Econ Lab · Markets
Profit maximisation
Picture one small cafe on a street full of them. It cannot set the price of a flat white. The street has already done that. So the only choice left is how many cups to brew. Make too few and it leaves money on the table. Make too many and the last cups cost more to produce than anyone will pay.
Slide the market price and watch the cafe's best quantity move with it.
A cafe with rivals takes the price as given, so every extra cup adds exactly the price to revenue. Read the rule one cup at a time. Brew a cup whenever the price beats the cost of making it, skip it when the cost beats the price. Follow that and you stop exactly where the price line crosses marginal cost.
At the baseline dollars a cup, the cost of one more cup catches the price at cups. One cup earlier, the price still beat the cost, so the cafe wanted that cup. One cup later, the cost would beat the price, so it does not. That tipping cup is the stop, and the shaded triangle is the gain on every cup up to it, about dollars. Slide the price and watch the stopping point ride along the cost curve.
Drag the slider to change the market price. The shaded triangle is the cafe's gain on the cups it sells.
Price is also marginal revenue
Because the cafe is one seller among many, it can sell as much or as little as it likes without nudging the price. Every extra cup it brews brings in exactly the same amount, the market price. That is what marginal revenue means here, the revenue from one more cup, and it is just the price.
On the figure that is the flat dashed line. It does not slope down the way a whole market's demand would, because this single cafe is too small to move the price.
Marginal cost rises
The other line is marginal cost, the cost of brewing one more cup. It climbs as the cafe gets busier. The first few cups are cheap, but pushing the machine and the staff harder makes each extra cup dearer than the last.
That upward slope is the whole reason there is a best quantity at all. If cost never rose, the cafe would never want to stop.
Stop where price meets cost
Keep brewing while the price beats the cost of one more cup. Each of those cups adds more to revenue than it adds to cost, so it adds to profit. The moment the cost catches the price, the gain on the next cup is gone. So the cafe stops exactly where the two lines cross.
The shaded triangle is the running gain, price minus marginal cost added up over every cup the cafe sells.
What you just did
You found the cafe's output rule without ever guessing. A higher price pushes the flat line up, the crossing slides out along the rising cost curve, and the cafe brews more. That rising marginal cost curve, read as a function of the price, is the cafe's supply curve. Profit maximisation is where a firm's supply comes from.