Duopoly Part 2 (stackelberg) Flashcards
Stackelberg feature
One firm chooses quantity first, then other follows
Firm 2’s thinking process
Firm 1’s thinking process
It knows q₁, so sets according to this
B) firm 1 chooses its output knowing it will influence q₂
How to solve pg 38-41: start backwards with firm 2’s profit max
B) for symmetric case, what would q1 and q2 be
1) Firm 2 profit max expression
Π2 = 𝑣 −(𝑞1 + 𝑞2)− 𝑐2)𝑞2 − 𝑓
2) FOC: rearrange to q₂ to get BR2
3) sub BR2 into firm 1’s profit max problem (same as above but c1 and q1 instead of c2 q2)
4) Then FOC and rearrange to q1 (BR1)
q1 = v+c2-2c1 /2
5) Sub q₁ into firm 2’s best response curve q₂
q2 = v-3c2+2c1/4
So cournot, sub BR2 into BR1 to get adjusted BR1. Whereas for stackelberg here, sub BR1 into BR2 to get adjusted BR2
For symmetric case, what would q1 and q2 be
B) What can we notice
q₁ = v-c / 2
q₂ = v-c / 4
B) Firm 1 (leader) supplies more than follower.
(sets quantity double of firm 2)
Note then find prices by subbing q1 and q2 into P= v-(q1+q2)
Pg 41
q1 = v+c2-2c1 /2
q2 = v-3c2+2c1/4
Final answer
P = v+2c1+c2 / 4
Or in symmetric case
P = v+3c/4 (lower than cournot price v+2c/3!)
Cournot vs monopoly comparison. pg 42 and 43
B) how could cournot maximise profit, and why can’t it
Cournot produces more than monopoly at a lower price, and total output is split between 2 firms.
B) if each could reduce its output to half to monopoly level (to in total reach Qm), but if no way of committing to Qm/2 each, there would be incentive for each firm to increase output
Best response diagram for all market structures pg 45
Green line is firm 2’s best response, if firm 1 produces Qpc, firm 2 produces nothing.
Still on green line, when firm 1 produces nothing, firm 2 produces qm!
Same intuition for firm 1’s best response curve in blue.
Dotted lines both mark collectively sharing monopoly quantity, or perfect competition quantity
So cournot lower price and higher output than monopoly.
So less welfare loss than monopoly, but still more than bertrand (perfect competition)
Why can cournot be also productively inefficient?
If a less efficient producer can still remain in the market alongside a more efficient producer
Effect of a rise in costs for both firms in cournot
what happens and why
Price increases but not as much as the cost increase, because P* = v + 2c /3
So only 2/3 of the increase is passed onto the consumer in the price
Effect of a fall in only firm 1’s costs in cournot
on quantity
b) price
Pg49
Firm 1 gains sales and market share as c1 falls
q*₁ = v -2c₁+c₂/ 3
firm 2 loses sales and market share as c1 falls
q*₂ = v+c₁-2c₂ / 3
b) Price changes less than cost change as mentioned!Only 1/3 of cost reduction will be passed onto consumer as P* = v+c₁+c₂/3
what if increased costs for all firms in bertrand
Price rises by full amount of the cost increase
(whereas cournot price changes less than cost change as just seen)
So a increase in cost for both firms in Bertrand means price increases by the full amount of increase.
What about a fall in costs for 1 firm in bertrand
A) symmetric case
B) asymmetric case
in symmetric case: now one c is lower, can lower price by ε in order to steal all sales
in asymmetric case (e.g if originally c₁<c₂), depends on who more efficient in first place.
if the more efficient firm’s costs fall, no change in price and all gains i.e firm 1 who continues to supply
if less efficient firm’s costs fall but still not as much as the efficient (still c₁<c₂) , entire fall in costs is passed through by price (but still not below C1, and so still doesnt supply at all)
if less efficient firm cost falls so now more efficient, (c₁>c₂) firm 2 can now supply the market at p=c₁-ε (they undercut by arbitrarily small)
Summary example: considering a increase in fuel prices (so a common cost change for both firms)
How can we tell which market structure it is
If price of flights rises by full extent of cost increase, suggest Bertrand competition
if price changes less, cournot
Why are both models criticised
Choosing prices seems more realistic (Bertrand) , however Bertrand makes zero profit which we don’t see in oligopolies
so in reality firms choose both quantity and price!
Bertrand paradox assumes a firm can supply the whole market.
In reality we can add capacity constraints. Assume firms have capacity k₁ and k₂ i.e max quantity they can sell
assume firm 2 sets P2 with D(P2) > k2 (excess demand than their capacity), and firm 1 sets P1>P2
What do firms do
b) what does this show us
Firm 2 sells its max capacity k2, but there is excess demand.
Firm 1 supplies the rest of market D(P1) - k2
b) how in real life, bertrand paradox is unlikely to hold, since capacity constraints mean an individual firm cannot supply whole market, thus stealing all sales upon price competition is not possible
What would firm 1 do if P₂(k₁+k₂) > c (pg57)
That is the equilibrium price when capacities have been fuly used.
So setting P1 < P(k1+k2) is less profitable, as already selling at max capacity k1, would just be selling at a lower price
So thus only consider if p1> p(k1+k2) can increase profits, because they won’t lose sales since firm 2 can’t supply the rest of market either, so it has the ability to charge higher!
So firm 1 only considers if it can set a higher price, since a lower price means selling same quantity k1 at lower price = less profit.
Draw diagram showing capacity constraints pg59
b) given the diagram, could firm 1 set a higher price or not
b) faces a new residual demand d1 (downward shift) and residual MR r1.
D - d1: is what firm 2’s supply (k2)
Since we assumed MC=0 MR>MC for any P1>P(k1+k2)
So every unit 0 to K1 (red bar), profit is increasing. So we supply at k1 but at P(k1+k2). If K1 capacity was beyond MR curve, then they could consider increasing price above it. (so answer to b is no, they can’t set a higher price since would sell less than k1. given where the capacity constraint k1 is drawn)
Price is determined by capacity:
So firm 1’s decision to increase price depends on its capacity! When does this not hold?
If capacity is very high, meaning both firms can serve the market, we end up with the Bertrand case, where firms undercut each other till P=MC.
Does analysis mean quantity/capacity is the crucial choice and accept cournot as right model?
Many industries consider capacity in long term and pricing short term, fits cournot
However in some sectors like banking, prices tend to be set first and quantity easily changed, so fits Bertrand better
