In this lecture what we're trying to do is introduce parameters to the model or try to see how a company can take into account other things that change while it's taking actions, which is in our example changing the price. So let's try to see how a company now can maximize its profit. One thing to remember is that revenue, which is how much money the company makes, does not automatically translate into profit because the company also has costs. Maybe some time it actually cost more to produce an item than the amount of money you make if you sell it too cheaply. So, in this case, the company would like to maximize it's profits, and he wants to make sure that after paying for all of the production costs of the item, it has money left in the bank. It actually can make money and continue operating. So now we must take into account the cost of the item. And the goal, as I said, is to maximize the profit, which is revenues minus the costs. So let's introduce cost to the model. Suppose each product in the same model we've discussed before, costs $2 to produce. What we can do is we can take the same graph of revenue we created before, and for each price point on the x-axis we now add a point of the total cost it would cost us to produce those amount of items we're setting. So, in this case, you can see the red dotted line, this is the total cost we're going to produce. Some interesting things we can see in this graph is first of all, that if we sell the product too cheaply, although the quantity may be high, the revenue is low, and it's actually revenue is lower than the cost. We're actually going to loose some money. One the left hand side, you can see that the red dot is higher than the blue does, where actually the profit there is negative. Now if you look at towards the right hand side of the graph when we're increasing price and the revenue, the blue dots, go up and up, costs also go down because the quantities we're setting go lower. And at some point, and this is where the green arrow is, we're maximizing our profit, which is basically the distance or the difference between the revenues and the cost. At that point, we're trying to find out what is the price we'd like to send, we would like to sell the product for. How can we find this maximum profit? In some cases, it's not very convenient to use the revenue and the cost graph I've shown you. And we can just take the difference of them. We can take the same table I've shown you in the previous lecture, and now calculate the cost, calculate the revenue, do revenue minus the cost, and you will get a profit. In this case, we get a profit graph. And we can see that the price of roughly $6.5 we're going to maximize the profit. However, it's a bit hard to tell from this graph where the profit is. And also it's going to be very hard to do it for a very big combination of prices. And one question I would like to ask is, can we find a principle? Can we find the way to find where the maximum profit is without having to actually calculate the profit and calculate those, and draw these graphs for every combination of prices? Now the answer is yes, and let's take a look at what kind of forces or trade-offs operate in this profit graph. So what we're trying to see so, if we can apply a principle, instead of drawing complex graphs or using the table, that will apply and help us find the maximum profit in any types of these problems. So let's take a look at the different forces operating in this type of problem I've shown you in the previous slide using the graph. One thing we notice is that selling too cheaply yields a negative profit, okay? There's actually a maximum amount you want to sell in order to make a profit from your company. The other thing is that the optimal price we found that generates the maximum profit is different than the price we found that generates the maximum revenue, right? Actually we needed to increase the price slightly from $5.5 to $6.5 to get a higher profit. And the question is can we actually use this principle? Do we actually need to increase the price, in order to increase the profit, when we include costs in our model, right? Because in the revenue example we assumed there's no cost, and when we introduced a profit and introduced cost, now we see that we need to actually increase the price to generate more profit. So let's try to understand what does optimization do? What does this maximize profit process actually do? And the idea is that we're actually trying to change the price and increase it slowly, until the benefit of the revenue doesn't go higher than the actual cost we're introducing, by producing less or more items. So let's take a look at an example. What happens when we increase the price from $1.5 to $2.0, okay? What we're going to see is, first of all, the revenue's going to increase from $13.17 to $16.66. So the difference is going to be $3.49. We actually achieved a gain in revenue, but also there is going to be a decrease in costs. And why? We increased the price a little bit, the quantity went down, and actually it costs us less to produce the product. In this case, the cost went down by $0.9, and the total gain in profit is $4.39. So actually we still have a positive gain in profit if we increase the price from $1.5 to $2.0. Now what happens if we increase the price too much? Let's take a look at what happens when we increase the price from $6.5 To $7.0. We do the same calculation as before. Now the revenue actually decreases. It decreases from $27.82 to $26.81, so actually we have a loss of revenue of $1.01. This is why we have a negative sign in this slide. And also the costs go down because, again, when we increase the price, the quantity goes down. In this case, the cost goes down by $0.9. When we sum them up together or we take the difference to find revenue minus the cost, we will see that actually we have a negative change in profit, and we actually lost 11 cents. So in this case, we see if we increase the price from a small price to a slightly higher price, we have a positive gain in profit. And if we increase the price from a very high price to even higher price, we have a negative gain in profit. And this leads us to the principle we were looking, right? What we saw is that we're slowly increasing the price in which we get a different increase or decrease in profit. And the question is how far should we increase the price and find optimal profit? How far or how high should we set the price to say stop here? This is the optimal profit. And the idea is that you just increase until the change in profit is 0, until it turns from positive to negative. And how can we find when this happens? We can look at the change in revenue, like we've seen before. And we can look at the change in profit. And when they're equal, when the increase in revenue equals the increase in cost, or the decrease in revenue equals the decrease in cost, then we say, we need to stop here and this is the optimal profit. This principle is called the marginal revenue equals the marginal cost principle. Marginal revenue means by how much are we increasing the revenue when we change the price of the item by one unit. And marginal cost means how much is the increase in the cost when we change the price of the item another one unit, and the change in those quantities increase of decrease, okay? And this principle applies to many, many types of profit maximization. So the previous slide, we've seen that increasing the price from the low price to the slightly higher price actually generates a positive profit and increases our profit. While increasing the price again from a very high price to even higher price actually generates a negative profit and decreases the profit. So the question is how do we know when to stop? Should we start increasing the price, and how can we find the optimal price? And the answer is well we need to increase the price until the gain in profit is 0, right? Until we're moving from a positive gain to a negative gain. Now how can we find that point easily without actually trying to do the profit calculations? The answer is we can look at the change in revenue, and see when it equals the change in cost. If the increase in revenue equals the increase in cost, then we're not going to have an additional profit. And also if the decrease in revenue equals the decrease in cost, we're also not going to have an additional profit. And this principle is called the marginal revenue equals the marginal cost principle. The marginal revenue is the additional revenue the company makes by selling one more item. By changing the quantity that it sells from, let's say, one to two or three to four, or four to five, and the marginal cost is the actual costs that the company needs to pay in order to produce one more item. And what we can do is we can look at the tables, we've seen a few slides before and actually take the differences between every row. And say how much would it cost me to sell one more item, or to sell two more items, or to sell three more items? And also by how much would the cost increase by sending one more item, or two more item, or three more items? When those two numbers are equal, this is when we find the optimal price and the maximum profit. And this called the marginal revenue equals the marginal cost principal.
