<p>Perfectly inelastic demand means quantity demanded is $q$ irrespective of the price. If producing quantity $q$ costs $c$ then the monopolist's problem is</p>

<p>$$\max_p \{pq-c\}.$$</p>

<p>This problem is not well-defined because the function $pq-c$ is increasing in $p$ and has no maximum. So there is no solution to the monopolist's problem. Intuitively, since the monopolist can sell $q$ units at <em>any</em> price, it would want to choose a price of $+\infty$.</p>

<p>You remark incredulously that we do not observe infinite oil prices. If the price of fuel increased to \$1000 per gallon then people would obviously drive their cars less and demand less fuel. So the demand for oil is not perfectly inelastic!</p>

<hr>

<p>One way to make the problem more well behaved, which is quite common in the literature, is to assume a so-called "choke-price" (a price above which demand drops to zero). Demand is therefore
$$D(p)=\begin{cases}q\text{ if }p\leq\overline{p}\\[0.3em]0\text{ if }p&gt;\overline{p}\end{cases}.$$</p>

<p>The monopolist's problem is then </p>

<p>$$\max_p \{pq-c\}\text{ s. th. }p\leq\overline{p}.$$</p>

<p>This is a well-defined maxmisation problem with a unique solition: $p=\overline{p}$ (assuming $\overline{p}q&lt;c$, otherwise the firm would just shit down).</p>

CC BY-SA 4.0
2018-09-18T11:07:16.473Z
so... the monopoly would reach the maximum possible price BUT nots its theoretical marginal profit or mr = mc...
Guy Louzon
CC BY-SA 4.0
2018-09-18T13:13:07.583Z
@GuyL Yes, the $MR=MC$ condition is a convenient shorthand that works well when the problem is well-defined and we can compute $MR$. But to compute $MR$ we need to be able to differentiate the revenue function. We can't differentiate revenue when demand is inelastic because revenue is not well-defined below $q$ and is discontinuous at $q$.
Ubiquitous
CC BY-SA 4.0
2018-09-18T09:45:46.997Z
2018-09-18T09:45:46.997Z
Ubiquitous
<p>What is the impact of an inelastic demand function on a monopoly production and pricing strategy ?</p>

<p>Theory states, that a monopoly will strive to maximize its profits, by arriving to a point where the difference between the revenues and costs is at a maximum</p>

<p>The potential revenues derive from quantity of sales at a certain price, whixh by turn, are affected by demand</p>

<p>If the demand is inelastic, that means that quantity will remain fixed, no matter the price</p>

<p>Does that narrow or broaden the monopoly's options ?
What's the impact of that type of demand on the equilibrium price ?</p>

CC BY-SA 4.0
2018-09-18T06:57:20.107Z
You write "quantity will remain fixed, no matter the price" and then ask "Does that narrow or broaden the monopoly's options?" The answer seems **very** straightforward. What is causing you difficulties?
Giskard
CC BY-SA 4.0
2018-09-18T07:19:28.570Z
Well, quantity wise you're right, price wise, the optimum is infinite... still we don't see infinite oil prices, for example
Guy Louzon
CC BY-SA 4.0
2018-09-18T07:19:57.347Z
beyond that, how does this affect the monopoly model?
Guy Louzon
CC BY-SA 4.0
2018-09-18T07:55:11.570Z
Take a look at Lerner's Index.
superhulk
CC BY-SA 4.0
2018-09-18T07:55:41.363Z
@superhulk do you havea link?
Guy Louzon
CC BY-SA 4.0
2018-09-18T08:21:49.517Z
Industrial Organisation: Markets and Strategies by Paul Belleflamme can be a good place to start. For some more advanced treatment in this area, you can also refer to 'The Theory of Industrial Organistaion' by Jean Tirole.
superhulk
CC BY-SA 4.0
2018-09-18T08:37:40.257Z
anything simpler than a book, model explanation level?
Guy Louzon
<microeconomics><elasticity><monopoly>
Monopoly equilibrium with a completely inelastic demand
Guy Louzon
# Monopoly equilibrium with a completely inelastic demand

(Asked by: username_0 on 2018-09-18)

What is the impact of an inelastic demand function on a monopoly production and pricing strategy ?

Theory states, that a monopoly will strive to maximize its profits, by arriving to a point where the difference between the revenues and costs is at a maximum

The potential revenues derive from quantity of sales at a certain price, whixh by turn, are affected by demand

If the demand is inelastic, that means that quantity will remain fixed, no matter the price

Does that narrow or broaden the monopoly's options ?
What's the impact of that type of demand on the equilibrium price ?

**Tags:** microeconomics, elasticity, monopoly

> **Comments:**
> **username_1**: You write "quantity will remain fixed, no matter the price" and then ask "Does that narrow or broaden the monopoly's options?" The answer seems **very** straightforward. What is causing you difficulties?  
> **username_0**: Well, quantity wise you're right, price wise, the optimum is infinite... still we don't see infinite oil prices, for example  
> **username_0**: beyond that, how does this affect the monopoly model?  
> **username_2**: Take a look at Lerner's Index.  
> **username_0**: @username_2 do you havea link?  
> **username_2**: Industrial Organisation: Markets and Strategies by Paul Belleflamme can be a good place to start. For some more advanced treatment in this area, you can also refer to 'The Theory of Industrial Organistaion' by Jean Tirole.  
> **username_0**: anything simpler than a book, model explanation level?  

---

## 1 answers

### Answer by username_3 on 2018-09-18. Score: 5 | Accepted answer

Perfectly inelastic demand means quantity demanded is $q$ irrespective of the price. If producing quantity $q$ costs $c$ then the monopolist's problem is

$$\max\_p \{pq-c\}.$$

This problem is not well-defined because the function $pq-c$ is increasing in $p$ and has no maximum. So there is no solution to the monopolist's problem. Intuitively, since the monopolist can sell $q$ units at *any* price, it would want to choose a price of $+\infty$.

You remark incredulously that we do not observe infinite oil prices. If the price of fuel increased to \$1000 per gallon then people would obviously drive their cars less and demand less fuel. So the demand for oil is not perfectly inelastic!

---

One way to make the problem more well behaved, which is quite common in the literature, is to assume a so-called "choke-price" (a price above which demand drops to zero). Demand is therefore
$$D(p)=\begin{cases}q\text{ if }p\leq\overline{p}\\[0.3em]0\text{ if }p>\overline{p}\end{cases}.$$

The monopolist's problem is then

$$\max\_p \{pq-c\}\text{ s. th. }p\leq\overline{p}.$$

This is a well-defined maxmisation problem with a unique solition: $p=\overline{p}$ (assuming $\overline{p}q<c$, otherwise the firm would just shit down).

> **Comments:**
> **username_0**: so... the monopoly would reach the maximum possible price BUT nots its theoretical marginal profit or mr = mc...  
> **username_3**: @GuyL Yes, the $MR=MC$ condition is a convenient shorthand that works well when the problem is well-defined and we can compute $MR$. But to compute $MR$ we need to be able to differentiate the revenue function. We can't differentiate revenue when demand is inelastic because revenue is not well-defined below $q$ and is discontinuous at $q$.  

---

