<p>Assume that the maxima/minima exists wherever referred (i.e., the necessary secondary conditions are satisfied). <span class="math-container">$p$</span> is the inverse-demand, <span class="math-container">$c_i(q_i)$</span> are the cost functions of the plants that a certain firm has, and <span class="math-container">$c(q)$</span> is the total cost function of the firm.</p>
<p>Suppose <span class="math-container">$q = q_1 + q_2$</span> and we have to maximize <span class="math-container">$\pi(q) = p(q)q - c_1(q_1) - c_2(q_2)$</span>. The maximization holds at <span class="math-container">$(q_1, q_2, q)$</span> which satisfies MR<span class="math-container">$(q)$</span> = MC<span class="math-container">$_1(q_1)$</span> = MC<span class="math-container">$_2(q_2)$</span> such that <span class="math-container">$q = q_1 + q_2$</span>.</p>
<p>I conjectured that at the point of optima, <span class="math-container">$c'(q) = c_1'(q_1) = c_2(q_2)$</span> holds where <span class="math-container">$c(q) = \min[c_1(q_1) + c_2(q_2)] \text{ subject to } q = q_1+q_2$</span>. In other words, MC<span class="math-container">$(q)$</span> = MC<span class="math-container">$_1(q_1)$</span> = MC<span class="math-container">$_2(q_2)$</span> at the optimal point.</p>
<p>How do I prove this?</p>

CC BY-SA 4.0
2022-11-03T09:15:18.317Z
Are you good with derivatives and maximization?
Giskard
CC BY-SA 4.0
2022-11-03T11:59:34.017Z
@Giskard Yes, I think I'll be able to follow your explanation if you provide one.
Harris
<microeconomics><monopoly><profit-maximization>
Show that the marginal cost of the total output equals the marginal cost of individual plant's outputs
Harris
# Show that the marginal cost of the total output equals the marginal cost of individual plant's outputs

(Asked by: username_0 on 2022-11-03)

Assume that the maxima/minima exists wherever referred (i.e., the necessary secondary conditions are satisfied). $p$ is the inverse-demand, $c\_i(q\_i)$ are the cost functions of the plants that a certain firm has, and $c(q)$ is the total cost function of the firm.

Suppose $q = q\_1 + q\_2$ and we have to maximize $\pi(q) = p(q)q - c\_1(q\_1) - c\_2(q\_2)$. The maximization holds at $(q\_1, q\_2, q)$ which satisfies MR$(q)$ = MC$\_1(q\_1)$ = MC$\_2(q\_2)$ such that $q = q\_1 + q\_2$.

I conjectured that at the point of optima, $c'(q) = c\_1'(q\_1) = c\_2(q\_2)$ holds where $c(q) = \min[c\_1(q\_1) + c\_2(q\_2)] \text{ subject to } q = q\_1+q\_2$. In other words, MC$(q)$ = MC$\_1(q\_1)$ = MC$\_2(q\_2)$ at the optimal point.

How do I prove this?

**Tags:** microeconomics, monopoly, profit-maximization

> **Comments:**
> **username_1**: Are you good with derivatives and maximization?  
> **username_0**: @username_1 Yes, I think I'll be able to follow your explanation if you provide one.  

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## 0 answers

