Abstract
This contribution provides a game theoretical derivation of market demand for status goods as a function of the level and distribution of income: if (1) the price is sufficiently low, everyone buys the good; if (2) the price is sufficiently high, only the rich buy the good (a status good in a narrow sense). If (3) the price is located in very high or in middle range, demand collapses. Thereby, we explain the critical price from which a status good acts as a distinctive signal. In addition, this approach shows the potential welfareimproving impact of conspicuous consumption.
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Notes
Today, Veblen’s continuing popularity among students stems less from his model of social evolution, but rather from Harvey Leibenstein’s article Bandwagon, Snob, and Veblen Effects in the Theory of Consumers’ Demand. There, “for want of a better term” (Leibenstein 1950, 203), he named the case of a partly upward sloping demand function the Veblen Effect, which can be found in almost each microeconomic textbook.
Corneo and Jeanne (1997) state that the prohibition of conspicuous consumption is always welfareimproving, whereas the welfare effects of taxation are ambiguous.
In addition, in their welfare analysis the Frank (1985) and Ireland (1994) compare a world with status signaling and income as private information with a world in which the income rank of each individual is common knowledge. This comparison leads to the result that a world in which the income rank of each individual is common knowledge is welfaresuperior, and so status signaling has to be seen as social waste. But: it is hardly surprising that conspicuous consumption as a signal makes no sense in a world of complete information which is definitely a part of Nirvana economics.
The present paper is also related to the contributions of Corneo and Jeanne (1998) as well as Van Long and Shimomura (2004) who analyze the intertemporal consumption decision in the framework of life cycle models. Cooper et al. (2001), Corneo and Jeanne (2001) and Rauscher (1997) focus on the impact of conspicuous consumption on economic growth. Haucap (2001) as well as Jaramillo et al. (2001) analyze conspicuous consumption in clubs. Ferrer (2005) presents a signaling game in which education acts as a status signal. Krähmer (2006) focuses on the connection between advertising and conspicuous consumption.
At a first view this assumption might be problematic. A second view disclosures that this assumption is straightforward. In the present mating game the only situation in which some individuals signal and some individuals do not is a separating equilibrium in a world with status signaling. This situation is characterised by individuals of type H who signal and individuals of type L who do not. So, if an individual does not signal in the separating equilibrium, the only conclusion can be that she is of type L. As a result, she has to expect that she will be seen by the others as being poor and with this be matched with an individual of type L for sure.
For deeper insights in fashion cycles see Pesendorfer (1995).
The set up that individuals purchase at most one unit of the status good is similar to the approach of Corneo and Jeanne (1997, 57). What differs to their seminal contribution as well to the approaches of Frank (1985, 103), Irland (1994, 93), and more recent Hopkins and Kornienko (2004, 1089) is that in our model neither status nor the consumption of the status good itself enters the utility function. From this perspective in our model conspicuous consumption is instrumental and without intrinsic motivation.
The single crossing property is the condition for the possible emergence of separating equilibria in signaling games. It is also well known as sorting condition, constant sign condition or SpenceMirrlees condition (Fudenberg and Tirole 1991, 259).
In his seminal contribution Haucap (2001) presents a similar welfare analysis. What differs in our contribution is the strict focus on the pricedependency of different welfare outcomes.
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Acknowledgments
The author would like to express his gratitude to the participants of the Annual Congress of the Verein für Socialpolitik (German Economic Association) in Graz, Austria, and of the 64th Congress of the International Institute of Public Finance in Maastricht, Netherlands, as well as to an anonymous referee for fruitful comments and helpful discussion.
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Appendix
Appendix
Note, that all parameter w _{ H }, w _{ L }, q _{ H }, α > 0.

(a)
The upper limit of the pooling case is given by
$$ p_{S}=w_{L}\root{\alpha}\of{{w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha} \left(\frac{ w_{L}+w_{H}}{2}\right)^{\alpha}\right)}}. $$Note, that in the pooling equilibrium the budget condition \(\alpha<\ln \left(\frac{1q_{H}}{q_{H}}\right)/\ln\left(\frac{w_{H}+w_{L}} {2w_{L}}\right)\) holds, so that \(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)>0\).
The first order derivatives of p _{ S } with respect to w _{ H }, w _{ L }, q _{ H } and α are given as follows:
$$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{H}}&=\frac{1} {2}q_{H}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha1}\\ &\quad\times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\left(\frac{1}{\alpha}1\right)}>0 \end{aligned} $$(25)$$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{L}}& =1\left(w_{L}^{\alpha1}+q_{H} \left(w_{L}^{\alpha1}\frac{1}{2}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha1}\right)\right)\\ &\quad\times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\left(\frac{1}{\alpha}1\right)}\lessgtr 0 \end{aligned} $$(26)If w _{ L } is low, \(\frac{\partial p_{S}}{\partial w_{L}}>0\) holds. If w _{ L } is high, \(\frac{\partial p_{S}}{\partial w_{L}}<0\) holds. One can construct parametric constellations for each case.
$$ \begin{aligned} \frac{\partial p_{S}}{\partial q_{H}}&=\frac{1} {a}\left(\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}w_{L}^{\alpha}\right)\\ &\quad \times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha} \left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\left(\frac{1}{\alpha}1\right)}>0\\ \end{aligned} $$(27)$$ \begin{aligned} \frac{\partial p_{S}}{\partial\alpha}& =\left(\frac{(w_{L}^{\alpha}+q_{H} (w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}))\ln(w_{L}^{\alpha} +q_{H}(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha})}{\alpha^{2} \left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)}\right.\\ &\quad\left.+\frac{\alpha\left(\left(q_{H}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\ln\left(\frac{w_{L}+w_{H}} {2}\right)\right)(1+q)w_{L}^{\alpha}\ln(w_{L})\right)} {\alpha^{2}(w_{L}^{\alpha}+q_{H} \left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2})^{\alpha}\right)\right)}\right)\\ &\quad \times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\frac{1}{\alpha}}>0 \end{aligned} $$(28) 
(b)
The lower limit of the separating case is given by
$$ p_{S}=w_{L}\root{\alpha}\of{{2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}}}. $$Note, that in the separating equilibrium the budget condition \(\alpha<\ln(2)/\ln\left(\frac{w_{H}+w_{L}}{2w_{L}}\right)\) holds, so that \(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}>0\).
The first order derivatives of p _{ S } with respect to w _{ H }, w _{ L }, q _{ H } and α are given as follows:
$$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{H}}&=\frac{1} {2}\left(\frac{w_{L}+w_{H}}{2}\right)^{(\alpha1)}\\ &\quad \times\left(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)^{\left(\frac{1}{\alpha}1\right)}>0 \end{aligned} $$(29)$$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{L}}&=1\left(2w_{L}^{(\alpha1)} \frac{1}{2}\left(\frac{w_{L}+w_{H}}{2}\right)^{(\alpha1)}\right)\\ &\quad \times\left(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)^{\left(\frac{1}{\alpha}1\right)}\lessgtr 0 \end{aligned} $$(30)If w _{ L } is low, \(\frac{\partial p_{S}}{\partial w_{L}}>0\) holds. If w _{ L } is high, \(\frac{\partial p_{S}}{\partial w_{L}}<0\) holds. One can construct parametric constellations for each case.
$$ \frac{\partial p_{S}}{\partial q_{H}}=0 $$(31)$$ \begin{aligned} \frac{\partial p_{S}}{\partial\alpha}&=\left(\frac{(2w_{L}^{\alpha} \left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha})\ln\left(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}\right)}{\alpha^{2}\left(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}\right)}\right.\\&\quad\left. +\frac{\alpha\left(\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}\ln\left(\frac{w_{L}+w_{H}}{2}\right)2w_{L}^{\alpha}\ln(w_{L})\right)}{a^{2}(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha})}\right)\\&\quad \times\left(2w_{L}^{\alpha}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha}\right)^{\left(\frac{1}{\alpha}\right)}>0\end{aligned} $$(32) 
(c)
If \(c_{A}\geq w_{H}^{\alpha}\left(\frac{w_{H}+w_{L}} {2}\right)^{\alpha}\) holds, the upper limit of the separating case is given by
$$ p_{S}=\left(\frac{w_{H}w_{L}}{2}\right) $$The first order derivatives of p _{ S } with respect to w _{ H }, w _{ L }, q _{ H } and α are given as follows:
$$ \frac{\partial p_{S}}{\partial w_{H}}=\underset{}{\frac{1}{2}}>0,\,\frac{\partial p_{S}} {\partial w_{L}}=\underset{}{\frac{1}{2}}<0,\,\frac{\partial p_{S}} {\partial q_{H}}=0\hbox{ and }\frac{\partial p_{S}}{\partial\alpha}=0 $$(3336)If \(c_{A}<w_{H}^{\alpha}\left(\frac{w_{H}+w_{L}}{2}\right)^{\alpha}\) holds, the upper limit of the separating case is given by
$$ p_{S}=w_{H}\root{\alpha}\of{{w_{H}c_{A}}}. $$Note that w _{ H } > c _{ A } holds.
The first order derivatives of p _{ S } with respect to w _{ H }, w _{ L }, q _{ H } and α are given as follows:
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Thomas, T. What price makes a good a status good? Results from a mating game. Eur J Law Econ 36, 35–55 (2013). https://doi.org/10.1007/s1065701192734
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DOI: https://doi.org/10.1007/s1065701192734