Dynamic Price Competitionwith Network Effects

Luıs Cabral

NYU and CEPR

1

Motivation: network effects

• Direct network effects:

− Typesetting software. Desktop OS.

• Complementary products or services:

− Handheld computers. Video-game consoles. Printers.

• Tariff mediated effects:

− Wireless telecommunications. ATM banking.

2

Motivation: literature

• Early research: simple models, specific points.

• Large theoretical and empirical literature, mostly based onstatic models. Limitations:

− Multiple equilibria / comparative statics.

− Adjustment to equilibrium (e.g., entry).

• Recent dynamic models.

• Related literatures: evolutionary dynamics, social networks.

3

Overview of model and results

• Model

− Two proprietary networks compete for consumers.

− Consumers die over time and are replaced by new consumers.

− In each period, profit and consumer surplus flows are afunction of network size.

• Results

− Price function: harvesting vs investment.

− Mkt share dynamics: market dominance vs rev. to the mean.

− Application: network access regulation in wireless telecoms.

4

Outline

• Introduction.

• Dynamic framework.

• Equilibrium characterization.

• Application: access charges.

5

Outline

• Introduction.

• Dynamic framework.

• Equilibrium characterization.

• Application: access charges.

6

Basic framework and timing

• Discrete time, infinite periods.

• Two firms (i, j), η consumers; i+ j = η.

• Timing within each period:

− One consumer is born and chooses a network.

− Firms and consumers receive profit, surplus flows.

− One consumer dies.

7

Consumers

...................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...........

........................................................................

....................................

...........................................................................................................

.............................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................................

........................................................................

............................................................................................................

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

• • • • • • • • •

• ••

• ••

• •

•

tt′

One birth, one death per period; constant population size η.

8

Consumers

• Upon birth, choose one of the networks.

• Pay p(i) to join network i. Receive one-shot payoff ζi.

• Remain with network i for life. (The meaning of “life.”)

• Receive λ(i) each period.

• Die at end of period with probability 1/η.

• Let u(i) be discounted expected value of future λ’s.

9

Firms

• Simultaneously set p(i) (price for newborn consumer).

• Gain new consumer with probability q(i).

• Receive profit flow θ(i) from installed base (aftermarket).

• At end of period, lose one consumer with probability i/η.

• Let v(i) be firm value.

10

Important assumptions

• Consumers remain locked-in to network for life.

− Similar to Beggs-Klemperer’s switching costs model.

• In each period, aftermarket payoffs are given by reducedform θ(i), λ(i).

− Implicitly assumes no commitment to future aftermarket prices(e.g., call prices in case of wireless telecoms).

11

Consumer demand

Choose network i iff

ζi − p(i) + u(i+ 1) > ζj − p(j) + u(j + 1)

or simply

ξi ≡ ζi − ζj > p(i)− p(j)− u(i+ 1) + u(j + 1) ≡ x(i)

Resulting demand

q(i) = 1− Φ(x(i)

)where x(i) ≡ and Φ is the c.d.f. of ξi.

12

Consumer’s value function

.................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................

................................................................

................................................................

...................

................................................................

................................................................

...................

...................................................................................................................................................

u(i)

u(i− 1)

0

u(i+ 1)

u(i)

i−1η

jη

1η

q(i− 1)

q(j)

q(i)

q(j − 1)

u(i) = λ(i) + δ

13

Firm’s value function

.......................................................................................................

.....................................................................................................................................................................................

......................................................................................................................

................................................................

......................................................

................................................................

......................................................

......................................................................................................................

v(i)

v(i+ 1)

v(i− 1)

v(i)

q(i)

q(j)

p(i) + θ(i+ 1) + δ

θ(i) + δ

i+1

η

j

η

i

η

j+1

η

v(i) =

14

Notation

p(i) Price (for new consumer).

q(i) Probability of a sale.

u(i) Consumer’s value.

v(i) Firm’s value.

λ(i) Consumer’s benefit flow.

θ(i) Firm’s profit flow.

Φ(ξi) Distribution of preference parameter (ξi ≡ ζi − ζj).

δ Discount factor.

(NB: Greek exogenous, Roman endogenous.)

15

Notation

P (i) ≡ p(i)− p(j)H(i) ≡ h(i)− h(j)

U(i) ≡ u(i+ 1)− u(j + 1)

W (i) ≡ w(i)− w(j)

16

Assumptions on Φ

• Continuously differentiable.

• Symmetric about zero: φ(x) = φ(−x).

• Positive density: φ(x) > 0, ∀x.

• Monotone hazard rate: Φ(x)/φ(x) strictly increasing.

17

Properties of θ, λ

Property 1 (INB). λ(i) is increasing.

Property 2 (IRS). θ(i) and θ(i+ 1)− θ(i) are increasing.

• Some analytical results depend on one or two of theseproperties.

• Application (wireless telecoms) will include conditions suchthat they hold; and an example where they don’t.

18

State transition

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

•

Firm i’snetwork size

Time

i+ 1

i

i− 1

t t+ 1

18

State transition

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

..........................................................

..........................................................

..................................................... q(i)

(j

η

)...............................................................................

prob. birth...............................................................................

prob. death

•

Firm i’snetwork size

Time

i+ 1

i

i− 1

t t+ 1

18

State transition

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

....................................................................................................................................................... q(i)(i

η

)+ q(j)

(j

η

)•

Firm i’snetwork size

Time

i+ 1

i

i− 1

t t+ 1

18

State transition

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

...

......................................................................................................................................................................... q(j)(i

η

)•

Firm i’snetwork size

Time

i+ 1

i

i− 1

t t+ 1

19

Equilibrium

• Markov perfect equilibrium (anonymous).

• State of game: i (or j = η − i).

• Firm strategy: p(i).

• Consumer strategy: x(i) (threshold of ξi).

• Equilibrium induces Markov process on state i withtransition matrix M and stationary distribution d(i).

20

Outline

• Introduction.

• Dynamic framework.

• Equilibrium characterization.

• Application: access charges.

21

Equilibrium characterization

• Method:

− Generic equilibrium characterization (analytical; partial).

− Equilibrium characterization for η = 2, δ ≈ 0, ψ ≈ 0(analytical; complete).

− Computation for other parameter values (numerical).

• Results:

− Existence and uniqueness.

− Pricing function (harvesting vs investing).

− Market share dynamics (network effect vs price effect).

− Profits and welfare.

22

Uniqueness

Lemma. Given {U(i),W (i)}, there exist unique {p(i), q(i)}satisfying equilibrium conditions; given {p(i), q(i)}, there existunique {u(i), v(i)} satisfying equilibrium conditions.

• Intuition: simultaneous vs sequential consumer choices.

• First part similar to Caplin and Nalebuff (1991).

• Used in uniqueness results below.

• Used in numerical computation algorithm.

23

Pricing: first-order condition

p(i) =q(i)−q′(i)

− w(i) = h(i)− w(i)

where

h(i) (“current profit from new consumer”) is increasing in U(i)w(i) (“continuation value from capturing new consumer”)

is approximately the “derivative” of δ v(i).

p(i)−(− w(i)

)p(i)

=q(i)

q′(i) p(i)that is,

p−MC

p=

1ε

24

Pricing: harvesting and investing

Taking differences,

P (i) = H(i)−W (i)

where H(i) is increasing in U(i).

Lemma. For a given W (i), there exists a U ′ such thatP (i) > 0 if and only if U(i) > U ′. For a given U(i), thereexists a W ′ such that P (i) < 0 if and only if W (i) > W ′.

25

Market share dynamics

• Weak market dominance: q(i) > q(j) iff i > j.

• Strong market dominance: q(i) > i/η iff i > j.

Lemma. In equilibrium, the higher U(i) +W (i) is, the higherq(i) is. Moreover, q(i) ≥ 1

2 if and only if U(i) +W (i) ≥ 0.

• Two sources of market dominance:

− U(i): willingness to pay or network size effect.

− W (i): pricing incentives effect.

26

Relation to Gilbert and Newbery

0

πD

πM

0 50 100

..............................................................................

.......................................................................

............................................................................................................................................................................................................

firm profit

marketshare(%)

0

50

100

0 50 100

v(i)

i...............................................................................................

......................................................

.....................................

.......................................................................................................................................................................................................

• In static model, monopolist has more to lose from becoming aduopolist than entrant has to gain from becoming a duopolist.

• In dynamic model, if v(i) is convex, then leading firm has more tolose from dropping to i− 1 than follower has to gain fromjumping to j + 1. That is, W (i) > 0.

27

The η = 2 case

Proposition. If η = 2, then there exists a unique equilibrium.Moreover, if Properties 1 and 2 hold, then q(1) > q(0) andv(1) > v(0). Finally, there exists a λ′ such that p(1) > p(0) ifand only if λ(2)− λ(1) > λ′.

28

The η = 2 case: naıve consumers

• Naıve consumers assume state will remain the same. Hence,u(i) = λ(i) + δ u(i).

• Let ∼ denote values corresponding to naıve consumer case.

Proposition. Suppose that Property 1 holds. If η = 2, thenq(0) < q(0) and v(0) < v(0).

• Intuition: rational consumers know that a small networkwon’t always be small.

• Difference can be “significant.”

29

The small δ case

Proposition. There exists a δ′ such that, if δ < δ′, then thereexists a unique equilibrium. Moreover,

(a) If θ(i+ 1)− θ(i) is constant and INB holds strictly, thenp(i) is strictly increasing;

(b) If λ(i) is constant and IRS holds strictly, then p(i) isstrictly decreasing.

30

The small δ case

Proposition. Suppose INB and IRS hold. There existδ′, λ′, θ′ such that, if δ < δ′, then:

(a) q(i) ≥ 12 if and only if i > i∗ (i∗ is the middle state);

(b) If i is close to zero or close to η − 1, then the state movestoward i∗ in expected terms;

(c) If i is close to i∗, and either λ(i∗ + 1)− λ(i∗) > λ′ orθ(i∗ + 1) + θ(i∗ − 1)− 2 θ(i∗) > θ′, then the state movesaway from i∗ in expected terms.

31

The small ψ case

Proposition. Suppose that λ(i) = ψ i and θ(i) = 0. Thereexists a ψ′ such that, if ψ < ψ′, then:

(a) Prices are increasing in network size;

(b) Larger networks are more likely to attract a new consumer;

(c) In expected terms, larger networks decrease in size;

(d) Firm value is increasing in network size;

(e) Industry profits are decreasing in the degree of networkeffects.

32

Outline

• Introduction.

• Dynamic framework.

• Equilibrium characterization.

• Application: access charges.

33

The Laffont-Rey-Tirole framework

...........................................................................................................................

..............................................................................................................................................................................................................................

......................

............................

.........................................................

............................................................................................................................................................................................................................................................................................................................................................

.....................................................

................................................................................................................................................

........................

............................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................... .................

......................................................................................................................

..

.......................

..........................

..............................

.....................................

................................................................

..................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................

................................................

..................................

.............................

.........................

.............. ..........................

.................•

•

•

•

A

B

C

Cost 2 c0 + c1Price p(i)

Cost 2 c0 + c1Price p(j)

Cost c0 + c1 + a(i)Price p(j)

Cost c0 + c1 + a(j)Price p(i)

Network i (size i) Network j (size j)

34

Dynamics

• In LRT model, all consumers choose a network at the sametime. By contrast, in my model consumer network choicesare staggered.

• In each period, firms set p(i), p(i) given their network sizes i(state variable). This results in particular values of θ(i) andλ(i).

• I assume a(i) are set by regulator and consider thecomparative dynamics of different rules for setting a(i).

35

Policy experiments

(A) Access charges regulated at marginal cost level:a(i) = c0.

(B) Access charges set at twice the marginal cost level:a(i) = 2 c0.

(C) Access charges inversely related to network size,specifically,

a(i) =(

2− i

η

)c0

36

Aftermarket profits, consumer surplus

0

50

100

0 50 100

θ(i)

i.........................................

........................................

....................................

....................................

...............................................

...................................

....................................

....................................

....................................

....................................

.....................

...............................................

..............................................

..............................................

..............................................................

0.0

0.5

1.0

0 50 100

λ(i)

i.....................................................

............................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................

• Uniform markups (red lines) imply

− Lower firm profits (double marginalization); convex.

− Lower consumer surplus; increasing.

• Asymmetric access charges imply

− Higher profits for small firms; concave function.

− Non-monotonic consumer surplus.

37

Representative results

−10

−5

0

0 50 100

p(i)

i

..................................................................................................................................................................................................................................................

.........................................................

.

............................................................................................................................................................................................................................................................................................................

.........................................................................................................................................

.....................................................................................................................................................................................................................................................................

0.0

0.5

1.0

0 50 100

q(i)

i........................................

......................................................

.....................................

...................................................................................................................................

.....................................................

...............................................

............................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................

0

5

10

0 50 100

s(i)

i................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0

500

1000

0 50 100

v(i)

i.............................................

..........................................

..........................................

......................................

......................................

....................................

..................................

...................................

..........................................................................................

.......................................

........................................

........................................

........................................

........................................

........................................

........................................

........................................

........................................

........................................

.

.............................................................

......................................

............................................

....................................................

......................................................

.................................................

...........................................

....................................

.......................

0

500

1000

0 50 100

cw(i)

i

.............................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................

0

500

1000

1500

2000

0 50 100

sw(i)

i

......................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0

25

50

75

100

0 5E09 1E10

i

t

.......

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................

....................................................................................................................................................................................

.......................................................................................................

................................................................................................................

..............

38

Comments on uniform markups

• At asymmetric states, firms compete less aggressively fornew customers.

• New customers are more likely to join large network: marketdominance.

• If network effects are sufficiently strong, then strong marketdominance holds and stationary distribution is bimodal.

• Social loss highest at 50%, when volume of off-net calls ishighest.

• Consumer welfare lower even at i ≈ 0 because firms are lessaggressive at attracting new customers.

39

Comments on asymmetric access charges

• In the short run, small firms are better off, large firms worseoff.

• At asymmetric states, firms compete very aggressively fornew customers: small firms have a lot to gain, large firms alot to lose.

• Social welfare uniformly lower, but consumer welfareincreases at i ≈ 0 (or i ≈ η) because firms compete veryaggressively for new customers.

• Firm value lower at i ≈ 0: market penetration is faster butmore costly.

40

Final remarks

• When network effects are important, dynamic models areparticularly useful.

− Multiple outcomes vs. multiple equilibria.

− Meaningful comparative statics.

• The power of combining analytical and numerical methods.

− Intuition for results.

− Robustness of “local” results.

− Calibration.

• Additional applications:

− Wireless: mergers, new wireless licenses.

− Antitrust treatment of aftermarket power.

− Understanding network tipping.

41

42