Why is a firm's profit maximized when MC=MR? Why not stop one unit before where it will still make a profit?

Question

It seems that an equilibrium where MC<MR is preferred to one where MC=MR since in the case of the latter the firm still makes a profit.

Put differently, why would a firm produce a unit at all if it will receive no profit from it?

Answer 1

Because profit is maximized at MC=MR. If MC<MR, then firm could still earn more profit by producing little bit more. Also firms do earn positive profit at point where MC=MR.

Practical example:

Firm profit is given by:

$$ \Pi = p(q) q - c(q)$$

Where $\Pi$ is profit $p(q)$ price which depends on quantity sold, $q$ is quantity, and $c(q)$ are costs.

Now lets assume $p(q)$ is given by $p=100-q$ and $c(q)=q^2$.

So we have:

$$\Pi = (100-q) q - q^2$$

To find profit maximizing quantity we take the derivative wrt quantity (and equate it to zero):

$$100-2q -2q= 0 \implies \underbrace{100 -2q }_{MR}= \underbrace{2q}_{MC} $$

So this first order condition for profit maximization literally says that $MR=MC$. Now lets continue, the profit maximizing quantity will be given by:

$$25 =q^*$$

This firm will produce at point where MC=MR when quantity is 25. Now lets compare profit where firm produces 25 and 1 unit less as you suggested (i.e. 24).

Profit at MC=MR

$$\Pi= (100-25) 25 - 25^2 = 1250 $$

Profit at 1 unit less (MC<MR)

$$\Pi= (100-24) 24 - 24^2 = 1248 $$

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Hence as you can clearly see the profit is maximized at point where MC=MR, despite the fact that there MC=MR. Literally by going one less from point where MC=MR firm would lose money. The reward for operating at MC=MR, as opposed to one unit less than MC=MR, is not zero. In the case above it is 2 extra dollars!

Answer 2

We note that until MC actually meets MB, all production is profitable (by logic you've already seen).

Since we agree you want to produce everything up until MC=MB, let's point out that might occur in the middle of a unit. So a firm would like to produce the first half of the unit - but not the latter half (by the logic of MC=MB). Since we are allowing for breaks in production like this, let's say this break happens nearly at the end of the unit. 99.999999999% of the way done with production, to be exact. That's indistinguishable from actually producing the unit.

Cases where your professor says "Oh, they just make this last unit" can get there by saying you are making the overwhelming majority of some fractional unit. And if you allow calculus along continuous production function, the problem disappears entirely, since there is only an instantaneous moment where MC=MB.

Answer 3

Since your question is posed in terms of a firm producing discrete units of a quantity, the answer has to take this discreteness into account. With discrete units, "marginal" refers to the next unit produced. For simplicity let's assume that the profit function has a unique maximizer. Generically, a quantity $Q$ with $MC(Q)=MR(Q)$ will not exist. However, profit is clearly maximized if the revenue from selling the last unit produced exceeded the costs of producing it, but the revenue from selling the next unit would already fall short of the costs of producing it. Therefore the profit maximizing quantity $Q$ is the one which solves

$$MC(Q-1)\le MR(Q-1)\quad\text{and}\quad MC(Q)\ge MR(Q).$$