SØK/ECON 535 Imperfect Competition and Strategic Interaction
ASYMMETRIC INFORMATION
Lecture notes 15.09.02
Introduction
Asymmetric information
Information exchange
Signalling
in order to influence expectations
Static price competition with uncertainty
Two firms, 1 and 2.
Differentiated products: Di ( pi , p j ) = 1 − pi + p j ; i , j = 1,2; i ≠ j
Constant unit costs:
c1 = c1L with probability
E c1 = xc1L + [1 − x ] c1H ;
x
and
c2 = c1H
with
probability
1-x,
c2 given.
i
Ex post profits: Π ( pi , p j ) = [ pi − ci ] 1 − pi + p j 
We look for the Bertrand equilibrium in a game in which firms choose prices
*
*
L
L
*
H
H
simultaneously: p2 , p1 = p1 if c1 = c1 and p1 = p1 if c1 = c1 .
Reaction functions
Firm 1: R1 =
1
2
[1 + p2 + c1 ], E R1 = 21 [1 + p2 + E c1 ] ;
Firm 2: R2 =
1
2
[1 + E p1 + c2 ]
L
H
Figure (reaction functions: R1 , R1 ,E R1, R2
Equilibrium:
3 + 2c1s + c2
, s = L, H,
3
3 + 2c2 + E c1
p2* =
3
p1s =
Conclusion:
prices decreasing in the probability x of low costs;
‘as if’ Firm 1 has costs E c1
Information exchange
Assume Firm 1 can provide verifiable information about its costs before
competition in prices take place.
Note that it would always be in Firm 1’s interest that Firm 2 chooses a high
price. Therefore:
H
if c1 = c1 , Firm 1 would inform about this;
L
if c1 = c1 , Firm 1 would like to conceal this information.
However, realising what the incentives of Firm 1 are, Firm 2 would treat no
information as evidence that costs are low (because, if costs were high, Firm 1
would have provided information about this).
We consequently have a ‘separating equilibrium’; the informed party’s
strategic choice depends on the realisation of events.
If it is not possible to provide verifiable information, Firm 1 may try to ‘signal’
via its choice of strategy.
Signalling
Milgrom-Roberts (Econometrica, 1982).
Two firms, 1 and 2.
Three stages:
1)
Firm 1 chooses its price p1 .
2)
Firm 2 chooses whether to enter the market or not.
3)
Firm 1 and (possibly) Firm 2 choose prices simultaneously.
2
Firm 1’s costs are low with probability x and high with probability 1-x. Firm 1’s
costs are private information ex ante, but become known to Firm 2 ex post
upon entry.
Firm 2’s costs are given (and common knowledge).
Define
M1t ( p1 ) is Firm 1’s monopoly profits at price p1 , t = L,H,
pmt is Firm 1’s profit-maximising (static) monopoly price, t = L,H,
( )
Mmt = M1t pmt is Firm 1’s maximum (static) monopoly profits, t = L,H,
Dit is Firm i’s duopoly profits, i = 1,2, t = L,H.
Note: because of the assumption that Firm 2 learns Firm 1’s costs upon entry
ex post price competition is independent of the price charged ex ante.
H
L
Assume D2 > 0 > D2 ; that is, entry is profitable only if Firm 2 has high costs.
The discount factor is δ .
Note: Firm 1 will want to give the impression that its costs are low in order to
keep Firm 2 out of the market. It can do so by charging a low price also when
costs are in fact high; this will lead to a loss in ex ante profits (since price does
not maximise static monopoly profits) but this is compensated for by a
monopoly market position, and hence higher profits, ex post. Firm 2 will, to the
extent that it understands Firm 1’s strategy, take this into account and hence
will not necessarily be ‘fooled’. Firm 1 understands this, and so on.
Two kinds of equilibria:
separating: the informed party’s strategy depends on its information (here
p1L ≠ p1H ). Hence, the uninformed party can infer the underlying
information from observing the informed party’s strategic choice;
pooling: the informed party’s strategy does not depend on its information
L
H
(here p1 = p1 = p1 ). Hence the uninformed party does not learn from
observing the strategic choice of the informed party.
Separating equilibrium
Equilibrium conditions include
H
Firm 1, when it has low costs, must not benefit from choosing price p1
L
instead of p1 , and vice versa;
3
Expectations upon out-of-equilibrium moves must be specified (here: what
L
H
Firm 2 believes if observing a price different from either p1 or p1 ).
Since Firm 2 will enter when believing that Firm 1 has high costs, Firm 1 may
H
H
as well choose p1 = pm in this case. Firm 1’s equilibrium profits is
H
H
consequently Mm + δ D1 .
L
Suppose Firm 1 where to choose p1 when its costs are high. Given that Firm
2 would not enter (believing that Firm 1 do have low costs), Firm 1’s profits
H
L
H
would be M1 p1 + δ Mm . A necessary condition for equilibrium is therefore
( )
( )
MmH + δ D1H ≥ M1H p1L + δ MmH
To make the problem interesting, we assume this condition is not satisfied for
p1L = pmL ; in other words, the low-cost type must charge an ex ante price below
the (static) monopoly price in order to signal convincingly that costs are low.
L
L
Note that Firm 1, when having low costs, could ensure a profit of Mm + δ D1 ,
and hence we must have
( )
M1L p1L + δ MmL ≥ MmL + δ D1L .
Under reasonable assumptions, these two conditions define an interval
< pL .

i ,p
p
p
m
 1 1  with 1
Example: Linear specification
Let
demand
be
Q m ( p1 ) = 3 − p1
in
the
duopoly
case
and
Qi ( p1, p2 ) = 1 − pi + pi in the duopoly case;
L
H
costs be c1 = 0, c1 = 1 and c2 = 0 ;
entry costs be f = 1; and
discount factor be δ = 1 .
Then we have
pmL = 1.5, pmH = 2, MmL = 9 4 and MmH = 1;
D1L = 1 and D1H = 4 9 ;
4
D2L = 0 and D2H = 7 9 .
Furthermore, we have
( )
;
D1H ≥ M1H p1L ⇒ p1L ≤ 1.25 = p
1
( )
i .
M1L p1L ≥ D1L ⇒ p1L ≥ 0.38 = p
1
Returning to the general model, equilibrium may be described as follows:
H
H
L
Firm 1: in the first period, choose p1 = pm if costs are high and p1 = pˆ1 ,
otherwise.
L
Firm 2: if p1 ≤ p1 expect entry to be unprofitable and stay out; otherwise,
enter.
Note:
out-of-equilibrium expectations is that entry is profitable if a high price is
L
observed (this is what deters Firm 1 from deviating from p1 when costs
are low);
L
there exists a continuum of similar separating equilibria, with p1 ∈  p1, pˆ1  .
Conclusion:
‘limit-pricing’, but Firm 2 becomes perfectly informed’;
limit-pricing is necessary in order not to be considered a high-cost firm;
welfare is greater than in the case of full information (since price is lower in
the low-cost event).
Pooling equilibrium
L
H
A pooling equilibrium does not exist if xD1 + [1 − x ] D1 > 0 . To see this, note
that Firm 2 would enter if Firm 1 played a pooling strategy (no information
revealed). However, given this, Firm 1 would rather want to charge the
relevant (cost dependent) monopoly price.
L
H
Assume xD1 + [1 − x ] D1 ≤ 0 .
For pooling to be part of an equilibrium requires
5
M1H ( p1 ) + δ MmH ≥ MmH + δ D1H
M1L ( p1 ) + δ MmL ≥ MmL + δ D1L
There generally exists an interval of prices p1 that satisfies these conditions. If
Firm 2 believes, upon observing (out-of-equilibrium) prices different from the
equilibrium (pooling) price p1 , that entry is profitable, a pooling equilibrium can
be supported.
Conclusion:
Firm 2 remains uninformed;
there is less entry than with symmetric information;
implications for welfare are not obvious.
Variations on the model
Alternative formulations:
if Firm 2’s costs are not known to Firm 2, but correlated with the costs of
Firm 1, Firm 1 will want to choose a high price in order to signal that costs
are high;
if demand, rather than costs, are unobservable to Firm 2, Firm 1 will want
to choose a low price to signal that demand is low.
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