So we want maximize profits. Well, let's write it out. We're going to draw a graph, and so fortunately, we've already figured out how to draw these graphs, so we just want to put them in the right place. We've got the tools in our toolbox. We just now have to put them in the right place and see how they can help us understand what's happening. We know the firm wants to maximize profits, which is equal to total revenue minus total cost. So let's see if we can draw this. So I'm going to draw an axis and put on the vertical axis dollars and cents and on the horizontal axis lowercase q representing individual firm output. We know that we have a total revenue function. The total revenue function is total revenue is equal to price times quantity. Notice, I put P_0 here because on the previous slide, I said that we are assuming the price is given to the firm exogenously, out of its control. Here's the price signal optimize. So I know what that price is, and therefore I can draw the total revenue, which is just a straight line. How do I know it's a straight line? Because I know the slope of that curve is P_0. I'm going to put one of my little thought bubbles here, but to do, and I'm just going to add an extra page to just to remind you that marginal revenue would be what? Well, as we've done throughout all of our videos, and we understand that economists use marginal to mean change in. So marginal revenue is the change in revenue. Marginal revenue is therefore just the slope of the total revenue function. Well, those of you who like calculus, marginal revenue is the derivative of total revenue with respect to output, which will be equal to the derivative of P_0 times q with respect to q, which would just be P_0. Again, if you don't like calculus, I bet somebody in one of your groups does like calculus, and could tell yeah, he didn't pulling a revenue out there and that's a simple straight-forward derivative. The price is the slope of the total revenue. We can talk about that intuitive. I'm going to go back to that picture, and ask you the question. If I increase output by one unit, how does my revenue change? Well, if I increase output, if I want extra q, where do I get an extra q? P_0. What if a increased output again by q, what do I get? P_0. What if I increase it again, what do I get? P_0. So every time I increase output, in every new unit I produced, my revenue goes up by exactly the same amount each time. That's basically a linear function. So now we've got total revenue. Let's think about the other side, total cost. What's the total cost curve look like? Well, we did that before. I'm not going to draw them all out there. You know there's a fixed cost, whatever that is, and on top of that there's a variable cost, and so we have a total cost curve that looks like this. That's the general form of the total cost function, we've shown it many, many times. So given those two, what you want to do is to maximize profit, which is total revenue minus total cost. So on this graph what you'd like to do is to figure out what is the largest vertical distance between the revenue function for the cost function, and we can eyeball that here, and it looks to me like that would be about right here. We'll call that q star. That's the output for the firm that gives the largest vertical distance between revenue minus cost that maximizes profits to the firm. So q star is clearly, just by eyeballing it, we can see that the output, that's the output that maximizes the vertical distance between revenue minus cost, but let's just get a little bit more formal about this. I'm going to draw another graph, and that graph will be really the same, it's going to be the same graph. I'm going to put dollars on the vertical axis, I'm going to put output on the horizontal axis. I've got a revenue function, which looks like this, and it is a straight line total revenue, and I've got a total cost curve, which has a fixed cost component, and then it goes up like this, and then like this. Now, again, you can eyeball this, and you can see well, it looks to me like very if you eyeball it this is probably the largest vertical gap, and that might be that q star. But I'm going to think about the different way. I'm going to ask you to do something for me. I'm going to ask you to construct a profit curve. So well Larry how would I do that? Well, what's the definition of profit? Profit is just equal to total revenue minus total cost. So if I wanted to actually plot that on here, what I would do if I was doing it the first thing I do is I would find the zeros. That is, where are the points of zero profit? You can say, "Well, look whether that's easy." Right here, at this point Alpha, and right here at this point Beta, total revenue is exactly equal to total cost, and that means that the profit if I was to plot profit, profit will be zero. Remember, the vertical axis major dollars and cents, and at these two output points profit is zero. Now, for output points in the intermediate, I can see that profits up in this range as I go, let's call this actually one on there be clever about this and call this q sub Alpha, and this will be q sub Beta, and for output points in the middle here, if I were to somehow increase my output passed q sub Alpha, I would start getting higher profit, and I'd see the profit get bigger, and bigger, and bigger, and bigger, and bigger, and then it would start getting smaller, and smaller, and smaller, and smaller, and so I can plot that and say that in the intermediate here, profit is going up, and then they're going down. In fact, if I were to go to the right up q sub Beta, where would I have profits be? Oh, very bad. To the right of q sub Beta, costs are screaming up whereas your revenue is just stuck on this straight line. The slope is constant on total revenue because you just get that price from the market. So here to the right, profits would be screaming down, and to the left of q Alpha, if I were to go in that general direction, you can see that in fact losses start growing, and loss grow, and grow, and grow, and then they start getting smaller and smaller, and so in fact if I were to go in this direction, I see that losses will go like that, and they would end up here, and what's this point? Well, this is obviously minus F_0 because if you produce nothing. This point here is obviously minus F_0, because if you produce nothing, you have zero revenue and you have zero variable costs, but you still have this fixed cost. So your losses would be that vertical distance. So that profit function tells us here's the output point that we would call q star, q star maximizes the profit function, and of course it's also the same thing that we saw just by using our eyeball earlier maximizing the vertical gap between revenue and total cost. Thanks.