<p>Your calculation is not wrong. First a more easy and more intuitive way of calculating marginal revenue is just to take derivative of total revenue.</p>
<p>Total revenue is price times quantity <span class="math-container">$TR=P (Q) Q$</span> where price is still function of quantity as at higher price people demand lower quantity of goods (I am expressing everything in terms of Q since that is the choice variable in your model). Substituting for quantity we get:</p>
<p><span class="math-container">$$TR=  (100/Q)Q \implies MR = \frac{dTR}{dq} = 0$$</span></p>
<p>Hence in this case they are indeed zero. Furthermore, in this case the profit maximization problem of monopolist does not even have a solution. Profit is given as total revenue minus total costs so:</p>
<p><span class="math-container">$$\Pi = TR-TC = (100/Q)Q -4Q$$</span></p>
<p>If we would try to derive the first order conditions for maximization we would just get that 0=4 which means there is no solution. To make that clear I even programmed a visualization of your problem in R, this is how it looks like where green curve is demand, red line is marginal revenue and blue line is marginal cost.</p>
<p><a href="https://i.sstatic.net/FA4iv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/FA4iv.png" alt="enter image description here" /></a></p>
<p><strong>So there are three options I see here:</strong></p>
<ol>
<li>This is a trick question and you teacher expects correct answer to be that there is no solution (no solution in standard non-negative real numbers that is as it could be argued that quantity produced should be as small as possible but not zero (infinitesimal) from right side - however thats not a standard real number).</li>
<li>You misread the problem or made mistake in part of the problem which asked you to first create cost function or demand from some text.</li>
<li>Your teacher made a mistake when making up the problem.</li>
</ol>
<p>Hence my advice is first make sure you read the set up thoroughly, if there is no mistake there then contact your teacher and ask if its possible that the teacher made a mistake since MR is zero in your case. If there is no mistake then its a correct solution.</p>
<hr />
<p>response to question in comments:</p>
<p>Price elasticity of demand is defined as:</p>
<p><span class="math-container">$$\epsilon = \frac{d Q}{dp}\frac{p}{Q}$$</span></p>
<p>So in your case it will be:</p>
<p><span class="math-container">$$\epsilon = -\frac{100}{p^2} \frac{p}{\frac{100}{p}}= -\frac{100}{p^2} \frac{p^2}{100} = -1$$</span></p>
<hr />
<p>Addendum: In response to comments I also added simulation for profits of the monopolist in this situation assuming that the profit function is:</p>
<p>a) <span class="math-container">$\Pi = \frac{100}{Q} Q -4Q$</span>  - in red color</p>
<p>b) <span class="math-container">$\Pi = 100 - 4Q$</span> - in blue color</p>
<p><a href="https://i.sstatic.net/Fgvff.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Fgvff.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-09-07T13:31:42.800Z
Hello, thank you for your reply! Whilst waiting for an answer, I read up a bit on internet and found out that demand curves of the form $PQ = C$, where $C$ is a constant, will be unit elastic everywhere i.e. $\epsilon = -1$. How is this derived, though, using the formula I provided above?
Ethan Mark
CC BY-SA 4.0
2020-09-07T13:38:19.283Z
@EthanMark I added an explanation for you at the bottom of the answer
1muflon1
CC BY-SA 4.0
2020-09-07T13:41:39.160Z
Hello, thank you for your reply! That was the solution I found as well, but I was wondering how I could use the $MR = P(1 + \frac 1 \epsilon) = MC$ to find $\epsilon$? Since in this case $MR$ and $MC$ both give different answers.
Ethan Mark
CC BY-SA 4.0
2020-09-07T13:45:56.233Z
oh I see, the problem here is that actually $P(1+\frac{1}{\epsilon} )= MC$ is not generally valid statement - only the first half of equality is generally valid. That is we can be only sure that $MR= P(1+\frac{1}{\epsilon})$. The second part of the equality requires that $MR=MC$ but that happens only in equilibrium when firm maximizes its profit since the first order conditions from a generic profit function $\Pi = TR-TC$ will normally give you result that $MR=MC$ however in this case there is no solution to the profit maximizing problem so you cant claim that $MR=MC$
1muflon1
CC BY-SA 4.0
2020-09-07T13:47:38.640Z
furthermore, note that $MR=MC$ holds only in profit maximizing equilibrium (if it exists) its not some universal truth. In fact at any other point then the one which maximizes profit $MR \neq MC$ assuming we have well behaved functions
1muflon1
CC BY-SA 4.0
2020-09-07T14:04:11.783Z
I see. Thank you :) I understand now! Also, I have looked hard at the question and I believe I copied it right. It is also a graded assignment so I do not think my professor made any mistakes. Perhaps the answer to the optimal quantity is $0$ i.e. the firm does not/will not operate!
Ethan Mark
CC BY-SA 4.0
2020-09-07T14:19:14.390Z
@EthanMark I would still recommend asking your teacher - I am PhD student but I still teach tutorials for undergraduates and honestly I few times made similar mistakes even when I was creating graded assignment. Also note here we just simply cannot determine optimal quantity because of the FOC's. Not being able to find solution is **not** the same as saying solution is 0
1muflon1
CC BY-SA 4.0
2020-09-07T14:30:26.893Z
You have a point there! I will double-check with my professor on that :) Thank you for your help on understanding this question!
Ethan Mark
CC BY-SA 4.0
2020-09-08T08:50:56.667Z
@Tomcat but how could that be? Let’s just take step back At $Q=0 $firm does not operate so $\Pi=0$ at $Q=1$ $\Pi=96$ - the problem here is that the profit would be maximized at the one sided limit of Q going to zero from positive side but not actually at zero as at precisely zero the profit will be zero as well. Actually it is even more complex as $\Pi$ is also not defined at $\Pi(q=0)$ but we can reason that at $\Pi(q=0)$ firm is not active. However, I honestly don’t think that their teacher wants a solution in terms of limit rather I think the correct answer here will be there is no solution
1muflon1
CC BY-SA 4.0
2020-09-08T10:03:21.567Z
@Tomcat or actually if you want to do it completely properly one should say there in no solution in non-negative real numbers as VARulle mentioned.
1muflon1
CC BY-SA 4.0
2020-09-08T15:21:50.143Z
@Tomcat I think you missed the second part of my comment or I did not explained it properly. At Q=0 profit is not defined - in demand function we have to divide by Q even though the Q there cancels because we have $(100/Q)*Q$ when we cancel the $Q$ in a denominator we have to write $Q\neq0$ this is important because that means that the maximum value is actually not 100 it only can be argued to approach it in the limit. So we simply can’t say anything about what the profit will be at $Q=0$. Then I said we can reason that at Q=0 firm shuts down and yes that is not rigorous but it’s a reasonable
1muflon1
CC BY-SA 4.0
2020-09-08T15:25:49.020Z
@Tomcat Assumption. But in any case I never meant to suggest that would be correct solution. Here the answer would be that some infinitesimal quantity $Q=\varepsilon$ would maximize the profit but if I remember my math lessons correctly infinitesimal do not exist in standard real  number system so here there would be no solution in positive real numbers.
1muflon1
CC BY-SA 4.0
2020-09-08T15:38:21.800Z
@Tomcat but only right sided limit is defined in this case, and can a function that is on open interval have any corner solution? Also where I see another problem is that actually solution to any optimization problem is not just x but also f(x) in this case for x=0 f(x) is not defined. Sure lim [f(x)] as x goes to zero will be defined but that’s not the same as claiming 0 is the solution.
1muflon1
CC BY-SA 4.0
2020-09-08T15:41:27.550Z
@Tomcat yes if the function was 100-4Q then I would agree solution is Q=0 because we have a corner solutions and function is defined at the corner but that is different function than function (100Q)/Q-4Q as these functions don’t share all points in common
1muflon1
CC BY-SA 4.0
2020-09-08T15:53:37.370Z
@Tomcat no they are not the same function. Their domains are different. For function $100-4Q$ the domain is that is the domain is all (positive) real numbers, whereas the function $(100Q)/Q -4Q$ has domain only on $(0, \infty)$ or all (positive) real numbers except for zero so they cannot be equivalent (I put postive into bracket because often it is assumed that Q must be non-negative but its not explicit in the problem)
1muflon1
CC BY-SA 4.0
2020-09-08T15:55:49.950Z
I mean just by definition of equal functions they are not same as "Two functions f and g are equal, if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, f = g if f(x) = g(x) for all x ∈ X, where f:X → Y and g:X → Y" https://en.wikipedia.org/wiki/Function_(mathematics). Also yes $y=a+bx$ and  $y=(ax)/x+bx$ are not the same - they are not the same functions at point x=0 even if everywhere else they are the same
1muflon1
CC BY-SA 4.0
2020-09-08T16:47:33.640Z
@Tomcat I added some numerical simulations to show you the functions are not the same. The numerical simulations are done over natural numbers for $Q$ starting from $0, 1, ... , 10$. I used natural numbers in order to highlight that the functions do not perfectly overlap each other (as using real numbers would make the point of no overlap too small to see it) but you can see that the two profit funcions are simply not the same at zero (even though everywhere else they overlap perfectly - the red function is imposed on top of blue one so where you can see only red they overlap)
1muflon1
CC BY-SA 4.0
2020-09-07T13:26:48.480Z
2020-09-09T11:11:25.570Z
2020-09-09T11:11:25.570Z
1muflon1
<p>I am currently starting on the topic of monopolies in school and I encountered the following scenario:</p>
<p>A monopolist faces a demand curve given by <span class="math-container">$Q = \frac {100} p$</span> and a cost function given by <span class="math-container">$C = 4Q$</span>. I need to find the price elasticity of demand (PED) as well as the optimal level of output for the monopolist.</p>
<p>I was taught that <span class="math-container">$MR = P(1 + \frac 1 \epsilon) = MC$</span>, where <span class="math-container">$\epsilon$</span> = PED.</p>
<p>I know that any producer produces at the point where <span class="math-container">$MR = MC$</span>, but before I could even proceed with solving for the PED, I am wondering how, in this case, can <span class="math-container">$MR = MC$</span>?</p>
<p>If I am not wrong, by my calculations, <span class="math-container">$MR = 0$</span> and <span class="math-container">$MC = 4$</span>. How do I interpret this?</p>
<p>Then, since <span class="math-container">$MR \neq MC$</span>, how can I proceed with this question?</p>
<p>And any intuitive explanations on how I can calculate the PED and optimal level of output will be greatly appreciated :)</p>

CC BY-SA 4.0
2020-09-08T07:34:12.297Z
Given that there is no solution in the nonnegative real numbers, you could just assume that there is some minimal positive quantity. Producing this minimal quantity is then a corner solution. Alternatively, assume there is some maximal amount of money in the world and set this as your price. At least both these cases are much more realistic than (implicitly) assuming that the good is infinitely divisible.
VARulle
<microeconomics><elasticity><monopoly>
Interpretation of demand and cost curves
Ethan Mark
# Interpretation of demand and cost curves

(Asked by: username_0 on 2020-09-07)

I am currently starting on the topic of monopolies in school and I encountered the following scenario:

A monopolist faces a demand curve given by $Q = \frac {100} p$ and a cost function given by $C = 4Q$. I need to find the price elasticity of demand (PED) as well as the optimal level of output for the monopolist.

I was taught that $MR = P(1 + \frac 1 \epsilon) = MC$, where $\epsilon$ = PED.

I know that any producer produces at the point where $MR = MC$, but before I could even proceed with solving for the PED, I am wondering how, in this case, can $MR = MC$?

If I am not wrong, by my calculations, $MR = 0$ and $MC = 4$. How do I interpret this?

Then, since $MR \neq MC$, how can I proceed with this question?

And any intuitive explanations on how I can calculate the PED and optimal level of output will be greatly appreciated :)

**Tags:** microeconomics, elasticity, monopoly
> **Comments:**
> **username_1**: Given that there is no solution in the nonnegative real numbers, you could just assume that there is some minimal positive quantity. Producing this minimal quantity is then a corner solution. Alternatively, assume there is some maximal amount of money in the world and set this as your price. At least both these cases are much more realistic than (implicitly) assuming that the good is infinitely divisible.  

---

## 1 answers

### Answer by username_2 on 2020-09-07. Score: 2 | Accepted answer

Your calculation is not wrong. First a more easy and more intuitive way of calculating marginal revenue is just to take derivative of total revenue.

Total revenue is price times quantity $TR=P (Q) Q$ where price is still function of quantity as at higher price people demand lower quantity of goods (I am expressing everything in terms of Q since that is the choice variable in your model). Substituting for quantity we get:

$$TR= (100/Q)Q \implies MR = \frac{dTR}{dq} = 0$$

Hence in this case they are indeed zero. Furthermore, in this case the profit maximization problem of monopolist does not even have a solution. Profit is given as total revenue minus total costs so:

$$\Pi = TR-TC = (100/Q)Q -4Q$$

If we would try to derive the first order conditions for maximization we would just get that 0=4 which means there is no solution. To make that clear I even programmed a visualization of your problem in R, this is how it looks like where green curve is demand, red line is marginal revenue and blue line is marginal cost.

[![enter image description here](https://i.sstatic.net/FA4iv.png)](https://i.sstatic.net/FA4iv.png)

**So there are three options I see here:**

1. This is a trick question and you teacher expects correct answer to be that there is no solution (no solution in standard non-negative real numbers that is as it could be argued that quantity produced should be as small as possible but not zero (infinitesimal) from right side - however thats not a standard real number).
2. You misread the problem or made mistake in part of the problem which asked you to first create cost function or demand from some text.
3. Your teacher made a mistake when making up the problem.

Hence my advice is first make sure you read the set up thoroughly, if there is no mistake there then contact your teacher and ask if its possible that the teacher made a mistake since MR is zero in your case. If there is no mistake then its a correct solution.

---

response to question in comments:

Price elasticity of demand is defined as:

$$\epsilon = \frac{d Q}{dp}\frac{p}{Q}$$

So in your case it will be:

$$\epsilon = -\frac{100}{p^2} \frac{p}{\frac{100}{p}}= -\frac{100}{p^2} \frac{p^2}{100} = -1$$

---

Addendum: In response to comments I also added simulation for profits of the monopolist in this situation assuming that the profit function is:

a) $\Pi = \frac{100}{Q} Q -4Q$ - in red color

b) $\Pi = 100 - 4Q$ - in blue color

[![enter image description here](https://i.sstatic.net/Fgvff.png)](https://i.sstatic.net/Fgvff.png)

> **Comments:**
> **username_0**: Hello, thank you for your reply! Whilst waiting for an answer, I read up a bit on internet and found out that demand curves of the form $PQ = C$, where $C$ is a constant, will be unit elastic everywhere i.e. $\epsilon = -1$. How is this derived, though, using the formula I provided above?  
> **username_2**: @EthanMark I added an explanation for you at the bottom of the answer  
> **username_0**: Hello, thank you for your reply! That was the solution I found as well, but I was wondering how I could use the $MR = P(1 + \frac 1 \epsilon) = MC$ to find $\epsilon$? Since in this case $MR$ and $MC$ both give different answers.  
> **username_2**: oh I see, the problem here is that actually $P(1+\frac{1}{\epsilon} )= MC$ is not generally valid statement - only the first half of equality is generally valid. That is we can be only sure that $MR= P(1+\frac{1}{\epsilon})$. The second part of the equality requires that $MR=MC$ but that happens only in equilibrium when firm maximizes its profit since the first order conditions from a generic profit function $\Pi = TR-TC$ will normally give you result that $MR=MC$ however in this case there is no solution to the profit maximizing problem so you cant claim that $MR=MC$  
> **username_2**: furthermore, note that $MR=MC$ holds only in profit maximizing equilibrium (if it exists) its not some universal truth. In fact at any other point then the one which maximizes profit $MR \neq MC$ assuming we have well behaved functions  
> **username_0**: I see. Thank you :) I understand now! Also, I have looked hard at the question and I believe I copied it right. It is also a graded assignment so I do not think my professor made any mistakes. Perhaps the answer to the optimal quantity is $0$ i.e. the firm does not/will not operate!  
> **username_2**: @EthanMark I would still recommend asking your teacher - I am PhD student but I still teach tutorials for undergraduates and honestly I few times made similar mistakes even when I was creating graded assignment. Also note here we just simply cannot determine optimal quantity because of the FOC's. Not being able to find solution is **not** the same as saying solution is 0  
> **username_0**: You have a point there! I will double-check with my professor on that :) Thank you for your help on understanding this question!  
> **username_2**: @Tomcat but how could that be? Let’s just take step back At $Q=0 $firm does not operate so $\Pi=0$ at $Q=1$ $\Pi=96$ - the problem here is that the profit would be maximized at the one sided limit of Q going to zero from positive side but not actually at zero as at precisely zero the profit will be zero as well. Actually it is even more complex as $\Pi$ is also not defined at $\Pi(q=0)$ but we can reason that at $\Pi(q=0)$ firm is not active. However, I honestly don’t think that their teacher wants a solution in terms of limit rather I think the correct answer here will be there is no solution  
> **username_2**: @Tomcat or actually if you want to do it completely properly one should say there in no solution in non-negative real numbers as VARulle mentioned.  
> **username_2**: @Tomcat I think you missed the second part of my comment or I did not explained it properly. At Q=0 profit is not defined - in demand function we have to divide by Q even though the Q there cancels because we have $(100/Q)*Q$ when we cancel the $Q$ in a denominator we have to write $Q\neq0$ this is important because that means that the maximum value is actually not 100 it only can be argued to approach it in the limit. So we simply can’t say anything about what the profit will be at $Q=0$. Then I said we can reason that at Q=0 firm shuts down and yes that is not rigorous but it’s a reasonable  
> **username_2**: @Tomcat Assumption. But in any case I never meant to suggest that would be correct solution. Here the answer would be that some infinitesimal quantity $Q=\varepsilon$ would maximize the profit but if I remember my math lessons correctly infinitesimal do not exist in standard real  number system so here there would be no solution in positive real numbers.  
> **username_2**: @Tomcat but only right sided limit is defined in this case, and can a function that is on open interval have any corner solution? Also where I see another problem is that actually solution to any optimization problem is not just x but also f(x) in this case for x=0 f(x) is not defined. Sure lim [f(x)] as x goes to zero will be defined but that’s not the same as claiming 0 is the solution.  
> **username_2**: @Tomcat yes if the function was 100-4Q then I would agree solution is Q=0 because we have a corner solutions and function is defined at the corner but that is different function than function (100Q)/Q-4Q as these functions don’t share all points in common  
> **username_2**: @Tomcat no they are not the same function. Their domains are different. For function $100-4Q$ the domain is that is the domain is all (positive) real numbers, whereas the function $(100Q)/Q -4Q$ has domain only on $(0, \infty)$ or all (positive) real numbers except for zero so they cannot be equivalent (I put postive into bracket because often it is assumed that Q must be non-negative but its not explicit in the problem)  
> **username_2**: I mean just by definition of equal functions they are not same as "Two functions f and g are equal, if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, f = g if f(x) = g(x) for all x ∈ X, where f:X → Y and g:X → Y" https://en.wikipedia.org/wiki/Function_(mathematics). Also yes $y=a+bx$ and  $y=(ax)/x+bx$ are not the same - they are not the same functions at point x=0 even if everywhere else they are the same  
> **username_2**: @Tomcat I added some numerical simulations to show you the functions are not the same. The numerical simulations are done over natural numbers for $Q$ starting from $0, 1, ... , 10$. I used natural numbers in order to highlight that the functions do not perfectly overlap each other (as using real numbers would make the point of no overlap too small to see it) but you can see that the two profit funcions are simply not the same at zero (even though everywhere else they overlap perfectly - the red function is imposed on top of blue one so where you can see only red they overlap)  

---

