I will now show you an example of what we just did and this example has two goals. The first goal is to show you how this is going to work if you have numbers and the second reason is that it will give me an opportunity to teach you some deeper things about competition. So, simple example, but with a very deep meaning. Let's go and bear with me, it's not going to be very difficult. It's the simplest it can get. Considering that we have the simplest possible demand curve, price equals one minus the total quantity. It can't get any simpler than that. So, it's a linear demand curve. Consider average costs, that they are just two numbers, c_1 and c_2, constant and the same. But now, let's make it so the two firms will not be equally productive. Meaning that, one of the two firms can have a smaller cost than the other one. Let's allow for this case in this model. So, the profit from one, as you have seen, is what? Is the price which is given by the demand curve, one minus the total quantity, q_1 minus q_2, times the quantity that this firm will sell, minus the cost to make this quantity, which is c_1 times q_1. So, this is the profit function. If I take this function and I maximize it, take the first derivative, this will give me one minus q_2, minus c_1, minus 2q_1 and this cost will be set equal to zero, in order to give me the maximum. If I solve these with respect to q_2, I could solve it with respect to q_1, but have I my reasons that you will understand a little bit what they are and I always solve these first order conditions in Cournot cases, with respect to the control variable of the second firm. So, q_2 is equal to one, minus c_1, minus two times q_1. I can do exactly the same for the firm two and I will take this condition, the final condition. Problems are very symmetric and you will see that of the two firms are very symmetric and you can easily get to this point. So, q_2 is equal to the fraction one minus c_2 over two, minus one-half q_1. Now, these reaction functions, these two conditions that we call reaction function and I will tell you a little bit why, they look to be a little bit different even if the problem is symmetric. Why? Because they are both solved with respect to the control variable of the second firm. If you solve the second condition with respect to q_1, you will take a very similar equation, like the one that you see that I've denoted there by R1. R1 is the name. Why do I call them R1 and R2? Because these are, we call them reaction functions. Actually, firm one, if the set their quanty so it will satisfy R1, this means that their profit will be maximum given what the other firm is doing. For firm two, if they set their quantity according to R2, you give them the quantity that the first firm will set and then automatically, this relationship shows you the quantity that they should produce in order to maximize the profit. Therefore, conditions R1 and R2 are called the optimal response or reaction functions. If you think about it, we are doing nothing different than what we did in the previous lecture when we had this matrix and we underlined the values, that they give us the maximum, the best outcome, for these player once the other player plays a specific strategy. So, my reaction function tells me what? Give me your quantity and I will plug it in this function, and this function will give me which quantity I should produce in order to maximize my profit. So, give me your quantity, I will plug it to my best response function, and I will take my best response. Your best response is when I give you my quantity, for every of my quantities, gives you your best response to my quantities. So, R1 gives how firm one responds to actions of firm two in the best way, and R2 shows how firm two responds to actions a firm one in the best way. This is very similar to when we underlined payoffs in the payoff matrix in simple static game theory. So, I can take R1 and R2, and this is a two by two system with two equations and two unknowns, and I can solve it with respect to q_1 and q_2. And once I do that, I will take these two formulas that they give me the equilibrium that we have here. And this equilibrium is nothing more than firm one doing its best given what firm two does and vice versa at the same time, simultaneously. So, simultaneously, both of the firms, they are doing the best they can according to what they expect the other firm is going to do. Let's observe these two equations and see what we can get. First of all, my own cost, as it goes up, it reduces my share. If I have higher cost, it means that I'm going to have a lower share in this market. I will sell less than the competition. My rivals cost, as my rivals cost increases, increases my share. This makes me sell more as the cost of my rival goes up. Finally, total quantity is more than in monopoly, so in Cournot, we sell more product than in monopoly, but less than perfect competition. If you have a perfectly competitive market, you price is equal to marginal cost, so your quantity will be much higher than the quantity of Cournot. But the quantity of Cournot will be higher than the quantity of monopoly. So, what you have to understand in this case and to remember very well, is that we do nothing different than the standard Nash Equilibrium process. We solve simultaneously the system of best responses and this gives us how the two firms behave at equilibrium. So, let's assume that we have n firms. The same model, but not two firms. We have M number of firms, that can be two, can be three, can be 11 can be one million. How is this going to affect our results? So, if we have n firms, but now they have the same common average cost and therefore marginal cost, constant to be C, let's say, then for each firm, let's assume firm i, one firm out of the n firms. So, quantity, price, and profit will be given by these formulas here. Now, this is not very difficult to prove, it's not very easy, too, but it needs a little bit of intermediate calculus in order to do it. But it's not that difficult. I will not give you the proof, it's beyond the scope of this course. This is not what we are doing here, but I want you to look at these relationships carefully. Be careful. A real economist can read those relationships. If you want to understand the business, you have to be able to at least read those relationships. So, put some effort and try to see what is behind those equations because the truth is behind the math. So, what happens if n, the number of firms, tends to one? If the n number of firms tends to one, I can replace n with one and I will still get my solutions there for quantity, price, and profit. And if you go back and you saw these collusion model or a monopoly model, you will see that this is the same. So, the collusion model is actually nested inside the Cournot model, by just setting n, the number of firms, equal to one. It's a very impressive result, because it means that the Cournot model can also accommodate without changes the solution for the collusion case or the monopoly case. What happens as n tends to be infinity? You cannot set the number equal to infinity, but for example in quantity, as n tends to infinity, quantity will tend to become very, very small. Price will tend to become, the fraction will become very small because n is in the denominator and it will get very big, so the fraction will tend to go to zero. So, price will go to marginal cost. I have a small quantity, price will go to marginal cost and then my profit there will tend to go to zero because n is in the denominator. This is nothing different than the solution of perfect competition. So, we can say that the Cournot model can nest not only the monopoly model, but also the perfectly competitive model. So, these are not three different case of markets, this is one single case of market in which a number of firms is different. This is what I wanted to tell you in the beginning, that today, we're going to take the models that you already know and we will see them under a new look, under a new prospect. You can read more about these specific nesting of models in the textbook, Church and Ware, there's a link in the description of today's lecture and this will be in page 243. Stay with us, because now, I'm going to show you competition that has a leader and a follower, and this will be very interesting.