Worked example of a profit maximization problem | Microeconomics | Khan Academy

We're told corn is used as food and as an input in the production of ethanol and alternative fuel. Assume corn is produced in a perfectly competitive market. Draw correctly labeled side-by-side graphs for the corn market and a representative corn farmer on your graphs. Show each of the following: the equilibrium price and quantity in the corn market labeled p sub m and q sub m respectively; the profit maximizing quantity of corn produced by the representative farmer earning zero economic profit labeled q sub f.

So, like always, pause this video and see if you can do this on your own before we work through it together.

All right, now let's work through it together. So we're going to do correctly labeled side-by-side graphs. So let me do this. This is going to be my horizontal axis for the market, and then this is going to be the horizontal axis for the farmer.

And this is going to be quantity in the market, quantity, quantity, and then this is going to be quantity for the farmer quantity, and then this is going to be price in the market. And whatever the market price is, that's also going to be the price that the farmer has to take because it says it's a perfectly competitive market. So the farmer is going to be a price taker here.

So let me make these axes. So this is price right over here, and this is price over here. So first, let's draw the corn market. So let me label this corn market, and we've done this multiple times already. Our demand curve might look something like this. This is our demand curve, as when price is high, low quantity demanded; when price is low, high quantity demanded.

And supply goes the other way around, so our supply curve would look something like this. And then this point, this helps us figure out this is going to be our equilibrium price, so that's p sub m, p sub m. And then this is going to be our equilibrium quantity, so q sub m.

Now this graph over here, we are going to draw the farmer. So this is going to be the farmer, the farmer's firm right over here. So the farmer is going to be a price taker, so whatever the equilibrium price in the market, that is going to be the price that the farmer is going to have to take. That market price is going to be the farmer's marginal revenue.

Now they say the profit maximizing quantity of corn produced by the farmer, by the represented farmer, earning zero economic profit labeled q sub f. We're gonna have some quantity right over here. It is the profit maximizing quantity, but it's also zero economic profit. So the zero economic profit tells us that the price must be equal to the average total cost at that quantity.

So I can make an average total cost curve that looks something like this, and I'm going to make its minimum point intersect that market price because we know from previous videos that the profit maximizing quantity happens where the marginal cost intersects the marginal revenue, which in this case would be the price that the farmer has to take from the market.

And we know that the marginal cost curve intersects the average total cost curve at this minimum point right over here. So I could draw a marginal cost curve; it might look something like this. So that is our marginal cost curve, and notice the marginal cost curve intersects the average total cost at that minimum point.

We explained that in multiple videos already, and we've explained in a previous video that the profit maximizing quantity is the quantity at which the marginal cost and the marginal revenue meet, and the price is the marginal revenue. Beyond that point, every incremental unit, the corn farmer is going to take a loss; it's going to take them more resources to produce that corn than they're going to be able to get in the market.

And we also mentioned that this has to be a situation of zero economic profit, so the average total cost has to be at that price, at that marginal revenue, right at that point. So this right over here would be our q sub f, and we're done.