In the second example I would like to show you, I will show you how you can use the same data from the descriptive part, and actually get a completely different conclusion, and actually make a different recommendation for the company. And this is the power of the prescription. You can use different, or the same data, but to achieve different goals and take different action. So let's say now we wanna maximize the revenue by the company. Most companies actually don't wanna maximize the amount of the product they sell, but they show you they want to make the most money, right, or generate the most revenue by selling a product. We can use a demand curve that Professor Wachter has shown you, to estimate actually, also what will be the revenue of the company, not only how much product will be sold. So let's see how we can do that. The first thing we would like to do, is define the goal, as we defined it before. And the goal is to maximize revenue, unlike in the previous example where we try to maximize quantity. The other thing we need to define is the action. And the action is again, the same action as before. Setting the price, or changing the price. And the question is, how does changing a price impact the revenue generated from selling a product? In this case, we need a model. We need a model to describe what is the relationship between setting prices and setting quantities of product, to generating revenue by a company. And now we need to think a little bit, and think what happens when we increase or decrease the price of a product, for the revenue of a company? Now one thing you're asking, how do we calculate revenue, perhaps? Well you take the amount of a product sold, the quantity, and you multiply it by the price, and this is the total revenue generated by the company. So in this case, we have two forces operating here. What happens, for example, when we decrease the price? When we decrease the price, as we've seen before, the quantity sold goes up, right? So we sell higher quantities of the product. But on the other hand, the amount of money were making for each item sold is lower, because we decreased the price. So when we multiply both, one side of the multiplication goes up, which is the quantity, and the other side, which is the price, goes down. And we can't really tell if the revenue will go up, or going down, by decreasing the price. The same thing happens when we increase the price. If you increase the price, the quantity sold goes down, but the price per item, with the revenue generated per item, goes up. And again, we can't really tell if it's better to increase the price, or decrease the price. Now what I would like to point out, is this is called a trade-off, where you change, you make one action, you set a price, but the revenue or the goal of the company might go up, or it might go down, depending on what the actual result will be. And the question is, how can we find the price that generates the maximum revenue for us? So let's see how we can do that. One thing we would like to do, is we can use Excel and try to say, for each price, we actually create a table, as you can see, the bottom of the slide. For each price, what would be the quantity sold, which is the column called Demand, and what would be the revenue, where we maximize the demand by the price. So for example, you could see that for a price of $3.00, the demand will be 7.43 items, and the total revenue will be $22.29. And if we increase the price more and more, the revenue goes up and up, but at some point the revenue goes down. And the question is, can we find the price that generates the maximum revenue? Now in this table, it's quite simple. We can just go by the order of the prices, and see which one generates the maximum amount. But in the general case, problems can be very hard to solve. A lot of variables to change, many actions to take, many it's not a discrete. One way we can do it and solve it, is try to take a graphical approach, and find where the maximum revenue is. So in this slide, what you will see is where I drew a graph that takes the previous table from the previous slide, and generates a graph where you can see the actual revenue on the y-axis, as a function of the price changing on the x-axis. And you can see, for example, that in this case, the maximum revenue is generated with approximately a price of $5.50. How do we find this maximum? We try to find the point in the graph which is maximized. Here it's highlighted by this arrow, and then we take the dashed line, basically going to the bottom, finding on the x-axis the price that generates this revenue. And this is the price we're saying is gonna maximize revenue, in this case $5.50. So what we've done so far, we've discussed what a problem is that we would like to solve, and get a prescription for. A problem is three parts. It has a goal to optimize, which can be to maximize, to minimize, to do something with the quantity. It has actions the company or the consumer can take. It can be changing the price, producing product, maybe showing an ad to someone, but different actions. And it has a model that links the action, and how they impact the different goals, or the different objectives we're trying to optimize. Using the same descriptive analytics that you've seen in the previous lectures, we arrived at two different conclusions. If you want to maximize the quantity sold, you should set a zero price. You should give your product for free. If you wanna maximize the revenue generated by selling a product, actually you wanna set a price of $5.50. The next thing I would like to do in the next lecture, is talking about parameters, and how different parts of the model impact the decision and recommendation we will give the company. And the next one after that, we will discuss market structure. Can we actually take the descriptive data, and find the optimal price, without really knowing how the customers interact with the product in the market?