Question

The geegaw industry consists of two Cournot competitors producing an identical product. The inverse demand equation is

P=591-4Q.

            The total cost equations of the two firms are:

TC1=15Q1;

TC2=31Q2.

            a.         Determine the total revenue equation for each firm.

            b.         What is the reaction function of each firm?

            c.          What is the Cournot-Nash equilibrium level of output?

            d.         What is the market-determined price of geegaws?

            e.         Calculate each firm’s total profit.

Answer 1

The geegaw industry consists of two Cournot competitors producing an identical product.

The inverse demand function is given as

P = 591 - 4.Q = 591 - 4.(Q1+Q2)

Where, Q1 is output of Firm 1 and Q2 is output of Firm 2.

(a) Hence, Total Revenue for firm 1 is

TR1 = P.Q1 = [591 - 4.(Q1+Q2)].Q1

or,

This is the total revenue equation for Firm 1.

And, Total Revenue for Firm 2 is

TR2 = P.Q2 = [591 - 4.(Q1+Q2)].Q2

or,

This is the total revenue euation for Firm 2.

(b) Now, we will calculate the Marginal Revenue equations from TR1 and TR2.

Marginal Revenue for Firm 1 is

MR1 = dTR1/dQ1 = 591 - 8.Q1 - 4.Q2........(1)

MR2 = dTR2/dQ2 = 591 - 4.Q1 - 8.Q2........(2)

Total Cost of Firm 1 is: TC1 = 15.Q1

Total Cost of Firm 2 is: TC2 = 31.Q2

Marginal Cost of Firm 1 and Firm 2 are,

MC1 = dTC1/dQ1 = 15.......(3)

MC2 = dTC2/dQ2 = 31.......(4)

Now, at Cournot competition the firms will maximize their profits. Hence, they will set their Marginal Costs equal to their Marginal Revenue to maximize profit.

Hence, for firm 1

MR1 = MC1

or, 591 - 8.Q1 - 4.Q2 = 15

or, Q1 = 72 - 0.5.Q2...........RF1

This is the reaction function for firm 1.

Also, fo Firm 2

MR2 = MC2

or, 591 - 4.Q1 - 8.Q2 = 31

or, Q2 = 70 - 0.5.Q1............RF2

This is the reaction function for firm 2.

(c) Now, solving the two functions RF1 and RF2, we get

Q1* = 49.33 and Q2* = 45.33

Hence, Cournot-Nash Equilibrum output for Firm 1 and Firm 2 is (Q1*, Q2*) = (49.33, 45.33).

Hence, Total Output at Cournot Nash Equilibrium is: Q* = Q1*+Q2* = 49.33+45.33

or, Q* = 94.66

(d) Hence, the market -determined price for geegaws is

P* = 591 - 4.Q* = 591 - 4×94.66

or, P* = 212.36

Hence, market determined price for geegaws is P*=212.36.

(e) The Total Revenue and Total Cost of Firm 1 are

TR1 = P*.Q1* = 212.36×49.33

or, TR1 = 10475.72

And, TC1 = 15.Q1* = 15×49.33

or, TC1 = 739.95

Hence, Profit of Firm 1 is

π1 = TR1 - TC1 = 10475.72 - 739.95

or, π1 = 9735.77

Profit of Firm 1 is 9735.77.

Also, Total Revenue and Total Cost for firm 2 are,

TR2 = P*.Q2* = 212.36×45.33

or, TR2 = 9626.28

And, TC2 = 31.Q2* = 31×45.33

or, TC2 = 1405.23

Also, Profit of Firm 2 is

π2 = TR2 - TC2 = 9626.28 - 1403.23

or, π2 = 8223.05

Profit of Firm 2 is 8223.05.

Hope the solutions are clear to you my friend.