Discussion Paper No Conformism and Structural Change. Takeo Hori, Masako Ikefuji and Kazuo Mino

Transcription

1 Discussion Paper No Conformism and Structural Change Takeo Hori, Masako Ikefuji and Kazuo Mino

2 Conformism and Structural Change Takeo Hori, Masako Ikefuji and Kazuo Mino March 17, 2013 Abstract Using a simple, multi-sector model of endogenous growth, we show that the presence of commodity-specific consumption externalities can be a source of structural change. When the degrees of consumption externalities are different between different goods, each sector grows at different rates, whereas the aggregate economy exhibits balanced growth in the sense that capital stock and expenditure grow at the same constant rate. Under the more restrictive condition such that the degrees of consumption externalities are the same, structural change does not arise. We also show that the dependence of the benchmark consumption levels on the past consumption is crucial for the divergent patterns of structural change across countries. Keywords: Structural change, Consumption externalities, Two-sector growth model, Kaldor facts JEL classification numbers: E21, E30, O10, O41 The authors are grateful to Ryoji Hiraguchi and other attendees of conference at EEA-ESEM Malaga 2012, the conference of Macroeconomics for Young Professionals in Kyoto, and seminar participants at Academia Sinica (Taiwan) and Kent University (the UK) for their helpful comments and suggestions. This study was financially supported by the Joint Usage/Research Center Project of ISER from the Ministry of Education, Culture, Sports, Science and Technology in Japan. Mino s research was also supported by JSPS Kakenhi Project (No ) as well as by JSPS Grandin-Aid Specially Promoted Research Project (No ). All remaining errors are ours. Corresponding author: College of Economics, Aoyamagakuin University, , shibuya, shibuya-ku, Tokyo, , Japan. t-hori@aoyamagakuin.jp. University of Southern Denmark, Niels Bohrs Vej 9-10, 6700 Esbjerg, Denmark, ikefuji@sam.sdu.dk Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan, mino@kier.kyoto-u.ac.jp 1

3 1 Introduction It has been long recognized that social comparison would affect consumption decision of agents. There is a renewed interest in this issue and several authors have conducted experimental or survey-data analyses to find that social comparison may yield significant effects on consumers behavior and welfare: see, for example, Alvarez- Cuardrado and Long (2011), Alpizar et al. (2005), Carlson et al. (2007), Frank (2005) and Maurer and Meier (2008). 1 Those studies may support the Easterlin paradox emphasizing that social comparison plays a key role in determining the relation between economic growth and happiness: see, for example, Easterlin (2001) and Clark et al. (2008). Along with the development in those behavioral economic studies, the role of social comparison has been well explored in macroeconomics as well. The basic assumption of this literature is that consumers felicity depends not only on their private consumption but also on the reference level of consumption that reflects the other consumers activities. The presence of such a psychological external effect may alter saving behaviors of consumers and thus dynamic property of the economy. Earlier studies such as Abel (1990) and Galí (1994) focus on the role of consumption externalities in asset pricing. The recent contributions to the role of consumption externalities in macroeconomics discuss various topics such as optimal taxation (Ljungqvist and Uhlig 2000), equilibrium efficiency (Alonso-Carrera et al. 2008, Garriga 2006, Liu and Turnovsky 2005, Nakamoto 2009 and Arrow and Dasgupta 2009), indeterminacy and sunspots (Weder 2000, Chen and Hsu 2007, Chen et al and 2012), and savings and long-term growth (Carroll et al and 2000, and Harbaugh 1996). The central purpose of this paper is to shed a new light on the role social comparison in the context of structural transformation of an economy. We construct a multi-sector model of endogenous growth in which each consumption good is associated with commodity-specific externaleffects. In our model, consumption ex- 1 The concept of social comparison has also attracted researchers in the neurosciences and the related fields. For example, Fliessbach et al (2007) examine the impact of social comparison on brain activity using functional magnetic resonance imaging (fmri), showing that not only the absolute level of payment but also relative level of payment similarly affect brain activity. In their paper, it is considered that neurophysiological evidence supports the importance of social comparison in the human brain. 2

4 ternalities are represented by outward-looking habit formation. Thus the stock of consumption external habits of each good is the reference level of consumption for the households. Since each habit stock may exhibit different pattern of evolution, the private preference structure continues changing during the transition, which causes structural transformation of the entire economy. More specifically, our baseline setting is a two-sector AK growth model with external habit persistence in consumption. In our model economy, one sector produces general goods used for consumption or investment, while the other sector produces pure consumption goods. Our key assumption is that consumption of each commodity is associated with commodity-specific external habit formation. We assume that preferences of households are homothetic in their private consumption, but they are non-homothetic from the social perspective due to the presence of asymmetric degree of external effects. In this paper we focus on the situation where conformism prevails among consumers. Namely, each commodity generates the catching up with the Joneses (CUJ) effect. We first demonstrate that in a two-sector model if the elasticity of substitution between the two goods is not unity, the economy converges to an equilibrium where outputs of the two sectors grow at the same rate when the same degree of consumption externalities is applied commonly to both types of goods. However, if the degree of conformism associated with each commodity is not the same, then the output with a stronger degree of conformism grows faster than the other, whereas the aggregate economy sustains the balanced growth equilibrium in the sense that aggregate capital stock and expenditure grow at the same constant rate. Moreover, we illustrate that the pattern of structural change depends on the initial levels of benchmark consumption as well as on the degrees of habit persistence and conformism of each commodity. When analyzing the model, we focus on the long-term equilibrium as well as on transition dynamics. Our investigation demonstrates that considering commodityspecific conformism of consumption provides us with empirically plausible patterns of structural transformation during the process of long-term economic growth: we show that the model economy satisfies Kaldor s stylized facts as well as empirically plausible patterns of structural change. We also show that our base model can be extended to the case of three sector economy, which exhibits well documented pattern structural transformation between the agricultural, manufacturing and service 3

5 sectors. In addition, although in the base model the relative price stays constant in the absence of exogenous productivity change, it is shown that the relative prices can be endogenized by a simple modification of the base model. Our study has two motivations. First, we intend to present an alternative driving force of transformation of industrial structure that can be reconciled with the Kaldor facts. In the foregoing literature, two approaches have been used to study structural change. The one is based on the demand-induced structural change caused by non-homothetic preferences. For example, Kongsamut et al. (2001) employ a Stone-Geary utility function to consider dynamics of sectoral labor reallocation. In their model, a knife-edge condition is required in order for their growth model to be reconciled with the Kaldor facts. Foellmi and Zwimüller (2008) introduce a hierarchic utility function to obtain nonlinear Engel curves for the various products, which generates consumption cycles. 2 The second approach is based on the assumption of unbalanced productivity change among production sectors. This approach is initiated by Baumol (1967) and a recent sample includes Ngai and Pissarides (2007), Acemoglu and Guerrieri (2008) and Duarte and Restuccia (2010). Despite the recent active research in the field of growth and structural change, Buera and Kaboski (2009) claim that the standard models often show empirically plausible behaviors only when a set of restrictive conditions are satisfied. 3 Unlike the existing studies, our base model assumes neither non-homothetic private preferences nor sectoral divergence in productivity growth. The driving force of structural change in our model is the relative degree of conformism rather than the exogenously specified nonhomothetic private preferences. Furthermore, we use an endogenous growth model and, hence, sustained growth is possible in the absence of exogenous productivity change. Since the mainstream literature in this field has employed neoclassical (exogenous) growth models, they fail to exploit the recent development of endogenous growth theory. 4 The second motivation of our investigation is to reveal that commodity-specific 2 Well-cited studies on structural change relying on non-homothetic preferences include Matsuyama (1992), Echevarria (1997) and Laitner (2000). 3 As to this point, see Section 6. 4 Recently, Guilló et al. (2011) construct a multisector overlapping generations model of endogenous technical change with the aim of showing that biased TFP growth and labor movement across sectors can be an endogenous response to the non-homothetic preference. 4

6 consumption externalities may play a pivotal role in the process of long-term economic growth. While the majority of the existing studies on the macroeconomic implication of consumption externalities utilize one-sector, neoclassical growth models, Carroll et al. (1997, 2000) introduce consumption external effects into an AK model of endogenous growth. They demonstrate that if there is outward-looking habit persistence in consumption, a one-sector AK growth model can have transition dynamics and it may exhibit empirically plausible patterns of growth. 5 analytical framework of this paper is a multisector version of the model studied by Carroll et al. (1997, 2000). Based on a simple extension of the AK model, we show that the presence of commodity-specific consumption externalities may yield substantial impacts on the patterns of economic growth associated with structural transformation. It is to be noted that the role of commodity-specific external habits has been examined by Ravn et al. (2006), Doi and Mino (2008) and Hori (2011). Those authors assume that consumers put the same importance on all types of consumption of other households. In such a situation, a consumer would care equally about consumption of bread and car of other consumers. However, using Spanish panel data, Browning and Collado (2007) show that food outside the home is habit forming, whereas food at home does not display any significant state dependence. Moreover, the recent studies of the behavioral economics provide the evidence that consumers put the different importance on the different types of consumption of other consumers. 6 Our formulation follows such a research agenda. The rest of the paper is organized as follows: The The next section sets up the base model. Section 3 examines the evolution of the aggregate economy and shows that the aggregate variables reconcile with Kaldor facts. Section 4 characterizes the stationary equilibrium and dynamic behavior of the economy out of the stationary state. Subsection 4.1 analyzes the dynamic system for the case that the degrees of consumption externalities of two goods are same and shows that structural change does not arise. Subsection 4.2 considers the dynamic system and the stability of the equilibrium when the degrees of consumption externalities of two goods are 5 Carroll et al. (1997, 2000) also consider the case of inward-looking habit formation. 6 See Solnick and Hemenway (1998), Alpizar et al. (2005), and Carlsson et al. (2007), for example. They investigate the positionality degree for different goods based on empirical or experimental approaches. 5

7 different and describes how structural change evolves. Subsection 4.3 illustrates the transitional dynamics. Section 5 presents two extensions of our basic model. In Subsection 5.1 we modify the model to endogenize the relative price. Subsection 5.2 extends the two-sector model to a three-sector model. Section 6 discusses our main results in comparison with the standard approaches to growth and structural change. Concluding remarks are given in Section 7. 2 The Baseline Model 2.1 Firms Time is continuous and denoted by t [0, ). We start with two sectors in a competitive economy. Sector 1 produces a good (called good 1), which can be either invested for capital accumulation or consumed. Sector 2 supplies a pure consumption good (called good 2). 7 Good 1 is chosen as numeraire. The price of good 2 at time t is denoted by p t. Firms in each sector are homogeneous. The number of firmsineachsectoris normalized to one. The representative firm in Sector i (= 1, 2) produces good i by using the following technology: Y i,t = A i Ki,t α (B i,tl i,t ) 1 α, for i =1, 2, where 0 < α < 1, Y i,t, K i,t, L i,t and A i > 0 are the output level, the capital input, the labor input and the productivity in Sector i, respectively. It is assumed that the two sectors have the same capital intensity. B i,t represents the sector-specific externalities. Let us denote the average capital and labor inputs of Sector i (= 1, 2) as K i,t and L i,t, respectively. We have K i,t = K i,t and L i,t = L i,t in equilibrium. We specify B i,t K i,t / L i,t so that the social technology of each production sector is given by a simple AK production function, Y i,t = A i K i,t. The reduced form of production structure is thus a two-sector extension of the one-sector AK growth model employed by Carroll et al. (1997). We assume that capital depreciation rate is equal to zero. Profit maximization 7 Ngai and Pissarides (2007) consider the economy with m sectors where one of the m sectors produces capital stock as well as consumption good, and the remaining m 1 sectors produce only consumption goods. The production side of our model is a special case of Ngai and Pissarides (2007). We will examine a three-sector model in Subsection

8 in both sectors yields: r t = αa 1 = p t αa 2, p t = A 1, A 2 K 1,t K 2,t w t =(1 α)a 1 = p t (1 α)a 2. L 1,t L 2,t (1a) (1b) The first line shows that due to our specification of the technology, the rental rate of capital, r, and the price of good 2, p, are both constant over time. In the following discussion, we omit the index t from r and p. The second line gives the wage rate. The capital and labor market equilibrium conditions are: K 1,t + K 2,t = K t, L 1,t + L 2,t = L t, (2a) (2b) where K t and L t are the aggregate capital stock and labor supply. There is free capital and labor mobility between the two sectors. The aggregate production of the economy is given by Y t Y 1,t + py 2,t = A 1 K t. 2.2 Households There is a continuum of identical households whose number is normalized to one. The representative household consumes goods 1 and 2. The household inelastically supplies one unit of labor, L t =1in each moment. We assume that the instantaneous utility of the representative household depends not only on her own consumption of the two goods but also on the benchmark levels of consumption that are determined through outward-looking habit formation. The instantaneous subutility of the representative household at time t is given by: h i θ γ(c 1,t h 1 1,t ) ε 1 θ ε +(1 γ)(c 2,t h 2 2,t ) ε 1 ε ε 1 ε, if ε 6= 1, u t = θ (c 1,t h 1 1,t ) γ θ (c 2,t h 2 2,t ) 1 γ, if ε =1. (3) where c i,t is consumption of good i (= 1, 2) at time t, ε > 0 is the elasticity of substitution between the two goods, and γ (0, 1) represents the importance of good 1. Here, h i,t denotes the benchmark level of consumption of good i at time t. The presence of h i,t in (3) represents the commodity-specific consumption externalities, as in Ravn et al. (2006), Doi and Mino (2008) and Hori (2011). As we discuss later, θ i is associated with commodity-specific consumption externalities. 7

9 Following Carroll et al. (1997, 2000), we assume that 0 θ i < 1. The benchmark levelofconsumption,h i,t is a weighted sum of the past average consumption of good i up to the time t: h i,t = φ i Z t e φ i (t u) c i,u du, φ i > 0, i =1, 2, where c i,u is the average consumption of good i at time u in the economy at large. Differentiating the both sides of the above equation with respect to t yields: ḣ i,t = φ i ( c i,t h i,t ), i =1, 2. (4) Concerning (3), the four points should be mentioned. First, when ε 6= 1holds, u t can be written as: u t = nγ[c 1,t 1 θ 1 (c 1,t /h 1,t ) θ 1 ] ε 1 ε o 1 θ +(1 γ)[c 2 2,t (c 2,t /h 2,t ) θ 2 ] ε 1 ε ε 1 ε. The subutility depends on the relative level of consumption (c i /h i ), as well as the absolute level of consumption. 8 When both θ 1 and θ 2 are equal to zero, only the absolute levels of consumption matter. If θ 1 = θ 2 =1, only the relative level of consumption is all that matters. For other values of θ i (0, 1), both are important. When θ i is strictly positive, an increase in h i,t negatively affects u t. The preference of each household exhibits jealousy toward consumption of others. Because the relative consumption of good i becomes more important as θ i increases, we interpret θ i as an indicator of how much each household cares about the relative consumption of good i. We call θ i the degree of consumption externalities of good i. It is also to be noted that if φ i =+, thenh i,t = c i,t holds so that external effects are only intratemporal. The second relevant point is the differences in the degrees of consumption externalities of good i. As discussed in Introduction, the previous studies on the commodity-specific consumption externalities assume that consumers put the same importance on the relative consumption of all good, which means θ 1 = θ 2. However, the recent studies of the behavioral and experimental economics suggest that the importance of the relative consumption varies depending on characteristics of goods. Therefore, we consider the case of θ 1 6= θ 2 as well as that of θ 1 = θ 2. Moreover, these 8 When ε =1holds, u t can be written as u t =[c 1 θ 1 1,t (c 1,t /h 1,t ) θ 1 ] γ [c 1 θ 2 2,t (c 2,t /h 2,t ) θ 2 ] 1 γ. This specification of subutility implies that the more individual consumes relative to the average of the society the larger the marginal impact of relative consumption on utility. 8

10 studies typically find that people tend to care more about the relative consumption of the more visible goods like cars than that of the less visible good. Therefore, in what follows, we will examine both cases. The third point is concerned with homotheticity. Notice that (3) is written as u = γ 1 (h 1,t ) c ε 1 ε 1,t + γ 2 (h 2,t ) c ε 1 ε 2,t ε ε 1, where θ 1 (1 ε) ε γ 1 (h 1,t )=γh 1,t, γ 2 (h 2,t )=(1 γ) h θ 2 (1 ε) ε 2,t. Thus the instantaneous utility function is homothetic as to private consumption and γ i (h,t ) represents the relative importance of consumption of good i. The existing literature has often employed the Stone-Geary utility function in which non-linear Engel curves stem from the non-homothetic functional form due to the presence of constant levels of basic consumption. In contrast, in our formulation non-linear Engel curves are derived by endogenous changes in the relative importance of the two goods. Adeficiency of the Stone-Geary utility function is that non-homotheticity ultimately disappears as consumption levels grow. In our formulation, private consumption of each good may not grow proportionally because the relative importance of each commodity continues changing. Therefore, as far as the degree of external effects are not symmetric (i.e. θ 1 6= θ 2 ), the non-linearity of the Engel curves persists even if consumption levels continue growing. Finally, it is to be noted that the relative importance of good i rises as the benchmark consumption h i,t increases if and only if θ i (1 ε) > 0. In this case a rise in the social level of average consumption of good i enhances the relative relevance of private consumption of that good and, hence, the household preferences exhibit conformism (CUJ effect) as to good i. Otherwise, a higher h i,t reduces the relative importance of good i, so that consumption of good i shows anti-conformism (Running Away from the Joneses (RAJ) effect). The welfare of the representative household is represented by: U = Z 0 e ρt ln u t dt, (5) where u t is given by (3) and ρ > 0 is the subjective discount rate. To ensure a positive growth rate in equilibrium, we assume αa 1 > ρ. 9 We use a logarithmic utility in (5) 9 As we will see later, the growth rate of the economy is given by αa 1 ρ in a long-run equilibrium. 9

11 because Ngai and Pissarides (2007) show that if the instantaneous utility function is a CES function, structural change is consistent with balanced growth of the aggregate economy only when the intertemporal elasticity of substitution is equal to one. The budget constraint is: ȧ t = ra t + w t E t, E t = c 1,t + pc 2,t, (6a) (6b) where a t denotes the asset holdings of the household at time t that is equal to K t in equilibrium. We solve the optimization problem of the household in two steps. In the first step, u t is maximized subject to (6b), which yields: pc 2,t c 1,t = µ Ã! 1 γ ε 1 ε θ ph 2 2,t. (7) γ h 1,t θ 1 The ratio of consumption expenditure on good 2 to consumption expenditure on good 1 is a weighted average of the ratio of the weights of each good in the utility function and of the relative prices of habit stock in the economy. Assume θ 1 > 0 and θ 2 > 0. Consider the case where the elasticity of substitution is large so that ε > 1 holds. c 1,t (resp. pc 2,t ). exhibits RAJ when ε > An increase in h 1,t (h 2,t ) reduces the relative expenditure on Therefore, in our specification, the preference of the household When ε < 1 holds, an increase in h 1,t (h 2,t )has apositiveeffect on the relative demand for c 1,t (resp. pc 2,t ). Then, CUJ prevails. Hence, a smaller ε represents a stronger degree of conformity among consumers. The intuition is as follows: As h 1,t increases, consumption of good 1 relative to its benchmark, c 1,t /h 1,t, decreases, which has negative effects on u t. When the elasticity of substitution is large, the household attempts to compensate the negative effects of h 1,t by substituting consumption of good 1 with that of good 2. Then, RAJ prevails. When the elasticity of substitution is small, the household attempts to compensate the negative effects of h 1,t by consuming good 1 more. Then, her preference exhibits CUJ. Most of the existing studies in the literature assume CUJ, rather than RAJ. Thecaseofε < 1 is more related to the literature although we also pay attention to thecaseofε > When the benchmark consumption includes the past consumption of the society, RAJ is sometimes called Falling Behind the Joneses. 10

12 From (6b) and (7), the following equations are derived: ³ c 1,t = E t 1+p 1 ε θ z h 2 θ 2,t /h 1 1 ε 1 1,t χ t E t, pc 2,t =(1 χ t )E t. (8) where z [(1 γ)/(γ] ε. Then, (3) can be written as: θ u t = E t γ(χ t h 1 1,t ) ε 1 ε 1 ε ε +(1 γ) ³(1 χ t )p 1 ε 1 θ h 2,t 2 ε. The second step of the household s problem is to maximize (5) subject to (6a) and the above equation. We then obtain: Ė t =(r ρ)e t. (9) The transversality condition (TVC) is given by lim t a t e rt =0. Since a t = K t holds in equilibrium, TVC is satisfied if lim t K t /K t <r(= αa 1 ). Before closing this subsection, we derive the next two equations for later use. Ã! ċ 1,t ċ2,t ḣ 1,t ḣ 2,t =(1 ε) θ 1 θ 2, (10) c 1,t c 2,t h 1,t ċ 1,t c 1,t = αa 1 ρ +(1 ε) µ 1 c 1,t E t h 2,t à θ 1 ḣ 1,t h 1,t θ 2 ḣ 2,t h 2,t!. (11) We obtain the first equation by differentiating both sides of (7) with respect to t. The derivation of (11) is presented in Appendix.1. Equation (11) shows that when θ 1 > 0 and θ 2 > 0 hold, if ε < (>) 1holds ḣ1,t/h 1,t has a positive (resp. negative) effect on ċ 1,t /c 1,t,whileḣ2,t/h 2,t has a negative (resp. positive) effect on ċ 1,t /c 1,t. This is because when ε < (>) 1holds, CUJ (resp. RAJ) prevails. 2.3 Goods Market Good 1 can be either invested for capital accumulation or consumed and good 2 is a pure consumption good. Thus the goods market clearing conditions are: K t = A 1 K 1,t c 1,t, c 2,t = A 2 K 2,t. The above two equations, together with (1a) and (6b), imply that the aggregate capital evolves according to: K t = A 1 K t E t. (12) 11

13 3 The Aggregate Economy Before examining the dynamics of the two sectors, we examine the evolution of the aggregate variables, K t and E t.define ξ t E t /K t. From (9) and (12), we obtain: ξ t = {ξ t [(1 α)a 1 + ρ]}ξ t. Figure 1 shows the graph of the dynamics of ξ. The path that ξ t converges to zero cannot be equilibrium because K t /K t converges to A 1 and TVC is not satisfied. The path that ξ t tends to + is not equilibrium either because K t converges to zero. Hence, the economy always stay at ξ (1 α)a 1 + ρ andthenwehave: E t =[(1 α)a 1 + ρ]k t, (13) for all t 0. In equilibrium, K t and E t grow at the same constant rate, g αa 1 ρ. The capital income share remains constant at α = rk t /Y t. The interest rate also remains constant. We then obtain the next proposition. Proposition 1 In equilibrium, ξ t remains constant at ξ (1 α)a 1 +ρ over time. K t and E t grow at g αa 1 ρ. The capital income share and the interest rate remain constant. Proposition 1 shows that the aggregate economy satisfies the facts pointed out by Kaldor. Several points should be mentioned. Because of our specification of the technology, the interest rate and the growth rate of K t always remain constant. If there are no sector-specific externalities and then the production function takes a standard Cobb-Douglas form, the transitional dynamics arise and hence the interest rate and the growth rate of K t no longer remain constant over time. Note that the studies about structural change, including Acemoglu and Guerrieri (2008) and Kongsamut et al. (2001), also focus on the steady state equilibrium where the interest rate remains constant and capital stock grows at a constant rate. Therefore, our analysis is in line with the literature. Second, as shown in Carroll et al. (1997, 2000), when the benchmark level of consumption depends on the past consumption, the transitional dynamics arise even if the production technology takes an AK form. Because of the logarithmic utility, the dynamics of the aggregate economy are independent of the benchmark level of consumption and hence no transitional dynamics arise in our model. Finally, the fact that K t and E t grow at the same constant 12

14 rate does not necessarily mean that the two sectors also grow at the same rate and there are no transitional dynamics of each production sector. In fact, the subsequent sections show that under reasonable conditions, the two sectors grow at different rates and hence structural change occurs. Because consumption externalities have no influence on the aggregate economy as mentioned just above, we can examine the direct consequence of consumption externalities on structural change. [Figure 1] 4 The Stationary Equilibrium and Transition Dynamics To examine the long-run behaviors of the two sectors, let us first define x 1,t c 1,t /K t, x 2,t pc 2,t /K t,andη i,t h i,t /K t (i =1, 2). In Appendix.2, we derive the dynamic system of the two sectors and show that the initial value of x 1,t is determined by state variables, h 1,0 and h 2,0. We now define a stationary equilibrium. Definition A stationary equilibrium (SE) is an equilibrium where ẋ 1,t = η 1,t = η 2,t =0holds and c 1,t, c 2,t, h 1,t and h 2,t grow at constant rates. It should be noted that c 1,t, c 2,t, h 1,t,andh 2,t do not necessarily grow at the same rate in an SE. An SE requires only that the grow rates of c 1,t, c 2,t, h 1,t,andh 2,t are constant over time. In the following discussion, we omit time index t from variables that are constant over time when we analyze an SE. When ε =1holds, the benchmark levels of consumption, h 1,t and h 2,t,donot affect the demand for both goods 1 and 2 (see (7)). Consequently, consumption externalities have no influence on x 1,t in equilibrium (see (.2.2a) and (.2.3)).Because we are interested in the effects of consumption externalities, we assume ε 6= 1in the following discussion. Before closing this section, we prove the next proposition concerning the existence of SE. Proposition 2 Suppose ε 6= 1and θ 1 6= θ 2. If there exists an SE, ċ 1,t /c 1,t 6= ċ 2,t /c 2,t must hold in the SE. 13

15 (Proof) See Appendix.3. When θ 1 6= θ 2 holds, there is possibility that the two sectors grow at different rates and hence structural change occurs. In the following discussion, we first consider thecaseofθ 1 = θ 2 and show that structural change does not occur. After that, we proceed to the case of θ 1 6= θ 2 and observe that the presence of consumption externalities is a source of structural change. 4.1 The Case of Symmetric Externalities We first examine the case that consumption external effects are symmetric between the two kinds of goods. Let us assume that θ 1 = θ 2 θ ( 0 θ < 1). In this case, x 1,t and x 2,t are expressed as functions of η 1 and η 2 from (.2.1) and (.2.3). After we eliminate x 1,t and x 2,t from (.2.2b) and (.2.2c) using (.2.1) and (.2.3), we obtain the dynamic system that is composed of two differential equations. The SE values of η 1 and η 2 are: η 1 (1 α)a 1 + ρ φ 1 φ 1 + g 1+p 1 ε z θ(1 ε), η 2 (1 α)a 1 + ρ φ 2 + g where [φ 2 (φ 1 + g )p ε z/φ 1 (φ 2 + g )] 1 1+θ(ε 1). 11 φ 2 p ε z θ(1 ε) 1+p 1 ε z θ(1 ε), Figure 2 describes the transitional dynamics. 12 The phase diagrams show that the SE is stable and the economy converges to the SE. [Figure 2] Because η i remains constant at η i (> 0), we have ḣ1,t/h 1,t = ḣ2,t/h 2,t = g. From (.2.3), we can see that x 1,t also becomes constant and strictly positive. Hence, we find that ċ 1,t /c 1,t = ċ 2,t /c 2,t = g. We then obtain the following proposition. Proposition 3 Suppose ε 6= 1and θ 1 = θ 2 = θ 0. The economy converges to the SE where ċ 1,t /c 1,t = ċ 2,t /c 2,t = ḣ1,t/h 1,t = ḣ2,t/h 2,t = g holds. When θ 1 = θ 2 holds, regardless of the presence of consumption externalities (whether θ is equal to zero or not), growth is asymptotically balanced in the sense that the 11 Setting η 1,t = η 2,t =0in (.2.2b) and (.2.2c) and using (.2.3) yield the SE values of η 1 and η The η 1 =0locusisgivenbyη 2 =(p 1 ε z) τ [φ 1 {(1 α)a 1 + ρ}/(φ 1 + g ) η 1 ] τ η τ+1 1. The η 2 =0locus is η 1 =(p 1 ε z) τ [φ 2 {(1 α)a 1 + ρ}/{p(φ 2 + g }) η 2 ] τ η τ+1 2, where τ θ 1 (1 ε) 1. 14

16 two sectors grow at the same asymptotic rate. 4.2 The Case of Asymmetric Externalities Let us now turn to the case of θ 1 6= θ 2. In this case, there is no SE where c 1,t and c 2,t grow at the same rate as shown in Proposition 2. However, the next proposition shows the existence of an SE where the two sectors grow at the different (asymptotic) rates. Proposition 4 (i) Suppose (1 ε)(θ 1 θ 2 ) > 0. There exists an SE where the followings hold: ċ 1,t c 1,t = ḣ1,t h 1,t = αa 1 ρ g, (14a) ċ 2,t c 2,t = ḣ2,t h 2,t = K 2,t K 2,t = 1+(ε 1)θ 1 1+(ε 1)θ 2 g ĝ (0 < ĝ <g ), (14b) ˆx 1 =(1 α)a 1 + ρ, ˆη 1 = φ 1[(1 α)a 1 + ρ] φ 1 + g, ˆη 2 =0, (14c) c 2,t h 2,t = A 2k 2,t η 2,t = φ 2 +ĝ φ 2. (14d) where ˆx 1 lim t x 1, ˆη 1 lim t η 1, and ˆη 2 lim t η 2. (ii) Suppose (1 ε)(θ 1 θ 2 ) < 0. There exists an SE where the followings hold: ċ 2,t c 2,t = ḣ2,t h 2,t = K 2,t K 2,t = αa 1 ρ g, (15a) ċ 1,t c 1,t = ḣ1,t h 1,t = 1+(ε 1)θ 2 1+(ε 1)θ 1 g ĝ (0 < ĝ <g ), (15b) ˆx 1 =0, ˆη 1 =0, ˆη 2 = φ 2[(1 α)a 1 + ρ] p(φ 2 + g, (15c) ) c 1,t = x 1,t = φ 1 + ĝ, (15d) h 1,t η 1,t φ 1 where ˆx 1 lim t x 1, ˆx 1 lim t η 1, and ˆη 2 lim t η 2. (Proof) See Appendix.4. When θ 1 6= θ 2 holds, growth is nonbalanced in the sense that the two sectors grow at the different (asymptotic) rates. We emphasize that because Proposition 1 holds, the aggregate economy exhibits balanced growth in the sense that K t and E t grow as the same rate. 15

17 Inthecaseof(1 ε)(θ 1 θ 2 ) > 0, both consumption and output of the two goods grow at different rates in the SE, while output of Sector i (= 1, 2) grows at thesamerateofk i,t. 13 However, this does not mean that Sector 2 produce nothing because the economy approaches the SE only asymptotically. At all points in time, Sector 2 produces positive amounts and grows at a positive rate, ĝ. At the same time, output of Sector 2 relative to that of Sector 1 decreases because Sector 1 grows at a higher rate than Sector 2. When ε < 1 holds, the condition for nonbalanced growth is θ 1 > θ 2. Equation (11) shows that when CUJ prevails, the growth rate of consumption of good 1 (ċ 1,t /c 1,t ) is positively affected by the growth rate of the benchmark consumption of good 1 (ḣ1,t/h 1,t ), whereas it is negatively affected by the growth rate of the benchmark consumption of good 2 (ḣ2,t/h 2,t ). An increase in the degree of consumption externalities, θ 1 enhances this positive effect. By contrast, an increase in θ 2 heightens the negative effect. Consequently, when θ 1 > θ 2 holds, c 1,t grows at a higher rate, which results in a higher growth in Sector 1. Then, nonbalanced growth of the two sectors arises. In Subsection 4.3, it will be observed that the shares of capital and labor allocated to Sector 2 gradually decreases along the transitional dynamics and they converge to zero as times goes. Inthecaseof(1 ε)(θ 1 θ 2 ) < 0, consumption of good 2 grows at a higher rate than that of good 1 in the SE. From the first equation of (15c), we know x 2,t = (1 α)a 1 + ρ (= pc 2,t /K t = A 1 K 2,t /K t ), which means that K 1,t /K t = α ρ/a 1 > 0 and K 2,t /K t =1 α+ρ/a 1 > 0. Wethenhave K 1,t /K 1,t = g = K 2,t /K 2,t. Because capital good is produced only in Sector 1, the demand of capital good for Sector 1 increases at the same rate as the demand for good 2 so as to provide capital input in the production of good 2. Thus, outputs in the two sectors grow at the same rate. We will show in the next subsection that the income share of each sector converges to a finite level, while x 1,t = c 1,t /K t converges to zero during the transition. The following proposition summarizes the stability properties in the SEs. Proposition 5 If ε 6= 1and θ 1 6= θ 2, thenthesesweconsiderarelocallystable. (Proof) in Appendix Note that ˆx 1 =(1 α)a 1 + ρ implies x 2,t =0so that K 2,t /K t =0and K 1,t /K t =1. Hence, K 1,t /K 1,t = g > ĝ = K 2,t /K 2,t holds in the SE. 16

18 4.3 Transitional Dynamics We first consider the case where ε < 1 and θ 1 > θ 2 hold. For simplicity, we normalize θ 2 to zero. Thus, the economy is characterized by (.2.2a) with θ 2 =0and (.2.2b). Figure 3 illustrates transitional dynamics in (x 1, η 1 ) space whereby an economy moves from three different initial points given K [Figure 3] AsshowninFigure3,wehaveẋ 1 < (>) 0in the region above (resp. below) the graph of η 1 = x 1,whereas η 1 < (>) 0in the region above (resp. below) η 1 =0locus. If the economy is initially located on Point A, the economy with lower levels of K 0 and h 1,0 monotonically converges to the SE. Because x 1,t monotonically increases to its SE value, expenditure and output shares of good 1 monotonically increase while those of good 2 monotonically decrease. From (1a) and (1b), we have L 2,t /L 1,t = K 2,t /K 1,t. Given A i K i (i =1, 2) production technology, the shares of capital and labor allocated to Sector 1 (Sector 2) increase (resp. decrease) as well. 15 Thus, structural change steadily progresses along the transition. We emphasize that while structural change progresses, the aggregate economy exhibits balanced growth in the sense that K t and E t grow at the same constant rate because Proposition 1 holds. Figure 3 provides the further implication. From Proposition 1, we know that if there are two countries that have the same technology and the same preference, the growth rates of outputs in the two countries become exactly the same. If the initial values of K t are the same in addition, the output levels of the two countries also become exactly the same over time and hence the two countries exhibit exactly the same performance of aggregate economy. However, if the initial values of h 1,t are different, the two countries follow the different patterns of structural change. Consequently, the two countries have different expenditure and production shares of the two goods, and different shares of capital and labor allocated to the two sectors at all points in time although these two countries have exactly the same 14 The ẋ 1,t =0locusisgivenbythethreelines: x 1 =0,x 1 =(1 α)a 1 + ρ, and η 1 = x 1. We restrict our attention to the region 0 x 1 (1 α)a 1 + ρ given (.2.1). The η 1 =0locusisgiven from (.2.2b) by η 1 = φ 1 x 1/(φ 1 + g ). Because g is strictly positive, the slope of η 1 =0locus is less steeper than that of η 1 = x Note that expenditure shares of good 1 is given by c 1,t/E t = x 1,t/{(1 α)a 1 +ρ} =1 pc 2,t/E t and production shares of good 1 is given by Y 1,t/Y t =[αa 1 ρ + x 1,t]/A 1 =1 py 2,t/Y t. 17

19 performance of the aggregate economy. The economy starting at Point A and the economy starting at Point C could be such a case. Our model suggests that the presence of consumption externalities might be a source of the divergent patterns of structural change across countries. The phase diagram for the case of θ 2 > θ 1 is depicted in Figure 4. Behaviors of the consumption expenditure share between the two sectors are opposite to those in Figure 3: the consumption expenditure share of the first sector ultimately converges to zero in this setting. [Figure 4] We finally show that the dependence of the benchmark consumption levels on the past consumption is crucial for the divergent patterns of structural change across countries. Consider the case of φ 1 =+ where the consumption externalities are only intratemporal. In equilibrium, we have h 1,t = c 1,t and η 1,t = x 1,t. From (11), (12), and (13), we obtain: ẋ 1,t = g (1 ε)θ 1 n1 1 (1 ε)θ 1 n1 o x 1,t (1 α)a 1 +ρ x 1,t (1 α)a 1 +ρ ox 1,t. Figure 5 presents the graph of the above equation. We can see in the case of φ 1 = +, the two countries endowed with the same level of K 0 havethesameinitialvalue of x 1,t. 16 Thus, the two countries that exhibit the same performance of aggregate economy follow exactly the same pattern of structural change. [Figure 5] 5 Extensions Judging from the empirical plausibility of the model analysis, our discussion so far has two drawbacks. First, our model assumes that the relative price between the two goods stays constant over time. Although this assumption has been often employed in the literature, it does not fit the observed long-run patterns of relative prices. Second, the standard empirical approach to the relation between growth and structural 16 Taking into account h 1,t = c 1,t and η 1,t = x 1,t, the initial value given by (.2.3) becomes x 1,0 = [(1 α)a 1 + ρ] 1+zp 1 ε θ x 1 (ε 1) θ 1,0 K 1 (ε 1)

20 change has used three-sector models rather than two sector models. Those studies focus on the structural transformation between the agricultural, manufacturing and service sectors. As is well known, the income share of manufacturing sector first increases and then declines as the economy develops. In addition, it has been confirmed that in the process of economic development, the income share of agricultural monotonically decreases, while that of the service sector monotonically rises. In this section, we show that simple extensions can make the baseline model fit better to the observed facts mentioned above. 5.1 Endogenous Relative Price A simple way to endogenize the relative price is to follow Rebelo s (1991) treatment of a two-sector AK growth model. As emphasized by Felbermayr and Licandro (2005), Rebelo s formulation is the simplest way to endogenize relative price in the endogenous growth setting. 17 The production function of the first sector is the same as before. The pure consumption good sector has a production function such that Y 2,t = A 2 K α 2,t(B 2,t L 2,t ) 1 α, B 2,t = K2,t λ /L2,t, 0 < λ < 1. Here, we assume that the marginal contribution of external effects created by capital in the second sector diminishes with the level of K2,t.This assumption means that diffusion of the average production knowledge is relatively weaker in the pure consumption good sector than in the general good sector. The resulting social production function of the second sector becomes Y 2,t = A 2 (K 2,t ) β, β = α +(1 α) λ < 1. and the thus the private rate of return to capital and the competitive real wage are respectively given by r t = αa 1 = p t αa 2 (K 2,t ) β 1, (16) K 1,t (K 2,t ) β w t = (1 α) A 1 = p t (1 α) A 2. (17) L 1,t L 2,t 17 It is to be noted that Rebelo (2001) considers a two-sector economy where one sector produces pure investment goods and the other produces pure consumption goods. Therefore, the balanced growth characterization is simpler than our model with two types of consumption goods. Also, Rebelo (2001) does not assume the presence of production externalities. 19

21 Consequently, the relative price satisfies p t = A 1. (18) β 1 A 2 (K 2,t ) Note that the aggregate income is given by Y t = Y 1,t + p t Y 2,t = A 1 (K 1,t+ K 2,t )= A 1 K t. Thus the aggregate production function is again expressed by an AK form. Using (7) and (18), we find that the relative share of consumption spending for two goods is expressed as µ Ã! p t c 2,t 1 γ ε 1 ε θ p t h 2 µ Ã 2,t 1 γ ε = = c 1,t γ γ h 1,t θ 1 h 2,t θ 2 h 1,t θ 1! 1 ε µa1 A 2 1 ε K (1 ε)(1 β) 2,t. As before, the economy is in the stationary state, when x 1,t = c 1 /K t,x 2.t = p t c 2,t /K t and η i,t = h i,t /K t (i =1, 2) stay constant over time. Provided that ε 6= 1, the balanced-growth condition for the general good sector is the same as before: ċ 1,t c 1,t = ḣ1,t h 1,t = K t K t = αa 1 ρ g. From (18) and (19) the balanced-growth rates of c 2,t, h 2,t and p t satisfy: (19) ċ 2,t c 2,t = ḣ2,t h 2,t = β K 2,t K 2,t, ṗ t p t =(1 β) K 2,t K 2,t. Hence, keeping in mind that ċ 2,t /c 2,t + ṗ t /p t = K 2,t /K 2,t in SE, we find that the balanced-growth rate of K 2,t and P t are respectively given by K 2,t K 2,t = 1+θ 1 (ε 1) 1+(ε 1) (βθ 2 +1 β) g = eg, (21) ṗ t = (1 β)[1+θ 1 (ε 1)] p t 1+(ε 1) (βθ 2 +1 β) g. Equation (21) means that sign (g eg )=sign [(1 ε)(θ 1 βθ 2 + β 1)]. Therefore, if (1 ε)(θ 1 βθ 2 +1 β) > 0, the first sector ultimately dominates the second sector so that the steady state value of x 2.t is zero. If (1 ε)(θ 1 βθ 2 +1 β) < 0, then the x 1,t ultimately converses to zero. As a consequence, provided that ε < 1 so that conformism prevails, it may hold that g > eg,evenifθ 2 is higher than θ 1. Conversely, in the case of anti-conformism, i.e. (1 ε) θ i < 0, can be higher than g, even if θ 2 < θ 1. Otherwise, the steady-state characterization is essentially the same as the model with constant relative price. 20

22 It is also to be pointed out that endogenizing the relative does not alter the behaviors of employment and value-added shares. relative price is fixed, it holds that From (1a) and (1b) when the L 2.t L 1,t = p ty 2,t Y 1,t = K 2.t K 1.t. It is easy to see that the above conditions hold in the case of endogenous relative price as well. Therefore, the behaviors of the employment and value-added shares exactly correspond to the behavior of the relative level of capital stock. 5.2 A Three-Sector Model We have so far examined the structural change in a two-sector model and shown that different degrees of consumption externalities of goods as well as the elasticity of substitution between the two goods are important determinants of structural change. In this section, we consider a three-sector model to illustrate how the presence of commodity-specific consumption externalities can generate structural changes in a more general situation. We focus on differences in the pattern of sectoral growth rather than differences in patterns of structural change across countries so that we consider the case that the consumption externalities are only intratemporal (φ i = + ). There are three sectors: Agricultural (a), Manufacturing (m), and Services (s). Following Kongsamut et al. (2001), we assume that manufacturing goods can be used either for consumption or for investment, while agricultural goods and services are consumed. The production technology remains the same in the basic model. Taking manufacturing goods as a numeraire, the relative prices of agricultural goods and services are p a = A m /A a and p s = A m /A s, respectively. The subutility of the households in (3) is replaced by: u t = ε 1 γ a c a,t ε + γ m ³c m,t h θ ε 1 m ε m,t + γ s ³c s,t h θ s s,t ε 1 ε ε 1 ε. When people have positional concerns, they place the most importance on the relative consumption of service good, while the least importance on the relative consumption of agricultural good. For example, higher education is more visible than car, and food is least visible good. Hence, we assume that θ a < θ m < θ s in this three-sector model. 21

23 From the optimality conditions for the instantaneous consumption choice, we obtain consumption expenditure on goods a and s relative to consumption expenditure on good m: µ à p j c ε j,t γj pj h θ! j 1 ε j,t = c m,t γ m h θ ω m j, for j = a, s. (22) m,t This conditions correspond to (7) in a two-sector model. The consumption spending is given by e E t =(ω a +1+ω s )c m,t. The instantaneous felicity can be expressed as a linear function of the consumption spending, e E t : u = e E t X i=a,m,s γ i à ω i,t h θ i i,t p i ( P i=a,m,s ω i,t)! ε 1 ε ε ε 1, where p m = ω m =1. We obtain the standard Euler equation, ėe t =(r ρ) e E t. Using the commodity market clearing conditions, K t = A m K m,t c m,t,c a,t = A a K a,t, and c s,t = A s K s,t and the relative prices, p a = A m /A a and p s = A m /A s, theresourceconstraintisgivenbyy t = p a Y a,t + Y m,t + p s Y s,t = E e t + K t = A m K t. Thus the growth rate of the economy is give by eg t = K t /K t = ėe t / E e t = αa m ρ and total expenditure is E e t =[(1 α)a m + ρ] K t, as derived in the previous two-sector model. The aggregate economy with three sectors reconcile with Kaldor facts. We now turn to the transformation of industrial structure. Here, we focus on the case where φ i =+, so that h i,t = c i,t = c i,t holds in the symmetric equilibrium. It is shown that even in this limit case, our three-sector model exhibits empirically plausible pattern of transformation among the industries. In the general case where 0 < φ i < + (i = a, s, m), dynamics of habit stocks, h i,t, may enhance reality of our model in the quantitative sense. Consumption expenditure on goods a and s in (22) can be rewritten as: ξ p j c j,t = ψ j c j m,t, (23) where ψ j ε/(1+θj (ε 1)) γ j /γ m and ξj [1 + θ m (ε 1)]/[1 + θ j (ε 1)], for j = a, s, and θ a =0. Aggregate consumption expenditure is given by: ξ ψ a c a ξ m,t + c m + ψ s c s m,t = E e t. Thus, c m,t / E e t = ψ a ξ a c ξ 1 m,t a +1+ψ s ξ s c ξ m,t s 1 1 > 0, which implies that when aggregate consumption expenditure increases, consumption expenditure on manufacturing goods increases. 22

24 We next analyze the behavior of consumption and output shares among three sectors. As shown in Proposition 2, when ε 6= 1and θ i 6= θ j for i, j = a, m, s, the growth rates can be different across sectors. Moreover, we have shown in Proposition 4 that when CUJ (RAJ) prevails, the growth rates of consumption and output of more (resp. less) visible good are higher in the SE than those of less (resp. more) visible good. Let us define the share of consumption on each good in aggregate expenditure as S a,t p a c a,t / E e t,s m,t c m,t / E e t, and S s,t p s c s,t / E e t. Using (23), the consumption shares are rewritten as S a,t = ψ a c ξ m,t a/ Et e, S m,t = c m,tet e,and S s,t = ψ a c ξ m,t s/ Et e. The effects of an increase in aggregate expenditure on the shares of consumption of agricultural good and services are, respectively given by: S a,t e E t S s,t e E t =( 1) ξ (1 ξ a )+(ξ s ξ a )ψ s c s 1 ψ a c ξ m,t a m,t ee t 2 c m,t e E t, = ξ (θ s θ m )+θ s [1 + θ m (ε 1)] ψ a c a 1 ª ψ s (1 ε) c ξ m,t s m,t ee t 2 [1 + θ s(ε 1)] c m,t E e. t When higher education is more visible than car and food is least visible good, it would be plausibly assumed that θ a < θ m < θ s, which implies that ξ a < 1 < ξ s, provided that ε < 1. Because c m,t / e E t > 0 as discussed above, as the economy grows, the consumption share for agricultural good decreases monotonically while the consumption share for service good increases monotonically, provided that ε < The effect of an increase in aggregate expenditure on consumption share for manufacturing good is: S m,t E e = t θ m (1 ε)ψ a (θ s θ m )(1 ε) 1+θ s (ε 1) ψ s c m,t ξ s ξ a cm,t ξ a ee 2 t c m,t e E t. The share of consumption on manufacturing good exhibits a hump-shaped development pattern and when c m,t = {θ m [1 + θ s (ε 1)ψ a ]/[(θ s θ m )φ s ]} 1/(ξ s ξ a ) it is maximized. Figure 6 shows a numerical example of consumption shares of the three sectors. [Figure 6] The output shares of the three sectors are defined as S Ya p a Y a /Y, S Ym (c m + K)/Y, and S Ys p s Y s /Y. Fromtheresourceconstraint,Y t = E e t + K t, these shares 18 As discussed in the previous sections, the Engel curves persists due to endogenous relative consumption concern. 23