The firms in a duopoly produce differentiated products. the inverse demand for firm 1 is:
p1 = 52 - q1 - 0.5q2.
the inverse demand for firm 2 is:
p2 = 70 - q2 - 0.5q1.
each firm has a marginal cost of m = $1 per unit. solve for the nash-cournot equilibrium quantities.
the cournot equilibrium quantities are:
q1 = and q2 = units. (enter your responses rounded to two decimal places.)

we get q1= 18 and q2 = 30

Explanation:

There is a standard way of solving for Q1 and Q2.

The profit function for firm 1 equals π1 = P1Q1-C1

The profit function for firm 2 π2 = P2Q2-C2

we know that

Determine the profit functions.

Determine the best response function for the firms.

Substitute Q1 or Q2 in the other profit function and solve.

we know that Marginal cost: MC = dC/dq

where C= total cost.

we have MC = 1

so dC=dq

integrate on both sides

we get C = q + a(constant)

marginal cost for both firms are same so we get two cost equations as

C1=q1+a

C2=q2+b

where a and b are integral constants.

π1 = P1Q1-C1

π2 = P2Q2-C2

substitute inverse demand function and cost function for both firms

we get

π1 = P1Q1-C1 = (52-q1-0.5q2)q1-(q1+a)

π2 = P2Q2-C2= (70-q2-0.5q1)q2-(q2+b)

The best response function can be determined by deriving the profit function of firm 1 w.r.t. Q1 and for firm 2 w.r.t. Q2 and set them equal to zero

so first derivative π1 = 52-2q1-0.5q2-1=0

sendond derivative π2  = 70-2q2-0.5q1-1=0

2q1+0.5q2=51

2q2+0.5q1=69

solving these both we get q1= 18 and q2 = 30