All right, here's a picture that you're familiar with because we've been using it several times. It's basically is we're trying to think about maximize profits. So on this graph we've got a total revenue function, which is a straight line function. It has a constant slope equal to that exogenously given price p sub 0 and we got a total cost function. And last time, we'd looked about the fact that you could eyeball where the profit maximization point is. You could say well you know, it looks like it's about right here. This looks like the largest vertical gap between revenue in excess of costs and we'll call that q star. And then, we also constructed a profit function to confirm that that was exactly the point where the profit hill was at it's peak if we drew that little curve. But we now need to think a little bit deeper about this issue and I'm going to start by reminding you of something. I'm going to start by reminding you a couple things here, that we want to maximize profit which is equal to total revenue minus total cost. And I want to recall, another one of my little bubbles here, I want to recall that the slope of total cost is marginal cost. And we know that the slope of total revenue is marginal revenue. Now these are important, that's why they're a memory thing. We're thinking about this. Now, we want to think about how do we maximize these two functions? In order to do that I'm going to make a step back into a world that you long since forgot. So we're going to introduce a graph here and I'm going to take you back to when when you first had a geometry class long time ago and you're thinking about geometry. And so if you were sitting in that geometry class, you probably had something that looked like this. And you had y and x and the instructor said suppose I give you two functions. A function that looks like this, Y is equal to f(x) and I gave you another function that looks like this, Y is equal to h(x). Now, h and f represent two different functional operators. There's one sort of algebraic equation that means the f and there's another algebraic equation that is the h and they generate these two different curves. Okay fine, and if you think back to those days the instructor says what value of x would maximize the difference between these two curves? And thinking about for a while you say it turns out that the vertical gap, the vertical gap between h(x) and f(x) is maximized at the x value where the slopes are equal. Now again, this is an easy thing to prove with calculus, but we're not doing calculus and you didn't do calculus back when you had your first geometry class a long long time ago. But you could think about this because you knew about slopes and you see look at values down here. Let's take these X values. You can see the slope of the slope of the f function is much steeper than the slope of the y function, which means those two curves are diverging. At low values of x, the two curves have different slopes. So they're diverging, which means the vertical gap is growing. However, after some point as you continue to add more x you can see the slope of f(x) is, that the two slopes are now converging. If they're converging, what's happening to that vertical gap? It's getting smaller. So at low values of x the vertical gap is still growing because the slopes are diverging and once the slopes get to be equal in value, you've found the peak value of the gap between them. And if you continue to go back on x, go farther out on x, the curves are going to start getting closer together. So the value, the key here is that the vertical gap between h and x is maximized at the point where the slopes are equal. Now again, it may have been a long time ago that you learned this sort of stuff. But even if you claim you never learned, it should be intuitively obvious here that if the slopes are different and going apart that gap is getting larger. Once you get to the point where the slopes are the same you've maximized the gap, because any farther point the slopes are getting closer, the curves are getting closer together. That vertical gap is getting smaller. Well, what's that tell us for over here? Look, we can go back to this picture. And we now know that we maximize profits by setting the slope of the total cost curve, that is marginal cost curve. The slope of the slope of the total revenue curve, which is this, equal to the slope of the total cost curve, which is this. Okay, and so by setting the point where the two slopes are equal we've maximized the largest vertical gap. Now, that's just straight intuition. I'm going to do one more picture. It's not even a picture. I'm just going to draw one more thing on here, and I'm going to give you a little bit of warning in advance of this. I'm going to give you a little calculus, you don't have to remember this. It's not required but it's really straightforward. For people who know calculus, this would be pretty straightforward. We say profit is equal to total revenue minus total cost and we know that if you were to take the derivative of profit with respect to output and if you wanted to maximize that, what do you do? Well again, if you haven't had a calculus course just ask your friends if I lied. With calculus, you would just take the derivative and set it equal to 0. Take the first derivative, set it equal to zero and you're going to maximize that function. So if we did the first derivative and set it equal to zero, that would be the same thing as taking the derivative of total revenue with respect to output minus the derivative of total cost with respect to output. So if those two, this derivative of total revenue minus derivative of total cost have to be equal to zero, that means they must be equal to each other. So this implies that this term ,which is marginal revenue, has to be equal to this term, which is marginal cost. So marginal revenue equals marginal cost will maximize profits for this firm given the fixed price. Okay, there were planer. And I'm just going to put a little a little orange circle around this because it's really important. Now, we know that we understand you've already understand what the marginal revenue curve looks like. We've spent a lot of time thinking about what the marginal cost curve looks like so we're now going to be able to use this to jump straight into profit maximization. Thanks.