Section 10.4 Platform price markups
Subsection 10.4.1 Reviewing the price markup and Lerner Index
Recall that a monopolist facing linear demand operates on the relatively elastic part of the demand curve, where \(\epsilon_D \lt -1\text{.}\) The degree of elasticity determines how drastically the monopolist can increase its price relative to its marginal cost. A perfectly competitive firm, on the other hand, charges a price exactly equal to its marginal cost! We can derive a formula for the monopolist’s price markup, defined as \(\frac{P}{MC}\text{,}\) the ratio of the price it charges to its marginal cost. Using the product rule to differentiate the monopolist’s total revenue function \(TR(Q) = P(Q)Q\text{,}\) then using the monopolist’s profit maximizing condition that \(MR = MC\) at \(Q^*\text{,}\) it can be shown that
\begin{equation*}
\frac{P}{MC} = \frac{1}{1 + \frac{1}{\epsilon_D}}
\end{equation*}
Notice that for a perfectly competitive firm who faces an (infinitely elastic) demand curve, the price markup is essential 1, i.e. no markup. Again, a monopolist will operate on the elastic part of a linear demand curve. 1 For very elastic demand, the price markup is high, while for moderately elastic demand, the price markup is still greater than one but smaller in absolute value.
The Lerner Index serves as another measure of monopolists’ market power, defined as
\begin{equation*}
\frac{P - MC}{P} = -\frac{1}{\epsilon_D}
\end{equation*}
The Lerner Index takes values from 0 (no market power, infinitely elastic demand) to 1 (maximum market power, the most inelastic but still relatively elastic demand, such as \(\epsilon_D = -1.1\)).
Subsection 10.4.2 Elasticity and platform pricing: a conundrum?
What is the relationship between the price elasticity of (quasi-)demand from buyers, the price elasticity of (quasi-)demand from sellers, and the optimal prices chosen by the platform? Since the individual demand functions are independent, we can compute price elasticities in the standard way as
\begin{equation*}
\epsilon_B = \frac{dq_B}{dP^B}\frac{P^B}{q^B}\text{ and } \epsilon_S = \frac{dq_S}{dP^S}\frac{P^S}{q^S}
\end{equation*}
When evaluated at equilibrium, they give the elasticities of buyers’ and sellers’ demands. In the symmetric linear demand case, recall that \(MC = c = \frac{1}{2}\text{,}\) and that \(P^{B*} = P^{S*} = \frac{1}{2}\text{.}\) On the individual demands, therefore, there is no price markup! Upon looking at the individual price elasticities, we can calculate that \(\epsilon_B = \epsilon_S = -1\text{.}\) Since individual demands are unit elastic at these prices, this is difficult to reconcile with the monopolist case described above because a monopolist must operate on the elastic part of the demand curve. 2
Now, consider the asymmetric demands case. Recall that the buyers have flatter (more relatively elastic) demand in general, while sellers have steeper (more relatively inelastic) demand in general. But what happens when we compute elasticities at the optimal prices? Interestingly, when we examine the different demands case, where \(c = \frac{1}{2}\text{,}\) \(P^{B*} = \frac{1}{6}\text{,}\) and \(P^{S*} = \frac{2}{3}\text{,}\) something odd seems to show up. With \(q^B = \frac{2}{3}\text{,}\) the price elasticity of (quasi-)demand from buyers is given by \(\epsilon_B = -\frac{1}{2}\text{;}\) demand from buyers is relatively inelastic, and buyers are charged a lower price! Conversely, given \(q^S = \frac{1}{3}\text{,}\) the price elasticity of (quasi-)demand from sellers is \(\epsilon_S = -2\text{:}\) sellers’ demand is relatively elastic and sellers are charged a higher price! This runs counter to our intuitive hypothesis from earlier, which seemed to be confirmed. What’s going on here?
An explanation can be found by going back to the original monopolist’s problem. Recall that the monopolist operates on the elastic part of the demand curve, and as a result, always has at least some (however small) price markup. And indeed, as we see here, the monopolist charges a markup to the demanders (here, the seller side of the market) whose demand is elastic! What about the buyers? Their demand is inelastic at the monopolist’s optimal prices: consequentially, the monopolist gives the buyers side of the market not a price markup, but a price markdown!
On the most straightforward level, this should make sense. Any successful transaction must include buyers, but if \(c = \frac{1}{2}\text{,}\) and the monopolist attempts to even charge a price equal to its marginal cost, the quantity demanded from buyers (\(q^B = 1 - 2P^B\)) would be zero! So, to get even one buyer on board, the monopolist must mark down to the buyers side of the market.
But let’s take a step back, and assume the marginal cost is just \(c\text{.}\) In the different demands case, \(P^{B*} = \frac{c}{3}\) and \(P^{S*} = \frac{c}{3} + \frac{1}{2}\text{.}\) From here, we can see that \(q^{B} = 1 - \frac{2}{3}c\) and \(q^S = \frac{1}{2} - \frac{1}{3}c\text{.}\) Or, rearranged,
\begin{equation*}
q^B = 2(\frac{3 - 2c}{6})\text{ and } q^S = \frac{3 - 2c}{6}
\end{equation*}
The buyers’ side will always have double the quantity demand of the sellers’ side, regardless of the value of \(c\text{.}\) On the price elasticities of (quasi-)demand,
\begin{equation*}
\epsilon_B = -2\frac{\frac{c}{3}}{2(\frac{3 - 2c}{6})} = -\frac{2c}{3 - 2c}
\end{equation*}
and
\begin{equation*}
\epsilon_S = -\frac{\frac{c}{3} + \frac{1}{2}}{\frac{3 - 2c}{6}} = -\frac{2c + 3}{3 - 2c}
\end{equation*}
First, notice that when \(c = \frac{1}{2}\text{,}\) we get the numerical elasticities above. Most importantly, though, sellers’ demand will always be more elastic than buyers’ demand at optimal platform pricing! And sellers will always face a higher price than buyers! So, this counterintuitive anomaly - that the more inelastic side of the market gets charged the lower price - is more pervasive than just our simple numerical example.
Subsection 10.4.3 An adjusted Lerner index for platforms
It turns out that this phenomenon is not the result of an ironclad pricing rule, but, rather, entirely by design! To see why, first let’s transform the general profit maximization problem for the platform by taking the log:
\begin{equation*}
\log{\pi} = \log{(P^B + P^S - c)} + \log{(q^B)} + \log{(q^S)}
\end{equation*}
Solving this profit maximization problem is identical to solving the original one, so we can differentiate to get first order conditions:
\begin{equation*}
\frac{d\log\pi}{dP^B} = \frac{1}{P^B + P^S - c} + \frac{1}{q^B}\frac{dq^B}{dP^B} = 0
\end{equation*}
and
\begin{equation*}
\frac{d\log\pi}{dP^S} = \frac{1}{P^B + P^S - c} + \frac{1}{q^S}\frac{dq^S}{dP^S} = 0
\end{equation*}
Let’s tackle the buyers’ side first. If we multiply both sides by \(P^B\text{,}\) we get that
\begin{equation*}
\frac{P^B}{P^B + P^S - c} = -\epsilon_B
\end{equation*}
where the right-hand side contains the price elasticity of demand from the buyers’ side of the market. Now flipping both sides, we see
\begin{equation*}
\frac{P^B + P^S - c}{P^B} = -\frac{1}{\epsilon_B}
\end{equation*}
which we can slightly rearrange to
\begin{equation*}
\frac{P^B - (c - P^S)}{P^B} = -\frac{1}{\epsilon_B}
\end{equation*}
This is essentially the Lerner Index on the buyers’ side of the market - with one minor change! It gives the platform’s markup on buyers relative to the platform’s opportunity cost of a price increase on the buyers’ side. The loss of a transaction due to an increase in buyers’ price \(P^B\) has opportunity cost \(c - P^S\) because the cost of the transaction \(c\) is partially offset by the \(P^S\) collected from the sellers’ side of the market. Similarly, it can be shown that under optimal platform pricing, it must be true that
\begin{equation*}
\frac{P^S - (c - P^B)}{P^S} = -\frac{1}{\epsilon_S}
\end{equation*}
On both sides of the market, relative to the true opportunity cost of a price increase, the standard Lerner Index relationship between price markup and price elasticity of demand holds!
In our different demands example, with \(P^B = \frac{1}{6}\text{,}\) \(P^S = \frac{2}{3}\text{,}\) and \(c = \frac{1}{2}\text{,}\) let’s interpret. On the sellers’ side, the opportunity cost of an increase in \(P^S\) is \(c - P^B = \frac{1}{3}\text{,}\) and the proportion of the price charged to sellers which is markup above opportunity cost is exactly one half - consistent with a price elasticity of demand for sellers of -2. On the buyers’ side, since the platform will more than cover the marginal cost of the transaction with the price it collects from sellers, the opportunity cost of an increase in \(P^B\) is negative: \(c - P^S = -\frac{1}{6}\text{.}\) 3 The difference between the buyers’ price and the opportunity cost of an increase in that price doubles the price itself, and links to the price elasticity of demand for buyers value of \(-\frac{1}{2}\text{.}\)
Subsection 10.4.4 Pricing conundrum solved
So, why does this confusion occur? Does the platform’s pricing strategy adhere to our intuitive understanding of subsidizing the elastic side and marking up the inelastic side?
In short, we can say that our intuition is confirmed.
There are what one might call feedback effects between price and price elasticity of demand. There is a common conception of demand elasticity relating to the slope of a linear demand function: flatter curves more elastic, steeper ones more inelastic. And, generally, it is true that for a monopolist platform facing two different demand curves, given a constant \(P = P^B + P^S\text{,}\) it generates more revenue to charge a lower price to the side with the flatter curve and a higher price to the side with the steeper curve.
But, upon further examination, we know that along the same linear demand function, higher prices correspond to higher values of \(\epsilon_D\text{,}\) while lower prices correspond to lower values of \(\epsilon_D\text{.}\) That is, elasticity changes along different points on the same linear demand curve! Sometimes, the calculation of an exact value of \(\epsilon_D\) at a given point along a demand curve is referred to as a point elasticity for precisely this reason.
Therefore, when the platform implements a pattern of optimal pricing which lowers the price to the side with the flatter demand curve, it slides downward along that curve, pushing the value of \(\epsilon_D\) to be more inealstic! Similarly, the platform increases the price to the side with the steeper curve, and in doing so, moves \(\epsilon_D\) to be more relatively elastic.
That is, the monopolist is still increasing the price on the side of the market “conventionally understood” to be more inelastic and decreasing the price on the side of the market understood to be more elastic. What we observe is the price on the inelastic side of the market driven high enough to render demand relatively elastic, and the price on the elastic side of the market driven low enough to render demand relatively inelastic. This result is well examined by Krueger (2009) 4
