 So today we are going to discuss a little bit more about the normative approach and which means that the question we ask is not how the system is, what is the state of the system, but rather how it should be. Okay, and the concept of Nash equilibrium describes the system as it is as we expect it to find it. So it's more of a positive approach. And as we have seen, for example, in the prisoner's dilemma, there are situations where the outcome of a Nash equilibrium is not really ideal. In the sense that if you remember the Nash equilibrium, you have that the players would be better off, each of them would be better off by cooperating. But instead, because the effecting is a dominant strategy, they end up with lower payoff because both of them they defect. Okay, so this leads me to define what is a notion by which you define an efficient outcome in an economic system, and this is what is called Pareto efficiency. So, in general terms, a system is Pareto efficient. So if it is not possible to improve the payoff of someone without decreasing, without decreasing the payoff for someone else. Okay, so this is a different notion as a Nash equilibrium. So in a Nash equilibrium, just to repeat, is a state in which no one can unilaterally improve his own payoff. So let's look at what happens to the payoff of the others. Okay, a Pareto efficient outcome instead is an outcome where essentially if you increase the payoff of someone, then someone else has to pay the cost. And essentially, the idea is that you would like an economic system from a normal approach, you would like an economic system to be Pareto efficient. Okay, because this does not mean that Pareto efficiency is a notion of efficiency is not a notion of fairness. Okay, so this is not because a Pareto efficient allocation of resources, for example, may also be very unequal. Okay, so there could be some agent that some individual that has a lot and some individuals that has very few or nothing. Okay, this is a system. And the only thing is that you cannot make this everybody more happy at the same time without making someone less happy. Okay. So today, I'm going to start moving from game theory to economics by describing a little bit of examples, a simple example that I do see that a number of concepts that have to do with this normative approach. One is the issue of public goods and property rights. Then, I'm going to discuss also welfare, welfare in an economy, in particular, with respect to oligopolies or monopolies. I will end up by discussing what is a competitive market. Okay, and then this is where we will start the last two lectures. Okay. Okay, so very good. So let's start by discussing this issue of these issues with a very simple problem, which is called the tragedy of the commons. So this is a very stark realization of how a national equilibrium, individuals just playing the national equilibrium may reach very inefficient outcomes. Okay. And so this is the situation where you have a plot of London and a number of farmers. Okay, we live close to this plot of London. And each of these farmers has his own farm. And they have, they buy goats. Okay, so farmer I gets GI goats. And then these goats go and graze on this common land. Okay. Each of these farmers sends these goats to graze on this land. And as a result, there will be a certain number of goats. And the grass will be eaten up by these groups of goats. And so the utility of agent I of farmer I so will depend both on the number of goats. This is the number of goats. So, of course, that he will buy that he will maintain and on the number of goats that the other people, the other farmers will will also decide to have. Okay, and this is will be proportional to the number of goats because at the end, these farmers will say sell the milk or sell the goats themselves. So the profit will be proportional to the number of goats. So, and then we also proportional to a function that let me write it in this way. And this function here depends. So a is the area of the land. And so a over G and G is the total number of goats. Let's say there are an individual farmers. So this is the total number of goats. And so, of course, the more goats you have the least this pasture. So if the total if the area of land per goat is larger than the profit of the farmer per goat will be larger. So, so this function P as a function of its argument, which is a over G will be generally say an increasing function. Okay, there will be some increasing function. And say, and see this is is a cost of maintaining each of these goals. Okay, the amount of work that it takes to maintain each of these goals. Okay, so what is the national equilibrium that we expect to find in this situation so well, in order to find a national equilibrium, then you have to take the derivative of this utility function with respect to GI and set it equal to zero. Okay, so Okay. Okay, so and so if you take this derivative, you will have that this is P of a divided by G minus C. The derivative of P, then you have P plus GI times the first derivative of P in a over G times the derivative of a over G with respect to GI and because of this. This is minus a over G squared. Okay. So this equation for GI. What you find is that, well, the optimal strategy. will be given by say G G squared G star squared. divided by a times P prime of a over G star. Times P of a over G star minus C. Okay. And so if you sum these up from all the agents. This will give you a factor and this will be equal to G star the total level of the total number of goals in the national equilibrium. Okay. So if you look at this equation, so this equation here is the equation that gives you a G star. And you can see that as any increases. So here you see that you have a term that increases within the. So you will have the G is an increasing function of Anna. So, as the number of farmers increases, you will have a lower and lower fraction of land for each of the goals. And in the limit when N goes to infinity and what you what you can find is that essentially the in the limit when N is very large G star will be very close to the situation where if this is C. This is a point. Let's call it G zero such that essentially the this the profit for each got is equal to zero. And what you can easily see is that when Anna goes to infinity. This G star will converge to this G zero as Anna goes to infinity. So now you understand why this is called the tragedy. It's called the tragedy because essentially if the number of farmers increases, then their profit becomes gets bigger and bigger gets smaller and smaller. And so the, the, and also the land becomes completely exploited. Okay, I mean, the grass in the land becomes completely exploited. So this is the reason why this is not an efficient outcome. And and this is a Nash equilibrium. So do you have questions on this. Okay, so looks like everything is clear. So let me have a question. Yes, please. I'm sorry this P. I couldn't understand what is this capital. So this capital P is essentially the productivity of each goal. So how what is the revenue that you extract from the farmer extract from a single goal. And this P this revenue depends on how much area per goat is available. Okay, the smaller the area per goat is a divided by G. The smaller is the productivity of a single goal. Is this clear. Does it make sense. Yeah, it makes sense. Thank you. Can I ask a question. Yes, please. Does a so the area of the land scales as an, or is it constant. No, in this case, a is constant. You can take it. Well, it's a, it's a, yes, it's a constant. It doesn't scale with them. I mean, in the real situation, you can think of the plot of land being there. Farmers can establish there. And they can. And can vary. It can be the same. Okay. Okay, thank you. Yes. Okay, very good. Yeah, excuse me, could you just quickly explain why does G star go to G zero mathematically I understand it from, from like an intuitive point of view, but could you explain how does the equation tell us that. Okay, so if you look at this equation, you'll see that this thing is constant. Okay. So, because the, the, the, the, the, the slow of this curve is essentially fine. Okay. This is also fine item because it cannot be larger than G zero. So when any increases, this has to go down. Okay, okay. Awesome. Okay. I understand. Thank you. You can have a finite G star when this goes as one over M. Okay. If you work the details of this model, a little bit more, you can find out that this utility of each of the farmers is proportional to and to the minus two. This thing is proportional to the area divided by N squared. Okay. So this. In the end, you will find that in the national equilibrium. This thing is proportional to the area divided by N squared. Okay. Okay, so let me now discuss what instead would be a social optimum. Imagine that now we have a social planner that steps in and say, look, I'm going to decide how many goods you can buy. Okay, and you will have to maintain these goods. And I will find out this number in such a way that the utility, the sum of the utility, the total welfare of the community of farmers is maximum. Okay. So how does this work? So, so let me so. Sorry. Okay, so how does this work? So, so now the welfare. This is a welfare function, which is the salary of the utility of each of the agents of each of the farmers. Okay. And so if you look at what this is, this is just a G times the P of a divided by G minus C. Okay. This is just a function of the total number of goals. Okay. And then, if you take a derivative of this welfare function with respect to the total number of goals. This is that this is your P of a over G minus C minus G times P prime of a over G times a over G squared. Okay. And then you said this is equal to zero. So what you find here is that now the G is equal to P. So it's P of a over C over G minus C divided by this P prime of a over G times a if I'm not mistaken, this is what it should be. Okay. So, so this is the social option. So you solve this equation. And so let me check that I did it right. So, then, so G should be equal to maybe this is to the minus one. Yes, also a I think multiplies P prime the minus one. Okay. So but say, this is similar to the equation before, but you see that it does not depend on and. Okay. So I think a is in the denominator. No, I think it is not. Okay, but say, sorry, let's do it. So, okay, so you have a P minus C times G minus the prime. Say P prime times a is equal to zero. Right. So G is equal to P prime times a divided by P minus C. Okay. Which is essentially. Okay, sorry, sorry, sorry. Yes. Okay, very good. So, let's look at this equation graphically. And so you have again your curve of a over G, which is the same as before this is P. And this is your C. So this is your point G zero. Now, the solution of this will be when it will be a point somewhere here is G bar. And and it's and it's the it will not depend on him. So G bar is going to stay the same, even if n goes to infinity. So this means that in this situation, the utility of each of these farmers will not be equal to one over N squared, but it will just be equal to essentially will be proportional to a divided by N. So it will be n times larger than the utility in the last equilibrium. Okay. So this is, this is what would be a Pareto optimal situation. Whereas the national equilibrium is clearly not Pareto optimal, because of course you can, there is another equilibrium which is this one where everybody's better off. Okay. Notice again that in this solution, we didn't say anything about how we are going to split this total G into the individual farmers. You can split it however you like. And, and this will be Pareto optimal allocation. And as long as so, and you can go from one part of Pareto optimal allocation to another Pareto optimal allocation. If you if you give some of the goods of one farmer, you give them to another farmer, then, but the total number of goods have to remain constant. Okay. So it's not a particularly reasonable. It is not a solution that individuals like very much. Do not like very much that central authority tells them what they should do. So it is not as so it is much better to try to find a decentralized solution, a solution where essentially the central authority fixes the rules. And then given these rules, the individuals, they maximize their profit. Okay. So one of these ways of achieving this Pareto optimal allocation is by introducing what are called property rights. So what is the idea. So you have this plot of land with all these farmers around. And essentially, what you do is that you decide that if this is a farmer I is going to be able to raise only in a fraction in this in this land AI. This guy is going to be able to grace on this. This is going to grace on this, but there are say barriers that prevents the gods of Mr. I to go into the plot of land of Mr. J. Okay. So, I mean, you introduce fences and you give rights to individuals to individual farmers to grace their goals only in their own plot of land. Okay. So what is the effect of this. So of course, then the total area now is split in smaller plots of land. Okay. So what is the effect of this that now the payoff of agent I is just a function of GI. Okay, and this function is just GI times the same function as before of AI divided by GI minus C. So what you can see very clearly is that say the ratio when the agent optimizes maximizes this function here. The optimal solution that you will find divided by AI will be equal to the social optimum. Okay. So in other words, each of these individuals will find a function of a say find a ratio of say AI divided by GI, which is exactly equal to to the social optimal. Okay. And then the, if everybody does this then the solution, the system as a whole will be socially optimal and will be a solution and maximize the welfare. Okay, so notice what we did with property rights. So we went from a situation where we had the Nash equilibrium where the utility of individuals also dependent on what other people do to a situation where instead the utility of an individual does not depend on what other people do. Okay, so the fact that the utility of an individual depends on actions of other individuals in economics is called an externality. So and essentially whenever, if you have a rational individual that maximize their own benefit, and there are no externality, then they will act as a social planner that is maximizing the total welfare. Okay, but if there are externalities, then this may not be true. Okay, because you may end up in the Nash equilibrium. So other questions on this part. Okay, so the, the other consideration. To be done in this respect is that essentially the property, the problem in this situation here is that you have this plot of land which is a public good. Okay, so that, so it's public in the sense that there are two ways that distinguish two things that may distinguish a good for me in public and private. The first one is that you can exclude others from exploiting it. Okay, and essentially, in this situation here, the fact that farmer I exploits this plot of land does not exclude does not prevent others from exploiting it. Okay, when you introduce instead property rights, when you put fences, then you are excluding other farmers to graze on your property. Okay, so what happens here is that the you go from these tragedy of the comments to a social welfare to a socially optimal solution to a Pareto optimal solution, because you turn a public good into a private good. Okay, the second characteristic of a public good, which is not does not apply in this case is that when you consume it, the public good is diminished. Okay, so if you think, for example, information of knowledge is a public good. So the fact that you know, so if you can think of say the laws of relativity is a public good, because if if you consume it if you learn it, that does not diminish the laws of relativity, I mean, also other people can essentially benefit in the same way as you did. Okay, instead, in this particular case, this public good is can be can be depleted. Okay, so if you graze on the land, then there will be less grass for me to graze on the same one. Okay. Okay, so there are a couple of questions. Is this a statement that the property rights still may not prevent the tragedy of commons. Since the middle may all all may end up in national to be the insider cross not well, if you say, if you when you set property rights, then the problem of each individual becomes just an optimization problem. It is not a game. It is not a game. So they are playing against themselves. So they are not playing against anyone else. They are just optimizing their pay off, even the constraints of how much land they have. Okay. So the, the, whereas a national equilibrium is not is the simultaneous optimization of different utility functions by different agents. Okay. Over different variables. Okay. Other questions. Okay, so. So, Okay, so, but say you can play really this game and do all the math and calculation and get convinced that this is what is going on. Okay. So let me turn to a different subject. Which is essentially instead the problem of modeling on an economy. So we will discuss this next, I'll say in two weeks and the last two lectures more formal details. Now, let me take a very simple description of one economy. And in this situation, I'm going to assume that there is just one good. Okay, and then there are say in an economy what you have is consumers. They have firms that produce this good. So the consumers, they want to buy these goods because they want to consume it, they want to eat it, and firms. They produce this good, and they want to maximize their profit. Okay. So, So what do consumers do well typically consumers as a utility function that tells you how much they like a certain amount of good G. And the problem they solve is that they want to maximize this utility function over G. And the problem is that the if the cost if the price of this good is being then the cost of the good must be less than how much they have to spend so this is the wealth and this is the price. This is telling you that if a if a consumer has $10 and the price is one, he will be able to buy only 10 units of good. Okay. And then say if you look at what is the solution of this problem. Imagine that this is G this is utility function imagine that this consumers are utility function like this. This is typically what happens it means that say the fact that this utility function is a concave means that the, the, the, when the consumers have a lot of good. Okay. They like less this good than when they have a very little good. So this is called the satiation. Okay, that essentially you, once you eat, say, half a kilo of pasta, maybe you don't like pasta any longer. Okay, so the solution of this problem is a constraint optimization problem and the way it works is that you draw a line, which has a slope P. And then you find the point where there is an intersect the line with slope P is tangent to this pointer and this gives you how much you would buy at this price P. Okay, so you may realize that if the price is higher. Imagine that the price is higher than you are going to buy less. Okay. And if the price is slower, you are going to buy more. Okay. So if you write what is the, the price is a function of G, you again will find the decreasing function. Okay, now let's. So this is for the consumers. So let's get to the firms. Okay, let me throw the consumers out of the of the picture. And let's focus instead on the firms. Okay, so the firms have a profit. So imagine now that there are say, there are n firms. Okay, which I love it with index from one to one. Okay. And the profit of firm I if she produces GI of goods and the others produce the other firms produce G minus I is proportional to how much they produce times the price at which they sell. Now the price at which they sell again will be will depend on the total number of goods. So let me call this G this is the total amount of goods that consumers will buy. This is minus a cost. This is a cost of production. Okay, this is C is a cost of production. Okay. Now this is apparently where essentially, again, this G is because is the total amount that is bought by consumer, it must also be the total number. The total amount that is produced by firms. Okay. Now you realize that this setting is very similar to the ones that to the one that we have described for example, of the tragedy of the commons. Okay. Now, so you can see what is the national equilibrium in this case. And the national equilibrium again, you have to take derivative of UI with respect to GI, keeping other things fixed. And here you have P minus C minus plus GI times the derivative of this function P of G prime times the derivative of G with respect to GI, which is equal to one. And then you have to set this equal to zero. So you have that the national equilibrium strategy will be equal to P of this G minus C divided by, well, there should be a minus sign but the absolute value of G, which means that if you compute what is the G star, the total production in the national equilibrium, then there will be an end. And, and so G star will satisfy this, this, this equation here. Okay. Now you see what happens, what happens is essentially the same as what was happening in the tragedy of the commons that if you have a C, which is essentially here, this is the cost of production. And this is the G zero, which is the point where the price of the good is equal to the cost of the unit cost of production. Then as n goes to infinity, this G star will go to converge to G zero. Okay. So, now this is a tragedy for the firms, but it's not a tragedy for consumers, because consumers would be able to buy the good at a cheaper price. Okay. So, indeed, this situation where, where, where the number of firms goes to infinity and the, the price is equal to the cost of production. This is a situation where, which is called a perfect competition, perfect competition, because it's a situation where essentially the firms compete and they cannot manipulate the price. So the price is, is fixed and it is equal to the cost of production. Okay. And so, indeed, in this setting, this perfect competition is also Pareto optimal from the point of view of welfare. Why is this so let me try to explain this. So, because welfare is usually measured as the total utility of, as the total utility of a society, the total utility of a society is equal to the total utility of the consumers. Because in the end, firms are owned by computer, by, by individuals, by consumers. So the, the profits that the, the, the, the, the firms make, end up in the budget set of the consumers, and they are used to buy the goods. Okay. So welfare is essentially equal to the sum of the utilities of consumers. Okay. So, and in this case, when you have a perfect competition, the price for consumers is as low as possible. And the, and as a result, this is the most efficient and Pareto optimal situation. Instead, what before was the social optimum. In this case becomes what is the situation when, when N is equal to one, when N is equal to one, you have what is called a monopolist. The monopolist will essentially solve this problem with n equal to one and essentially will fix a price, which is much higher than the price that you would have in competitive markets. And this is very inefficient. This is not very efficient for the consumers. And essentially the amount of the monetary loss for the society as a whole is essentially given by the area under this curve here, because this is essentially how much revenue is lost because of lack of perfect competition. Okay. So the idea, what I wanted to, the idea of today's lecture was essentially to show that there may be different ways in which even the same model can be interpreted and then can be different stories behind the different, the same model. And, and that there is a type of In certain cases you are interested in describing the system, finding what is the national equilibrium predicting what is going to happen. In some other cases you are interested in measuring how efficient your allocation is in terms of the welfare of the individuals. Okay, so from this point of view, this is why essentially in most western countries monopolist are not allowed. And also, what is not allowed is also to form cartels. Cartels are situations where firms strike an agreement on how much they should produce in order to manipulate prices. So the, as we will see in the next lecture, the perfect situation in perfect competition is one where the first maximize profit, consumers maximize their utility, and the market fixes the prices. But the non, nor the utilities, nor the, nor the firms, nor the consumers, manipulate prices can affect prices by their behavior. Okay. So questions. I have a question. You mentioned that Pareto optimality for the consumers is Z zero. What would be the Pareto optimal outcome for the firms. Is it G bar is which is a monopoly. So, okay, so the Pareto optimal for the consumers is when P of G is equal to C. Okay, so is this point here. Okay. What if you want when this is equal to zero or the profit of the firms is equal to zero. Okay. And the reason why you talk about Pareto optimality is because in an economy firms are owned by individuals and individuals are consumers and the profits that individuals make out of the share of owning company owning shares of companies is essentially spent in buying goods. Okay. So in a, I mean in this picture of an economy, all you care is about the welfare of consumers. Okay, there is no great sense to consider the profit of firms. Okay. Okay. So, the other thing is, say, Yes, so for the firms, but if you just think about, say, what would be Pareto, what would be Pareto optimal for the firms. Then you're right. It is this point here, where they agree on producing this much. And then they share, say, the profits. Okay. Professor, so for a for a monopolist system. So could you say the case, so could you say that it's the case that a monopolist could decrease G arbitrarily, in order to diminish the supply and thus ramp up the prices. And that's the reason why the area of what do you call the area. The shaded area is kind of maximized. Because I assume that in a monopolist system, the area that denotes how much value is lost. The shaded area is maximized. No, no, no. Oh, okay. I mean, the G bar is only determined by the maximization of the profit of the monopolist doesn't really care about inflicting as much as much pain as possible on the consumers. Okay. Okay. Okay. Yes. So you need to say the monopolist would also choose to produce this one. This is G here. Okay. And charge this price here. But in this case, he would inflict a loss to himself also. He would inflict a damage to consumers because he would sell a very small amount of this wood at a very high price. But at the same time would also make damage to himself. Oh, okay, because because the consumers wouldn't be willing to pay the price. No, no, no. So this is the price. This curve is tensile for a particular for a particular amount G. What is the price at which consumer will be willing to buy. Okay. Okay. I think at this in this way so the consumers will buy an infinite amount of this good if the price. So, okay, so if if G. So this is the price that consumers will agree to pay. For this quantity G. Okay. So this is a question. So we are assuming that consumers as individuals, old firms, they own firms. Yes. If yes, where is a bonus enters in the calculation. No, it does not enter into this calculation. This is a very simple, say, example that I choose just to illustrate a number of subject. We are going to discuss this next time when we discuss a more general model of an economy, which is called the general equilibrium tier. Okay. Can I ask another question. Yes, please. Yeah, so I was thinking if monopoly is not an efficient market neither is the spirit optimal condition of the zero and efficient market as far as forms are concerned, will then efficient market be where there is a oligopoly, a few forms dictating everything. Yeah, so essentially, so this thing corresponds to n equal one. If you take n equal to you will have another point here. If you take n equal to you will get another point here and as you increase and you will converge to this point G zero. Okay, so this is a situation where you have a finite number of firms. So, only go fully, only go fully. Okay, which means the same tyranny of the few. Okay, only goes few or say, Yes, so then you had a question on what on efficiency, which I didn't understand. I was asking the G bar is not an efficient outcome for consumers G zero is not an efficient outcome for firms. So I was thinking maybe a oligopoly condition would be efficient outcome for all. Yes, yes, no but as I was trying to mention, yes, so you could say if you consider a game where you have say is computer, the firms are different players than consumers. Then you would be around. Okay, but what I'm saying is that firms in real life are owned by consumers. And so they are not really different players. Okay, okay. Yeah, but it is not really realistic that. So, look, I think every CEO on this planet on the phone. Sorry, no, the rationale here is just that any say owner of a firm has to eat as to put gasoline in the car as to as to consumers any other individual. Okay, as to buy goods. Okay, so he's also a consumer. Okay, I get it from where you're coming from. Okay, so I think we should stop here and take a few minutes of break before going back to neuroscience.