# How does demand-supply equilibrium interact with profit maximisation?

My understanding of supply and demand is that at higher prices sellers are more willing to supply and buyers will demand less, and the total transaction volume will be supply or demand, whichever is less. At some finite price the two are equal, given an equilibrium; and whereas at a higher price demand and hence transaction volume would fall, at a lower one supply and hence transaction volume would fall. Thus the equilibrium price should also maximize transaction volume.

But here's where I get confused. Excuse my inventing notation. Suppose at selling price $P$ there is a transaction volume $V(P)$, maximised at equilibrium price $P=P_E$ so $V'(P_E)=0$. If the unit cost to the seller is $c$, their profit is $\Pi=(P-c)V(P)$. Hence $$\dfrac{d\Pi}{dP}=V(P)+(P-c)V'(P),$$ which at $P=P_E$ is $V(P_E)>0$. Thus the profit-maximising price exceeds $P_E$. This logic seems to imply profit maximization won't occur at the equilibrium price.

So what does microeconomics actually predict happens in a market?

• From your formulation it seems that the quantity traded (which you call transaction volume) is maximised at equilibrium. Why would this be the case? In most standard models, an equilibrium by definition entails profit maximisation by sellers (along with utility maximisation by buyers, and the requirement that supply equals demand). – Theoretical Economist Dec 4 '17 at 13:46
• @TheoreticalEconomist As I explained, my reason for thinking the equilibrium price maximises transaction volume is because I define it as the price at which supply equals demand, and if the price moves away from $P_E$ either supply or demand falls, reducing transaction volume. – J.G. Dec 4 '17 at 15:08
• You’re right; I didn’t read your question very carefully. The short answer is that you’ve set up a model where the seller has market power — their decisions impact the quantity demanded by buyers. It is well known that the appropriate notion of equilibrium in these models is not where “supply” equals demand, precisely because this isn’t where sellers maximise profits. I’m a bit busy at the moment, but I’ll try to write up a more detailed answer later. (Of course, I’m happy for anyone else to take this comment and turn it into an answer...) – Theoretical Economist Dec 4 '17 at 15:17

I think one source of confusion in your question is that you are piecing together assumptions from different models that are incompatible. In particular, you've implicitly referred to a supply curve, but attempted to write down the optimisation problem of a monopolist. This doesn't work because a monopolist has no supply curve: a supply curve describes how much a seller would be willing to sell for a given price, taking prices as given. However, monpolists do not, by definition, take prices as given. Supply curves only make sense when the seller faces perfect competition.

Hopefully I can make this clearer by giving you an overview of the two different models.

One standard assumption in basic models is that markets are perfectly competitive. For sellers, this means that they take prices as given, and nothing they do can affect the price of the good that they are selling. (Alternatively, they solve their profit maximisation problem while ignoring any effects their choices have on the market price.)

Formally, a seller chooses the output quantity $q$ to maximise

$$pq-c(q),$$

where $p$ is the price of the good, and $c$ is the (convex) cost of producing $q$ units of the good. Hence, the seller's profit-maximising level of output is given by

$$p = c^\prime(q).$$

This gives us the seller's supply curve

$$q^s(p) = (c^\prime)^{-1}(p).$$

Given some demand curve $q^d$, the relevant equilibrium condition is now

$$q^d\left(p^*\right)=q^s\left(p^*\right) = q^*,$$

where $p^*$ and $q^*$ are, respectively, the equilibrium price and quantity. (This treatment needs to be slightly modified when $c(q)$ is linear, as in your question, but this can be done.)

We can, of course, relax the assumption that the seller is a price-taker. Suppose now the seller realises that their choice of $q$ can affect the equilibrium price. Here, we say that the seller is a monopolist. They now choose $q$ to maximise

$$p(q)q - c(q),$$

where $p(q)$ is the inverse demand curve. That is,

$$p (q) = (q^d)^{-1}(q).$$

(Equivalently, as in your question, they would choose $p$ to maximise

$$p q^d(p)-c\left(q^d(p)\right). )$$

The equilibrium quantity is then given by the solution to

$$p^\prime (q^m) q^m + p(q^m) = c^\prime (q^m).$$

Note that it now doesn't make sense to ask what the seller's supply curve is. The seller's profit-maximising level of output isn't determined by a given price, because their choice of output affects the price.

One can show, as you've found above in the case with constant marginal costs, that (assuming demand is downward sloping)

$$q^m < q^*$$

and

$$p(q^m) > p^*.$$

In other words, when the seller is a monopolist, the seller's profit-maximising output (which they will sell in the equilibrium where the seller is, in fact, a monopolist) is smaller than the equilibrium output under perfect competition.

• Thanks for this. One question: in the monopolist's case, there is no supply curve with which to define a demand-supply equilibrium; so why do we call the profit-maximising $q$ an "equilibrium" in this case? – J.G. Dec 5 '17 at 18:45
• @J.G. There is a lot more to equilibria than the equality of supply and demand. Perhaps the best way to think of the outcome in the setting with a monopolist as a (Nash) equilibrium of an appropriately defined game. – Theoretical Economist Dec 6 '17 at 19:35
• Ah; this makes much more sense. Then in the case of perfect competition, I suppose we again have a Nash equilibrium because neither buyer nor seller would benefit from being the sole party to deviate from what is expected from comparing demand & supply curves. – J.G. Dec 6 '17 at 19:56
• @J.G. It depends. There is a way to model the perfect competition case as a game, but you will typically have to model it as a situation with many buyers and many sellers. However, that isn’t typically how these models are taught in an intermediate (or even intro) microeconomics class. It’s usually just enough to introduce utility and profit maximisation problems where both buyers and sellers take prices as given, yielding supply and demand curves. Equilibrium is then defined as the price and quantity that clears the market. – Theoretical Economist Dec 8 '17 at 23:15
• Fair enough. If you can recommend any resources on this I'd be grateful. – J.G. Dec 9 '17 at 0:16