Platform competition and price discrimination

Abstract

We study price discrimination strategy of an Incumbent platform that faces competition from an Entrant. We introduce heterogeneous agents on the consumer side, where buyers assign different values to the interaction benefits generated on a platform. We analyse a dominant firm equilibrium, where the incumbent platform offers two versions of its service to the consumers. The results are compared across two scenarios. The first is when sellers are allowed to multi-home and the second is when they can only join a single platform. We find that in case of multi-homing, the platform cannot charge any consumer group more than the marginal cost of the service provided to them. In the singe-homing case, the strength of indirect network effects determines the side from which platform extracts a positive surplus.

8 Appendix

1.1 Proof of Proposition 2

We will again consider the best response for E for a given $P^I$ such that all agents hold favourable beliefs for I and $N^I=(1,1,1)$. Once again, E can give small cash subsidy to the sellers to attract them away from I. In addition, it will also have to set ${r_h}^E$ and ${r_l}^E$ such that buyers receive higher utility on E. The maximum profit that E can get will again be given by Eq. (7).

In this case, it is optimal for I to offer both the premium and the standard service at prices ${r_h}^I=t$ and ${r_l}^I=0$, respectively. If for instance, I does not offer the premium service such that $r_h=\emptyset$, all high type buyers get a utility of $\alpha n_s-r_l$ on I. Here, E can offer premium services at a cost of t per buyer and set ${r_h}^E$ such that

$$\begin{aligned} (\alpha +u_{h(A)})n_s-{r_h}^E > \alpha n_s -{r_l}^I \end{aligned}$$

(13)

This means any price set by E for the high type buyers such that ${r_h}^E-{r_l}^E<u_{h(A)}$ would result in higher utility for high type buyers. This would lead to E making positive profits. Therefore, not offering the premium service is not a best response for I.

1.2 8.1 Intermediate cases for the extent of indirect network effects

1.3 Case III: $\varvec{u_{h(A)}+\alpha >\pi _h}$ & $\varvec{\alpha <\pi _l}$

To give a complete picture, we also consider the case, where a high type buyer’s benefit from the participation of sellers is higher than the benefit sellers get from their participation. On the other hand, the low type buyer’s net benefit from the participation of sellers is lower than the value that sellers derive from them. We first discuss the scenario when sellers multi-home following which we present the results when both agents can only join a single platform.

Buyers single-home and sellers multi-home:The optimal strategy for I is laid out in the following proposition.

Proposition 8

When ${u_{h(A)}+\alpha >\pi _h}$ & ${\alpha <\pi _l}$, there exists a dominant firm equilibrium such that ${P^I}^*=(\pi _h+\pi _l, t, \alpha -\pi _l)$ and ${N^I}^*=(1, 1, 1)$, where I makes a profit of $\pi _h-\alpha$.

Suppose E gives a small cash subsidy to the sellers to guarantee their participation, it cannot attract the buyers, because it will be unable to offer them prices lower than I. On the other hand, lets us say it considers subsidising the buyers to attract sellers. First, E cannot offer an amount of subsidy to high type buyers which it can recoup on the seller side. This is because high type buyers earn a utility that is greater than the benefit sellers derive from their participation net of the costs of offering a premium service. Neither can E attract a low type buyer, because the utility they get from joining I is equal to $\pi _l$, the maximum amount of subsidy it can sustain on the seller side.

Both buyers and sellers single-home: The following proposition lays down the optimal strategy for I.

Proposition 9

When $u_{h(A)}+\alpha >\pi _h$ & $\alpha <\pi _l$, there exists a dominant firm equilibrium, where I sets ${P^I}^*$ such that $w=min\{-\pi _h-\pi _l,\pi _h+\pi _l-u_{h(A)}-2\alpha +t\}$, $r_h= u_{h(A)}+\alpha$ and $r_l=\alpha$. Here, I captures the whole market and earns profit equal to $u_{h(A)}+2\alpha +min\{-\pi _h-\pi _l,\pi _h+\pi _l-u_{h(A)}-2\alpha$}.

The strategy for I is identical to the one specified in Proposition 3. Even though $\pi _l>\alpha$, I will not give a subsidy to the low type buyer, because the incentive compatibility constraint of the high type will be violated. Therefore, I sets its prices so as to extract the entire surplus of both buyers.

Here, any subsidy given by E to attract the the low type buyer will also be taken by the high type buyer, because it gives them positive utility. Therefore, to prevent defection of buyers to E, I guarantees sellers the value they derive from the buyers net of the cost of premium service irrespective of whether buyers participate or not.

1.4 Case IV: $\varvec{u_{h(A)}+\alpha <\pi _h}$ & $\varvec{\alpha >\pi _l}$

Here, sellers value the participation of high type buyers more than the high type buyer value the sellers’ participation. On the other hand low type buyers derive a higher value from the sellers than the sellers get from the participation of low type buyers.

Buyers single-home and sellers multi-home: Let $\dfrac{\pi _h+\pi _l-t}{2}-u_{h(A)}-\alpha =\phi$. We derive the equilibrium conditions when $\phi >0$.

Proposition 10

When ${u_{h(A)}+\alpha <\pi _h}$ & $\alpha >\pi _l$, there exists a dominant firm equilibrium such that ${P^I}^*=(\pi _h+\pi _l,-\phi ,-\phi )$, where ${N^I}^*=(1,1,1)$ and ${{\Pi }^I}^*=\pi _h +\pi _l-t-\phi$

I subsidises the high type buyer in such a way that it is not possible for E to offer a higher subsidy. In addition, the low type buyer is offered a subsidy by I even though it generates lower benefits for sellers than what the sellers generate. This is because I cannot sustain a negative price for the high type without offering an equivalent or lower price to the low type buyer. E has two strategies to consider which are, (i) to subsidise the buyers and earn profits from the sellers or (ii) to subsidise the sellers to earn profits from the buyers.

To attract a high type buyer E has to offer it a subsidy greater than $u_{h(A)}+\alpha +\phi$ and also offer the same amount of subsidy to the low type buyer. This is amount is greater than $\pi _h+\pi _l$, the amount it can recover from the sellers. In addition, if E decides to offer a subsidy to the sellers, it cannot attract the buyers with lower prices, because the amount of utility they earn on E will be lower than I. When $\phi <0$, the equilibrium conditions are given by the following proposition.

Proposition 11

There exists a dominant firm equilibrium such that ${P^I}^*=(\pi _h+\pi _l, t,0)$, where ${N^I}^*=(1,1,1)$ and ${{\Pi }^I}^*=\pi _h +\pi _l$.

The proof follows a similar line of argument as explained in the above proposition.

Both buyers and sellers single home: Proposition 6 and 7 determine the equilibrium conditions for this case. This is because given that the platform offers subsidies to a high type buyer, it will also offer the subsidy to the low type buyer to satisfy their incentive compatibility constraints.

1.5 8.2 Numerical examples

Example 1

Let us assume specific values for the parameters. These examples provide a better exposition of the results. For our first example, we assume the following values to explain the results given in Proposition 1.

$$\begin{aligned} u_{h(A)}=10, \alpha =10, t=3, \pi _h=10, \pi _l=5. \end{aligned}$$

(14)

Given these parameters, suppose I follows a pricing strategy such that $P^I=(15,20, 10)$ which captures the entire surplus of the agents. Lets assume that all buyers and sellers join the platform. Hence, the utility of all the agents when I sets $P^I$ would be

$$\begin{aligned} {U_h}^I= (u_{h(A)} + \alpha )n_s-{r_h}^I=(10+10)\cdot 1-(20) \ge 0,\nonumber \\ {U_l}^I=\alpha n_s-{r_l}^I=(10)\cdot 1-10=0,\nonumber \\ {V_s}^I=(\pi _ln_{bl}+\pi _hn_{bh})-w^I=(10\cdot 1+5\cdot 1)-15=0. \end{aligned}$$

(15)

The high type buyer receives zero utility by purchasing A at $r_h$. On the other hand, a low type buyer consumes the basic version by paying the price $r_l$. She does not purchase A, because doing so gives her a negative utility.

What would be a profitable strategy for E here? It can set $w^E$ slightly less than 0, say $w^E=-1$, and offer premium service A at a per buyer cost of $t=3$. The sellers will join E irrespective of the buyers’ decision to join, because they can multi-home. They earn a positive utility of one by joining E. E can then charge high type and low type buyers, ${r}^E_h=19$ and ${r}^E_l=9.5$, respectively. The utility of all the agents, if they join platform E, would be

$$\begin{aligned} {U_h}^E= (u_{h(A)} + \alpha )n_s-{r_h}^E=(10+10)\cdot 1-19= 0.5,\nonumber \\ {U_l}^E=\alpha n_s-{r_l}^E=(10-9.5)\cdot 1=0.5,\nonumber \\ {V_s}^E=(\pi _ln_{bl}+\pi _hn_{bh})-w^E=(10\cdot 1+5\cdot 1)+1=16. \end{aligned}$$

(16)

All the buyers would join E as they get a higher utility than by joining I. This would leave I with zero profits and E would make a profit of

$$\begin{aligned} {\Pi }^E= 19+9.5-1-3=24.5. \end{aligned}$$

(17)

It is easy to check that ICCs of both types of buyers will be satisfied for $P^E=(-1,19,9.5)$. Therefore, for a given ${P^I}$, E needs to charge sellers a negative price, almost equal to zero, and undercut the consumer prices offered by I to enter successfully.

Given E’s threat of entry, the optimal strategy for I is to set ${P^I}^*=(\pi _h+\pi _l,t,0)=(15,3,0)$. Here, I is offering A at ${{r_h}^I}^*=3$ and the basic version at ${{r_l}^I}^*=0$, which is the marginal cost of servicing a high type and a low type buyer, respectively. It subsequently charges the sellers their full surplus and sets ${w^I}^*=15$. At ${P^I}^*$, even if E sets $w^E<0$, it cannot offer more competitive prices to both types of buyers. This is because I is already charging both the buyers their marginal cost, and offering them lower prices would generate losses for E. Hence, all agents join I and earn the following level of utilities.

$$\begin{aligned} {U_h}^I= (u_{h(A)} + \alpha )n_s-{r_h}^I=(10+10)\cdot 1-3= 17,\nonumber \\ {U_l}^I=\alpha n_s-{r_l}^I=10\cdot 1-0=10,\nonumber \\ {V_s}^I=(\pi _l n_{bl}+\pi _h n_{bh})-w^I=(10\cdot 1+5\cdot 1)-15=0. \end{aligned}$$

(18)

It is easy to see that a high type buyer will purchase A, because it gives her a higher utility. On the other hand, a low type buyer would never register for A as it would earn a utility of $10-3=7$ which is less than 10. Thus, ${P^I}^*$ ensures that incentive compatibility constraints of both types of buyers are satisfied. The profit made by I is

$$\begin{aligned} {\Pi }^I=w\cdot n_s +p_h\cdot n_b-t\cdot n_b=15\cdot 1+3\cdot 1-3\cdot 1=15. \end{aligned}$$

(19)

The platform extracts the entire surplus of the sellers. Conversely, it offers premium service A at the marginal cost of servicing a high type buyer. Moreover, it provides its basic service for free.

Example 2

We assume the same values as we did in Eq. (14) but consider that both sides single-home. This example corresponds to the results in Proposition 3.

The optimal pricing strategy for I will be ${P^I}^*=(-12,20, 10)$, where the incumbent platform offers premium service as well. The utility of all the agents by joining platform I is given by

$$\begin{aligned} {U_h}^I \ge 0,\quad {U_l}^I\ge 0,\quad \text {and}\quad {V_s}^I=27. \end{aligned}$$

(20)

To attract sellers away from the incumbent platform, the entrant has to offer a subsidy greater than 27 to the seller. This subsidy is greater than what can be recouped on the buyer side (the maximum utility that buyers derive due to the presence of sellers on the platform is 30 and subtracting t from it leaves a surplus of 27). If on the other hand, the entrant tries to attract buyers with a negative subsidy, it will be unable to induce sellers to join, because the utility that sellers will earn because of the presence of buyers on E will be equal to 15, smaller than the subsidy offered by the incumbent platform.

In this case, the platform extracts all the surplus from buyers, i.e., the side that receives higher benefits from the participation of the other side. Unlike in the multi-homing case, it is the sellers that are subsidised as they generate higher value for the buyers than what buyers generate for the sellers. The profit of the platform is equal to

$$\begin{aligned} {\Pi }^I=20\cdot 1+10\cdot 1-12\cdot 1-3\cdot 1=15. \end{aligned}$$

(21)