<p>Let's go for steps:</p>
<p>Profit function : <span class="math-container">$\Pi(Q) = P(Q)Q -C(Q)$</span></p>
<p>Where <span class="math-container">$\Pi(Q)$</span> are profits and <span class="math-container">$P(Q)$</span> is the inverse demand function, i.e., the price at which <span class="math-container">$Q$</span> can be sold given the existing demand, and <span class="math-container">$C(Q)$</span> is the cost function.</p>
<p>Then, profit max implies: <span class="math-container">$\Pi'(Q) = P(Q) + \frac{dP}{dQ}Q - C'(Q)=0$</span></p>
<p>Note that <span class="math-container">$P(Q) + \frac{dP}{dQ}Q$</span> is marginal revenue(MR), and</p>
<p><span class="math-container">$C'(Q)=0$</span> is marginal cost.</p>
<p>Profit maximization implies MR=MC</p>
<p>Then, MR can also be written as <span class="math-container">$P(1+\frac{dP}{dQ} \frac{Q}{P}$</span>)</p>
<p>but the second term in parenthesis is just the reciprocal of the price elasticity of demand ( call it <span class="math-container">$\epsilon_p$</span>). You can rewrite MR as:</p>
<p><span class="math-container">$P(1+ \frac{1}{\epsilon_p})=P(\frac{1+\epsilon_p}{\epsilon_p})$</span></p>
<p>Then, equating MR to MC and solving for P:</p>
<p><span class="math-container">$P=(\frac{\epsilon_p}{1+\epsilon_p})MC$</span></p>
<p>Then, letting <span class="math-container">$\eta$</span> be the reciprocal of the price elasticity of demand:</p>
<p><span class="math-container">$P=(\frac{1}{1+\eta})MC$</span></p>
<p>Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule (<strong>markup rule</strong>).</p>
<p>Now, if a monopolist faces isoelastic demand, this means that the price elasticity of demand is constant and also <span class="math-container">$\eta$</span> is constant (because it is the reciprocal of the price elasticity of demand). Indeed, the term in parenthesis,<span class="math-container">$\frac{1}{1+\eta}$</span>, is simply the markup (and it is constant over the MC).</p>

CC BY-SA 4.0
2022-09-01T20:04:37.813Z
2022-09-01T20:04:37.813Z
Maximilian
<p>Why if the monopolists face isoelastic demands, profit maximization(for monopolists) implies that each of them sets a constant markup over the marginal cost?</p>
<p>ADD DETAILS:</p>
<p>This is a common situation when you deal with an endogenous R&amp;D growth model. You have a unique homogeneous final good produced competitively. The production of this final good is described by a CES production function or simply a Cobb-Douglas aggregator. This homogenous final good is produced by a unit continuum of intermediates (or a range of intermediates evolving endogenously) produced in monopolistic competition. Then, the demands faced by the monopolists (producing each of them an intermediate) are isoelastic. Why does it imply constant mark up over the MC ?</p>

CC BY-SA 4.0
2022-08-31T08:16:27.470Z
Please provide some more details
Papayapap
<macroeconomics><microeconomics><endogenous-growth>
Isoelastic demands and constant markup
John M.
# Isoelastic demands and constant markup

(Asked by: username_0 on 2022-08-30)

Why if the monopolists face isoelastic demands, profit maximization(for monopolists) implies that each of them sets a constant markup over the marginal cost?

ADD DETAILS:

This is a common situation when you deal with an endogenous R&D growth model. You have a unique homogeneous final good produced competitively. The production of this final good is described by a CES production function or simply a Cobb-Douglas aggregator. This homogenous final good is produced by a unit continuum of intermediates (or a range of intermediates evolving endogenously) produced in monopolistic competition. Then, the demands faced by the monopolists (producing each of them an intermediate) are isoelastic. Why does it imply constant mark up over the MC ?

**Tags:** macroeconomics, microeconomics, endogenous-growth

> **Comments:**
> **username_1**: Please provide some more details  

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## 1 answers

### Answer by username_2 on 2022-09-01. Score: 2 | Accepted answer

Let's go for steps:

Profit function : $\Pi(Q) = P(Q)Q -C(Q)$

Where $\Pi(Q)$ are profits and $P(Q)$ is the inverse demand function, i.e., the price at which $Q$ can be sold given the existing demand, and $C(Q)$ is the cost function.

Then, profit max implies: $\Pi'(Q) = P(Q) + \frac{dP}{dQ}Q - C'(Q)=0$

Note that $P(Q) + \frac{dP}{dQ}Q$ is marginal revenue(MR), and

$C'(Q)=0$ is marginal cost.

Profit maximization implies MR=MC

Then, MR can also be written as $P(1+\frac{dP}{dQ} \frac{Q}{P}$)

but the second term in parenthesis is just the reciprocal of the price elasticity of demand ( call it $\epsilon\_p$). You can rewrite MR as:

$P(1+ \frac{1}{\epsilon\_p})=P(\frac{1+\epsilon\_p}{\epsilon\_p})$

Then, equating MR to MC and solving for P:

$P=(\frac{\epsilon\_p}{1+\epsilon\_p})MC$

Then, letting $\eta$ be the reciprocal of the price elasticity of demand:

$P=(\frac{1}{1+\eta})MC$

Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule (**markup rule**).

Now, if a monopolist faces isoelastic demand, this means that the price elasticity of demand is constant and also $\eta$ is constant (because it is the reciprocal of the price elasticity of demand). Indeed, the term in parenthesis,$\frac{1}{1+\eta}$, is simply the markup (and it is constant over the MC).


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