Macroeconomic Implications of Conspicuous Consumption: A Sombartian Dynamic Model

Abstract

This chapter 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 à la Corneo and Jeanne (J Public Econ 66:55–71, 1997a). Following the literature, social norms are given as either bandwagon type or snob type. I then 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 (Liebe, Luxus und Kapitalismus (reprinted 1967). Deutscher Taschenbuch Verlag, Munchen, 1912).

The original article first appeared in the Journal of Economic Behavior & Organization 67, 322–337, 2008. A newly written addendum has been added to this book chapter.

Appendices

Appendices

1.1 Appendix 1

This appendix proves Proposition 1. When $K \in (\underline{K},\bar{K})$ the wealth evolution in the snobbish economy is governed by

$$\displaystyle{ K_{t+1} =\alpha \{ K_{t} +\theta K_{t}^{\beta }N(K_{ t})^{1-\beta }\}, }$$

(6.16)

and

$$\displaystyle{ \mathit{bN}_{t} =\theta (1-\beta )\left (\frac{K_{t}} {N_{t}}\right )^{\beta }. }$$

(6.17)

From these two equations, it is obtained that

$$\displaystyle{K_{t+1} =\alpha \{ K_{t} +\theta ^{ \frac{2} {1+\beta } } \frac{(1-\beta )} {b} K_{t}^{ \frac{2\beta } {1+\beta } }\}.}$$

As is apparent from that equation, this is a strictly concave function of K _t .

It is convenient to investigate the characteristics of dynamics in the snobbish economy globally by inquiring into two points. One is the relative position of $\bar{K}$ to K ^∗. When $\bar{K} < K^{{\ast}}$ holds, K ^∗ must be the steady state, although there might be another steady state in $K \in (\underline{K},\bar{K})$. Indeed, because (6.16) is strictly concave and, from the continuity argument of Eqs. (6.12) and (6.13) at $\bar{K}$, there might be another steady state in $K \in (\underline{K},\bar{K})$ if $K_{t+1} - K_{t} < 0$ holds at $\underline{K}$. The other is the condition to generate the steady state in N ∈ (0, 1). If there is a steady state of N in (0, 1), then it must be the one with $K \in (\bar{K},\underline{K})$.

For the first condition, it is simply obtained that $\bar{K} < K^{{\ast}}$ holds if and only if

$$\displaystyle{ b <\theta ^{ \frac{1} {1-\beta }}(1-\beta )( \frac{\alpha } {1-\alpha })^{ \frac{\beta }{ 1-\beta }}. }$$

(6.18)

As for the other condition, the steady state level of N, I impose the steady state condition on (6.16) and (6.17) to obtain the following.

$$\displaystyle\begin{array}{rcl} N& =& \left (\frac{1-\alpha } {\alpha } \right )^{ \frac{\beta }{ 1-\beta }}\theta ^{ \frac{1} {1-\beta }}\frac{1-\beta } {b} {}\\ & & \equiv \Lambda (b) {}\\ \end{array}$$

The last condition indicates that the number of steady states with N less than one is, at most, one. That is, when $\Lambda (b) < 1$, there will be a steady state of capital, $K \in (\underline{K},\bar{K})$. This condition excludes the case in which there are two steady states in $K \in (\underline{K},\bar{K})$ when condition (6.18) is violated.

The last question is whether or not $\Lambda (b) < 1$ and $\bar{K} < K^{{\ast}}$ simultaneously hold and two steady states are generated: one is for $K \in (\underline{K},\bar{K})$ and the other is K ^∗. It is, however, readily apparent that (6.18) and $\Lambda (b) < 1$ are contradictory, so that, in any case, only one steady state generates globally. To sum up, there is a steady state without conspicuous consumption, K ^∗, when $b <\theta ^{ \frac{1} {1-\beta }}(1-\beta )( \frac{\alpha }{ 1-\alpha })^{ \frac{\beta }{ 1-\beta }}$ holds (Fig. 6.1). Otherwise, conspicuous consumers exist in the steady state with a lower level of capital (Fig. 6.2). Q.E.D.

1.2 Appendix 2

The assumption in Sect. 3.2 is derived using the following three conditions:

• (i) The condition for tangency

  $$\displaystyle{d =\theta \beta (1-\beta )\left (\frac{\bar{K}^{B}} {\bar{N}^{B}}\right )^{\beta }(\bar{N}^{B})^{-1},}$$

• (ii) The condition for equilibrium

  $$\displaystyle{c - d\bar{N}^{B} =\theta (1-\beta )\left (\frac{\bar{K}^{B}} {\bar{N}^{B}}\right )^{\beta },}$$

  and

• (iii) The condition for inner solution

  $$\displaystyle{\bar{N}^{B} < 1.}$$

  Q.E.D.

1.3 Appendix 3

The demand curve is upward sloping in the modest schedule. Consider then the ruin schedule, in which the equilibrium level of N increases with K, and assume that p decreases with K. In this case, J decreases as K increases. The decline of J implies a drop in the signaling value of conspicuous consumption in the bandwagon economy. This, however, contradicts the equilibrium condition given by Eq. (6.9). For Eq. (6.9) to be satisfied, p must increase with K in the bandwagon economy. Hence, it is shown that the demand curve is upward sloping in the ruin schedule, too. Q.E.D.

1.4 Appendix 4

This appendix proves Proposition 2. To analyze how K evolves in the bandwagon economy, it is sufficient to investigate three aspects: (i) the sign of $K_{t+1} - K_{t}$ in the ruin schedule when K is close to zero; (ii) the sign of $K_{t+1} - K_{t}$ at $\bar{K}^{B}$; and (iii) the relative position of $\bar{K}^{B}$ to K ^∗.

As to the first point, it is shown that

$$\displaystyle\begin{array}{rcl} K_{t+1} - K_{t}\mid _{K_{t}\rightarrow 0}& =& \alpha K_{t} -\theta K_{t}^{\beta }N_{ t}^{1-\beta }- K_{ t}\mid _{K_{t}\rightarrow 0} {}\\ & =& -(1-\alpha )K_{t} - \frac{b} {\beta (1-\beta )}N_{t}^{2}\mid _{ K_{t}\rightarrow 0} {}\\ & <& 0. {}\\ \end{array}$$

This indicates that the ruin line lies below the 45^∘ line near the origin. Notice also that the positive capital level is ensured by Eq. (6.11).²³ Furthermore, it can be seen that the ruin line is a monotonously increasing curve because N increases with K and the price of conspicuous good and the number of conspicuous consumers decline with K along the ruin schedule.

With respect to the second matter, remembering that the ruin line and the modest line intersect at $\bar{K}^{B}$, a little algebraic treatment leads to

$$\displaystyle\begin{array}{rcl} K_{t+1} - K_{t}\mid _{K_{t}\rightarrow \bar{K}^{B}}& =& \alpha K_{t} -\theta K_{t}^{\beta }N_{ t}^{1-\beta }- K_{ t}\mid _{K_{t}\rightarrow \bar{K}^{B}} {}\\ & =& \bar{K}^{B}\left [\theta \frac{\bar{K}^{B}} {N_{t}} ^{\beta -1} - (1-\alpha )\right ] {}\\ & =& \bar{K}^{B}\left [\frac{c^{\frac{\beta -1} {\beta } }\theta ^{ \frac{1} {\beta } }(1-\beta )^{\frac{1-\beta } {\beta } }} {(1+\beta )^{\frac{\beta -1} {\beta } }} - (1-\alpha )\right ]. {}\\ \end{array}$$

Hence, $K_{t+1} - K_{t}\mid _{K_{t}\rightarrow \bar{K}^{B}} \geq 0$ if and only if

$$\displaystyle{ \frac{c^{\frac{\beta -1} {\beta } }\theta ^{ \frac{1} {\beta } }(1-\beta )^{\frac{1-\beta } {\beta } }} {(1+\beta )^{\frac{\beta -1} {\beta } }} - (1-\alpha ) \geq 0, }$$

(6.19)

and vice versa.

Finally, as to the third point, $\bar{K}^{B} > K^{{\ast}}$ holds if and only if

$$\displaystyle\begin{array}{rcl} \bar{K}^{B} = \frac{\beta c^{\frac{1+\beta } {\beta } }} {(1-\beta )^{\frac{1} {\beta } }(1+\beta )^{\frac{1+\beta } {\beta } }\theta ^{ \frac{1} {\beta } }d} > \left ( \frac{\alpha \theta } {1-\alpha }\right )^{ \frac{1} {1-\beta }} = K^{{\ast}}.& &{}\end{array}$$

(6.20)

Under the condition that $\frac{c\beta } {d(1+\beta )} < 1$, (6.19) and (6.20) do not hold simultaneously.

Now the proposition will be best understood by graphical expositions. Figures 6.5–6.7 illustrate three possible patterns of capital accumulation. Figure 6.5 is a case in which $\bar{K}^{B} \geq K^{{\ast}}$ and $K_{t+1} - K_{t}\mid _{K_{t}\rightarrow \bar{K}^{B}} < 0$ hold. Hence there is a unique steady state (denoted as K ^a in the figure). The local stability of the steady state depends on the slope evaluated at K ^a. The figure depicts a stable case.

Figure 6.6 depicts a case where $\bar{K}^{B} < K^{{\ast}}$ and $K_{t+1} - K_{t}\mid _{K_{t}\rightarrow \bar{K}^{B}} < 0$ hold, whereas Fig. 6.7 illustrates the case where $\bar{K}^{B} < K^{{\ast}}$ and $K_{t+1} - K_{t}\mid _{K_{t}\rightarrow \bar{K}^{B}} \geq 0$ hold. For both cases, there is a stable steady state of K ^∗, in which conspicuous consumption disappears.

For the case of Fig. 6.6, there is a steady state denoted as K ^b in the figure. The stability of K ^b is ambiguous (the figure shows an unstable case). Finally, the case given by Fig. 6.7 shows a steady state K ^c, which is always unstable. Q.E.D.

Addendum: A Caveat

Based on Clark et al. (2008)’s review of the empirical literature of social preferences, recent trends in the field continue to grow.²⁴ Theoretical contributions also continue to emerge, including Ravn et al. (2010) and Di Pace and Faccini (2012). In this note, I suggest a caveat regarding how we should interpret various types of social preferences; in particular, I have found that some studies use inappropriate definitions of preference externality related to consumption.

A classic view of social preference is found in Veblen (1899). Veblen started his discussion with a perspective of pecuniary emulation among citizens. When citizens compete with others in terms of monetary achievements, they must explicitly demonstrate that they are superior to their peers in that regard. Thus, what matters when researchers think about the effects of pecuniary emulation on economic decisions is the information structure they introduce to their analyses. Also, it is important to remember that preference externality can be defined over several variables, including consumption, asset holdings, and income levels. In the literature on happiness, these three variables significantly affect happiness levels with the expected signs for coefficients.

Preference externality defined over consumption is the most convenient for theoretical analyses but it requires some caution. It is natural for researchers to introduce such effect in a reduced form utility function, such as $U(c,\bar{c})$ where c is own consumption and $\bar{c}$ is reference consumption. Behind this specification is the information structure in which citizens can recognize their peers’ consumption levels. While such an assumption seems acceptable, one common misconception in the theoretical literature of social preferences is that $U(c,\bar{c})$ captures the effect of conspicuous consumption. It indeed captures only consumption externality, which includes not only the influence of consumption of others in the current period, but also the effects of habit formation due to past own consumption and future aspirations regarding own consumption levels.

So, what is conspicuous consumption? It actually appears when preference externality is defined over asset/saving levels. It is possible for researchers to introduce preference externality via saving as $U(c) + V (s,\bar{s})$, where c is own consumption, s is own saving, and $\bar{s}$ represents reference saving. The functional form of V would be the author’s choice, but a recent experimental study by Ono and Yamada (2012) shows that a difference specification such as $V (s,\bar{s}) = v^{1}(s -\bar{ s})$ fits the experimental data better than a ratio specification expressed as $V (s,\bar{s}) = v^{2}(s/\bar{s})$. Here again, an information structure plays a crucial and implicit role, which is that economic agents can observe the asset/saving levels of peers perfectly. Obviously, this is a strong assumption and it will make theoretical analyses more straightforward when introduced.

In real life, it is plausible that asset/saving levels of peers constitute private information. When information is private regarding $\bar{s}$, economic agents somehow must inform their peers that they are indeed better off than others. In this case, a greater amount of standard consumption, which would be visible to others, does not work as a device to advertise wealth levels. This is because even when the amount of standard consumption is greater than that of others, it is possible that people’s asset/saving levels, over which social preference is defined, are smaller than their peers’. Such an observation can be explained by the differences in propensity to consume among people.

According to Veblen, this is where conspicuous consumption plays its role. By definition, people cannot derive utility from consuming conspicuous goods. Put differently, the definition of conspicuous goods includes that they do not provide the consumer with any value by consuming them. So, why do consumers spend money on seemingly useless goods? Veblen suggested that it is because they can signal their level of wealth in such a way that peers’ inferences about their asset levels will be valid. A neoclassical economics theory validates this, as the marginal utility from standard consumption is smaller for greater amounts of consumption levels, just as Corneo and Jeanne (1997a) and Yamada (2008) showed. A typical example of consumption of conspicuous goods would be the use of aristocratic names. As Sombart argued, dropping aristocratic family names in conversation would signal wealth levels but would not increase utility levels by itself. Obviously, the signaling effect of conspicuous consumption is quite different from consumption externality, which is expressed as $\bar{c}$ in $U(c,\bar{c})$.