Avoiding the Bertrand Trap

Avoiding the Bertrand Trap

Part I: Differentiation and other strategies

The Bertrand Trap

Recall the model’s assumptions: 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

Q

P

Dpmin

firm demand

mkt. demand

An “Easier” Bertrand Model

Q

P

D

pmin

firm demandmkt. demand

v

Avoiding the Bertrand Trap

Avoiding the trap means altering these assumptions; that is, doing at least one of the following:

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

Lowering your costs. 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

Question is sustainability of cost advantage: 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).


Limiting capacity 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

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

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 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. 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 …

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

Quantity

Price

Market demand

qr

Your demandc

MRqo

p

If Rival Produces More

Quantity

Price

Market demand

qr

Your demandc

MRqo

p

Your quantity goes down

P

rice

falls

Insights

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

Quantity

Price

Market demand

qr

Yourdemand

cMR

qo

p

In equilibrium, must play mutual best responses. Given assumed symmetry, this means qo = qr .

Comparison with Monopoly

Quantity

Price

Market demandc

qo

Monopolist’s MRQm

Cournot

price

Monopolyprice

More Insights 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. Again, the danger of entry and emulation.

Summary of Method 2 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:

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 Raise consumer search costs

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

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

How to keep search costs high? Must prevent price advertising. Must ensure comparison shopping hard (or

pointless). Preventing price advertising.

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

Limit store hours Detroit automobile dealers Closing laws (more gov’t lobbying)

Do not readily supply price information automobile dealers again use multiple prices (extras on cars,

supermarkets) Make it pointless

guarantee lowest price meeting competition clauses


Raise consumers switching costs 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 Consider “easier” model of Bertrand. Assume, first, that w ½(v - c). An equilibrium exists in which both firms

charge monopoly price, v: 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 less.

Raise Consumers Switching Costs

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):

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 Although equilibrium difficult to calculate,

we thus know positive profits made in it.

Method 5: Product Differentiation

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.

E.g., Assume customers each want one unit.

Technical details: See the product differentiation handout on the website.

0 1dry sweet

Equilibrium with Great Differentiation

0 0

Firm

0’s

pric

e

Firm

1’s

pric

e

Firm 0’s quantity Firm 1’s quantity

MC

D0(p0|p*) D1(p1|p*)p*

MR0 MR1

Equilibrium with Modest Differentiation

0 0

Firm

0’s

pric

e

Firm

1’s

pric

e


MC

D0(p0|p*) D1(p1|p*)p*

MR0 MR1

Equilibrium with Even Less Differentiation

0 0

Firm

0’s

pric

e

Firm

1’s

pric

e


MC

D0(p0|p**) D1(p1|p**)

p*

MR0 MR1

p**

An Experiment In this experiment, you need to decide where to locate

in a differentiated market. The market works as follows:

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: Entry costs $20. Marginal cost is $0.

Experiment continued Rules

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.

Analysis of Experiment(This slide intentionally left blank for you to write your notes. For “full” version of slides, download them after

4:30pm, April 8.)

Conclusions You can avoid or escape the Bertrand Trap

if You can achieve a cost advantage (Method 1) You can limit capacity (Method 2)

Cournot competition You can raise search costs (Method 3)

Sneaky benefits to price matching guarantees You can raise switching costs (Method 4) You can differentiate your product (Method 5)

But … Some of these solutions can be vulnerable

to lack of market discipline or entry/emulation: 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

Importance of maintaining discipline: Topic for next time – Method 6 – tacit

collusion. Importance of deterring entry:

Topic for later in term.

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Avoiding the Bertrand Trap