This is the second part of a multi part series that is studying the fallout from a recent court case that is about to drastically change the way Realtors are compensated. Specifically we are going to be studying the incentive structures involved in the real estate market with a key focus those related to the commission structure set by the Realtors.
In the first part of this series we had a primer on several key economic concepts that we are going to be using to conduct our study.
In this part we will create a simple economic model that will be used as the foundation for all of our future analysis. This model will be overly simplistic on purpose. The goal is to create a clean pallet void of unnecessary complexity to give us a solid starting point.
In part 3 will assess the components of the aspects of the market that are most pertinent to this particular study and enhance our simple model to enable us to better observe those components as we progress through our analysis.
Further parts will dive into the various arguments presented by key commentators and analyze them through the lens of our model. In the end, we will coalesce what we have learned at form our own conclusions where we can and highlight the areas that we still remain uncertain.
Key Players
In order to create a full model we need to develop a utility function for each of the key players involved in the sale of a house. While in reality there are many more people involved than those I list here, the key players that are pertinent to our study here are:
- the Seller,
- the Buyer,
- the Seller’s Agent,
- the Buyer’s Agent, and
- the Lender
Seller’s Utility Function
\Large
U_{s} = \text{?}Starting with the obvious, the sale price of their house is very important to the seller. They want to get the highest price for their house they can possibly get.
\Large
U_{s} = P_{h} \\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~P_{h} &= \text{Sale price of house}
\end{aligned}Similarly, the seller wants to keep their costs low. They know they will have to spend something during the course of the sale – repairs, closing costs, realtor commissions, … – but the lower they can keep the costs, the more money in their pocket at the end of the day.
\Large
U_{s} = P_{h} - C_s \\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~P_{h} &= \text{Sale price of house,} \\
C_s &= \text{Costs incurred by the seller}
\end{aligned}While money is the most obvious part of the utility function for a seller, they care about other things too. For one, they don’t want to wait forever for their house to sell. So the time it takes to sell the house (T) is important.
Not everyone is in a rush, though. Some sellers are fine waiting a few weeks or even months for their house to sell. So we need to include a term that represents how much the time it takes to sell the house matters to the seller, (\alpha).
We combine those two terms into \alpha(T).
Since this is a negative effect on their utility we add it to the utility function as so.
\Large
U_{s} = P_{h} - C_s - \alpha_s(T_s) \\
\phantom{U} \\
\footnotesize
\begin{aligned}
P_{h} &= \text{Sale price of house,} \\
\text{Where,} ~C_s &= \text{Costs incurred by the seller,} \\
\alpha_s &= \text{Seller's sensitivity to time on market }(T_s) \\
\end{aligned}Throughout the process of selling their home, the seller will have to do some amount of work. They will have to clean their house to make it show worthy. They will have to make various minor repairs or coordinate contractors to do major repairs. They have to deal with agents and lawyers. They will have to warp their schedule around showings. The list goes on. For simplicity’s sake we will lump all of this into a single term, E_s.
As before, every seller will have a different preference for the amount of effort they put into the sale. Some will be fine getting their hands dirty and putting a ton of effort into the sale while others will want to streamline the process and make it as effortless as possible. So we add a new term to represent this preference, \beta_s, and combine it with effort to get \beta_s(E_s).
\Large
U_{s} = P_{h} - C_s - \alpha_s(T_s) - \beta_s(E_s)\\
\phantom{U} \\
\footnotesize
\begin{aligned}
P_{h} &= \text{Sale price of house,} \\
\text{Where,} ~C_s &= \text{Costs incurred by the seller,} \\
\alpha_s &= \text{Seller's sensitivity to time on market }(T_s) \\
\beta_s &= \text{Seller's sensitivity to effort required }(E_s)
\end{aligned}At this point we have a utility function that covers just about everything that effects the seller’s utility. There is a lot of nuance that is not specifically itemized in this model that may be worth adding later to make specific questions we will be asking easier to answer, but everything is covered.
Buyer’s Utility Function
\Large
U_{b} = \text{?} \\
The two biggest and most obvious things that go into a buyer’s utility function are the how much they like the house and how much the house costs.
There is a lot that goes into how much a buyer likes a house. The size of the house, the layout of the house, school districts, distance from work, etc. For now we are going to lump that into a single term of value, V_b.
The price of the house is much more straight forward and we can even use the exact same term that we used in the seller’s utility function. However, since this is money the buyer is paying out, it has a negative impact in the buyer’s utility function
\Large
U_{b} = V_b - P_{h} \\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~P_{h} &= \text{Sale price of house,} \\
V_b &= \text{Value of the house to the buyer}
\end{aligned}Just like the seller, the buyer also has costs associated with the purchase that we can lump into a single term, C_b, a preference for how much time they are willing to spend looking for the right house, \beta_b(T_b), and a preference for how much effort they are willing to put into their house search and purchase, \gamma_b(E_b)
\Large
U_{b} = V_b - P_{h} - C_b - \alpha_b(T_b) - \beta_b(E_b) \\
\phantom{U} \\
\footnotesize
\begin{aligned}
P_{h} &= \text{Sale price of house,} \\
V_b &= \text{Value of the house to the buyer} \\
\text{Where,} ~C_b &= \text{Costs incurred by the buyer} \\
\alpha_b &= \text{Buyer's sensitivity to time searching }(T_b) \\
\beta_b &= \text{Buyer's sensitivity to effort required }(E_b)
\end{aligned}Agent’s Utility Function
The agent’s that represent the buyer and seller also have utility functions that guide their decision making. As we expand on our model the two agents will have unique utility functions, but at this stage they are identical so we will develop their utility function as one.
\Large
U_{a} = \text{?} \\
Just as with the buyer and seller we can start with the most obvious part, the money. Agents get paid for the services they provide. The most common method of payment is a percentage of the sale price as a commission and could be expressed very deliberately to reflect that, (\omega_a(P_h), but for our simple model we will leave this term generic as a simple wage, W_a.
\Large
U_{a} = W_a \\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~W_a &= \text{Agent's wages} \\
\end{aligned}Obviously every single sale will require some amount of work. All else held equal, a sale that requires less work on the behalf of an agent is preferred over one that requires more work. Also, agents have different preferences around how much work they are willing to do for a single client.
\Large
U_{a} = W_a - \phi_a(E_a)\\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~W_a &= \text{Agent's wages,} \\
\phi_a &= \text{An agent's aversion to effort }(E_a)
\end{aligned}Another aspect of value that agents get out of a sale is in future work. If they do a good job for their client they are more likely to be hired by them again in the future or to be recommended to others.
\Large
U_{a} = W_a - \phi_a(E_a) + \sigma_a(F_a)\\
\phantom{U} \\
\footnotesize
\begin{aligned}
W_a &= \text{Agent's wages,} \\
\text{Where,} ~\phi_a &= \text{An agent's aversion to effort }(E_a) \\
\sigma_a &= \text{How much an agent values future work }(F_a) \\
\end{aligned}Lastly, agents value the satisfaction that comes with a job well done. Even if it didn’t come with the potential to get rehired in the future, it just feels good to guide a client through such a big transaction and see them happy at the end. We’ve already created definitions of how satisfied the buyer and seller are so we can plug those directly into the agent’s utility functions.
\Large
U_{a} = W_a - \phi_a(E_a) + \sigma_a(F_a) + \tau_a(U_{b/s})\\
\phantom{U} \\
\footnotesize
\begin{aligned}
W_a &= \text{Agent's wages,} \\
\text{Where,} ~~~\phi_a &= \text{An agent's aversion to effort }(E_a, \\
\sigma_a &= \text{How much an agent values future work }(F_a) \\
\tau_a &= \text{How much agent values client satisfaction }(U_{b/s})
\end{aligned}Lender’s Profit Function
The lender is simpler than the others. Firstly, since they are a profit seeking business, they get a profit function instead of a utility function. Functionally the same thing, just indicates that they are less susceptible to amorphous things such as personal preferences and that their function is much more rigidly attached to the all mighty dollar. By convention economists use the Greek letter \pi to represent profit so we will use that here.
\Large
\pi_{l} = ? \\At the most basic level, the lender’s profit is the interest they collect on the mortgage.
\Large
\pi_{l} = \rho(M) \\
\phantom{U} \\
\footnotesize
\begin{aligned}
\text{Where,} ~\rho &= \text{Interest rate} \\
M &= \text{Value of the mortgage }\\
\end{aligned}On a more practical level, the lender has to consider a lot of other factors when deciding if they should issue a mortgage. They consider things such as the time value of money, current and future inflation rates, and so on. While these things are important to a lender, they are not important to our model so we are going to leave them out.
One additional factor they consider is important to us, risk of default. Risk of default itself is a very complex factor that could be expanded on. However for our use case we will keep it simple as a single term.
\Large
\pi_{l} = \rho(M) - R\\
\phantom{U} \\
\footnotesize
\begin{aligned}
\rho &= \text{Interest rate} \\
\text{Where,} ~M &= \text{Value of the mortgage }\\
R &= \text{Risk of default}
\end{aligned}Putting It All Together
Pulling it all together, we have a complete model of the utility and profit functions for all of the key players involved in the sale of a house.
In its pure mathematical notation it looks quite intimidating. But it’s just a way to write down concepts that we intuitively understand. If you and I were to sit down and have a non-technical conversation about how a seller would consider if they want to paint their house or leave it as it is we would naturally we would discuss it using the exact same concepts our functions represent. All these models do is put those concepts down on paper in a way that makes studying them easier.
\LARGE
\underline{\text{Seller}}\\
\Large
\phantom{U} \\
U_{s} = P_{h} - C_s - \alpha_s(T_s) - \beta_s(E_s)\\
\phantom{U} \\
\footnotesize
\begin{aligned}
P_{h} &= \text{Sale price of house,} \\
\text{Where,} ~C_s &= \text{Costs incurred by the seller,} \\
\alpha_s &= \text{Seller's sensitivity to time on market }(T_s) \\
\beta_s &= \text{Seller's sensitivity to effort required }(E_s)
\end{aligned}
\phantom{U} \\
\phantom{U} \\
\LARGE
\underline{\text{Buyer}}\\
\Large
\phantom{U} \\
U_{b} = V_b - P_{h} - C_b - \alpha_b(T_b) - \beta_b(E_b) \\
\phantom{U} \\
\footnotesize
\begin{aligned}
P_{h} &= \text{Sale price of house,} \\
V_b &= \text{Value of the house to the buyer} \\
\text{Where,} ~C_b &= \text{Costs incurred by the buyer} \\
\alpha_b &= \text{Buyer's sensitivity to time searching }(T_b) \\
\beta_b &= \text{Buyer's sensitivity to effort required }(E_b)
\end{aligned}
\phantom{U} \\
\phantom{U} \\
\LARGE
\underline{\text{Agents}}\\
\Large
\phantom{U} \\
U_{a} = W_a - \phi_a(E_a) + \sigma_a(F_a) + \tau_a(U_{b/s})\\
\phantom{U} \\
\footnotesize
\begin{aligned}
W_a &= \text{Agent's wages,} \\
\text{Where,} ~\phi_a &= \text{An agent's aversion to effort }(E_a, \\
\sigma_a &= \text{How much an agent values future work }(F_a) \\
\tau_a &= \text{How much agent values client satisfaction }(U_{b/s})
\end{aligned}
\phantom{U} \\
\phantom{U} \\
\LARGE
\underline{\text{Lender}}\\
\Large
\phantom{U} \\
\pi_{l} = \rho(M) - R\\
\phantom{U} \\
\footnotesize
\begin{aligned}
\rho &= \text{Interest rate} \\
\text{Where,} ~M &= \text{Value of the mortgage }\\
R &= \text{Risk of default}
\end{aligned}Up Next in Part 3
We intentionally kept our models simple as we built them in this part. We could use them as they are now in order to conduct our study, however we can do better by specifically expanding on the parts of our model that are most important to our study.
For instance, we will do a better job at defining the wage, W_a that the realtors earn by making it much more specific. \phi_{sa}(P_h), where \phi_{sa} is the commission rate of the seller’s agent, is a much more useful representation of the financial compensation the seller’s agent receives. And the costs of both the buyer and seller will benefit from being broken down a bit further.
In part 3 we will begin to look at the various arguments being made around this debate and consider how our simple model can be expanded upon in order to best analyze the merits of those arguments.