Macroeconomic implications of conspicuous consumption: A Sombartian dynamic model

Journal of Economic Behavior & Organization 67 (2008) 322–337

Macroeconomic implications of conspicuous consumption: A Sombartian dynamic model Katsunori Yamada ∗ Japan Society for the Promotion of Science and Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043, Japan Received 27 September 2005; received in revised form 19 June 2007; accepted 19 June 2007 Available online 23 June 2007

Abstract This paper presents a dynamic general equilibrium model in which consumers have status preference. I investigate the manner in which capital accumulation is impeded by conspicuous consumption a` la Corneo and Jeanne [Corneo, G., Jeanne, O., 1997a. Conspicuous consumption, snobbism and conformism. Journal of Public Economics 66, 55–71]. Following the literature, social norms are given as either bandwagon type or snob type. I show that when the economy is characterized by a bandwagon type social norm, capital accumulation exhibits interesting patterns. Those patterns include, for example, an oscillating convergence path: the rise of the economy feeds its decay through conspicuous consumption and that decay suppresses conspicuous consumption and engenders prosperity, as predicted by Sombart [Sombart, W., 1912. Liebe, Luxus und Kapitalismus, Deutscher Taschenbuch Verlag, Germany (reprinted 1967)]. © 2007 Elsevier B.V. All rights reserved. JEL classification: E10; Z13; E30 Keywords: Status preference; Conspicuous consumption; Capital accumulation

1. Introduction This paper presents an investigation into how capital accumulation is impeded by conspicuous consumption in the framework of a dynamic general equilibrium model. In the model, motivation of conspicuous consumption is related to the signaling of status, which is consistent with the original definition of conspicuous consumption by Veblen (1899). The framework of the signaling game is given by the microeconomic study of Corneo and Jeanne (1997a) (henceforth, CJ), and status utility is generated by rank utility. It is shown that the capital accumulation path might exhibit oscillating convergence as well as polarization caused by multiple equilibria when the social norm is of the bandwagon type. It has been well recognized that consumer decisions might not be explained adequately by merely using utility derived from consumption. Rather, economic agents have been considered to derive additional utility from their social status, as highlighted, for example, by Smith (1759), Hume (1739) and Veblen (1899). Many empirical analyses support ∗

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the hypothesis of status preference through tests of Veblen’s views of pecuniary emulation. Microeconomic evidence includes that presented by Clark and Oswald (1996), McBride (2001), Blanchflower and Oswald (2004) and Saito et al. (2005), among others. Basmann et al. (1988), Hayes et al. (1988, 1992) and Slottje (1992) provide evidence for Veblen’s view of conspicuous consumption with data from the U.S., North America, European countries, and Japan, respectively. Frijters and Leigh (2005) examine conspicuous leisure in the U.S. and find strong effects of conspicuous leisure on work hours. Furthermore, quantitative approaches also exist to test the validity of status preference, such as by Abel (1990), Gali (1994) and Bakshi and Chen (1996), who argue that observed asset price volatility can be explained through inclusion of the motivation to keep up with the Joneses. Thus, it seems that status preference has many consequences at the macroeconomic level as well as on individual decision problems. Along with the empirical studies mentioned above, theoretical studies have examined how external effects of status preference affect the equilibrium and especially the economic growth rate. These studies propose that analyses with status preference might engender markedly different outcomes from those obtained through analyses in which utility depends only on consumption.1 In contrast to previous macroeconomic studies of the literature such as by Cole et al. (1992), Konrad (1992), Zou (1994), Corneo and Jeanne (1997b) and Futagami and Shibata (1998), who investigate the effects of status preference among consumers on the economic growth rate, this paper analyzes the disturbing effects of status preference through conspicuous consumption on the accumulation of capital. I use a model that has the same property as the well-known Solow model to extract the effects of relative concern with clarity. This is the same strategy as that used by Konrad (1992), Zou (1994) and Corneo and Jeanne (1997b), who based their analyses on the well-known Ramsey economy.2 The intuition underlying the analysis is as follows. When status utility is obtained by conspicuous consumption, this type of preference might cause serious distortion in capital accumulation because conspicuous consumption behavior, by Veblen’s definition, works effectively when the resource is devoted to economically meaningless activities. Hence, as argued by Frank (1985), conspicuous consumption might cause serious inefficiencies in the form of downward distortions in the demand for non-conspicuous goods. In turn, this distortion might badly affect the accumulation of an economically valuable good, such as productive capital.3 A simple and tractable model with conspicuous consumption behavior is constructed to clarify the dynamic implications of this distortion. The model has the same properties as the textbook Solow model if there is no status utility. I introduce status preference by adopting the framework of CJ; fundamentally, this paper can be considered as a first attempt to extend the static work of CJ into a two-sector general equilibrium model with dynamics.4 I note here that this paper is a first attempt in the dynamic analysis of conspicuous consumption because full analytical solutions require the consideration of a special case in which the preference function for substantial expenditure converges to linearity. Due to this condition, I will neglect a level effect of changing wealth distribution on economic decisions by consumers. It will be shown that the conspicuous consumption motivation can cause rich types of capital accumulation paths in the model. I propose that the model has consequences related to a prediction by Sombart and a convergence controversy summarized, for example, in Galor (1996). The rise of the economy might feed decay through conspicuous consumption and the decay might suppress conspicuous consumption, and bring prosperity, as predicted by Sombart. Simultaneously, quests for social status can engender polarization of two economies with the same economic fundamentals, initial conditions, and social norms. This paper is organized as follows. The next section describes the model. Explanations of CJ economy are included. Section 3 analyzes the equilibrium paths. Section 4 contains discussions of the relevance of the model to the modern economy. The last section concludes the paper.

1

See, for comprehensive surveys and discussions of theoretical studies of social status (Hayakawa, 2000; Easterlin, 2001; Hollander, 2001). Although the propensity to save in the model is constant, as it is in the Solow model, here is a micro-foundation to explain it. 3 It is intuitively plausible that wealth accumulation is impeded by human vanity; history shows that the rise and fall of aristocratic lineages might be explained by rat races in a quest for ever greater social status. For a comprehensive survey of human vanity and desire for social status, especially among the aristocratic class and wealthy merchants, see the third chapter of Sombart. 4 In one feature of the model, I illustrate the status good as a marketable good, whereas Cole et al. (1992, 1995) stress that analyses of status utility are meaningful when utility from some non-market action has consequences related to market decisions. Nevertheless, I adopt the present strategy following work by Bagwell and Bernheim (1996), Corneo and Jeanne (1997a,c) and Becker (1991), in which the demand for conspicuous goods is determined in the market in accordance with some social norms. 2

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2. The model Time is discrete and extends to infinity. The economy is one in which people derive utility from social status as well as from the consumption of intrinsic goods and granting bequests. The economy is populated by a continuum of agents who belong to lineages; each agent lives for one period and has one offspring.5 Each individual of the lineage has index i ∈ [0, 1]. Within the lineage, generations are connected by bequest motives. All lineages in this economy are homogeneous except for the amount of bequests inherited from ancestors, although it is assumed that no lineage is endowed with the same level of wealth in the initial period. Aside from inherited wealth, agents are endowed with one unit of labor that is inelasticly supplied to production sectors and earns the wage income. 2.1. Consumers and social status Each agent’s utility is given as Ui,t = u(qi,t ) + νi,t , where q is called substantial expenditure, which comprises consumption ci,t and bequest ωi,t+1 . Also, ν reflects a social reward from conspicuous consumption and is determined endogenously. As is apparent in the equation, total utility consists of u(q) and ν in an additively separable form. With respect to utility from consumption and bequests, I assume that standard assumptions hold: it is strictly concave and satisfies the Inada condition for each argument. Following CJ, consumption of the conspicuous good is limited for only one unit: the conspicuous good is indivisible, and agents cannot buy more than one unit. In the dynamic model, it is also assumed that agents cannot bequest conspicuous goods to offspring and that they perish within the period.6 The information structure of this economy is as follows. Agents cannot see the level of wealth, the amount of consumption, and the amount of bequest of others; the relative wealth position of each lineage and economic decisions are private information. On the other hand, the way in which wealth is distributed in the population is socially known. I assume that a firm that produces conspicuous goods has enough information to distinguish those who will be rich from those who will not. Information superiority of the firm is attributable to its infinite life span: agents live only one period whereas the firm lives forever. Lastly, purchasing behavior of conspicuous goods is observable among consumers and denotes δi,t by a dummy variable taking value 1 if agent i buys a conspicuous good at time t; otherwise, it is zero. With this information structure, conspicuous consumption behavior provides, in a Bayesian way, some value to agents in the economy with special social norm. The social norm adopted in CJ and this paper is called wealth rank utility: agents in front of the norm are assumed to extract utility from their relative wealth position in the economy.7 Because they cannot directly compare their wealth rank with each other because of imperfect information, they must signal their wealth to extract status utility. For that reason, they might spend for conspicuous goods, which is by definition a meaningless but observable activity. That is, agents spend for conspicuous goods merely because they want to advertise their wealth and extract rank utility rather than utility from consumption (and bequests). Define here the rank utility function a(·), which maps the wealth rank in the society into rank utility.8 Next, denote Ωt (i) : [0, 1] → [0, 1] by the function mapping family index i (at the beginning of time t) into wealth order at the time. If a lineage with index k ∈ [0, 1] is the wealthiest in the society, then Ωt (k) = 0. Hereafter, I consider the mapped family index j(= Ωt (i)) in the analysis below. Because the wealth rank of a family is unobservable, status utility ν is obtainable as the expected rank utility, which is conditional on the observable purchasing behavior of the conspicuous good (δ = 0, 1) as νt = νt (δi,t ) = E[a(j)|δi,t ]. 5

(1)

Hence, no population growth occurs in the model. From the standpoint of Veblen’s view, it is essential to conspicuous consumers that they themselves purchase conspicuous goods rather than inherit them from their ancestors. Abstention from conspicuous consumption by a member of a lineage will give an impression of family decay. 7 Section 2.1 in CJ for a more detailed discussion. 8 It is assumed here that the rank utility function is continuous on the interval [0, 1] and monotonously decreasing, with a finite lower and upper bound. 6

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By this formulation, ν(·) reflects the collective value aspect of conspicuous consumption under the condition that the wealth position of each lineage is unobservable. Now the optimal behavior of agents can be depicted. First, the budget constraint of an agent reads as ci,t + ωi,t+1 + δi,t pt ≤ wt + (1 + rt )ωi,t , where prices (the relative price of conspicuous good, pt ; the wage, wt ; and the rental price of capital, rt ) are given competitively. Define here yi,t = wt + (1 + rt )ωi,t by the total wealth for agent i at time t. Agents determine their purchasing behaviors with respect to the conspicuous good such that the decisions are optimal, given their inferences related to their social status. With the additively separable utility function, this condition can be written as νt (1) − νt (0) = E[a(j)|1] − E[a(j)|0] ≥ u(yi,t ) − u(yi,t − pt ).

(2)

In the analysis below, st ≡ v(1) − v(0) is designated as the signaling value of conspicuous goods at time t. The value of conspicuous consumption, in turn, can be determined in the Bayesian way after purchasing decisions of all lineages as to the conspicuous good are made. Specifically, through Eq. (1), st is given by the difference of expected rank utility between the conspicuous consumers and others: st = σ(Jt ) =

 Jt 0

a(j) dj − Jt

1 Jt

a(j) dj

1 − Jt

1 = Jt (1 − Jt )



J

{a(j) − a¯ } dj,

(3)

0

where a¯ is the average rank utility over the whole population and Jt is the number of conspicuous consumers.9 σ(·) is called the signaling value function, and the signaling value of conspicuous consumption depends only on the number of conspicuous consumers. Here are three remarks on the signaling value function. First, the social norm in the model economy is regulated by the shape of σ(·). When dσ(·)/dJ < 0 holds, the market demand for a conspicuous good decreases because others are purchasing the same good; consequently, the economy is “snobbish”. On the other hand, when dσ(·)/dJ > 0 holds, the demand increases because others are buying the same good. Therefore, the economy is a “bandwagon”. Following the literature, I will consider two social norm types in the dynamic analyses below: snobbish and bandwagon. Second, the shape of the signaling value function is determined by the shape of the rank utility function through Eq. (3).10 That is, any continuous and differentiable function σ(·) can be rationalized by a rank utility of the form of a(x) = a¯ + (1 − 2x)σ(x) + x(1 − x)σ  (x), where x denotes the wealth rank of a lineage. It is also noteworthy that, if the rank utility function is time invariant, then the signaling value function is unchanged through time, which I assume throughout the paper; the social norm is assumed to be fixed for a society. In the application below, the rank utility function, the social norm, is assumed to be quadratic. Consequently, the assumption gives linear σ(·) functions (see the r.h.s. of Eqs. (14) and (15)). Third, the domain of σ(·) is considered. The function is defined on the open interval of J ∈ (0, 1): the case of the pooling equilibrium (either J = 0 or J = 1) for the conspicuous good market can be excluded. When J = 0 and no conspicuous consumers exist, the economy merely reduces to the Solow regime (see Eq. (13)). Hence, there is no need to define the signaling value function when J = 0. Furthermore, it will be shown that Lemma 2 in the snobbish economy excludes the case in which everyone buys the conspicuous good; in the bandwagon economy, J might converge asymptotically to one but never equals one. All in all, σ(·) is defined on J ∈ (0, 1).

9

Throughout this paper I assume that st is strictly positive and finite. As to how the social norm is regulated by the rank utility, CJ establishes that the snobbish society is attributable to the convexity of a(·), whereas the bandwagon economy corresponds to the case in which a(·) is concave. For an explanation of the intuition behind the argument, refer to CJ (61). 10

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2.2. Producers The economy has two production sectors: sector 1 produces the intrinsic good and sector 2 produces the conspicuous good. In sector 1, the technology is given by a homogeneous function with two inputs, capital and labor, as Yt = F (Kt , Nt ). Competitive prices are given as rt =

∂F (Kt , Nt ) , ∂Kt

wct =

∂F (Kt , Nt ) , ∂Nt

where rt is the rental price of capital and wct is the wage paid in sector 1. The production of conspicuous goods requires only a labor force.11 For that reason, it is written as Yˆ t = G(nt ), where nt is the labor force supplied to sector 2 and the competitive wage is given as wt − pt G (nt ) = 0, p

p

where wt is the wage paid in sector 2 and pt is the price of the conspicuous good. In equilibrium, Yˆ t coincides with the number of conspicuous consumers Jt because each lineage can buy the conspicuous good with one unit. 2.3. Equilibrium p

Now the equilibrium of this economy can be defined. The equilibrium at time t is the set of prices (wt , wct , rt , pt , st ) p and the set of allocations (cj,t , ωj,t+1 , δj,t ) and (nt , Nt , Kt ) so that (i) given (wt , wct , rt , pt , st ), (cj,t , ωj,t+1 , δj,t ) p c solves the consumers’ maximization problems; (ii) given (wt , wt , rt , pt ), (nt , Nt , Kt ) solves the firms’ maximization problems; (iii) the market for labor, capital and the conspicuous goods markets clear. Here functional forms must be specified to obtain explicit solutions. For production technology of the intrinsic good, I assume that β

1−β

Yt = θKt Nt

,

where θ > 0 is a productivity parameter and β is the capital share. As for sector 2, I assume that Yˆ t = nt . p

Assuming competitiveness and an interior solution, the first-order condition of the firm in sector 2 gives pt = wt . p In turn, the competitive market also implies that wt = wct (≡ wt ) must hold in equilibrium. As for the utility from 12 substantial expenditure, I assume that η

1−α α u = u(q(cj,t , ωj,t+1 )) = (cj,t ωj,t+1 ) ,

where α ∈ (0, 1)

and

η ∈ (0, 1).

Here, the bequest motive is given by the impure altruism form adopted by Banerjee and Newman (1993), among others, and u(·) is strictly concave in q. From the specification described above, the prices are obtained as  β−1  β  β ∂Yt Kt ∂Yt Kt Kt rt = = θβ , wt = = θ(1 − β) , pt = wt = θ(1 − β) . (4) ∂Kt Nt ∂Nt Nt Nt In addition, the signaling value, st , is given by the signaling value functions discussed later. 11 The assumption that production in sector 2 depends only on labor is not essential to the following analysis, but it dramatically simplifies the analysis. 12 Here I might consider that ω is a term representing a value from investment: agents are now assumed to live forever and to derive substantial utility from today’s consumption and from consumption plans onward. With this interpretation, the details of a full-fledged inter-temporal decision problem with rational expectations can be avoided.

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Market-clearing conditions of labor, capital and the conspicuous good market are given respectively as Nt + nt = 1,  Kt = ωj,t dj,

(5)

Jt = nt .

(7)

(6)

Furthermore, from Eq. (2), the conspicuous consumption goods market clears such that the marginal gain from conspicuous consumption and the marginal gain of substantial expenditure are equal for the lineage with index j = J 13 : νt (1) − νt (0) = st = u(yJ,t ) − u(yJ,t − pt ).

(8)

In order to obtain maximum analytical scope in the problem, I must consider the special case of lim η → 1. By this condition and the functional form of utility from the substantial expenditure, it is readily apparent that u(yj,t ) − u(yj,t − pt ) converges to (1 − α)1−α αα pt irrespective of the lineage index j. It is not needed to know explicitly the level of total wealth that lineage J has: yJ,t . Thereby, the equilibrium condition given by Eq. (8) reduces to (1 − α)1−α αα pt = st = σ(Jt ).

(9)

As described above, I can choose the shape of σ(·) through an appropriate choice of the rank utility function. In this paper, the signaling value function σ(·) is assumed to be linear on (0, 1) (equivalently, the rank utility function is −1 assumed to be quadratic). Moreover, here I designate a new function of S(N) such that S(Nt ) = [(1 − α)1−α αα ] σ(Jt ) for all Nt = 1 − nt = 1 − Jt ∈ (0, 1).14 This is the signaling value function defined on N with market clearing conditions. Hence, Eq. (9) reduces to the following equilibrium relationship between capital and the number of workers in sector 1:  β Kt S(Nt ) = pt = θ(1 − β) . (10) Nt From Eq. (10), Nt (and Jt ) will be determined for a given level of capital as Nt = N(Kt ) = 1 − Jt . With the impure altruism utility function and the condition that η → 1, the following optimal allocation rules for each lineage can be obtained: cj,t = (1 − α){(1 + rt )ωj,t + wt − δj,t pt }, δj,t = 1

iff j ∈ [0, Jt ],

δj,t = 0

ωj,t+1 = α{(1 + rt )ωj,t + wt − δj,t pt },

iff j ∈ (Jt , 1].

(11)

Eqs. (4)–(7), (10) and (11) determine the equilibrium of the economy at time t. Furthermore, the dynamics of capital are also obtained because the utility function allows the aggregation of the optimal allocation rule of ω for all lineages j ∈ [0, 1]: the social resource constraint is given as    ωj,t+1 dj = α{(1 + rt )ωj,t + wt − δj,t pt } dj = α{(1 + rt ) ωj,t dj + wt − pt Jt }. Because Kt ≡



ωj,t dj, it is obtained that β

Kt+1 = Ψ (Kt ) = α{(1 + r(N(Kt ), Kt ))Kt + w(N(Kt ), Kt )−p(N(Kt ), Kt )J(Kt )} = α{Kt + θKt N(Kt )1−β }. (12) 13

In the economy, every agent can afford to buy a conspicuous good and the solution is always interior: from the individual budget constraint, it can be seen that the budget is always satisfied because pt = wt in equilibrium. By this property in the general equilibrium model, I can concentrate on the analysis of the accumulation path of the aggregate level of capital. This exhibits a contrast with CJ in which the corner solutions are possible. 14 S(·) is continuous, linear, bounded on (0, 1) and positive by construction.

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It is noteworthy that if no status utility pertains in the economy so that N = 1 for all t, Eq. (12) reduces to β−1

Kt+1 = ψ(Kt ) = α{(1 + θβKt

β

β

)Kt + θ(1 − β)Kt } = α{Kt + θKt }.

(13)

Apparently, the property of the dynamics in Eq. (13) is essentially identical to that of the Solow model. That is, Eq. (13) has a unique globally stationary point of K∗ = (αθ/(1 − α))1/1−β because dKt+1 /dKt > 0, d2 Kt+1 /dKt2 < 0, limKt →0 (dKt+1 /dKt ) = ∞, limKt →∞ (dKt+1 /dKt ) = α ∈ (0, 1) and (dKt+1 /dKt )|K=K∗ = α + β − αβ ∈ (0, 1) when α ∈ (0, 1) and β ∈ (0, 1). On the other hand, when the conspicuous consumption motive is present in society, capital accumulation is disturbed by two causes and the dynamics are governed by Eq. (12). One cause is the outflow of the labor force from sector 1 to sector 2, which results in the reduction of productivity of capital. The other is the resource devoted to economically meaningless activity: conspicuous consumption. Before closing the discussion in this section, I would note that when I do not consider the special case of η → 1, then it is necessary that the dynamic evolutions of the total wealth of all the lineages in the economy be tracked because Jt and the other endogenous variables depend on the wealth distribution at time t. That proposition is, of course, analytically intractable. Because this study is intended to show analytically that conspicuous consumption motivation has the potential to affect the capital accumulation path in interesting ways, I proceed with this assumption throughout the paper. I note here that an important implication in CJ will become completely invalid when I consider the case of η = 1. In this situation, the situation renders it ambiguous who in the economy actually spends for the conspicuous good, although J can still be determined, as in the case of η → 1. This is because the willingness to purchase the conspicuous good becomes the same regardless of the wealth level. That situation, in turn, indicates that agents who buy the conspicuous good cannot advertise that their wealth positions are high. Nonetheless, I might be able to proceed innocuously in the analyses with η = 1 because I assume that the conspicuous good firm has all the information related to the wealth of agents; I can assume that the firm sells only to the rich. The firm has information superiority because it lives forever and has some screening skills that are acquired. That is, although all agents might have the same level of willingness to buy the conspicuous good and can indeed afford to do so, those who purchase are those who are qualified by the firm. Although this strategy might seem to be an arbitrary device to allocate conspicuous goods among agents, it has a great merit in that J and the other endogenous variables are calculable without information of wealth distribution, as in the case of η → 1. 3. Dynamics This section presents a description of the patterns of the equilibrium dynamics of the economy, given by Eqs. (12) and (13), for the snobbish economy (where dS(·)/dN > 0) and the bandwagon economy (where dS(·)/dN < 0). 3.1. Snobbish economy In the snobbish economy, dS(·)/dN > 0 (equivalently, dσ(·)/dJ < 0) holds because an agent feels reluctant to abstain from buying the conspicuous goods that he sees others buying. A linear signaling value function defined on N with zero intercept, S(N) = bN (b > 0 and N ∈ (0, 1)), provides analytical solutions.15 Equilibrium conditions for conspicuous consumption given by Eq. (10) are rewritten as the following:  β Kt bNt = θ(1 − β) . (14) Nt It can be readily seen from Eq. (14) that Nt ∈ (0, 1) is uniquely determined for a given Kt . Notice also that when Kt is sufficiently high so that Nt in Eq. (14) is greater than or equal to 1, there is no inner solution for the conspicuous good ¯ by K ¯ = {K|limN→1 S(N) = market and capital accumulation is governed by Eq. (13) rather than Eq. (12). Here define K 15

Qualitative implications obtained below will be unchanged as long as S(N) is increasing monotonously on (0, 1) and Nt is uniquely determined for every Kt from Eq. (10).

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¯ = {b/(θ(1 − β))}1/β . This is the level of K, above which the demand for the limN→1 θ(1 − β)Kβ N −β } so that K conspicuous good disappears. A simple algebra shows that Nt and pt are strictly increasing in Kt in the snobbish economy. This provides the next lemma. Lemma 1. The demand curve of the conspicuous good is downward sloping in the snobbish economy. The equilibrium can be interpreted as follows. When K is small, the price of the conspicuous good is low so that the opportunity cost of conspicuous consumption is small. A smaller opportunity cost directly implies a smaller signaling value, which in the snobbish economy implies more numerous conspicuous consumers in equilibrium. As K increases ¯ the price of the conspicuous good rises so that the number of conspicuous consumers must decrease to in (K, K), compensate for higher opportunity cost. Because the demand curve of the conspicuous good is downward sloping in the snobbish economy, an important proposition proposed by CJ applies. Proposition 2.2 in CJ argues that the demand for the conspicuous good becomes negligible when the price of the conspicuous good is sufficiently low and the number of the conspicuous consumers approaches asymptotically to full population, in the economy where the demand curve of the conspicuous good is downward sloping. The following lemma replicates Proposition 2.2 in CJ. Lemma 2. There exists a price pˆ > 0 such that the demand is negligible if p < p. ˆ Proof. See the proof in CJ.



In this study, such a low price of conspicuous goods will be realized when K approaches zero.16 Denote K as the level of capital below which the demand for the conspicuous good is nil. Now I consider the phase diagram of K. Eq. (12), in a relevant range, is readily apparent as strictly concave (see ¯ Eq. (13) determines the evolution of capital. Finally, I obtain Appendix A).17 On the other hand, when K ∈ / (K, K), ¯ that Eqs. (12) and (13) are continuous at K. The next proposition summarizes possible patterns of capital accumulation in the snobbish economy. Proposition 1. Two types of wealth accumulation paths are generated in the snobbish economy, depending on the value of b. For both cases, the economy has a unique steady state that is globally stable. In one case, conspicuous behavior disappears in the steady state, and in the other case, the steady state is characterized by a lower level of capital and a positive number of conspicuous consumers. Proof. See the proof in Appendix A.



Figs. 1 and 2 illustrate how the level of capital evolves in the snobbish economy. Realization depends on four parameters: (α, β, b, θ). When the marginal decrease of the signaling value of the conspicuous good for an additional conspicuous consumer is sufficiently high, b ≥ θ 1/1−β (1 − β)(α/1 − α)β/1−β , conspicuous consumption remains in the steady state. On the other hand, when b < θ 1/1−β (1 − β)(α/1 − α)β/1−β holds, the steady state is that without conspicuous consumers. The result indicates that, in the snobbish economy, the disturbing effects of conspicuous consumption are weak; capital evolution is, despite the presence of conspicuous consumers, monotonic, as in the textbook Solow model. 3.2. Bandwagon economy The analysis of the bandwagon economy is rather complicated because the demand for the conspicuous good might not be determined uniquely. To see this with clarity, define the signaling value function in the bandwagon economy as

16 It is readily apparent that the convergence of K to zero occurs more rapidly than that for N. Consequently, prices are finite and p become close to zero when K approaches to zero from Eq. (4). 17 All of the appendicies are available on the JEBO website.

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Fig. 1. Phase diagram in the snobbish economy: no conspicuous consumers in the steady state.

S B (N) = c − dN, where c > 0, d > 0, and N ∈ (0, 1). Hence, Eq. (10) now reads as  β Kt c − dNt = θ(1 − β) . Nt Fig. 3 depicts the equilibrium condition in the labor market given by Eq. (15).

Fig. 2. Phase diagram in the snobbish economy: the steady state with conspicuous consumers.

(15)

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Fig. 3. Labor market equilibrium in the bandwagon economy.

The figure suggests that when K is sufficiently small, there will be a unique intersection in N ∈ (0, 1) for Eq. (15). Intuitively, as capital increases and r.h.s. of Eq. (15) shifts up for any level of N, another intersection will be generated. Hence, multiple equilibria are generated in the bandwagon economy. ¯ B , at which S B (N) and θ(1 − β)Kβ N −β are tangent. For later In the bandwagon economy, there is a point of K B ¯ reference, K is calculated here as ¯B = K

βc(1+β)/β . (1 − β)1/β (1 + β)(1+β)/β θ 1/β d

¯ B , there are no inner solutions for Eq. (10); consequently, Eq. (13) governs the dynamics. For a higher level of K than K Note here that, in contrast to the snobbish economy, the number of conspicuous consumers jumps to zero discontinuously ¯ B , N will jump to one, which suggests that Eqs. (12) and in the bandwagon economy. That is, when K overcomes K B ¯ (13) are discontinuous at K The other cases, in which a unique equilibrium is always realized in the labor market for Eq. (10), might occur. That is, if S B (N) is tangent with θ(1 − β)Kβ N −β for some given K at the level of N > 1, it is seen that there are no multiple equilibria. I impose a parametric restriction to exclude the equilibrium dynamics without multiple equilibria to concentrate on the more interesting cases. Condition. ¯B = c β N < 1, d (1 + β) ¯ B is the equilibrium value of N when the economy has a capital level of K ¯ B . The derivation of the condition where N is explained in Appendix B. With that assumption, two equilibrium schedules are obtained for the determination of N, as is apparent from Fig. 3. In one equilibrium schedule, an equilibrium level of N(J) decreases (increases) with K. Call this schedule modest. It is β generated when K is sufficiently high: denote KB such that KB = {KB |limN→1 S B (N) = limN→1 θ(1 − β)(KB ) N −β }. This is the level of K above which the modest schedule is generated. ¯ B ) (and For the modest schedule, it is easy to see that the price of the conspicuous good increases with K ∈ (KB , K hence, with J) from Eq. (4). That is, the demand curve of the conspicuous good is upward sloping in this regime.

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Fig. 4. Demand curve of the conspicuous goods in the bandwagon economy.

Indeed, the possibility of an upward sloping demand curve is suggested by CJ for the bandwagon economy. As can be readily inferred, Eqs. (12) and (13) are continuous at KB and Kt+1 is strictly decreasing in Kt for the modest schedule (see Figs. 5–7). For the other equilibrium schedule, the equilibrium level of N(J) increases (decreases) with K. Call this schedule ¯ B ). As for the the ruin schedule. From Fig. 3, it can be deduced that the ruin schedule is generated when K ∈ (0, K demand curve in the ruin schedule, it might seem ambiguous that the demand curve is also upward sloping, as in the modest schedule, because both K and N rise in Eq. (4). The next lemma, however, shows that the demand curve is upward sloping in this regime as well. Lemma 3. The demand curve of the conspicuous good in the bandwagon economy is upward sloping. Proof. See the proof in Appendix C.



Fig. 4 portrays the demand curve of conspicuous goods in the bandwagon economy. As apparent there, the curve consists of two schedules that have a kink at the point of J¯ B defined as ¯B =1− J¯ B ≡ 1 − N

cβ . d(1 + β)

In the bandwagon economy, the rationale of the multiple equilibria is given as follows. When the number of the conspicuous consumers increases, the price of the conspicuous good, which is equal to the opportunity cost of conspicuous consumption, rises (see Lemma 3). This higher opportunity cost will be compensated by a higher signaling value driven by the larger number of conspicuous consumers. The economy then will attain the equilibrium. On the other hand, when the number of the conspicuous consumers decreases, the price of the conspicuous good, the opportunity cost of conspicuous consumption, declines. The signaling value is also reduced because of the smaller number of conspicuous consumers. Therefore, the economy will attain the equilibrium. Both of these situations are possible, so multiple equilibria are generated in the bandwagon economy. Now I consider the phase diagram of capital. In the bandwagon economy, multiple equilibria of conspicuous ¯ B ]. On this regime, agents choose, consumption are generated when the economy has capital level of K ∈ [KB , K taking the level of capital at the top of the period as given, which of two equilibria will be realized. The determination

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Fig. 5. Phase diagram in the bandwagon economy: an oscillating convergence path.

¯ B , ∞) whereas the economy depends on the expectation of agents. The economy is governed by Eq. (13) when K ∈ (K B is on the ruin schedule when K ∈ (0, K ). In the following analyses of the phase diagrams in (Kt+1 , Kt ) plane, I call the line for the ruin schedule the ruin line, and the line for the modest schedule the modest line. The following proposition classifies possible capital accumulation paths in the bandwagon economy. Proposition 2. There are three types of capital accumulation paths in the bandwagon economy under the assumption of cβ/d(1 + β) < 1.18 For one case, a unique steady state, which might not be locally stable, exists (Fig. 5). For another ¯ B ] (Fig. 6). The local stability of the case, there are two steady states, one of which is K∗ . The other lies in K ∈ [KB , K latter steady state is ambiguous. In the third and final case, there are two steady states, one of which is K∗ . The other ¯ B ] (Fig. 7). is locally unstable and lies in K ∈ [KB , K Proof. See the proof in Appendix D.



Fig. 5 depicts the case in which there is a unique steady state. When the steady state is stable as in the figure, then the economy can exhibit oscillating convergence to Ka along the modest line. This is the situation illustrated in chapter 3 of Sombart: the rise of the economy feeds the decay, and that decay subsequently brings prosperity. However when the economy follows the ruin line, it shrinks monotonously. Figs. 6 and 7 show cases in which two steady states generate. One steady state is K∗ ; there are no conspicuous consumers in the steady state. Regarding the case of Fig. 6, when the other steady state of Kb is unstable, it might be seen that no conspicuous consumers exist in the steady state even if there are some in transition. On the other hand, when Kb is stable (not shown), it can be deduced that if conspicuous consumers exist in the initial period, conspicuous consumers must exist not only in transition but also in the steady state. The economy will be caught in a poverty trap once capital becomes less than KB . 18

I exclude the origin, the poverty trap, from the definition of the steady state in this paper.

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Fig. 6. Phase diagram in the bandwagon economy: polarization of the economy (a).

Finally, the case given by Fig. 7 is examined. Because there is a steady state Kc that is always unstable in this case, it can be suggested that the economy will converge to K∗ if the initial level of capital is greater than Kc . In this case, conspicuous consumers disappear even if some exist in transition. Otherwise, the economy will be caught in the trap.

Fig. 7. Phase diagram in the bandwagon economy: polarization of the economy (b).

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Among all the cases, it is most interesting that the economy might have an oscillating convergence path along ¯ B = cβ/d(1 + β) < 1 and Kt+1 − Kt |K →K¯ B < 0 are satisfied. These the modest line.19 This will happen when N t conditions require that c be smaller than d in the signaling value function and that the level parameter in the production function θ should be small (see Appendix D). Because these parameters are mutually independent, I might say that the situation of oscillating convergence will be indeed plausible. In that case, the rise of the economy feeds the decay through conspicuous consumption, and the decay suppresses conspicuous consumption and engenders prosperity in the next period. Sombart predicted this situation. Polarization of two economies with identical economic fundamentals, initial levels of capital, and social norms might occur. The realization depends on households’ expectations. Cole et al. (1992) argue that two economies with identical economic fundamentals and initial conditions can have different growth rates when social norms of the two societies are different. The novelty here is that even if two economies have the same social norm, they can polarize. The results will have theoretical consequences on the convergence controversy, as summarized in Galor. I close this section by briefly discussing social welfare in the case where there are multiple equilibrium paths. Constructing adequate social welfare functions is indeed far beyond the scope of this paper. Nonetheless, the social welfare function imposing equal weights to all the agents in the economy will be the simplest one and facilitates discussion. In the bandwagon economy, two paths of capital accumulation can be generated and the realization depends on the expectations of agents in the model. However, as to social welfare measured using an equally weighted social welfare function, it is easy to see that the modest schedule provides higher welfare than the ruin schedule. Then, it can be suggested that coordination failure occurs if the economy evolves along the ruin schedule when an alternative path of the modest schedule pertains. To see this, merely note that the aggregate social rank utility is given as a¯ , irrespective of the number of conspicuous consumers: aggregate social rank utility is same in the ruin schedule and in the modest schedule. Therefore, utility from the substantial expenditure distinguishes the two schedules. It is readily apparent that the modest schedule dominates the ruin schedule in terms of the substantial expenditure because, in the modest schedule, the stock of capital becomes higher and the number of workers in sector 1 is larger. 4. Discussion Above theoretical results show that when the social norm is of bandwagon type, conspicuous consumption seriously changes the pattern of capital accumulation from that of the Solow model. This outcome is indeed consistent with Hirschman (1984) and Basu (1989), who argue from philosophical points of view that the bandwagon effect in consumption is important, especially in the modern economy.20 For further illustrations, here I will relate the results to anecdotal evidence that might be thought of as macroeconomic consequences of conspicuous consumption.

Example 1. The first example is from an anthropological study of Rao (2001) on rural India villages. Marriot and Inden (1977) suggest that in rural India, personhood is defined entirely in terms of one’s relationships to others. For those individuals in those areas, it might be said that the bandwagon effect should be strong. Rao then investigates why very poor households in rural India spend large sums on celebrations and festivals. The study shows that their behaviors are explainable by their quest for higher social status within their villages. This anecdotal evidence might explain why underdeveloped regions in India have remained underdeveloped for a long time: poor households in rural India spend as much as six times their yearly incomes on celebrations. As a result, capital accumulation will be impeded seriously. This situation might be explained using the ruin line or by an oscillating convergence path along the modest line to a steady state with conspicuous consumers and lower levels of capital.

19

Mino (2007) shows that the oscillating convergence paths can be generated when we introduce the effect of keeping up with the Joneses into an overlapping-generations model. 20 Mason (1998) is an excellent and comprehensive survey of the conspicuous consumption hypothesis. See chapter 9 of Mason for a discussion of the contemporary importance of bandwagon effects.

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Example 2. A second example is that of inefficient allocations of resources by Japanese firms.21 Some economists argue that cooperate governance of Japanese firms was weak.22 Especially during the bubble period, Japanese firms bought up Old Masters’ paintings at high prices.23 Because art will not function as capital, this could be conceptualized as conspicuous consumption by Japanese firms. Instead, their resources would have been better used for R&D activities to enhance productivity or for capital accumulation. As is well known, Japan’s economic growth stagnated during the 1990s, which has come to be referred to as the lost decade. It might then be plausible that conspicuous consumption by firms engendered inefficient allocation of resources and caused business instability. An oscillating convergence path along the modest line might demonstrate this situation. 5. Concluding remarks In this paper, I described a macroeconomic model of capital accumulation with conspicuous consumption behaviors. The analyses should be regarded as a first attempt at investigating the macroeconomic implications of conspicuous consumption because I consider the special case where the felicity function for the substantial expenditure converges to linearity. Nonetheless, the results obtained in the paper are rich and suggest novel implications on the literature of status preference theory. Especially if the social norm is given by the bandwagon type, it is shown that the economy will be characterized by non-monotonic evolution of capital as well as by a multiplicity of equilibrium paths. The former result states that a Sombartian economy is depicted with the model: the rise of the economy feeds decay through conspicuous consumption, and decay suppresses conspicuous consumption and brings prosperity. Hence, I might say that conspicuous consumption is a cause of business fluctuation. The latter outcome indicates the possibility of polarization of two economies with identical economic fundamentals, initial conditions and social norms. Hence, the results imply that the literature of status preference theory will affect the convergence controversy of Galor. Acknowledgements The author is grateful to Akihisa Shibata, Koichi Futagami and two anonymous referees for their many helpful comments. The author also would like to thank Oded Galor, Giacomo Corneo, Kazuo Mino, Takashi Unayama, Takuma Kunieda, Keisuke Okada and seminar participants at Osaka University and Brown University. Any remaining errors are the sole responsibility of the author. This research was partially supported by a Ministry of Education, Culture, Sports, Science and Technology (MEXT) Grant-in-Aid for the 21st Century COE Program “Interfaces for Advanced Economic Analysis”. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.jebo.2007.06.003. References Abel, A.B., 1990. Asset prices under habit formation and catching up with the Joneses. American Economic Review 80, 43–47. Bagwell, L.S., Bernheim, D.B., 1996. Veblen effects in a theory of conspicuous consumption. American Economic Review 86, 349–373. Bakshi, G.S., Chen, Z., 1996. The spirit of capitalism and stock-market prices. American Economic Review 86, 133–157. Banerjee, A.V., Newman, A.F., 1993. Occupational choice and the process of development. Journal of Political Economy 101, 274–298. Basmann, R., Molina, D., Slottje, D., 1988. A note on measuring Veblens’s theory of conspicuous consumption. Review of Economics and Statistics 70, 535–540. Basu, K., 1989. A theory of association: social status, prices and markets. Oxford Economic Papers 41, 653–671. Becker, G., 1991. A note on restaurant pricing and other examples of social influences on pricing. Journal of Political Economy 99, 1109–1116. Blanchflower, D., Oswald, A.J., 2004. Well-being over time in Britain and the USA. Journal of Public Economics 88, 1359–1386.

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Slottje supports the existence of conspicuous consumption motivation in Japan with aggregate data for the pre-bubble period of 1974–1988. See, for discussion of cooperate governance of Japanese firms (Osano, 2001). For example, it is well known that Yasuda Insurance Inc. bought a van Gogh painting for 5.8 billion yen.

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