Question

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A market demand function is P= 100 - Q. MC = 40. Total revenues =P*Q = (100 - Q)*Q= 100Q – Q2. Therefore Marginal Revenue = dTR/dQ = 100 – 20. a. At P=MC, what is the price and quantity sold? b. What is the profit-maximizing price and quantity for a single firm? Imagine there are two identical firms, selling the same product and with the same MC = 40. The resulting market price is P = 100 – 21-22. Use this information to find the Cournot solution to both firms' profit maximizing output and price.

Answer 1

P=100- Q and dre=yo price equation At Р= мс 100-Q = 4o Q = 100-40 P=60 units. Aituting a into P = loorbo P =$40 TR= p.Q = (10orQQ TR = 100 Q- Q² MR = 2TR - 100(1) -20 MR = 100-20 Monopoly profit-maximizing condition MR = MC

100-20 = 40 20 = 100-40 Q = 60 2 c Q = 30 units. Substituting a into price eqreation p = loor30 P = $70 Cournot model: From I BRI TR=PQi = [100-Q,-2₂] Qi TRI = 1000,- Q-Q, Q, d TR1 = 100 (1) -20, -Q (1) MP1 =100 -20, - Q, profit-maximizing condition: MRI=MC, 100-20, - Q2 = 40 20, +ag = 100-40 29, +2 = 60 (BRI) dQ

Firma (BR2) The= Pla, = [100-Q- Qe Q2 TR₂ = 100 - QQ - Q² STR 2 clooll) - a (0) - QQ2 MR₂ = 100 - Q - 20 profit-maximizing condition MR₂ =Mę 10orp, -20= 40 a+224 = 100-40 Pi+2 = 60 multiplying by a 28, +uQ =10 solving (BR1) and (BR2) (BR) 29, tha = 20 20+ Q =60 302 = 60

Q = 20 units. substituting az into BR, 20, +20=60 Q =60-20 2 a = un Q = 20 units. substituting Q, and a 2 into p equiti O equation. P=100-20-20 p = 100 - no p = $60