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Suppose a monopolist faces the following demand curve: $𝑃(𝑄) = 120 βˆ’ 3𝑄$, where $𝑄 = π‘ž_1 + π‘ž_2$. The monopolist has two factories. Factory 1 and factory 2 have the following marginal costs: $$ 𝑀𝐢_1(π‘ž_1) = 10 + 20π‘ž_1 \\ 𝑀𝐢_2(π‘ž_2) = 60 + 5π‘ž_2 $$ Find the monopolist's optimal total quantity, price, and the optimal division of output between the two factories (i.e., the optimal choice of $π‘ž_1$ and $π‘ž_2$)

So far, ive found the revenue function which is $R(Q) = Q \cdot P(Q) = -3Q^2 + 120Q$. The Marginal revenue is then given by $R'(Q) = -6Q + 120$.

I think the optimal plan can be found by solving the set of equations $$ -6Q + 120 = 10 + 20π‘ž_1 \\ -6Q + 120 = 60 + 5q_2 \\ Q = q_1 + q_2 $$ Which gives the solution $Q = 7, q_1 = 3.4, q_2 = 3.6$ and price $p = P(7) = 99$.

Is this correct or should i use another approach?

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2 Answers 2

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To produce $Q$ units at the minimum cost, we can obtain the optimal values of $q_1$ and $q_2$ using $\text{MC}_1$ and $\text{MC}_2$ and get: \begin{eqnarray*} q_1 = \min\left(2 + \frac{1}{5}Q, Q\right) \\ q_2 = \max\left(\frac{4}{5}Q-2, 0\right)\end{eqnarray*} So, the (combined) marginal cost is: \begin{eqnarray*} \text{MC} = \begin{cases} 10+20Q & \text{if } Q < 2.5 \\ 50+4Q & \text{if } Q \geq 2.5\end{cases}\end{eqnarray*} Marginal Revenue is \begin{eqnarray*} \text{MR} = 120-6Q\end{eqnarray*} $\text{MR}=\text{MC}$ condition yields the profit maximising choice of quantity as $Q^m=7$. Therefore, $q_1^m = 3.4, q_2^m = 3.6$ and the price $p^m = 99$.

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That is not correct. Your solution results in more than the optimal Q since the firm solves the monopolists profit maximization problem for each plant.

If $Q^\star$ is the the output that maximizes profit, the firm wants to produce each unit up to $Q^\star$ at the plant with the lowest marginal cost. Looking at the $MC$ equations, we can see that plant 1 has lower marginal costs at low levels of output, and plant 2 has lower marginal costs at high levels of output. Solving $MC_2 - MC_1 = 0$ for $q$, we can find the $q$ at which the firm will switch to plant 2.

Then solve $MC(Q)=MR(Q)$ to get the optimal overall output.

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