichever sign T has; hence, the BGP is an indeterminate equilibrium. Now, see Figure 1. From (19), the slope of F(ω) is ( ) 1 1 (2 α)λαω 1 α +(1 α)(1 α g)λω α. The slope of G(ω) is (1 )τ+ρ. At the BGP, the following equality holds by using (19). (1 )τ +ρ = ( ) 1 1 αλω 1 α +Λ(1 α g)ω α + τ ω. Thus, the slope of F(ω) minus the slope of G(ω) at the BGP is {( ) } 1 Λαω α 1 (1 α)ω (1 α g) τω α 1. (26) Λα When (26) is negative, (25) is positive. Thus the cross point at ω d (ω i ) in Figure 1 is a determinate (indeterminate) equilibrium. 3 IfthesignofT changesfrompositivetonegativewiththesignofthedeterminantunchanged, the signs of two eigenvalues become negative, implying that bifurcation arises. Imposing the condition g < 1 α excludes the possibility of bifurcation for a simple analysis. 11

12 3.2. Phase diagram and global indeterminacy Here, we depict the phase diagram of the economy in Figure 2 and show that the economy exhibits global indeterminacy. From (17) and (18), the χ = 0 locus and the ω = 0 locus are as follows: χ = 0;χ = ( 1 g α ) Λω 1 α + ρ+τ, ω = 0;χ = (1 α g)λω 1 α + τ ω +τ +(1 α g)λω α. In the χ - ω plane, it is easy to see that χ = 0 locus is upward-sloping and concave. The analysis of the shape of the ω = 0 locus is rather complex. We call the RHS of ω = 0 locus h. When ω tends to zero, the value of h goes to negative infinity when τ < 0 and 1 α g > 0. The derivative value of h with respect to ω is (1 α g)λω 1 α [(1 α)ω α] τ/ω 2. (27) When ω is small, the value of Equation (27) is positive. It means that the graph of h is upward sloping when ω is small. When (1 α g)λω 1 α [(1 α)ω α] = τ, the derivative value of h is equal to 0. Taking the horizontal axis as ω, we can depict the (1 α g)λω 1 α [(1 α)ω α] curve and the τ(< 0) line as in Figure 3. 4 Consequently, the graph of ω = 0 is upward sloping when ω is small; after it reaches its peak, it becomes downward sloping. The analysis in previous 4 The value of Λω 1 α [(1 α)ω α] goes to zero (infinity) as ω tends to zero (infinity). The derivative value of it is Λ(1 α)ω α [(2 α)ω α]. Thus, it takes on the minimum value when ω is equal to α/(2 α). 12

13 subsection implies that there are always two intersections between the χ = 0 locus and the ω = 0 locus. The equilibrium with the low ω is indeterminate and the other equilibrium is determinate. The economy exhibits global indeterminacy, as it has two BGPs and both dynamic variables are jumpable. 4. Growth rate and welfare comparison Since both dynamic variables, χ and ω, are jumpable, both equilibrium are attainable. In this section, we compare the growth rate and welfare at each BGP equilibrium. The growth rate at BGP is represented by ċ c = 1 ( Λαω 1 α τ ρ ). (28) The multiple equilibria emerge for the same value of τ. Thus, the larger ω is, the larger the growth rate is. That is, the growth rate at the determinate BGP equilibrium is larger than the one at the indeterminate BGP equilibrium. Comparison of welfare is more complicated. Defining the growth rate at a BGP as θ, the level of consumption at time t is more formally represented by c(t) = c(0)e θ t, (29) where c(0) is the level of consumption at time zero. Substituting Equation (29) into Equation (5), the lifetime utility becomes c(0) 1 U = (1 )((1 )θ ρ) 1 (1 )ρ. (30) 13

14 The lifetime utility is governed by the level of initial consumption and the subsequent growth rate. From (17) c(0) = { 1 [ Λαω 1 α τ ρ) ] } +(1 g)λω 1 α k(0), (31) where k(0) is the level of physical capital at time zero. Substituting (28) and (31) into (30), we get ] 1 U = k(0)1 [ Λω 1 α( α +(1 g)) + τ+ρ 1 ( ) (1 ) (1 ) Λαω 1 α τ ρ ρ (1 )ρ Differentiating it with respect to ω, we have U k(0)1 ω = 1 1 Ω [ (1 )X Λ(1 α)ω α( α +(1 g) ) ( (1 ) Λαω1 α τ ρ X 1 (1 ) 1 (1 α)λαω α ] ) ρ [ ( = k(0)1 X (1 α)λω α α ) ( ) Ω +(1 g) (1 ) Λαω1 α τ ρ ρ X 1 ] α [ ( = k(0)1 X (1 α)λω α α ) ( Ω +(1 g) (1 ) Λαω1 α τ ρ (Λω 1 α( α ) +(1 g) + τ +ρ ) 1 ] α = k(0)1 X (1 α)λω α Ω [( α ) ( ( )) +(1 g) αλω 1 α τ ρ + (1 ) ρ where Ω is τ +ρ 1 ] α, ( ) 2 (1 ) Λαω1 α τ ρ ρ and X = Λω 1 α ( α +(1 g)) + τ+ρ. α +(1 g) > 0bytheconditionweassumedinfootnote(3)and( (1 ) τ ρ ρ ) can be assumed to be minus if is near to 1. Thus, the sign of above equation ) ρ 14

15 is positive and the welfare is larger at the determinate BGP than at the indeterminate BGP. 5. Concluding Remarks In this paper, we show that multiple equilibria and global indeterminacy occur when the government subsidizes (imposes a negative tax) on asset. We reveal that both the growth rate and the welfare are always higher at the determinate balanced growth path equilibrium than the ones at the indeterminate balanced growth equilibrium. Our results provide an insightful policy implication. Suppose that the government wants to enhance capital accumulation and subsidizes asset holdings. If the government finances it by newly money issuing, multiple equilibria occur and the economic condition might be worse after the subsidy is introduced. Two remarks are follow: First, we used a Cobb Douglas production function that helps us to find multiple equilibria and global indeterminacy, in the case where 1. However, it is a restrictive model. A more general form of the MIP may yield more general results. Second, though we show that there are two attainable equilibria and one of them are better than the other in terms of the growth rate and welfare, quantitative difference between them has not been examined yet. We can simulate our model to see how low the growth rate and the welfare are numerically when the economy stays at the bad equilibrium. These will be subjects of future research. 15

16 Appendices Appendix A. On the transversality condition From Equations (4) and (10) in the BGP, the household s budget constraint can be rewritten as follows: ȧ = (r τ)a c. (A-1) To satisfy the transversality condition, differentiating Equation (11) with respect to t must be negative. Using Equations (7) and (A-1), the condition gives λ λ + ȧ a ρ = c a. From Equations (9) and (A-1) and the no-ponzi game condition, we have c a ( 1)(r τ)+ρ =. Thus, when 1, the transversality condition is always satisfied. Appendix B. The figure of (19) The equilibrium value of ω must satisfy Equation (19). We call the left (right)- hand side of Equation (19) F(ω) (G(ω)). Differentiating F(ω) with respect to ω, we get ( ) 1 1 (2 α)λαω 1 α +(1 α)(1 α g)λω α. We imposed the condition 1 α g > 0 in footnote (3). This implies that the slope of F(ω) is positive for all small ω and is negative for all large ω. As G(ω) 16

17 is linear with a positive intercept if τ < 0, there are two equilibria as long as g is not so large, as in Figure 1. 17

18 G(ω) F(ω) τ ωi ω d Figure 1: ω at BGPs when τ < 0 ω χ χ = 0 ω = 0 ω i ω d ω Figure 2: the phase diagram when τ < 0 18

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