Econ 515 Essay

The dancing machine industry is a duopoly. The two firms, Chuckie B Corp. and Gene Gene Dancing Machines, compete through Cournot quantity-setting competition. The demand curve for the industry is P = 100 – Q, where Q is the total quantity produced by Chuckie B and Gene Gene. Currently, each firm has marginal cost of $40 and no fixed cost. Show that the equilibrium price is $60, with each firm producing 20 machines and earning profits of $400.
Pc= 100 – (Qc + Qg), MCc= 40, Q= Qc + Qg
Profit= TRc – TCc = PcQc – TCc = (100 - Qc - Qg) Qc - 40 Qc
= 100 Qc – Qc2 – QgQc – 40 Qc
From Chuckie B Corp’s view and assumption
Maximum profit:
100 – 2Qc – Qg – 40 = 0
60 – 2Qc – Qg = 0
Qc = 30 - .5Qg
Pg= 100 – (Qc + Qg) MCg= 10 Q= Qc + Qg
Profit= TRg – TCg = PgQg – TCg = (100 - Qc - Qg) Qg - 40 Qg
= 100 Qg – Qg2 – QgQc – 40 Qg
From Gene Gene Dancing Machines’ view and assumption
Maximum profit:
100 – 2Qg – Qc – 40 = 0
60 – 2Qg – Qc = 0
Qg = 30 - .5Qc
When Qc = 30 - .5Qg and Qg = 30 - .5Qc
Qc = 30 - .5Q2 = 30 - .5(30 - .5Qc) = 30 – 15 + 0.25Qc
0.75Qc = 15
Qc = 20
So by symmetry Qg = 20
Thus, equilibrium price P = 100 – Q = 100 – Qc – Qg = 100 – 20 – 20 = $60
And finally each firm’s profits:
Profits C = (60*20)–(40*20) = 1200–800 = $400
So by symmetry
Profits G = (60*20)–(40*20) = 1200–800 = $400

Consider a market with two horizontally differentiated firms, X and Y. Each has a constant marginal cost of $20. Demand functions are:
Qx = 100 – 2Px + 1Py
Qy = 100 – 2Py + 1Px
Calculate the Bertrand equilibrium in prices in the market. πx = Qx ( Px – ATC) = (100 – 2Px + 1Py) * ( Px – 20)
= 100Px – 2Px2 + PyPx – 2000 + 40Px – 20Py
= 140Px – 2Px2 + PyPx – 2000 – 20Py holding on Py
Mπ= 140 + Py – 4Px =0
4Px = 140 + Py
Px = 35 + 0.25Py

Do the same process: πy = Qy ( Py – ATC) = (100 – 2Py + 1Px) * ( Py – 20)
= 100Py – 2Py2 + PyPx – 2000