4820–7
Strategic
commitments
Geir B.
Asheim
Strategic commitments
Introduction
Short- and long-term commitments
4820–7
Short-run –
Price
competition
Short-run –
Quantity
competition
Geir B. Asheim
Long-run –
Capacity
competition
Department of Economics, University of Oslo
ECON4820
Spring 2010
Last modified: 2010.03.02
Why model repeated strategic commitments?
4820–7
Strategic
commitments
Geir B.
Asheim
Problem with repeated games
No possibility for strategic commitments
No change in fundamentals
Introduction
Markov
perfect
equilibrium
Outline
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Problem with two-stage competition
The second stage is a static game
No response to actions in the second stage
Need for models with
Dynamic rivalry
Competition over a longer time horizon
Actions change fundamentals
Possible to make strategic commitments
Markov strategies simplify the analysis
State-dependent strategies – depending on payoff-relevant history
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Markov
perfect
equilibrium
Outline
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Definition (State)
The state at two nodes in a game tree is the same if and only if
the subgames defined by the nodes are identical.
Example (An infinitely repeated game)
The state is the same at all nodes.
Example (A finitely repeated game)
The state is the same at all nodes at a given time.
Example (A two-stage game)
The state in stage 2 dep. 1-to-1 on 1’s choice of K in stage 1.
Definition (Markov strategy)
A Markov strategy determines the same action in the same state
Markov-perfect equilibrium
4820–7
Strategic
commitments
Geir B.
Asheim
Definition (Markov-perfect equilibrium – 1)
A Markov-perfect equilibrium is a subgame-perfect equilibrium
where players use Markov strategies.
Introduction
Markov
perfect
equilibrium
Outline
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Definition (Markov-perfect equilibrium – 2)
A Markov-perfect equilibrium is a strategy profile where, in all
subgames, each player uses his best Markov strategy, given the
Markov strategies of his opponent.
Result (The definitions are equivalent)
A player’s set of best responses contains a Markov strategy if the
opponents use Markov strategies.
Comparison of . . .
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Markov
perfect
equilibrium
Outline
repeated games
No strategic commitments
only one state (no state variable)
Tactical competition is studied
‘Bootstrapping’
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
dynamic games studied by means of Markov strategies
Strategic commitments change state variables
Tactical competition are not studied
Reduced form Π-function where
profit depends on the state
Outline
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Markov
perfect
equilibrium
Outline
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Short-run commitments
Price competition
Maskin, Tirole, A theory of dynamic oligopoly,
II: Price competition, kinked demand curves, and
Edgeworth cycles, Ecma 56 (1988) 571–599
Quantity competition
Maskin, Tirole, A theory of dynamic oligopoly,
III: Cournot competition, EER 31 (1987) 947–968
Long-run commitments
Capacity competition
Fudenberg, Tirole, Capital as commitment:
Strategic investment in continuous time,
J Econ Theory 31 (1983) 227–256
Short-run commitments
Price competition
4820–7
Strategic
commitments
Geir B.
Asheim
Motivation: Foundation for “kinked-demand curve”
Menu costs explain that
price can be short-run commitments
Introduction
Short-run –
Price
competition
Markovperfect
equilibrium
Examples
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Model: Firms alternate at setting prices
⎧
⎪
⎨ D(pi )
1
Di (pi , pj ) =
D(pi )
2
⎪
⎩
0
if pi < pj
if pi = pj
if pi > pj
Per period profit: Πi (pi , pj ) = (pi − c)Di (pi , pj )
Intertemporal profit at time t:
∞
s=0 δ
s Πi (p
i,t+s , pj,t+s )
A Markov-perfect equilibrium (R1 , R2 ) satisfies:
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Markovperfect
equilibrium
Examples
Short-run –
Quantity
competition
Long-run –
Capacity
competition
There exists functions V1 , W1 , V2 and W2 such that
1
V1 (p2 ) = maxp Π (p, p2 ) + δW1 (p)
Discounted profit given best choice now
and the players follow (R1 , R2 ) later.
R1 (p2 ) = arg maxp Π1 (p, p2 ) + δW1 (p)
The reaction function specifies a best choice.
W1 (p1 ) = Π1 (p1 , R2 (p1 )) + δV1 (R2 (p1 ))
Discounted profit when the opponent acts
and the players follow (R1 , R2 ).
and likewise for V2 and W2 .
R1 and R2 may specify mixed actions;
if so, W1 and W2 are found by taking expectations.
Example 1: Kinked demand curve
Example 2: Edgeworth cycles
4820–7
Strategic
commitments
Stage game: D(p) = 3 − p and c1 = c2 = c = 1
p(x) =
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Markovperfect
equilibrium
Examples
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Πi (x) =
x+3
3
where x ∈ {0, 1, 2, 3, 4, 5, 6}
⎧ x(6−x)
⎪
⎨ 9
x(6−x)
⎪ 18
⎩
0
if pi = p(x) < pj
if pi = p(x) = pj
if pi = p(x) > pj
Ex. 1: p
p(6) p(5) p(4) p(3) p(2) p(1) p(0)
R(p) p(3) p(3) p(3) p(3) p(1) p(3)
p(3)
p(1)
Ex. 2: p
p(6) p(5) p(4) p(3) p(2) p(1) p(0)
p(5)
R(p) p(4) p(4) p(3) p(2) p(1) p(0) p(0)
Firms react to defend market shares,
not to punish (as in repeated games)
Short-run commitments
Quantity competition
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Short-run –
Quantity
competition
Markovperfect
equilibrium
Properties
Long-run –
Capacity
competition
Motivation: Dynamic reactions
Firms alternate at being Stackelberg leaders
Production lags (time between
change in production plans and change in output)
explain that quantity can be short-run commitments
Model: Firms alternate at choosing quantity
Per period profit: Πi (qi , qj )
Πij < 0 — Increased quantity is aggressive
Πiii < 0 — Second-order condition
Πiij < 0 — Strategic substitutes (e.g., Cournot competition)
Intertemporal profit at time t:
∞
s=0 δ
s Πi (q
i,t+s , qj,t+s )
A Markov-perfect equilibrium (R1 , R2 ) satisfies:
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Short-run –
Quantity
competition
Markovperfect
equilibrium
Properties
Long-run –
Capacity
competition
There exists functions V1 , W1 , V2 and W2 such that
1
V1 (q2 ) = maxq Π (q, q2 ) + δW1 (q)
Discounted profit given best choice now
and the players follow (R1 , R2 ) later.
R1 (q2 ) = arg maxq Π1 (q, q2 ) + δW1 (q)
The reaction function specifies a best choice.
W1 (q1 ) = Π1 (q1 , R2 (q1 )) + δV1 (R2 (q1 ))
Discounted profit when the opponent acts
and the players follow (R1 , R2 ).
and likewise for V2 and W2 .
Properties
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Short-run –
Quantity
competition
Markovperfect
equilibrium
Properties
Long-run –
Capacity
competition
The dynamic reaction curves
are downward sloping if Πiij < 0.
The dynamic reaction curves
coincide with the static response curves if δ = 0
The dynamic reaction curves
are outside the static response curves if δ > 0
Long-run commitments
Capacity competition
4820–7
Strategic
commitments
Geir B.
Asheim
If capital does not depreciate and cannot be sold,
then capital accumulation is a long-run commitment
Introduction
Short-run –
Price
competition
If capital stocks are strategic substitutes,
then the firm will engage in an investment race
The firm that is ahead will preserve its lead,
trying to limit the other’s mobility
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Model
Are capital stocks strategic substitutes?
Yes, if capital stocks determine capacity,
and the firms compete in prices in the short-run
Model
4820–7
Strategic
commitments
Geir B.
Asheim
Introduction
Short-run –
Price
competition
Short-run –
Quantity
competition
Long-run –
Capacity
competition
Continuous time and an infinite horizon
Ki (t): i’s capital at time t. K1 (0) ≥ 0. K2 (0) = 0.
Ii (t) = K̇i (t) ≡
dKi (t)
dt
∈ [0, Ī ]
Per period profit: Πi (Ki , Kj )
Πij < 0 — Increased capacity is aggressive
Πiii < 0 — Second-order condition
Πiij < 0 — Strategic substitutes (e.g., Cournot competition)
Model
Intertemporal
at time t:
∞ −rt profit
(Πi (Ki (t), Kj (t)) − I (t))dt
s=0 e
Consider the case where r → 0