 Okay, very good, so this course is a course on, which is introductory course on mathematical economics and so the idea of this course is to give you a sense of how you can use or how economists have been using mathematics to model economic behavior and so let me tell you how this course will work so first of all you have material on this website where you will find some background material and some video lecture for some short video for each of the topics that we will cover in this course so today every lecture that we will do we will deal with one of these topics and the last lecture we will maybe make a summary or cover some extra material so the idea is that you look at this material and the lectures, the video lectures before the lecture every day every time and then we go during the lecture we go over this material I will summarize this content and then you will have the question we will have the question to discuss this material we will have a possibility to ask questions and to discuss them okay so today we are going to discuss the introductory part and individual rationality okay so the first comment I would like to make is that economics is very different from physics so when you try to model an economic system a system where there are agents rational or intelligent agents then you can think that you have to realize that some of the features that we are used to when we deal with natural systems with physical systems do not work so the first one is that economic actors have a forward-looking behavior so in the sense that in any particular situation so an intelligent actor will try to forecast or to predict what will happen in the future and take actions in such a way as to achieve a particular goal this means that when we write down equations in physics these equations are typically forward equations that is you specify the state of a system at a given time and then you have equations that tell you how the behavior of the system will evolve from that state onward in an economic situation sometimes you can have a situation where there is an economic agent that wants to satisfy a certain goal at a certain time and then the type of equations that you solve are backward equations, backward in time to understand what is the action that you need to do now in order to achieve a result later so in this sense economic theories or type of theories that you have in economics are not, there is no causality and also there is a non-locality of say interactions the second big thing that one has to realize is that in physics we are used to characterize the behavior of a system as an optimization problem over a function be it the Hamiltonian, the actions, the free energy or whatever so you have a function and then the state of the system can be defined in terms of the maximization so each degree of freedom maximizes the same function in economics instead you have the different agents maximize different actions and the resulting behavior, the resulting state of the system is characterized by the combined maximization of different objective functions and this may result in very different type of behavior and the result can also be quite different from say what would we call the social optimal in particular in physics because we know that say for example if the interaction is an interaction that only depends on say the difference of the positions for example in physics and you know that there is a law of reciprocal reactions that the force that is acting on particle 2 because of particle 1 is the inverse of the force that is acting on particle 1 because of particle 2 okay so this is, this will not be true in economics in general the last what I want to make about this is that in physics we are interested in finding mathematical model that describe what is going on you know on the system that you are studying so this is called a positive approach or a descriptive approach and in economics instead of course one wants to one is interested in economic system because of a positive approach but also one is also interested in understanding how the system should be in order to achieve certain objectives of say fairness or efficiency etc. so this is called a normative approach okay and say if you want the normative part of physics is what is called engineering okay so in economics you have both these two type of emphasis on both approaches okay so let me start by saying why do we insist on rationality so on the website you find a paper that gives you a background on what does really this assumption of rationality means what are the are we really thinking that the Russian the economic actors are behaving in a rational way is this an assumption that psychology or what so this is a very interesting discussion but we will not enter into this discussion we just treat rationality as a working hypothesis because essentially it is very so it is a it is a hypothesis that allows us to model individual behavior and to model the behavior of interacting individuals and there is a single way to be rational there are many many ways to be irrational so at least the theory of economics under the assumption of rationality gives you a benchmark over which you can then compare real behavior with what would be predicted by rationality so we will consider rationality as a working hypothesis so then why is this rationality so important for economists so it is important because of what economists call micro foundation that is in any economic phenomenon they want to understand what is the underlying cause in the individual behavior that results in an aggregate behavior so let me make this point by discussing the problem of the wealth inequality okay you may know that the so wealth is very unequally distributed in society so there is a a small part, a small fraction of people that own a large fraction of the total wealth in an economy so that if you look at the distribution of how much wealth do say how many individuals have a certain level of wealth you find what is called a Pareto distribution which is essentially a power of low distribution it's a very broad distribution and it's essentially if you plot it in a log scale you find a straight line so this finding was first derived by Wilfredo Pareto and at his time he had a big problem in trying to convince his colleagues that this was an important finding and the reason was that at that time economics was not really a mathematical science it was mostly part of I mean economics was part of low and so the normative aspect of economics was very much prominent and so when economists read Pareto's paper on these new theories of economics where he was finding this distribution very broad distribution of wealth they were thinking of what are the moral underpinning I mean they were thinking this was a moral statement about society but he was just pointing out a fact a fact which had to do with measurement with an empirical measurement and that needed to be understood and so there's been a lot of work on trying to understand what this wealth distribution what could give rise to this wealth distribution particularly physicists have been working a lot on this and there is a book which essentially summarizes what physicists have done and they were able to reproduce the empirical distribution of wealth in different countries with very very simple model which just more or less assumed that people meet at random and when they meet they will trade for something so they will exchange some money and someone will get a little bit more rich someone will get a little bit more poor but you need to make very little assumptions in order to recover this empirical finding now this finding didn't really have a lot of impact in economics and the reason is precisely because well they don't tell you anything on what are the incentives what is the mechanism in the behavior of people in the behavior of the agents that gives rise to this particular behavior here instead this subject became very much a hot topic when this book by Thomas Piketty came out and this is the capital in the 21st century and well first of all it came out because inequality is on the rise rising a lot and second because it was explaining what are the mechanisms that are responsible for this behavior and how what type of micro-economic measures could tame this type of problem and so so this should give you an example of what is the difference between a physicist approach and an economics an economics approach okay having said this so I would like to just pose for questions from any of you on this first part so this is the way I want to organize this lectures by discussing reviewing a little bit of material and then see whether everybody is with me or if you have questions just turn off your mic and ask a question if you have any okay so okay so so why do we see this inequality within wealth distribution as a problem okay so this is a very important question so one main idea is that essentially in an economy what you care about is that essentially there is a circulation of wealth and goods okay and so and this is what is called GDP which is how many goods are produced and exchanged in an economy okay so when the economy when the financial when the finance of an economy when the wealth in an economy becomes concentrated in very few individuals then this wealth does not circulate in the economy and then the economy does not work properly in the sense that there are a lot of people who would like to buy stuff but they do not have money and there are people who who want to buy and this is a problem because the economy gets stuck and you get a recession so this is one argument against inequality the other argument which is maybe more sociological is that the more unequal society the more you have the risk of serious unrest like revolutions or things like this and of course the real question is well how much is when does inequality becomes too large when it becomes when it is okay and this question is a quantitative question that for example we have addressed in one of our papers and what we find is that when the slope of this when the exponent of this Pareto distribution becomes so such that the expected value of the wealth distribution of the wealth under this distribution diverges then essentially the economy stops economic exchanges they come to a halt excuse me professor can I ask a question yes please so you can be like poor in a poor environment and still be like kind of cool like kind of okay then you could also be relatively well off in a very wealthy place and still be unhappy so that was one thing and then also regarding the perspective do you know any instances maybe like is it a law that maybe is kind of that holds in all the sectors where maybe creative production of people is the main factor because something that comes to mind is like best sellers a lot of people write books but only a small proportion of them are successful and then of those successful only a small proportion of them do you know anything about this opens Carlo a very very long discussion on what are the feedbacks in different systems and what are the returns in different sectors it's a very interesting discussion we cannot go into it so there are other questions in the chat let me go through this and then we'll go back to the lectures because otherwise so in the first slide we have seen economics is not equal to physics but economics has a huge role to play in society so today are going to study about the impact of economics in the day-to-day life or the adjacent role with the physics so I don't know if I understand well this question but essentially we are not going to solve any real problem but I think it is important to understand what is the theory of economics in order to think about these problems in a more disciplined manner so then there is another questions are there any alternatives to assuming agent rationality when trying to mathematically model the economical system there are many alternatives and there is a lot of research going on there are fields that are called behavioral economics and there is also a field that is trying to relate economic behavior with neuroscience which is called neuroeconomics so it's there's a lot of stuff going on yes but doesn't that make money worthless if everyone possesses approximately the same wealth not necessarily okay so there is another big issue here what is really money and I think we will discuss about this we should understand what money is in order to answer this question so physics can be categorized in terms of normative approach yes if you want to make an airplane fly then you need a normative approach and and that is called engineering essentially thus probably the distribution has a heavy or fat tail it has a fat tail so parallel distribution is a fat or heavy tail distribution yes okay so then there are also theories on networks in economics or say network theories more and more used in economic theory I don't think we are going to enter into this thing there are also people thinking about introducing altruistic behavior in economics but we are not going to talk about this and so for the sake of time I think I should go on with the presentation so let me go back to my slides I'm happy thank you very much for all these questions and I think this makes our spring college a little bit interactive okay so let's think about modeling yeah can I so there's the last there's the last question and I'd be I'd be kind of interested in knowing if you think it's relevant to our further discussion it's could you please explain normative approach in short if it's relevant to our discussion could you please do that yes we are mostly going to deal with positive approach we are mostly dealing with describing economic behavior so the normative approaches is mostly related with how should the society be okay so what is a good measure of welfare okay how do you measure efficiency in a society okay and so these are subjects that you deal with in a normative approach okay so let me go back to modeling rational individual so we start by saying that so being rational means making rational choices okay and there is one way to so the first thing you have to decide is what are you choosing and so the first element is you have to define what is the set of alternatives and we will call this X and then on this set or actually on the direct product of this set with itself we define a preference relation in this way of writing we mean that X is at least as good as Y so there are other relations that you can define binary relations so one is the strict preference which means which is equivalent to saying that X is at least as good as Y but Y is not at least as good as X and this preference relation is reflexive in the sense that X cannot be strictly preferred to X and then there is the indifference relation which means that if X is at least as good as Y and Y is at least as good as X then this individual under this preference relation is indifferent between X and Y and this relationship is reflexive in the sense that X is equivalent to X so then what we say is that a preference relation is rational if it is complete which means that for any two alternatives either X is at least as good as Y or Y is at least as good as X or both and so that for any two choices for any two alternatives you know what is your preference relation the second hypothesis that makes a preference relation rational is transitivity is if you have three choices X, Y and Z then if X is at least as good as Y and Y is at least as good as Z then X is also at least as good as Z now these two properties of the preference relation of course imply similar properties for the strict preference relation and for the indifference relation so for example the indifference relation is also transitive so one has to realize that these two hypotheses are rather strong in the sense that they look like natural but they are rather strong and in real life well the fact that you an individual has a complete preference relation of all possible alternatives he can choose that's not very clear I mean think of when you go to the supermarket and you have to choose a brand of a certain good then maybe you have to think hard about whether you prefer X to Y is not immediately apparent and also transitivity is not is a strong assumption when you compare X to Y you do that consider a certain number of factors and when you compare Y to Z you may consider a different set of factors and when you compare X to Z you may consider a yet different set of factors so the fact that the individual preference relation is transitive is a strong assumption also there is another aspect by which transitivity is a big assumption which is essentially what is called just perceivable differences that is when you have to choose a real number in a continuum spectrum then of course between any two points which are very close you may be indifferent and then if you are indifferent between close points then if you apply transitivity and also between the extremes but of course between the extremes you are not indifferent so think of when you want to choose a colour to paint your room there are many shades of that colour that go from black to white and you may be indifferent there are slightly different shades but definitely you are not indifferent between black and white so this was the other questions on this subject on this part an example when a preference relation is not complete so when a preference relation is not complete we will discuss this so you have to think that this preference relation and we will come back to this is an unobserved quantity is something that we assume is in the back of the mind of the economic agent of the individual is actually not something that is directly measurable so again you can think at yourself when you go to a supermarket and think about what type of particular wood you want to buy when there are many brands and so it's if you think about it if you compare two brands one with the other and you can do the exercise and ask yourself whether your preference is transitive and and this is this is why completeness is a strong assumption especially when the number of alternatives becomes very large okay sorry sir I still don't understand this I mean two alternatives either they will be equally good I mean there is some preference relation so I mean complete would mean you have all these choices there is some preference between X and Y or there is no there is indifference completeness means that this relation between this pair Y relation is not defined for all pairs there is some pair X and Y for which the preference relation is not defined but okay but that seems I mean you can always have some model which will define this is preferred than the other alternative how are you saying that it's not defined no no I mean say now you are thinking about modeling an individual makes choices and we have to take into account the fact that this individual may not have a complete preference relation on all possible alternatives and this example of the supermarket when you enter into the supermarket you see all the alternatives that you have but you enter into the supermarket without having say hard like in your hardware like a preference relation over all these alternatives so I mean if you work a little bit maybe you come out with a preference relation over all these alternatives but even the choice that you make at the supermarket is a choice that well you may choose to buy but you don't care about establishing whether B is better than C or not because you just want to buy A okay okay okay okay okay okay so isn't preference relation implicitly dependent on some set of factors why should we worry about transitivity of some fixed preference relation being factor dependent or not so I'm not sure I understand the question but say probably we will understand better from a mathematical point of view why transitivity can be lost when we discuss about social choice because yes so for example when when you want to decide about a trip imagine that now we are in the pandemic we cannot go on holiday so you have to decide whether you want to go in city A or in city B and this is a complex decision because well there is the trip how are you going to go there there is a travel imagine that you are taking a flight and whether there are say good sightseeing in one place there is the food there are many aspects and maybe you based on the food you may prefer A to B but based on say sightseeing you may prefer B to A okay so how do you decide between A and B and B and C you could weight these kind of things differently okay I don't know whether this answers the questions but so let me go ahead with the next with the next part okay so one important thing is that when you have a preference relation you can also in some cases represent it with a function and this is called a utility function utility function is a function so you say that a function U is a utility function that represents a certain preference relation if whenever X is at least as good as Y then U of X is larger or equal to U of Y for all X and Y okay so this function is a function that is defined on all the alternatives so that so that this it is clear that if if a preference relation is represented by utility function then this preference relation is rational okay so any preference relation which is represented by utility function is rational because it is complete because the utility function is defined on all values of X and because if X if U of X is larger than U of Y U of Y is larger than U of Z then U of X is clearly larger than U of Z okay so it's also transitive okay so now the converse is not always true okay so which means say if you have a utility function then does this always represent a rational preference relation one can prove that if the number of alternatives is finite then the converse is also true okay that a utility function represents a rational preference okay now essentially this restriction of I mean there are counter example where you have a accountably infinite set of choices when essentially say transitivity can break down but these are rather say pathological cases okay so then this means that essentially we can represent rational preference in terms of utilities and so while preferences are ordinary type of relation between the choices so you can rank if you have a preference relation then you can rank your choices what is the first so what is the second etc etc the cardinal the utility function is essentially a cardinal property cardinal property so this means that it is expressed by real number you could think that if the utility of X is U of X then U of X tells you how much you like X okay so this is one way of putting but this also means that the same preference relation rational preference relation can be represented so for example if you take a utility function which is an increasing function of a utility function then these two utility functions are going to represent the same rational preference relation okay so other questions on this I think this is a very simple thing but so we have well practical example of infinite choices is when say for example the set of alternatives is a continuous set for example is the real is a point on the real axis okay or say it's a yes you have to decide yes for example what shade of what color you want to paint your room then a color can be represented by three numbers R, G and B and then the choice of a color is a choice between a set which is say the positive real say to the cube and so cardinal preference again so what I will say is that utility function is a cardinal property cardinal property means that it is related to a number a real number whereas ordinal properties are related to just a ranking say an ordering of the preferences and you may express this by saying that utility function expresses how much you like a particular alternative okay yes it is correct to say that utility function is a function to turn our preference into a numerical value that can be that we can work with that's okay is a very practical way but let me go to the final point excuse me you mentioned earlier that for say two elements of X and Y of in the set of alternatives you can use different framings to evaluate your preference so you can use different set of factors to evaluate X and Y how does that tie in with this function you okay so the utility function as we have said represents a rational preference okay so it represents a rational preference where these problems of intransitivity are not present let me get to the final point that I want to discuss today which is essentially if you want inverse approach to choice behavior so up to now what we have discussed preference relation and utility function are unobservable properties so you cannot how you cannot observe the utility function of a person you cannot really measure it but what you can measure what you can observe is actually how it behaves and what are the choices that an individual makes in different circumstances okay so this is a different say a specular approach to individual rationality that starts from observed behavior okay so now here we have our set of alternatives X and between this set of alternatives we may consider a particular problem where the individual has to choose between a subset B of choices and when we observe this behavior the choice that the individual makes what we see is that among these B alternatives these subset C is the set of choices that he has chosen okay so we can formalize this by introducing what is called the choice structure so the choice structure is made of two elements so one element is this B which is a family of subsets of X and you can think at this B as being an observation of an experiment a choice experiment okay so these B are also called budget set that is the set of alternatives which are available in a particular instance in a particular situation okay so the set of alternatives is a function that maps each budget set each B into a subset of B which is the set of things the set of alternatives which are chosen in this particular experiment and of course this set must be non-empty for example and the idea is that essentially among all these options these are the ones that are chosen then somehow we want to infer that the individual prefer these green alternatives here contains in C to alternatives which are in B but not in C okay so these are the two possible choice structures so here you have three elements this is X and you can think at a situation where in your budget set you have either a possibility to choose between X, Y and Z or the possibility to choose between X and Y and the first choice behavior the first choice structure where essentially in this case what is chosen is just X and also here what is chosen is just X whereas here in the first case you choose X and Y in this case you just choose X okay so now the issue is what is the relation between choice structures and the idea is that the choice behavior actually may reveal preferences but in order to do so the choice structure has to satisfy some consistency requirement what does consistency means so this is sort of this is the subject of the weak axiom of revealed preferences so the weak axiom of revealed preferences says essentially that if we observe X X is chosen when Y is also available then X should be chosen whenever X and Y are available and Y is chosen okay you have to think a little bit about this statement so if X and Y are available and I choose X then whenever X and Y are also available and I choose Y then I also choose X so if a choice structure satisfies this weak axiom of revealed preferences then you can infer a preference relation out of it so which is called the revealed preference relation and you say that X is revealed preferred to Y if there is one budget set that contains X and Y such that X is chosen okay so now of course if you have a rational preference relation then even if you choose a subset of budget sets a family of budget sets and if you look at what is the scene that is induced by this choice behavior this choice structure will satisfy the weak axiom of revealed preferences okay the interesting thing is whether the converse is also true or not so that is imagine that you have a choice behaviors that satisfies the weak axiom of revealed preferences then does this correspond to a revealed preference relation that is rational and the answer is yes but in order for it to be true then the set B should include all subsets with at least three elements okay so to make this content of this weak axiom let's go back to these two examples and let's see whether these two choice structures satisfy the weak axiom of revealed preferences so so in this case if you consider C1 you see that there is a set which is this B here where X is preferred is in C belongs to C and there is another set where X and Y are also present and X is also there so this C1 is a structure that obeys the weak axiom of revealed preference this one instead is not because so because essentially think of Y so in this case X was also available then it should be that whenever X and Y are available and X is chosen then Y should also be chosen but that is not true so C2 does not obey the weak axiom of revealed preference so of course this means that there is a revealed preference relation in this choice structure C1 there is no revealed so this choice behavior here is not consistent with the preference relation in the case of C2 so questions can we say in other words that rational behavior is a fixed rule behavior at any choices there is a fixed rule for individual in choosing fix in which sense in the sense that you are considering time I mean different choices in time preference could change in time or something like that so yes we are considering a very simple case where as the preference relation is we want to model the choice behavior in a particular instance and for that we assume that there is a preference relation so of course all these things why we consider subset inside X because X is the set of alternative so there is we are only considering alternatives in X X is the set of all possible alternatives let's ask should the utility function always be a continuous function no it should not there is no need for it to be a continuous function is it correct to say that T of X is yes this we already answered so what if C of B is equal to B this is a very interesting question from Fabio so what would you for example if all elements if one is indifferent between all elements of B then C of B will be equal to B okay but again the issue about choice structure is that we describe the third behavior and then from this we want to infer preferences okay so our time is over so tomorrow we will discuss the choice under uncertainty I recommend you to go and look at the lectures on the website so that we can have a similar we can go through this material tomorrow excuse me sorry for the silly question but we as PGS students have exams at the end of the spring college right what kind of exams because this is not of interest to anyone