# Can marginal revenue be increasing?

As per my calculations, the derivative of the MR curve is given by: $$MR'(q)=2p'(q)+qp''(q)$$, where $$p(q)$$ is the inverse demand curve. Therefore,for a downward sloping demand curve, if the second derivative is large and positive, MR should actually be increasing. However, intuitively, if demand falls, then marginal revenue should fall as well, and by a greater amount than demand. The two don't seem to match up, and that's my concern.

It is perfectly consistent for the marginal revenue to increase in $$q$$, even if the demand curve decreases.

Marginal revenue is $$p(q)+ q p'(q).$$

• The first term says "if I sell one extra unit then I will receive an extra $$p$$ in revenue". The larger is this effect, the higher is the MR.
• The second term says "in order to sell one extra unit, I will have to reduce my price by $$p'(q)$$, which means that much less revenue on each of the $$q$$ units I would have sold anyway". The larger is this effect, the lower is the MR.

If $$p''(q)>0$$ it means the price cut needed to sell an extra unit (the second effect above) gets smaller as we move down the demand curve. So the negative part of the marginal revenue is diminished, which can cause the marginal revenue overall to increase.

Note 1: although the marginal revenue function can be increasing, it can never rise above the demand curve. Indeed, from the expression above we see that the marginal revenue is strictly below the demand curve whenever $$p'(q)<0$$.

Note 2 (slightly more technical): the possibility of an increasing marginal revenue implies a potential non-uniqueness of the solution to the profit-maximisation problem (even when marginal costs are constant). This is because there may be more than one point at which MR=MC, as the below figure shows (this figure was generated with the demand function $$p(q)=1-q+3q^2-3q^3$$):

We only have to think about points where MR=MC and MR is decreasing (the one in the figure where MR is increasing is a local minimum for profits rather than a local maximum). The other two are both local maxima, but typically only one will be the global profit-maximising point.

Becuase dealing with cases where there are multiple potential solutions is messy, people often make an assumption to guarantee a unique solution. One such assumption is that "demand is log-cocave", meaning $$ln(p(q))$$ is concave. Let's see why this works. Demand is log concave if

$$\frac{d^2\ln(p(q))}{dq^2}<0$$ $$\frac{p''(q)}{p(q)}-\frac{p'(q)^2}{p(q)^2}<0$$ So if demand is log-concave we know $$p''(q)<\frac{p'(q)^2}{p(q)}.$$

We can use this information to see how large we can make the derivative of the MR curve that you calculated: $$\frac{d MR}{dq}=2p'(q)+qp''(q)\leq 2p'(q)+q\frac{p'(q)^2}{p(q)}.$$ Can this be positive? The answer is yes if $$2p'(q)+q\frac{p'(q)^2}{p(q)}>0$$ i.e. if $$2p(q)+qp'(q)=MR+p(q)<0.$$ (in the last line I changed the direction of the inequality because I divided by $$p'$$, which is negative).

We find that, given log-concave demand, $$\frac{d MR}{dq}$$ can be positive only if $$MR<0$$. So the marginal revenue curve will never slope up in the region the firm would consider choosing.

• Is there a demand function that actually shows this type of multiple intersection? Is there an utility function that does generate it? I don't know if the answer is trivial, but if it isn't I will probably ask this as a separate question. – erik Oct 6 '18 at 5:59
• @erik Finding such a demand function is quite easy. For example, the function $P(q)=1-q+3q^2-3q^3$ is a downward-sloping demand whose marginal revenue, is non-monotonic. I updated the plot above to show this function, along with the associated MR. Finding a utility function to generate such demand is non-trivial, and might merit a separate question. – Ubiquitous Oct 6 '18 at 6:49
• Thank you. Upvoted for the edit. May I please ask what you used to generate the plot? And thank you for the suggestion about the question. Will ask. – erik Oct 6 '18 at 6:55
• @erik The plot is generated with PGFplots. If you interested in learning more about it, the relevant community is tex.stackexchange.com – Ubiquitous Oct 6 '18 at 6:57