WARNING: This model is significantly more advanced than the rest of this course. I’ve included it here as additional background and for those students curious about the structure of more sophisticated models of firm theory.
Section 10.3 Two-sided platforms: a theoretical model
In Depth 10.3.1.
We introduce the multisided platform firm model here. In this benchmark multisided platform model, the platform is two-sided and the firm is a monopolist. 1 The multisided platform here will be in the business of producing transactions, denoted by \(Q\text{.}\) Both buyers (\(B\)) and sellers (\(S\)) will demand these transactions via the platform. For each transaction, the platform incurs a per-unit cost of \(c > 0\text{.}\) This is the unit cost of Uber in facilitating a driver-passenger connection, the cost of Amazon posting and linking a sale, or the cost of OpenTable to fill a seat at a restaurant. The platform can charge one price to the buyer side of the market, \(P^B\text{,}\) and a different price to the seller side of the market, \(P^S\text{.}\)
Buyers demand transactions according to the buyers’ demand function, \(q^B(p^B)\text{,}\) while sellers demand transactions according to the sellers’ demand function, \(q^S(p^S)\text{.}\) Rochet and Tirole (2003) calls these individual demands quasi-demand functions. For simplicity here, we will model the number of actual transactions as the product of the (assumed independent) demand functions from each side of the market. 2 That is,
\begin{equation*}
Q(p^B, P^S) = q^Bq^S
\end{equation*}
The platform makes \(P^B\) per transaction from each buyer, \(P^S\) per transaction from each seller, and incurs cost of \(c\) per transaction. Therefore, the (monopolist) platform maximizes profit
\begin{equation*}
\pi(P^B, P^S) = P^BQ + P^SQ - cQ = (P^B + P^S - c)q^Bq^S
\end{equation*}
In Depth 10.3.2.
Demand in percentage terms
We will use demand functions in the model below, and typically, demand functions give the number of units demanded at a given price. For example, if \(Q_D = 200 - P\text{,}\) then at a price of 40, there are 160 units demanded. At a price of 0, there are 200 units demanded.
We will need a minor but important re-interpretation of demand in our model. Instead of measuring a quantity as a number of units, we will measure quantity as the percentage of all possible units demanded. For example, in the demand function above, the maximum number of units which could be demanded is 200, since this is the number of units demanded when the good is free! Therefore, it is equivalent to make the following two statements:
- At a price of 40, there are 160 units demanded;
- At a price of 40, 80% of the market is demanded;
To emphasize this interpretation, we will write demand functions in small terms. For example, we will see demand written like
\begin{equation*}
Q_D = 1 - \frac{1}{4}P
\end{equation*}
Therefore, at a price of 3, \(Q_D = 0.25\text{.}\) This is not saying that there is one fourth of a unit demanded, but, rather, that 0.25 = 25% of the market is demanded at the price of 3. Do not be thrown off by the decimals as quantities demanded - they are percentages!
Subsection 10.3.1 Linear case (symmetric)
As a first pass, let us solve a linear demand version of the model. Let \(q^B = 1 - P^B\) and \(q^S = 1 - P^S\text{.}\) This gives platform profit as
\begin{equation*}
\pi(P^B, P^S) = (P^B + P^S - c)(1 - P^B)(1 - P^S)
\end{equation*}
How does the platform simultaneously choose the pair of prices \((P^B, P^S)\) to maximize its profit? Optimization dictates that the optimal prices must satisfy the following two equations 3 :
\begin{equation*}
q^B - (p^B + p^S - c) = 0
\end{equation*}
and
\begin{equation*}
q^S - (p^B + p^S - c) = 0
\end{equation*}
Substituting in quasi-demands and solving this system of equations yields optimal prices as \(P^{B*} = P^{S*} = \frac{c + 1}{3}\text{.}\) If, for example, the unit cost is \(c = \frac{1}{2}\text{,}\) then \(P^{B*} = P^{S*} = \frac{1}{2}\text{,}\) \(q^B = q^S = \frac{1}{2}\text{,}\) and \(Q^* = \frac{1}{4}\text{.}\) Given identical demands, the platform charges identical prices to each side of the market.
Subsection 10.3.2 Linear case (asymmetric)
What if demands differ? Let \(q^B = 1 - 2P^B\) and \(q^S = 1 - P^S\text{.}\) If we look at these two demand functions, the buyers’ quasi-demand function is flatter than the sellers’ quasi-demand function. This allows us to formulate a hypothesis based on our earlier discussion: given the differences across sides of the market, we expect the buyers’ side to be the subsidy side and the sellers’ side to be the money side.
Now the profit maximizing platform chooses prices to maximize
\begin{equation*}
\pi(P^B, P^S) = (P^B + P^S - c)(1 - 2P^B)(1 - P^S)
\end{equation*}
Here, the optimal pricing conditions are given as 4 :
\begin{equation*}
q^B - 2(p^B + p^S - c) = 0
\end{equation*}
and
\begin{equation*}
q^S - (p^B + p^S - c) = 0
\end{equation*}
Substituting in quasi-demands here yields the system
\begin{equation*}
1 - 4P^B - 2P^S + 2c = 0
\end{equation*}
and
\begin{equation*}
1 - 2P^S - P^B + c = 0
\end{equation*}
This gives optimal prices as \(P^{B*} = \frac{c}{3}\) and \(P^{S*} = \frac{c}{3} + \frac{1}{2}\text{.}\) With \(c = \frac{1}{2}\text{,}\) we see that \(P^{B*} = \frac{1}{6}\) and \(P^{S*} = \frac{2}{3}\text{.}\) Then, \(q^B = \frac{2}{3}\text{,}\) \(q^S = \frac{1}{3}\text{,}\) and \(Q^* = \frac{2}{9}\text{.}\) Given different demands, different optimal prices are offered to different sides of the market. Notice that since the per unit cost is \(\frac{1}{2}\text{,}\) the sellers’ side of the market is charged a price above marginal cost, while the buyers’ side of the market is charged a price below marginal cost! The buyers’ side in effect subsidizes the sellers’ side and our hypothesis is confirmed.
In Depth 10.3.3.
Platforms ... and more? Continued.
We have seen how platforms maximize profit by subsidizing one side of the market and marking up prices on the other. But many platforms engage in input-to-output production as well. We’ll look at two examples of how input-to-output production from platforms can augment platform profit maximization.
Case 1: iPhone and platform bundles. In the iPhone, Apple produces both the platform (the software iOS is a platform which is used to run applications) and the hardware - the phones themselves. In its platform capacity, Apple develops and supports the platform iOS, whose applications serve to connect iPhone users (buyers) with app developers (sellers.) Apple earns money from the developers’ side (it costs $99 to register as a developer who can submit apps to their App Store), as well as from any transactions which occur (reports suggest Apple earns 30% of each app sale.)
But importantly, Apple sells iPhone hardware as well. In order to use iOS, a user must own an iPhone, and Apple is a monopolist in the production of iPhones. In this sense, Apple bundles their platform with a physical consumer product it sells. Apple’s market power in the sale of iPhones can allow it to markup its price to consumers, but the tie-in with the platform is critical. Since there is one-to-one monopoly power between platform and hardware - iOS only functions on iPhones, and iPhones can only operate via iOS - consumers’ demand for one requires consumer demand for the other. Apple is able to augment its hardware monopoly by pairing it to its iOS monopoly: consumers are more willing to buy an iPhone when its iOS has apps they like. As a result, Apple successfully uses a combination of platform theory and traditional input-to-output production in a complementary way.
Case 2: Netflix original content and subsidy substitutes. Netflix operates as a video streaming platform, where buyers pay for subscriptions to watch content - movies and TV shows, primarily - produced elsewhere. Netflix pays content creators to get their content on the platform (subsidy side), and charges individual users for monthly subscriptions (the money side.) Recently, however, Netflix has generated more original content, including TV shows, miniseries, documentaries, and more. The platform prioritizes this Netflix original content, running previews upon user login.
Platform theory explains the rationale. Netflix is a two-sided platform, and faces pressure to maintain a healthy subsidy side of the market. But, by definition, the subsidy side of the platform is the side where Netflix loses money. So, in an effort to compensate for these losses, Netflix looks to substitute its own original content on the subsidy side of the market. Netflix both maintains the platform but also becomes a seller of content. Rather than subsidize itself, Netflix invests in content creation, which is an input-to-output production process. The process might allow it to decrease the size of its subsidy, while also making an investment in its own content production 5 . If the investment pays off (shows like House of Cards and Orange is the New Black indicate it may), Netflix can further diminish its subsidy to external content producers!
