<p>To produce <span class="math-container">$Q$</span> units at the minimum cost, we can obtain the optimal values of <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span> using <span class="math-container">$\text{MC}_1$</span> and <span class="math-container">$\text{MC}_2$</span> and get:
<span class="math-container">\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*}</span>
So, the (combined) marginal cost is:
<span class="math-container">\begin{eqnarray*} \text{MC} = \begin{cases} 10+20Q &amp; \text{if } Q &lt; 2.5 \\ 50+4Q &amp; \text{if } Q \geq 2.5\end{cases}\end{eqnarray*}</span>
Marginal Revenue is
<span class="math-container">\begin{eqnarray*} \text{MR} = 120-6Q\end{eqnarray*}</span>
<span class="math-container">$\text{MR}=\text{MC}$</span> condition yields the profit maximising choice of quantity as <span class="math-container">$Q^m=7$</span>. Therefore, <span class="math-container">$q_1^m = 3.4, q_2^m = 3.6$</span> and the price <span class="math-container">$p^m = 99$</span>.</p>

CC BY-SA 4.0
2023-04-19T13:26:12.853Z
2023-04-19T13:43:32.963Z
2023-04-19T13:43:32.963Z
Amit
<p>That is not correct. Your solution results in more than the optimal Q since the firm solves the monopolists profit maximization problem for <strong>each</strong> plant.</p>
<p>If <span class="math-container">$Q^\star$</span> is the the output that maximizes profit, the firm wants to produce each unit up to <span class="math-container">$Q^\star$</span> at the plant with the lowest marginal cost. Looking at the <span class="math-container">$MC$</span> 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 <span class="math-container">$MC_2 - MC_1 = 0$</span> for <span class="math-container">$q$</span>, we can find the <span class="math-container">$q$</span> at which the firm will switch to plant 2.</p>
<p>Then solve <span class="math-container">$MC(Q)=MR(Q)$</span> to get the optimal overall output.</p>

CC BY-SA 4.0
2023-04-18T17:13:06.517Z
2023-04-19T12:38:50.730Z
2023-04-19T12:38:50.730Z
A. Miller
<p>Suppose a monopolist faces the following demand curve: <span class="math-container">$() = 120 − 3$</span>, where <span class="math-container">$ = _1 + _2$</span>. The monopolist has two factories. Factory 1 and factory 2 have the following marginal costs:
<span class="math-container">$$
_1(_1) = 10 + 20_1 \\
_2(_2) = 60 + 5_2
$$</span>
Find the monopolist's optimal total quantity, price, and the optimal division of output between the two factories (i.e., the optimal choice of <span class="math-container">$_1$</span> and <span class="math-container">$_2$</span>)</p>
<p>So far, ive found the revenue function which is <span class="math-container">$R(Q) = Q \cdot P(Q) = -3Q^2 + 120Q$</span>. The Marginal revenue is then given by <span class="math-container">$R'(Q) = -6Q + 120$</span>.</p>
<p>I think the optimal plan can be found by solving the set of equations
<span class="math-container">$$
-6Q + 120 = 10 + 20_1 \\
-6Q + 120 = 60 + 5q_2 \\
Q = q_1 + q_2
$$</span>
Which gives the solution <span class="math-container">$Q = 7, q_1 = 3.4, q_2 = 3.6$</span> and price <span class="math-container">$p = P(7) = 99$</span>.</p>
<p>Is this correct or should i use another approach?</p>

<production-function><monopoly>
Optimal production plan in monopol
Fjeeds Arcade
# Optimal production plan in monopol

(Asked by: username_0 on 2023-04-18)

Suppose a monopolist faces the following demand curve: $() = 120 − 3$, where $ = \_1 + \_2$. The monopolist has two factories. Factory 1 and factory 2 have the following marginal costs:
$$
\_1(\_1) = 10 + 20\_1 \\
\_2(\_2) = 60 + 5\_2
$$
Find the monopolist's optimal total quantity, price, and the optimal division of output between the two factories (i.e., the optimal choice of $\_1$ and $\_2$)

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
$$
-6Q + 120 = 10 + 20\_1 \\
-6Q + 120 = 60 + 5q\_2 \\
Q = q\_1 + q\_2
$$
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?

**Tags:** production-function, monopoly

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## 2 answers

### Answer by username_1 on 2023-04-19. Score: 2

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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### Answer by username_2 on 2023-04-18. Score: 1

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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