Real Estate Agents, House Prices, and Liquidity

Abstract

Comparing agent-owner with agent-represented home sales illustrates that commission contracts lead to external agent moral hazard. Real estate developers are sophisticated sellers who can either use external agents or hire internal agents. The theory shows that neither scheme eliminates agent moral hazard. The empirical study of how the seller-agent relationship affects both price and liquidity in a simultaneous system concludes that external agents enjoy superior selling ability that offset moral hazard effects.

Appendix

This appendix derives several properties of the equilibria and propositions used in the paper. The efficient effort e ^∗(w) satisfies the marginal condition

$$ w{f}_e(e)=1 $$

(21)

Implicit differentiation yields the slope of the e ^∗ trajectory or function in (w, e) space as

$$ \frac{\partial {e}^{\ast }(w)}{\partial w}=\frac{f_e}{-w{f}_{ee}}>0 $$

(22)

The e* curve is upward sloped as depicted in Figs. 1 and 2.

The outside agent with ability a has equilibrium effort e ^o(w, a, c) satisfying the marginal condition

$$ cwa{f}_e(e)=1 $$

(23)

Implicit differentiation yields the properties of this function as

$$ \frac{\partial {e}^o\left(w,a,c\right)}{\partial \omega }=\frac{f_e}{-w{f}_{ee}}>0 $$

(24)

$$ \frac{\partial {e}^o\left(w,a,c\right)}{\partial a}=\frac{f_e}{-a{f}_{ee}}>0 $$

(25)

$$ \frac{\partial {e}^o\left(w,a,c\right)}{\partial c}=\frac{f_e}{-c{f}_{ee}}>0 $$

(26)

The e ^o curve is upward sloped in (w,e) space as in Figs. 1 and 2. Increases in ability a or the commission rate c induce parallel upward shifts in the curve as depicted in Fig. 1.

The inside agent in the asymmetric developer-agent information environment supplies sales effort e ⁱ(w, B, q) satisfying the marginal condition

$$ Bw{f}_e(e){V}_{\upsilon}\left(q- wf(e)\right)=1 $$

(27)

Focusing on results used in the paper, implicit differentiation yields the slope and bonus shift properties

$$ \frac{\partial {e}^i\left(w,B,q\right)}{\partial w}=\frac{V_{\upsilon }{f}_e-{V}_{\upsilon \upsilon}w{f}_ef}{-{V}_{\upsilon }w{f}_{ee}+{V}_{\upsilon \upsilon}{w}^2{f}_e^2} $$

(28)

$$ \frac{\partial {e}^i\left(w,B,q\right)}{\partial B}=\frac{V_{\upsilon }{f}_e}{-{V}_{\upsilon }w{f}_{ee}}>0 $$

(29)

(29) establishes that increasing the bonus increases the effort supplied by the inside agent at each level of productivity, shifting the insides agent effort supply curve e ⁱ upward in (w, e) space.

Finally, note that the application of the implicit function theorem in all three models yields continuous effort functions over the relevant regions. The continuity property plays a role in the proofs below.

The first result establishes that the inside agent cannot replicate the efficient outcome when V is a single-peaked symmetric distribution. The result holds for other distributions, but this particular case is a popular form in incentives theory that is easy to work with and, in this application, is sufficient to establish the property that efficiency can only be attained in special situations; efficiency is not a general property of the inside agent equilibrium and indeed is not attainable in most cases (as here). Here we establish that e ⁱ(w,B,q) = e*(w) can hold at most two realized w; that is, the inside agent is inefficient almost everywhere.

Proposition 1

For the single-peaked symmetric distribution V(υ), e ⁱ(w,B,q) = e*(w) with measure zero.

Proof

Consider the bonus B set such that the inside agent replicates the efficient effort. Define w′ such that Bw ′ f _e V _υ (q − w ′ f) = 1. In this case (27) satisfies (21) so that e ⁱ(w ′, B, q) = e ^∗(w ′). Thus w′ represents an intersection of the inside agent and efficient effort functions in (w, e) space. To prove the proposition, show that there exist at most two such w′. Specifically, (i) there exists at most one w′ such that w ′ f(e ⁱ(w ′)) < q (i.e., υ > 0) and (ii) there exists at most one w′ such that w ′ f(e ⁱ(w ′)) > q (i.e., υ < 0). For the single-peaked symmetric distribution V(υ), by the law of probability V _υ(υ) ≥ 0 for all υ within the support, and V _υυ (υ) > 0 for all υ < 0 and V _υυ (υ) < 0 for all υ > 0 (recall E[υ] = 0), so that

$$ \begin{array}{ccc}\hfill {V}_{\upsilon \upsilon}>0\hfill & \hfill \mathrm{for}\ wf>q\hfill & \hfill \left(\mathrm{i}.\mathrm{e}.,\kern0.5em \upsilon <0\right)\hfill \end{array} $$

(30)

$$ \begin{array}{ccc}\hfill {V}_{\upsilon \upsilon}<0\hfill & \hfill \mathrm{for}\ wf>q\hfill & \hfill \left(\mathrm{i}.\mathrm{e}.,\kern0.5em \upsilon >0\right)\hfill \end{array} $$

(31)

Substituting Bw ′ f _e V _υ (q − w ′ f) = 1 from (27) into the slope of the e ⁱ function (28) at any w′ and simplifying yields

$$ \frac{\partial {e}^i(w)}{\partial w}=\frac{f_e-B{V}_{\upsilon \upsilon}w{f}_ef}{-w{f}_{ee}+B{V}_{\upsilon \upsilon}{w}^2{f}_e^2} $$

(32)

Together with (22) and (30)–(31), this yields

$$ \frac{\partial {e}^i\left(w\prime \right)}{\partial w}\gtrless \frac{\partial {e}^{\ast}\left(w\prime \right)}{\partial w}\ \mathrm{for}\ {V}_{\upsilon \upsilon}\left(\upsilon \right)\lessgtr 0,\kern0.5em \mathrm{i}.\mathrm{e}.,\kern0.5em \mathrm{for}\kern0.5em \upsilon \gtrless 0 $$

(33)

Result (33) establishes that the e ⁱ function is shallower than the e* function at any intersection in (w, e) space for υ < 0; therefore, since both functions are continuous there can be at most only one such intersection for all υ < 0 (i.e., wf > q). This establishes claim (i). Result (33) also establishes that the e ⁱ function is steeper than the e* function at any intersection in (w, e) space for υ > 0; therefore, there can be at most only one such intersection for all υ > 0 (i.e., wf < q). This establishes claim (ii). Together, the claims establish the proposition. ■

A similar result pertains to the effort of inside and outside agents. The following proposition shows that there is no inside agent bonus scheme that will induce the agent to replicate outside agent behavior in the developer-agent asymmetric information environment. The proposition is proved by showing that whenever the two effort curves intersect at a low realized w value, e ⁱ must cut e ^o from below and whenever the two effort curves intersect at a high realized w value, e ⁱ must cut e ^o from above, as depicted in Fig. 2.

Proposition 2

Consider the bonus B set such that the inside agent replicates the effort of the outside agent. Define ω′ such that Bw ′ f _e V _υ (q − w ′ f) = ca, i.e., e ⁱ(w ′, B, q) = e ^o(w ′, a, c); ω′ represents an intersection of the inside and outside agent effort functions in (w, e) space. For the single-peaked symmetric distribution V(υ) there exist at most two such w′. Specifically, (i) there exists at most one w′ such that ω ′ f(e ⁱ(w ′)) > q (i.e., υ > 0) and (ii) there exists at most one w′ such that ω ′ f(e ⁱ(w ′)) > q (i.e., υ < 0).

Proof

Substitute Bw ′ f _e V _υ (q − w ′ f) = ca into the slope of the e ⁱ function (28) at any w′ to obtain

$$ \frac{\partial {e}^i(w)}{\partial w}=\frac{ ca{f}_e-B{V}_{\upsilon \upsilon}w{f}_ef}{- caw{f}_{ee}+B{V}_{\upsilon \upsilon}{w}^2{f}_e^2} $$

(34)

Together with (24) and (30)–(31), this yields

$$ \frac{\partial {e}^i\left(w\prime \right)}{\partial w}\gtrless \frac{\partial {e}^o\left(w\prime \right)}{\partial w}\kern0.5em \mathrm{for}\ {V}_{\upsilon \upsilon}\left(\upsilon \right)\lessgtr 0,\kern1em \mathrm{i}.\mathrm{e}.,\kern0.5em \mathrm{for}\ \upsilon \gtrless 0 $$

(35)

Result (35) establishes that the e ⁱ function is shallower than the e ^o function at any intersection in (w, e) space for υ < 0; therefore, applying continuity there can be at most only one such intersection for all υ < 0 (i.e., wf > q). This establishes claim (i). Result (35) also establishes that the e ⁱ function is steeper than the e ^o function at any intersection in (w, e) space for υ > 0; therefore, there can be at most only one such intersection for all υ > 0 (i.e., wf < q). This establishes claim (ii) and the proposition. ■