Profit Max Cartel

Question

2 Companies both selling the same luxury good. Company A cost to produce < Company B cost to produce, price for the good is the same (P). How do I address the issue of profit maximization if the output of Company A is < the Output of Company B. (Even though Company A has a lower cost of production), ie they both exhibit increasing and convex cost functions.

Answer 1

Letting $Q_i$, $TC_i(Q_i)$, $P_i(Q)$ be the quantity, cost function, and inverse demand for i, we can write the joint profit maximization as:

$max_{Q_A,Q_B} $

$Q_A (P_A(Q_A,Q_B)) + Q_B (P_B(Q_B,Q_A))- TC_A(Q_A) -TC_B(Q_B)$

With FOCs:

$P_i(Q_i,Q_j) = d/dQ_i TC_i - Q_i (d/dQ_i P_i) - Q_j (d/dQ_i) P_j$

If they’re selling the same good it would be that $P_B(Q_B,Q_A) = P_A(Q_A,Q_B)$, which simplifies the problem a bit to:

$P= d/dQ_A TC_A - Q_A (d/dQ_A P) - Q_B (d/dQ_A ) P = d/dQ_B TC_B - Q_B (d/dQ_B P) - Q_A (d/dQ_B ) P $