Okay, so far, we have seen many games, like a location game, prisoner dilemma, and coordination game. All those games are very simple, and we were able to solve those games just by inspection but if the situation is complicated, you need the power of math to find out an equilibrium. So in this lecture we are going to peek into the power of math and I'm going to show that by using the power of mathematics, you can drive really interesting law in social science. Okay, so let's look at the market for pencils, 'kay, and the days that demand craft for pencil is the price is high demand is low, if price is low demand is high, and let's suppose that unit cost of production is ten cents. So, it takes ten cents to produce one pencil, and what is the supply of this market, okay. And if there are lots of firms, and if price is above ten cents, okay, it's profitable to produce a pencil, so lots of firms would like to produce pencils. So supply is infinite if price is strictly above ten cents, and if price is below ten cents, strictly below ten cents it doesn't make sense to produce pencil, so supply is zero, and if price is exactly equal to ten cents, then you are indifferent between, you don't mind producing pencil. So therefore, economics textbooks say that supply is a flat curve, 'kay. If price is equal to ten cents, you have lots of supply. This is the supply curve when there are lots of firsts. Okay, and economics textbook says that price and quantity in perfectly competitive market where there are lots of firms, okay, perfect competitive equilibrium is given by the intersection between demand and supply, okay. So price and quantity is given by this intersection point, okay. This is what is called the perfectly competitive outcome or market equilibrium. So economics textbooks says that large number of firms implies perfect competition. So let's examine by using game theory if this is actually true. So, let's calculate the part of the competitive equilibrium intersection between demand and supply. The quantity of perfectly competitive market partly depends on the unit cost of production of pencil. 'Kay, so if cost is c by a simple calculation, the total quantity of pencil is given by a minus c over b, okay? Remember this value, it's going to play an important role. Okay, so game theory can show what's going to happen if you have N firms, if na is equal to 3, game theory predicts what's going to happen if you have three firms. Okay, so let's find Nash equilibrium. So I'm going to assume that firms chooses their output, how much to produce, and this situation was originally analyzed by a French economist in 19th Century, Cournot. So this game is sometimes called the Cournot game, okay, so let's examine your profit. Your profit partly depends on how much you, how much is your profit per pencil. Okay, so it's price minus quantity and it's going to be given by this formula. So, let me explain the notation. Q, small q here, is your output, okay, and large Q here is other firms' total output. Okay, and this part is equal to market price because price is given by the demand curve, and the demand is equal to a minus b times total production, total output, and total output, X, is a summation of new output and the other firms' total output. So this is your profit if you produce one pencil, okay? So therefore if you, if your output is q, this times your quantity is your total profit. Is it clear? Okay, so you can just rewrite this formula in this way. So this is your profit. So let's draw a diagram of your profit, okay, so your profit is a function of your output, q, small q. So this is equal to constant times your profit, times your output, okay, so your pro, out, your profit is equal to constant times q minus constant times q times q. Okay, this is what is called quadratic equation, and the graph of quadratic function looks like this, 'kay. It, it's a very simple graph with a single peak, and it's symmetric. Okay, so this graph partly depends on other firms' total output. So, therefore this graph shows that if this is other firms' total output, your best reply is given by the peak of your profit function. Is it clear? If other firms upload this in large Q, this is your best reply, best production, okay. So let's try to calculate your best reply. So to calculate your best reply it's useful to find other points where your profit becomes 0. So, your profit is 0 if your output is zero, okay, that's clear, and let's look at this part. If your output is equal to this amount here, then this part here becomes 0 and your por, profit is also 0, okay? So if your output is equal to this yellow number then your profit is 0, is it clear? Okay, and the your best reply is in between 0 and it is quantity, and since the graphic is symmetric, we have the same reply, it's just the half of this yellow quantity. Okay, so now I have computed your best reply. If other firms output this in the large Q, this is the optimal point of the output. Okay, so given this observation, let's calculate the Nash equilibrium. Since all firms are identical they have the same unit cost and Nash equilibrium has the property that every firm produces the same amount, okay? So let's say q star is the Nash equilibrium output of each firm and the definition of Nash equilibrium says that this should be mutual best reply. So if other firms are producing q star it's your best reply to produce the same amount, q star. So let's write down the best reply, relationship, by means of equation. So you are producing q star, that should be a best reply to other people producing q star, and we have already calculated the best reply function. It's half of this number here, a minus c over b minus total output of the remaining firms, okay, so since there are n firms, the number of other firms is n-1, and each of those n-1 firms are producing q star of Nash equilibrium, so therefore this is what we denoted by large Q, the total quantity of other firms, and this is your best reply, and at the Nash equilibrium everybody's best responding each other, okay? So this is a simple equation. One equation and you have one known q star, so you can solve for q star. So, let's do that. So let's multiply both sides by 2. So 2 times q star is equal to a minus c divided by b minus n minus 1 and q star, okay, so you move this term from right to left, and what you have here is N plus 1 times q star, equals to this number here, a minus c divided by b, okay. So therefore q star is equal to 1 over n plus 1 times this number here, a minus c, divided by b, okay. So therefore, the total output at the Nash equilibrium was N firms is N times this number, so N over N plus 1 times this number here, a minus c divided by b, okay? So let me just rewrite this part, N over N plus 1, so N is equal to N plus 1 minus 1. That's the numerator. The denominator is N time, N plus 1, so this is equal to, say, 1 minus, 1 over N plus 1. Okay, so if you have N firms, total production is this type this quantity times this number here, so by using the power of math we have identified that the total output when N firms present in the market, and let's compare this result with perfectly competitive market. Again, this is a picture of perfectly competitive market, and in the perfect competition, total output is this yellow number here, and on the other hand oh, in contrast, if you have N firms, we have calculated the total output, and it's all, it's equal to this number, okay, and as you can see, as N increases, this part here, 1 over N plus 1, decreases sharply. So everything is quickly brought into this yellow number. So let me show you several cases. If N is equal to 1, the quantity is small and the price is higher, but an, as N increases, the total output quickly converges to competitive equilibrium total output, 'kay? So Nash Equilibrium sets the following things. If there are few firms in the market, price is very high and quantity is small, but as the number of firms increase, price goes down and eventually it converges to competitive, perfect competitive market equilibrium, okay? So, large number of firms actually implies perfect competition. This is the prediction by Nash equilibrium. So by using the concept of Nash equilibrium, and by using the power of math to find Nash equilibrium, we have derived the Law of Market Competition in economics.