A Case for Percentage Commission Contracts: The Impact of a “Race” Among Agents (P.7)

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We first consider the equilibrium under full information where the seller can observe the agents’ intensity levels. Under full information, the seller would choose the optimal intensity level, λ**, that maximizes his expected payoff in Eq. 10 subject to U = φ. Given that we have assumed n identical agents, we will focus on symmetric Nash equilibria where each agent chooses the same intensity level, λ j = λ. Thus, the seller’s choice of λ** can be written as:

$$\lambda * * = \frac{{r\left( {P - c\prime \left( {\lambda * * } \right)} \right) + nc\left( {\lambda * * } \right)}}{{nc\prime \left( {\lambda * * } \right)}}.$$
The total value of the agency contract is given by the summation of the seller’s and all the agents’ payoffs,

$$\begin{array}{*{20}l}   {{W{\left( {\lambda _{1}  \ldots \lambda _{n} } \right)} = V{\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)} + {\sum\limits_j {U_{i} {\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)}} } = \frac{{{\left( {1 - \alpha } \right)}P{\sum\limits_j {\lambda _{j} } }}}{{r + {\sum\limits_j {\lambda _{j} } }}} + \frac{{{\sum\limits_j {{\left\{ {\alpha P\lambda _{j}  - c{\left( {\lambda _{j} } \right)}} \right\}}} }}}{{r + {\sum\limits_j {\lambda _{j} } }}},or} \hfill}  \\   {{W{\left( {\lambda _{1}  \ldots \lambda _{n} } \right)} = \frac{{{\sum\limits_j {{\left\{ {P\lambda _{j}  - c{\left( {\lambda _{j} } \right)}} \right\}}} }}}{{r + {\sum\limits_j {\lambda _{j} } }}} = \frac{{n{\left( {P\lambda  - c{\left( \lambda  \right)}} \right)}}}{{r + n\lambda }}.} \hfill}  \\ \end{array} $$

The efficient level of agent intensity would maximize the joint payoffs in Eq. 14. The social planner’s choice of intensity incorporates the externalities imposed by agent i’s intensity level on other agents and the seller. Note that the commission rate drops out of the planner’s problem because it is merely a transfer from one player to the other in the economy. It may appear that the objective function in Eq. 14 differs from the seller’s objective function under full information, as given in Eq. 10, where the seller’s aim is to maximize his individual payoff subject to agents’ participation constraint. However, the participation constraint dictates that the seller incurs a commission cost that is increasing in the cost of intensity by the agents. In fact, the participation constraint will force the seller under full information to choose a level of agent intensity that internalizes the externalities and yields the efficient outcome. This can be seen by substituting the participation constraint, $$U_i \left( {\alpha ,\lambda _1 \ldots \lambda _n } \right) = \frac{{\alpha P\lambda _i - c\left( {\lambda _i } \right)}}{{r + \sum\limits_j {\lambda _j } }} = \varphi $$, into the seller’s objective function in Eq. 10, which yields $$V\left( {\alpha ,\lambda _1 \ldots \lambda _n } \right) = W\left( {\lambda _1 \ldots \lambda _n } \right) - n\varphi $$. Thus, the seller’s and the planner’s objective functions are the same except for the constant, . This result is stated in the following remark.

Remark 1   With agent competition and n identical agents, the efficient intensity level is the same as the seller’s choice of agent intensity under full information.
Proof   Since agents are identical, in the symmetric equilibrium all the agents will be choosing the same intensity level (λ i = λ). Rewriting Eq. 14 in terms of n and λ taking the derivative with respect to λ yields,

$$\lambda * * = \frac{{r\left( {P - c\prime \left( {\lambda * * } \right)} \right) + nc\left( {\lambda * * } \right)}}{{nc\prime \left( {\lambda * * } \right)}},$$

which is identical to the full information intensity level λ** in Eq. 13.

It is also worth noting that total differentiation of Eq. 15 yields a positive relationship between λ** and P.

Remark 2   The efficient intensity level is higher for higher priced assets.

The question of primary importance is how the agent’s choice of intensity level, when the seller cannot observe the agent’s intensity level, compares to the efficient intensity level. The following proposition states the main result of the paper.

Proposition 2   When there is more than one agent in the market, the efficient percentage commission rate, α*, is less than 100%.
Proof   To arrive at the commission rate which procures the efficient intensity level, notice that the efficient intensity, λ**, satisfies:

$$P - W\left( {\lambda _1^{ * * } \ldots \lambda _n^{ * * } } \right) = c\prime \left( {\lambda * * } \right).$$
The agent’s choice of intensity level, λ*, as derived in Eq. 12, solves

$$\alpha P - U\left( {\alpha ,\lambda _1^ * \ldots \lambda _n^ * } \right) = c\prime \left( {\lambda * } \right).$$

Since the total surplus exceeds the expected surplus of an individual agent, W(λ 1λ n ) > U(α, λ 1λ n ), for any α, λ 1λ n , and since c(λ) is a strictly convex function, it is clear that λ* = λ** if and only if α* < 1.11

In other words, if α = 1 so that each agent treats the property as one of her own, we would get too much intensity level from agents who compete against each other to sell the property and obtain the commission. A less than 100% commission mitigates this inefficiency.

Using Eqs. 16 and 17, the commission rate that induces agents to exert the first-best intensity levels can be derived as:

$$\alpha * = \frac{{rP + \left( {n - 1} \right)c\left( {\lambda * * } \right)}}{{rP + \left( {n - 1} \right)P\lambda * * }} <1.$$
Proposition 3   Although the percentage commission system with multiple agents can produce optimal intensity levels, a uniform commission rate is not compatible with efficiency.

The intuition for Proposition 3 is straightforward. The efficient intensity level in Eq. 15 varies depending on price levels, size and costs of a particular market. Since the percentage commission implements the efficient intensity level given each agent’s utility function, it too varies across markets.

Proposition 3 is significant because the observed commission rates in the industry display uniformity across assets of different prices. Our results indicate that while the percentage commission rate can induce first-best intensity levels, it does so at a different rate for different asset prices and market sizes.

We finally note that

Remark 3   The efficient intensity level per agent is lower for a larger numbers of competing agents.
Proof   From Eq. 13 we obtain

$$\frac{{d\left( {\lambda * * } \right)}}{{dn}} = \frac{{c\left( {\lambda * * } \right) - \lambda * * c\prime \left( {\lambda * * } \right)}}{{c\prime \prime \left( {\lambda * * } \right)\left( {r + n\lambda * * } \right)}} <0.$$

From the first order condition in Eq. 16 we can show that the numerator is negative, and therefore, efficient intensity levels decline in larger markets.

We have shown in Proposition 2 that in the presence of competition among agents the efficient commission rate is less than 100%. This does not, however, necessarily mean that the current commission rates in the industry, typically 5–6%, are efficient. This question has been tackled empirically by two recent studies by Rutherford et al. (2004) and Levitt and Syverson (2005). These studies utilize large data sets of single-family home transactions where a portion of the homes sold were homes that were owned by real estate agents. Their data sets enable the authors to test for agency problems in brokerage industry by investigating whether the owner-agents sell their own properties more quickly and/or obtain a higher price for their own properties relative to their clients’ properties. The empirical results in these papers confirm the presence of agency problems; they find that agent-owned houses sell at a price premium of 3.7% to 4.8% in Levitt and Syverson (2005) and 4.5% to 7.0% in Rutherford et al. (2004). While agent-owned properties in Levitt and Syverson (2005) stay on the market about 10% (9 days) longer, they sell no faster or slower than the client-owned properties in Rutherford et al. (2004). In a related work, Anglin and Arnott (1999) simulate their theoretical model and report that the efficient commission rate is significantly lower than the prevailing 6% rate.

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