Question

1. There are two firms, A and B producing differentiated products. Their demand curves are:

_{qA=100-2PA+3PB}

_{qB=120-2PB+2PA}

and both have MC=5. Note that demand curves are not symmetric. Assuming that firms are engaged in Bertrand price competition:

(a)Write down the profit function of firm A and find its price response function

Hint:
_{πA=(PA-5)(100-2PA+3PB)}

(b) Write down the profit function of firm B and find its price response function

(c) Find equilibrium prices P_A and P_B; equilibrium quantities q_A and q_B; and profits for firms A and B.

Answer 1

Given: qA = 100 - 2PA + 3PB

          qB = 120 - 2PB + 2PA

             MC = 5

(a)

For Firm A:

Total Revenue(TRA) = PA x qA

Profit for firm A( πA) = TRA - TCA

Since MCA =

we have TCA = MCA x qA

Rewriting the profit equation we have:

πA = PA x qA - MCA x qA

       = (PA - MCA) x qA

     = (PA - 5)( 100 - 2PA + 3PB)

  

Now, we'll calculate

Since at equilibrium, profit will be maximized, the above first derivative will be zero:

=> 0 = 110 - 4PA+ 3PB

=>    -----------------------------------------------------(i)

This is the price response function for firm A.

(b)

For Firm B:

πB = PB x qB - MCB x qB

       = (PB - MCB) x qB

     = (PB - 5)( 120 - 2PB+ 2PA)

     =

     =

Now, we'll calculate:

Since at equilibrium, profit will be maximized, the above first derivative will be zero:

=> 0 = 130 - 4PB+ 2PA

=>    -------------------------------------------------------(ii)

This is the price response function for firm B.

(c)

Substituting the value of PB in eq(i) to obtain the equilibrium prices:

Substituting the value of PA in eq(ii), we have:

Substituting the equilibrium prices in demand functions, we obtain:

Similarly,

Substituting the equilibrium prices in profit functions we have: