 Okay, so today what I want to do is to start discussing what is called general equilibrium theory, which is essentially the theory that describes economies as a, say, the behavior of economies, starting from the behavior of individuals, which is what economies called micro foundations. It's a little bit like what we do in statistical mechanics, where we discuss the properties of matter, starting from the fundamental equations of motion for particles, okay, and their interaction. Okay, so this is a general theory that has been developed in the last century, mostly, say, around the 50s. And so, and I hope you have seen some videos on the web page, and there is a lecture of, given introductory lecture to microeconomics that essentially explains a little bit the general framework. And I think this is very instructive to talk about to think about, to realize how economists have been discussing about economies. So what I want to do, of course, in this hour is to just give you a very brief sketch of the main ideas, going through the different, the main results, okay? And, okay, so I hope I'm trying to share my screen. Let me see, okay, very good. So, okay, so, okay, so, okay, so what I want to describe is what is called general equilibrium theory. So how do you describe in general an economy? So in an economy, there are three main actors. One actor is what are called consumers, and these are essentially the household, the individuals, okay, and the other actor is what are called firms, and the third actor is markets. Okay, so essentially these three elements interact with each other. So for example, say consumers, they work in firms, so they receive salaries, wages from the firm they work on, but they also own the firms, okay? They own shares of the firms, okay? Then essentially consumers, what they do, what this name tells is they consume goods that they buy in the markets, okay? So this is what gives rise to what is called demand, and, but also they also sell what they have in the markets. So in the end, in this economy, in this view of an economy, everything is owned by the consumers. So even, so consumers are born with their endowments, which is what they own to start with, and then they trade in markets in order to buy goods to buy goods, okay? To buy goods, to consume and to maximize their utility, okay? So, okay, and what do firms do? Well, they essentially buy inputs from the markets. They transform these inputs, I mean inputs are like raw material, and then they produce an output, which is essentially what is the supply to the market, okay? So among the inputs, of course, there is also work, there is also labor, so it is essentially the consumers, they go to the market, there is a market for labor, and then as a result of this, they sell their labor to firms, and they get wages, okay? So let me go a little bit more in detail on each of these elements, okay? So let's look at what firms do, okay? So firms, let's focus on firms. So firms, they essentially maximize profit, okay? What is the profit of a firm? So the profit of a firm is essentially a firm F, if they produce some output Y, is essentially equal to the profit that they make, which is just the price, the market price at which they sell these outputs minus the cost of production, okay? The production costs. And this depends on, well, depends on many things, but essentially let me just say that it depends on how much they produce. So essentially what a firm has to do is to find the optimal production plan, okay? So which is what maximizes their profit, okay? So now generally, so you can see that this problem, if this is Y, and this is the cost of production, so when you say, let's imagine that the cost of productions are something like this, then when you look at this problem and you maximize, then you take a first derivative of this and you find out that the price, the condition at the maximum is that, sorry, the price will be equal to the first derivative of Y. This is what is called the marginal cost. So what firms will do is that they will set the price, the production schedule in such a way that the marginal cost are equal to the price, okay? Now, so if say the price, let's say that the price is something like this, so this first term, say P times Y is essentially, this first term P times Y is essentially a linear term, okay? So if the price is more, this is like a line with this slope, then the optimal production will be given by this one, okay? If instead the price is higher, so it's something like this, then the production will be higher, okay? So from this, you can see that what the output of a firm, Y as a function of P would be a function of the price. And well, in this case, you can see that it is an increasing function, so if the price is low, then the output is more, if the price is high, it is, so it will be something like this, okay? So this is what is essentially the supply of the firms to the market, okay? Are there questions on this? If there are no questions. Do we get the equation of price from the first equation of the profit? So it equals to C prime of Y, yeah. Yeah, so this comes to the derivative of pi with respect to Y, so if you take- In the maximum? Yeah, if you take this equation for pi, if you take a derivative, then this is the first order condition, right? Okay. So here I'm describing a very simplified problem. I mean, indeed, what you can imagine that the cost of production depend on the prices of the inputs and on how much inputs the firm is taking. The firm is taking, so I'm just simplifying all this just to give you the main idea. Is this clear? Yes, yes. Okay, very good. So let's go then on the other side, which is consumers. So what do the consumers do? Well, the consumers, what they do, what we said is that they maximize the utility of consumption, okay? So the consumers, they maximize the utility of consumption. So it means that there are many consumers as there are many firms. So each of them essentially decides how much to consume given a certain value of the price by maximizing a utility function that tells them how much, what is the satisfaction that they get from consuming an amount XC of goods, okay? Now this maximization is constrained by essentially what agent C, what consumer C can afford. And this is encoded in this budget set. So the budget set is essentially the set of all consumption bundles such that the cost of this bundle is less or equal than the wealth of consumer C, okay? So this is the wealth. And what is the wealth? Well, the wealth is essentially what consumers get by selling their initial endowment, let me call it W. So these are endowments of consumer C, plus whatever they get from the shares of firms of the profit of the firms, okay? So as I told you, the consumers also own the firms and firm F, a fraction of F of firm F is owned by consumer C, so this should be also depending on C. And so this is essentially the amount of revenue dividends that they get from the firms, okay? Okay. So notice one thing that here the utility of each consumer does not depend on what other consumers consume. Say XC prime, okay? So it means that I'm assuming that there is no externality. So the fact that my neighbor consumes its meat every day does not affect in any way my utility, okay? Now this may not be true because say maybe if you consume meat every day, it does barbecue and then there is some say noise or some pollution. So this might not be true, but this is what is assumed in this framework. So I see there are some questions. Okay, is the price of firms selling the product different from the one I'm coming to discuss about prices? And that is the next question, the next issue when we will discuss markets, okay? In this example, we are always talking about a single type of product with fixed price, right? We want to discuss a more complex situation in which there are lots of different goods. I suppose that the multimedia description would be a lot more difficult. Indeed it is a lot more difficult. So if you go on the website, the lectures which you will find on the website describe the general situation where essentially you have many goods, many consumers and many firms, okay? And for each good you have a market. So for each good you have a price, okay? For this moment, what I wanted to do is just to describe, give you the main concepts of the main ideas, okay? And so that you'll get a general picture of how an economy works, okay? Or how it is described in these theories, okay? Okay, so now the output of what consumers do is essentially a demand function. So how do you get this? Well, essentially, again, this is an optimization problem. If the utility function of a consumer is like this, then and if the prices are, if there is a certain vector of prices, then essentially the solution for this consumer will be a certain amount of goods at given prices. And then you can derive what will be the demand function, how much goods will consumers, so this will be for one particular agent, okay? But then you can compute what will be the aggregate demand which is the sum of all consumers, okay? And so now this is what is the demand which is submitted to the market, okay? And what do the markets do? Well, they take these two functions, so they take, so this function here, and this function here, and they determine what the prices are, okay? So if you plot one side as a function of the price, this is the supply, y of p, and this is the demand, x of p. So then this will be the price, okay? This will be the price of the market. So you see, in the general description of an economy, you have consumers that solve an optimization problem at fixed prices, first that solve an optimization problem at fixed prices, and the market will match the demand from one side with the supply from the other side and determine at what prices the demand is equal to the supply, okay? And this will happen for any good, okay? So this theory does not describe how are, what is the mechanism that fixes the prices. It just described based on the properties of the consumers and based on the properties of the firms, what are the prices that you expect to see in the market and how this market also, how these prices will change as you change the characteristics of individuals, okay? So one point that I want to make is that sometimes the equilibrium may not always exist. So for example, even in this simple case, you may realize that say, for example, you may have a consumption as a marginal, say the cost of production, which are something like this and then like this, okay? Sorry, maybe it's not that are something like this and then like this, okay? So in this case, you will see that for the same price, okay? You can have two solutions, okay? One solution which is here and one solution which is, sorry. So for this price, you can have this solution but you can also have this solution, okay? So you can have two values of Y for the same price, okay? So this means that the curve, the supply can have these features here, okay? And if you have a supply function that has these properties, when you match it with the demand, you can get into a situation where the demand and the supply do not meet each other, okay? And so the equilibrium does not exist, okay? So generally this problem of existence of equilibrium is related to non-convexities. So if you have convex, so what you can show is if you have sufficient convexity property on the utility functions and on the production functions, then the equilibrium exists, okay? Then another issue is whether it is unique or it is not unique. So in order for it to be unique, you have to introduce further assumptions, okay? Okay, so are there other questions? Isn't a bit counterintuitive that the quantity of supply goods increases when their price increases? No, well, yeah, so this is just an example. It can, you can have a different function for the cost of production and this will result in a different curve, okay? So it can also be a decreasing curve or it can be flat curve. So yeah, so in the real world, you have, I mean, this issue of non-convexities, it is something that is applies to the real world because this is essentially what is called increasing returns to scale. So the fact that your, the cost of production, the marginal cost of production maybe in some part it can be increasing with the production level. In some part it can be decreasing, okay? So in this type of non-convexities cause problems if you want to find for the existence of a solution, okay? Okay, so very good. Now let's, so the type of equilibria that you describe with this theory are what are called competitive equilibria, okay? And we have already seen, okay, so we have already seen what competitive means when we discussed the Carnot, Cournot, Oligopoly. So essentially in this competitive equilibria you have a situation where essentially neither the firms nor the consumers, they can manipulate the price by their own choices, okay? So there as I was describing, so both the consumers and the firms solve a problem at fixed prices, okay? And so this is very interesting because it means that essentially in an economy described in this way every agent, every individual, both firms and both consumers are completely de-coupled from each other. So the solution of the problem of Mr. I is totally independent of the solution of the problem of Mr. J. The only dependence between these problems is that they solve a problem with the same, at the same prices, okay? But otherwise there is no strategic issue here, also like in game theory, no? In game theory, game theory are complex because the utility function of player I also depends on the strategy of player J. Here instead the utility of consumer C does not depend on the utility of consumer C prime nor on the profit of firms, okay? So why is this, when is this assumption of competitive equilibria justified? Well, in one case we already saw when we started the Cournot game, the Cournot oligopoly. And what we saw there is that if you have a situation where the number of firms is very, very large then the dependence of the price on the choice of each of the firm is of order one over M. So the price moves by a very little in response to each individual player or to each individual firm. And in the limit when the number of firms goes to infinity essentially the prices do not move at all, okay? And what we saw in that case is that when firms optimize their output then you tend to converge to a situation where the output adjusts in such a way that the marginal cost of production is equal to the prices, okay? You can go back and check this case, okay? So now what are the consequences, the general consequences of competitive equilibrium of this setting, okay? So if you have enough assumptions on say convexity properties then essentially what you can generally show are some general results. Well, one is, but the main ones are what are called welfare theorems. And this is what I would like to discuss. So there are two theorems. So the first one says that say the equilibrium allocations in competitive equilibria are Pareto efficient. Okay, so this is essentially a very powerful example. What it essentially says is that if under these conditions this system where you have consumers and firm that optimize their utilities and profits under competitive markets then will reach a situation where you cannot have anyone being better off without having someone else worse off, okay? Which is essentially a Pareto optimal allocation of goods. Okay, so this is sometimes what goes also under the name of invisible and so you see that essentially you reach these optimal allocation of resources just by letting markets do their job, okay? Just by allowing trading into markets and if markets are competitive then prices are adjusted in the right way, then you will get this invisible hand will give to each participant in the economy an optimal allocation, okay? So the second result is what is called well, the second welfare theorem. It says that every Pareto optimal allocation can be realized as a combination of competitive equilibrium with the transfers. Okay, so what does it mean? So maybe as we told, as we discussed, so Pareto optimal allocation, they are optimal in the sense that no one can be better off without someone being worse off but they may not be fair in the sense that some individual may get a lot and some other individual may get very little. So maybe you would like as a planner or as a social planner, you would like the society to converge towards Pareto optimal allocations that are more fair or say less unequal in society, okay? What the second welfare theorem tells you is that essentially you can do that if you take some of the endowments of some of the consumers and give it to some other consumer. If you can transfer the endowments of the consumers from one consumer to the other, then you can achieve any Pareto optimal allocation. So this is the principle essentially of taxation, of redistribution. So that says that if you find, so for any possible Pareto optimal allocation that you want to achieve, there is an optimal way to introduce, say taxes on properties or endowments that will enforce, that will be such that if you then let markets go, if you can let just people interact in competitive markets, then you will reach that target Pareto optimal allocation, okay? So is this clear? With at least the general concept, okay? So these are, I mean, you can see these are really very powerful results. And you can also see how much they've been influential in the political economics of the last century at least, okay? Because essentially these have led to the idea that essentially you don't need, you can let just market do their job and markets by themselves, they will reach say an optimal allocation for everyone, okay? Now, of course, to show this result in full generality is really not feasible. So what I want to do is just to discuss this result in a very, very, very simple setting that is called a very simple economy where essentially we have just two consumers and two goods, okay? So this is just, so there are no produce, there are no firms here. So this is what is called exchange economy, okay? So it's an economy where essentially, yes, you don't have markets firms, there are no firms. So everything that happens is that consumers will exchange goods among themselves and in markets, okay, at fixed prices, okay? Okay, so now, how do we describe these two, this situation? Well, let's look at agent individual one, okay, sorry. And what I'm going to draw is a space of possible consumption sets for individual one, okay? So imagine that this is say X one, amount of good one, this is amount of good two, and let's say that this is the initial endowments, okay? So this point is omega one, this is omega one, one is how much agent one has at the beginning, and this is how much agent one has of good two at the beginning, okay? So you can think at this, you can think at X one and X two as apples and banana, and so consumer one has a certain number of trees of apples that give them, give him these many apples and a lower number of trees of banana that gives him a lower endowment of bananas, okay? Okay, so now we have to describe what are the preferences of this agent, okay? And these preferences are described by utility function. Okay, let me draw it a little bit better. So this utility function, you can draw it in the third direction of the plane, but here what I am drawing is just the set of X such that the utility, sorry, the utility of X is equal of X one, the utility of agent one for is equal to the utility of his initial endowment, okay? So this is called the indifference curve. So it means that all points up here are points that agent one prefers with respect to his endowment and all these points here are points that are points that for which he prefers his endowment to X for all the points that are on the other side of this line, okay? So it means that this agent would be very willing to get any of these points up here, okay? In this upper region, okay? So now let's imagine that there is a market and what the market does is it allows this consumer to trade his goods, okay? So he can change an amount delta X one of good one at price P one with an amount delta X two of good P two and so he can sell delta X one, imagine this is negative and he can buy a certain amount of good two at this fixed price. So but this must be, well, must be non zero. So non cannot be say positive, okay? So essentially if we consider trades of good one for good two at this fixed prices here, then essentially this is equivalent to finding, sorry, to finding them all the, so as a result of this agent can reach all the points that are on this line, okay? And you can see that there are some points here that are preferred by him to his initial endowment, okay? So if he can go to this market and buy an exchange his goods at this prices, what is the point where this guy will, what is the, how much he will buy of X two in exchange of X one? Well, the answer to this is that this will correspond to the point where there is indifference curves are tangent to the line of the prices, okay? Because you can see that so there will be no point that is better, that allows him to get a better a bundle of goods that he prefers with respect to this one, okay? And now you can also see that if you change the prices, if prices change, then eventually these amount of good will also change, okay? So as a result of this as a function here, you can see essentially that there will be a curve, okay? Passing through all these points that describe what is the behavior of the agent as a function of the prices. As these prices move, as these prices move, then his optimal point will be essentially on this line, okay? Very good. So now let's get to, no, this is just agent one. Now what, well, consumer one, what consumer one has to do is to essentially whatever he gets, this is a closed economy, whatever he gets, it should be given by agent two, okay? So of course you can do the same picture also for agent two, for a consumer two, okay? So also for him, you can draw these two, lines and you can draw the indifference curves. Imagine that he starts from an initial endowment which is up here. And then, well, you can do exactly the same picture and then there will be another, say line that describes what is the optimal consumption of, what is the optimal consumption of agent two as a function of the prices, okay? Now essentially, this is, what you have to do is to find what is the, to combine these two pictures here, okay? And the way to do this is essentially what is called the Edgeworth box, okay? So let me try to explain this. And the idea is that now you take, so let me remove this. So you take this plot for consumer two and you turn it around and you put it on the same plot of consumer one. In such a way, so what you do is you put this plot exactly here, okay? You say, turn it around, so you turn it around and in such a way that this point, so that this point here coincides with this point here, okay? So in other words, that this one is omega one is omega two two, so this distance here is omega two two and this distance here is omega two one, okay? Now, if you put on this plot also the indifference curve of agent two, this one, then essentially what you get is a picture like this one, okay? Well, now you see that all the points inside this region would be all points that both agent one and agent two prefer with respect to their initial endowment. Okay, so we have a question on parade of distribution. Let me go back to this and finish this, okay? So now you can also put this optimal curve for agent for consumer two on the same plot and this will essentially be say a curve that will be essentially going like this, okay? If I just mirror this thing here, okay? So you see now that you have a point here which is exactly the intersection of these two lines. So this is when the demand of consumer one matches exactly the demand of consumer two and what you can realize also is that by the way which I've been constructing these curves, this is a point where the indifference curves of agent one and agent two are both tangent to the same line, okay? So they have the same derivative, okay? So the same tangent, okay? So this is essentially the competitive point of equilibrium in this case. And you can see that it is also part of optimal because essentially no of the two agents can get a better deal without having the other one having a worse deal, okay? Now, so there are many other Pareto optimal allocations in this economy and these are all the points that as this point here share this property that the indifference curve of agent one of consumer one is tangent to the indifference curve of consumer two, okay? So for example, you may have another point like this up here, okay? Where the two curves, indifference curves, they are tangent to each other, okay? And you can have another point which is, I don't know, back here, okay? Now, if you join all these points, okay, so if you join all these points, this gives you the essentially what is the set of all Pareto optimal allocation or what you can call the Pareto set, okay? Now, this describes all the possible equilibrium, okay? So essentially, so the fact that you see by introducing these markets and letting the prices adjust in such a way that it coincides with the demand of both consumers, you get essentially the first welfare theorem, okay? Essentially, the first welfare theorem, I won't say welfare theorem. So that tells you that the competitive equilibrium is Pareto efficient. So the second welfare theorem tells you essentially that now instead of starting from these endowments, imagine that you want to realize, say, this particular equilibrium here. So what this tells you is that there is a transfer of endowments that you can make between agents one and agent two. In such a way that the equilibrium that you get, the competitive equilibrium with these initial endowments goes exactly through this Pareto optimal allocation, okay? And so the second welfare theorem tells you that the second welfare theorem tells you that any competitive equilibrium can be realized with transfers. Transfers are essentially changes in the initial endowments, okay? So this is essentially what I mean, just gives you a very sketchy idea of general equilibrium theory. There is a little bit more details on the website in the lectures that you find on the website and even more details on the book of Masculand. But I hope that this can be sort of a map for you to understand what are the main concept and the main ideas. So let me get to this question. So if wealth is Pareto distributed, wouldn't this mean that the assumption of consumers first having non-externality be avoided, some big players may influence others? No. So no matter how say endowments are distributed, if markets are competitive in the sense that firms cannot manipulate prices no matter how big they are, then the allocation will be Pareto efficient, okay? So in the real economy, this is not so in the sense that there are many situations in which big firms have a sense of monopoly power on particular markets. But a different solution is Mike, which is open. So, but if the conditions of this wealth of the general equilibrium are satisfied, then essentially even if you have a very unequal distribution of wealth, then that's not going to be the case. So can you show the analytical format to get the Pareto set? Okay, so as you see the Pareto set is a set which is, so determined by the fact that say the gradient of the utility of agent one at those prices must be equal to minus, what this is with respect to conditions of agent one should be minus the gradient with respect to second player that coordinates of the second player of U2 of P, okay? So what this tells you is that the utility of agent one increases in this direction, and the utility of agent two increases in this direction, and these are opposite directions, okay? Because both these two directions are orthogonal to the line of prices, okay? Is this clear? You know what I'm saying? Okay, very good. So I couldn't get second welfare theorem. Ah, you couldn't get the second welfare theorem, okay? So, okay, so now this has become a very messy picture, okay? So this, so, but within this picture, what I have time to show is that if you start from a certain endowment, there is a unique point which is the competitive equilibrium that can be enforced by having a market with these specific prices, which is such that at this point, the vector say this line of, this straight line is tangent to both indifference curves, okay? Now, you can define a set of points where this condition holds, where the two indifference curves of consumer one and consumer two are both tangent, okay? And these identify all these yellow sets, okay? Now, the second welfare theorem tells you that if you have, if you pick another point on this set, okay? So there is a, there is at least one possible way to change the initial endowment to another point in such a way that the competitive equilibrium of a market start with this initial endowment will exactly be this point here, okay? Is this clear? So that for any point on the Pareto set, you can find a transfer of initial endowment that will lead you to this point, such that competitive markets will lead you to these points. Is this anything more clear? Okay, okay, thank you. So other question, you didn't find anything on start make of general equilibrium theory. Yes, this I was planning to give you a little bit of some elements of this. Maybe what I will do is I will discuss this in the next lecture, okay? Which is the day after tomorrow, okay? So because now time is over, I think there was already a lot of material in this lecture. What do you think? Okay, so it's not tomorrow, it's the day after tomorrow, I'll call you Benjamin. Okay, so maybe we take 10 minutes of break and then recombine for the next lecture, okay? Okay. Okay.