Monopolies are just a mathematical misunderstanding

Question

A little head-scratcher (and a good example why we should be careful with notation).

Consider a profit maximizing monopoly, that solves over price

$$\max \pi = PQ(P) - C(Q(P)) \tag{1}$$

Following the routine steps (see this post)

we arrive at the important result that, at the profit maximizing price, the price elasticity of demand should be higher than $1$ in absolute terms, or lower than $-1$ in algebraic terms. Namely at the profit-maximizing price we have

$$\eta^* = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} <-1 \Rightarrow \frac {\partial Q }{ \partial P}P <-Q$$

$$\Rightarrow \frac {\partial Q }{ \partial P}P +Q <0 \tag{2}$$

But $\frac {\partial Q }{ \partial P}P +Q$ is the derivative of $PQ(P)$ and $PQ(P) = TR$, Total Revenue. So $\frac {\partial Q }{ \partial P}P +Q = MR$, Marginal Revenue and we just obtained that at the profit maximizing price and in order to have elasticity greater than $1$ in absolute terms, we must have $MR^* <0$.

But we also now that at the profit maximizing point we have $MR^*=MC^*>0$.

So a solution does not exist, and therefore we conclude that monopolies are just a mathematical misunderstanding.

Now, I went into the trouble(?) to write this smirking post, I hope somebody will go into the few dozens of seconds required to write a clear answer to point out where the trick lies.

Answer 1

$PQ(P)=TR$, Total Revenue.

$\frac{∂Q}{∂P}P+Q$ is the derivative of $PQ(P)$ with respect to $P$.

$MR$, Marginal Revenue, is the derivative of $TR$ with respect to $Q$.

So in general $\frac{∂Q}{∂P}P+Q \neq MR$

Answer 2

To complement @AdamBailey to-the-point answer, the purpose of this post was to alert interested readers to the consequences of changing decision-variables in our thinking.

We are accustomed to think of Demand as either "price depending on quantity" or "quantity depending on price". But on the production-cost side, we automatically tend to think of cost depending on quantity, not on selling price.

Therefore, being even a bit tediously explicit with notation pays off (ask the guys over dynamic optimization, e.g. Caputo's book). In the specific example, the symbols $TR$, $MR$, $MC$, do not reveal the decision variable, and this is where the ruse was based. But if, we wrote

$$\max \pi = TR[Q(P)] - C[Q(P)]$$

we would clearly signaled that our ultimate decision variable is price, and so

$$f.o.c: \;\;\;MR(Q)\cdot \frac {\partial Q}{\partial P} - MC(Q)\frac {\partial Q}{\partial P} =0 $$

$$\implies (MR(Q) - MC(Q))\cdot \frac {\partial Q}{\partial P} =0 \implies MR(Q) = MC(Q)$$

while also we would clearly see that

$$\frac {\partial TR}{\partial P} = MR(Q)\cdot \frac {\partial Q}{\partial P} = \frac {\partial Q}{\partial P}Q + Q$$

and so that the requirement on the price elasticity of demand leads to

$$\frac {\partial TR}{\partial P} = MR(P) = \frac {\partial Q}{\partial P}Q + Q < 0 \implies MR(Q)\cdot \frac {\partial Q}{\partial P} < 0 \implies MR(Q) >0$$

(since $\frac {\partial Q}{\partial P} <0$). So at the optimal point, marginal revenue with respect to quantity should be positive, but marginal revenue with respect to price should be negative.