What price makes a good a status good? Results from a mating game

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 welfare-improving impact of conspicuous consumption.

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 $\alpha<\ln \left(\frac{1-q_{H}}{q_{H}}\right)/\ln\left(\frac{w_{H}+w_{L}} {2w_{L}}\right)$ 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:

  $$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{H}}&=\frac{1} {2}q_{H}\left(\frac{w_{L}+w_{H}}{2}\right)^{\alpha-1}\\ &\quad\times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}-\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\left(\frac{1}{\alpha}-1\right)}>0 \end{aligned} $$

  (25)
  $$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{L}}& =1-\left(w_{L}^{\alpha-1}+q_{H} \left(w_{L}^{\alpha-1}-\frac{1}{2}\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha-1}\right)\right)\\ &\quad\times\left(w_{L}^{\alpha}+q_{H}\left(w_{L}^{\alpha}-\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)\right)^{\left(\frac{1}{\alpha}-1\right)}\lessgtr 0 \end{aligned} $$

  (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:

  $$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{H}}&=\frac{1} {2}\left(\frac{w_{L}+w_{H}}{2}\right)^{(\alpha-1)}\\ &\quad \times\left(2w_{L}^{\alpha}-\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)^{\left(\frac{1}{\alpha}-1\right)}>0 \end{aligned} $$

  (29)
  $$ \begin{aligned} \frac{\partial p_{S}}{\partial w_{L}}&=1-\left(2w_{L}^{(\alpha-1)}- \frac{1}{2}\left(\frac{w_{L}+w_{H}}{2}\right)^{(\alpha-1)}\right)\\ &\quad \times\left(2w_{L}^{\alpha}-\left(\frac{w_{L}+w_{H}} {2}\right)^{\alpha}\right)^{\left(\frac{1}{\alpha}-1\right)}\lessgtr 0 \end{aligned} $$

  (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:

  $$ \frac{\partial p_{S}}{\partial w_{H}}=\underset{}{\frac{1}{2}}>0,\,\frac{\partial p_{S}} {\partial w_{L}}=\underset{}{-\frac{1}{2}}<0,\,\frac{\partial p_{S}} {\partial q_{H}}=0\hbox{ and }\frac{\partial p_{S}}{\partial\alpha}=0 $$

  (33-36)

  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:

$$ \frac{\partial p_{S}}{\partial w_{H}}=1-\frac{\root{\alpha}\of{{w_{H}-c_{A}}} }{{\alpha\cdot(w_{H}-c_{A})}}<0 $$

(37)

$$ \frac{\partial p_{S}}{\partial w_{L}}=0,\quad \frac{\partial p_{S}}{\partial q_{H}}=0 $$

(38–39)

$$ \frac{\partial p_{S}}{\partial\alpha}=\frac{\root{\alpha}\of{{w_{H}-c_{A}} \ln(h-c)}}{\alpha}>0 $$

(40)

$$ \frac{\partial p_{S}}{\partial c_{A}}=\frac{\root{\alpha}\of{{w_{H}-c_{A}}}} {\alpha\cdot(w_{H}-c_{A})}>0 $$

(41)