Transcript Lecture 5 - Faculty Directory | Berkeley-Haas

Avoiding the Bertrand Trap
Part I: Differentiation and other
strategies
Administrative Matter
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In section this Friday, the GSI will go
over Exercises #2, #3, #4, #5 (parts
(a)-(c) only), #6, and #7 from
“Introduction to Game Theory & The
Bertrand Trap.”
This information is also posted on the
course web site.
The Bertrand Trap
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Recall the model’s assumptions:
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they produce a homogeneous product
they have unlimited capacity
they play once (alternatively, myopically, or w/o
ability to punish)
customers know prices.
customers face no switching costs
the firms have the same, constant marginal
cost
Bertrand Model
P
firm demand
mkt. demand
pmin
D
Q
An “Easier” Bertrand Model
P
firm demand
mkt. demand
v
pmin
Q
D
Avoiding the Bertrand Trap
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Avoiding the trap means altering these
assumptions; that is, doing at least one of
the following:
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don’t produce a homogeneous product
don’t have unlimited capacity
don’t play myopically (facilitate tacit collusion)
make it difficult for customers to learn prices
make it difficult for customers to switch from
one firm to the other
lower your costs
Avoiding the Trap: Method 1
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Lowering your costs.
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Lower your MC to k < c, where c is your rival’s MC.
Equilibrium: you charge po = c - , where  is a
very small amount and your rival charges pr = c.
Proof: An equilibrium p > c would lead to Bertrand
undercutting, so p c in equilibrium. Your rival will
never charge less than c, so you can get away with
charging c - .
Potential Problems with
Method 1
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Question is sustainability of cost
advantage:
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Could fail the “I” test in VRIO.
Care that cost-cutting today does not result
in negative long-run consequences.
Could make firm vulnerable to fluctuations
in trade policy (if cost advantage gained by
“exporting” jobs).
Avoiding the Trap: Method 2
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Limiting capacity
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Let K1 and K2 be the capacities of the two firms.
For convenience, assume a flat demand curve
(i.e., easier model).
If K1 + K2  D, then no problem: equilibrium is p
= v (i.e., monopoly pricing); there is no danger of
undercutting on price because neither rival can
handle the additional business.
Limiting Capacity
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If K1 + K2 > D, but Kt < D for t = 1,2; then
monopoly price (i.e., v) cannot be sustained
because of undercutting.
However, each firm is guaranteed a profit of
at least (D - Kr)(v - c) > 0, where Kr is the
rival’s capacity.
Equilibrium in this simple model involves
complicated mixed strategies.
But positive profits made!
Choosing Capacities
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It turns out that the game in which
firms first choose their capacities and
then play a Bertrand-like game is
equivalent to Cournot competition.
Cournot Competition
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Firms simultaneously choose quantity
(capacity).
If Q is total quantity, then price is such that
all quantity just demanded; that is, so D(p) =
Q.
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Note we are abstracting away the firms ability to
set their own prices, but this turns out to be
without consequence in equilibrium and it vastly
simplifies the analysis.
Cournot Competition
continued …
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Assume two identical competitors.
Each has a constant marginal cost of c.
If you think rival will produce qr , then
your demand curve is D(p)-qr .
Your Best Response
Price
qr
p
Your demand
Market demand
c
MR
qo
Quantity
If Rival Produces More
 Price falls
Price
qr
p
Your demand
c
MR
qo
 Your quantity goes down
Market demand
Quantity
Insights
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Despite competition, you make a
positive profit (price > unit cost).
You produce less if you think rival will
produce more (have less capacity if you
think rival will have more).
Your profits decrease with the output
(capacity) of rival.
Equilibrium of Cournot Game
Price
In equilibrium, must play mutual
best responses. Given assumed
symmetry, this means qo = qr .
qr
p
Your
demand
Market demand
c
MR
qo
Quantity
Comparison with Monopoly
Price
Monopoly
price
Cournot
price
Market demand
c
Monopolist’s MR
qo
Qm
Quantity
More Insights
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Relative to monopoly, Cournot competition
results in more output and lower prices.
That is two means a lower price and more
output than one.
Logic continues: Three Cournot competitors
results in a lower price and more output than
with two.
In general, prices and firm profits fall as the
number of Cournot competitors increases.
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Again, the danger of entry and emulation.
Further on Cournot
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On the class web site there is an
additional reading on Cournot.
Summary of Method 2
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Limiting capacity is a way to escape or avoid
the Bertrand Trap.
Competition in capacity is like the Cournot
model.
Lessons of the Cournot model:
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Firms charge lower price than monopoly, so still
room for improvement through tacit collusion or
other strategies.
The more competitors, the lower will be price.
Avoiding the Trap: Method 3
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Raise consumer search costs
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Return to basic assumptions, except
assume that it costs a consumer s > 0 to
“visit” a second firm (store).
Let pe be the equilibrium price. That is,
the price consumers expect to pay. Then
each firm can charge p = min{pe + s,v},
because a customer would not be induced
to visit a second store.
Raise Consumer Search Costs
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Since customers expect both firms to charge
pe, customers are evenly divided between the
firms.
There is no benefit to undercutting on price,
since if rival is not charging more than
min{pe+s,v}, you won’t attract any of its
customers.
Pressure now is to raise prices.
Equilibrium is pe = v; i.e., the monopoly price.
Issues with Implementation
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How to keep search costs high?
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Must prevent price advertising.
Must ensure comparison shopping hard (or
pointless).
Preventing price advertising.
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Lobby gov’t to make illegal (liquor stores)
“Gentlemen’s agreement” (a form of tacit collusion)
Have professional association prohibit (generally
found to be violation of antitrust laws)
Making Comparison Shopping
Hard
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Limit store hours
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Do not readily supply price information
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Detroit automobile dealers
Closing laws (more gov’t lobbying)
automobile dealers again
use multiple prices (extras on cars,
supermarkets)
Make it pointless
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guarantee lowest price
meeting competition clauses
Avoiding the Trap: Method 4
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Raise consumers switching costs
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Return to assumptions of basic model,
except now consumers are initially
allocated equally to the two firms and must
pay w to switch to another firm.
Consumers know the prices at both firms.
Raising Switching Costs
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Consider “easier” model of Bertrand.
Assume, first, that w  ½(v - c).
An equilibrium exists in which both firms charge
monopoly price, v:
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To steal rival’s customers must charge
v – w – .
Profits from stealing:
( v – w –  – c) D .
Profits from not stealing:
(v – c)D/2,
which is greater.
Raise Consumers Switching Costs
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If w < ½(v - c), then complicated equilibrium
in mixed strategies.
We know, however, that each firm can charge
at least c + 2w (which is less than v):
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To profitably undercut a price of c + 2w, a firm would have
to drop price to below c + w. But
(c + 2w – c ) D/2 > (c + w -  - c)D
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Although equilibrium difficult to calculate, we
thus know positive profits made in it.
Method 5: Product
Differentiation
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Two firms with identical, constant MC = c.
Customers differ in their preferences.
Imagine that customers are uniformly
distributed along the unit interval with respect
to taste.
dry
sweet
E.g., 0
1
Assume customers each want one unit.
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Technical details: See the product differentiation handout on the
website.
D0(p0|p*)
D1(p1|p*)
Firm 1’s price
Firm 0’s price
Equilibrium with Great Differentiation
p*
MC
MR0
MR1
0
0
Firm 0’s quantity
Firm 1’s quantity
D0(p0|p*)
D1(p1|p*)
Firm 1’s price
Firm 0’s price
Equilibrium with Modest
Differentiation
p*
MC
MR0
MR1
0
0
Firm 0’s quantity
Firm 1’s quantity
D0(p0|p**)
Firm 1’s price
Firm 0’s price
Equilibrium with Even Less Differentiation
D1(p1|p**)
p*
p**
MC
MR0
MR1
0
0
Firm 0’s quantity
Firm 1’s quantity
An Experiment
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In this experiment, you need to decide where to locate in a
differentiated market.
The market works as follows:
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Consumers are located on a number line from 1 to 63.
There is one consumer at each location.
Every consumer will pay $1 to buy one unit of the product, but only
from the nearest store.
If there is a tie, then a consumer buys fractional units from all the
equally distant stores.
A monopolist can locate anywhere and make $63 because all
consumers will buy from the monopolist and pay $1 each.
Costs:
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Entry costs $20.
Marginal cost is $0.
Experiment continued
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Rules
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I will invite people (as individuals or teams
of 3 or fewer) to enter.
You must choose a location that is a
counting number between 1 and 63
inclusive (i.e., 3.5 is not a valid location).
When people cease to be willing to enter, I
will collect the entry fees and return profits
according to location.
Where to “Locate”
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Basically you want to locate far-away from
your rivals—remember this is driving not
football.
Caveats
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If population is concentrated in “one place,” may
need to get close to that place.
When entering, sometimes pays to enter where
the trade is (e.g., near other restaurants).
Where to locate (continued …)
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Note that product differentiation is not
a panacea if there is no market
discipline.
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Brand proliferation.
If can’t block entry or emulation.
How would we evaluate differentiation
strategy in Cramer?
Conclusions
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You can avoid or escape the Bertrand Trap if
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You can achieve a cost advantage (Method 1)
You can limit capacity (Method 2)
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You can raise search costs (Method 3)
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Cournot competition
Sneaky benefits to price matching guarantees
You can raise switching costs (Method 4)
You can differentiate your product (Method 5)
But …
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Some of these solutions can be vulnerable to
lack of market discipline or entry/emulation:
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Others may be able to cut costs too.
Others may attempt to capture business by
lowering search or switching costs.
Others may not be disciplined about capacity.
Entry can erode benefits of limited capacity.
Others may not be disciplined about maintaining
brand distinctions.
Entry can erode benefits of differentiation.
… which points to
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Importance of maintaining discipline:
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Topic for next time – Method 6 – tacit
collusion.
Importance of deterring entry:
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Topic for later in term.