Cyclical Determinants of Brokerage Commission Rates

Abstract

This study evaluates the determinants of percentage commission rates in the residential brokerage industry. The model developed here predicts that market share of the brokerage firm, property selling difficulty, and prevailing market conditions influence commission rates. Empirical analysis using a sample of 14,891 condominium transactions from the Fort Lauderdale, Florida area provides support for the model’s predictions.

Appendices

Appendix A. Proofs

Proof of Result 1

Substitute equation (3) into (4) and take the partial derivative with respect to σ, to get

$$ \frac{{\partial {C_{{ij}}}*}}{{\partial \sigma }} = \frac{\partial }{{\partial \sigma }}\left[ {\frac{{{\pi_i} + {F_i}\left( {{{\widehat{t}}_j}\left( \sigma \right)} \right)}}{{{\delta^{{{{\widehat{t}}_j}\left( \sigma \right)}}}}}} \right] - {\phi_i}\frac{\partial }{{\partial \sigma }}\left[ {\frac{{{\pi_i} + {F_i}\left( {{{\widehat{t}}_j}\left( \sigma \right)} \right)}}{{{\delta^{{{{\widehat{t}}_j}\left( \sigma \right)}}}}}} \right]. $$

(A1)

Take the partial derivative of equation (1) with respect to σ, to get

$$ \frac{{\partial {{\widehat{t}}_j}}}{{\partial \sigma }} = - \frac{{{\alpha_{\sigma }}}}{{{{\left( {\Delta {\lambda_B}\widehat{\alpha}\left( \sigma \right)} \right)}^2}}} > 0, $$

(A2)

because ${\alpha_{\sigma }} < 0$. Next, consider that

$$ \frac{\partial }{{\partial \sigma }}\left[ {\frac{{{\pi_i} + {F_i}\left( {{{\widehat{t}}_j}\left( \sigma \right)} \right)}}{{{\delta^{{{{\widehat{t}}_j}\left( \sigma \right)}}}}}} \right] = \frac{{{F_{{\widehat{t}}}}{t_{\sigma }}}}{{{\delta^{{\widehat{t}}}}}} + \frac{{{t_{\sigma }}\widehat{t}\left( {\pi + F} \right)}}{{{\delta^{{\widehat{t} + 1}}}}} > 0. $$

(A3)

Equation (A3) is positive because ${F_{{\hat{t}}}} > 0$ and ${t_{\sigma }} > 0$ (as shown in equation(A2)). Combined with the assumption that $0 < {\phi_i} < 1$, it is evident that ${{{\partial {C_{{ij}}}*}} \left/ {{\partial \sigma }} \right.} > 0$.

Proof of Result 2

Substitute equation (3) into (4) and take the partial derivative with respect to A, to get

$$ \frac{{\partial {C_{{ij}}}*}}{{\partial {A_j}}} = \frac{\partial }{{\partial {A_j}}}\left[ {\frac{{{\pi_i} + {F_i}\left( {{A_j},{{\widehat{t}}_j}\left( {{A_j}} \right)} \right)}}{{{\delta^{{{{\widehat{t}}_j}\left( {{A_j}} \right)}}}}}} \right] - {\phi_i}\frac{\partial }{{\partial {A_j}}}\left[ {\frac{{{\pi_i} + {F_i}\left( {{A_j},{{\widehat{t}}_j}\left( {{A_j}} \right)} \right)}}{{{\delta^{{{{\hat{t}}_j}\left( {{A_j}} \right)}}}}}} \right]. $$

(A4)

Next, consider that

$$ \frac{\partial }{{\partial {A_j}}}\left[ {\frac{{{\pi_i} + {F_i}\left( {{A_j},{{\widehat{t}}_j}\left( {{A_j}} \right)} \right)}}{{{\delta^{{{{\widehat{t}}_j}\left( {{A_j}} \right)}}}}}} \right] = \frac{{{F_A}{F_{{\widehat{t}}}}{F_{{\widehat{t}A}}}{t_A}}}{{{\delta^{{\widehat{t}}}}}} + \frac{{{t_A}\widehat{t}\left( {\pi + F} \right)}}{{{\delta^{{\widehat{t} + 1}}}}}. $$

(A5)

Equation (A5) is positive iff ${F_A}{F_{{\widehat{t}}}}{F_{{\widehat{t}A}}} > \widehat{t}{\delta^{{ - 1}}}\left( {\pi + F} \right)$ because ${F_A} > 0$, ${F_{{\hat{t}}}} > 0$, ${F_{{\widehat{t}A}}} < 0$ and ${t_A} < 0$. Combined with the assumption that $0 < {\phi_i} < 1$, it can be seen that equation (A4) is positive when the same condition exists.

Appendix B

Table 7 Correlations: Time-on-market