<p>In <a href="https://economics.stackexchange.com/questions/1726/price-elasticity-of-demand-for-positive-price-increases/1728#1728">this post</a> you can find the algebraic steps that lead to the (standard) result mentioned in Varian's book.  </p>

<p>Now, let's assume that, in a specific market, the consumer's preferences are such that they lead to a constant elasticity demand curve, with elasticity lower than unity in absolute terms, $|\eta| &lt; 1$, for example</p>

<p>$$Q^d = AP^{\eta}, -1 &lt;\eta &lt; 0$$ 
Also, let's assume that for historical or institutional reasons this market is a monopoly. From the post mentioned above we have that profit maximization by the monopolist requires that
$$P^* =  \frac {|\eta|}{|\eta|-1} MC \tag{1}$$</p>

<p>where</p>

<p>$$\eta = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} \Rightarrow \frac {\partial Q }{ \partial P} = \eta \cdot \frac {Q}{P} \tag{2}$$
and $MC$ is marginal cost.
Obviously, this price is negative in our case, and so meaningless. We don't need to go into sophisticated constrained maximization procedures to see what happens here: the profit function is
$$\pi = P\cdot Q(P) - C(Q(P)) \tag{3}$$
and its derivative with respect to price is
$$\frac {\partial \pi}{\partial P}  = Q + P\frac {\partial Q }{ \partial P} - MC\cdot \frac {\partial Q }{ \partial P}  \tag{4}$$</p>

<p>Using $(2)$ we get</p>

<p>$$ \frac {\partial \pi}{\partial P}=Q + P\cdot \eta \cdot \frac {Q}{P} - MC\cdot \eta \cdot \frac {Q}{P} $$</p>

<p>$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 + \eta  -  \eta \cdot \frac {MC}{P}\right]$$</p>

<p>$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 - |\eta|  +  |\eta| \cdot \frac {MC}{P}\right] \tag{5}$$</p>

<p>From $(5)$ we see that</p>

<p>$$|\eta| &lt; 1 \implies \frac {\partial \pi}{\partial P} &gt; 0,
\;\; \forall P &gt;0 \tag{6}$$</p>

<p>So a profit maximizing monopolist would theoretically have the tendency to increase the price to "infinity" sending the quantity supplied to zero. Note that the Revenue function here is</p>

<p>$$R = P\cdot Q^d = P\cdot AP^{\eta} = AP^{1-|\eta|},  \uparrow \text{in} \;P$$</p>

<p>while Costs are decreasing in $Q^d$. So indeed profits would tend to infinity by selling less and less for higher and higher price.  </p>

<p>What markets could be described by such tendencies?</p>

CC BY-SA 3.0
2015-04-08T17:14:09.537Z
Does it make a difference if we partially differentiate profit function with P than Q? When I differentiated above profit function with Q, I got negative first derivative, which supports the argument by another way, i.e., by reducing output indefinitely the monopolist could keep on increasing profits.
Dhruv Goel
CC BY-SA 3.0
2015-04-08T17:17:02.757Z
@DhruvGoel You should get the same essential result indeed.
Alecos Papadopoulos
CC BY-SA 3.0
2015-04-08T14:38:44.967Z
2015-04-08T14:38:44.967Z
2017-04-13T12:51:38.253Z
Alecos Papadopoulos
<p>A constant elasticity demand function has the form $q=p^{-\epsilon}$. Let's check this indeed gives us a constant elasticity...
$$\frac{d q}{dp}=-\epsilon p^{-\epsilon-1}$$
so, as we'd hoped, the elasticity is constant:
$$\frac{d q}{dp}\frac{p}{q}=-\epsilon p^{-\epsilon-1}\frac{p}{p^{-\epsilon}}=-\epsilon p^{-\epsilon-1+1+\epsilon}=-\epsilon p^{0}=-\epsilon.$$</p>

<hr>

<p>Now, suppose a monopolist has constant marginal cost $c\geqslant0$ (and no fixed cost). It's profit is
$$q(p-c)=p^{-\epsilon}(p-c).$$
The effect on the firm's profits of a small increase in its price is
$$\frac{d}{d p}p^{-\epsilon}(p-c)=p^{-\epsilon -1} [c \epsilon -p\epsilon+p].$$
Now we can answer the question: if $\epsilon&lt;1$ then $c \epsilon -p\epsilon+p$ is definitely positive <strong>for every $p$</strong>. That means, no matter how high the price is, the firm would still like to increase the price further—the optimal price is $+\infty$!</p>

<hr>

<p>Intuitively, constant  $\epsilon&lt;1$ means that the firm can always more than double its price by foregoing less than half its current sales. Provided marginal cost isn't very quickly decreasing, this is always an attractive prospect for the firm.</p>

CC BY-SA 3.0
2015-04-08T14:34:49.777Z
2015-04-08T14:34:49.777Z
Ubiquitous
<p>While reading Ch. 24-Monopoly from Intermediate Microeconomics by Hal Varian (8 <sup>th</sup> edition), on pg. 441, he writes that a monopolist will never choose to operate where the demand curve is inelastic. I understand the argument, but, if we have constant elasticity demand curve with</p>

<p>$|\epsilon| &lt; 1 $</p>

<p>then how this effect the monopolist's choice?</p>

CC BY-SA 3.0
2015-04-12T21:36:08.850Z
Exercise:  Show that it is impossible for a demand curve to have a constant elasticity with absolute value less than one.  Hint:  Think about the budget constraint.
Steven Landsburg
<elasticity><monopoly><producer-theory>
Monopoly pricing under constant elasticity of demand
Dhruv Goel
# Monopoly pricing under constant elasticity of demand

(Asked by: username_0 on 2015-04-08)

While reading Ch. 24-Monopoly from Intermediate Microeconomics by Hal Varian (8 th edition), on pg. 441, he writes that a monopolist will never choose to operate where the demand curve is inelastic. I understand the argument, but, if we have constant elasticity demand curve with

$|\epsilon| < 1 $

then how this effect the monopolist's choice?

**Tags:** elasticity, monopoly, producer-theory

> **Comments:**
> **username_1**: Exercise:  Show that it is impossible for a demand curve to have a constant elasticity with absolute value less than one.  Hint:  Think about the budget constraint.  

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## 2 answers

### Answer by username_2 on 2015-04-08. Score: 1 | Accepted answer

In [this post](https://economics.stackexchange.com/questions/1726/price-elasticity-of-demand-for-positive-price-increases/1728#1728) you can find the algebraic steps that lead to the (standard) result mentioned in Varian's book.

Now, let's assume that, in a specific market, the consumer's preferences are such that they lead to a constant elasticity demand curve, with elasticity lower than unity in absolute terms, $|\eta| < 1$, for example

$$Q^d = AP^{\eta}, -1 <\eta < 0$$
Also, let's assume that for historical or institutional reasons this market is a monopoly. From the post mentioned above we have that profit maximization by the monopolist requires that
$$P^\* = \frac {|\eta|}{|\eta|-1} MC \tag{1}$$

where

$$\eta = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} \Rightarrow \frac {\partial Q }{ \partial P} = \eta \cdot \frac {Q}{P} \tag{2}$$
and $MC$ is marginal cost.
Obviously, this price is negative in our case, and so meaningless. We don't need to go into sophisticated constrained maximization procedures to see what happens here: the profit function is
$$\pi = P\cdot Q(P) - C(Q(P)) \tag{3}$$
and its derivative with respect to price is
$$\frac {\partial \pi}{\partial P} = Q + P\frac {\partial Q }{ \partial P} - MC\cdot \frac {\partial Q }{ \partial P} \tag{4}$$

Using $(2)$ we get

$$ \frac {\partial \pi}{\partial P}=Q + P\cdot \eta \cdot \frac {Q}{P} - MC\cdot \eta \cdot \frac {Q}{P} $$

$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 + \eta - \eta \cdot \frac {MC}{P}\right]$$

$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 - |\eta| + |\eta| \cdot \frac {MC}{P}\right] \tag{5}$$

From $(5)$ we see that

$$|\eta| < 1 \implies \frac {\partial \pi}{\partial P} > 0,
\;\; \forall P >0 \tag{6}$$

So a profit maximizing monopolist would theoretically have the tendency to increase the price to "infinity" sending the quantity supplied to zero. Note that the Revenue function here is

$$R = P\cdot Q^d = P\cdot AP^{\eta} = AP^{1-|\eta|}, \uparrow \text{in} \;P$$

while Costs are decreasing in $Q^d$. So indeed profits would tend to infinity by selling less and less for higher and higher price.

What markets could be described by such tendencies?

> **Comments:**
> **username_0**: Does it make a difference if we partially differentiate profit function with P than Q? When I differentiated above profit function with Q, I got negative first derivative, which supports the argument by another way, i.e., by reducing output indefinitely the monopolist could keep on increasing profits.  
> **username_2**: @username_0 You should get the same essential result indeed.  

---

### Answer by username_3 on 2015-04-08. Score: 2

A constant elasticity demand function has the form $q=p^{-\epsilon}$. Let's check this indeed gives us a constant elasticity...
$$\frac{d q}{dp}=-\epsilon p^{-\epsilon-1}$$
so, as we'd hoped, the elasticity is constant:
$$\frac{d q}{dp}\frac{p}{q}=-\epsilon p^{-\epsilon-1}\frac{p}{p^{-\epsilon}}=-\epsilon p^{-\epsilon-1+1+\epsilon}=-\epsilon p^{0}=-\epsilon.$$

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Now, suppose a monopolist has constant marginal cost $c\geqslant0$ (and no fixed cost). It's profit is
$$q(p-c)=p^{-\epsilon}(p-c).$$
The effect on the firm's profits of a small increase in its price is
$$\frac{d}{d p}p^{-\epsilon}(p-c)=p^{-\epsilon -1} [c \epsilon -p\epsilon+p].$$
Now we can answer the question: if $\epsilon<1$ then $c \epsilon -p\epsilon+p$ is definitely positive **for every $p$**. That means, no matter how high the price is, the firm would still like to increase the price further—the optimal price is $+\infty$!

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Intuitively, constant $\epsilon<1$ means that the firm can always more than double its price by foregoing less than half its current sales. Provided marginal cost isn't very quickly decreasing, this is always an attractive prospect for the firm.


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