Profit maximization | APⓇ Microeconomics | Khan Academy

We've spent several videos talking about the costs of a firm, and in particular, we've thought about how marginal cost is driven by quantity and how average total cost is driven by quantity. We think about other average costs as well.

Now in this video, we're going to extend that analysis by starting to think about profit. Now profit, you're probably already familiar with the term, but one way to think about it very generally, it's how much a firm brings in. You could consider that its revenue minus its costs, minus its costs.

A rational firm will want to maximize its profit. To understand how a firm might go about maximizing its profit or what quantity it would need to produce to maximize its profit based on its cost structure, we have to introduce revenue into this model here. In particular, we are going to introduce the idea of marginal revenue, and we're going to assume that this firm is in a very competitive market, and so it is a price taker.

So, regardless of how much this firm produces, the incremental revenue per unit of what it produces—maybe this is a donut company—the incremental amount per donut is going to stay the same regardless of how much this firm in particular produces. So let's say that the marginal revenue in this industry, in this market, is right over here. One way to think about it is this would be the unit price in that market.

Let me put this right up there: marginal revenue. Once again, for every incremental unit, how much revenue you're going to get, so it would just be the price of that unit. So how much would a rational firm produce in order to maximize its profit? If the marginal revenue is higher than the marginal cost, well, that means every incremental unit it produces it's going to bring in some net money into the door. So it's rational for it to do it.

So it would keep producing, keep producing, keep producing, keep producing. Now it gets interesting as the marginal cost starts to approach the marginal revenue. As long as the marginal revenue is higher than the marginal cost, it's rational for the firm to produce. But right at that unit where the marginal cost is equal to the marginal revenue, well, there on that incremental unit the firm just breaks even, at least on the margin.

It might be able to utilize some of its fixed costs a little bit, but then after that point, it makes no sense at all for it to keep producing. Why is that? Well, if the marginal cost is higher than the marginal revenue, that would be like saying, "Hey, I'm going to sell a donut for a dollar even though that incremental donut costs me a dollar ten to produce." Well, no rational person, if they want to maximize their profit, would do that.

So a rational firm that's trying to maximize its profit will produce the quantity where marginal cost intersects marginal revenue. It will produce this quantity right over there. Now a natural question might be how much profit will it make from producing that quantity.

Well, all you have to do is think about this is the marginal revenue that it gets, and another way you could think about it—because this is constant—it's also going to be the average revenue that it gets per unit. And this right over here is the average total cost per unit.

So what you could do is this: this is how much it's getting on average per unit, and then multiply that times the number of units, and what you get is the area of this rectangle. For those of you who are more visually inclined, one way to think about it is a profit-maximizing firm—a rational profit-maximizing firm—would want to maximize this area.

Think about what would happen if they only produced this much. Well, then they're giving up a ton of area. Then the rectangle would only be this big. This would be the profit that the firm is going to be making from those units.

Then if it decides, for some irrational reason, to produce more than this quantity that we settled on before—let's say this right over here—notice even though the base of this rectangle is longer, the height is less, and this would actually have a lower area.

The reason why I feel very confident that this will have a lower area is because in this situation, the firm is losing money on all of these incremental units where the marginal cost is higher than the marginal revenue.

So, big takeaway: a rational firm that's trying to maximize its profit will produce the quantity where marginal cost and marginal revenue are equal to each other.