<p>If MC increases, profits need to go down. Let <span class="math-container">$\pi (q(c))$</span> and <span class="math-container">$\pi' (q(c'))$</span> denote profits before and after the cost increase. Observe that producing <span class="math-container">$q(c')$</span> was feasible when the marginal cost was low and, therefore, it must be the case that</p>

<p><span class="math-container">$$ \pi (q(c))=[p(q(c))-c]q(c)\geq[p(q(c'))-c]q(c').$$</span></p>

<p>In addition, we have that</p>

<p><span class="math-container">$$[p(q(c'))-c]q(c')&gt;[p(q(c'))-c']q(c')=\pi' (q(c')),$$</span></p>

<p>which gives us <span class="math-container">$\pi (q(c))&gt;\pi' (q(c')).$</span></p>

CC BY-SA 4.0
2019-01-01T18:29:27.827Z
2019-01-01T18:29:27.827Z
Patricio
<p>Let a monopolist have constant MC = c, with no fixed costs. If <span class="math-container">$p(c)$</span> and <span class="math-container">$q(c)$</span> are the profit maximising price and output as a function of MC and the market demand schedule is given by <span class="math-container">$q = D(p)$</span>, how does the monopolists' profit vary with a small change in MC? This is what I have so far: </p>

<p>Maximised profits is given by:</p>

<p><span class="math-container">$$
π = p(c)q(c) - cq(c)
$$</span></p>

<p>Differentiating w.r.t c:</p>

<p><span class="math-container">$$
\frac{dπ(c)}{dc} = q(c)[p'(c) - 1] + q'(c)[p(c) - c]
$$</span> </p>

<p>The first term <span class="math-container">$q(c)[p'(c) - 1]$</span> is clearly positive because <span class="math-container">$q(c) &gt; 0$</span> and for a monopolist <span class="math-container">$[p'(c) - 1]$</span> is also positive because: </p>

<p><span class="math-container">$$
p(c) = \frac{c}{1+\frac{1}{Ɛ}}
$$</span></p>

<p><span class="math-container">$$
p'(c) = \frac{1}{1+\frac{1}{Ɛ}}
$$</span></p>

<p>and for there to be a profit maximising level of output, <span class="math-container">$Ɛ &lt; -1$</span></p>

<p><span class="math-container">$$
p'(c) =  \frac{1}{1+\frac{1}{Ɛ}}&gt; 1
$$</span></p>

<p>For the second term, <span class="math-container">$q'(c)$</span> is clearly negative since higher MC leads to higher price and therefore lower quantity. <span class="math-container">$[p(c) - c] &gt; 0$</span> since for a monopolist <span class="math-container">$p &gt; MC = c$</span>.</p>

<p>At this stage, the effect on profits of a rise in MC is ambiguous since both terms have opposing effects (one is positive, other is negative) and how profits change depends on the magnitude of the two terms. How can we quantify the effects or at least know the relative magnitudes of both terms to find how profits will change to an increase in MC?</p>

<monopoly><profit-maximization>
Effect of changing MC on monopolists' maximised profits
Robin Liao
# Effect of changing MC on monopolists' maximised profits

(Asked by: username_0 on 2019-01-01)

Let a monopolist have constant MC = c, with no fixed costs. If $p(c)$ and $q(c)$ are the profit maximising price and output as a function of MC and the market demand schedule is given by $q = D(p)$, how does the monopolists' profit vary with a small change in MC? This is what I have so far:

Maximised profits is given by:

$$
π = p(c)q(c) - cq(c)
$$

Differentiating w.r.t c:

$$
\frac{dπ(c)}{dc} = q(c)[p'(c) - 1] + q'(c)[p(c) - c]
$$

The first term $q(c)[p'(c) - 1]$ is clearly positive because $q(c) > 0$ and for a monopolist $[p'(c) - 1]$ is also positive because:

$$
p(c) = \frac{c}{1+\frac{1}{Ɛ}}
$$

$$
p'(c) = \frac{1}{1+\frac{1}{Ɛ}}
$$

and for there to be a profit maximising level of output, $Ɛ < -1$

$$
p'(c) = \frac{1}{1+\frac{1}{Ɛ}}> 1
$$

For the second term, $q'(c)$ is clearly negative since higher MC leads to higher price and therefore lower quantity. $[p(c) - c] > 0$ since for a monopolist $p > MC = c$.

At this stage, the effect on profits of a rise in MC is ambiguous since both terms have opposing effects (one is positive, other is negative) and how profits change depends on the magnitude of the two terms. How can we quantify the effects or at least know the relative magnitudes of both terms to find how profits will change to an increase in MC?

**Tags:** monopoly, profit-maximization

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## 1 answers

### Answer by username_1 on 2019-01-01. Score: 1 | Accepted answer

If MC increases, profits need to go down. Let $\pi (q(c))$ and $\pi' (q(c'))$ denote profits before and after the cost increase. Observe that producing $q(c')$ was feasible when the marginal cost was low and, therefore, it must be the case that

$$ \pi (q(c))=[p(q(c))-c]q(c)\geq[p(q(c'))-c]q(c').$$

In addition, we have that

$$[p(q(c'))-c]q(c')>[p(q(c'))-c']q(c')=\pi' (q(c')),$$

which gives us $\pi (q(c))>\pi' (q(c')).$


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