Caillaud Jullien (2003) - Chicken And Egg

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Caillaud, Bernard and Bruno Jullien (2003) "Chicken & Egg: Competition among Intermediation Service Providers", The RAND Journal of Economics, Vol. 34, No. 2. (Summer), pp. 309-328. link pdf


We analyze a model of imperfect price competition between intermediation service providers. We insist on features that are relevantfor informational intermediation via the Internet: the presence of indirect network externalities, the possibility of using the nonexclusive services of several intermediaries, and the widespread practice of price discrimination based on users' identity and on usage. Efficient market structures emerge in equilibrium, as well as some specific form of inefficient structures. Intermediaries have incentives to propose non-exclusive services, as this moderates competition and allows them to exert market power We analyze in detail the pricing and business strategies followed by intermediation service providers.

The Model

The basic set-up is:

  • There are two types of agents [math]i={1,2}\,[/math] who want to be matched to each other.
  • If there are matched the total gains from trade are 1
  • Type 2 agents have a better bargaining position so that [math]u_2 \ge \frac{1}{2} \ge u_1\,[/math], where [math]u_1+u_2 = 1\,[/math]
  • Each agent has a unique match in the opposite population
    • Probability of finding a match without an intermediary = 0
    • Probability of finding a match without an intermediary = [math]\lambda \le 1\,[/math] if both parties are registered with that intermediary
  • Two matchmakers will compete: [math]k \in \{I,E\}\,[/math] (Incumbent and Entrant)
    • Each matchmaker uses the same technology [math]\lambda\,[/math] (at least to start with)
  • Matchmakers earn revenue:
    • Registration fees [math]p\,[/math]
    • Transaction fees [math]t\,[/math] (charged only if trade takes place)
  • and have costs [math]c\,[/math] per registration
  • A type 2 agent seeking a type 1 agent on a matchmaker with [math]n_1\,[/math] type 1 agents has a match probability of [math]n_1 \lambda\,[/math]
  • It is assumed that intermediation is efficient: [math]\lambda \gt c = c_1 +c_2\,[/math]

Sequence of events:

  1. Matchmakers simultaneously (and noncooperatively) set prices
  2. Agents on both sides choose matchmakers to register with, if any
  3. Matches are made and payoffs realized

Divide and Conquer

To take market share from a 100% share incumbent and entrant must subsidize one side of the market (i.e. divide) and leverage network externalities to recruit the other side (conquer).

Equilibrium Concepts

The expected surplus from trade must be weakly positive, so for prices ([math]p_1^k,p_2^k,t^k\,[/math]):

[math]\lambda u_i(1-t^k) - p^k \ge 0 \quad \forall i\,[/math]

Users must maximize thier utility, according to prices [math]P\,[/math], an action [math]k \in \{I,E,\emptyset\}\,[/math] and a distribution of users [math]N\,[/math]:

[math]n_i^k \gt 0 \Rightarrow U_i(P,N,k) = \max_{h \in \{I,E,\emptyset\}} U_i(P,N,h)\,[/math]

The utility of the agents is:

[math]U_i(P,N,k) = n_j^k \lambda u_i(1-t^k)-p_i^k\,[/math]

The profit function of the firms is:

[math]\pi^k(P,N) = \sum_{i=1,2} n_i^k(p_i^k - c_i) + \lambda n_1^k n_2^k t^k\,[/math]

The authors use a monotonicity restriction on market allocations with respect to price.


The dominant firm equilibrium is the most efficient (if inefficiency exists, one firm could undercut prices and serve the whole market).However, the dominant firm can only sustain its position is an entrant can earn zero market share and zero profits (irrespective of its pricing strategy).

If the following condition holds, users will have no incentives to register with E when they expect all others to register with I:

[math]\lambda u_i(1-t^I)-p_i^I \ge -p_i^E \quad \forall i\,[/math]

To get a positive market share E must adopt a divide and conquer strategy and subsidize one group. Dividing i users:

[math]p_i^E \lt p_i^I - \lambda u_i(1-t^I)\,[/math]

This gives [math]n_i^E = 1\,[/math]. Note that prices are negative, so [math]p_i^E\,[/math] is more negative (i.e. the Entrant pays more to the user).

E then conquers j users through the externality:

[math]p_j^E + \lambda u_j t^E \lt \lambda u_j + \inf \{ p_j^I , 0 \} \quad \mbox{as presented in the paper}\,[/math]
[math]p_j^E + \lambda u_j(1-t^E) \lt \lambda u_j(1-t^I) + \inf \{ p_j^I , 0 \} \quad \mbox{as understood by Ed}\,[/math]

And sets [math]t^E = 1\,[/math].

With exclusive intermediation services, the only equilibria are dominant-firm equilibria where [math]t^I=1\,[/math] and registration is subsidized to make zero profits:

[math]\underbrace{p_1^I+p_2^I}_{\mbox{aggregate price}} = \underbrace{c -\lambda}_{\mbox{aggregate surplus}}\,[/math]

Note that both sides are negative as [math]\lambda \gt c\,[/math], so the intermediary pays the customers.


A user has two incentives to add a second provider:

  1. It increases the probability of a match by [math](1-\lambda)\lambda\,[/math]
  2. For double matches the user can pay the lowest transaction fee - he can save on transaction fees with probability [math]\lambda^2 Denoting \lt math\gt n_i^k\,[/math] as the number of i users registered only with k, and [math]n_i^M\,[/math] as the number registered with both there are two efficiency possibilities:

If [math]\lambda(1-\lambda) \lt c\,[/math] then [math]n_i^I = 1\,[/math]

If [math]\lambda(1-\lambda) \gt c\,[/math] then [math]n_i^M = 1\,[/math]

Best Response Analysis

Let [math]r_i^k \equiv p_i^k + \lambda u_i t^k\,[/math] denote the maximum revenue that can be extracted by k from i users.

If all users register with I, then an i user will only multihome if [math]p_i^E \lt 0\,[/math] (as otherwise he would face a loss), and get a subsidy. If only one type of users register with E then he will face a loss (because of the subsidy pricing) as he won't be able to recoup on the transaction fees.

There is some price pair [math](P^I, P^E)\,[/math] such that with [math]n_j^I=1\,[/math] and [math]n_i^M=1\,[/math] that is an equilibrium if:

[math]r_j^E \ge r_j^I\,[/math]
[math]r_j^E \ge \lambda(1-\lambda)u_j + \lambda^2 u_j \max \{t^I,t^E \}\,[/math]

So if E were to try to conquer j users (dividing i users is almost costless) then the most surplus E could get from j users is:

[math]r_j^E \lt \max \{r_j^I; \lambda(1-\lambda)u_j + \lambda^2 u_j \max \{t^I,t^E \} \}\,[/math]

If this (and [math]p_i^E \lt 0\,[/math]) holds then all users will register with E, whether or not they still register with I depends on E's pricing strategy:

  1. Become a second source: [math]t^E \ge t^I\,[/math], so only users that can't match with I will perform transactions with E
  2. Become a first source: [math]t^E \lt t^I\,[/math], so only users that can't match with E will perform transactions with I
  3. Become a sole source: At least one population must not register with I.

The profit when there is multihoming has an upper bound of [math]\underbrace{\lambda(1-\lambda) -c}_{\mbox{Multihoming Agg. Surplus}}\,[/math].

Multihoming is a market allocation if no user of type h prefers registering with I only:

[math]r_h^I \le \lambda(1-\lambda)u_h + \lambda^2 u_h \max \{t^I,t^E\} \quad \forall h\,[/math]

The the minimal surplus for using I as a second source is:

[math]z^I \equiv \min_h \{\frac{ \lambda(1-\lambda)u_h + \lambda^2 u_h t^I - r_h^I}{\lambda^2 u_h} \}\,[/math]

Then users engage in multihoming iff:

[math]\max \{t^I,t^E\} \ge t^I - z^I\,[/math]

If E is a first source then [math]t^E \le t^I\,[/math] and so it must be that [math]z^I \lt 0\,[/math]. If E is a sole source then [math]t^E \le t^I - z^I\,[/math].

Therefore (under pessimistic beliefs) E's best response to prices [math]P^I\,[/math] (if E sells) is either:

  1. If [math]z^I \ge 0 E\,[/math] becomes a first-source with [math]t^E = t^I\,[/math] or uses a second source strategy
  2. If [math]z^I \lt 0 E\,[/math] becomes a sole source with [math]t^E \le t^I - z^I\,[/math] or uses a second source strategy

The profit as a sole source or first source is:

\pi^F = \lambda(1-\lambda)u_2 + \lambda(u_1 + \lambda u_2)t^E - c\,</math>

The profit as a second source is:

[math]\pi^SS = \lambda(1-\lambda) -c\,[/math]

[math]\pi^F \gt \pi^SS\,[/math] iff [math]t^E \gt \frac{\lambda(1-\lambda)u_2}{\lambda(u_1 + \lambda u_2)}\,[/math].

Thus the best response is determined by which of the following is highest:

  1. [math]t^I\,[/math]
  2. [math]t^I-z^I\,[/math]
  3. [math]\frac{\lambda(1-\lambda)u_2}{\lambda(u_1 + \lambda u_2)}\,[/math]

In the first two cases set [math]t^E\,[/math] equal to this level. In the last case [math]t^E\,[/math] is maximal.

Pure Strategies

According to the paper, the market allocation of a pure equilibrium is efficient (everyone from the same population makes the same decision). The paper characterizes the profits in this (multihoming) equilibrium.

Mixed Strategies

For the details on this see pp318-320 of the paper.