Hello,In a) I get qi=2, price=3 and profiti= 4. 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 equilibrium of pricing game. Hence, the "optimal price = market-clearing" price is a Nash Equilibria as in Lemma 1.Pls, can you help me with (b)?Thanks 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 protitalleto 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 1. Consider a Cournot oligopoly with three firms i = 1, 2, 3. All firmshave the same constant marginal cost c = 1. The inverse demandfunction of the market is given by P = 9-Q, where P is the marketprice, 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: theNE is symmetric).(b) Now suppose these three firms play a 2-stage game. In stage1, they produce capacities 71, 72 and 73, which are equal to theNash equilibrium quantities of the Cournot game characterisedby part (a). In stage 2, they simultaneously decide on theirprices p1, P2 and pP3. The marginal cost for each firm to sell upto capacity is 0. It is impossible to sell more than capacity. Theresidual demand for firm i is9 – Pi - EjtiIj if p; > P; for all j # iif pi = Pjif p; < P; for all j # iD; (pi, P-i) =9-pi3for 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 stagesubgame that each firm charges the market clearing price p* =9 — G1 — Ф2 — Ҫз-

Answered: 1. Consider a Cournot oligopoly with…

ENGR.ECONOMIC ANALYSIS

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Chapter1: Making Economics Decisions

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Hello,

In a) I get q_i=2, price=3 and profit_i= 4.

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 equilibrium of pricing game. Hence, the "optimal price = market-clearing" price is a Nash Equilibria as in Lemma 1.

Pls, can you help me with (b)?

Thanks

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.

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-

Definition Video Definition Point where the demand and supply curves intersect. At the equilibrium price the quantity supplied of a good is equal to the quantity demanded. When the market is not in equilibrium, market surplus or market shortage occurs. The region that is above the price equilibrium indicates market surplus and the region below the price equilibrium indicates market shortage. Video

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