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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 :)

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    $\begingroup$ 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. $\endgroup$ – VARulle Sep 8 '20 at 7:34
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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

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

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  • $\begingroup$ 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? $\endgroup$ – Ethan Mark Sep 7 '20 at 13:31
  • $\begingroup$ @EthanMark I added an explanation for you at the bottom of the answer $\endgroup$ – 1muflon1 Sep 7 '20 at 13:38
  • $\begingroup$ 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. $\endgroup$ – Ethan Mark Sep 7 '20 at 13:41
  • $\begingroup$ 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 $MR=MC$ $\endgroup$ – 1muflon1 Sep 7 '20 at 13:45
  • $\begingroup$ 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 $\endgroup$ – 1muflon1 Sep 7 '20 at 13:47

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