ECO 562 Assignment: Cournot Duopoly - Optimal Output and Price

This assignment provides a detailed solution to a Cournot duopoly problem, specifically addressing ECO 562 requirements. It begins by defining the market demand and cost functions for two firms (s and t). The solution then derives the reaction functions for each firm by equating marginal revenue and marginal cost. Subsequently, it calculates the profit-maximizing output levels for both firms by solving the system of equations. Finally, it determines the profit-maximizing price based on the derived output quantities. The solution adheres to the format specified in the assignment brief and likely includes an Excel attachment demonstrating the calculations.

Running head: ECONOMICS ASSIGNMENT
Economics Assignment
Name of the Student
Name of the University
Course ID

1ECONOMICS ASSIGNMENT
Given market demand function
P=2000−2 QS −3Qt
The respective total cost function of firm ‘s’ and firm ‘t’ are given as
TCS=10000+ 4 QS + 0.45QS
2
TC t=60,000+ 2Qt + 0.25Qt
2
For firm s,
Total revenue is given as
TRS=QS × P
¿ QS × ( 2000−2 QS −3 Qt )
¿ 2000 QS −2QS
2 −3 Qt QS
From the total revenue, marginal revenue for firm s is obtained as
MRs= d ( TRs )
d Qs
¿ ∂ ¿ ¿
¿ 2000−4 Qs−3 Qt
From the given total cost function, the marginal cost for firm s is obtained as
MCs =d ( TCs )
d Qs

2ECONOMICS ASSIGNMENT
¿ ∂(10000+4 QS +0.45 QS
2 )
∂ Qs
¿ 4 +0.90 Qs
Profit maximization condition of firm s requires,
MRs=MCs
¿ , 2000−4 Qs −3 Qt=4+ 0.90Qs
¿ , 2000−4 Qs −3 Qt−4−0.90 Qs=0
¿ , 1996−4.90 Qs−3 Qt =0
¿ , 4.90 Qs =1996−3 Qt
¿ , Qs= 1996−3 Qt
4.90
¿ , Qs=407.35−0.61Qt
The reaction function of firm s thus is given as
Qs =407.35−0.61Qt
Qs =407.35−0.61Qt ………………………………………………………………..(i)

3ECONOMICS ASSIGNMENT
For firm t,
Total revenue is given as
TRt =Qt × P
¿ Qt × ( 2000−2 QS −3Qt )
¿ 2000 Qt −2Qt QS−3 Qt
2
From the total revenue, marginal revenue for firm t is obtained as
MRt= d ( TRt )
d Qt
¿ ∂(2000 Qt −2Qt QS−3 Qt
2 )
∂ Qt

4ECONOMICS ASSIGNMENT
¿ 2000−2Qs−6 Qt
From the given total cost function, the marginal cost for firm t is obtained as
MCt = d ( TCt )
d Qt
¿ ∂(60,000+2 Qt +0.25 Qt
2 )
∂Qt
¿ 2+0.50 Qt
Profit maximization condition of firm t requires,
MRt=MCt
¿ , 2000−2 Qs −6 Qt =2+ 0.50 Qt
¿ , 2000−2 Qs −6 Qt −2−0.50 Qt =0
¿ , 1998−2 Qs −6.50 Qt=0
¿ , 6.50 Qt=1998−2 Qs
¿ , Qt =1998−2Qs
6.50
¿ , Qt =307.38−0.31 Qs
The reaction function of firm t thus is given as
Qt =307.38−0.31Qs…………………………………………………………………………(ii)

5ECONOMICS ASSIGNMENT
Putting the value of Qt in equation (i), optimal output for firm s is obtained as
Qs =407.35−0.61Qt
¿ , Qs=407.35−0.61 ( 307.38−0.31 Qs )
¿ , Qs=407.35−187.50+0.19 Qs
¿ , Qs−0.19Qs=219.85
¿ , 0.81 Qs=219.85
¿ , Qs= 219.85
0.81
¿ , Qs=271.12
Using the value of Qs, the optimal output for firm t is obtained as

6ECONOMICS ASSIGNMENT
Qt =307.38−0.31Qs
¿ 307.38− ( 0.31× 271.12 )
¿ 307.38−84.05
¿ 223.33
After obtaining the profit maximizing output for firm s and firm t, the profit-maximizing price is
obtained as
P=2000−2 QS −3Qt
¿ 2000− ( 2× 271.12 ) − ( 3 ×223.33 )
¿ 2000−542.24−670
¿ 787.76

7ECONOMICS ASSIGNMENT