Question

Joint profit maximizing output and Cournot model

Consider two firms facing the demand curve P=75-5C, where Q Q1 +Q2. The firms' cost functions are Cl (Q1) = 15 +10Q1 and C2 (Q2)-10 + 20Q2. Suppose that both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? Combined, the firms will produce units of output, of which Firm 1 will produce units and Firm 2 will produceunits. (Enter a numeric response using a real number rounded to two decimal places.) What is each firm's equilibrium output and profit if they behave noncooperatively? Use the Cournot model. If the firms compete, then Firm 1 will produce produce units of output. units of output and Firm 2 will

Answer 1

a) The demand curve of duopolist firms is

P = 75- 5Q= 75-5(Q₁ +Q₂ )..............(i)

The cost function of firm 1 is

C(Q₁ ) = 15 +10Q₁ ........(ii)

The cost function of firm 2 is

C(Q₂ ) = 10+20Q₂ .........(iii)

When both firms jointly produce the output their cost function will be

C(Q) = C(Q₁ ) +C(Q₂ ) = 15+10Q₁ +10+20Q₂ = 25+10Q+10Q₂ ........(iv)

Therefore, marginal cost of both firms is

MC= d(C(Q)/dQ= 10

Also using eqtn (i), the marginal revenue of both firms is

MR = d(PQ)/dQ= 75-10Q

The profit maximizing condition of a firm is

                     MR =MC

        i.e., 75-10Q=10 or, Q=6.5

Therefore, profit maximizing output of both firms = 6.5 units and Firm 1 will produce 6.5/2=3.25 units and Firm 2 will produce 6.5/2=3.25 units

b) If both firms will compete with each other,

then the marginal profit of firm 1 (d$\pi$₁ /dQ₁) = MR₁ -MC₁ = 75-10Q₁ -5Q₂ - 10=65-10Q₁ -5Q₂ =0

and the marginal profit of firm 2 (d$\pi$₂ /dQ₂ ) = 75-10Q₂ -5Q₁ -20= 55-10Q₂ -5Q₁ =0

Therefore, cournot nash equilibrium conditions are,

         65-10Q₁ -5Q₂ =0

      or, 10Q₁ +5Q₂ =65 or, 2Q₁ + Q₂ =13............(v)

and 55-10Q₂ -5Q₁ =0

   or, 2Q₂ +Q₁ =11.......................(vi)

Solving (v) and (vi), we get Q₁ = 5 and Q₂ = 3

Therefore, if both firms compete, then firm 1 will produce 5 units and Firm 2 will produce 3 units.