A Case for Percentage Commission Contracts: The Impact of a “Race” Among Agents (P.5)

The Model

It is important to note that objective of residential home sellers is likely a combination of minimizing time to sale and maximizing sale price. Various models in the literature address one or both of these objectives. For the purposes of this paper, we will take the expected sale price of the property as given and focus on the time to sale as the criteria for maximizing seller utility.4 This simplifies the problem and allows us to focus on the optimality of the commission structure. It is also in line with the argument that the price is determined by the supply and demand forces in the market. We assume that the seller and all agents are risk neutral and have a common discount factor, r.

First, we consider the game between one seller and one agent. The two players enter into a listing contract which grants the agent exclusive rights of sale, meaning that the agent receives the full sales commission upon sale of the property. While the outcome of this case will follow the insights of standard principal-agent theory, it is useful to develop this case for the purposes of comparison to a model with multiple agents.

Let λ be the intensity of search that the agent selects at time 0 and maintains until the buyer is found. The agent searches continuously until a buyer is found. Although a strategy for the agent is to determine the optimal search intensity time path, we only consider the non-cooperative Nash equilibrium of stationary pure strategies.5 Let X be the stochastic time until the agent procures a buyer for the listed real estate, where $$X \sim \exp \left( \lambda \right)$$, $$E\left( X \right) = \frac{1}{\lambda }$$, and λdt is the probability that an agent procures a buyer in the short time interval dt. Search is costly and the agent incurs a cost, c(λ), for the chosen intensity level. The cost per unit of time is increasing and strictly convex in intensity λ, c′(λ) > 0, c″(λ) > 0. Supply and demand forces in the market determine the sales price of the real estate, P. The seller will pay a commission to the agent at the time of sale equal to αP, where 0 < α ≤ 1. The commission rate is taken as exogenous by the players.6

The solution concept imposed is the non-cooperative Nash equilibrium, and we only consider pure strategies.

Given a sale at time t the present value of the payoffs for the seller and agent, respectively, are

$$\pi _s = \left( {1 - \alpha } \right)Pe^{ - rt} ,$$


$$\pi _b = \alpha Pe^{ - rt} - c\left( \lambda \right)\frac{{\left( {1 - e^{ - rt} } \right)}}{r}.$$
The present value of unconditional expected payoff to the seller at time zero is,

$$\begin{aligned}& V\left( {\alpha ,\lambda } \right) = \int\limits_0^\infty {\pi _s \lambda e^{ - \lambda t} dt} ,\quad or \\& V\left( {\alpha ,\lambda } \right) = \frac{{\left( {1 - \alpha } \right)P\lambda }}{{r + \lambda }}. \\ \end{aligned} $$
Similarly, the present value of unconditional expected payoff to the agent at time zero is,

$$\begin{array}{*{20}l}   {{U{\left( {\alpha ,\lambda } \right)} = {\int\limits_0^\infty  {\pi _{b} \lambda e^{{ - \lambda t}} dt} },\quad or} \hfill}  \\   {{U{\left( {\alpha ,\lambda } \right)} = \frac{{\alpha P\lambda  - c{\left( \lambda  \right)}}}{{r + \lambda }}.} \hfill}  \\ \end{array} $$
The agent’s problem is to choose λ to maximize Eq. 4. The source of the agency problem is evident here since Eq. 4 does not account for the effects of the agent’s search intensity on the seller’s expected profits. As Eq. 3 reveals, the seller’s expected payoff is strictly increasing in the agent’s intensity level:

$$\frac{{\partial V\left( {\alpha ,\lambda } \right)}}{{\partial \lambda }} = \frac{{\left( {1 - \alpha } \right)rP}}{{\left( {r + \lambda } \right)^2 }} >0$$
Therefore, under full information, the seller would demand that the agent exerts the maximum possible intensity level subject to the agent’s participation constraint being satisfied,

$${U{\left( {\alpha ,u} \right)} = \varphi }$$

where φ > 0 is the reservation payoff of the agent.7

Substituting Eq. 4 into Eq. 5 and solving for α, we obtain the full information commission rate:

$$\alpha = \frac{{\varphi \left( {r + \lambda } \right) + c\left( \lambda \right)}}{{P\lambda }}.$$
From the seller’s point of view, the optimal search intensity by the agent, subject to the agent’s participation constraint and 0 < α ≤ 1, can be found by substituting Eq. 6 into Eq. 3 and maximizing with respect to λ. This assumes for the moment that λ is observable and enforceable. Rewriting Eq. 3 yields,

$$V\left( {\alpha ,\lambda } \right) = \frac{{P\lambda - c\left( \lambda \right)}}{{r + \lambda }} - \varphi .$$
Taking the derivative of V(α, λ) and setting it equal to zero, we obtain the full information intensity level λ** as

$$\lambda \, * * = \frac{{r\left( {P - c\prime \left( {\lambda * * } \right)} \right) + c\left( {\lambda * * } \right)}}{{c\prime \left( {\lambda * * } \right)}}.$$
The intensity level λ** is also the efficient, or first-best, intensity level. This is due to the fact that V(α, λ) incorporates the agent’s cost via the full information commission rate. To see this, note that the joint payoffs of the seller and the agent are given by:

$$W\left( \lambda \right) = U\left( {\alpha ,\lambda } \right) + V\left( {\alpha ,\lambda } \right) = \frac{{P\lambda - c\left( \lambda \right)}}{{r + \lambda }}.$$

The efficient intensity level that maximizes the joint payoffs above is the same as the intensity level in Eq. 7 that maximizes the seller’s payoff under full information.

When effort intensity is not observable, the agent has incentives to exert effort up until the point that his marginal cost of effort intensity equals his expected marginal payoff. Taking the derivative of Eq. 4 and solving for the optimal intensity level from the agent’s perspective, λ*, yields,

$$\lambda * = \frac{{r\left( {\alpha P - c\prime \left( {\lambda * } \right)} \right) + c\left( {\lambda * } \right)}}{{c\prime \left( {\lambda * } \right)}}.$$
Proposition 1   When a single agent is involved in selling an asset, the optimal level of agent effort intensity from the seller’s perspective and from the efficiency perspective is obtained when α* = 1.
Proof   Comparing Eqs. 7 and 8 reveals that λ* = λ** if and only if α* = 1.

Proposition 1 above states the standard result in the literature that the percentage commission structure where the agent receives a portion of the transaction price (α < 1) cannot perfectly align the interests of the agent and the seller. In fact, the proposition suggests that net listing is the only commission structure that would result in efficient intensity level from the agent.8 The following section will show that when there are multiple agents involved in selling an asset, the percentage commission structure can yield efficient intensity levels.

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