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Statement: If a monopoly is producing a quantity for which its MR is negative, its total profit cannot be positive.

Is this statement correct or incorrect, and why?

I've read a lot about this, but sources tend to discuss maximizing profitability, while the question above presumes that MR < 0 (and obviously smaller than MC) but wonders about profitability in this scenario.

My hunch is that yes, it is possible, but I'm struggling to prove it.

Edit: this getting downvoted is the reason why SE is deteriorating at the rate it is. I’ll go look for a more welcoming community.

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    $\begingroup$ Hint: Suppose there's no cost, so that profit equals revenue. Is total revenue non positive when MR is negative? $\endgroup$ – Herr K. Jan 22 '19 at 13:32
  • $\begingroup$ @HerrK. I guess it’s not, but P > MR in monopolist markets? $\endgroup$ – zerohedge Jan 22 '19 at 13:54
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    $\begingroup$ Please clarify any constraints on the problem. Is this a question about a regulated monopolist? Price discriminating? Is this a question about equilibrium profit maximization? Of does it just have to be possible, in the sense of something in the choice set of the producer? Or does it have to be possible while profits are positive? $\endgroup$ – BKay Jan 22 '19 at 14:25
  • $\begingroup$ @BKay: if there were any additional constraints I would have specified them. $\endgroup$ – zerohedge Jan 22 '19 at 16:49
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    $\begingroup$ Given zerohedge's response, @HerrK's should write up his response as an answer. He provides an example that satisfies the question. $\endgroup$ – BKay Jan 22 '19 at 17:24
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As a counterexample to the statement, consider the following:

Let demand be $p=10-q$, so that $MR=10-2q$. Suppose there’s no cost, so that profit equals revenue. If the monopolist produces at $q=6$, $MR=-2<0$, but $TR=(10-6)6=24>0$, thus contradicting the statement.

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  • $\begingroup$ thanks for answering. But am I allowed to just assume there are no costs? (context: this is a correct/incorrect question from a 2016 microeconomics exam) $\endgroup$ – zerohedge Jan 26 '19 at 21:11
  • $\begingroup$ @zerohedge: You can incorporate costs into this setup and the conclusion is not going to change. $\endgroup$ – Herr K. Jan 26 '19 at 22:59

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