Real Estate Brokers and Commission: Theory and Calibrations

Abstract

This paper has two goals: (a) To model an inherent conflict of interest between a seller of a house and the real estate broker hired by the seller. In this environment, the pressure brokers exert on sellers to reduce prices generates faster sales and hence reduces sellers’ expected profit. (b) To calibrate the brokers’ commission rates that would maximize sellers’ expected gain. The calibration results may hint whether the ongoing uniform commission rate reflects collusion among real estate agencies, or should be viewed as competitive.

Appendix: Derivation of Equation 6

The right side of Eq. 6 can be written as

$$ \begin{array}{rll} &&{\beta}\left\{ \left[ 1+(1-{\beta})+(1-{\beta})^2+\cdots \right] + \left[ (1-{\beta})+(1-{\beta})^2+\cdots \right] \right. \\ &&\left.\left[ (1-{\beta})^2+(1-{\beta})^3+\cdots\right] + \left[ (1-{\beta})^3+(1-{\beta})^4+\cdots\right]\right\} . \end{array} $$

Since each bracket term constitutes a converging sequence, the above equals to

$$ {\beta}\left\{ \frac{1}{1-(1-{\beta})}+ \frac{1-{\beta}}{1-(1-{\beta})}+ \frac{(1-{\beta})^2}{1-(1-{\beta}) }+\cdots \right\} =\sum_{t=0}^\infty (1-{\beta})^t =\frac{1}{{\beta}} . $$