Duopoly (Cournot and Bertrand)

Question 1

Key feature of oligopoly

Answer 1

Strategic interdependence - firms consider rivals decision

Question 2

Cournot: what do firms do

Answer 2

Firms simultaneously (at same time) set quantity, taking other firms as given

Question 3

Bertrand: what do firms do

Answer 3

Firms simultaneously sets price taking other firms price as given

Question 4

Stackelberg

Answer 4

One firm (leader) chooses quantity before the other, then other responds.

(Leader gains an advantage)

Question 5

Cournot general model :

Firm 1’s profit max problem

Answer 5

Max P(q1+q2)q1 - c1(q1)]

Differentiate with respect to q1

Rearrange to find price

Question 6

What do we find about price compared to monopoly

Answer 6

Price is above MC, but below monopoly price as firms ignore how their choice decreases rivals profit

(RMB in trade drawing 3 out, Pc had middle profits between Bertrand and monopoly)

Question 7

So how do we display how to find an optimal quantity (q₁) for a given q₂?

Answer 7

Best response diagram - where they intersect is the quantity each produces in equilibrium

Pg 12

Question 8

Linear case:

Inverse demand function is P(q1,q2) = v - (q1+q2)

what is profit max for firm 1 + whole calculations, then find P and Q

b) what about when firms have same MC, what is P and Q now

Answer 8

see working pg 14,15,16

B) just treat c1=c2=c

Question 9

Bertrand: how does demand work

Answer 9

If one firm charges less, takes all demand.

Tied price, evenly split

Question 10

Recall Bertrand paradox with MC’s

a) symmetrical costs

b) asymmetrical costs

Answer 10

if equal MC’s both set P=MC (perfect comp)

b) if one firm more efficient, they can charge slightly under the other firms MC and steal all demand

Question 11

assume symmetric costs.

Best response if p2 > c

B) what if p2<=c

Answer 11

Best response to p2 is to set the smallest price below p₂ i.e p₂ - ε where ε is a arbitrarily small number to steal market

but with symmetric costs, both end up setting price=c (MC)

b) p1 = C is best response

Question 12

Bertrand best response diagram pg 25

b) then draw best response of firm 2, how can we prove both firms end up setting P=MC

Answer 12

3 sections to firm 1’s best response curve (blue)
1st: if P2 > Pm, set p1 = pm (vertical as don’t go right/higher than Pm)

2nd: if c<p2<=Pm, set p1 = p2 - ε (undercut by the tiniest bit)

3rd: if p2<=c set p1=c (vertical as p1 doesn’t go left of C)

Similar for firm 2

B) where the 2 BR intersect, is where they set P=MC

Question 13

Symmetric costs mean both P=MC.

What about asymmetric costs, e.g if c1<c2. What is firm 1’s best response to

a) p2>c1
b) p2<=c1
c) what is important to note about result from b?

Answer 13

Undercut firm 2 by tiny amount P2 - ε.

b) best response is to set p1=c1 (it cannot sell below c1 otherwise a loss)
c)
firm 1 is still constrained even though firm 2 does not produce, it prevents firm 1 from charging the monopoly price! since if they charge above c, firm 2 will just supply since p2<= c !

Question 14

Best response diagram now, with asymmetric costs pg 28

Answer 14

at intersection we see P₂ sets price = c₂
while P₁ sets price c₂ - ε (undercuts)

note: at top of diagram, Firm 1 has a lower monopoly price than firm 2 since lower cost, hence why best response function goes below Pm₂.

Question 15

Symmetric costs = both set p=mc (perfect comp)
Asymmetric costs = profits but only one firm sells (the one with lower cost)

How to get both firms sell profits (3)

Answer 15

Remove assumption of no capacity constraints i.e now means neither firm can supply the whole market

Product differentiation

Repeat interactions

Question 16

Why no fixed costs in this Bertrand model

Answer 16

with common MC, if both enter, they suffer losses equal to fixed costs

hence only one enters first, and other will not follow to avoid losses for both (first mover advantage)