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.
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Consumers
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• • • • • • • • •
• ••
• ••
• •
•
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
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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
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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
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
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•
Firm i’snetwork size
Time
i+ 1
i
i− 1
t t+ 1
18
State transition
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..................................................... q(i)
(j
η
)...............................................................................
prob. birth...............................................................................
prob. death
•
Firm i’snetwork size
Time
i+ 1
i
i− 1
t t+ 1
18
State transition
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....................................................................................................................................................... q(i)(i
η
)+ q(j)
(j
η
)•
Firm i’snetwork size
Time
i+ 1
i
i− 1
t t+ 1
18
State transition
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......................................................................................................................................................................... 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
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firm profit
marketshare(%)
0
50
100
0 50 100
v(i)
i...............................................................................................
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• 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
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•
•
•
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)
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0.0
0.5
1.0
0 50 100
λ(i)
i.....................................................
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• 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
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0.0
0.5
1.0
0 50 100
q(i)
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0
5
10
0 50 100
s(i)
i................................................................................................................................................................................................................................................................................................
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500
1000
0 50 100
v(i)
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0
500
1000
0 50 100
cw(i)
i
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0
500
1000
1500
2000
0 50 100
sw(i)
i
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0
25
50
75
100
0 5E09 1E10
i
t
.......
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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