We're working with a firm. We're trying to figure out how the firm maximizes profits. We've now taken a great deal of time to construct cost curves. We have to think about how the firm is going to take these cost curves and work with them to optimize. So, in this particular module, we're thinking about how firms optimize. Our goal is going to be to understand individual firm behavior. The firm wants to maximize profits, we know that. So, for this lesson, we're going to say the firm knows the following. One, its own set of cost curves associated with production, and two, it also knows an exogenously given price, and we'll call that P sub 0. So, right now, the main goal for the course is to think about figuring out price. But right now, we're stepping along the way, and we want to think of how a firm will optimize for any given price. Once we know that, then we'll start to move price around and see how they respond. Way back in the very first lesson, we looked at how firms output decision, and I mean firms into grand total. Market output changed as prices were changed. Raising prices cost firms to put forth more output, lowering cost firms to put forth lowers levels of output across the entire market. That was our supply curve. Well, today, we want to start a lesson where we're thinking about pushing forward on how that one individual firm were optimize for one particular price. Let's just pick a price out of thin air and work out the optimization problem. Okay? Great. So, the firm wants to maximize its own profit. So, firm I will maximize the profit. So, I'll denote that profit as the profit of one individual firm, firm I. Now, the profit for firm I is equal to total revenue minus total cost. If we could draw a graph for total revenue and draw a graph for total cost, we can do that. We can figure the output level for that particular firm that would maximize profits by just taking the difference between those two curves. Well, let's start with total revenue. Total revenue is pretty easy. Total revenue is just price times quantity. In this case, price is given parametrically times quantity. Price for this lesson is some fixed price, and we're spending the whole lesson thinking about how the firm will make its decision of production based on what that fixed price is, understanding its cost curves. What's that total revenue function looks like? It's not too difficult. It looks like this. It's a straight line. How do we know it's a straight line? Well, the definition of a straight line is a graphical construction that has constant slope. A linear function has a constant slope. That's what this is. What is that constant slope? Well, that constant slope is just this. I have exogenously given this price. Suppose it's three dollars a unit, three dollars per jar of mayonnaise. No matter how many jars of mayonnaise you produce, you're going to get three dollars for each one. That means that along this line, the total revenue function will be a straight line. Marginal revenue is defined as the slope of the total revenue function, which is just basically the change in total revenue for the change in output. The slope then, which would be the marginal revenue, is equal to P sub 0 in this case. We know that because that's the constant slope along that straight line. Okay. So, to maximize profits, we need to take this type of curve and add our cost function. We know what that looks like as we built this before. So, let's draw another graph. We know that our total cost is a fixed cost component plus a variable cost component, which gives us total cost that looks like this. Now, my total revenue price times quantity, price is parametric. Price is given exogenously, okay? So, it's a fixed number. It means it's a straight line, which I'm going to draw for the total revenue, and it looks something like this. So, my total cost curve and my total revenue curve look like this, and how I maximize profit? I want to maximize profit which is the same thing as saying maximize the difference between revenue minus cost. On this picture, that looks like it's about right here. So, we'll say that at this particular output point, this vertical gap is maximized, and so we'll label this point as q star. Given what the firm knows about its costs, we've spent the last lesson slogging it through, figuring out what these cost curves look like and how they come from the production function in the first place, we know what cost look like, and given that the firm is faced with an exogenously given price, this would be the output level that maximizes profits for the firm. Now, there's probably a better way to think about this. So, what I want to do is before I switch on this page, just so you understand. We'll call this point alpha and we'll call this point beta. Profit at q star is equal to alpha minus beta, right? That vertical gap is the profit. It's probably a little bit more satisfying instead of just eyeballing it. Why don't we build a profit function? Why don't we build a profit function on this graph and in fact, it's pretty easy to do that? If we looked at that picture again and I asked you to build a profit function, the easiest way to do this is think about the zero points to begin with. We know that at this particular output point, what's profit? Zero. When revenue at this particular output point, total cost is exactly equal to total revenue profits is zero. At this particular output point, we know profits are equal to zero. Now, for outputs between these two, interior to this range, what are profits? Well, they're positive. Revenue is everywhere above cost in this range. Outputs to the right of this point, costs exceed revenue. Outputs to the left of this point, cost exceed revenue. So, now that we know our zero points, let's draw this graph. The graph would have positive segment here, and to the left of this point, it's going to be negative, and to the right of this point, it's going to be negative. In fact, it's going to screen down negative. As you can see up here on this graph, total cost curve is really arching up quite rapidly because we're at that point where marginal returns has really set in. What's it look like down here at the end? Well, you know in fact it's going to go down here, and then it's going to picture go back up and it's going to end about right here. We know this is equal to minus fixed costs, right, because we know that if the firm produces zero, it has zero revenue, it has zero variable costs. But even if it's producing zero, it still has to pay a fixed cost because we're in the short run. The short run can't get out from underneath that. You've got that brick and mortar you got to pay for even if you're producing zero of your output. That means we could find our profit maximizing point by finding the peak of this hill. Right at the peak would be Q star. It's the same thing as eyeballing a link. Now, a little bit more systematic by putting this in terms of a graph.