1. Consider a Cournot oligopoly with three firms i = 1,2,3. All firms have the same constant marginal cost c = 1. The inverse demand function of the market is given by P = 9-Q, where P is the market price, and Q-19; is the aggregate output. (a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits (Hint: the NE is symmetric). (b) Now suppose these three firms play a 2-stage game. In stage 1, they produce capacities 1, 2 and 3, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices P1, P2 and P3. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for firm i is Di (Pi, p-i) = 9-Pi-ji 9-Pi if pi > p; for all ji if pip; for all ji. 3 9-Pi if pip; for all ji (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p* = 9-91 92-93.
1. Consider a Cournot oligopoly with three firms i = 1,2,3. All firms have the same constant marginal cost c = 1. The inverse demand function of the market is given by P = 9-Q, where P is the market price, and Q-19; is the aggregate output. (a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits (Hint: the NE is symmetric). (b) Now suppose these three firms play a 2-stage game. In stage 1, they produce capacities 1, 2 and 3, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices P1, P2 and P3. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for firm i is Di (Pi, p-i) = 9-Pi-ji 9-Pi if pi > p; for all ji if pip; for all ji. 3 9-Pi if pip; for all ji (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p* = 9-91 92-93.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Hello,
In a) I get qi=2,
I had a question on (b), I attached my answer in handwritten, is it correct to do it by showing Lemma 1 result? It is so confusing.
So, based on this with q(bar) as capacity constrained, and they are not gigantic capacity, I assumed it is played under pure strategy
Pls, can you help me with (b)?
Thanks
Transcribed Image Text:ar or suppose Pi = Po =P;> PC qT9, + q3)
+ The oprimal price is higher , and at legst 4 tirm canneT sell up to the copacity
(a) Let's luok
+ so, ir is protitalle
to tle capacity , hene hishe, protir.
* Sine ir is protitalle to deviate , this is not Nash Eghilibia.
to charge p-s (undar cut slighty) ana manase to bell up
(6) Now suppo se P,=Pq=Ą<p(š,rš, rž,)
the oprimal pice lowe than market chearig
prices.
- with these condition, all firm will retion their Custoned
→ ha the oprimel prie stil lowe then the marker price
, and they are not selliy up
to the cageacity sir is mtiralle to deviate by increas'y a vir its pice → Fall up
To the capacity and earn highe, profi.
→ Since ir iB protitalle
to deviate ,this is alse neT N.E
ine firm is price is lowe, than ote firms, firm i wants to aise its ria it it
is capacity constrain eel ·
- it firm i is nor sppanity constrajted , then firm i is a monopals tirm cand te
firm
took al the de mand, so zero protir for tirm i. It, they hhodercap the
pice by Pins i thog will makse ponitive protir retler then sero prhit.
: Sa by the se 3 onditions ot nor Nash Bgmimm, me proot thet P, =P=P3 = p(5,p5)
They will charge markey clearing pice.
Transcribed Image Text:1. Consider a Cournot oligopoly with three firms i = 1, 2, 3. All firms
have the same constant marginal cost c = 1. The inverse demand
function of the market is given by P = 9-Q, where P is the market
price, and Q =E19i is the aggregate output.
(a) Solve for the Nash equilibrium of the game including firm out-
puts, market price, aggregate output, and firm profits (Hint: the
NE is symmetric).
(b) Now suppose these three firms play a 2-stage game. In stage
1, they produce capacities 71, 72 and 73, which are equal to the
Nash equilibrium quantities of the Cournot game characterised
by part (a). In stage 2, they simultaneously decide on their
prices p1, P2 and pP3. The marginal cost for each firm to sell up
to capacity is 0. It is impossible to sell more than capacity. The
residual demand for firm i is
9 – Pi - EjtiIj if p; > P; for all j # i
if pi = Pj
if p; < P; for all j # i
D; (pi, P-i) =
9-pi
3
for all j + i .
9 Pi
(Note, here we assume that the efficient/parallel rationing ap-
plies). Prove that it is a Nash equilibrium of the second stage
subgame that each firm charges the market clearing price p* =
9 — G1 — Ф2 — Ҫз-
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