[MUSIC] Hi, in a previous tutorial, I discussed with you a linear function and how to plot those linear functions in the graph. So we plotted the first function by finding the intersection with axis y and axis x. And by combining those two points with a line, we managed to find the exact location of this function on the axis, in the same way we done for the second function. Now the interesting point is this intersection point, and why is it interesting? So in order to give some sort of content to this example, let's pick up a more realistic scenario, which talks about two competitive firms that compete in the market on either quantities or prices. And each of those firms can be represented by a graph like that. And the reason for that is that they would like to take a larger share of the market, and a way of looking at their competitors and how they react to the situation of the market. The thing that we have to remember about the competitors is that they move simultaneously. In other words, they don't have a preexisting information about a competitor's decision to produce, and therefore they produce simultaneously a quantity. Now if we translate those two functions into reaction function in duopoly, we can write instead of x, we can assign a quantity that is produced by firm a, if we talk about two firms a and b. And if we would like to translate what will be the quantity purchased by the second firm, we will say that this will be quantity of qb. So imagine two firm that produce quantity qa and qb respectively. And therefore, we will assign those values into x and y, in those two functions. In order to give some content that relates to the original functions, we will assign the value of 4 to the qa production and value of 1 to the qb production. And both of them will be equal to 6. The same thing we will do with the second function. For x we will assign a quantity of qa, and for y we will assign a quantify of qb, and both them equal 5. So let's try to read those two function again. This is the reaction function of qa, and this can be the reaction function of qb. In order to better understand that, one has to put all the variables of qb on one side of the equation, and all of the variables of qa on the other side of the equation. So let's try to do that. So let's start with this function. If we say, this is the reaction function of firm a. So firm a will produce quantity a, which will be equal to 5-2qb. If this is the reaction function of b, we will say that qb = 6-4qa. Let's try to read them and understand what they mean. Because, I remind you, in duopoly firms youth simultaneous decision to produce a certain quantity, we'll say that the quantity that is produced by firm a, qa, will be equal to 5, to some independent level of production, -2qb. In other words, for each and every level of production by competitor of qb, the production of qa will be reduced by 2 units. And the reason is for that is that the more qa and qb are being produced in the market, the prices will start to decline. And if the prices start to decline, the revenue might decline with that as well. So therefore we will say that the production of the competitor will affect negatively on the production of the original qa. And the same story will apply here, as you can see. qb, sorry, the production by firm b, will be again negatively affected by the production of qa, the competitor. Now why this and how this relates to the intersection, the intersection point, you will find it out later on, represent equilibrus situation in the market. And what does it mean, equilibrus situation in the market? If both of us, or both competitors, move simultaneously, the equilibra is what decision they will reach given the knowledge of the other competitor's decision. No decision to produce, but decision of how to produce, in other words, the reaction function. So in order to find out what is the real equilibria here, one can imagine it to be solved graphically, so actually finding the corresponding levels of x and y. Or, if you don't plot it correctly, as I cannot plot correctly on this graph, the best way to do that is to do it algebraically. So in order to do that, one can imagine that those qb and this qb are identical. Therefore, we can plug, Instead of this value of qb, the value of the reaction function of the competitor b. So what we have is the following function, 5- 2, open brackets, because b's qb is represented by expression 6- 4 qa. So we'll plug it here, 6-4qa. And now we see, instead of having two unknowns, qa and qb, we only have one unknown, qa, here and here, which we'll be able now to solve the expression. If you solve it correctly, let's try to solve it together, so qa = 5-12+8qa. Let's get rid of this information for now. One can say that 5-12, so let's rewrite this again. qa = 5-12, which is -7+8qa. And it's not very hard to see that the actual expression, if we put this on the other side of the equation, we get -8qa + qa = -7, or in other words, qa = 1. Just to remind you, qa was our x. So the value of x as corresponds to our equilibria is actually x=1, or in our case, qa = 1. Which results immediately in the value of y, because as you can see, the value of x on both lines will be exactly the same, as well as value of y will be the same. So let's try to find the value of y, and the value of y corresponds to the value of qb, which is exactly that. This is where we find this expression originally. So if we find the value of qb, we'll be also in a position to say what is the corresponding value of y in this case. So 6-4, now we know the value of qa = 1. And we can say easily that y, or qb in this case, is equal 2. So actually the corresponding point, or intersection point, is a point where x, or qa, is equal 1, and y, or qb, is equal 2. So what does it mean? It means that in the duopoly market that we have here with those reaction functions, both qa and qb will produce an equilibrium value of production over of output, which for firm a will be value one, or one unit of production, and for production of b will be two units of production. [MUSIC]