Revenue maximization

Question

We have two firms with identical cost structure compete in a market

Demand function = $p=a-bq$

And $q=q_1+q_2$

They are identical in every way. However, firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative.

Both firms have constant and equal marginal cost c. So I want to find the quantities that they will choose.

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What I did is...

For firm 1,

$$\pi_1=max[(a-b(q_1+q_2))q_1-cq_1]$$

FOCs for $q_1$

$$a-2bq_1-bq_2-c=0$$

So $$q_1={a-bq_2-c\over 2b}$$

For firm 2,

$$max [(a-b(q_1+q_2))q_2]$$

FOCs $$a-bq_1-2bq_2=0$$

$$q_2={a-bq_1\over 2b}$$

So, $$q_1={a-b({a-bq_1\over 2b})-c\over 2b}$$

$$q_1^*={a-2c\over 3b}$$

And $$q^*_2={5a+2b\over 6b}$$

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The question says that “* firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative.*”

Because of this sentence, I am exactly not sure about my solution. Especially for firm 2.

I’m confused at this point. Please tell my mistakes. Thank you.

Answer 1

Firm 1's objective is to maximize profits : $$\max_{q_1\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_1 - cq_1$$

Solving the above problem, we get the best response function of firm 1 as $$q_1 = \dfrac{a - c - bq_2}{2b}$$

Firm 2's objective is to maximize revenue subject to the constraint that its profits are non-negative : $$\max_{q_2\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_2$$ $$\text{s.t. }\left(a - b(q_1+q_2)\right)q_2 - cq_2 \geq 0$$

Solving the above problem, we get the best response function of firm 2 as \begin{eqnarray*} q_2 =\begin{cases} \dfrac{a - bq_1}{2b} & \text{if } a - bq_1 \geq 2c \\ \dfrac{a - c - bq_1}{b} & \text{if } a - bq_1 < 2c \end{cases}\end{eqnarray*}

Solving the best response functions for $q_1$ and $q_2$ yields \begin{eqnarray*} (q_1^*, q_2^*) = \begin{cases} \left(\dfrac{a-2c}{3b}, \dfrac{a+c}{3b}\right) & \text{if } a \geq 2c\\ \left( 0, \dfrac{a-c}{b}\right) & \text{if } a < 2c \end{cases} \end{eqnarray*}