Social status, human capital formation and super-neutrality in a two-sector monetary economy

Economic Modelling 28 (2011) 785–794

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Social status, human capital formation and super-neutrality in a two-sector monetary economy☆ Hung- Ju Chen ⁎ Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei 100, Taiwan

a r t i c l e

i n f o

Article history: Accepted 25 October 2010 JEL classification: C62 E52 O42 Keywords: Cash-in-advance economy Endogenous growth Social status

a b s t r a c t In this paper, we study how social status affects the impact of monetary policy on the long-run growth rate in a two-sector monetary economy with human capital accumulation, and find that the super-neutrality of money, with regard to the growth rate of the economy depends on the formation of human capital. In an economy with Lucas-type human capital formation, money is super-neutral; however, for an economy in which both physical and human capital are used as inputs for human capital accumulation, the money growth rate will have a positive effect on the long-run economic growth rate. The existence, uniqueness and saddlepath stability of balanced-growth equilibrium are also examined. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Economists have demonstrated considerable interest over recent years in the effects of social status on economic performance within a dynamic general equilibrium framework. The concept of social status can be traced back both to the ‘spirit of capitalism’ of Weber (1958) and the ‘wealth effects’ of Kurz (1968), with social status since then having invariably been represented within economic models as the pursuit of wealth.1 The presence of wealth-enhanced social status motivates agents to pursue the accumulation of physical capital, which in turn affects other macroeconomic variables, such as consumption, savings and economic growth. Early examinations of the macroeconomic effects of social status within a standard optimal growth model can be found in the studies of

☆ The author would like to thank Juin-Jen Chang, Jang-Ting Guo, Ming-Chia Li and participants at the CEANA session of the 2009 AEA annual meetings and the Macroeconomics conference of the Program for Promoting Excellent Universities for helpful comments and suggestions. The financial support provided by the Program for Globalization Studies at the Institute for Advanced Studies in Humanities at the National Taiwan University is also gratefully acknowledged (grant number: 99R018). The usual disclaimer applies. ⁎ Tel.: +886 2 23519641x535; fax: +886 2 23582284. E-mail address: [email protected]. 1 The concept of relative wealth was adopted by Corneo and Jeanne (1997b) and Clemens (2004) to represent social status. Instead of using physical capital stock, Rauscher (1997) and Corneo and Jeanne (1997a) used conspicuous consumption to represent social status, while social status in both Fershtman and Weiss (1993) and Fershtman et al. (1996) was represented by a person's occupation. 0264-9993/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2010.10.010

Zou (1994) and Wirl (1994).2 Modern growth theory, which followed on from the seminal works of Romer (1986), Lucas (1988) and Rebelo (1991), modifies the traditional optimal growth model in such a way that the economic growth rate is endogenously determined.3 This spurred on subsequent studies on social status which analyze the ways in which social status affects economic performance in an endogenous growth model. The impact of social status in a one-sector AK model was analyzed by Zou (1994) and Corneo and Jeanne (1997b), while Chang et al. (2004) and Chang et al. (2008) went on to examine the effects of social status in a two-sector model with human capital formation. The more recent studies on social status have switched from a real economy to a monetary economy, characterized by the use of the cash-in-advance (CIA) model to examine the ways in which social status affects the super-neutrality of money,4 since the macroeconomic effects of the money growth rate have long been an important issue in macroeconomics. Based on a descriptive aggregate model, the pioneering paper of Tobin (1965) demonstrated that a higher money

2 Social status has also been studied in many other types of models; for example, Bakshi and Chen (1996) analyzed the impact of social status on stock-market prices, while Fisher and Hof (2005) examined the effects of social status for a small open economy. Cole et al. (1992) had earlier shown that the introduction of social status may cause multiple equilibria. 3 The transitional dynamics of Lucas (1988) were studied by Benhabib and Perli (1994) and Xie (1994). 4 In this paper, super-neutrality of money refers to the proposition that permanent, exogenous changes to the growth of the money supply do not affect the long-run economic growth rate.

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H.-J. Chen / Economic Modelling 28 (2011) 785–794

Table 1 Related literature. Optimal growth model

Optimal growth model with social status

CIA model

CIA model with social status

Panel A: One-sector model y = Akα

Ramsey (1928) King and Rebelo (1990)

Stockman (1981) Abel (1985) Suen and Yip (2005)

Gong and Zou (2001)

y = Ak

Zou (1994) Wirl (1994) Zou (1994) Corneo and Jeanne (1997b)

Chang et al. (2008)

Marquis and Reffett (1991)

This paper

Chang et al. (2004)

Mino (1997)

This paper

Panel B: Two-sector model Lucas-type human capital formation Generalized human capital formation

Benhabib and Perli (1994) Xie (1994) King et al. (1988) Rebelo (1991) Bond et al. (1996) Mino (1996)

growth rate can positively affect the accumulation of physical capital due to the reduction in the real interest rate, which has since become known as the Tobin effect.5 The CIA model with liquid constraints on consumption can be found in Clower (1967), Lucas (1980) and Stockman (1981). Stockman (1981) showed that the money growth rate does not affect the steady-state value of physical capital when consumption has liquid constraints.6 Based upon their development of a one-sector CIA model with an AK production function aimed at examining the superneutrality of money, Suen and Yip (2005) showed that indeterminacy may occur. A two-sector CIA model with Lucas-type human capital formation was also constructed by Marquis and Reffett (1991), within which only human capital was required for the accumulation of human capital. Following on from this, generalized human capital formation was subsequently introduced into a CIA model by Mino (1997), with both human and physical capital being required for human capital accumulation. Both studies demonstrated that if the cash-in-advance constraints were to apply only to consumption, then money is super-neutral. Social status was introduced into the CIA models by Gong and Zou (2001) and Chang et al. (2000) and Chang and Tsai (2003) in order to examine the macroeconomic effects of monetary policy; although Chang et al. (2000) displayed endogenous growth, Gong and Zou (2001) and Chang and Tsai (2003) did not.7 Thus, it is clear that the prior studies relating to the impact of social status in a monetary economy have tended to focus on the analysis of physical capital accumulation, thereby ignoring the role of human capital. In this study, we develop a two-sector CIA model with human capital formation to examine the ways in which social status affects the impact of the money growth rate on long-run economic performance. Three types of production functions and human capital formations are considered in CIA models with social status: the Lucastype human capital formation and the production function without or with human capital externality, and the generalized human capital formation.8 As shown in Table 1, this paper completes the study of social status in a CIA model by providing comprehensive analysis of both the short-run and long-run effects of monetary expansion under different endogenous growth monetary economies.9 5 However, the Tobin effect was subsequently challenged by Sidrauski (1967) who demonstrated that money growth does not affect the steady-state value of physical capital based on an infinite-horizon, representative agent model. 6 If the cash-in-advance constraint applies to both consumption and investment, an increase in the money growth rate will, however, lower the steady-state value of physical capital. A study of the transitional dynamics of the Stockman (1981) model was provided by Abel (1985). 7 See also Chen and Guo (2009, forthcoming) for studies of the impact of social status in a monetary economy with endogenous growth. 8 A two-sector optimal growth model with generalized human capital formation was studied by King et al. (1988), King and Rebelo (1990), Bond et al. (1996) and Mino (1996). 9 Since Table 1 is not intended to be an exhaustive literature review, many important contributions may not have been included.

Chang et al. (2000)

We find that for two-sector endogenous growth models, the super-neutrality of money with respect to the growth rate of the economy is dependent on the formation of human capital.10 When a permanent increase in the money growth rate occurs, it causes an increase in the inflation rate which reduces the real money balance and future consumption through the cash-in-advance constraint. Hence, the agent tends to use current consumption to substitute for future consumption and investment to the detriment of physical capital accumulation decreases. On the other hand, the real interest rate becomes lower due to an increase in the inflation rate and this encourages investment in physical capital (the Tobin effect). The presence of the desire for social status reinforces the second effect. In a two-sector model with Lucas-type human capital formation, the two effects cancel each other out so that an increase in the money growth rate does not affect the long-run economic growth rate. Although money is super-neutral in an economy with Lucas-type human capital formation, within an economy with generalized human capital formation, the money growth rate will positively affect the economic growth rate. This is because with generalized human capital formation, an increase in the investment in physical capital is beneficial to both physical and human capital accumulation. Hence, the presence of social-status seeking will strengthen the motivation of physical capital accumulation and an increase in the money growth rate will raise the economic growth rate. Therefore, money is not super-neutral when the desire for social status is present. We also show that a two-sector CIA model can be represented by a four-dimensional dynamic system. We then go on to examine the existence, uniqueness and saddle-path stability of the balancedgrowth equilibrium for each of the models in this study. Our analysis provides a simplified version of the Routh theorem applied to the study of transitional dynamic property for a four-dimensional dynamic system. The remainder of this paper is organized as follows. The basic model is developed in the next section. In Section 3, we examine the macroeconomic effects of monetary policy under Lucas-type human capital formation and generalized human capital formation. The final section presents the conclusions drawn from this study. 2. The model We begin our analysis by considering a two-sector CIA model with human capital formation, with the economy comprising of a representative, infinitely-lived agent. Following Kurz (1968), we assume that the representative agent cares about both consumption

10 Chang et al. (2000) showed that for a one-sector economy with an AK production function, an increase in the money growth rate will raise the long-run growth rate when consumption is liquidly constrained.

H.-J. Chen / Economic Modelling 28 (2011) 785–794

(ct) and social status, which is represented by the level of physical capital (kt). The discounted lifetime utility is: ∞

∫0 ½logðct Þ + β logðkt Þexpð−ρt Þdt where β ≥ 0 denotes the degree (desire) for the spirit of capitalism, and ρ ∈ (0, 1) is the discount factor. We consider an endogenous growth monetary economy comprising of two sectors, one of which produces outputs used for consumption and investment, while the other produces human capital (ht). Labor supply, which is inelastic, is normalized to unity. Let st and ut respectively denote the fractions of physical and human capital devoted to the production of output. The rest of physical and human capital is used for the human capital accumulation. Both output production function and human capital formation are constant-returns-to-scale. The production function is represented by: α

1−α

yt = Aðst kt Þ ðut ht Þ

;

ð1Þ

where A N 0 denotes the total factor productivity and α ∈ (0, 1) is the capital share to output. The human capital accumulation function is11:

dht = B½ð1−st Þkt ϕ ½ð1−ut Þht 1−ϕ ;

ð2Þ

Let λmt, λk and γt respectively represent the co-state variables of Eqs. (5), (6) and (2) and ξt represents the multiplier of Eq. (4). The first-order necessary conditions for the representative agent's optimization problem are: 1 = λmt + ξt ; ct

ð7Þ

λmt = λkt ;

ð8Þ α

ð1−st Þkt ð1−ut Þht

as the ratio of capital in the output production and bt = physical capital to human capital in the human capital accumulation function. In period t, the government injects money into the economy by giving nominal lump-sum transfers to the representative agent. Let Mt and pt denote the nominal money balance and the common price in period t. We assume that the nominal money supply grows at the rate of μ N 0. Let mt =

Mt pt

dp

represent the real money balance, and πt = ptt

represent the inflation rate. Then the law of motion of the real money balance is governed by:

d

mt = ðμ−πt Þmt

ϕ

λmt Að1−αÞat = γt Bð1−ϕÞbt ; α−1

λmt Aαat

ϕ−1

= γt Bϕbt

ð9Þ

;

ð10Þ

ϕ−1 dλkt = ðρ + δÞλkt − β −λmt Aαst aα−1 −γt Bϕð1−st Þbt ; t

ð11Þ

dλmt = ðρ + πt Þλmt −ξt ;

ð12Þ

dγt = ργt −λmt Að1−αÞut aαt −γt Bð1−ϕÞð1−ut Þbϕt ;

ð13Þ

kt

plus the transversality conditions: −ρt

lim e

t→∞

where B N 0 represents the technology parameter for the human capital accumulation. Besides, ϕ ∈ (0, 1) and (1 − ϕ) are the elasticities of human capital accumulation with respect to the physical capital and human capital. Following King and Rebelo (1990), we assume that ϕ b α. We define at = ustt khtt as the ratio of physical capital to human

787

λkt kt = 0;

lim e

−ρt

t→∞

λmt mt = 0;

−ρt

lim e

t→∞

γt ht = 0

As is common in the literature, we assume that CIA constraint is strictly binding. The goods market clearing condition implies: ð14Þ

ct + it = yt

We define three stationary variables as wt = kt 1λ , zt = kctt and kt xt = hktt . Bringing together Eqs. (9) and (10), we obtain the relationship between at and bt: bt =

ð1−αÞϕ a: αð1−ϕÞ t

ð15Þ

Combining Eq. (15) and the fact that ut at + ð1−ut Þbt = leads to: at =

ð3Þ

αð1−ϕÞ xt ½ðα−ϕÞut + ð1−αÞϕ

kt ht

=

1 xt

ð16Þ

Eqs. (15) and (16) imply that The representative agent uses the real money balance in period t, brought forward from the previous period, t − 1, to buy goods for consumption; hence, the cash-in-advance constraint for the representative agent is: ct ≤mt

ð4Þ

The budget constraint for the representative agent is:

d

ct + it + mt = yt −πt mt + τt

ð5Þ

where it denotes investment and τt = μmt is the real lump-sum transfers that households receive from the monetary authority. The law of motion of capital follows:

dkt = it −δkt

ð6Þ

where δ ∈ [0, 1] is the depreciation rate of physical capital.

11 In this paper, we assume that human capital does not depreciate. However, assuming that human capital depreciates will not change our results.

dat at

=

dbt bt

dxt

=−

xt

−

d

ðα−ϕÞ ut ½ðα−ϕÞut + ð1−αÞϕ

ð17Þ

The law of motion of the real money balance is governed by Eq. (3). Substituting for yt and it in Eq. (14) by using Eqs. (1) and (6), we obtain:

dkt kt

α

= Aut xt at −zt −δ

ð18Þ

From Eq. (2), the human capital growth rate can be written as:

dht ht

ϕ

= Bð1−ut Þbt

ð19Þ

The inflation rate is endogenously determined by Eqs. (7), (8), (11) and (12): α−1

πt = −1 + δ−Aαat

−βwt +

wt zt

ð20Þ

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H.-J. Chen / Economic Modelling 28 (2011) 785–794

From Eqs. (8), (11) and (18), the evolution of wt can be expressed by:

d

wt α−1 = −ρ + ðα−ut xt at ÞAat + βwt + zt wt

ð21Þ

The dynamics of zt can be derived by using Eqs. (3), (4), (18) and (20):

and   a11  Rð J Þ =  a21 a 31

   a11 a13   a23  +  a21 a a33  41

a12 a22 a32

zt

= 1 + μ + βwt +

α−1 ðα−ut xt at ÞAat

xt

ϕ

α

ð23Þ

Substituting Eq. (10) into Eq. (13) derives the constant growth rate of γt: = ðρ−BÞ

γt

ð24Þ

dat at

dγt dλkt

=

γt

−

a23 a33 a43

3

2

 a24  a34  a  44

ð28Þ

The local property of the dynamic behavior around the steady state is determined by the signs of the real parts of ηi. Since the model contains three jump variables (wt, zt and ut) and one non-jump variable (xt), the BGP equilibrium displays saddle-path stability if there are three eigenvalues with positive real parts and one eigenvalue with a negative real part. The following theorem gives a simple method to study the stability property around the equilibrium for a four-dimensional dynamic system: Theorem 1. The number of roots of the polynomial with positive real parts in Eq. (A1) is equal to the number of variations of the signs in the scheme:   Δ Γ 1; TRð JÞ;  ; ; Detð JÞ ; TRð JÞ Δ

Combining Eqs. (7) and (9), we can then derive: ðα−ϕÞ

   a22 a14   a34  +  a32 a a44  42

ð22Þ

= δ + Bð1−ut Þbt −Aut xt at + zt

dγt

a13 a33 a43

P ðηi Þ = ηi −Trð J Þηi + Bð J Þηi −Rð J Þηi + Det ð J Þ = 0

w + zt − t zt

Combining Eqs. (18) and (19), the dynamics of xt is governed by:

dxt

   a11 a14   a24  +  a31 a a44  41

Let ηi (i = 1,.., 4) represent the characteristic roots of the associated characteristic equation: 4

dzt

a12 a22 a42

ð25Þ

λkt

Substituting Eqs. (11), (17), (23) and (24) into Eq. (25), the dynamics of ut can be expressed as: Þϕ dut = ½ðα−ϕÞut + ð1−α Γt ; 2

ð26Þ

ðα−ϕÞ

where

ð29Þ

where Δ = − Tr(J)B(J) + R(J) and Γ = − R(J)Δ − (Tr(J))2Det(J). Proof. This is an application of the Routh Theorem for a fourdimensional dynamic system (see Gantmacher, 1960). Q.E.D. 3. Two types of human capital formation In this section, we examine macroeconomic effects of the money growth rate under two different human capital formations: the Lucastype human capital formation and generalized human capital formation.

ϕ

Γt = ð1−α + ϕÞδ + ½1−α + ðα−ϕÞut Bbt α−1

+ ½−α + ðα−ϕÞut xt at Aat

3.1. Lucas-type human capital formation

−ðα−ϕÞzt −βwt :

The economy can be represented by a four-dimensional dynamic system of equations comprised by Eqs. (21)–(23) and (26) in wt, zt, xt and ut.

We first consider a simple case where human capital is the only input for the accumulation of human capital and all physical capital is used for output production. That is, the human capital formation of Eq. (2) is reduced to:

2.1. The BGP equilibrium and the transitional dynamics

dht = Bð1−ut Þht

Along the balanced-growth path (BGP) equilibrium, ct, yt, kt, ht and mt grow at a common growth rate (g*), while the common growth rate for λkt and λmt is (− g*); hence, along the BGP equilibrium,

dw

t

wt

=

dz

t

zt

=

dx

t

xt

=

du

t

ut

⁎

⁎

⁎

d d d d

a12 a22 a32 a42

 a12  + a22 

α

1−α

yt = Akt ðut ht Þ

ð31Þ

= 0. Let w , z , x and u represent the steady-

a13 a23 a33 a43

32 3 a14 w−w⁎ ⁎ 7 6 a24 76 z−z 7 7= a34 54 x−x⁎ 5 a44 u−u⁎

2

3 w−w⁎ 6 z−z⁎ 7 7 J6 4 x−x⁎ 5 u−u⁎

ð27Þ

where J is the Jacobian matrix. We use Tr(J) and Det(J) to represent the trace and the determinant of matrix J and define:  a Bð J Þ =  11 a21

The production function of output becomes:

⁎

state values of wt, zt, xt and ut. To study the local property of the dynamic behavior around the steady state, we linearize Eqs. (21)–(23) and (26) evaluated at the steady state: 2 3 2 a11 w 6z7 6 6 7 = 6 a21 4x5 4 a31 a41 u

ð30Þ

 a  11 a 31

 a13  + a33 

 a  11 a 41

  a a14  +  22 a44  a32

 a23  + a33 

 a  22 a 42

 a24  + a44 

 a  33 a 43

 a34  ; a44 

This model is a simplified version of the model of generalized human capital formation with ϕ = 0 and st = 1 (that is, the agent does not need to make optimal decision on st). The dynamic behavior of the economy now is represented by the following equations in wt, zt, xt and ut:

dwt wt

dzt zt

dxt xt

1−α

= −ρ−Að1−αÞðut xt Þ

+ βwt + zt ;

1−α

= 1 + μ + βwt −Að1−αÞðut xt Þ

=

dht dkt ht

−

kt

ð32Þ

+ zt −

1−α

= δ + Bð1−ut Þ−Aðut xt Þ

wt ; zt

+ zt ;

ð33Þ

ð34Þ

H.-J. Chen / Economic Modelling 28 (2011) 785–794

dut ut

=

ð1−αÞδ B β + −Bð1−ut Þ−zt − wt ; α α α

ð35Þ

From Eqs. (32)–(35), the steady-state conditions are:  1−α + z⁎ = ρ−βw⁎ ; −Að1−αÞ u⁎ x⁎

789

that the BGP equilibrium is a saddle point. Since Tr(J) N 0 and Det(J) b 0, the fourth case is present if Δ N 0 and R(J) N 0. Computing the value of B(J), we can obtain: Bð J Þ =





 1−α w⁎   w⁎ β w⁎   Bu⁎ + ⁎ βw + z + Bu⁎ ⁎ −Að1−αÞ u⁎ x⁎ Bu + w⁎ + ⁎ + z⁎ : α z z z

ð36aÞ

ð38Þ

ð36bÞ

From Eq. (36a), we have βw* + z* = ρ + A(1 − α)(u⁎x⁎)1 − α. Substituting βw* + z⁎ in Eq. (38), by using Eq. (36a), we can get:

   1−α + z⁎ = 0; δ + B 1−u⁎ −A u⁎ x⁎

ð36cÞ

Bð J Þ = Bu⁎

  ð1−αÞδ B β + −B 1−u⁎ −z⁎ − w⁎ = 0: α α α

ð36dÞ

1+μ+



βw⁎ −Að1−αÞ

u⁎ x⁎



1−α

+

w⁎ z⁎ − z⁎

= 0;



Furthermore, using Eqs. (36a)–(36d), we can derive w⁎, z⁎, x⁎ and ⁎ u as: αρ + ð1−αÞðB + δÞ ; α + βð1 + μ + ρÞ

z⁎ = w⁎ x⁎

= ð1 + μ +

ð37aÞ

ρÞz⁎ ;

ð37bÞ ð37cÞ

ρ ; B

ð37dÞ

where q = 1 + β(1 + μ + ρ). We assume that ρ b B so that u⁎ ∈ (0, 1). This implies that qz* − ρ N 0 and thus, x* N 0. To examine the stability of the BGP equilibrium, we calculate the Jacobian matrix: a111 = βw⁎ ; a112 = w⁎ ; a113 = −Að1−αÞ2 w⁎ u⁎ 1−α x⁎ −α ; a114 = a113 1

1

a21 = βz⁎ −1; a22 = z⁎ +

⁎ ⁎ w⁎ 1 1 z 1 1 x ; a = a13 ; a24 = a23 ; z⁎ 23 w⁎ u⁎

x⁎ ; u⁎

   1−α 1 −α 1−α 1 1 1 ; a31 = 0; a32 = x⁎ ; a33 = −Að1−αÞ u⁎ x⁎ ; a34 = −x⁎ B + Að1−αÞw⁎ u⁎ x⁎ 1

a41 = −

β ⁎ 1 1 1 u ; a42 = −u⁎ ; a43 = 0; and a44 = Bu⁎ : α

Using Eqs. (36a)–(36d), we can calculate the trace, Tr(J), and the determinant, Det( J), of the Jacobian matrix: Trð J Þ = ρ + Bu⁎ +

w⁎ N 0; z⁎



 2−α  ⁎ 1−α βw⁎ x 1+ b0: Det ð J Þ = −ABð1−αÞw⁎ u⁎ αz⁎ Since we have derived Tr(J) N 0 and Det(J) b 0, there can be only four possibilities of the signs in scheme (29): (1)  TRðΔ JÞ N 0 and ΔΓ N 0, (2)  TRðΔ JÞ b0 and Γ Δ b0.

Γ Δ

Computing the value of R(J), we obtain: Rð J Þ =

 

    1−α β ⁎ w⁎ β ⁎ z⁎  Bu⁎ βw + z −Að1−αÞ u⁎ x⁎ w + z⁎ + Bu⁎ z +1+ z⁎ w⁎ α α

ð40Þ Substituting βw* + z⁎ in Eq. (40), by using Eq. (36a), we can get: Rð J Þ =





  1−α β ⁎ w⁎ z⁎ w + z⁎ Bu⁎ ρ−Að1−αÞ u⁎ x⁎ 1 + Bu⁎ ⁎ ⁎ α z w ð41Þ

  1 B qz⁎ −ρ 1−α = ; ρ Að1−αÞ

u⁎ =



 1−α β ⁎ w⁎ w⁎ w + z⁎ +ρ +ρ −Að1−αÞ u⁎ x⁎ ð39Þ α z⁎ z⁎

N 0, (3)  TRðΔ JÞ N 0 and

Γ Δ b0,

and (4)  TRðΔ JÞ b0 and

Eq. (41) implies that R(J) N 0 if, and only if,



 1−α β ⁎ z⁎ w + z⁎ 1 + Bu⁎ : Bu⁎ ρ N Að1−αÞ u⁎ x⁎ α w⁎ Since u⁎, z⁎, x⁎ N 0, we know that if R(J) N 0, then:



 1−α β ⁎ z⁎ w + z⁎ Bu⁎ ρ N Að1−αÞ u⁎ x⁎ 1 + Bu⁎ α w⁎

 ⁎ ⁎ 1−α β ⁎ w + z⁎ : N Að1−αÞ u x α This indicates that B( J) N 0 if R( J) N 0. Using Eqs. (37d), (39) and (41), we can compute: 

w  Δ = −  + ρ + Bu Bð J Þ + Rð J Þ z "

( )

#  1−α β ⁎   w⁎ 2 + ρ + Bu⁎ Bð J Þ = − Bu⁎ 2 + Að1−αÞ u⁎ x⁎ w + z⁎ z⁎ α

This implies that Δ b 0 if B( J) N 0. Hence, Δ b 0 if R( J) N 0. We can then exclude the possibility of the fourth case and conclude that the BGP equilibrium exhibits saddle-path stability. We summarize our result in the following proposition: Proposition 1. When the motive for social status is present and a cashin-advance constraint is applied to consumption in a monetary economy with a Lucas-type human capital formation, a unique BGP equilibrium exists which displays saddle-path stability.

There are three variations of the signs in scheme (29) in the first

three cases, whereas in the fourth case, there is only one variation of the signs. Because there are three jump variables and one non-jump variable in the model, the BGP equilibrium displays saddle-path stability if the signs in the order of Eq. (29) change three times; that is, there is a BGP equilibrium which exhibits saddle-path stability in the first three cases, while indeterminacy is presented in the fourth case.12 Therefore, if the fourth case can be excluded, we can conclude 12 This is because in the fourth case, there is one root with a positive real part, and three roots with negative real parts.

Eqs. (37a), (37c) and (37d) imply that z*, x* and u* are all independent of μ . The BGP growth rate can be derived as:  1−α ⁎    g = A u⁎ x⁎ −z −δ = B 1−u⁎ = B−ρ N 0:

ð42Þ

Eq. (42) indicates that there exists a positive, constant growth rate in the long run which is independent of the presence of social status and the money growth rate. The finding of the super-neutrality of money in an economy without the motive of status seeking is in line with the result presented by Marquis and Reffett (1991).

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H.-J. Chen / Economic Modelling 28 (2011) 785–794

The following proposition gives the effect of money growth rate on the long-run economic growth for an economy with the desire for status seeking: Proposition 2. When the motive for status seeking is present and a cash-in-advance constraint is applied to consumption, money is superneutral in a monetary model with Lucas-type human capital formation since an increase in the money growth rate will not affect the BGP growth rate. Furthermore, the BGP growth rate equals B − ρ. Proposition 2 demonstrates that there is a constant, positive BGP growth rate which is dependent only on the technology parameter for human capital accumulation and the discount factor. Note that there are two channels through which monetary expansion can affect the BGP growth rate. First, a permanent increase in μ reduces the real money balance, which depresses future consumption through the cash-in-advance constraint; hence, the agent tends to use current consumption to substitute for future consumption, thereby reducing investment to the detriment of physical capital accumulation. On the other hand, an increase in the inflation rate reduces the real interest rate, thereby encouraging investment in physical capital, which is beneficial to economic growth. Chang et al. (2000) showed that in a one-sector monetary model with an AK production function, these two effects cancel each other out, so that money is super-neutral in the long run when there is no desire for social status (β = 0). When the agent does care about the social position (β N 0), this desire for social status will reinforce the second effect; hence, an increase in μ will raise the BGP growth rate. That is, the presence of the desire for social status will motivate the agent to accumulate more physical capital, which will in turn increase the long-run growth rate; hence, money is not super-neutral due to the presence of social-status seeking. However, the presence of the desire for social status affects both the accumulation of physical and human capital in a two-sector model. To see this, we take the partial derivatives of z⁎ and x⁎ with respect to β: ∂z⁎ ð1 + μ + ρÞz⁎ b0; =− α + βð1 + μ + ρÞ ∂β ∂x⁎ ð1 + μ + ρÞx⁎ z⁎ b0: =− ½α + βð1 + μ + ρÞðqz⁎ −ρÞ ∂β

ð43aÞ

ð43bÞ

affect g*, although it will have level effects, since it will reduce both z⁎ and x⁎. Our model can easily be extended by incorporating human capital externality into the production function of output, as studied by Lucas (1988), Benhabib and Perli (1994) and Xie (1994). The analysis shown in Appendix demonstrates that the consideration of human capital externality within the production function does not change the results shown in Propositions 1 and 2. 3.2. Generalized human capital formation We go on to examine the effect of money growth rate on the longrun economic growth rate in an economy with generalized human capital formation. From Eqs. (21)–(23) and (26), the steady-state values w⁎, z⁎, x⁎ and u⁎ are determined by the following equations: 

  α−1 + z⁎ = ρ−βw⁎ ; α−u⁎ x⁎ a⁎ A a⁎

   α−1 w⁎ + z⁎ − = 0; 1 + μ + βw⁎ + α−u⁎ x⁎ a⁎ A a⁎ z⁎

ð45aÞ ð45bÞ

  ϕ  α δ + B 1−u⁎ b⁎t −Au⁎ x⁎ a⁎ + z⁎ = 0;

ð45cÞ

−ð1−α + ϕÞδ + ðα−ϕÞz⁎ + βw⁎

  ϕ

  α−1 : = 1−α + ðα−ϕÞu⁎ B b⁎ + −α + ðα−ϕÞu⁎ x⁎ a⁎ A a⁎

ð45dÞ Using Eqs. (45a)–(45d), we can express x⁎, b⁎, a⁎ and z⁎ as functions represented by u⁎:   x⁎ u⁎ =   b⁎ u⁎ =

 1   ⁎ B u −ϕ ϕ ð1−αÞϕ ; ðα−ϕÞu + ð1−αÞϕ ρ 1   ϕ ρ αð1−ϕÞ ⁎  ⁎  b u ; ; a⁎ u⁎ = Bðu⁎ −ϕÞ ð1−αÞϕ

ð46aÞ



ð46bÞ

     α    ϕ −δ−B 1−u⁎ b⁎ u⁎ ; z⁎ u⁎ = Au⁎ x⁎ u⁎ a⁎ u⁎

ð46cÞ

    w⁎ u⁎ = ð1 + μ + ρÞz⁎ u⁎ :

ð46dÞ

From Eqs. (46a)–(46c), we can determine that:

Eqs. (43a) and (43b) indicate that in a two-sector model with Lucas-type human capital formation, the presence of the desire for social status will increase not only the accumulation of physical capital (a decrease in z*), but also the proportion of human capital used for output production (since physical and human capital are complements in the production function), while also reducing the proportion of human capital devoted to human capital accumulation (a decrease in x⁎). Although the former is beneficial to economic growth rate, the reverse is the case for the latter. These two effects cancel each other out so that an increase in μ does not affect the longrun economic growth rate. The partial derivatives of z⁎ and x⁎, with respect to μ, are: ∂z⁎ βz⁎ b0; =− α + βð1 + μ + ρÞ ∂μ

ð44aÞ

∂x⁎ βx⁎ z⁎ b0: =− ½α + βð1 + μ + ρÞðqz⁎ −ρÞ ∂μ

ð44bÞ









∂x ∂b ∂a ∂z N 0; b0;  b0; and N 0: ∂u ∂u ∂u ∂u Substituting Eqs. (46a)–(46d) into Eq. (45d), we have:     Φ u⁎ = Ω u⁎ ;

ð47Þ

Þ ⁎ ⁎ α−1 and Ω(u⁎) = β(1 + μ + ρ) where Φðu⁎ Þ = δ + ρuð1−ϕ  −ϕ −Aαða ðu ÞÞ ⁎ ⁎ z (u ). Eq. (47), together with Eqs. (46a)–(46d), can be used to determine the steady state; however, due to the complexity of the model, we are unable to solve for the analytical solution of the steady state; thus, in the following, we graphically prove the existence and uniqueness of the steady state. Taking the first-order derivatives of Φ(u⁎) and Ω(u⁎), we have:

   α−2 ∂a ∂Φ ρð1−ϕÞ −Aαðα−1Þ a⁎ u b0:  = −  2 ∂u ∂u ðu −ϕÞ 

Eqs. (44a)–(44b) imply that an increase in the money growth rate or an increase in the desire for social status will lower both z⁎ and x⁎. Therefore, when the desire for social status is present, an increase in the money supply, or an increase in the desire for social status, will not

∂Ω ∂z = βð1 + μ + ρÞ  N 0: ∂u ∂u Hence, Φ(u⁎) is a monotonically decreasing function in u⁎, while Ω(u⁎) is a monotonically increasing function in u⁎. Note that Eqs. (46a)

H.-J. Chen / Economic Modelling 28 (2011) 785–794

791

and (46b) show that the non-trivial solutions of x⁎, a⁎ and b⁎ exist if u⁎ N ϕ. Therefore, we impose the restriction of u⁎: ϕ b u⁎ ≤ 1. The boundary values of Φ(u⁎) and Ω(u⁎) are:

u * is determined by Eq. (47) when the desire for social status is present. The first-order partial derivates of Ω(u⁎) with respect to β and μ are:

  Φ1 = lim þ Φ u⁎ = ∞;

Φ2 = Φð1Þ = δ + ρ−Aαx⁎2

  Ω1 = lim þ Ω u⁎ = −∞;

 1−α Ω2 = Ωð1Þ = βð1 + μ + ρÞ Ax⁎2 −δÞ;

    ∂Ω u⁎ ∂Ω u⁎ N 0 and N 0: ∂β ∂μ

u⁎ →ϕ

u⁎ →ϕ

1−α

;

 1 ð1−αÞϕ ð1−ϕÞB ϕ where = αð1−ϕÞ . Functions Φ(u⁎) and Ω(u⁎) are ρ illustrated in Fig. 1. A sufficient condition for the existence and uniqueness of the steady state is given in the following proposition: x⁎2

Proposition 3. When the desire for social status is present and a cash constraint is applied solely to consumption in a two-sector monetary model with generalized human capital formation, a unique BGP equilibrium exists if: A ðβÞ = ANX

ρ + ½1 + βð1 + μ + ρÞδ ⁎ α−1 x2 : ½α + βð1 + μ + ρÞ

Proof. Since Φ(u⁎) is a monotonically decreasing function in u⁎, while Ω(u⁎) is a monotonically increasing function in u⁎, a unique solution exists for u⁎ if Φ2 b Ω2 (see Fig. 1). This implies that A N AðβÞ. Q.E.D. P Note that when there is no desire for social status, Eq. (48) is reduced to:   Φ u⁎ = 0:

ð48Þ

Then when β = 0, the unique BGP equilibrium exists if Φ2 b 0, δ ⁎ α−1 which implies that A N P Aðβ = 0Þ = ρ + . Since Φ(u*) is α x2 independent of μ, Eqs. (48) and (46b) indicate that changes in μ will not affect the steady-state values of u* and b*(u*). According to Eq. (19), money is super-neutral since changes in μ will not affect g*. This result reconfirms the finding of Mino (1997). To examine the impact of monetary expansion when the desire for social status is present, we graphically illustrate the impact of an increase in the money supply rate. Note that the steady-state value of

Fig. 1. The determination of u⁎.

This indicates that an increase in β or μ will shift the curve of Ω(u⁎) in Fig. 1 upwards. Hence, it reduces u⁎ and increases (1 − u⁎) which, according to Eq. (46b), will increase b*(u*); therefore, Eq. (19) implies that an increase in μ will increase g* since both (1 − u⁎) and b*(u*) become higher. This is because in a two-sector model with generalized human capital formation, although the presence of social-status seeking will strengthen the motivation for physical capital accumulation, leading to a relative reduction in the motivation for human capital accumulation, as indicated by the increase in b* in Eq. (19), the increase in the investment in physical capital is beneficial to both physical and human capital accumulation. Therefore, money is not super-neutral when the desire for social status is present. Proposition 4. When the desire for social status is present and there is a cash constraint applied solely to consumption in a two-sector monetary model with generalized human capital formation, the money growth rate will positively affect the BGP growth rate; hence, money is not superneutral in the long run. Results in Proposition 4 are different from those stated in Proposition 2 because money growth rate will affect the accumulation of physical capital differently. Under generalized human capital formation, the BGP growth rate becomes g* = B(1 − u⁎)[b(u⁎)]ϕ. Combining Eqs. (8) and (9), we have: α

Að1−αÞat =

γt ϕ Bð1−ϕÞbt : λkt

ð49Þ

The left-hand side of Eq. (49) represents the marginal benefit of raising ut while the right-hand side represents the marginal cost. Under the Lucas-type human capital formation, the BGP growth rate is g* = B(1 − u⁎). Furthermore, Eq. (49) is reduced to: α k γ −α Að1−αÞ t ut = t B: ht λkt

ð50Þ

In the presence of social status, an increase in the money growth rate encourages physical capital accumulation and raises the ratio of kt/ht (the inverse of xt). Thus, the marginal benefit increases. However, this is counteracted by an increase in the marginal cost due to a decline in the shadow price of physical capital (λkt). Therefore, an increase in the money growth rate will affect u* and money is superneutral under Lucas-type human capital formation. However, Eq. (49) shows that under generalized human capital formation, an increase in the money growth rate will not only affect physical capital accumulation, but also the allocation of physical capital stock between output production and human capital accumulation. Thus, money is not super-neutral. Comparing our results with those obtained by Mino (1997) who showed that money is superneutral in a two-sector CIA model with generalized human capital formation, our results demonstrate the important role of social status in concerning the impacts of monetary policy. With three jump variables (wt, zt and ut) and one non-jump variable (xt), the BGP equilibrium exhibits saddle-path stability if the signs in the order of Eq. (29) change three times. However, because of the complication of the model, we are not able to pin down the signs of Tr(J), B(J), R(J) and Det(J). Therefore, we resort to numerical methods to examine the stability property of the equilibrium under a reasonable setting of parameter values. We assign α = 1/3, ϕ = 0.2, ρ = 0.025, A = 0.1, B = 0.1 and δ = 0.05.

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H.-J. Chen / Economic Modelling 28 (2011) 785–794

Table 2 The effects of β and μ.

Table 3 The effects of β and μ on z⁎, x⁎ and g⁎ for endogenous growth models.

β

μ

u*

b*

z*

g*

Roots

0.8 1.0 1.2 0.8 0.8

0.05 0.05 0.05 0.07 0.10

0.5478 0.5433 0.5399 0.5475 0.5469

0.1918 0.2047 0.2154 0.1928 0.1944

0.0516 0.0445 0.0391 0.0510 0.0501

0.0325 0.0333 0.0338 0.0326 0.0327

−+ −+ −+ −+ −+

AK

+ + + + +

+ + + + +

β=0

βN0

Increase in β μ

Impact on z* N/A – 0 –

Increase in β μ Increase in μ

Impact on x* N/A N/A N/A N/A Impact on g* 0 +

Lucas (v = 0)

Lucas (v ∈ (0, 1))

King et al. (1988)

β=0

βN0

β=0

βN0

β=0

βN0

N/A 0

– –

N/A 0

– –

N/A 0

– –

N/A 0

– –

N/A 0

– –

N/A 0

– –

0

0

0

0

0

+

Table 2 presents the values of u*, b*, z* and g* with varying β and μ; rows 2, 3 and 4 present the effects with an increase in β from 0.8 to 1.2 under μ = 0.05, while rows 2, 5 and 6 present the effects with an increase in μ from 0.05 to 1 under β = 0.8. Our numerical results reveal that an increase in β or μ will lower u⁎, while raising b⁎ and g*. We also compute eigenvalues of the linearized dynamic system to examine the stability property of the BGP equilibrium. There are three eigenvalues with positive real parts and one eigenvalue with a negative real part in all cases presented in Table 2, indicating that the BGP equilibrium exhibits saddle-path stability in all cases.13

ct ≤mt ;

ðA3Þ

dkt = it −δkt ;

ðA4Þ

4. Conclusions

dht = Bð1−ut Þht ;

ðA5Þ

This study examines the effects of the money growth rate on long-run growth in two-sector monetary models with the presence of the desire for social status. We find that money is super-neutral in an economy with Lucas-type human capital formation (with or without human capital externality), even when the desire for social status is present, although it has level effects on the values of consumption and on physical and human capital. However, for an economy in which both physical and human capital are used as inputs for human capital accumulation, money is not super-neutral; thus, an increase in the money growth rate will raise the economic growth rate. Table 3 summarizes the impact of money growth rate on the key macroeconomic variables for endogenous growth models. The existence, uniqueness and saddle-path stability of the balancedgrowth equilibrium are also examined. Our paper concludes with the suggestion that this study could easily be extended and applied to a variety of issues which would seem to be ripe for future study, pointing out two possible directions. First, we can consider cash-in-advance constraints applied on consumption, physical capital investment and human capital investment.14 We believe that it is important for studies to be undertaken on the ways in which the cash constraint affects physical and human capital accumulation. Second, we have focused in this paper on the effects of social status represented by physical capital; however, in addition to physical capital, social status can be also represented by the real money balance and human capital. Hence, it would be interesting to extend our model to the consideration of other types of social status.

yt = Akt ðut ht Þ

Appendix A

α

∞

d

13

ðA6Þ

1 = λmt + ξt ; ct

ðA7Þ

λmt = λkt ;

ðA8Þ α

−α

λmt Að1−αÞkt ðut ht Þ

ν

Ht = γt B;

ðA9Þ

dλkt = ðρ + δÞλkt − β −λmt Aαktα−1 ðut ht Þ1−α Htν ;

ðA10Þ

dλmt = ðρ + πt Þλmt −ξt ;

ðA11Þ

−α ν dγt = ργt −λmt Að1−αÞkαt u1−α ht Ht −γt Bð1−ut Þ; t

ðA12Þ

kt

together with the transversality conditions. The resource constraint is:

dkt = Akαt ðut ht Þ1−α Htν −ct −δkt :

ðA1Þ

subject to ct + it + mt = yt −πt mt + τt ;

ν

Ht ;

where Ht is the average human capital and ν ∈ (0, 1) represents the degree of human capital externality in the production function. Following Xie (1994), we assume that ν b α. The first-order necessary conditions for the interior solution are:

In this Appendix, we consider a cash-in-advance model within which the production function contains human capital externality. The representative agent's problem is given as follows: max∫0 ½logðct Þ + β logðkt Þ expð−ρt Þdt;

1−α

ðA2Þ

We also conduct sensitivity analysis regarding the stability of the equilibrium by varying the settings of parameter values. The results indicate that the stability property of the equilibrium is not sensitive to parameter settings. 14 Studies of the effects of money growth rate on the long-run economic growth rate in a two-sector monetary economy with a cash-in-advance constraint applied to consumption and investment can be found in Chen (2010, 2011).

ðA13Þ

The law of motion of the real money balance is governed by:

d

mt = ðμ−πt Þmt :

ðA14Þ

Since agents are homogeneous, we have ht = Ht at the equilibrium. ht is not a Note that with human capital externality, the variable kt stationary variable; hence, we redefine the stationary variable xt as xt =

1−α + + ν 1−α

ht

kt

. From Eqs. (A7), (A8), (A10) and (A11), the inflation rate

can be calculated as: 1−α

πt = −1 + δ−Aαðut xt Þ

−βwt +

wt : zt

ðA15Þ

H.-J. Chen / Economic Modelling 28 (2011) 785–794

793

Combining Eqs. (A8), (A10) and (A13), the evolution of wt can be expressed by:

To examine the stability of the BGP equilibrium, we calculate the Jacobian matrix:

dwt

a11 = βw⁎ ; a12 = w⁎ ; a13 = −Að1−αÞ w⁎ u⁎

wt

A

1−α

= −ρ−Að1−αÞðut xt Þ

+ βwt + zt ;

ðA16Þ

A

A

1−α

= 1 + μ + βwt −Að1−αÞðut xt Þ

w + zt − t ; zt

a34 ðA17Þ

xt

Bð1−α + νÞ 1−α ð1−ut Þ−Aðut xt Þ + zt : 1−α

=δ+

w⁎ ; z⁎

A

A

ðA18Þ

Substituting Eq. (A9) into Eq. (A12), we know that γt grows at a constant rate:

2

1−α ⁎ −α A x ; a14

z⁎ ; w⁎  ⁎ 1−α

a23 = a13

A

= a13

A

2

a24 = a23

 A A a32 = x⁎ ; a33 = −Að1−αÞ u⁎ x ;

ð1−α + νÞB −α 1−α + Að1−αÞw⁎ u⁎ x⁎ = −x⁎ ; 1−α

a41 = −

Furthermore, from Eqs. (A5) and (A13), the dynamics of xt is governed by:

dxt

A

a22 = z⁎ +

2

a31 = 0; A

zt

A

a21 = βz⁎ −1; Using Eqs. (A13)–(A15), the dynamics of zt is governed by:

dzt

A

β ⁎ u ; α

A

a42 = −u⁎ ;

A

a43 = 0

and

A

a44 =

x⁎ ; u⁎

x⁎ ;; u⁎

α−ν ⁎ Bu : α

The trace, Tr(J), and the determinant, Det(J), of the Jacobian matrix can be calculated as: α−ν ⁎ w⁎ Bu + N 0; α z⁎

 2−α  1−α βw⁎ x⁎ 1+ b0: Det ð J Þ = −ABð1−αÞw⁎ u⁎ αz⁎

Trð J Þ = ρ +

From Theorem 1, if we can exclude the possibility where  TRðΔ JÞ b0 and ΔΓ b0, then the BGP equilibrium will display saddle-path stability.

dγt γt

= ðρ−BÞ:

ðA19Þ

Bð J Þ =

From Eq. (A9), we can derive that the growth rate of ut is:

dut ut

=

dht dγt !

dkt

dλmt

1 + α + ðν−αÞ − : α λmt kt ht γt

To examine the transitional dynamics of the system, we then calculate:



 1−α β ⁎ α−ν ⁎ w⁎ w⁎ Bu w + z⁎ ; +ρ +ρ −Að1−αÞ u⁎ x⁎ α α z⁎ z⁎ ðA26Þ

ðA20Þ

Rð J Þ =





  1−α β ⁎ w⁎ ðα−νÞρ ⁎ z⁎ 1 + Bu⁎ : Bu −Að1−αÞ u⁎ x⁎ w + z⁎ α α z⁎ w⁎

ðA27Þ From the inclusion of Eqs. (A5), (A8), (A10), (A13) and (A19) into Eq. (A20), the dynamics of ut can be expressed by:

dut ut

=

ð1−αÞδ B Bðν−αÞ β + − ð1−ut Þ−zt − wt : α α α α

ðA21Þ

Therefore, the dynamic system is represented by Eqs. (A16)–(A18) and (A21). Along a balanced-growth path, ct, yt, kt and mt grow at a 1−α  common growth rate (g*), ht grows at a growth rate of 1−α + ν g and λkt, λmt and γt grow at a common growth rate of (− g*). Hence, along

dw

dz

⁎

zt ⁎

the BGP, wtt = ⁎

t

=

dx

t

xt

=

du

t

ut

ðα−νÞρ + ð1−α + νÞB + ð1−αÞδ ; α + βð1 + μ + ρÞ

w⁎ = ð1 + μ + ρÞz⁎ ;   1 B qz⁎ −ρ 1−α x = ; ρ Að1−αÞ ⁎

u⁎ =

ρ ; B

where q = 1 + β(1 + μ + ρ) and qz* − ρ N 0.





 1−α β ⁎ ðα−νÞρ ⁎ z⁎ Bu N Að1−αÞ u⁎ x⁎ w + z⁎ 1 + Bu⁎ : ⁎ α α w Under this condition, B(J) N 0 as indicated by Eq. (A26). With a few calculation steps, we can obtain: ( Δ = − Bu⁎

" )



#   1−α β ⁎ 2α−ν w⁎ 2 α−ν ⁎  + Að1−αÞ u⁎ x⁎ + ρ+ w + z⁎ Bu Bð J Þ : z⁎ α α α

ðA23Þ

This implies that Δ b 0 if B(J) N 0. Hence, Δ b 0 if R( J) N 0; therefore, we can exclude the possibility where  TRðΔ JÞ b0 and ΔΓ b0, and conclude that the BGP equilibrium exhibits saddle-path stability. When the desire for social status is not present, Eqs. (A22), (A24) and (A25) imply that z*, x* and u* are all independent of μ. Then the BGP growth rate of human capital equals B − ρ, while the BGP growth + ν rate of physical capital equals g  = 1−α 1−α ðB−ρÞ. Therefore, money is super-neutral. When the desire for social status is present, using Eqs. (A22) and (A24), we can derive that:

ðA24Þ

∂z⁎ ð1 + μ + ρÞz⁎ ∂x⁎ ð1 + μ + ρÞx⁎ z⁎ b0; b0; =− =− α + βð1 + μ + ρÞ ½α + βð1 + μ + ρÞðqz⁎ −ρÞ ∂β ∂β

= 0.

⁎

Let w , z , x and u represent the steady-state values of wt, zt, xt and ut. Using Eqs. (A16)–(A18) and (A21), we can compute that:

z⁎ =

Eq. (A27) indicates that R(J) N 0 if, and only if,

ðA22Þ

∂z⁎ βz⁎ ∂x⁎ βx⁎ z⁎ b0; b0; =− =− ½α + βð1 + μ + ρÞðqz⁎ −ρÞ α + βð1 + μ + ρÞ ∂μ ∂μ

ðA25Þ

In order to calculate the BGP growth rate, Eq. (A25) indicates that u⁎ is independent of μ. With a few calculation steps, we can derive

dh

t

ht

= B−ρ and

dk

t

kt

=

1−α + ν 1−α ðB−ρÞ

at the BGP equilibrium and find

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H.-J. Chen / Economic Modelling 28 (2011) 785–794

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