Question

Consider an oligopoly with 2 firms. The inverse demand curve is given by P = 100 – Q₁ – Q₂. Firm 1’s total cost function is TC₁ = 30Q₁. Firm 2’s total cost function is TC₂ = 20Q₂. Assume now that the firms compete by choosing their prices simultaneously, so it is a Bertrand Oligopoly model. Assume that firms choose prices in 0.01 in intervals. (i.e. A firm can choose to charge $10.00 or $10.01, but not $10.005).

a) Consider the case where firm 1 chooses P₁ = 30 and firm 2 chooses P₂ = 20. Argue whether this is or is not a Nash Equilibrium.

b) Consider the case where firm 1 chooses P₁ = 30.01 and firm 2 chooses P₂ = 30. Argue whether this is or is not a Nash Equilibrium.

Answer 1

Marginal cost of firm 1, C1 is 30 and marginal cost of firm 2, C2 is 20.

a. If firm 1 chooses a price 30 and firm 2 chooses a price 20, firm 2 is charging a price equal to its marginal cost and gence getting 0 profit. If firm 2 deviates to charging a price above 20 and less than 30, say 29, he captures all of the market (ie. q1 = 0) and gets a profit (29-20)q1=9q1 >0 (as q1=100-29=71). Therefore, firm 2 has a profitable deviation and hence, this cannot be a Nash equilibrium.

b. This is a Nash equilibrium. when P1=30.01 and P2 =30, firm 1 is not able to sell anything ie. Q1 = 0 and firm 1's profit is 0 due to price competition and firm 2's profit = (P2- C2)Q2 = (30-20)(100 - 30) = 10*70= 700.

Considering possible deviations for firm 1:

If P1 =30, firm 1's profit will be 0 as P1=C1=30.

If P1 >30, firm 1's profit will be 0 as P1>P2.

If P1<30, firm 1's profit will be negative as P1<C1.

Therefore, firm 1 cannot unilaterally deviate to get a positive profit and hence, has no profitable deviations.

Now consider, firm 2:

If P2>30.01, firm 2's profit will be 0 as P2 > P1

If P2<30, firm 2's profit will be < 700 (current profit) as profit is an increasing function of price in this range.

To show this,

Profit of firm 2 when P2<P1(=30.01)= (P2-20)(100-P2)

120-2P2>0 if P2 <60

Hence, profit is an increasing function of price when P2<60.

If P2 = 30.01, the market will be split between firm 1 and firm 2 and each of the firms will get a profit close to 700/2=350 which is less than 700.

Therefore, firm 2 cannot unilaterally deviate to get a profit greater than 700 and hence, has no profitable deviations.

Therefore, this is a Nash Equilibrium.